TEACHING NEGATIVE NUMBER OPERATIONS

This is a reconstructed copy of the doctoral thesis of RL Hayes. The original is available from the library of the University of Melbourne. RL Hayes is the sole author. The thesis has been uploaded by Kaye Stacey with permission.

TEACHING NEGATIVE NUMBER OPERATIONS

A comparative study of the neutralisation model using integer tiles

Robert Leslie Hayes B.Sc. (Melb.), B.Ed. (Monash), M.Ed. (Nott.)

A thesis submitted to complete the requirements for the degree of Doctor of Education

July, 1998

Department of Science and Mathematics Education Faculty of Education University of Melbourne

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ABSTRACT

The thesis describes an attempt to improve the teaching and learning of negative number concepts and operations. A teaching method based on the neutralisation model and using counters in the form of ‘integer tiles’ as the manipulative materials was developed and evaluated. The thesis begins with a review of the history and development of negative number. The influence that authors have exerted in embedding the use of the number line in signed number teaching practice was shown by examination of popular textbooks. Testing of classes across Years 8 to 11 revealed negative number skill deficiencies resulting from normal teaching methods. The major component of the research compared the integer tile method with normal teaching methods, in terms of learning outcomes, using a series of four tests of signed number operational skills and applications. Qualitative descriptions of student and teacher reactions and behaviour based on observation, interviews, conversations and examination of students’ work by the researcher are also given. The experimental and control groups were shown to be well-matched on the pre-test. The experimental group performed significantly better on the posttest. The difference between the groups widened during the year following initial teaching. However no significant difference between the groups was found two years after the initial teaching. The evidence suggests that the integer tile method was more effective and for a wider range of student ability levels at the time of teaching than the normal method. Subsequently middle ability students in the control group were capable of catching up following application and practice in the use of negative numbers in later topics. However weaker students in both groups who did not fully master operational skills at the time of teaching continued to display skill deficiencies. The conclusion reached is that the integer tile teaching method could be a more effective initial negative number teaching method than normal methods based on the number line for most students. Further research is required to investigate the effectiveness of the integer tile method with weaker students.

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DECLARATION

Except where due acknowledgment has been made this thesis contains only the original work of the author and contains no material which has been submitted for examination in any other course or accepted for the award of any other degree.

R. L. Hayes

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ACKNOWLEDGMENTS

This study would not have been possible without the help, cooperation and patience of the principals, mathematics coordinators, mathematics teachers and students in the three schools where the research was undertaken. I particularly thank; •

the coordinators who facilitated the school-based research activities, arranged access to classes and students for interviews and generally assisted in the administration of the project,

•

the teachers who participated enthusiastically in the project, allowed the researcher unrestricted access to their classes, provided valuable feedback and constructively critical comments with regard to the experimental materials and administered the tests,

•

the students who accepted me as part of the classroom scene and responded courteously and frankly to in-class questions and in out-of-class interviews and patiently submitted to the required series of tests.

I also thank my supervisor, Professor Kaye Stacey for her patient and thorough guidance throughout the duration of the project, and Dr Mollie MacGregor who also provided helpful comments and constructive criticisms whenever approached. Others who provided help from time to time included my former Mathematics Department colleagues at the Hawthorn Institute of Education, Michael Barraclough and Russell Keam, and some other ‘non maths’ members of staff who patiently worked through draft versions of tests whilst I watched and questioned them. Secondary level students and neighbours, Andrew and Peter Zajac also kindly assisted by working through tests and providing useful feedback and ideas on a few occasions. I am also indebted to John Haley, formerly of HIE, who suggested the use of and cut the integer tiles. He patiently made several thousand. Finally I thank my loving and devoted wife for her encouragement and support throughout what became a longer task than originally intended. Elaine put me back on track several times and this thesis is dedicated entirely to her.

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TABLE OF CONTENTS

Chapter 1

Introduction

Chapter 2

A Brief History of Negative Number

1 10

Development of negative number from ancient to modern times 10 Implications for teaching 25

Chapter 3

Textbook Approaches to Negative Number

27

Nineteenth century approaches 28 Textbooks used in Australia prior to 1960 36 Textbooks after the introduction of New Maths 41 Summary and conclusions 47

Chapter 4 50

Outcomes of Normal Teaching Approaches (Study P) Introduction 50 Normal teaching 50 Study P design and method 50 Results 53 Findings from Test P8 62 Findings from Test P9/11 69 Conclusion 71

Chapter 5

The Neutralisation Model

73

History and theory 73 Research studies 79 Chapter summary 85

Chapter 6

The Integer Tiles Teaching Method

87

Description of the method 87 Teaching materials 88 Comparison of number line and integer tile approaches 96 Expected learning outcomes 98

Chapter 7

Experimental Design and Method

99

Aims 99 The sample involved 100 Tests 102 Teaching 106 Data 107 Limitations 108 Strengths 112 A major assumption 113

Chapter 8

Quantitative Outcomes of the Integer Tiles Teaching Experiment (Study E) An overview of test results 114 Detailed analysis of selected test items and questions 116 Pre-test analysis 117

114

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Post-test analysis 119 Retention test analysis 134 Long-term retention test analysis 141 Recollections of negative number teaching methods 146 Chapter summary and findings 152

Chapter 9

Outcomes within Schools

153

School A 154 School B 164 School C 169 Video-taped interviews 171 Comparison of some Study P and Study E results 175 Further observations and comments 178 Conclusion 180

Chapter 10

Conclusion

183

Discussion and summary of research findings 183 Findings from history 183 Influence of textbooks 184 Findings from Study P 185 Findings from Study E 186 Recommendations for curriculum developers, teacher educators and teachers 188

References and Bibliography

190

Appendices

195

1 2 3 4 5 6 7 8 9 10 11 12

Test P8 (Preliminary Study Year 8 test) 195 Test P9/11 (Preliminary Study Years 9-11 test) 197 Test U1 (Pilot Study E pre-test) 200 Test U2 (Pilot Study E post-test) 205 Test W (Study E pre-test) 210 Test X (Study E post-test) 211 Test Y (Study E retention test) 215 Test Z (Study E long-term retention test) 217 Sample of experimental class student work (Work Booklet One) 218 Sample of experimental class student work (Work Booklet Two) 228 Sample of control class student work 239 Samples of item analysis spreadsheets (Excel) 254

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LIST OF TABLES Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 3.5 Table 3.6 Table 4.1 Table 4.2 Table 4.3 Table 4.4A Table 4.4B Table 4.5 Table 4.6 Table 4.7 Table 4.8 Table 4.9 Table 4.10A Table 4.10B Table 4.11 Table 4.12 Table 4.13 Table 4.14 Table 6.1 Table 7.1 Table 7.2 Table 8.1 Table 8.2 Table 8.3

Approaches to introducing and teaching negative number operations in Borchardt (1946) and Durell (1954) Approach to introducing and teaching negative number operations in Lockwood and Down (1954) Approaches to introducing and teaching negative number operations in Wilson & Ross (1938) and Wilson (1946) Approaches to introducing and teaching negative number operations in Math Book 1 (1972) and Math Book 2 (1973) Recent SMP approach to introducing and teaching negative number operations Approaches to introducing and teaching negative number operations in Maths 8 and Mathematics Today Year 8 Tests P8 and P9/11. Mean scores and standard deviations on 24 integer operations items Tests P8 and P9/11. Number and percentage of students correct on addition and subtraction items Tests P8 and P9/11. Number and percentage of students correct on multiplication and division items Tests P8 and P9/11. Number and percentage of students correct on equations items Tests P8 and P9/11. Number and percentage of students correct on equations items Test P8. Number and percentage of students correct on brackets items Test P8. Number and percentage of students correct on sequences items Test P8. Number and percentage of students correct on ordering number pair items Test P8. Number and percentage of students correct on temperature items Test P9/11 Year level mean scores and standard deviations for test subsections Test P9/11. Number and percentage of students correct on substitution and evaluation items Test 9/11. Number and percentage of students correct on substitution and evaluation items Test 9/11. Number and percentage of students completely correct on table of values questions Test 9/11. Number and percentage of students correct on expansion and simplification items Test P9/11. Number and percentage of students correct on linear equations items Test P9/11. Number and percentage of students attempting and correct on simultaneous equations items Operational comparison of number line and integer tile teaching models Study E (Pilot Study (1994)) Testing and Teaching Activities Study E (Main Study (began 1995)) Testing and Teaching Activities Test W: Group mean scores and standard deviations Test X: Group mean scores and standard deviations Test Y: Group mean scores and standard deviations

37 39 40 43 45 46 54 54 56 56 57 57 59 60 61 63 64 65 67 67 68 69 97 102 102 114 115 115

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Table 8.4 Table 8.5 Table 8.6 Table 8.7

Test Z: Group mean scores and standard deviations Test W: Percentage of students correct on addition items Test W: Percentage of students correct on subtraction items Test W: Percentage of students correct on multiplication items

116 117 118 118

Table 8.8

Test X: Q.1 Integer operations mean scores and standard deviations Test X: Percentage of students correct on addition items Test X: Percentage of students correct on subtraction items Test X: Percentage of students correct on multiplication and division items Test X: Percentage of students correct on equation items Test X: Percentage of students correct on equation items Test W: Percentage of students correct on items containing brackets Test X: Scramble table mean scores and standard deviations Test X: Percentage of students correct on mixed operations items Text X: Total integer calculations mean scores and standard deviations Test X: Percentage of students at performance levels shown in ordering signed numbers Test X: Percentage of students correct on word problems Test Y: Percentage of students correct on addition and subtraction items Test Y: Percentage of students correct on multiplication and division items Test Y: Percentage of students correct on equations items Test Y: Percentage of students correct on brackets items Test Y: Percentage of students giving the correct sign on addition and subtraction items Test Y: Percentage of students giving the correct sign on multiplication and division items Test Y: Percentage of students correct on substitution and evaluation items Test Z: Percentage of students correct on substitution and evaluation items Test Z: Percentage of students correct on expansion and simplification items Test Z: Percentage of students correct on completing tables of values items Test Z: Percentage of students correct on quiz show problem items Test Z: Group distributions of numbers of students using types of operation explanation Percentage of students in Study P and Study E classes correct on basic integer operations items

119

Table 8.9 Table 8.10 Table 8.11 Table 8.12A Table 8.12B Table 8.13 Table 8.14 Table 8.15 Table 8.16 Table 8.17 Table 8.18 Table 8.19 Table 8.20 Table 8.21 Table 8.22 Table 8.23 Table 8.24 Table 8.25 Table 8.26 Table 8.27 Table 8.28 Table 8.29 Table 8.30 Table 9.1

120 121 124 125 125 126 128 128 129 130 131 134 135 135 136 138 139 140 142 142 143 144 146 175

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LIST OF FIGURES Figure 1.1 Figure 1.2 Figure 2.1 Figure 2.2 Figure 4.1 Figure 4.2 Figure 8.1 Figure 8.2 Figure 8.3 Figure 8.4 Figure 8.5 Figure 9.1 Figure 9.2 Figure 9.3 Figure 9.4 Figure 10.1

Graph comparing percentages of students in Years 8, 9, 10 & 11 correct on sum and product of negatives items Integer tile faces Chinese positive and negative rod method of solving systems of linear equations Diagram representing Hamilton’s ‘moments in time’ Addition Scramble Table Multiplication Scramble Table Comparison of experimental and control group mean percentage scores across the four tests Test X: Addition Scramble Table Test X: Multiplication Scramble Table Test X: Subtraction Scramble Table Test X: Division Scramble Table Comparison of experimental and control class test mean percentage scores in School A Comparison of experimental and control class test mean percentage scores in School B Comparison of experimental and control class test mean percentage scores in School C Comparison of performances of Year 8 and Year 10 classes on integer operations Comparison of percentage of experimental and control group students correct on items of the form 8 − -4 across four tests

2 7 12 22 58 58 116 128 128 128 128 164 168 171 176 187

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Chapter 1

Introduction

“I would say that when you add a negative and a negative you get a positive so therefore -5 + -3 = 8.” “Two negatives make a positive so for -5 + -3 it would be 8 positive.” “Because it (-5 - -3) is take away and a negative number follows this number must be added. To remember this rule pretend that the two dashes (minus signs) will form a (+) plus signal if they are beside one another.” “When you have to (sic) negatives facing each other it (3 - -5) becomes positive.” “2 negatives = a positive you cant (sic) subtract a negative number.” (for 0 - -3) “You have 4 positive cubes and 6 negative cubes (drawn). One positive cube & one negative cube when added equals zero. Therefore +4 + -4 (drawn) equals zero, but you still have 2 negative cubes, therefore answer is -2.” (for 4 + -6) (A sample of responses of Year 10 students, when asked to explain examples of negative number operations to ‘a friend’.)

Aim The major aim of this research is to describe the development, implemention and evaluation of a teaching method and teaching materials to improve the learning of negative number concepts and performance of operational skills. The teaching method will be based on the use of a neutralisation model using integer tiles.

Need for the study Difficulties with negative number operations are common. For example some students (and adults) think that adding two negative numbers produces a positive number. Many seem to recall that subtracting a negative is equivalent to adding a positive but few can explain why. Fewer understand why the product of two negatives is a positive. Lack of understanding and reliance on misinterpreted versions of sign rules becomes a major impediment to progress in mathematics. Fawcett and

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Cummins comment on the “... deadening of learning in algebra ...” resulting in the emphasis of rules to show ‘how’ but neglecting ‘why’. 1 The experimental teaching method developed for the purpose of this study is designed to help students find out ‘why’ and, as far as possible, articulate ‘how’ themselves. Ideally students completing secondary level school should emerge with a thorough knowledge and understanding of the complete real number system which, of course, includes negative number. The contention underpinning this study is that the teaching approaches appearing in popular textbooks, commonly using the number-line as the initial operational model, do not lead to sufficient negative number understanding and skills for many students. The small preliminary study of negative number skills (Study P) in which students in Years 8, 9,10 and 11 were tested for the purpose of gathering base-line comparative data for the experimental study (Study E) to follow, supports this contention. As mentioned earlier, a common error is the giving of a positive answer for the sum of two negatives. The initial study confirmed this revealing that, for the samples tested, more students in years 9, 10 and 11 gave the correct product of two negatives than gave the correct sum of two negatives. Figure 1.1 is a graph comparing year-level performances on such items.

% correct

100 80 60

-4 + -2

40

-2 x -7

20 0 Year 8

Year 9

Year 10

Year 11

Figure 1.1 Graph comparing percentages of students in Years 8, 9, 10 & 11 correct on sum and product of negatives items.

1

H. P. Fawcett & K. B. Cummins (1970). The Teaching of Mathematics from Counting to Calculus. Columbus, Ohio. p. 185.

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The Year 11 result is particularly disturbing considering that the students tested were studying VCE Mathematical Methods. This example is just one piece of evidence suggesting that the development of more effective teaching methods is necessary.

Teaching and learning difficulties The formal study of negative number concepts and operations usually begins during early studies in algebra when it becomes necessary to extend knowledge of number beyond that needed for day-to-day arithmetic (natural numbers, zero, place value, the four processes, fractions, and decimals). Since mathematics is used to quantitatively describe ‘the world’, a number system that efficiently and elegantly deals with signed quantities, directional change and opposing qualities (e.g. opposites, inverses and reversals) is needed. Introductory examples in textbooks use signed (or directed) numbers to model situations such as; •

movements or positions; up/down, left/right, above/below, north/south, east/west, clockwise/anticlockwise,

•

changes; speeding up/slowing down, gains/losses,

•

properties such as; oversize/undersize, before/after, attract/repel, credit/debit, for/against,

•

temperature measurement; above and below freezing,

•

scores in some games; under/over par in golf, plus/minus scores in indoor cricket.

Popular Year 8 Mathematics textbooks, (e.g. Maths 8, 2 Moving through Maths Year 8, 3 Mathematics Today Year 8 4) all begin the treatment of negative number, within the topic of signed or directed number, with an extensive selection of directional examples, based on themes such as those listed previously. The intention is to demonstrate and justify the need for 2

B. Lynch, R. Parr, L, Picking & H. Keating (1991). Maths 8. (2nd edn), Melbourne. D. Blane & L. Booth (1991). Moving through Maths Year 8. (rev. edn), North Blackburn, Victoria. (There is also a Year 9 textbook, Moving through Maths Year 9.) 4 T. Daly, J. Ardley, J. Buruma, M. Cody, L. Mottershead & P. Tomlinson (1993). Mathematics Today Year 8. (2nd edn), Sydney. 3

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signed number usage. However students often easily get required answers for the introductory textbook problems, without using the signed number processes intended to be applied. For example in one of the test items used in this research students were required to find the difference in altitude between a location 5400m above sea-level and a location 250m below sealevel. 5 Similar examples can be found in most Year 8 mathematics textbooks. Just over half (53.6%) of the sample of the 145 students tested managed the correct answer. A majority of the students who were correct simply wrote 5400 + 250 = 5650. Some students included a vertical line diagram with their calculation. Several of the students who were incorrect wrote 5400 − 250 = 5150. Only a small number of students appeared to interpret the problem as 5400 − -250. When questioned, most considered this to be an unnecessary (written) intermediate step. This example illustrates how situations which can be modelled using signed numbers can be easily reduced to basic arithmetic by students. The validity of such problems as signed number exercises therefore becomes questionable. The aforementioned textbooks make little or no use of word problem contexts that can be modelled by multiplication and division operations. 6 Although introductory examples like the above are considered by textbook writers and teachers as necessary to introduce negative numbers they fail to convince students of their need for the purpose of calculation. Later however when real-life situations and problems are modelled mathematically and studied generally and algebraically using equations, formulae and index notation, or graphically using coordinates and functions and relations, the use of negative numbers becomes unavoidable. Following Year 8, students are increasingly expected to manipulate algebraic expressions, deal with positive and negative coefficients, solve linear and quadratic equations, use indices and generally extend into mathematical areas relying on negative number skills. 5

Test X, Q.9 (Appendix 6) Daly et al., p. 18, provide two introductory multiplication model situation examples but no practice exercises. Lynch et al. provide no such examples and no exercises. Blane & Booth defer multiplication and division until their Year 9 textbook and provide one model situation exercise for each of multiplication and division (p. 52).

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5

To progress in algebra, and the many natural and social science fields in which algebra is applied, it is essential that students achieve mastery of negative number concepts and operations. Unfortunately this does not occur for many students and it appears that difficulty with the topic may be a contributing cause of students becoming disenchanted with mathematics. Sheila Tobias draws particular attention to difficulties arising due to “The Many Meanings of Minus”. 7 There are, in fact, three uses for the minus sign: designation of the operation of subtraction (e.g. 5 − 3), as part of numerals for negative numbers (e.g. −3 or -3), and to signify the additive inverse of a pronumeral or variable (e.g. −a or -a). There are also other complications. Subtracted terms in algebraic expressions and equations and formulae are considered to have negative number coefficients. Students may also be confused by the meaning of ‘subtract’. The most common and popular interpretation is probably ‘take away’ or ‘deduct’. 8 However in mathematics classes ‘subtract’ may be given the meaning ‘find the difference’. 9 As a consequence uncertainty may arise particularly in the context of operations involving negative numbers. Tobias’ concerns remain valid and changes in approach and more effective teaching methods are needed to overcome the weaknesses and confusion that continue to be displayed by many students.

Lessons from history Negative numbers first appeared as coefficients, due to subtracted terms (positive quantities) in early Chinese and Greek equations and formulae. Since then negative numbers have appeared and been used in a variety of contexts similar to those still used in the journey through school mathematics. It took around 2500 years for negative numbers to gain full recognition and acceptance as legitimate, necessary and useful entities within the real and complex number system. An outline of the history of negative number is included in the thesis to highlight conceptual difficulties

7 8

S. Tobias (1980). Overcoming Math Anxiety. Boston. This is the usual meaning given in English dictionaries (e.g. Pocket Oxford Dictionary).

6

expressed and encountered in their development and provide some insight as to why many students find them intuitively difficult. Such insight should assist appropriate sequential development of the teaching method and process.

Popularity of the number line An extensive examination of past and present-day textbooks, conducted and reported as part of this research, shows that the horizontal number line has been adopted as the fundamental model of negative number used for introducing integer addition and subtraction. Allan White asked, “Who put them on the directed number line?” 10 He did not identify the culprit. However the present author’s investigations suggest that textbook authors have probably been responsible for keeping ‘them’ fixed to it and have asserted a strong influence in determining classroom teaching method. Some texts also make limited use of the number line to introduce multiplication. The number line appears to be rarely (if ever) used for modelling division.

An alternative teaching method The major component of this study is an experiment designed to investigate the potential of an alternative negative number teaching method. The alternative teaching method developed uses sets of small (2 cm. square) double-sided tiles which will be referred to as integer tiles. It is a development of the neutralisation model using two colours of counters. Freudenthal 11 calls it the annihilation model. Despite reappearing in some form or another from time to time in journal articles and occasional textbooks, 12 the model is seldom used in secondary-level classrooms. The fact that it is a structured aids approach requiring some materials

9

This is the meaning given in mathematics dictionaries (e.g. E. Borowski & J. Borwein (1989). Dictionary of Mathematics. London.) 10 A. White (1994). Hurdles: who put them on the directed number line? The Australian Mathematics Teacher, 50, 1, 14-18. 11 H. Freudenthal (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht.

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preparation, organisation and handling may deter some teachers. Lack of knowledge and understanding of the model and lack of in-depth attempts at classroom research and evaluation of methods based on the model are other probable reasons for its neglect. Having successfully used various versions of the model for several years with secondary and TAFE trainee teachers recommencing studies in mathematics, the author 13 believes that the model provides a more basic and meaningful introduction to negative number concepts and signed number operations than the number-line approach. Hence the decision to develop and evaluate the model for more general use in secondary school classrooms. The model developed for this study uses two kinds of integer tile. One type is labelled ‘1’ and ‘-1’ on opposite faces and the other is labelled ‘0’ on both faces (Figure 1.2). 14

+

1 -1 0

Figure 1.2 Integer tile faces

The intention was to contrast learning and operational skill outcomes for (experimental) classes using the integer tiles with outcomes for comparable ability (control) classes taught using number-line based methods in conjunction with normal textbooks and teaching.

12

J. J. Del Grande, P. Jones, L. Morrow, D. Kennedy, I. Lowe, I & D. Wirth, (1972). Math Book I. Hawthorn, Victoria. 13 R. Hayes (1992a). An Excursion to the Land of Integers, unpublished D.Ed. seminar project report, University of Melbourne. 14 A colleague suggested that [2]/[-2], [3]/[-3], [4]/[-4], ... tiles (and perhaps +/- fraction tiles) may be useful. The tiles would then have become merely substitutes for written numbers and the practical benefit and visual imagery gained from building and manipulating the integers reduced. The author considers that integer heaps proportional in size to integer absolute values modelled may assist the visualisation process.

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Hypothesis The general hypothesis to be tested is that students experiencing teaching of negative number in either Year 7 or Year 8, taught initially using the integer tiles method developed for this study, will display better levels of performance in short-term and long-term negative number skills than students taught by popular number-line and textbook based methods.

Thesis structure and content Structure The thesis is structured on the following themes and components; •

history of negative number,

•

the influence of textbooks on negative number teaching practice,

•

a study of learning outcomes following normal teaching practice,

•

evolution of the neutralisation model and previous research,

•

development of an integer tile embodiment of the neutralisation model as a teaching method,

•

an experimental study comparing the learning outcomes of the newly developed integer tile method with the outcomes of normal practice.

Chapter contents Chapter 2 contains a review of relevant secondary sources of historical literature and traces the evolution of the development of negative number. Chapter 3 provides a comprehensive overview and comparison of the interpretations and teaching approaches contained in secondary school text books over the last century or so. Chapter 4 deals with the results and findings of a preliminary study (Study P) showing deficiencies in learning outcomes from normal number line and textbook based teaching. Chapter 5 outlines the history of the neutralisation model and reviews the relevant teaching theory and research literature. There has been very little research directly focussed on classroom use of the model. The conceptual basis and a description of the experimental integer tile teaching and learning materials and methods are contained in Chapter 6.

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The experimental design and method, including details of the tests and methods of analyses used, are the content of Chapter 7. The quantitative outcomes of the experimental study (Study E) are provided in Chapter 8. The results of the main testing program comparing the across school experimental and control groups are tabulated, analysed and discussed. A qualitative discussion of within school and classroom events during the teaching component of Study E is the subject of Chapter 9. The final chapter concludes the study by giving a summary of the major findings and recommendations for curriculum developers, teacher educators, teachers and further research. Copies of the tests, experimental teaching materials (work booklets), samples of student work and item analyses spreadsheets are attached as appendices.

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Chapter 2 A Brief History of Negative Number The negative number story is fascinating, controversial and somewhat turbulent. The debate on what negative numbers are, what they mean and how they should be interpreted and used, raged and was not closed until a little over a hundred years ago. The aim of this chapter is to provide an overview of the evolution, development and use of negative number. Secondary sources in the form of published historical works and articles are used. There is no intention to advance scholarship in this historical area. Before attempting to improve the teaching of negative number it seems necessary to examine the history and didactical development of the topic. First rank mathematicians took over 2000 years to become intellectually comfortable with intuitively difficult aspects of the topic − the product of two negatives being the outstanding example. Appreciation of historical difficulties may assist in the understanding of why many secondary-level students have difficulties with the topic. Specific questions to be answered are; •

How and why did negative number develop?

•

What forms do negative numbers take?

•

Why did it take so long for negative numbers to gain acceptance?

•

What are the intuitive difficulties associated with negative numbers?

•

What historical factors need to be considered with regard to the teaching of negative number?

Development of negative number from ancient to modern times The historical development of negative number is traceable mainly through materials written for instructional purposes (tablets, manuscripts, articles and textbooks) from ancient to modern times. From about 300BC until around the seventeenth century AD negative numbers sometimes resulted as meaningless by-products during the process of solving equations.1 Throughout roughly the same period equations with negative coefficients 1

V. J. Katz, (1993). A History of Mathematics An Introduction. New York.

11 (due to subtracted positive terms) were used. Medieval Indian mathematicians, who were mainly algebraists, showed rules for calculating with positive and negative numbers in their works. 2 It became practice to use negative numbers to signify commercial losses in Europe about 800 years ago.3 This idea evolved into the use of signed numbers to designate directional properties associated with numbers similar to those used in junior secondary level textbooks (e.g. upward/downward, left/right, profit/loss, anticlockwise/clockwise, over/under (par), after/before, above/below, ...).4 Developments in formal algebra in the nineteenth century eventually resulted in the acceptance of negative numbers as the additive inverses of positive numbers so that all linear equations could have valid solutions. Formulation of the real number field laws resulted in subtraction becoming addition of the additive inverse (e.g. 0 - 2 = 0 + -2) and division becoming multiplication by the multiplicative inverse (e.g. 6 ÷ 2 = 6 × 2-1). Negative numbers were then recognised as having equal abstract entity status with positive numbers. Negative solutions and values could be interpreted and applied in the context of the mathematically described models or situations in which they had arisen. For example if a ball is thrown vertically up, its acceleration whilst moving upwards, is negative because it will be slowing down. If a projected profit formula produces a negative value it is predicting a loss. 5 In summary, historically, negative numbers arose in several contexts. Initially they were coefficients resulting from subtraction operations in equations and formulae. They were used to signify below expected value and directional or opposing properties relating to measured quantities. At various times they have appeared as incidental by-products of equation solving processes. Later they were shown to have useful application in analysis, calculus and coordinate geometry. In the modern era they have become essential abstract elements of the real number system defined as numbers less than zero, generated by subtracting a larger G. G. Joseph, (1992). The Crest of the Peacock. London. J. D. Barrow, (1993). Pi in the Sky. London. 4 A. Hooper, (1951). The River Mathematics. Edinburgh. 5 E. E. Kramer, (1964). The Mainstream of Mathematics. New York. 2

3

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unsigned (or positive) real number from a smaller such number (e.g. 0 - 1, 4 - 6, 1.3 - 1.4, ...). Negative numbers are used in all of these contexts in secondary school mathematics. Chinese counting rods Of particular relevance to this study is a procedure developed in China over two thousand years ago. Counting rods (small bamboo rods about 10cm long) were arranged and manipulated on counting boards for the purpose of performing calculations. During the process of solving some (practical) problems producing systems of linear equations it was found necessary to distinguish between positive and negative numbers. Red rods were used for positive numbers and black rods for negative numbers. 6 The method was essentially a matrix strategy, analogous to Gaussian elimination in more recent times, with red rods used to represent positive coefficients and black rods for negative coefficients.7 Figure 2.1 demonstrates possible number rod moves for solving a simple three equations and three unknowns linear system. The Chinese arranged the rods representing the coefficients for each equation in columns.

° °

° °

° °

→

…............................

°°°°° °°° °°°°

° ☻

° °

° °

→

.…........................

° °°° °°°°

☻☻

° °

° °

…............................ ☻☻

°°° °°°

°

→

°

☻

….................... ☻

°° °°°

x + y =5 y −z = 1 − 2z = −2 z = 1, y = 2, x = 3 y + z= 3 y + z=3 y + z= 3 x + z= 4 x + z= 4 x + z= 4 Figure 2.1 Chinese positive and negative rod method of solving systems of linear equations (° = red rod = 1, • = black rod = -1). The modern equivalent notation is shown below each stage.

In recent times it has become more usual to use black to designate positive and red for negative. 7 Katz, p. 17, describes the method used to manipulate the rods for three equations in three unknowns. The method was also used for larger systems up to five equations in six unknowns. Single linear equations and 2 by 2 systems were solved by algorithms not requiring the rods. See also Joseph, p. 174. 6

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Addition and subtraction rules for positive and negative quantities almost identical to modern textbook versions were also given: For subtraction−with the same signs, take away one (quantity) from the other; with different signs add one to the other; positive taken from nothing makes negative, negative from nothing makes positive. For addition−with different signs subtract one from the other; with the same signs add one to the other; positive and nothing makes positive; negative and nothing makes negative. 8 The rules provided instructions for manipulating the rods. This becomes clear if we replace ‘signs’ with ‘colours’ and ‘quantity’ with ‘set of rods’. The problems used (based on practical situations, e.g. grain bundles and measures) always produced positive solutions. The Chinese obviously realised that the same colour rods appearing on both sides of an equation at the completion of the calculating process produced a positive solution. Considering that the Chinese did not have the benefit of concise mathematical symbolism, and all problems and solutions were written out in words, the rods provided a sophisticated and elegant means of codifying and systematically simplifying the systems. It was a remarkable achievement pre-dating equivalent Western mathematics by at least 2000 years. Joseph suggests that reliance on rod computational techniques by the Chinese may have inhibited their development of abstract algebra. 9 The Chinese rods have properties in common with the integer tiles used in the experimental teaching component of this study. For example a red rod plus a black rod produces nothing (zero). Coloured rods could be used instead of the tiles for the purpose of the experiment. On the other hand the tiles could be used, in the manner shown above, for solving systems of linear equations. However in the modern context it is anticipated that the experimental activities will enhance rather than hinder the development of algebra knowledge and skills.

Greek algebra 8 9

Katz, p. 17. Joseph, p.176.

14

In the third century AD, Diophantus of Alexandria wrote a book entitled Arithmetica which actually dealt mainly with algebra. It was an organised collection of problems that produced indeterminate equations to be solved using rational numbers. He used an abbreviated symbol system including a symbol similar to Λ, possibly an abbreviation for lepsis meaning wanting or negation, for ‘minus’.10 A polynomial expression such as, x3 − 3x2 + 3x − 1, would have been written something like ΚΥαγςΛ∆ ΥγΜ°α.11 Because negative (subtracted) terms were always collected, a single Λ symbol was sufficient for all terms following it. Thus its written form was equivalent to, (x3 +3x) − (3x2 +1), in modern symbolism. Diophantus knew the correct unlike sign rules which he used for multiplying algebraic expressions involving subtractions and assumed that students using his text would already know the rules for adding and subtracting positive and negative terms. Multiplication of pairs of negative terms was unnecessary and did not arise. For example, using modern symbolism, a product such as, (x − 2) × (x − 3), would be treated as; x(x − 3) − 2(x − 3) = (x2 − 3x) − (2x − 6) = x2 − 3x − 2x + 6 = x2 − 5x + 6. Thus the multiplication of two negative terms is avoided by regarding all numbers as positive (unsigned) and subtraction of negative terms is used instead.12 In fact all multiplications involving negative numbers can be performed as subtractions using distributive rules. Negative numbers did not exist for Diophantus so isolated negative terms did not occur. Equations could contain negative terms (e.g. x2 - 4x = 5) but a solution could not be negative. Thus an equation such as, x2 - 4x = 5, would become x2 - 4x + 4 = 9 and (x - 2)2 = 9. Thus, x - 2 = 3 and x = 5.

Indian mathematics

Katz, pp. 163-164. According to Katz, ΚΥα is equivalent to x3 × 1, γς = 3x, Λ means minus, ∆ Υγ = x2 × 3, and Μ°α represents an invariable element × 1. 12 This is similar to the method that will be used in conjunction with the integer tiles for the teaching of the product of two negatives.

10 11

15

Sixth-century Indian mathematicians mentioned negative numbers and one practice used was to signify negative numbers by encircling the number symbol (as is practice in the 48-card pinochle game). 13 By the middle ages, Indian mathematicians, who were mainly algebraists, included rules for calculating with positive and negative numbers and fractions along with an extensive range of algebraic and computational techniques in their texts. Although the rules shown in their texts are correct, indications of how they were arrived at are generally missing. Emphasis was placed on students memorising the rules needed for solving problems. Brahmagupta (seventh century AD) who worked extensively on indeterminate equations, gave the following rules for operating with positive and negative numbers. The sum of two positive quantities is positive; of two negative quantities is negative; of a positive and a negative is their difference; or if they are equal, zero ... . In subtraction, the less is to be taken from the greater, positive from positive; negative from negative. When the greater, however, is subtracted from the less, the difference is reversed ... . When positive is to be subtracted from negative, and negative from positive, they must be thrown together. The product of a negative quantity and a positive quantity is negative; of two negative, is positive; of two positive, is positive ... . Positive divided by positive or negative by negative is positive ... . Positive divided by negative is negative. Negative divided by positive is negative. 14 Although generally treating zero like any other number, Brahmagupta made the mistake of saying that zero divided by zero was zero and that a positive or negative number divided by zero was a fraction with zero as the denominator. Indian mathematicians solved quadratic equations using a quadratic formula virtually in modern form. However their problems always produced at least one positive solution. For example, a problem used by Bhaskara (1114 - 1185) was; The fifth part of a troupe of monkeys less three, squared, had gone to a cave; and one monkey was in sight having climbed on a branch. How many were there?15 Barrow, p.90. Katz, p. 212. 15 Katz, p. 213. 13 14

16

In modern form the equation used to solve the problem can be written as, (1/5 x - 3)2 + 1 = x or x2 - 55x + 250 = 0, and the roots, 50 and 5, were found. However the second root was recognised as invalid and discarded because it is impossible to subtract three monkeys from one-fifth of five. A negative number could come from an equation but not from a problem . At this stage Indian mathematicians recognised that negative numbers could appear as meaningless by-products of the solution process. Islamic mathematics Arabs used a dot over the number symbol for a negative number. 16 The Islamic mathematician, Al-Samaw’al (1125-1180), dealing with algebraic manipulation, introduced negative coefficients and gave rules for dealing with their subtraction in his algebra text Al-Bahir fi’l-hisab (The Shining Book of Calculation). The rules could be used to add and subtract polynomials by combining like terms. 17 Developments in Europe and Britain In Italy, Leonardo of Pisa (Fibonacci) (c. 1170-1240) used negative numbers to represent financial losses.18 By the late fourteenth century, the correct rules of signs were being written down in words and sometimes justified by numerical arguments. For example Pacioli, in 1494, provided the following justification for the product of two negative numbers (‘m’ was used as the symbol for subtraction): 10 m 2 equals 8, thus if 10 m 2 is multiplied by 10 m 2, the result is 64. If however the cross multiplication is applied, we obtain 10 multiplied by 10, namely 100, then 10 twice multiplied by m2, which gives m40, which together give 60; thus it becomes evident that m2 multiplied by m2 should give the number 4.19 Cardano (1501-1576) in his Ars Magna, included the use of negative numbers as solutions to problems and provided the first appearance of Barrow, p. 90. Katz, pp. 236-237. 18 Barrow, p. 90. 19 Y. Thomaidis (1993). Aspects of Negative Numbers in the Early 17th Century. Science & Education, 2, 69-86, p. 77. 16 17

17

complex numbers in the context of a quadratic problem. 20 The initial problem posed was the division of 10 into two parts to produce a product of 40. Cardano gave the solutions as 5 + √ -15 and 5 − √ -15 which he verified by substitution. However he was not comfortable with this ‘useless’ outcome of ‘arithmetic subtlety’. By this period negative numbers were appearing as roots of equations but they were referred to as ‘false’ roots in contrast to ‘true’ positive roots. As for Indian mathematicians several centuries earlier, they were regarded as meaningless by-products of the equation solving process. In 1591, Viete initiated the beginning of a new era in symbolic algebra. He provided a general justification for the product of two negative numbers (actually two subtracted positive numbers) producing a positive number by using what we would recognise as a distributive law approach equivalent (in modern symbolism) to; (a − b) × (c − d) = a(c −d) − b(c − d) = (ac − ad) − (bc − bd) = ac − ad − bc + bd

(a, b, c, d ≥ 0)

This formalised Pacioli’s numerical ‘proof’ a century earlier. 21 Viete argued that by subtracting the quantities ad and bc from ac, too much will have been taken away, so −b × −d must produce a positive number bd. If a = c = 0, the identity reduces to −b × −d = bd.22 The disassociation of symbolic forms from concrete interpretations and the emergence of negative numbers in algebra appeared gradually through the development of the theory of equations. Negative solutions to equations sometimes occurred in medieval mathematics in, for example, problems involving exchanging money but in general meaningful interpretation of such solutions was rare.23

Katz, p. 334. In modern notation Pacioli’s numerically based argument is that (10-2)(10-2) = 8.8 = 64. Expanding first, produces 100 - 20 -20 + ? = 100 - 40 + ?. Thus the missing value must be 4, so -2.-2 = 4. As mentioned earlier Diophantus also used the same idea when multiplying algebraic expressions containing negative terms. However Diophantus did not perceive a need to find the product of two negative numbers. 22 Thomaidis, p. 76 23 J. Sessiano, (1985). The Appearance of Negative Numbers in Medieval Mathematics. Archive for History of Exact Sciences, 32, 105-150. 20 21

18

Recognising the fact that every equation can have as many distinct roots as the dimension of the unknown quantity, Descartes (1596-1650) introduced a negative solution (false root) by deliberately building an equation which had a negative (false) root. In essence (in modern symbolism) the process used can be demonstrated thus; If x − 2 = 0 or x − 3 = 0, x2 − 5x + 6 = 0, will have solutions x = 2 or x = 3. If x − 2 = 0 or x − 3 = 0 or x − 4 = 0, x3 − 9x2 + 26x − 4 = 0, will have solutions x = 2, x = 3 or x = 4. If x − 2 = 0 or x − 3 = 0 or x − 4 = 0 or x + 5 = 0, x4 − 4x3 − 19x2 + 106x − 120 = 0, is an equation having four roots, namely three true roots, x = 2, 3, and 4, and one false root, -5.24 Although making use of the reasoning above, Descartes appears still to have interpreted the roots as needing to represent quantities. He saw symbolic meaning existing in the expression x + 5 = 0, but not in a (meaningless) number that when added to 5 would produce zero. In common with predecessors mentioned beforehand, negative numbers still had the status of meaningless artifacts of algebraic processes. In a practical sense Descartes endeavoured to re-formulate his analytic geometry problems to avoid the occurrence of negative numbers. There is an opinion that he dedicated his third book of Geometrie to ‘the art of getting rid of false roots’.25 Although Descartes and Fermat (1601-1665) independently conceived the idea of using distances from lines to locate points on a plane, neither considered using negative values for such distances and their works contain no references to orthogonal Cartesian coordinate systems in the modern sense. Newton (1643-1727) and Liebniz (1646-1716) introduced the concept of negative distance. 26 Wessel (1745-1818), a surveyor, provided “A method whereby from given right lines to form other right lines by algebraic operations: and how to designate their directions and signs”, in a paper entitled On the Analytic Representation of Direction in 1797 and appears to have invented the orthogonal coordinate system using Thomaidis, p. 77. Ibid., p. 78. 26 J. Gulberg, (1997). Mathematics: From the Birth of Numbers. New York. 24 25

19

both positive and negative numbers.27 Wessel considered that certain geometrical concepts could be more clearly understood if there was a way to represent both length and direction of a line segment in a plane by a simple algebraic expression. Such expressions could be manipulated algebraically and arbitrary changes of direction could be shown more generally than simply using negative signs to indicate the opposite direction.28 By the eighteenth century negative number usage was becoming more acceptable and applicable. Maclaurin (1698-1746) wrote A Treatise of Algebra in Three Parts. To him algebra was not abstract but simply generalised arithmetic, founded on the same principles as arithmetic. He showed how the basic operations can be applied to negative numbers. Positive and negative quantities can enter algebraic computations as increments or decrements (excess/deficit, money due/money owed, line to right/line to left, elevation above/depression below etc.). He considered that a greater quantity could be subtracted from a lesser quantity if such an operation has contextual meaning (e.g. Have $50, spend $60, owe $10; 50 − 60 = −10). Thus a negative quantity is no less real than a positive one. Maclaurin demonstrated how to calculate with positive and negative quantities and in particular provided the following justification for the rule of signs in multiplying such quantities; If a, n > 0, since a − a = 0 then n(a − a) = na − na = 0. The first term of the product, na, is positive. The second term, −na, must therefore be negative. Similarly, since −n(a − a) = 0 and the first term of this product is negative, the second term, (−n)(−a), must be positive and equal to na. 29 Maclaurin’s justification would be criticised by modern mathematicians for its confused use of binary (subtraction) and unary (negative) signs. In 1767, Euler (1707-1783) published a textbook entitled Vollständige Anleitung zur Algebra (Complete Introduction to Algebra). Starting with a discussion of the algebra of positive and negative quantities, his explanation of multiplication was less formal than that given by H. Midonick, (1965). The Treasury of Mathematics, Volume 2. Harmondsworth, Middlesex. p. 320. 28 Katz, p. 666. 29 Katz, p. 552.

27

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Maclaurin. For −a × 3, the quantity −a can be considered as a debt. A debt taken three times gives a total debt of −3a. Thus in general −a × b is −ba or −ab. For the case of the product of two negative numbers he said that −a times −b cannot be the same as −a times b or −ab and therefore it must be +ab.30 In England in the nineteenth century, developments in algebra saw renewed interest in symbolic manipulation and its relation to mathematical truth. George Peacock (1791-1858) explained his symbolic approach in his Treatise on Algebra (1830) which was enlarged and revised in the period 1842-1845. He explored questions on the meaning of negative and imaginary numbers raised by several English mathematicians in the late eighteenth century. A concern was that although negatives were considered necessary and had been freely used to obtain all sorts of algebraic results, mathematicians had been unable to explain their meanings in any way apart from physical analogies. Perceived lack of adequate foundation for negative and imaginary numbers led Francis Maseres (1731-1824) and William Frend (1757-1841) to write texts renouncing their use. However negatives and imaginaries had been shown to be useful in the study of solutions of equations so this radical view was generally unacceptable. 31 Peacock and his colleagues resolved conceptual difficulties with negatives and imaginaries by distinguishing between two types of algebra −‘arithmetical algebra’ and ‘symbolical algebra’. Arithmetical algebra was based on universal arithmetic using the basic principles of arithmetic and the non-negative real numbers using letters to represent the numbers. Thus in arithmetical algebra, a − (b − c) = a − b + c, but this is subject to the conditions that c < b and b − c < a, so that subtractions can in fact be performed using only non-negative reals (e.g. 8 − [5 − 3] = 8 − 5 + 3). In symbolic algebra however, the symbols (letters) used need not have a particular interpretation. Symbol manipulations analogous to those used in arithmetic can be performed without restriction to their range of applicability. The identity a − (b − c) = a − b + c is universally applicable in symbolic algebra. Ibid., p. 554. Ibid., p. 611.

30 31

21

Peacock interpreted negative numbers as symbols of the form −a. Since (a − b)(c − d) = ac − ad − bc + bd in arithmetic, subject to a > b and c > d, (e.g. (8 − 3)(5 − 2) = 40 − 16 − 15 + 6 ) the same rule applies in symbolic algebra without that restriction. Using this identity it can be shown that (a) (−d) = −ad, (−b)(c) = −bc, and (−b)(−d) = bd.32 Peacock justified this as an application of the principle of the permanence of equivalent forms. According to Katz, “... any law of arithmetic, expressible as an equation, determines a law of symbolic algebra by the removal of any limitations on the symbols involved.” 33 This signalled a shift in the focus of algebra away from the meaning of the symbols to the laws of operations on the symbols.34 Thus during the latter half of the nineteenth century, the extension of the number system to include negative numbers and complex numbers took place. Having been started by Peacock, mathematicians of the period such as De Morgan (1806-1871) and Hamilton (1805-1865) continued the formalisation process. De Morgan created algebraic axioms for symbols that could represent things other than quantities and magnitudes and set out the rules he believed to be essential for algebraic processes. 35 Hamilton pushed the formalisation process considerably further in developing an algebra for completely abstract entities known as quaternions (ordered quadruples). Seeking to justify the use of negatives in algebra, Hamilton used ‘moments’ in time as his model. 36 If A and B are moments in time (B after A thus B > A), then α = B −A is the time step from A to B and will be a positive number. (His ‘moments’ in time also seem to be analogous to ‘positions’ on a number-line.) The time step θα (his notation for -α) is given by A − B and will be a negative number (see Figure 2.2) α → •→ 0 A B time ← θα Katz, p. 612. Ibid., p. 613. 34 Ibid., p. 614. 35 Katz., p. 614. 36 Ibid., p. 616. 32 33

Figure 2.2 Diagram representing Hamilton’s ‘moments in time’.

22

To Hamilton this idea seemed to overcome conceptual difficulties caused by negative numbers being regarded as ‘less than nothing’ (e.g. If negative numbers are less than nothing how can the product of two negatives be positive?) Using time step multiples Hamilton demonstrated the standard rules for arithmetic processes. In essence the method is an ordered pair approach. Positive numbers, negative numbers and zero, α and β, can be generated as ordered pairs (a, b) and (c, d) where α = b − a and β = d − c (a, b, c, d ≥ 0). 37

Assuming the laws of arithmetic the formal operational rules are; α + β = (a + c, b + d), α − β = (a − c, b − d), α.β = (ad + bc, ac + bd) and α/β = (a/(d − c), b/(d − c)). For example; for ordered pairs (5, 2) = −3 and (7, 3) = −4, −3 × −4 = ((2.7 + 5.3), (2.3 + 5.7)) = (29, 41) = 12. Hamilton developed his time steps model further and used it to interpret complex numbers as real number ordered pairs without the need for ‘imaginary’ numbers (e.g. the complex number, 2 + 3i can be defined as an ordered pair (2, 3) and i as (0, 1)).38 For complex numbers (a, b) and (c, d), where a, b, c, d ∈R, the rules for addition and multiplication are (a + c, b + d) and (ac - bd, ad + bc) respectively. Hamilton’s extension of complex number theory from 2-space to 3space (ordered triplets) led to the invention of the algebra of ordered quadruplets (quaternions).39 His work, starting with a concrete idea to overcome a conceptual difficulty with negative number ended with quaternions, abstract algebraic entities independent of concrete More recent authors use the reverse interpretation, i.e. α = a − b and β = c − d. See P. G. Scopes, (1973). Mathematics in Secondary Schools - A Teaching Approach. London. p. 92. Scopes also suggests using vertical ordered pairs. 38 Katz, p. 617. 39 Ibid., p. 618.

37

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interpretation. Application to vectors by Maxwell (1831-1879) and Gibbs (1839-1903) followed later. The formalisation of number was completed by Hermann Hankel (1839-1873) in 1867. The change involved moving completely from the concrete to the formal viewpoint. Thenceforward numbers could be introduced in a purely formal manner without need for the concept of magnitude.40 Starting with fractions, Hankel linked formal operation with content interpretation, introducing them formally as solutions of equations of the form x.a = b where a and b are natural numbers. To facilitate unrestricted division the domain of numbers was expanded by introducing numbers of the form b/a, interacting with earlier numbers in accordance with the definition (b/a).a = b. Similarly negative numbers were necessary to ensure solutions to all equations of the form ax + b = c. For the equation, x + n = 0, (n ∈N), the solution is x = −n so (−n) + n = 0. In this way the set of negative integers is obtained. Addition and multiplication is defined so that specified calculation rules will continue to hold to the extent possible without contradictions. Rules for addition and multiplication can be proved thus: For (−a) + (−b) (−a) + a = 0 (−b) + b = 0 [(−a) + (−b)] + [a + b] = 0 Thus

[(−a) + (−b)] = −[a + b]

For (−a) × b and (−a) × (−b) Postulating the rule, 0.a = a.0 = 0 and assuming the permanence of the distributive law (−a + a).b = 0 (−a).b + a.b = 0 Thus

(−a).(−b + b) = 0 (−a).(−b) + (−a).b = 0

(−a).b = −(a.b)

(−a).(−b) + −(a.b) = 0 Thus

(−a).(−b) = a.b

L. Hefendehl-Hebeker, (1991). Negative Numbers: Obstacles in Their Evolution from Intuitive to Intellectual Constructs. For the Learning of Mathematics, 11, 1, 26-32. p.30. 40

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Although Hankel and his followers extended the real number system within the framework of formal mathematics, independent of the basic content notions of quantity and magnitude and physical models, this did not mean that the algebraic systems needed to be kept isolated from content meanings. Prior to the formalisation process negative numbers and concepts had already become working elements in areas of mathematics including; •

index notation and logarithms,

•

directed distances and analytic geometry,

•

calculus and analysis.

Further applications soon followed including; •

introduction of the vector concept

•

more useful descriptions and simpler and more elegant computations in physics and physical sciences, eg. applications to velocities, forces, and electromagnetic and particle theory.

Thus the requirement of numbers needing to be associated with quantities and magnitudes had been removed. Henceforth numbers could be considered as abstract entities, governed by a small set of formal binary operational laws, derived from the extension of the intuitive laws applying to natural numbers. Implications for teaching Today, well over a hundred years since Peacock and Hankel took the development to the final stage outlined above, many students continue to be haunted and impeded by misgivings and misunderstanding regarding the meaning and reality of negative numbers. History indicates that it took negative numbers a long time to gain acceptance owing to perceived lack of need. Finally when needs, applications and usefulness were demonstrated opposition ceased. Learning and appreciation of negative number may be improved if students are allowed to pass carefully through an adapted form of the

25

historical developmental sequence. The following is the teaching sequence suggested by history;41 1.

negative coefficients42 in the context of arithmetic algebra using only unsigned numbers. The implication is that students are taught, for example, that a − 2(b − 3c) = a − 2b + 6c is a logical consequence of a subtraction process reasoned in the manner of early mathematicians. 43

2. use of signed numbers for the purpose of measuring and labelling of directional quantities and measures (e.g. temperature, altitude, share price fluctuations, measurement tolerances, ... ); locating signed numbers on horizontal and vertical number lines and order conventions; negatives and positives as opposites; the use of coordinates to locate and plot points on the Cartesian plane. 3. extension of the set of arithmetic numbers (i.e. unsigned integers, fractions and decimals) to include negatives to facilitate the solution of all linear equations and provide values for all subtractions (e.g. 0 − 1 = -1, 2 − 5 = -3, ... ); integer addition, subtraction, multiplication and division; signed rational number operations; formulae and expression evaluations involving negative numbers. 44 4.

symbolic algebra;45 application and use of negatives in coordinate geometry (graphing linear functions, finding slopes and intercepts (e.g. What is the slope of the graph of y = 6 − 3x? Where does the graph of y = 2x + 8 cut the x axis?); solving quadratic equations and graphing quadratic functions; index notation and indicial operations ... . However, in the modern context, because it has become standard

practice to use negative numbers for applications such as temperature measurement, and also the likelihood for students to have discovered The development would not fit within the three weeks usually allotted to the topic in either Year 7 or Year 8. The author’s view is that the teaching of negative number should be more carefully integrated with other topics on the basis of need and application.

41

Actually subtracted positive terms. Because there are no negative numbers in arithmetic algebra the given expression is valid if and only if a>=2 (b − 3c) >= 0 and b − 3c >= 0. Encouraging students to read (verbalise) expressions and equations before expanding or solving may be a useful teaching strategy. 42

43

44 At this stage numbers are considered to be and treated as abstract entities in their own right as is the case when students are given ‘sums’ to evaluate and equations to solve.

For students who achieve reasonable competency in arithmetic algebra the transition to symbolic algebra should be virtually seamless. The rules of arithmetic algebra become the rules of symbolic algebra but calculations can now include negative numbers. 45

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negative numbers through calculator and counting backwards activities in primary school, some aspects of step 2 may be more naturally and easily taught and understood before step 1. This is the suggested major modification to the historical sequence outlined above. The teaching of negative numbers, integrated where possible with other appropriate topics, should span the first four years of the secondary school mathematics curriculum. 46 The experimental teaching attempted in this study is within stages 2 and 3 outlined above.

46

Negative numbers should not be taught as a topic in Years 7 or 8 and then neglected.

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Chapter 3 Textbook Approaches to Negative Number This chapter contains an overview of textbook approaches to negative number and their likely impact on teaching practices and learning. In particular the role that textbook authors have played in establishing the number line as the preferred and most popular initial operational teaching model will be discussed. Methods of explanation, notational conventions, models and applications, and types of examples used in exercises and the use made of negative number in following topics will be compared. An article by Helena Pycior will be used to gain insights into the nature of negative number teaching in the nineteenth century. 1 Although the article deals with textbooks written or translated for use in the USA, the nature of British and French textbooks, on which they were based, is also described. Also useful for the purpose of this study is the comparison of the synthetic and analytic styles of teaching algebra. Direct reference to the actual textbooks used for the teaching of algebra in Victorian schools in the twentieth century will be used for the remainder of the chapter. From the latter half of the nineteenth century until the era of New Maths, beginning in the 1960s, algebra textbooks used in Victoria tended to be either from England (e.g. Hall & Knight, 2 Borchardt, 3 Durell, 4 Lockwood and Down 5) or be produced locally and follow an English type content and approach (e.g. Wilson & Ross, 6 Wilson. 7) Before New Maths it was customary for algebra to be taught as an isolated subject. The topic of negative number (in many books introduced as directed or signed numbers) was usually included among the middle

1

H. M. Pycior (1989). British Synthetic vs. French Analytic Styles of Algebra in the Early American Republic in Rowe, D. E. & McCleery, J. (eds.). The History of Modern Mathematics Ideas and Their Reception. San Diego. 2 H. S. Hall & S. R. Knight (1887). Elementary Algebra for Schools. London. (in print until after 1964). 3 W. G. Borchardt (1946). A First Course in Algebra (4th Edn). London. (originally published c1900) 4 C. V. Durell (1954). A New Algebra for Schools. London. (reprinted 28 times, 1930-54) 5 E. H. Lockwood & D. K. Down (1954). Algebra. London. 6 R. Wilson & A. Ross (1951). The First Two Years’ Arithmetic and Algebra (Rev.Edn). Melbourne. 7 J. Wilson (1946). A Junior Algebra for Technical Schools Parts I and II. Melbourne.

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chapters of junior secondary algebra textbooks 8 following introduction to arithmetic algebra. Examination of books, such as those mentioned above, reveals similar basic content and order of presentation. From the beginning of the New Maths period there was a much more eclectic mix of textbooks and curriculum materials with locally authored books becoming the norm, supplemented by texts and materials from Britain (e.g. SMP), the U.S.A. (eg. Creative Publications, SRA and NCTM materials) and Canada (e.g. Del Grande et al. 9 - a cooperative Canadian/Australian effort also published in Australia).

Nineteenth century approaches Whilst in Australia the early practice was to adopt British textbooks, mathematicians in the USA were more active in either adapting British or French materials and writing their own books. Pycior discusses examples of textbooks used in the Early American Republic and in doing so compares British synthetic and French analytic styles of algebra. A typical synthetic teaching style begins with statements of definitions and rules followed by examples and applications. It places little or no emphasis on detailed development of structure and reasons for the definitions and rules. An analytic style is a discursive and reasoning approach intended to lead students to the discovery of ‘truths’. 10 The dichotomy of synthetic and analytic teaching styles, particularly with regard to the teaching of negative number, provides a useful framework for discussing the textbooks to be compared in this chapter. By the turn of nineteenth century the British and French had developed distinctly different ways of presenting the subject and Americans were considering the relative merits of the two algebraic styles. Some textbooks (eg. Jeremiah Day, Introduction to Algebra, first published in 8

It was common practice to use the same introductory algebra text in both Years 7 and 8. J. Del Grande, P. Jones, L. Morrow, D. Kennedy, I. Lowe, I., & D. Wirth (1972). Math Book I. Hawthorn, Victoria. J, Del Grande, P. Jones, L. Morrow, D. Kennedy, I. Lowe, & D. Wirth (1973). Math Book 2. Hawthorn, Victoria. 10 H.P. Fawcett, & K. B. Cummins (1970). The Teaching of Mathematics from Counting to Calculus. Columbus, Ohio. pp. 18-26. 9

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1814) followed the British synthetic style whilst Charles Davies, Elements of Algebra, which initially appeared in 1835, was basically a translation of a French textbook by M. Bourdon, Elémens d’algèbre, incorporating some British ‘practical methods’. Benjamin Peirce’s, Elementary Treatise on Algebra of 1837 generally ignored both styles. Of the four texts considered in the article only John Farrar’s translation of S. F. Lacroix, Elémens d’algèbre, published in 1818, was a purely French based method textbook. The negative number content of the first three of the textbooks, is outlined below.

Jeremiah Day, Introduction to Algebra Day (adopting the British synthetic style), acknowledging the difficulty that ‘young minds’ have in attaching meaning to the abstraction, ‘negative quantity’, started with a definition, and then offered examples from everyday life. 11 He defined ‘negative quantity’ as ‘one which required to be subtracted’ and gave as usual examples - loss/gain, debt/credit, backward motion/forward motion, and descent/ascent as illustrations needed to “catch the students attention and aid conceptions”. He also said, No quantity can really be less than nothing. It may be diminished, till it vanishes, and gives place to an opposite quantity. and qualified his definition with, The terms positive and negative, as used in mathematics, are merely relative. They imply that there is, either in the nature of the quantities, or in their circumstances, or in the purposes which they are to answer in calculation, some such opposition as requires that one should be subtracted from the other. 12 Pycior maintains that British mathematicians shaped Day’s algebraic work. Day shared their view of mathematics including “... their concern for the problem of the negative numbers, and even their frustration when the reality algebra failed to match the synthetic ideal.” 13 In writing his textbook

11

Pycior, p. 128. Ibid. 13 Ibid., p. 129. 12

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Day used the work of British mathematicians 14 pre-dating Peacock and Hankel who finally legitimated negative numbers with the development of symbolic algebra. 15 Thus Day would have been attempting to use negative numbers in the context of arithmetic algebra.

John Farrar’s translation of S. F. Lacroix, Elémens d’algèbre Farrar believed that the French style of algebra was more natural and beneficial to students than the rival British approach. 16 In translating Lacroix, Farrar thus adopted the French analytic style, which regarded algebra as a part of analysis and an instrument of discovery (a means of discovering new truths). Emerson, a colleague of Farrar, considered that the pedagogical merits of the analytic approach were especially evident in its development of negative quantities. All the apparent absurdity of the usual statement of the doctrine of plus and minus both in addition and multiplication is entirely avoided. 17 Lacroix had used careful explanations to arrive at solutions to problems prior to formulating and solving them in algebraic form. The message of the long, careful introduction was thus clear: algebra developed naturally from a desire to abbreviate or shorten, and then generalise, simple reasoning. 18 A significant feature was that negatives were approached by a gradual route and did not start with definitions. The analytic approach involved two stages; Stage I. Initially + and − signs are exclusively operational. Methods of performing operations on simple quantities that are represented by letters are referred to as those which consist only of one term (e.g. +2a, −3ab, etc.) and the use of the names ‘positive’ and ‘negative’ is avoided. The rules for operating with such quantities are developed. Terms preceded by a ‘−’ sign 14

Pycior, p. 128. See Chapter 2. 16 Pycior, p. 129. 17 Ibid., p. 132. 18 Ibid., p. 134. 15

31

are considered exclusively as terms of a polynomial rather than as isolated terms. The subtraction of −c from a is first considered as subtracting b − c from a which leads to the formula; a − (b − c) = a − b + c, assuming b ≥ c ≥ 0 (e.g. 10 − (6 − 4) = 10 − 6 + 4). The general implication is that subtraction of algebraic quantities leads to a change of signs. The rule of signs for multiplication was approached in similar fashion. 19 Stage II. Isolated negatives were treated much later in the textbook in a separate section on ‘Questions Having Two Unknown Quantities, and of Negative Quantities’. A problem (concerning payments made to a labourer, wife and son for their work) that generated a negative solution was given. The equations generated are 12x + 7y = 46 and 8x + 5y = 30 with solutions x = 5 and y = −14/7, (7y = −14). The question posed is, “How are we to interpret the ‘−’ sign which affects the insulated quantity 14?” It is known what to do when two quantities are separated by a − sign but how can a quantity be subtracted when it is not connected with another number when it is found? 20 The first consideration was that negative solutions occur due to contradictions in the formulation of problems. In the above case the wife and son were incurring expenses not covered by payments for their work. A ‘solution’ is to re-formulate the problem to reconcile the contradictions. In this case the wife and son could be seen as incurring an expense to the labourer not covered by the payments. Cases such as the above show that negative solutions to problems can occur so Lacroix felt no need to define or legitimate them. The major need was to ‘settle’ the rules governing their manipulation. The rules obtained in Stage I are insufficient because they do not refer to ‘insulated’ quantities. He had avoided the direct problem of subtracting −c from a, by using a − (b − c) but he considered that it should not be necessary to go through the intermediate step of making b = 0.

19 20

Pycior, p. 134. Ibid., p. 135.

32

Referring to the history of algebra Lacroix ‘... reconstructed the moment at which the negatives first arise; worked through to “the idea ... attached to a negative quantity,” which he expressed in an equation; and used the equation to derive the rules governing the negatives.’ 21 The greatest subtraction from a quantity, is the quantity itself, which produces zero, (a − a = 0). However when a greater quantity is to be subtracted from a lesser quantity, to indicate that we cannot subtract the greater entirely, negative quantities are introduced. The negative quantity indicates what is needed to complete the subtraction. For example for 3 − 5 = −2, the ‘−’ preceding the 2 indicates that it is necessary to add 2 for the subtraction to be completed ‘normally’ (3 + 2 − 5 = 0). 22 The meaning of −a can be inferred from the equations , a − a = 0, b − b = 0, ... , which can all be regarded as equivalent to zero. Thus since a = a + b − b, a − (+b) = a + b − b − (+b) = a − b and a − (−b) = a + b − b − (−b) = a + b. For multiplication, (a − a) × b = 0 so since a × b = ab, −a × b = −ab, which is necessary for the ‘first term to be destroyed’. Present day mathematicians may criticise this argument for its confused use of binary and unary signs. Similar arguments are provided to justify the remaining rules for multiplication. 23 Farrar’s translation of Lacroix’s textbook, was considered to have brought a new style of algebra, and algebra education, to America. Key methods and concepts (including negative numbers) were approached slowly and through the history of the subject.

21

Pycior, p. 136. Ibid. 23 Ibid. 22

33

Davies and Bourdon’s Elements of Algebra 24 Davies revised and adapted Bourdon’s Elémens incorporating a synthesis of the ‘practical methods of the English’ and the ‘scientific discussions of the French’. Bourdon’s two stage treatment of negatives resembled that of Lacroix with + and − signs introduced as signs of operation. In Stage I, polynomials are decomposed into two types of terms; those preceded by + are called positive terms and those by − negative terms. The applicability of the standard rules for addition and multiplication of such terms are demonstrated. Also, as for Lacroix, Stage II follows later in the book in a separate section entitled ‘Problems Which Give Rise to Negative Results. Theory of Negative Quantities.’ The following admission is made: The use of algebraic signs in the resolution of problems often leads to results which at first view are embarrassing; on reflection however, it will appear, not only that they are capable of explanation, but that by their means the language of algebra may be further generalised. 25 Examples of problems which give negative results follow. The initial problem used is finding a number which when added to b gives the sum a. The formula is written as x = a − b. If a = 24 and b = 31, then x = −7, which is called a negative solution. What does this mean? Two interpretations are suggested; 1. A negative value indicates some inconsistency in the statement of the problem (i.e. there is no number added to 31 which gives 24). 2. The value, independent of its sign may be regarded as the solution to the problem (i.e. x = 7 is the solution to the problem; ‘Find a number which subtracted from 31 gives 24.’). 26 Bourdon used the negatives to stress the role of convention in algebra. The names ‘positive’ and ‘negative’ are used in Stage I (earlier than Lacroix)

24

According to Pycior, p.137, Davies, Charles (1835) Elements of Algebra: Translated from the French of M Bourdon. Revised and Adapted to the Course of Mathematical Instruction in the United States. The textbook was the most popular American algebra translation with over forty printings from 1835 to 1910. 25 Pycior, p. 139 26 Ibid., pp. 139-140.

34

although it is stated that the use of these names is ‘improper’ but ‘sanctioned’ by custom. By appealing to convention for the explanation of negatives it is proposed that; •

some of the rules of algebra cannot be rigorously proved but come from an extension of other results,

•

algebra concerned ‘subjects’ that are often imaginary,

•

by convention ‘negative expressions’ were considered as quantities. 27 Results for multiplication and division involving negatives were

obtained by applying to simple quantities the rule of the signs established for the multiplication and division of polynomials. Bourdon argued that convention rather than demonstration should take over: We say then, that the rule for the signs, established for polynomial quantities, is extended to simple quantities, in order to interpret the peculiar results to which algebraic operations lead. 28 Davies’ modified the approach of Bourdon. He appended material from Stage II to Stage I. After determining the rules for addition and subtraction of polynomial terms he distinguished between arithmetical and algebraic sums as well as arithmetical and algebraic differences. In arithmetic, addition conveys the idea of augmentation, whilst in algebra, −b may be added to a to produce a numerical result that is less than a. Thus an algebraic sum may lead to a negative result. Davies version of Stage II opened with some general remarks on algebraic signs leading quickly to a definition of negative numbers. If a particular sign stood for a particular operation, that operation should be performed on every quantity before which the sign is placed. Pycior considers the text to be a ‘hodgepodge of synthetic and analytic styles’ lacking the elegance of synthetic textbooks that defined negative numbers at the very beginning as algebraic entities and lacking the pedagogical effectiveness of analytic textbooks that worked slowly from

27 28

Pycior, p. 140. Ibid.

35

polynomial terms to isolated negatives through examples. 29 Nevertheless, as mentioned previously, the textbook was popular.

Concluding comments. In general Pycior’s article provides useful insight into negative number teaching approaches and emphases in Britain, France and the USA in the nineteenth century. The books were written for the teaching of college algebra. Algebra was a required course in most American colleges in this period. No reference is made to the use of the number line. Nor is the number line used for the teaching of negative number operations in the English textbook written by Bowman, 30 in typically synthetic style, which first appeared early in the twentieth century and was intended for a similar type of undergraduate student as the American texts discussed. The number line first appears in Bowman’s text in an introduction to real and imaginary numbers. “We can think of the whole set of real numbers as being represented by the whole set of points on a straight line ...” 31 Use of negative numbers in the context of coordinates and linear and non-linear graphs follows over 200 pages later in the book. 32 The use of the number line for the teaching of signed number operations probably first appeared for the purpose of teaching secondary school level algebra. Both Bowman and the American textbooks used the intuitive results of arithmetic and arithmetic algebra to justify or develop the negative number operational rules. It may be useful to consider the textbooks used in Australian schools during the twentieth century from the perspective of the synthetic (rule-based) and analytic (exploratory) teaching style dichotomies used by Pycior in her discussion. This is particularly relevant because the experimental teaching method and materials developed for the purpose of this study tend toward the analytic style.

Textbooks used in Australia prior to 1960 29

Pycior, p. 142. F. Bowman, (1930). Elementary Algebra Part I (New impression). London. 31 Ibid., p. 155. 32 Ibid., p. 360. 30

36

English textbooks As mentioned previously three examples of textbooks, popular in England and used in some Australian schools were those written by Borchardt, Durell, and Lockwood and Down. Prior to New Maths a common practice was to teach Algebra, Geometry and Arithmetic as separate subjects (often with different teachers) up to Year 10 (Form 4) level. Older teachers, still involved in mathematics, may have used and been influenced by such texts. The books were quite common in private schools and selective high schools and were intended for a narrower range of high and middle ability students than those attending and studying mathematics in modern secondary schools. Also a greater proportion of teaching time was devoted to mathematics subjects; around 8 periods per week compared with about 5 now. Negative number chapters contained carefully worded justifications and explanations, formal treatment of the topic, statements of rules and extensive sets of practice exercises. The approaches used to introduce negative numbers and teach the four operations in Borchardt and in Durell are compared in Table 3.1 and Table 3.2 outlines the approach used in Lockwood and Down. A common feature of Borchardt, Durell, and Lockwood and Down is the thorough treatment of arithmetic algebra before the introduction of negative number. Thus students would have met and practised expansions such as a - (b - c) and 5(t + 1) - 3(t - 1), before meeting negative number. 33 Later, following careful introductory considerations of the need for negative number, the arithmetic laws are then assumed to be extendable to apply to negative number. 34 This method utilises the historical sequence of symbolic algebra development by Peacock and Hankel in the latter half of the nineteenth century.

33 34

Borchardt, pp. 28-30, Durell, pp. 81-85, Lockwood and Down, pp. 51-56. Borchardt, p. 94, Durell, p. 111, Lockwood and Down, p. 52.

37

Table 3.1 Approaches to introducing and teaching negative number operations in Borchardt (1946) and Durell (1954) Textbook Borchardt Chapter 11 Negative Numbers (Changes title of chapter to Directed Numbers following the introductory section.) Avoids stating (+/-) sign rules until students have done plenty of examples. Gives an abrupt treatment of multn. and division. Durell Chapter 7 Directed Numbers States sign rules after introductory explanation but before student work any examples. Precedes numerical and algebraic exercises with sets of imaginative application word problems.

35

Introduction In algebra meet expressions such as (3 - 8) i) If man has -$5 what is meant? Owe $5. ii) directed distances (number-line) and order convention. iii) Centigrade thermometer iv) positive and negative rotations v) Cartesian plane

Addition Uses number-line diagrams to show; (+5) + (+3) (-3) + (+5) (+3) + (-5) (-5) + (-3) N.B. (+5) + (-5) = 0 and generally (+a) + (-a) =0 Exercises contain numerical examples, word problems, summing algebraic terms.

Subtraction Uses ‘shop method’ (complementary addition) and numberline. e.g. (+5) - (-3) What must be added to (-3) to get (+5)? (-3) + x = (+5) ∴x = (+8) ∴(+5) - (-3) = (+8) ... “Several general results may be deduced from the typical examples ...”

Begins with signless numbers and explains need for signed numbers for directed quantities.

Uses the vertical scale to add a debt (-5) and an asset (+2). (-5) + (+2) [start at -5 and move up 2 steps] or (+2) + (-5) [start at +2 and move down 5 steps] = (-3)

Also uses the ‘ladder’ and subtn as opposite of addn first and follows with; (-3) - (+2) [start at -3 and move down 2 steps] = (-5) (-3) - (-5) [start at -3 and move up 5 steps]

In general; +(+N)= +N +(-N) = -N.

In general; -(-N) = +N -(+N) = -N

Imaginative set of directed number exercises using tables. Uses Centigrade scale as an example of ‘the number scale’ (ladder).

Multiplication Multn. by +ve numbers is repeated addn. Define multn by -ve numbers as repeated subtraction. e.g. (-9)×(-3) = -(-9)-(-9)-(-9) = +9+9+9 = +27 States sign rules symbolically and in words (like signs). “... results for positive numbers ... true for negative numbers, ... c(a+b) =ca+cb c(a-b)=ca-cb for all values of a, b, c ...” Explains using water temp. changes in a boiler with mid-day as zero hour. n is time (directed). e.g. If temp. incr. at 5oC/hr. At 2pm, n=+2, (+5) × (+2) =(+10) (hotter) If temp. decr. at 5oC/hr. At 9am, n=(-3) (-5) × (-3) =(+15) (hotter) “Argument can be applied to any directed numbers; we therefore make the following rule of signs ... (stated symbolically)”

The identity may have been better if given as x × y ÷ y = x.

Division Uses identity x ÷ y × y = x to show sign rules. 35 e.g. (-ab) ÷ (-b) = (+a)×(-b) ÷ (b) = +a “... we have the same rule of signs as for multiplication, ...” “... (ca+cb)÷c = a+b (ca-cb)÷c = a-b for all values of a, b, c, positive or negative.”

Treated as opposite of multiplication using numerical examples. Includes note about mult. and divn. involving zero concluding with, “... we shall never speak of dividing a number by zero.” States (in words) the like and unlike sign rules for multiplication and division. Finishes with a note about squares and square roots.

38

Number line (or scale) approaches were used in the three textbooks for addition and subtraction explanations, however each used a different context with Borchardt using the horizontal version still being used in modern books. Durell used vertical (temperature scale) interpretations. Lockward and Down used a variety of vertical (e.g. lifts and ladders), horizontal (e.g. forward and backward pacing) and other (temperature, altitude, banking) contexts. Durell and Lockwood and Down, in particular, contain quite imaginative application contexts and problems. For example; Two cars P, Q are travelling in the same direction along a road at u and v miles per hour, respectively. At noon, Q is s mile ahead of P. At what time will P pass Q? What does your answer become if u =20, v = 24, s = 2, and what does it mean? 36 (The solution produces t = -1/2 which is interpreted as P passing Q at 11.30AM.) The speed of a ship is 12 knots in still water. She steams along a channel along which there is a tidal current of x knots in the direction of her motion. Find in terms of x her time for a distance of 2 sea miles. Find the current when this time is (i) 8 minutes, (ii) 12 minutes. 37 ((i) produces x = 3 (a 3 knot current in the direction of the ship’s movement) (ii) x = -2 (a 2 knot current in the opposite direction to the ship’s movement))

However the standard of algebra required, for examples such as the above, may be beyond many of the students in current day classes of mixed ability Year 8 students. In general Durell used a synthetic teaching approach, presenting and generally developing rules and laws with little emphasis on student exploration and discovery. Borchardt allowed for a good deal of student working and exploration of integer addition and subtraction in the lead up to the rules. In contrast his approach to multiplication and division was somewhat trite. Lockwood and Down was quite an innovative textbook with the use of context based teaching exercises useful for class teaching and discussion preceding formal material and statement of rules and laws. The discursive presentation was generally analytic and exploratory in style

36 37

Durell, p. 114. Lockwood and Down, p. 107.

39

opening up the rules and laws. However it stopped short of allowing the students to articulate the rules for themselves. Table 3.2 Approach to introducing and teaching negative number operations in Lockwood and Down (1954) Textbook Lockwood Chapter 7 Directed Numbers Places +/signs over numbers In Borchardt, Durell and Lockwood, expanding brackets involving unsigned numbers is developed earlier (Ch.5) e.g. 10 - (6 - 3) = 10 - 3 = 10 - 6 +3 ... leading to a - (b - c) =a-b+c

Introduction Introduces each of the four operations with a Teaching Exercises (context word problems)

Addition Starts with lift movements. e.g. 6 floors up, followed by 4 floors down is equivalent to ... ? We will now use shorthand ... +6 + -2 = +4 Other themes used are temp. altit. bank. “Express ... statements in shorthand.” Follows with numerical ex. Slips in 3×+2, 3×-2, ... without explanation.

Subtraction Again uses lift. e.g. Starts at ground, goes down 2 floors, how far to 5th floor? (7 fl.) Can write as, -2 + ... = +5 Answering ‘What must be added to -2 to get +5?’ is subtraction. +5 - -2 = +7. Also explores with other themes including a +/- number rung ladder. Leads to useful rules e.g. ... subtracting a ‘minus number’ is the same as adding the corresponding ‘positive number’.

Multiplication Directed distances and times for train travel are used to quickly establish the sign rules. Other themes used are tank filling and emptying and heating and cooling temperature changes over time. Includes solving equations using negatives and giving negative solutions in exercises. e.g. 7(-x - 1) = -3(2x - 8) (x = -31)

Division Again uses train travel. e.g. “A train is going south at 30mph (-30). When will it be 60 miles south (-60)? Common sense says in 2 hours time (+2). In shorthand, -60 ÷ -30 = +2.” Sign rules are thus developed and then stated in words. Special rules for zero are explored using the train. e.g. If speed is 0mph position of train 2hr ago (0×-2 = 0) With a speed of 60mph how long will it take to cover 0 miles? (0÷60 = 0)

Victorian textbooks prior to ‘New Maths’ Two textbooks written specifically for and used widely in Victorian high schools and junior technical schools up until the introduction of ‘New Maths’ were Wilson & Ross, and Wilson. The negative number content of each is outlined in Table 3.3.

40

Table 3.3 Approaches to introducing and teaching negative number operations in Wilson & Ross (1938) and Wilson (1946) Textbook

Introduction

Addition

Subtraction

Wilson & Ross Algebra Section, Ch. 2, Like and unlike terms: meaning of a negative number. Ch. 8, Directed Numbers.

Introduced in Ch.2, Numbers less than zero. Interpreted in the context of money owed. Ch.8, Number-lines (horiz. & vert.) Directed quantity exercises.

Uses number-line. Includes adding simple algebraic expressions, e.g. Find the sum of 3a - 6b + 2c and a - 2b - 5c.

“subtracting (-3) is the same as adding (+3)” (explained as a cancelled debt) “subtracting 3 is the same as adding (-3)” 38 For subtraction of algebraic expressions the instruction given is to change the signs and add.

Vertical setting out for such examples. Wilson, Part II, Ch. 9 Positive and Negative Numbers. (The chapter title changes to Positive and Negative Quantities in the body of the text.)

Discusses nature, need and contexts of zero. Positive and negative scales (Centigrade thermometer) Definition of positive and negative numbers. “Numbers on the opposite side of zero are negative numbers and are distinguished by prefixing the minus sign.” 41

Uses horizontal number scale. Movements in a right direction are positive, left are negative. Includes numerical and algebraic examples in exercises and simple substitutions and evaluations. Also uses vertical setting out.

Explained using diagrams in terms of temperature changes; final temp. - initial temp. “ ... obvious that the subtraction of a term is equivalent to the addition of that term with the sign changed.” 42 The above process is then applied to numerical and algebraic examples.

Multiplicatio n The given sign rules “ ... should be carefully noted and remembered. Do not worry at present about any explanation of them. Learn to apply them accurately.” 39 Exercises merely apply the rules. “... method of adding together a number of equal terms.” 43 Sign rules shown using ‘profit’ and ‘loss’ arguments for 2 × 3 and 2 × (-3). 44 Uses commut. law for (-2) × 3. For (-2) × (3)

Division “The rules for division follow immediately from the rules for multiplication.” 40 Multiplication and division exercises included in the same set. Practice sets include numerical and algebraic examples and the use of indices.

Shows that mult. rule of signs applies to division. Extensive set of numerical and algebraic exercises and problems follow some including application of all four operations. 45

lets -3 = x. (-2) × x = 2 × (-x) = -x + -x = -x -x = -(-3) - (-3) = +(+3) + (+3) =6

Includes expansions, simplifications and equations in exercise set. 38

Wilson & Ross, p. 335 Ibid, p. 337 40 Ibid, p. 338 41 Wilson, p. 134 42 Ibid, p. 143 43 Ibid, p. 147 44 Ibid, p. 147. Text actually uses 4 × 3, 4 × (-3), ... . ‘2’ is used here to fit table space. 45 Ibid, pp. 152-160 39

41

Compared with the English texts discussed previously, the teaching content is far more abbreviated with Wilson and Ross providing minimal explanations for operations apart from addition. Rote learning of rules and completing copious sets of repetitive practice and application exercises was the general approach used. Wilson provided explanations for addition, subtraction and multiplication but reverted to the common approach of treating division as the inverse of multiplication and obeying identical sign rule behaviour. Similarly to Borchardt, both Durell and Lockwood & Down preceded the treatment of positive and negative operations with coverage of arithmetic algebra. Wilson and Ross (like the English texts) included expansions and simplifications of expressions with brackets and negative terms (e.g. Simplify 2p(p + 3) − 5(1 − 2p)) before teaching negative numbers. 46 In contrast Wilson inserted only plus signs in front of brackets before treating negative numbers. Thus expansions of brackets preceded by a minus were deferred and considered later as negative number multiplications.

Textbooks after the introduction of ‘New Maths’ New Maths textbooks The following selection of texts samples those written after the beginning of the ‘New Maths Era’. Apart from content change (e.g. new topics such as sets, mappings, motion geometry, metrication) and emphasis change (e.g. the integration of arithmetic, algebra, geometry into the single subject of mathematics at junior levels) a characteristic of many of the texts, written from around the mid 1960s on, was to make the books appear less formal. Some textbooks produced were more conversational in style and included cartoon characters, balloon text, colourful marginal notes, and in some cases, suggested the use of structured aids and game-type activities. Innovative examples were the companion textbooks, Math Book 1 47and

46 47

Wilson & Ross, p. 281-284. Del Grande, et al. (1972).

42

Math Book 2. 48 Negative number is introduced in Book 1 (intended for Year 7 in Victoria), treating addition and subtraction. Book 2 (Year 8) revised addition and subtraction and followed with multplication and division. A particular feature was the multi-embodiment approach adopted in the text with a wide range of models and strategies used for teaching and reinforcing the operations. The negative number content of the two books is outlined in Table 3.4. In contrast with the almost exclusive reliance on number-line approaches for negative number addition and subtraction used in most textbooks, Math Book 1 and Math Book 2 includes two versions of the neutralisation model used in the experimental component of this study. Book 1 uses ‘up and down arrows’ and ‘black and white ants’ as representations of positive and negative integers to teach addition and subtraction. Reversible ‘squares’ are used in Book 2 and the model was extended to cover multiplication and division. In general the books fully utilised ‘new maths’ language, terminology and methods such as sets and set notation, reflection images and Papygrams (mappings). The number-line and Mira mirror were used to complete the treatment of rational number (Book 2, Chs. 4 & 5). A discursive analytic approach 49 was used, attempting to lead students to the discovery of patterns and rules. With regard to negative number content, in particular, it was the most adventurous maths textbook available at the time. The books were not widely used as class textbooks in Victorian schools.

48 49

Del Grande et al. (1973). See Chapter 2.

43

Table 3.4 Approaches to introducing and teaching negative number operations in Math Book 1(1972) and Math Book 2 (1973) Textbook

Introduction

Addition

Subtraction

Del Grande, Math Book 1 (Year 7) Ch. 4 Integers (addn and subtn) Math Book 2 (Year 8) Ch. 2 revises integer addition and subtraction Chapter 3 Integers (multn. and division)

Examples of opposites (e.g. up/down, happy/sad, east/west, ... ) Integers are numbers and each integer has an opposite. Walks on the integer line starting at zero and later serial walks. (e.g. +3 walk, -2 walk, ... ). Order convention on integer line. Applications; temperature, banking. Examples of opposite integers (e.g. +$100 (dep.) -$100 (withd.) Up arrows, down arrows and black and white ants games used as intro. to cancelling (annihilation) of opposites.

Integer addition explored using integer walks, up (+) and down (-) arrows and black (+) and white (-) ant combinations . Cancelling and annihilation process used to obtain answers.

a) as take away Uses ants model. “... sometimes have to make adjustments first.” (Matched pairs of ants added to make ‘take away’ possible.) Shows process using 2 - -1 and 0 - -4. b) as difference Uses integer line diagrams;

Nomogram method of integer addition and subtraction is also demonstrated .

e.g. -3 - 1 = What must be added to 1 to get -3? (arrow from 1 to -3) Uses the ant model to show that subtraction is equivalent to adding the opposite. (This method overcomes the need for ‘necessary adjustments’.)

Multiplicatio n Adapts black and white ants model to paper squares with B on one side, (+1), and W on the opposite side (-1). Multiplicatio n equivalent to repeated addn. with flips used to form opposites. e.g. -2 × -3 is interpreted as ‘flip three lots of -2.’ (WW → BB BB BB) Through carefully graded exercises students expected to discover and complete sign rule statements themselves.

Division Treated as reverse of multiplication using black and white squares game and then with numbers. e.g. -6 ÷ -3 is interpreted as flip and divide the number of squares by 3. (WWWWWW → BBBBBB → BB) 6 ÷ -3 is modelled as (BBBBBB → WWWWWW → WW) -6 ÷ 3 (WWWWWW → WW) Again the students were expected to find the sign rules.

In general the preference in Victoria was for less challenging and more conventionally presented texts such as Modern Mathematics 50, On Course Mathematics 51 and Contemporary Mathematics 52 all of which used number-line and rule-based approaches with minimal emphasis on opportunity for student discovery. The negative number content of such texts generally consisted of re-formatted material from pre-New Maths, sometimes expressed in set language. 50

L, Dawes, G. Bail, B. Daffey, R. Harrison, & J. McLeod (eds) (1966). Modern Mathematics 2. Melbourne. 51 A. McMullen & J. Williams (1962). On Course Mathematics, Form 1, Numbers. Melbourne. 52 E. A. Byrt, (1970). Contemporary Mathematics II. Sydney.

44

For example; Simplify the solution set for (a) x + +12 = +18 + +9 ... (j) x + +52 = −(-40 − +12)

53

Graph the following sets on integer number lines (a) {a: a ≥ +2} ... (e) {e: e < -4} ... (k) {k: k < -4} ∪ {k: k > +2} ... Solve the following equations for x, where a, b and c are different elements of J, the set of integers. (a) x + a = b ... (h) − x = a − b ... (l) x + a − b = 0 54

Recent textbooks Popular current textbooks include SMP 11-16, Book R1, 55 Maths 8, 56 and Mathematics Today, Year 8. 57 SMP is commonly used in Britain whilst the other two are used in Victorian schools (including two of the schools participating in the experimental teaching in this study). Since the beginning of the ‘New Maths Era’ SMP has produced and refined many series of secondary school textbooks. The approach to negative number in SMP is summarised in Table 3.5. The presentation is clear, concise and uncluttered. A vertical number line is used only for demonstrating adding on and subtracting positive numbers. Pattern-based reasoning is used for adding on and subtracting negative numbers. Operational rules are concluded from the patterns. There is no number line pacing for addition and subtraction of negative integers, as is the case in many other recent textbooks. An innovative feature is the use of a paired number line enlargement model for integer multiplication. In general the book is presented in direct synthetic style with rules developed or given and worked examples leading to student exercises. Students using the text would not discover much for themselves.

53

Byrt, p. 30. Ibid. p. 32. 55 SMP 11-16, Book R1. (1994). Cambridge. (First published 1986, printed nine times to 1994) 56 B. Lynch, R. Parr, L. Picking & H. Keating (1991). Maths 8 (2nd edn). Melbourne. (First published 1982). 57 T. Daly, J. Ardley, J. Buruma, M. Cody, L. Mottershead & P. Tomlinson (1993). Mathematics Today, Year 8 (2nd Edn.). Sydney. (First published 1988) 54

45

Table 3.5 Recent SMP approach to introducing and teaching negative number operations Textbook

Introduction

Addition

Subtraction

SMP 11-16 Book R1 Chapter 11 Negative numbers

“Positive and negative numbers can be marked on a number line which extends in both directions.”

Uses pattern based approach to show that as the number you add goes down so the answer goes down.

Similar pattern approach used to show that as the number you subtract goes down so the answer goes up. Pattern shows that, “Subtracting -n is the same as adding n.”

Concludes chapter with brief explanation of handling negative numbers (including decimals) on a calculator (change sign key etc.)

Asserts that number line (shown vertically) can be used for simple operations, e.g. -2 + 5, -3 − 2 (i.e. adding or subtracting positive numbers) but not for adding or subtracting a negative number, e.g. 4 + -3, 7 − -3.

Continuing the pattern shows that; “Adding -n is the same as subtracting n.”

Good and bad behaviour points story used to show the rule leads ‘to reasonable results’.

Multiplicatio n Enlargement model shows multiplication of positive and negative integers by positive integers. Commutative law applies. Model extended for negative by negative. “Notice that multiplication by -2 changes the sign of a number.” Rule of signs given in table form.

Division Treated as inverse of multiplication. “... rules of signs ... the same as in multiplication.” Given in table form.

Temperature difference formula (a − b) (inside/outside )included in application examples.

The final two textbooks examined were Maths 8 and Mathematics Today, Year 8. The negative number content of the texts is compared in Table 3.6. Maths 8 contains large sets of repetitive exercises. ‘Running figure’ number line illustrations are used to assist the explanation of addition and subtraction operations and ‘blue box’ summaries are used to highlight rules and calculation hints; e.g.

++

is +

+−

is −

−+

is −

− − is +

.

Number line diagrams are used to explain positive by positive and positive by negative multiplications. The commutative law is used when the lefthand number is negative to facilitate number line use (e.g. -2 × +6 is changed to +6 × -2). However it is “... necessary to resort to number patterns to evaluate multiplication of two negative numbers.” 58 The text

58

Lynch et al., p. 61

46

carefully explains the use of like and unlike sign rules in the calculation process. Table 3.6 Approaches to introducing and teaching negative number operations in Maths 8 and Mathematics Today Year 8 Textbook

Introduction

Addition

Subtraction

Lynch Maths 8, Ch.3, Directed Numbers

Temp., dir. dist., altitude, opposites, labelling numbers.

Number-line pacing.

Number-line pacing.

To add; positive face east and pace forward, negative face east and walk backwards.

To subtract; positive face west and pace forward, negative face west and walk backwards. Movements show that adding positive gives same result as subtracting negative. (given as rule) ‘Rings’ pairs of signs as visual aid to simplifications . Indoor cricket scoring. Subtraction as inverse of addition. Pacing on the number-line. Observation of patterns leads to, “Subtracting a directed number is the same as adding its opposite.” and “Taking away a debt is like gaining number.”

Thermometer used to intro. number-line. Considerable emphasis on ordering using the number line

Daly Mathematics Today, Year 8. Unit 1, Directed Numbers Includes investigative projects (e.g. building an integer wall) and puzzles, e.g. magic square using positive and negative integers. Calculator usage explained including no need for + sign for positives.

Distance and direction exercises. Need for directed numbers established and use of +/labels and zero as origin. Opposites contexts and examples. Avoid use of raised signs or brackets for directed numbers, (e.g. use +3, -2, ... )

Number line using various themes (e.g. banking, temp., lift levels, ... ) “Make up a ... story about ... temp. for; -3 + 4 = 1 ...”

Multiplicatio n Serial counting forwards and backwards. Mult. a form of addition. Commutative property. Multiple hops on numberline. (“Always start from zero.”) Neg. by neg. evaluated by by continuing pattern to complete full table. ‘Same’ and ‘different’ sign rules then follow. Repeated addition. e.g. three 5c. losses = -5 + -5 + -5 = 3 × -5 = -15 “Therefore when different signs are multiplied the answer is negative.” No mention of commutativit y although exercises inc. negs. on either side. Patterns used to find and write neg. by neg. rule. (“Show your teacher.”) Findings checked using calculator.

Division “The same rules may be used for division of directed numbers.” Inverse of multiplication. Change to multiplication and invert second number. eg. −10 ÷ −5 = −10 × −1/5 = +(10 × 1/5) = +2, so −10 ÷ −5 = +(10 ÷ 5) = +2

Treated as the inverse of multiplication. ‘Cloud’ beside exercises shows “So negative ÷ negative is positive”. Complete ‘division rules’ exercise is included. Again the calculator is used as a check.

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Compared with the text by Lynch et al., Mathematics Today, Year 8, is a more imaginative and attractively presented textbook. The directed number labels are not raised as in most other texts (i.e. -2 is used instead of -2). The number line is used for addition and subtraction but not for any cases of multiplication. The compact negative number exercise sets are scattered with student exploratory examples and some attention is given to leading students to sign rules through observation of patterns before seeing them stated in the text. In general Mathematics Today displays some analytic characteristics although some rules are ‘flagged’ alongside ‘discovery’ exercises. 59 The application puzzles and projects and inclusion of the use of calculators (and in later chapters spreadsheets) makes the presentation appear more interesting and up-to-date than that found in Maths 8.

Summary and Conclusion The article by Pycior provided insight into the contrasting teaching approaches used in Britain and France in the nineteenth century and their influence on mathematics teaching styles adopted in the USA. In particular the characteristics of synthetic (rule-based and structured) and analytic (discursive and exploratory) approaches on the teaching of negative number were outlined. The negative number content of a sample of textbooks used in the teaching of junior secondary level mathematics during the twentieth century was examined. From early in the century the number line became, and remained, the most common teaching model used for explaining directed (signed) number addition and subtraction. Concerns expressed by some mathematics educators and researchers 60 over the last twenty years regarding the limited effectiveness of the number line model as a teaching strategy, have seemingly been ignored. A variety of strategies have been used for teaching multiplication, particularly the product of two negatives. Early texts, which generally provided a thorough grounding in arithmetic algebra before introducing 59 60

Daley et al. p.14 and p.18. See Chapter 5.

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negative number, deduced the rules numerically and then symbolically used the laws of algebra. Before New Maths kinematics (directed speed, time and distance examples) or water tank filling or heating models were sometimes used to develop and justify the sign rules. Recent textbooks tend to use consistency patterns and tables to justify the rules. Practically all textbooks stress like and unlike sign rules for multiplication and division. The danger is that such rules (often quoted by both teachers and students in abbreviated form), in the absence of understanding, easily leads to misapplication and operational confusion. Various forms of the neutralisation model have appeared in mathematics education literature since the seeds for the teaching idea appeared in the article by Haner in 1947. However teaching strategies based on the model appear in only a minority of textbooks. Of the textbooks examined versions of the model appeared in only the companion pair Math Book 1 and Math Book 2. Blane and Booth 61 use the model as a supplementary integer addition and subtraction investigation project in their Year 8 text and for multiplication and division in their Year 9 text. However the model appears as the fourth strategy following the use of the number line, patterns and calculator investigations. There seems to be little point in using a model, intended to help students to explore and discover rules for themselves, if they have previously been given the rules. In conclusion it appears that few, if any, textbook authors have adopted the neutralisation model as the initial integer operations teaching strategy. This could be a major reason for its apparently rare usage in mathematics classrooms. General lack of understanding and awareness of the model is another likely reason for its neglect. A colleague suggested that there may be something wrong with the model as a teaching method. Perhaps there is. The intention of this research is to identify both the strengths and the weaknesses of the neutralisation model as the basis for

61

D. Blane, & L. Booth (1991a). Moving Through Mathematics Year 8, North Blackburn, Victoria. D. Blane, & L. Booth (1991b). Moving Through Mathematics Year 9, North Blackburn, Victoria.

49

teaching negative number concepts and operations. The author’s view is that the number line should also be used, but not for the initial teaching of operations. The number line is essential for such things as order, rational number, directed distances, directional measures and graphing - the things that cannot be taught using the neutralisation model.

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Chapter 4 A Study of Outcomes of Normal Teaching Approaches Introduction This chapter reports a small preliminary study (Study P) conducted for the purpose of investigating learning outcomes of normal negative number teaching practices. Study P took place in one of the three schools (School B) involved in the major experimental study of the neutralisation model (Study E). 1 Normal teaching As discussed in Chapter 3, it has become common practice, due to the influence of popular textbooks, to use the horizontal number line and pattern tables as the basis for teaching negative number operations and justifying the sign rules. Some modern textbooks also include the use of calculators in the teaching of the topic. For the purpose of this study popular textbook-based classroom teaching will be defined as ‘normal teaching’.

Study P design and method To obtain an indication of learning outcomes of normal negative number teaching and to gather base-line data for comparison purposes later, Year 8, 9, 10 and 11 classes were tested in August, 1994. The single Year 8, 9 and 10 classes tested were not streamed and were considered by the mathematics coordinator to be reasonably representative of their respective year levels with regard to mathematics ability. The two Year 11 classes tested consisted of students who had selected Mathematics Methods as a subject in their VCE courses. 2 The mathematics coordinator selected and

1

There were three schools (A, B and C) involved in Study E. Study P testing was done in School B, because School A and School C were engaged in trialing of experimental and test materials in preparation for Study E in the following year (1995). School A also assisted with trialing of the tests used in Study P. 2 Students can opt out of studying mathematics in Year 11. Mathematics Methods (including algebra, functions, probability and calculus) is for students af average ability and above and is necessary for continuing studies in mathematics. The classes tested may have been ‘above average’ in mathematics ability compared with the more junior classes.

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arranged for the classes to be tested. The mathematics teachers administered the tests. The numbers of students tested at each year level were; Year 8 (23), Year 9 (19), Year 10 (22), Year 11 (39). In the context of the overall research project the testing described and discussed in this chapter was a minor component. It facilitated a ‘snapshot’ case study of negative number skills in the types and levels of classes to be involved in Study E in the following years. The author did not observe any of the teaching but did meet and talk with teachers, some students and the mathematics coordinator after the tests were corrected and analysed.

Aims The specific aims of the tests were to; •

assess negative number skills at Year 8 several (5) months after the conclusion of the initial teaching described below,

•

assess negative number related skills at Years 9, 10 and 11 levels,

•

compare performances between year levels,

•

explore the relationship between negative number skills and basic algebra manipulative skills,

•

identify common weaknesses and attempt to relate and explain such weaknesses in terms of possible deficiencies in negative number teaching practices used.

Negative number teaching Year 8. At School B the initial teaching of integers and negative numbers was based on Unit 1 (Directed Numbers) of the textbook Mathematics Today Year 8. 3 An outline of the content and approach used in the book was given in Chapter 3. The teaching covering this material took place over a three week period early in Term 1, 1994 from late February to mid March. Later in the same book negative numbers are used in the context of Coordinates and Graphs (Unit 4) and Numbers and Exponents (negative indices for very small numbers) (Unit 5). By the time of the test (August)

3

Daly et al., (See Unit 1, Directed Numbers, pages 1-27)

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the latter topics had been taught. The Year 8 class tested contained twenty three students. Years 9-11. According to the coordinator and teachers any teaching of negative number after Year 8, tends to be of an incidental nature as part of topics requiring the application of such skills e.g. indices, standard form and scientific notation, algebraic manipulations (factorising, expansions, simplifications), solving equations (linear and quadratic), coordinate geometry (slopes, intercepts, linear and non linear functions). None of the teachers consulted considered that reference to a model (e.g. number line) and spending time reteaching the topic served much purpose after Year 8. The usual reaction, when sign errors are observed, is to remind students (orally or in writing) of the rules (e.g. ring pairs of signs and write ‘?’, comment “signs” or “you are adding not multiplying”, ...). The emphasis appears to be on correctly applying rules rather than on understanding processes. Investigating the general effectiveness of the teaching approaches outlined above is the purpose of this part of the study.

Tests Two tests were designed by the author to gather the data and provide the insights required for this component of the study. The reason for using two tests was that the Year 8 class was only tested on subject content that had been taught prior to the test being given, whilst the Years 9, 10 and 11 test included application skills beyond Year 8. 4 However there were some questions and items (on integer operations) common to both tests. The Year 8 test will be referred to as Test P8 and the Years 9, 10 and 11 test as Test P9/11. Versions of the tests were trialed with classes in School A and necessary modifications and corrections were made before administration in 4

Originally it was hoped to use a more imaginative and general form of test containing items with simple teaching frames containing examples (e.g. vector and matrix addition and scalar multiplication). Students could then have used and demonstrated their signed number knowledge and integer operational skills in a more interesting and imaginative context. However the coordinator and principal did not want ‘unfamiliar items and things that had not been taught’ included in the test. The grade six child of a colleague managed the proposed items correctly and without difficulty.

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School B. Usual class teachers administered the tests which were intended to be completed comfortably within a 40 minute lesson. Test P8 5 included; •

positive integer calculations,

•

integer operations,

•

sequence completion,

•

ordering,

•

word problems.

The test was designed to cover the range of negative number skills initially taught in Year 8. Test P9/11 6 included; •

integer operations,

•

substitution and evaluation,

•

completing tables of values for given linear functions,

•

expansions and simplifications,

•

solving linear and simultaneous equations.

Results The results of both tests will be discussed together, where possible, to facilitate between year comparisons. For the purpose of this study total test performance is less relevant than performance on particular questions and items. However total test scores between Years 9, 10 and 11 can be compared and the relationship between integer skills and algebra performance will also be explored.

Positive number calculations In Test P8 addition and multiplication scramble tables (Q.1 & Q.2) were used to assess knowledge of basic number facts involving only unsigned numbers. Each table required students to fill 14 and 15 blank spaces respectively. For the addition table the mean number of correct responses 5

See Appendix 1.

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was 13.8 (s.d. = 0.4) and for the multiplication table 14.3 (s.d. = 1.4). Most of the errors that did occur appeared to be careless slips. Thus the students appeared to have a good mastery of addition and multiplication facts.

Integer operations Q.3 in Test P8 contained 30 integer operations items 24 of which were also included in Q.1 of Test P9/11. The mean scores on the 24 items for each year level are shown in Table 4.1. Table 4.1 Tests P8 and P9/11. Mean scores and standard deviations on 24 integer operations items Year level N Mean S.D. 8 23 14.3 6.0 9 19 18.1 6.5 10 22 17.1 5.9 11 39 22.1 3.2

To facilitate analysis of the strengths and weaknesses across the year levels the details of class performance on each of the items follows. The data has been sorted into operational skill categories.

Addition and subtraction The year level performances on the addition and subtraction items are compared in Table 4.2. Table 4.2 Tests P8 and P9/11. Number and percentage of students correct on addition and subtraction items Year Item 5+2 8+ -8 -4 + -2 -2 - -5 -6 - -4 -6 - 8 5 - -2 -3 - -3 8 # correct 19 21 17 10 11 12 11 16 (n = 23) % correct 82.6 91.3 73.9 43.5 47.8 52.2 47.8 69.6 9 # correct 16 18 14 11 13 15 12 12 (n = 19) % correct 84.2 94.7 73.7 57.9 68.4 78.9 63.2 78.9 10 # correct 21 19 14 12 13 13 12 17 (n = 22) % correct 95.5 86.4 63.6 54.5 59.1 59.1 54.5 77.3 11 # correct 38 39 35 36 35 35 35 36 (n = 39) % correct 97.4 100 89.7 92.3 89.7 89.7 89.7 92.3

0 - -4 15 65.2 13 68.4 15 68.2 36 92.3

Incorrect values given for -4 + -2 were 6, -2 and 2 with a total of nine students (Year 8 (2), Year 9 (2), Year 10 (4), Year 11(1)) 7 giving 6. “Because two minuses make a plus!” were reasons offered by Years 8, 9 6 7

See Appendix 2. Henceforward these statistics will be given in the form (2, 2, 4, 1).

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and 10 students. The Year 11 student’s incorrect answer appeared to be a slip as it was inconsistent with the quality of other work on the test. Many errors were made in each of the subtraction items with a range of incorrect responses being given for the more difficult items. This appears to indicate inability to interpret expressions, lack of understanding and probably confusion with rules. For example 31 students (13, 6, 9, 4) could not manage -6 - -4 resulting in the incorrect responses -10 (8, 3, 6, 3), 2 (4, 2, 2, 1) and 10 (1, 1, 1, 0). Questioning some students indicated that the most frequent incorrect answer (-10) was due to students interpreting and evaluating the expression as -(6 - -4) = -(6 + 4). Thirty four students (13, 8, 10, 3) were incorrect on -2 - -5 with -

7 (6, 4, 7, 3), being the most prevalent error. According to a Year 10

student “Two minuses make a plus, so 2 + 5 = 7 and the minus in front makes the answer minus”. The student evaluated the expression as -(2 - -5). Similar faulty reasoning appears to have caused many of the errors on other items. In summary the item analysis revealed that many students including some in Year 11 have difficulties with integer subtractions. Only around half of the Year 8 students tested appeared to be competent at subtraction following teaching of the topic and skill weaknesses were shown to persist in subsequent years. Some students who appear to know sign rules make errors because they are unable to correctly read expressions. 8

Multiplication and division Table 4.3 shows the year level performances on multiplication and division items. Students giving incorrect answers almost unanimously gave the correct digit but the wrong sign. Ten students (5, 3, 1, 1) responded with 14 for the item -2 × -7. The weakest performance was for the item 6 ÷ -2 where thirteen students (5, 4, 2, 2) gave 3 as the answer.

8

This type of weakness recurs frequently throughout the study.

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Table 4.3 Tests P8 and P9/11. Number and percentage of students correct on multiplication and division items Year level Item 3 × -2 2×5 2 × -7 12 ÷ 4 6 ÷ -2 8 # correct 17 18 14 16 12 (n = 23) % correct 73.9 78.3 60.9 69.6 52.2 9 # correct 17 16 16 17 15 (n = 19) % correct 89.5 84.2 84.2 89.5 78.9 10 # correct 19 21 18 20 17 (n = 22) % correct 86.4 95.5 81.8 90.9 77.3 11 # correct 38 38 37 37 36 (n = 39) % correct 97.4 97.4 94.9 94.9 92.3

The evidence suggests that many students do not master the sign rules in Year 8 and continue to display similar weaknesses in later years. Comparison of the data in Tables 4.2 and 4.3 shows that subtraction tends to be more difficult than multiplication and division. Subtraction involving negative numbers seems to provide more interpretational possibilities and thus a greater range of incorrect responses.

Equations Eleven of the integer operations items required students to fill empty boxes on the left hand side of equations. Tables 4.4A and 4.4B show the year level outcomes. Year 11 students were generally more proficient at this task but several made errors in some of the items. Many Year 8, 9 and 10 students found the items difficult. Collectively the equations involving subtraction operations produced the largest number of errors with a variety of incorrect values given for several of the items. Table 4.4A Tests P8 and P9/11. Number and percentage of students correct on equations items Year level 8 (n = 23) 9 (n = 19) 10 (n = 22) 11 (n = 39)

Item # correct % correct # correct % correct # correct % correct # correct % correct

-

4 + [ ] = -7 3 + [ ]= -2 -3 + [ ]= 4 11 16 16 47.8 69.6 69.6 16 16 14 84.2 84.2 73.7 13 19 17 59.1 86.4 77.3 38 38 36 97.4 97.4 92.3

3-[]=8 13 56.5 11 57.9 16 72.7 33 84.6

-

5 - [ ]= 2 10 43.5 12 63.2 14 63.6 34 87.2

57 Table 4.4B Tests P8 and P9/11. Number and percentage of students correct on equations items Year level 8 (n = 23) 9 (n = 19) 10 (n = 22) 11 (n = 39)

Item # correct % correct # correct % correct # correct % correct # correct % correct

-

3 - [ ] = -9 [ ] - -8 = 2 -2 × [ ] = 8 -20 ÷ [ ]= -5 -30 ÷ [ ]=10 9 8 14 10 16 39.1 34.8 60.9 43.5 69.6 12 12 12 17 13 63.2 63.2 63.2 89.5 68.4 14 10 13 19 15 63.6 45.5 59.1 86.4 68.2 36 30 37 37 32 92.3 76.9 94.9 94.9 82.1

For items of this nature success depends on being able to read and interpret the equation and on selecting and performing the correct operation. As the previous results show many students made errors on integer operations such as 5 - -2 = [ ]. The seemingly extra demands of filling a space on the left could perhaps be expected to lead to lower levels of performance on a comparable item (e.g. 3 - [ ] = 8). However this did not always happen. In Year 10 four students who could not manage the former item were able to do the latter. Some students in Year 9 (2) and Year 11 (3) who made errors on -4 + -2 completed -4 + [ ] = -7. Errors such as ‘two minuses make a plus’ may have become less likely in the context of ‘What number added to -4 gives -7?’ Brackets The Year 8 test included six items containing brackets and mixed operations. 9 Students were expected to perform a bracketed addition or subtraction followed by a multiplication or division if the simplest approach was recognised and adopted. Thus each item provides at least two opportunities for error using this method of calculation. Some students attempted to mentally expand the brackets. The results are provided in Table 4.5. Table 4.5 Test P8. Number and percentage of students correct on brackets items Year level Item 5 × (3-5) -2 × (6-9) (-3 + -6)÷3 (3-5) ÷ -2 -12 ÷ (1-4) -4 × (2 - -5) 8 # correct 14 12 12 11 10 7 (n = 23) % correct 60.9 52.2 52.2 47.8 43.5 30.4

9

Following trialing in School A, these items were omitted from the Years 9, 10 and 11 test which was found to be too long for completion in a 40 minute period.

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Overall the performances were poor. Four of the items contained similar bracketed operations ((3 - 5), (6 - 9), (1 - 4)). Judging by the pattern of responses for such items some students appear to have evaluated the brackets as 2, 3 and 3 respectively (i.e. they have reversed the subtraction) A common incorrect response for the easiest item (5 × (3 - 5)) was 10, due to evaluation as 5 × 2. One student spoken to was found to have incompletely expanded the bracket obtaining 15 - 5 = 10. The three most difficult items each elicited several incorrect responses. The item, -4 × (2 - -5), produced eight wrong values (28, -12, -13, -

30, -7, -1, -40, 12). Five incorrect values were given for the items, 12 ÷ (1- 4), (-4, -16, -36, 36, -9), and (3 - 5) ÷ -2, (4, -4, -1, 0, 3).

-

Misinterpretation of operation signs provided another source of error. Students found (invented) several ways of making errors in the items and incompetence in negative number operations was just one of them.

Scramble tables Test P8 also tested integer operations skills in the context of addition and multiplication scramble tables. Q.5 and Q.6 required completion of addition and multiplication scramble tables. (See Figures 4.1 and 4.2). + 1 -2

2 3

-4 -3

0

3 4 1

0 -1 Figure 4.1 Addition Scramble Table

× 1 2 -3

2 2

-4

0

3

4 Figure 4.2 Multiplication Scramble Table

Completing the addition table required supplying 12 missing integers. The mean number of correct responses obtained was 8.7 (s.d. = 1.3). The weakest responses were in the last row which required the students to firstly provide the number to which 3 is added to produce -1. The multiplication table contained 15 missing integers. The mean number of correct responses was 11.7 (s.d. = 1.5). The table format appeared to enable weaker students to provide correct responses to types of calculations they could not manage in question three. For example more

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students correctly completed the whole of row 3 than gave correct responses for similar items (pre-multiplying by a negative number) in question 3. Most errors occurred in row 4. Students first had to solve, ? × -4 = 4 and then use the answer (-1) to complete the rest of the row. The most common error was ‘1’ (7 students).

Other positive and negative number contexts and applications (Year 8) Test P8 also included; the use of integers in sequences (forwards and backwards serial counting), positive and negative number ordering, and the solving of simple word problems.

Sequences Q.4 contained five items requiring students to provide three terms missing from given arithmetic sequences. The latter four items included the need to provide negative integers. The mean score obtained was 4.0 (s.d. = 1.4). Table 4.6 shows the results. Table 4.6 Test P8. Number and percentage of students correct on sequences items Year level Item 1, 4, 7, ... 8, 6, 4, ... . , 3, 0, -3, .. .. , 2, 7 , 12, . 11, 6, 1, ... 8 # correct 21 20 20 17 15 (n = 23) % correct 91.3 87.0 87.0 73.9 65.2

The poorest response was for the sequence, 11, 6, 1, _ , _ , _ , for which the most common incorrect answer ( -5, -10, -15), was due to carelessness rather than an inability to count with negative numbers. Five students gave the correct values (-8, -3,) to the left of 2, 7, 12, but suggested -17 as the right hand term. One student said he had written minus because the other missing numbers were minus. Inability to count using negative numbers did not appear to be a general skill deficiency in the class.

Ordering Q.10 and Q.11 involved ordering of numbers. The majority of students successfully answered the first of the questions containing six number pairs requiring insertion of either greater than (>) or less than ( -1/5, ∴-0.6 > -0.5) 10 Q.11 required the placement of ten given numbers (integers or oneplace decimals) ranging from -3 to 4 in order from smallest to largest. Only nine of the 23 achieved the task correctly. Three students had two of the numbers out of order, seemingly due to slips and three students gave the numbers in correct descending order. The rest had more than two numbers out of order. Some appeared to have made errors due to misreading (e.g. omitting some negative signs) and leaving numbers out resulting in the need for alterations. Confusion over the meaning and interpretation of lowest and highest in this context has been suggested however this is unlikely because no student attempted to write the numbers in order of absolute value. It has been shown that many students have difficulties with the ordering of (unsigned) decimals. 11 The inclusion of negatives introduces a further complication so that the low level of competency found here is not surprising. Examination of the textbook used in School B 12 shows that ordering of integers is taught and included in exercises but the ordering of negative decimals is not given direct attention and practised.

Word problems

10 K. Moloney & K. Stacey (1997). Changes with Age in Students’ Conceptions of Decimal Notation. Mathematics Education Research Journal, 9, 1, 25-38. 11 Ibid., pp. 36-37. 12 Daly et al.

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Q. 7 used positive and negative numbers in the context of daily maximum temperatures. During a particular week in winter in the town of Fridgeberg the maximum daily temperature (0C) is shown in the following table: Da Sun Mon Tues Wed Thur Fri Sat y 0 C 5 3 1 0 -4 -7 -5 a) How many degrees difference was there between the highest and lowest maximum temperatures for the week? b) Between which two successive days was there the biggest change in maximum temperature?

The results are shown in Table 4.8. Table 4.8 Test P8. Number and percentage of students correct on temperature items a) Highest and lowest difference Year level Item 8 # correct 9 (N = 23) % correct 39.1

b) Successive days biggest change

12 52.2

Inability to correctly interpret the question handicapped many students. For example, eight students used the first and last day temperatures to answer (a) and four students considered Sunday-Saturday as successive days. Most students did not write down their calculation and the few that did used direct addition (e.g. 5 + 7 = 12) and avoided using negative numbers to find the required difference. No student wrote 5 - -7 for (a). Three students drew diagrams (mini thermometers or ‘vertical’ number lines). In general errors were mainly due to students selecting incorrect temperature pairs before giving an answer rather than being unable to, at least mentally, perform calculations.

Q.8 involved giving the altitude difference between two locations. The top of Mt Whitney in California, the highest point in the USA, is 4450 metres above sea level. Not far away Dantes View in Death Valley, the lowest point in the USA, is 86 metres below sea level. What is the difference in altitude between the two places?

Thirteen of the twenty three students (56.5%) gave the correct answer. Of these, nine students showed the addition, 4450 + 86 = 4536, and three wrote 4450 - -86 = 4536 (two placed a ‘+’ sign over the two ‘-’ signs). The most common error was to directly interpret the problem as a subtraction

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(i.e. 4450 - 86 = 4364). Eight students did this. None of the students appeared to make the error 4450 - -86 = 4450 - 86. The indications are that, for the type of word problems used in the test (typical of those appearing in Year 8 textbooks) many students can avoid using negative numbers and still obtain required answers. Inability to correctly interpret problems and write valid expressions was a cause of difficulty for some students.

Findings from Test P8 The students appeared to have a good grasp of basic positive number addition and multiplication facts. However at the time of testing (five months after initial teaching and without revision) many students (at least 30%) did not display a mastery of integer operations and related skills. The test items sampled a range of basic skills mastery of which would seem to be essential for advancing in algebra beyond Year 8. Performance deficiencies were revealed in using all four operations with subtraction being the weakest. Performance was also affected by other non-negative number related mathematical skill deficiencies (e.g. order of operations and ability to read and operationally interpret mathematical expressions and problems prior to making calculations). The contention is that the revealed deficiencies are at least partially due to the number-line based teaching method and emphasis on giving rules without providing understanding. Although the textbook used asks students to find rules using the number line and patterns (e.g. “What did you notice? Explain your answers.”) a highlighted box sits adjacent to the questions telling the students the answer. 13 Thus the need to think is discouraged! Lack of understanding of subtraction may also be due to the difficulty that students have in interpreting the process on the number-line using the ‘pacing and facing’ method of explanation given in the textbook. The actions representing the unary (label) and binary (operation) signs are easily confused. The text book uses ‘facing’ for the operation (face right to add, left to subtract) and ‘pacing’ for the label (walk forwards for positive,

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backwards for negative). 14 The events at the front of rooms or in schoolyards when teachers are attempting to explain and demonstrate this may be highly amusing to an observer but often frustrating for both students and teachers. 15 Also the process is clumsy and hard to draw in workbooks. This was the only model of the operation used and it failed to generate understanding for many students. Lacking understanding the students thus rely on rules. Subtraction as ‘difference’ is much easier to illustrate and explain on the number line. However this was not attempted. Use of negative number in algebraic contexts (Years 9, 10 and 11) Test P9/11 tested negative number knowledge and skill in several algebraic contexts. The purpose was to investigate the relationship between negative number skills and ability to do basic algebra (i.e. the type of algebra included in the test). The year level mean scores for each subsection of the test are shown in Table 4.9. Table 4.9 Test P9/11 Year level mean scores (standard deviations) for test subsections Year level

Integer Substitution operations & evaluation Q.2 Q.1 (/24) (/16) 9 18.1 9.4 (n = 19) (6.5) (5.1) 10 17.1 9.9 (n = 22) (5.9) (5.8) 11 22.1 13.1 (n = 39) (3.2) (4.0)

Tables of values Q.5,6,7 (/3) 1.2 (1.1) 1.1 (1.3) 2.5 (0.9)

Expansion & Linear Simplification Equations Q.8 Q.9,10 (/4) (/6) 1.1 2.6 (1.6) (1.7) 1.5 2.5 (1.8) (1.9) 1.7 3.7 (1.5) (1.8)

Algebra Total Q.2-Q.10 (/29) 14.3 (8.6) 15.0 (10.0) 21.0 (7.3)

The results show very little difference in performance between the Year 9 and Year 10 classes. Given the basic nature of the test the results of both classes seem poor and indicate generally low skill levels. The Year 11 (Mathematics Method) students did reasonably well across the test however some skill weaknesses were indicated in algebraic manipulations and equation solving. Students’ responses on the algebra items will be probed to 13

Daly et al, p.14. Daley et al., pp. 10-14. 15 The author has witnessed such events whilst supervising teaching practice for trainee teachers. 14

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determine whether weaknesses are due to lack of negative number skill or other causes such as faulty algebra and/or arithmetic (e.g. ab interpreted as a + b, incorrect order of operation). The item by item analysis that follows is an attempt to explore the relationship between negative number skills and ability to do algebra. There is also the possibility that lack of general arithmetic awareness (beyond knowing basic number facts and tables) and the ability to understand and do arithmetic algebra will hinder the development of negative number skills.

Substitution and Evaluation Q.2 in Test P9/11 contained 16 substitution and evaluation items. The year level results are shown in Table 4.10A and Table 4.10B. The correct answer together with common incorrect responses (CIRs) given by students for each item are also included and given in decreasing order of frequency. The incorrect responses are useful for analysing misunderstandings and weaknesses. The errors that seem mainly due to faulty integer operational skills are highlighted. (The latter procedure will be used throughout the analysis of the algebra applications items.) Table 4.10A Test P9/11. Number and percentage of students correct on substitution and evaluation items (a = 2, b = -3, c = -4) Year

Item

9

# correct

(n = 19) % correct 10

a-b

0-a

3+ab

0-ac

-2c

b2

15

14

13

16

13

6

12

11

73.7

68.4

84.2

68.4

31.6

63.2

57.9

16

14

15

17

14

12

15

14

72.7

63.6

68.2

77.3

63.6

54.5

68.2

63.6

34

34

33

36

33

31

31

31

87.2

87.2

84.6

92.3

84.6

79.5

79.5

79.5

5

-2

-3

8

2

-9, -20, -15

-8, -6, -2

# correct

(n = 39) % correct

abc

78.9

# correct

(n = 22) % correct 11

a+b+c

Answer

-5

24

CIRs

-9,

-24, -5,

9,

3

234

-5, -1,

6

8 -8,

6, -6,

9 -9, -6,

6

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Table 4.10B Test 9/11. Number and percentage of students correct on substitution and evaluation items (a = 2, b = -3, c = -4) Year

Item

3ab

b(c-a)

a-(b+c)

a-bc

a+c2

-2c2

c÷a

9

# correct

12

8

8

8

11

9

14

9

63.2

42.1

42.1

42.1

57.9

47.4

73.7

47.4

(n = 19) % correct 10

# correct

(n = 22) % correct 11

# correct

(n = 39) % correct Answer CIRs

2ab÷c

16

14

10

9

13

13

17

15

72.7

63.6

45.5

40.9

59.1

59.1

77.3

68.2

35

31

28

25

32

32

34

32

89.7

79.5

71.8

64.1

82.1

82.1

87.2

82.1

-18

18

9

14

18

-32

-2

-9,-12,

3

-18,-5,

24

-5,

14,-14 10,-24,-12 10,-14,-18 32,-18, 16 2, 6,- 8

3 -3,-6,

6

Errors were made in all items by students at each of the year levels. The highlighted incorrect responses show that for 14 of the 16 items the most commonly occurring errors are attributable to faulty integer operational skills. The other errors seem to be caused by misinterpretation of algebraic expressions (e.g. abc = a + b + c thus producing the incorrect response 5) or carelessness (several incorrect responses given defy logical analysis). Some students interpreted -2c as -2 + c = -2 + -4 producing the incorrect answers -6 or 6. A Year 10 student, consistently substituted directly for the letters without performing multiplication operations. The value of abc was given as -234, -2c as -24 and 3ab as -323. Some errors were due to incorrect substitutions and order of operation difficulties. For example the value of -

20 for 3 + ab was due to substitution of the value for c instead of b,

producing 3 + 2(-4), which became 5 × -4 = -20. (The student did, at least, manage to get the sign correct in the last step.) The easiest item overall was 0 - a, which became 0 - 2. The only incorrect response provided was 2. Some students (including five in Year 10) made this mistake. An extensive array of notational misinterpretations emerged across the items. Such errors are well known and documented in algebra research. 16 17 18

16

L. Booth (1984). Algebra: Children’s Strategies and Errors. Windsor. C. Kieran (1992). The Learning and Teaching of School Algebra, in Grouws, D. A. (Ed.) (1992). Handbook of Research on Mathematics Teaching and Learning. New York. 18 L. English & G. Halford (1995). Mathematics Education Models and Processes. New Jersey. (Chapter 7) 17

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The four most difficult items in Q.2 were; •

b(c - a) = -3(-4 - 2) = -3 × -6 = 18; incorrect responses included -18 (sign rule error), -7 + 2 = -5 (added -3 and -4), and 12 - 2 = 10 (expanded brackets but multiplied only the first term). (24.7% of all students incorrect)

•

a - (b + c) = 2 - (-3 + -4) = 2 - -7 = 9; errors included 2 - 7 = -5, -(2 + 7) = -9 and careless misreading of signs producing 2 × 7 = 14, and 2 × -7 = -14. (24.7% incorrect)

•

a - bc = 2 - -3 × -4 = 2 - 12 = -10; with errors made including 12 - 2 = 10, 5 × -4 = -20, -(2 × 12) = -24, 7 - 2 = 5, and 5 - 4 = 1. (24.7% incorrect)

•

0 - ac = 0 - 2 × -4 = 0 - -8 = 0 + 8 = 8 (found to be the most difficult of the sixteen items); giving errors including 0 - 8 = -8 (14 students), and -

6, -2, and 6 (all due partly to misreading of signs). (26.2% incorrect)

In general Year 11 performed much better than Years 9 and 10 on each of the 16 items. For several of the items there was little difference between the performance of Year 9 and Year 10 with Year 9 showing higher proportions correct on some of the items. Summing up, substantial negative number weakness was revealed at all three year levels. This combined with poor algebra skills produced very low performance levels on several of the items. Completing tables of values Q.5, Q.6 and Q.7 required students to complete tables of values for given linear functions. The results are given in Table 4.11. Also shown in the table are the given values of x, the corresponding values of y and the most common cause of error or incorrect values given.

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Table 4.11 Test 9/11. Number and percentage of students completely correct on table of values questions Year Item y = -2x y=2-x y = 7 - 3x (x values) (-3, -1, 0, 2, 5) (-5, -1, 0, 2, 7) (0, 2, 3, -1, -3) 9 # correct 11 9 2 (n = 19) % correct 57.9 47.4 10.5 10 # correct 10 7 7 (n = 22) % correct 45.5 31.8 31.8 11 # correct 36 33 27 (n = 39) % correct 92.3 84.6 69.2 Correct y values 6, 2, 0, -4, -10 7, 3, 2, 0, -5 7, 1, -2, 10, 16 Common errors 0, 8, 12, -4, -12 signs, adding 3, 1, ,5 (i.e. 7 - 3x = 4x) (i.e. -2x = -2 + x)

Overall Years 9 and 10 performed very poorly on these questions and interpretational difficulties were also apparent for some Year 11 students. Again it was evident that weak algebra skills (similar to those discussed previously) combined with negative number skill deficiencies to produce low performance levels.

Expansions and simplifications The four items involving expansions and simplifications were poorly handled by many students at each of the year levels. Table 4.12 shows the year level results. Table 4.12 Test 9/11. Number and percentage of students correct on expansion and simplification items Year Item 7 - 2(x+3) 6 + 3(x-2) 8 - 4(x-2) 3(x-2y)-2(x-y) 9 # correct 5 6 4 6 (n = 19) % correct 26.3 31.6 21.1 31.6 10 # correct 8 10 6 10 (n = 22) % correct 36.4 45.5 27.3 45.5 11 # correct 12 18 13 24 (n = 39) % correct 30.8 46.2 33.3 61.5 Answer 1-2x 3x 16-4x x-4y CIRs 9x-18 5x+15, 13-2x 4x-8, 4x, -4x 5x-4y, x-8y

The most common error with the first three items was to do the arithmetic preceding the bracket before expanding the bracket, thus 7 - 2(x + 3) became 5(x + 3) which was then simplified to 5x + 15. Thirteen Year 11 students and one in each of Years 9 and 10 consistently made this error across the items. The highlighted responses are those that seem to be caused

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by faulty integer skills. Some students (5 in Year 9, one in Year 11) confused the items with solving equations. More success was achieved with the fourth item because the students saw no easy simplification ‘up-front’ thus forcing them to expand the brackets first. However inability to correctly apply the distributive and sign laws produced errors. For example a Year 10 student showed the following incorrect working; 3(x - 2y) - 2(x - y) = 3x - 6y - 2x - y* = x - 7y. (*sign and distribution error) Three Year 11 students produced 3x - 6y - 2x - 2y* = x - 8y. (*sign error) The Year 11 VCE students did no better than Year 10 students.

Linear Equations Q.9 was intended to test the use of integer operations in the context of simple linear equations. The results are shown in Table 4.13. Table 4.13 Test P9/11. Number and percentage of students correct on linear equations items Year 9 (n = 19) 10 (n = 22) 11 (n = 39)

Item # correct % correct # correct % correct # correct % correct Answer CIRs

3x = -15 13 68.4 15 68.2 31 79.5 5 5

*Results for a similar item given as -5 × brackets.

-

6x = 12* 12 (12) 63.2 (63.2) 12 (13) 54.5 (59.1) 30 (37) 76.9 (94.9) 2 2

9 + 2x = 3 11 57.9 11 50.0 29 74.4 3 3, 6, -6

6 - 3x = 9 5 26.3 9 40.9 26 66.7 1 5, -5, 1, 3

= 10 in Q.1 (Integer operations) shown in

The results are poor for all year levels. In the first two items many students applied the appropriate operation but made sign errors. The comparison with the results on an earlier item shown in the table is consistent for Years 9 and 10 but at Year 11 the figures differ. This may indicate that some of the Year 11 students were relying on informal strategies and could ‘see’ the answer for “-5 times ‘what?’ equals 10”, but could not solve -6x = 12. 19 The fourth equation produced the lowest results with several incorrect 19 The response differences were not noticed at the time so students were not questioned about solution strategies used.

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answers given. As found previously wrong answers were caused by both poor algebra and integer operation errors. Examples of faulty working are; 6 - 3x = 9 -3x = 15* x = -5 (or 5**)

6 - 3x = 9 3x* = 9 x =3

6 - 3x = 9 -3x = 3 x = 1**

(Years 9, 10, 11)

(Year 11)

(Year 9)

(*algebra error, **integer operation error) Simultaneous Equations The percentages shown in Table 4.14 are based on the number of students attempting each item. There was little difference between Years 10 and 11 both of whom were ahead of Year 9. The elimination method of solution was used by most students. Faulty algebra, sign errors and careless slips were the main causes of incorrect answers. Table 4.14 Test P9/11. Number and percentage of students attempting and correct on simultaneous equations items Year Item x+y=0 2x -3y = 13 x - y = -6 * x + y = -6** 9 # correct 5 3 (n = 19) % correct 45.5 37.5 10 # correct 5 4 (n = 22) % correct 71.4 57.1 11 # correct 17 11 (n = 39) % correct 60.7 57.9 *# attempting item: Year 9 (11); Year 10 (7); Year 11 (28) **# attempting item: Year 9 (8); Year 10 (7); Year 11 (19)

Findings from Test P9/11 Test P9/11 sampled knowledge and skills at around the minimum level required to indicate mastery of integer and negative number concepts and skills adequate for middle to upper secondary school mathematics and beyond. The number content was Year 8 level and the algebra content a combination of Years 8 and 9 levels. On number operation items competent students should have been capable of almost perfect scores. However although students in Years 9, 10 and 11 were shown earlier to have generally performed better than Year 8 on integer operation items some students at the upper levels made similar errors to those observed at Year 8. The implication is that lack of understanding and misapplication at the initial teaching stage continues for some students unchecked through the

70

later years. This indicates the need for more effective teaching at Year 8 level. The attempt to assess integer operation skills in the context of basic algebra was somewhat thwarted by the poor arithmetic and algebra skills of many students. Lack of mastery of order of operations and arithmetic algebra (which history suggests should precede and assist entry into and understanding of negative number concepts and operations 20) made measurement of ability to use negative numbers in performing algebraic tasks difficult. A major drawback for some students at each of the levels tested was inability to read and interpret given expressions. If students are unable to learn, for example, what abc or a2 means or are unable to operationally interpret a formula or expression then the teaching of further algebra and the use of negative number skills seems pointless. Historically negative numbers as worthwhile entities were validated by demonstrated usefulness and application in algebra (e.g. as solutions to equations), coordinate geometry and calculus. These days the assumption is made that negative (or directed) number skills are needed before advancing very far in algebra. In fact this is false. In performing algebraic manipulations the use of negative numbers is unnecessary. Negative numbers only become necessary when expressions are being evaluated. The symbols may represent negative numbers but the numbers (coefficients) in front of the symbols are always added or subtracted positives. This is how they should be treated. Thus a key step, that must be included on the pathway to teaching negative number, is the ability to handle and understand the behaviour of subtracted positives. This should be included early in the teaching of algebra in Year 7 and precede the teaching of negative number operations in Year 8. The author’s assertion is that algebra and negative number skill weaknesses may become less prevalent if students learned, understood and used expansions such as the following before being taught negative (integer) number operations; a + (b − c) = a + b − c,

a − (b + c) = a − b − c,

a − (b − c) = a − b + c,

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a + 2(b − c) = a + (2b − 2c) = a + 2b − 2c (i.e. the ‘− 2’ is not obtained by multiplying 2 by -1 and the writing of the middle step is essential), a − 2(b − 3c) = a − (2b − 6c) = a − 2b + 6c,

2 = 0 − 2,

-

a = 0 − a.

-

This tended to be the practice used before the introduction of New Maths. In short the teaching sequence (over several years) should be arithmetic, arithmetic algebra (including manipulation of subtracted positives (i.e. negative coefficients), simple linear algebra, negative number operations, simple coordinate geometry, symbolic algebra (harder linear equations, quadratic equations), ...

Conclusion The results of Study P demonstrate shortcomings of normal teaching approaches to the teaching of negative number. Students at each of the four levels tested showed substantial weakness in both integer operational skills and in ability to do basic algebra. In Year 8 the number line model used as the basis for teaching addition and subtraction operations and rules, and the pattern-based method of teaching multiplication rules without the use of a model, failed to produce satisfactory negative number skill levels for many students. Similar weaknesses were clearly apparent among some Year 9, 10 and 11 students. The findings support the criticisms of negative number teaching by Kuchemann, 21 Lytle, 22 and Sfard. 23 Sfard suggests that although students may become quite skilful at using negative numbers when they are initially taught, lack of understanding means that negative numbers never become fully-fledged objects and the ability to manipulate them becomes purely mechanical. It is thus “fragile and quick to disappear”. 24

20

See Chapter 2. D. Kuchemann (1981). ‘Positive and negative numbers’, in Hart, K. M. (Ed.) Children’s Understanding of Mathematics: 11-16. Newcastle. pp. 82-87. 22 P. Lytle (1994). Investigation of a Model Based on the Neutralization of Opposites to Teach Integer Addition and Subtraction, in Proceedings of the 18th International Group for the Psychology of Mathematics Education, Vol. 3, Montreal, 1994. pp. 192-199. 23 A. Sfard (1994). ‘Mathematical Practices, Anomalies and Classroom Communication Problems’, in Ernest, P. (Ed.) Constructing Mathematical Knowledge: Epistemology and Mathematical Education. London. pp. 248-273. 24 Sfard, p. 268. 21

72

Kuchemann and Lytle advocate the development of a simpler and more user-friendly discrete-element integer model. Developing and evaluating such a model is the purpose of the major experimental component of this research.

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Chapter 5

The Neutralisation Model

This chapter begins by reviewing the history and theory of the neutralisation model 1 for the teaching of negative number and outlines various embodiments and interpretations used since the idea was first introduced. The results and outcomes of relevant negative number research studies are then considered.

History and theory Lytle 2 considers that the earliest published description of the neutralisation model was given in 1947 by Haner 3 who identified signed numbers as, “... one of the principal breeding grounds for student maladies in algebra.” 4 In particular Haner was concerned by students’ inability to comprehend; • that adding a negative number is achieved by subtracting a positive number, • that subtracting a negative number is achieved by adding a positive number, • explanations interpreting subtractions as reversals of directions on number scales (lines), • why gaining is equivalent to ‘losing a loss’, • why signs of operations can be interchanged with signs of direction. He therefore decided to use an ‘active surplus’ concept for explaining integer subtraction. An active surplus consists of the integer ‘units’ that extend beyond the region of ‘deadlocked’ positives and negatives. For example 7 + -10 = -3, can be illustrated as; +++++++ −−−−−−− −−− deadlocked +ves & -ves active surplus

1

Other names used include annihilation model and cancellation model . Lytle (1994). 3 Haner, W. (1947). Teaching the Subtraction of Signed Numbers, School Science and Mathematics, 47, 656-658. 4 Ibid., p. 656. 2

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Intuitively difficult (or seemingly impossible) subtractions 5 are facilitated by introducing a region of deadlocked positives and negatives to allow the required number of either positives or negatives to be subtracted (removed). For example 2 − -5 = 7, can be shown as; new active surplus

++

+++++ (− − − − −) introduced deadlocked +ves & -ves

Haner identified the advantages of the ‘active surplus’ method as, •

readily observable positives and negatives in place of mystifying manipulations among calibrations on a number scale,

•

can be pictured quickly on a chalkboard to illustrate operations,

•

promotes observation, discussion and clear understanding of the subtraction process,

•

avoids premature abstraction and isolation of signed numbers from improperly understood contexts,

•

provides a meaningful diagram system to assist adolescent adventures along a road to mastery in mathematics.

An interesting feature of Haner’s approach was his use of the description ‘region of deadlocked positives and negatives’ and not actually identifying ‘the region’ as equivalent to zero in the article.

Use of counters Freudenthal 6 attributed the idea of using two colours of counters for the teaching of integer addition and subtraction to Gattegno (possibly in the 1950s). He categorised it as a non-geometric ‘new model’ 7 and referred to it as an ‘annihilation model’. Letting black counters represent positive integers and red counters negative integers, black and red pairs annihilate one another (i.e. 1 + -1 = 0). Alternatively black and red pairs can be created 5

The calculation may seem ‘impossible’ if the ‘take away’ definition of subtraction is used. The other meaning of subtraction is ‘finding the difference or missing part’. 6 H. Freudenthal (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht. 7 The number line using directed line segments (magnitudes) is a geometric approach.

75

from zero (i.e. 0 = 1 + -1). Theoretically the method is based on considering integers as ordered pairs of natural numbers satisfying the equivalence relation; [a, b] ∼ [c, d] ⇔ a + d = b + c. ‘Easy’ addition and subtraction operations (e.g. 3 + -5, -6 − -4, ...) are performed using the counters with simplifications achieved by the annihilation strategy. Calculations such as 4 − 6 and 3 − -5 are facilitated by the ‘anti-annihilation’ property of the model which enables generation of integers from zero (e.g. 0 = 1 + -1, 0 = 5 + -5, ...). Freudenthal likens the strategy to familiar methods used in column arithmetic calculations replacing 10 units by a ten and decomposing a ten into ten units. 8 A major strength of the model is that it gives integer operations a concrete and easily manipulative physical and visualisable representation using the ‘combine’ interpretation of addition and ‘take away’ interpretation of subtraction. Bell 9 considered a weakness of the model to be due to subtraction not always being possible without the intellectual leap of introducing zero pairs. Bell regarded this as a ‘luxury’ with respect to its contribution to directed number learning10 since it does not relate closely to any of the normal operations as applied in commonly used situations. He considered electrons and positrons to be the only obvious natural ‘annihilator’ model. He suggested that most common contexts are more satisfactorily modelled as displacements (directed line segments). 11 Grady, 12 who considered the number line to be, “... sometimes as confusing as it is helpful.”, turned to the neutralisation model using two colours of poker chips as the manipulative material. He asserted that the neutralisation technique provided a concrete representation of a very abstract notion and a workable alternative to the number line as a model for integer operations.

8

Freudenthal, p. 441. A. W. Bell (1983). Directed Numbers and The Bottom Up Curriculum. Mathematics Teaching, 100, pp. 28-32. 10 Bell appears to have not recognised that the neutralisation model is intended to teach signed number as distinct from directed number for which the number line seems essential. 11 Bell seems to be talking about vectors rather than numbers. 9

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Rowland 13, experimenting with primary teachers and a group of four 11 year old students, used red and black multilink cubes to teach integer addition and subtraction. He used an ordered pair notation 14 to record modelled calculations and moved through the following sequence of learning steps; 1. evaluation of mixed (positive and negative) piles, e.g. (3, 4) = -1 2. identification of equivalent piles, e.g. (4, 7) = (5, 8) = (97, 100) = -3 3. addition (combining) of piles, e.g. (2, 3) + (1, 4) = (3, 7) = -4 4. reduction to smallest pile, e.g. (1, 0) + (0, 3) = (1, 3) = (0, 2) = -2 5. easy subtraction, e.g. -8 − -5 = (0, 8) − (0, 5) = (0, 3) = -3 6. difficult subtraction (requiring addition of a neutral pair), e.g. 3 − -5 = (3, 0) − (0, 5) = (3, 0) + (5, 5) − (0, 5) = (8, 5) − (0, 5) = (8, 0) = 8. Rowland considered that modelling subtraction with the cubes avoids confusions caused by number line approaches to subtraction and assists understanding that subtraction is equivalent to adding the inverse. e.g. A possible setting out and steps for evaluating -5 − 2 is shown as, ooooo o o o o o (o o • •) ooooooo ---------- → ---------------------- → •• • • (• = black = 1, o = red = -1) 15 In ordered pair notation this would represent; (0, 5) − (2, 0) = (0, 5) + (2, 2) − (2, 0) = (2, 7) − (2, 0) = (0, 7) = -7. Battista 16 described what he claimed to be ‘A Complete Model for Operations on Integers’. He used two colours of counters (white and red), labelled ‘+’ and ‘−’ to represent positive and negative charges. Collections of charges (tipped into clear jars) model the integers. The integer value of a collection is determined by the difference in the numbers of positive and

12

M. Grady (1978). A Manipulative Aid for Adding and Subtracting Integers. Arithmetic Teacher, 26, 3, p. 40. 13 T. Rowland (1982). Teaching Directed Numbers: An Experiment. Mathematics in School, 11, 1, pp. 24-27. 14 Rowland uses vertical ordered pairs. However for ease of typing horizontal (+ve, -ve) notation is used for explanations here; e.g. (5 black, 3 red) = (5, 3) = 2, (4 black, 7 red) = (4, 7) = -3, (3 black, 3 red) = (3, 3) = 0 15 This the opposite ‘colour’ representation to that used for the Chinese positive and negative rods described in Chapter 2. 16 M. T. Battista (1983). A Complete Model for Operations on Integers. Arithmetic Teacher, 30, 9, pp. 115-133.

77

negative charges contained. If there are equal numbers of positives and negatives the value is zero, if more positives than negatives the value is positive, if more negatives than positive the value is negative. In Battista’s embodiment addition is modelled by combining collections and subtraction by removing counters from a collection. The model works clearly for some cases of multiplication using repeated addition (e.g. 2 × -3 = -3 + -3 (two lots of three reds). However it could be considered to lack rigour for cases such as -2 × -3, which is modelled as subtracting two lots of negative three charges from zero with no apparent explanation given as to why the first ‘-’ means subtract and the second ‘-’ means negative. It thus inconsistently mixes the unary and binary meanings of the sign. 17 The model becomes quite convoluted and confusing for division. Where the signs are the same in both numerator and denominator, for example for -24 ÷ -6, the interpretation is; How many times must -6 be added (to an empty jar) to get -24 (in the jar)? Answer is 4, thus -24 ÷ -6 = 4. 18 Where the signs differ in the numerator and the denominator, eg. 24 ÷ -6, the interpretation given would be well beyond the comprehension of most junior secondary level students. 19 Battista claimed that as well as modelling the integers and the four operations, the model can be used to illustrate important structural properties such as the commutative and associative properties, the existence of additive and multiplicative identities and the existence of additive inverses. He saw it as a useful (concrete and manipulative) aid for introducing students to integer operations and superior to the (usual) pictorial use of the number line. Each of the versions of the neutralisation model considered so far can be regarded as an ordered pair approach and operations could be 2 × -3 has been modelled as 0 − 2 × -3 = 0 − -6 = 6 + -6 − -6 = 6 + 0 = 6. An easier and more obvious interpretation would be the number of times -6 could be subtracted from a jar initially containing -24. 19 How many times must -6 be subtracted (from zero) to leave +24 in the jar? Four subtractions are needed, thus 24 ÷ -6 = -4. 17 18

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recorded in a manner similar to that used by Rowland. The integer values resulting from operations are obtained by neutralising to reduce collections to their simplest form in which case only one colour or type of counter remains. The absolute value of the integer is represented by the quantity of counters and the sign is indicated by the single colour or type of counter contained in the final collection. Recently some other ordered pair approaches have been used. Streefland 20 describes an ordered pair approach to teaching integers in the context of bus occupancy and people boarding and alighting. For example; five people on and three people off = (5, 3) = 2. The number of people on the bus has increased by 2. Two on and four off = (2, 4) = -2. The model has been used with primary level students for the teaching of integer addition. Linchevski & Williams, 21 in a teaching experiment with year 6 students, used a double abacus to model integers as ordered natural number pairs in the context of people entering and leaving a disco. The teaching experiment used the children’s intuitive understanding of the disco situation and their manipulations of the abacus to keep track of movements in and out of the disco gates to facilitate construction of the integers and to develop strategies for performing integer addition and subtraction operations. In the latter embodiments the neutral element is given by equal numbers entering or leaving. The integer values are modelled by the changes in bus occupancy or disco attendance. The actual numbers on the bus or in the disco are not considered. There have been several other articles dealing with versions of the neutralisation model including those by Weissglass 22 (black and red

20

L. Streefland (1993). Negative Numbers; Concrete and Formal in Conflict? in Contexts in Mathematics Education, Proceedings of the 16th Annual Conference of MERGA, Brisbane, 1994. pp. 531-536. 21 L. Linchevski & J. Williams (1996). Situated Intuitions, Concrete Manipulations and the Construction of Mathematics Concepts: the Case of Integers in Proceedings of the 20th International Group for the Psychology of Mathematics Education, Vol. 3, University of Valencia, Valencia, Spain, July, 1996. pp. 265-272. 22 J. Weissglass (1979). Exploring Elementary Mathematics. New York. pp. 126-130.

79

counters), Whimbey and Lochhead 23 (used a charged particle ‘chemical reaction’ model as a remedial teaching strategy for college students) and Hollingsworth 24 (jelly beans). The usual reason given for turning to some form of the neutralisation model is dissatisfaction with learning outcomes of the number line model. However, as indicated in Chapter 2, the neutralisation model appears in a minority of textbooks and the number line has remained as the preferred model.

Research studies

Study by Kuchemann Kuchemann, a member of the CSMS project team at Chelsea College, University of London, tested 13-15 year-old students’ understanding of negative number. 25 A finding was that older students performed significantly worse than younger students on negative number operations. He contended that the fall-off in performance was due to older students’ tendency to forget the meaning given to integers when they were first introduced. It was this result that prompted Sfard’s suggestion that negative numbers do not become fully-fledged mathematical objects for students. Consequently knowledge may quickly disappear. 26 Kuchemann recommended abandoning the number line teaching strategy and advocated use of the neutralisation model. For addition the (number line) model is extremely straightforward and effective... . However, for subtraction the (number line) is far more difficult to use, not only because the operation is not seen as a simple sequence but also because the meanings given to the integers (may) differ and are not (may not be) consistent with the simple meaning used for addition. 27

23

A. Whimbey & J. Lochhead (1981). Boxes of Charged Particles for the Teaching of Signed Number Rules. Focus on Learning Problems in Mathematics, 3, 4, pp. 1-4. 24 C. Hollingsworth (1992). Integers as Jelly Beans. The Australian Mathematics Teacher, 48, 1, pp. 32-34. 25 Kuchemann (1981). pp. 82-87. 26 Sfard (1994). pp. 248-273. 27

Kuchemann, p. 87. (parentheses inserted by present author)

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The difficulty with meaning that Kuchemann refers to may arise from the conventions chosen by textbook writers and teachers to explain the processes. Integer (number) values may sometimes be given by position on the number line and sometimes by directed length measures. Whilst addition can be modelled on the number line as a sequence of combined forwards (positive number) and backwards (negative number) moves, subtraction may also involve consideration of the direction faced. For example subtraction of a negative may be explained as facing left (west) and moving backwards. 28 The ‘take away’ definition of subtraction can only be used in simple cases (e.g. 5 − 3, -5 − -3). For more complicated cases (e.g. -3 − -5, 5 − -3) it is necessary to consider subtraction as either ‘finding the difference’ or ‘adding the opposite’. He continued with the following recommendation; This change in meaning suggests that the number line should be abandoned despite its proven effectiveness for addition, in favour of a more consistent approach, for example one in which the integers are regarded as discrete entities or objects, constructed in such a way that the positive integers cancel out the negative integers. The clear advantage of such a model is that the same meaning can be used for the integers both within and across the operations of addition and subtraction, and it seems likely that this would enhance children's understanding of subtraction in particular. 29

Study by Fary There have been practically no in-depth classroom research and evaluation studies of teaching using the neutralisation model. Barbara Fary30 included 28

Lynch et al., p. 55-56; Daly et al., p. 13-14. Kuchemann p. 87. 30 B. A. Fary (1980). Children’s use of Imagery in the Learning of a Mathematical Skill. (Unpublished M.Ed. thesis), Monash University. 29

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the use of the model as one of the teaching strategies whilst researching other issues in a study of imagery in the learning of negative number skills. She contended that teaching a mathematical skill using concrete generalisable models and diagrams would ... enable the learner to store the skill in memory as part of a complex network of related skills, concepts, images and episodes, and that this would lead to better retention and transfer of the skill. 31

Research was conducted on the topic of directed number teaching (addition, subtraction and multiplication of integers) with year 7 students at two high schools. The teaching was done using specially prepared linear programmed teaching booklets. Students were expected to work through the booklets at their own pace. The study used three treatment groups; •

a rules group (R) - emphasised rules of operation,

•

an abstract diagrams group (D) - used abstract diagrams complemented by materials (number lines and counters),

•

a realistic imagery group (RI) - diagrams and models of ‘real-life’ concrete situations (exercises using thermometers and temperature changes, ‘walks’ east and west on the number line (for addition and subtraction) and films of moving objects with the projector running forwards and backwards (for multiplication) were used to provide meaning for the operational skills.)

The abstract diagrams and materials treatment group was first taught addition and subtraction using the horizontal number line, and they then used white and black counters to represent positive and negative ‘integers’ for addition, subtraction and multiplication. The realistic imagery group also engaged in some practical activities such as marking out and pacing along number lines in the playground and comparing day-to-day capital city temperature differences using newspaper cuttings. In terms of the learning model it was expected that rules treatment would encourage procedural memory coding of the skills, abstract diagrams and materials would encourage imaginal as well as procedural memory of 31

Ibid., p. iv.

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the skills, whilst realistic imagery would encourage imaginal and procedural memory storage as well as storage of episodes in both imaginal and linguistic memory and develop richer and more varied storage in long term memory. The results provided only tentative support for expectations. On the immediate post-test of the skills, the rules group performed better than the other groups. On the retention test the realistic imagery group was better able to remember the skills. There was no significant difference between the groups with regard to skill transfer when tested six weeks after the initial treatments. The test task was to explain ‘to naive learners’ familiar and unfamiliar integer questions. Regardless of treatment method, students tended to rely on rules (either taught or self-developed) to perform and explain the skills. Rule-based behaviour emerged as the favoured way of coping with mathematical tasks. Fary provides no detailed description of the use made of, and student classroom reactions to, the counter-based component which was about 50% of the teaching provided in the abstract diagrams and materials group. Only 15 out of 67 students in the latter group (22%) chose to explain addition and subtraction skills in terms of positive and negative counters, with the rest preferring to use the number line embodiment. The fact that the students used the number line before the counters may have influenced student behaviour. Students who have successfully learned from the first embodiment may regard the second embodiment as unnecessary and therefore not take it seriously. Of the 15 students mentioned previously, only four could satisfactorily explain subtraction in terms of the counters. During the teaching students seemed able to use the counters “without too much difficulty”, but after a six week break students did not appear to remember how to handle the operations using the embodiment. The suggestion made is that students would need more time to “abstract the essential meanings of the skills”. 32

Study by Lytle

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Motivated by observed deficiencies in performance resulting from number line teaching methods, Lytle did a small case study investigating the use of the neutralisation model for teaching integer addition and subtraction. She used different coloured bingo chips labelled with a ‘+’ or a ‘−’ with four grade seven students (2 strong, 1 average and 1 weak student). Observations and semi-standardised interviews were used to evaluate learning outcomes. The students appeared to intuitively cope with integer additions and ‘trivial’ subtractions using the material. However difficulties emerged for subtractions where there was insufficient quantity of material for removal of the subtrahend. Lytle reports that such non-trivial subtractions, “did not appear to be demystified by the neutralisation model.” 33 The procedure of adding neutral pairs to facilitate subtraction was a non-intuitive, demanding task, and appeared to be a ‘rule without reason’ for the students. As mentioned earlier Bell expressed similar reservations. An intuitive obstacle is finding meaning for the fact that subtractions involving two negative numbers produce a positive result (e.g. -2 − -5 = +3). To some students this seems absurd. Lytle suggested that more careful teaching needs to be given to subtraction than was provided in her study. An interesting observation was that when students were given a mixture of addition and subtraction tasks students were uncertain about whether to ‘neutralise’ or ‘add neutrals’. Counter-intuitively, addition can involve removal (subtraction) of neutral pairs and subtraction the insertion (addition) of neutral pairs. The small scale and limited nature of the investigation permitted only very tentative findings and in particular the relative effectiveness of the model for teaching difficult integer subtractions (i.e. those requiring the addition of neutrals) was not clearly determined.

Study by Mukhopadadhyay

32 33

Fary, p. 229-230. Lytle, p. 197.

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Although the study by Swapna Mukhopadadhyay34 did not research the use of the neutralisation model it is of relevance to the present study because of the emphasis given to verbalisation of integer tile activities as part of the learning process. Her investigation dealt with students’ ability to construct stories to suit given addition and subtraction operation involving negative numbers. The actual task was to solve given equations orally (e.g. (-11) + (+15) = _ , (-3) + (-4) = _ ) explain the procedure and create a story to fit the equation. The students were encouraged to draw pictures and diagrams and to use concrete manipulatives to explain their thinking. It was found that students who could offer only algorithmic methods (rules) to obtain answers and were unable to attach meaning to equations with a model (e.g. number line movements, tally marks and cancelled tally marks) were also unable to provide a valid story and connect the equation to a ‘real world’ situation. The students who could describe the solution process in terms of a model were more able to provide confident and sensible narratives. Her theory is that the model facilitates understanding of the mathematics through the development of language and descriptive skills. The experimental teaching materials prepared for the purpose of the present study encourage the use of verbalisation and description of the integer operations modelled by the integer tiles. It is anticipated that such use of language will assist in developing understanding of the operational processes.

Study by Shui Another study relevant to this investigation is one conducted by Christine Shui around 1980. 35 The study is relevant because of its finding with regard to the value of using a single effective embodiment. Three directed number teaching methods were compared;

34

S. Mukhopadhyay (1995). Story Telling as Sense-Making: Children’s Ideas about Negative Numbers. In Hunting, R., Fitzsimons, G., Clarkson, P. & Bishop, A. (Eds) Proceedings of ICMI Conference, Regional Collaboration in Mathematics Education 1995, Monash University, Melbourne. pp.519-532. 35 C. Shui (undated). Teaching the Addition and Subtraction of Directed Numbers, A Shell Centre for Mathematics Education research project report, University of Nottingham.

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a) ‘systems and strategies’ explorations using directed number pairs (coordinates) and displacements on the number line (basically a single embodiment), b) guided discovery of addition and subtraction rules based on SMP work cards using several embodiments including horizontal and vertical number lines, coordinates and journey problems, (c) a more direct expository class lesson version of a combination of (a) and (b) followed by teacher led generalisation of the operation rules. The main conclusion from the study was that systematic embodiment of a single embodiment reinforced by intensive practice of skills (i.e. teaching method (a) above) produced greater immediate and more sustained learning than multi-embodiment approaches. This finding contrasts with the views of previous mathematics educators who advocated the use of several embodiments for the development of understanding of concepts and processes. 36 37 The experimental teaching method evaluated in the present study is also based on thorough teaching of a single embodiment at the initial teaching stage.

Chapter summary Dissatisfaction with the number line as a model for teaching signed number operations motivated a small number of teachers and mathematics educators to try for an alternative approach. Some turned to using the neutralisation model using counters of two colours to represent positive and negative integers. Despite the fact that the embryonic idea for the model was first suggested over 50 years ago and concerns are still being expressed about the effectiveness of the number line model, teaching based on the neutralisation model has not been widely adopted in classrooms. One reason is that the model seldom appears in popular textbooks. If it does appear it is usually given the status of an optional alternative embodiment or as a supplementary classroom or homework investigation activity. Another reason is an almost total lack of research attention to the model. 36 37

Z. P. Dienes (1971). Building up Mathematics (4th Edn). London. T. Leddy (1977). Mis-directed Number, Mathematics Teaching, 78, pp. 26-28.

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The small amount of research that has been reported is inconclusive and no where near persuasive enough to tempt teachers to try it in their classrooms. Research evidence supporting the value of an effective teaching model that enables the use of language skills to assist learning and understanding was described. There is also supportive evidence to suggest that thorough use of a single teaching model may produce better and more sustained learning than multi-embodiment approaches.

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Chapter 6 The Integer Tiles Teaching Method Description of the method As described in Chapter 5 previous interpretations of the neutralisation (annihilation, cancellation) model have used equal quantities of positive and negative counters (chips, multilink cubes etc.) to form zero. The embodiment developed for this study uses tiles. They are called ‘integer tiles’ because groups of tiles model the integers 1. A modification to the neutralisation model, for the purpose of this experiment, is the introduction of zero (neutral) tiles. The conjecture is that the use of the zero tile may assist students overcome difficulties mentioned by some authors (e.g. Bell, Lytle) with regard to the need to introduce zero to facilitate difficult subtractions. When confronted with an ‘impossible’ calculation (e.g. 2 − -3), a zero tile can be introduced first and then swapped for a balanced collection of positive and negative tiles. The author considers that the presence of a zero tile may help cue such a strategy during the learning process. For example the evaluation of 2 − -3 could be performed in this manner using the tiles: [+1] [+1] → [+1] [+1] [0] → [+1] [+1] [+1] [+1] [+1] → [+1] [+1] [+1] [+1] [+1] [-1] [-1] [-1] (2)

(2 + 0)

(2 + 3 + -3)

(2 + 3 + -3 − -3 = 5)

It is necessary for students to be given an appreciation that the zero tile has a value of ‘0’ equivalent to any chosen balanced set of positive and negative tiles. Thus when it is necessary to perform the neutralisation process a zero tile can be used to replace a balanced collection of positive and negative tiles. The intention is to provide a concrete, manipulable, embodiment to assist student understanding through practical interpretation and 1

The absolute value of non-zero integers is modelled by the size of simplified tile collections. In practice small absolute valued integers (say ≤ 10) are represented by small collections (e.g. 3 is modelled by three [+1] tiles and -2 by two [-1] tiles). For large valued integers students are encouraged to imagine the heaps. It is considered that such visualisation assists the learning process. For this reason it was decided not to use tiles with values such as [+2], [-2], [+3], [-3}, ... which would diminish the discrete counter-based properties of the model. Such tiles would be merely substitutes for written numbers.

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visualisation of integer concepts and operations. Addition is modelled by combining groups of tiles, subtraction by removing tiles from a group (take away), multiplication by repeated addition of equal sized groups and division as either repeated subtraction (quotition) or sharing (partition).

Teaching materials The only teaching materials provided for the experimental classes were sets of integer tiles and two accompanying work booklets. The integer tiles For the experimental teaching, it was decided to use double-sided, two centimetre square, white tiles cut from large sheets of three millimetre plastic by a workshop technician 2. There are two types of integer tile; •

a unit tile labelled ‘+1’ on one side and ‘-1’ on the other side,

•

a zero tile labelled ‘0’ on both sides.

Each student received a set containing at least ten unit tiles and two zero tiles. For group work and discussions the students were able to pool their materials.

The work booklets Experimental class students worked from and in two work booklets written by the author. 3 The booklets were intended to maximise the learning impact of the tiles. Copies of the booklets, are included as Appendix 9 and Appendix 10. The booklets covered the development of essentially similar operational competencies to those developed in current Victorian year 8 mathematics textbooks and to at least the performance levels appropriate to year 8 (i.e. Level 5 in Mathematics CSF 4).

2

Some early pilot study classes used sets of blank playing cards labelled ‘+1’, ‘-1’ or ‘0’. These proved to be impractical for extended use in the classroom situation due to damage, loss and cost. The plastic tiles were cheap and relatively easy to have produced in a suitable workshop. Alternatively the double-sided tiles can be made by carefully aligning and pasting computer produced ‘+1’, ‘-1’ and ‘0’ sheets onto cardboard or system board. The latter method has been used successfully for teacher in-service workshop presentations. 3 Hayes, B., ‘Integer Operations’, Work Booklets One and Two. (see Appendices 9 & 10) 4 Board of Studies (1995). Mathematics: Curriculum and Standards Framework. Carlton, Victoria.

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Tile-based activities were used as the initial strategy for developing negative number operational understanding and competency. Extension to the number line embodiment followed later for the teaching of ordering and coordinate plane exercises. The carefully sequenced teaching steps based, wherever possible, on use of the integer tiles were as follows:

Work Booklet One 1. Positive integer addition and subtraction. This is intended to provide an easy introduction to using the tiles. Following examples showing how tile collections can be used to represent integers, addition is modelled as combining tile collections and subtraction as removing tiles from a given collection. e.g. 3 + 4 [+1] [+1] [+1]

→ [+1] [+1] [+1] [+1] [+1] [+1] [+1]

[+1] [+1] [+1] [+1]

e.g. 7 − 4

[+1] [+1] [+1] [+1] [+1] [+1] [+1]

→

[+1] [+1] [+1]

For larger values (e.g. 151 + 237, 1000 - 643) students are required to imagine and describe what would happen using the tiles. This practice is used throughout the workbooks and students are continually encouraged to visualise and describe the operations in terms of tile manipulations.

2. Integer addition and subtraction. Showing only the negative side of the tiles students are first required to provide values for various negative collections. Simple cases of negative integer addition and subtraction examples follow. e.g. -3 + -4 = -7, [-1] [-1] [-1]

[-1] [-1] [-1] [-1] →

[-1] [-1] [-1] [-1] [-1] [-1] [-1]

e.g. -7 − -4 = -3 [-1] [-1] [-1] [-1] [-1] [-1] [-1]

→

[-1] [-1] [-1]

90

3. Using zero. The nature of zero and its various representations is explored. Some tile operations leave no tiles remaining. ‘Emptiness’ or ‘nothing’ can be represented by zero (0). e.g. -5 − -5 = 0 [-1] [-1] [-1] [-1] [-1]

→

[0]

The effect of adding zero to a given number (e.g. 6 + 0, -8 + 0,) or adding a number to zero (0 + 7, 0 + -75) is discussed. The neutralisation process is introduced by the following workbook explanation; Zero can also be represented by a collection of tiles containing equal numbers of [+1] and [-1] tiles. For example take a collection of tiles containing four [+1] tiles and four [-1] tiles. Form pairs of [+1] and [-1] tiles. Each pair has the value zero. Such a pair can be replaced by a zero tile. So the value of the whole collection is zero. We can write this as 4 + -4 = 0. Another way of looking at this is to say that pairs of [+1] and [-1] cancel or 'zero' each other. So 1 + -1 = 0. 5

Collections containing equal numbers of positive and negative tiles follow. Form (or imagine) six different tile collections that have the value zero. Write down the mathematical equation for each of your collections. 6

4. Addition involving both positive and negative integers. Tile collections containing unequal numbers of positives and negatives are evaluated by replacing the matched positive and negative pairs with zero. The unmatched tiles provide the positive or negative value of the original collection. e.g. -5 + 2 = -3 [-1] [-1] [-1] [-1] [-1] [+1] [+1]

→

[-1] [-1] [-1] [0]

→

[-1] [-1] [-1]

5. Integer subtraction including the use of zero to make subtraction possible. This is a key step in the teaching of integer operations using the 5 6

Work Booklet One, p. 5. Ibid.

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integer tile strategy. Zero is used to facilitate intuitively difficult calculations such as subtracting numbers from zero or subtraction in cases where there is a deficiency in the number of positives or negatives available for subtraction. When necessary zero tiles can be replaced by suitable positive and negative number tile pairs after which the required number of positive or negative tiles can be removed. The remaining tiles provide the answer. e.g. 0 − -4 = 4 [0]

→

[+1] [+1] [+1] [+1] [-1] [-1] [-1] [-1]

→

[+1] [+1] [+1] [+1]

e.g. -2 − -4 = 2 [-1] [-1] →

[-1] [-1] [0]

→

[-1] [-1] [-1] [-1] → [+1] [+1]

[+1] [+1]

The students use the tiles to effectively perform the operational steps without, at this stage, being required to write down the formal steps. However the students are encouraged to verbalise the process.

6. Review. Following the five steps above, students were required to complete addition and subtraction tables using the integer sequence from -4 to 4, based on their findings from the tile activities, and to look for patterns in their results that would lead to articulating addition and subtraction sign rules for themselves. 7

7. Positive integers multiplied by positive integers. Multiplication can be considered as repeated addition. For example 3 × 2 can regarded as three lots of two and illustrated with the tiles by forming three lots of two [+1] tiles 8.

7

Work Booklet One, p. 12. There seems to be no fixed convention for the additive interpretation of multiplication. Some textbook authors (Wilson, Daly et al., Lynch et al.) interpret 3 × 2 as 2 + 2 + 2, whilst others (Borchardt, Del Grande et al. use 3 + 3. The two interpretations are probably due to language usage. “Three times two” may become 2 + 2 + 2 whilst “three multiplied by two” becomes 3 + 3. The workbooks prepared for this experiment use the former interpretation.

8

92

8. Positive integers multiplied by negative integers. This operation was initially presented as repeated addition of equal sized negative tile collections. 3 × -4 can be regarded as three lots of four [-1] tiles.

9. Multiplication using different tile collections to represent numbers. For example; Make a collection containing three [-1] tiles and one [+1] tile. What is its value? Make another of the same collection. Combine the two collections. What is the total value of the combined collections? 9

[-1] [-1] [-1] [+1]

→

[-1] [-1] [-1] [+1] 2 × (-3 + 1)

[-1] [-1]

→

[-1] [-1] =

2 × -2

[-1] [-1] [-1] [-1] -

=

4

Examples of the type, are also used to show that; 2 × (-3 + 1) = (2 × -3) + (2 × 1) = -6 + 2 = -4, illustrating the distributive property. [-1] [-1] [-1] [+1] [-1] [-1] [-1] [+1]

→

[-1] [-1] [-1]

[+1]

[-1] [-1] [-1]

[+1]

→

[-1] [-1] [-1] [+1] [-1] [-1] [-1] [+1]

10. Multiplication of zero is modelled using the zero tiles and also as multiple lots of zero collections. e.g. 2 × 0 = 0 + 0 = 0 and 2 × (-3 + 3)

In the booklets each of the preceding steps is demonstrated and justified by the tile ideas and activities. It is necessary for the students to accept that the tiles model the integers and that tile manipulations can model the operations. However, in the steps that follow the tile activities are augmented by some formality and the number line model to complete the coverage of integer operations.

11. Negative numbers multiplied by positive numbers. In preparation for this operation the distributive property is extended to include cases of right

9

Work Booklet One, p. 14.

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hand distribution by using tile activities to interpret and expand examples such as (4 − 2) × 3, e.g. Four lots of three [+1] tiles minus two lots of three [+1] tiles leaves two lots of three [+1] tiles. i.e. (4 − 2) × 3 = (4 × 3) − (2 × 3) = 12 − 6 = 6 Extension is then made (in the set exercises) to cases with the left hand integer negative, e.g. -3 × 2. Earlier tile activities showed that 0 − 3 = -3, so -3 can be replaced by 0 − 3. Thus -3 × 2 = (0 − 3) × 2. Using the tiles, this may be described as no lots of two [+1] tiles minus three lots of two [+1] tiles, which becomes 0 − 6 which, in earlier tile activities, was shown to be -6. Therefore -3 × 2 = (0 − 3) × 2 = (0 × 2) − (3 × 2) = 0 − 6 = -6. Work Booklet Two 12. Negative numbers multiplied by negative numbers. The method is similar to that used in the previous step, e.g. -3 × -2. 3 × -2 = (0 − 3) × -2. Using the tiles the right hand side can be interpreted

-

as, ‘no lots of two [-1] tiles minus three lots of two [-1] tiles’ producing 0 − -6. Earlier tile activities showed that 0 − -6 = 6 In formal terms -3 × -2 = (0 − 3) × -2 = (0 × -2) − (3 × -2) = 0 − -6 = 6. Students worked through further similar examples and were asked to comment on a ‘curious feature’ of their answers.

13. Review of multiplication findings. Students are required to compile a multiplication table using the integer sequence from -4 to 4, using the outcomes of their tile activities, looking for patterns that suggest general rules. As for addition and subtraction earlier the intention is that the students will find and be able to express and write the multiplication sign rules for themselves. When pairs of integers are multiplied there are three possible outcomes - the result may be positive, negative or zero.

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a) A positive result may be produced in two ways. Describe how and give an example of each. b) A negative result may also be produced in two ways. Describe how and give an example of each. c) A zero result can also occur in two ways. Describe how and give an example of each. 10

14. Introduction of the number line. For the purpose of extending the discussion of negative numbers to rational numbers, order properties and plotting planar coordinates the number line is introduced. Also included at this stage are examples involving the addition, subtraction and multiplication of positive and negative rational numbers, applying the sign rules discovered and established using the tile activities.

15. Positive integer division. Initially division is treated as repeated subtraction or quotition. (Recall that previously multiplication was treated as repeated addition.) For example, because 10 - 2 - 2 - 2 - 2 - 2 = 0 then 10 ÷ 2 = 5. In terms of the tiles, how many lots of two [+1] tiles can be subtracted from a collection of ten [+1] tiles? Cases leaving remainders are included and division of zero by a positive integer is also explored.

16. Negative integers divided by negative integers. Again repeated subtraction is used. For example, since -8 − -2 − -2 − -2 − -2 = 0, -8 ÷ -2 = 4. In terms of the tiles, how many lots of two [-1] tiles can be subtracted from a collection of eight [-1] tiles?

17. Negative numbers divided by positive numbers. At this stage the relationship between multiplication and division is considered and the interpretation of division as 'sharing' or 'dividing' (i.e. partition) is discussed. For example 4 × 8 = 32, so 32 ÷ 8 = 4. Thirty two [+1] tiles divided into eight equal piles, means there will be four [+1] tiles in each pile.

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Thirty two [-1] tiles divided into eight equal piles, produces four [-1] tiles in each pile, so -32 ÷ 8 = -4. 11

18. Positive numbers divided by negative numbers. There is no simple, direct tile method for showing this operation. The inverse relationship between multiplication and division is used. e.g. Since -4 × -2 = 8 then 8 ÷ -2 = -4. Because 6 × -3 = -18, -18 ÷ -3 = 6. The division findings are then reviewed. What is the sign of the number obtained in each of the following cases of integer division; Positive number divided by a positive number? Negative number divided by a negative number? Negative number divided by a positive number? Positive number divided by a negative number? 12

19. Overall review. The findings and sign patterns for each of the four operations are examined using compilation of tables, general questions and sign recognition exercises. 13 The recommendation is for this to be done in the form of class discussion with the students talking about their findings and the operational rules discovered. (The teachers had been requested to avoid directly telling the students any of the rules as they worked through the work booklets.)

20. Extensions and applications of negative numbers. Topics included are; calculations using a mixture of operations, formulae evaluations and number substitutions in algebraic expressions, use in solving linear equations, the cartesian plane and graphs, simple algebraic bracket expansions and simplifications.

Comparison of number line and integer tile approaches 10

Work Booklet Two, p. 4. This was the approach contained in Work Booklet Two, Worksheet 14, pp. 9-10. However one of the examiners pointed out a semantic problem. Earlier 4 × 8 was defined as 8 + 8 + 8 + 8 whilst here 32 = 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4. To be consistent the worksheet item should have been given as 8 × 4. Before future use this part of the Work Booklet requires revision. 12 Work Booklet Two, p. 10 13 Ibid., pp. 11- 14. 11

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The experimental classes were taught using the work booklets containing the integer tiles based teaching sequence outlined above. The control classes were taught using textbooks (either Mathematics Today Year 8 14 or Maths 8 15) that used the number line as the integer operation model for addition and subtraction. It is useful to compare the manner in which the respective models are used (or can be used) to teach operations. The textbooks do not exploit the number line model fully, particularly with regard to multiplication and division. An objective of both experimental and control teaching was for the students to learn and have an appreciation of the operational sign patterns and rules, at least in verbal form. For example students were expected to learn that ‘the sum of two negative numbers is negative and the answer is given by adding the ‘values’ and that the product of two negative numbers is positive. Table 6.1 provides an operational comparison of the models and the corresponding evaluation rules in symbolic form.

14 15

Daly, et. al. Lynch, et. al.

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Table 6.1 Operational comparison of number line and integer tile teaching models Operation (a, b > 0) Number line Integer tiles Rule Addition a + b Combining of directed Combining of tile a+b line segments tail to collections. a + -b head. Total displace- Neutralised pairs (a + b) ment from zero gives removed when . a + -b required positive necessary. Counting a−b or negative value of tiles remaining gives a + b the sum.a value. b−a Subtraction Modelled by removal a − b Tail to tail directed a−b line segments. of tiles. (take away) Neutral (zero) pairs a − -b (Distance between b−a heads in the direction added to facilitate removal when a+b a − -b of the minuend.) or can add inverse of necessary. (a + b) a − b subtrahend. (tail to head

Sign +ve -ve +ve or -ve +ve or -ve +ve or -ve +ve or -ve +ve -ve

addition)

Multiplication

Division

a × b Multiple directed line segments (hops)b a × -b Some early textbooks use kinematics (d = vt) a × b Multiple directed line segments (hops).

Multiple equal-sized collections. Use -a = 0 - a and distributive law. Multiple equal-sized collections. Use -a = 0 - a and a × b Commute to b × a and use multiple hops. distributive law. a ÷ b Could use partition or Repeated subtraction quotition.c from collection. (quotition) Could use partition. a ÷ b Could use quotition. Quotition (repeated subtraction of -b) -

a ÷ b Could use partition.

Partition (divide into b parts)

-

ab

+ve

ab

+ve

-

ab

-ve

-

ab

-ve

a÷b

+ve

a÷b

+ve

(a ÷ b)

-ve

-ve a ÷ -b No simple number line No simple tile (a ÷ b) representation. representation. a Both textbooks chosen used ‘facing and pacing’ on the number line to explain addition and subtraction. Interpretations used were; addition (face right or east), subtraction (face left or west), positive number (move forward), negative number (move backward). Subtraction was therefore treated as the inverse of addition. b Maths 8 used this method for multiplication. Table patterns were used to justify negative by negative. Mathematics Today Year 8 uses equal addition and table patterns. c No textbook appears to use the number line for division. Maths 8 turns division into multiplication by a fraction and uses the multiplication sign rules. Maths Today treats division as the inverse of multiplication.

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Expected learning outcomes On thorough completion of the work booklets it was anticipated that students would have achieved the following knowledge and skills; •

a basic understanding of integer concepts and the ability to use the integer tiles and the neutralisation property for small integer calculations,

•

awareness of and ability to use the neutral pair properties of zero as a basis for understanding difficult aspects of integer subtraction,

•

sound knowledge of the rules of integer operations and some appreciation of how the rules can be justified,

•

ability to automatically apply the rules correctly in performing positive and negative number calculations. 16

The strength of the model lies in the simplicity of the integer tile embodiment and the manner in which the operations of addition and subtraction can be demonstrated and explained using physical manipulation. The physical manipulations are extendable, in a visualisable or imaginable sense, beyond the realm of practical demonstration. Historically the use of counters and counter-based calculating devices (e.g. pebbles, bones, notches, knots, grooves in clay tablets, tally marks on slates, and various forms of abacus) underpins the development of natural number calculations. The discovery and final acceptance that positive numbers alone were insufficient for the purpose of mathematically describing the real world and universe meant that the old positive number models required extension. Apart from the early Chinese, with their red and black rods, no description of a manipulative model incorporating concrete positive and negative integer representations reappeared until the middle of this century. The integer tile embodiment extends the neutralisation model by introducing the use of a zero tile as a cue to overcome a possible teaching and learning problem when the need arises to introduce neutral pairs to facilitate difficult calculations.

16 Freudenthal (1983) asserts that the development of automatic responses is an objective of all such teaching models (p. 458).

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Chapter 7

Experimental Design and Method

This chapter describes the experimental design and method used to collect data for the classroom evaluation of the neutralisation model using the integer tiles method of teaching described in Chapter 6. This component of the research project is referred to as Study E.

Aims The aims were to; •

implement the integer tiles approach as an alternative negative number teaching strategy to the normal textbook and number line based teaching methods used,

•

compare the short and long-term learning outcomes following teaching using the integer tiles (experimental) and normal (control) methods.

A series of four tests was used with the specific aims of determining short and long term differences (if any) between experimental and control groups with regard to integer operation skill performance. Answers to the following questions were sought. •

Does it appear that the integer tile method is superior (or inferior) to the control method with regard to operational skills?

•

Has the integer tile method successfully overcome any of the weaknesses found in Study P? 1

•

How did students react to and use the tiles and the accompanying work booklets?

•

How did teachers use the tiles and work booklets?

•

Is there evidence to suggest that the integer tile method has (or has not) provided pupils with a more meaningful understanding of negative number and related concepts and operations?

•

What are the apparent advantages and disadvantages resulting from using the integer tiles as an initial negative number teaching strategy?

1

See Chapter 4.

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•

How much direct teaching, demonstration and discussion is needed to supplement the practical 'hands-on' approach of the integer tile approach?

•

How sustained is learning resulting from the integer tile method? How does this compare with the outcomes from the normal method?

The sample involved Schools Three schools, two government (Schools A and B) and a private coeducational school (School C), were approached and agreed to participate in the teaching experiment. The government schools involved begin the teaching of negative number early in Year 8. The private school introduces the topic in term three of Year 7. Schools A and C provided classes for Study E activities (teaching, testing, interviewing) during the period 19941997. School B participated in Study E from 1995-1997. 2 Pilot studies were conducted in Schools A and C in 1994 to trial the experimental teaching materials and versions of the pre-test and post-test. 3 The full teaching and testing program for Study E began in 1995 and took place in Schools A, B and C. The final test was given in mid-1997.

Teachers and classes In each of the schools there were experimental and control classes. The students in the experimental classes were supervised by their usual teachers who were willing to adopt the experimental approach. Students worked, substantially at their own pace, exclusively from and in the especially designed work booklets described in Chapter 6. Teachers organised their classes and provided explanations and assistance when required. Control classes continued with the normal methods used by their teachers. In each of the control classes the teaching was generally based on the approach

2

Testing for Study P was conducted in School B in 1994. School A assisted with the trialing of the tests used in Study P. See Chapter 4. 3 Described in Chapter 6.

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contained in the textbook chosen by each school. 4 In School B the same teacher taught both the experimental and control classes. The teachers in Schools A and C who were involved taught either an experimental or a control class. In each school it was necessary to use existing classes as timetabling constraints made random allocation of students to teaching methods impractical. However the classes involved were all non-streamed and considered by their teachers to be of approximately comparable ability ranges.

Testing and teaching program The students, in each class involved, were pre-tested in the lesson immediately prior to beginning the three weeks of lessons (one per day) devoted to the teaching of the topic. Classes were post-tested in the mathematics lesson after completing the topic. The first retention test was administered approximately six months later. A long term retention test was given to as many as possible of the 1995 cohort in mid 1997. Tables 7.1 and 7.2 summarise the details and sequence of the testing and teaching activities for each of the classes involved in the pilot and main components of Study E. The numbers of students (N) shown for each class in the tables were the numbers on the class lists. Some students were not present for all tests. Pseudonyms have been given to the teachers by the author.

4

Schools A and B used Mathematics Today Year 8 (Daly et al.). School C used Maths 8 (Lynch et al.).

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Table 7.1 Study E (Pilot Study (1994)) Testing and Teaching Activities School Year level Class Teacher N Test U1 or Teaching Test U2 or Test W Test X A 8 Expt. Helen 11 * Cards a * b 8 Cont. Irene 14 * Normal * C 7 Expt. Kath 17 * Tiles a * 7 Expt. Norma 11 * Tiles * 7 Cont. Michelle 17 not tested 5 Normal * *indicates tested Test U1 = Pre-test (School A) Test U2 = Post-test (School A) Test W = Pre-test (School C) Test X = Post-test (School C) Table 7.2 Study E (Main Study (began 1995)) Testing and Teaching Activities School Year Class Teacher N Test W Teaching Test X Test Y Test Z A 8 Expt. Helen 19 * Tiles * * * 8 Cont. Irene 19 * Normal * * * B 8 Expt. Jack 25 * Tiles * * * 8 Cont. Jack 24 * Normal * * * C 7 Expt. Kath 19 * Tiles * * * 7 Expt. Laurie 20 * Tiles * * * 7 Cont. Michelle 14 6 * Normal * * * Test W = Pre-test Test X = Post-test Test Y = Retention test Test Z = Long term retention test (Same tests used in each school) a b

Cards/Tiles = Integer cards/integer tiles and work booklets Normal = number line and textbooks

Tests Pilot study pre-tests and post-tests Similar tests (Tests U1 and U2), with only the numerical values changed were used for the pre-test and the post-test in the pilot study conducted in School A early in 1994. The fifty item test included; •

positive number facts and calculations,

•

a full range of integer operations (involving completing equations, compiling tables and evaluating expressions),

•

completing integer sequences,

•

word problems (themes were temperatures and altitudes),

•

locating integers on the number-line,

•

order.

Copies of the tests are attached as Appendix 3 and Appendix 4.

5

Teaching of the topic began in the week before the test was available.

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Test U1 was found to be unsatisfactory as a pre-test. It did provide an indicator of students’ ability levels with basic number facts and calculations and evidence that a proportion of students showed some knowledge of the negative number prior to the commencement of formal teaching. However the use of such a long pre-test, containing much unfamiliar content, was later judged to be unwise and may have, at least temporarily, affected some students’ attitudes to the topic. Although students had been told that the test “didn’t count” as far as their personal results were concerned, many felt that they had failed. The Test U1 did show that most students appeared to know the basic positive number facts and could perform calculations at least to the level likely to be met in the context of the negative numbers topic. Following discussion with the teachers it was decided that future pre-tests, used in Study E, should be of shorter duration (10-15 minutes) consisting of items covering only basic number facts and simple integer operations to assess acquired and intuitive knowledge of the negative numbers prior to beginning teaching the topic. Test U2 proved to have shortcomings as a post-test. It was found to be too long and contained some redundant items which produced no useful data. In particular, the basic number facts and skills items were unnecessary in the post-test, and the integer sequences items were either too easy (forward or backwards counting arithmetic sequences) and practically everyone got them right, or too subtle and confusing (geometric sequences with negative common ratios) and most got them wrong. Thus the items did not satisfactorily discriminate between strong and weak students. A new short pre-test (Test W) was prepared for the pilot study in School C, conducted in the second half of 1994. The post-test was also modified to overcome the deficiencies mentioned above. The new pre-test and revised post-test (Text X) were found to be satisfactory as measures of negative number knowledge and skills and also as indicators of student

6

Although this control class was smaller in size than the two experimental classes in School C, the pre-test results showed that with regard to entry-level mathematical ability the classes did not differ.

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weaknesses and misunderstandings. The latter tests were then used for all classes involved in 1995.

Main study tests Pre-test For the reasons outlined above, the pre-test (Test W) was limited to three questions and one page in length and took 10-15 minutes to complete. It was administered by the class teachers, in the lesson before commencing the teaching of negative numbers. Knowledge of basic number facts (addition and multiplication) was tested in the first two questions in the form of completing ‘scramble tables’. The third question contained 14 simple integer operation items. The latter items were intended as indicators of entry level knowledge and intuitions (or ability to guess correctly) with regard to such operations. A copy of the test is attached as Appendix 5.

Post-test The post-test (Test X), administered in the lesson following completion of the topic, consisted of a comprehensive, period-length (35-40 minute) ten question test covering a range of integer knowledge, operational skills and applications. It included; •

a range of integer operations (involving completing equations, compiling tables and evaluating expressions),

•

word problems (themes were temperatures and altitudes),

•

order.

A copy of the test is included as Appendix 6.

Retention test The retention test (Test Y) consisted of four questions. A copy is attached as Appendix 7. Question one was basically identical to that used as question one in Test X to facilitate direct performance comparisons for particular operational skills. Only the numerical values were changed for each of the 30 items. Question two tested knowledge and application of the sign rules

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in the context of binary operations between pairs of large integers. The items in question three involved making integer substitutions in and then evaluating a range of simple algebraic expressions. Question four, which was not scored, surveyed students’ ability to write a sentence or two describing a practical situation or real-life application for each of four simple binary integer expressions. The intention was to gather student impressions regarding the application and use of integers. Two neighbouring secondary level students worked through the first three questions before Test Y was used in the schools. Test Y was administered to each of the classes involved several months (8 months for Schools A and B, 5 months for School B) after Test X. No attempt was made by the researcher to influence any teaching relating to the use of integers and negative number during the interim period. It was expected that all students would be applying negative numbers and their extended knowledge of the real number system to other topics that had followed (e.g. solving equations, algebraic manipulations, substitutions in and evaluation of formulae and algebraic expressions, and graphing).

Long-term retention test In June, 1997 a long-term retention test (Test Z) was given to as many as possible of the 1995 student cohort involved in the study. The test is attached as Appendix 8. The intention was to look for evidence of longterm residual effects of the initial negative number teaching strategies used. It is acknowledged that much will have happened during the two year period that had elapsed. In each school students had been regrouped and, in some cases, been taught by several mathematics teachers during the intervening period. Some students had left and were not available for testing. Some teachers had also changed schools and were not available for consultation. To make the test suitable for year nine and ten level students the questions were presented in an algebraic form with each item focussing on particular

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essential negative number skills required for continuing further in mathematics. The test included; •

substitution in and evaluation of simple algebraic expressions,

•

use of sign rules and ability to make bracket expansions,

•

simplifications requiring collection of like terms preceded by plus or minus signs,

•

simple formulae evaluations covering the various sign possibilities when multiplying and subtracting positive and negative integers,

•

a word problem involving scoring positive and negative points in a quiz show.

The final question (Q.7) attempted to detect recollection of integer teaching strategies experienced. Students were asked to show or explain how they would teach a friend to do and understand a given simple integer calculation. Each student was given one such calculation to explain. Five examples of such calculations were used and randomly distributed among the test papers. This question was not scored but the responses were categorised with regard to the type of explanation provided (i.e. model, rule, logical, no attempt).

Teaching Experimental classes The work booklets and integer tiles were described in Chapter 6. The work booklets were first trialed in the pilot study in School A. School A also trialed the first versions of the work booklets. Following extensive examination of students’ work in the books, observation of classroom behaviour and reactions, and verbal and written comments, suggestions and corrections from Helen, the booklets were revised and reprinted. The new versions were then used for the Pilot Study in School C, found to be satisfactory, and continued to be used thereafter.

Control classes No special materials were prepared by the researcher for the control classes. The only request was that teachers teach the topic of negative numbers in

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their normal manner. ‘Normal manner’ in each of the schools turned out to be teaching mainly based on the chosen texts (as mentioned earlier) using the horizontal number line as the initial model for explaining integer (or directed number) addition and subtraction.

Data Quantitative data To facilitate analysis of the tests, item data for all classes were entered into Excel spreadsheets. 7 This enabled statistical tests to be made comparing the performance of the experimental and control classes. As well as whole test performance comparisons, the spreadsheets made it possible to compare performances on each test item and group of items. Where possible students’ incorrect responses were also recorded for the purpose of comparing and analysing item error patterns. Examples of the spreadsheets are included as Appendix 12. Frequency counts of correct and incorrect responses were made using Excel. Excel and SPSS software 8 was used for statistical calculations. Where possible performance comparisons will be drawn between the student experimental and control classes involved in Study E and the classes tested in Study P discussed in Chapter 4.

Qualitative data Qualitative data and information was obtained by the researcher, throughout Study E, from the following sources; •

conversations with the class teachers and mathematics coordinators before, during and after the class teaching of the topic and when discussing test outcomes,

•

classroom observations of student and teacher behaviour and reactions during the teaching of the topic,

• 7 8

student work booklets, MS Excel 5.0 Statistical Package for the Social Sciences (SPSS for MS Windows Release 5.0)

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•

video-taped interviews with small samples of both experimental and control class students.

The qualitative data is discussed in the context of particular school and classroom events and outcomes in Chapter 9.

Limitations The following influences and factors may have affected the generality of the outcomes and findings of Study E and therefore need consideration; a) Experimental and Hawthorne effects All experimental and control students knew they were involved in a teaching and learning experiment 9. The extent to which this may have affected outcomes is difficult to assess. No evidence emerged to suggest that any students tried either more or less merely because they were involved in the study. Having an extra person (the researcher) in the room during some experimental and control lessons, observing and sometimes talking to students about their work, and also occasionally helping them, may have had some effect. However this was reasonably balanced across the classes. b) Use of available and potentially un-matched classes It was not possible to randomly allocate students to experimental and control classes and thus available classes were used. Furthermore the classes were not randomly allocated to treatments. The teachers in Schools A and C offered to take either experimental or control classes. In School B the same teacher taught both experimental and control classes. The seven classes involved contained non-streamed and students with a range of abilities. The School C classes appeared to be slightly skewed toward the higher ability ranges but also contained some weak students. The pre-test provided an indication of entry level number knowledge and a relative measure of class comparability.

9

An Ethics Committee requirement was that parental permission was necessary preceding student involvement in the study. Explanatory letters, together with consent forms for parents to sign, were taken home by all students. No parent refused permission. Only one parent contacted the researcher with an inquiry. The parent wanted to know if it was better

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c) Student behaviour and attitudes Students’ attitudes and behaviour (including moods and preferences) were impossible to control. An assumption made is that the experimental and control classes and groups were large enough to provide comparable ranges of favourable and unfavourable student characteristics. Some experimental class students actually rejected making physical use of the integer tiles (“as unnecessary” or “stupid”). However they could not avoid using the work booklets provided, which required ‘tile-based’ thinking, for the major proportion of the topic content. Test result reliability may have been affected by a small number of instances of student capriciousness. Apart from the post-test, which most teachers decided to include as an assessment component, the students were made aware (if they asked) that the other three tests “didn’t count”. Thus there were isolated instances of student under-performance due to not attempting some test items. As this behaviour seemed to occur mainly among weaker students, it is likely that it was fuelled by difficulty and lack of understanding. The more competent students appeared to generally ‘do their best’. d) Sample decay By the time of the long-term retention test some students had left the schools. Thus the original experimental and control class groups became somewhat smaller at the final observation stage. It has been assumed that the groups remained big enough and reasonably representative for comparison purposes with regard to residual effects of initial teaching strategies experienced. To check this a subsidiary analysis of some of the data involving only students who completed all four main study tests was made.

e) Uncontrolled teacher characteristics and behaviour The teachers involved all appeared to be competent, conscientious and experienced. Each cooperated fully and completed every aspect of the study for the child to be in the experimental or control group and was told that existing whole classes would be allocated teaching methods on the basis of teacher preference.

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required. However, as was to be expected, their teaching and classroom behaviour and relationships with students varied and was highly individualistic. The experimental class teachers were fully briefed by the researcher with regard to the nature and use of the integer tiles and associated work booklets. The control class students were given no instructions and apart from administering the tests as required they were free to conduct their classes in their normal manner. The amount of direct ‘up-front’ teaching varied ranging from a lot, to practically none. The individualised nature of the experimental integer tiles and booklets tended to require less direct class teaching than teaching using textbook based approaches. Whilst the experimental class students did their written work directly into the especially prepared and provided work booklets, the control class students worked from their textbooks into their normal workbooks. The full comparison therefore being made is between integer tiles/work-booklets teaching versus no-structuredmaterials/textbooks/normal-workbooks teaching. The assumption is that the use of the integer tiles provides the most significant difference between the teaching approaches thus overshadowing the other factors mentioned. Approaches to corrections and feedback differed. Some teachers collected booklets and workbooks for detailed correction after each lesson (“If they take them home they (the work booklets) will never come back!” commented one teacher). One rarely looked at them out of class and let the students take the booklets home to complete assigned sections. A detailed interpretation of the impact of the various teaching styles on performance outcomes is given Chapter 9. The pilot study, during which the work booklets and tests were being developed, and where necessary modified and corrected, gave opportunity for two of the experimental class teachers (Helen and Kath) to gain experience and rehearse the integer tile method in readiness for the main study the following year. However since all teachers involved were very experienced the effects of teacher practice effect has been assumed to be minimal. f) Schools and school events

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The nature of the schools and the general expectations transmitted to students by parents differed between the schools. The independent school is strongly academic, with all parents expecting good results. One of the government schools (with a good academic tradition) is large with a high demand for places and comparatively large classes. The other government school caters for a very broad ability range with a comparatively large proportion of lower ability and less academically inclined students. In each of the schools there were minor interruptions to the scheduled teaching and testing program including curriculum days, special holidays, school sports, drama rehearsals, teacher absences and student absences. However students were generally expected to catch up on work missed (e.g. homework, extra teacher assistance in or after class). 10 The overall effects of the interruptions have been regarded as minimal compared with the other factors considered. g) Subsequent teaching and other experiences No attempt was made to control or closely monitor teaching practices after the initial three weeks teaching period. None of the experimental teachers claimed to have attempted to reinforce or revise the tile method. Apart from the three weeks of teaching and learning activities done in the experimental and control classes, the students will have been exposed to a variety of in and out of class experiences that may have helped or hindered their performance.

Strengths Whilst conceding that the study outcomes need to be considered in the context of the possible limitations outlined above there also appears to be the following strengths; a) Size and nature of the study 10 The researcher assisted a small number of students who had missed work through absence and actually supervised or assisted in the supervision of a few classes in Schools A and C when the usual teachers were absent. This occurred in both experimental and control classes.

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In comparison with the study of Lytle (1994) which used only four students to study the potential of the neutralisation model as an alternative to the number line model for the teaching of integer addition and subtraction, this investigation used substantial samples of students in normal sized classes conducted by the usual teachers. b) Diversity of schools The diverse nature of the three schools participating produced samples of students that could be considered to be reasonably representative of the general population of students within the community. c) Teacher differences Whilst the effects of teacher behaviour characteristics, as discussed previously, could place some limitations on the validity of the study (teacher rather than model instrumental in producing better performances producing mixed results favouring neither model), teacher differences may also provide opportunity for indication of the relative effectiveness of the teaching models. In the latter case the model rather than the teacher would be instrumental in producing the better performances (i.e. if results consistently favoured one model). The involvement of Jack in both methods provided one school situation where teacher effect was, at least to some extent, controlled. 11 d) Pilot study The participation of two of the schools in the pilot study enabled necessary careful revision of the experimental integer tile teaching materials and also the trialing of pre-tests and post-tests. Whilst the student work booklets which were quickly produced before the trial could not be considered to have reached optimum standard and did not have the polish of the textbooks used by the control groups, they did however work very well in the classroom situation. This was largely due to feedback from teachers and observations student work and reactions. The pilot study also quickly

11

Prior to the experiment Jack expressed a preference for the normal teaching methods and was concerned that the integer tiles method focussed too strongly on one embodiment (i.e. the neutralisation model). Nevertheless he was prepared to “give it a go”. The author’s classroom observations revealed no evidence of personal preference in the approach to teaching adopted with either class.

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indicated the initial folly of using blank playing cards for integer materials and the need for more robust material. This resulted in the idea of using the reversible integer tiles.

A major assumption Considering both the strengths and the limitations listed it has been assumed that across the three schools with six teachers, seven classes, and 140 students involved, valid and reliable data would emerge from the testing and observation processes to enable a worthwhile comparison of the respective teaching methods. The major assumption made is that the use of the integer tiles and work booklets provides the most influential difference between the teaching approaches experienced thus overshadowing the other, possibly confounding and limiting factors. The overall effects appear to be reasonably balanced across the experimental and control classes and groups.

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Chapter 8 Quantitative Outcomes of the Integer Tiles Teaching Experiment This chapter provides an analysis and discussion of the quantitative results of the sequence of four classroom tests (described in Chapter 7) given to the experimental and control classes in each of the three schools during Study E. The aggregated (across school) classes will be referred to as the experimental and control groups. For within school comparisons the terms experimental and control classes will be used. Group mean scores on each of the tests are provided first. This is followed by a detailed breakdown and analysis of experimental and control group performances on selected test questions and items. The overall test results showed that; •

the experimental and groups were closely matched on the pre-test (Test W),

•

the experimental group performed significantly better than the control group on both the post-test (Test X) and the first retention test (Test Y),

•

the performances of the groups converged on the long-term retention test (Test Z) two years later and the difference in favour of the experimental group was then not significant.

An overview of test results Pre-test The mean scores on Text W given in Table 8.1, show that the experimental and control groups were closely matched in terms of mastery of positive number addition and multiplication number facts and with regard to entrylevel knowledge of integer operations. Table 8.1 Test W: Group mean scores and standard deviations Group Positive number facts Integer operations (/30) (/14) Mean SD Mean SD Expt. 29.1 2.6 7.8 3.4 (N = 84) Cont. 28.4 2.6 7.6 3.9 (N = 58) t-test (t, d.f., p)

Total score (%) Mean SD 83.1 9.6 82.6

11.2

0.08, 143, 0.934 (n.s.)

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Post-test The results of Test X shown in Table 8.2, indicate that the experimental group performed significantly better (p < 0.01) than the control group. Table 8.2 Test X: Group mean scores and standard deviations Group Integer operations (/30) Mean SD Expt. 23.1 6.3 (N = 82) Cont. 20.0 7.8 (N = 57) t-test (t, d.f., p)

Total score (%) Mean 77.4

SD 18.3

66.5

24.2

3.03, 137, 0.003 (sig.)

Retention test Table 8.3 shows that the experimental group also performed significantly better (p < 0.01) than the control group on Test Y administered to each class between five and eight months after Test X. The 30 integer operations items were similar with only the numerical values changed. The results on the two tests were extremely consistent. Table 8.3 Test Y: Group mean scores and standard deviations Group Integer operations (/30) Mean SD Expt. 24.4 6.3 (n = 74) Cont. 19.8 7.7 (n = 58) t-test (t, d.f., p)

Total score (%) Mean 79.2

SD 21.8

63.9

23.9

3.83, 130, 0.0006 (sig.)

Long-term retention test Test Z was given to the students who had taken the three previous tests and who were still attending the same schools two years after the teaching experiment. 1 Table 8.4 shows that the mean scores on integer operations were almost identical. The difference between the mean percentage test scores was not significant. 1

During the elapsed two-year period the number of students involved had diminished in size particularly in the government schools. Students had either left the schools or were unavailable for the test. However referring back shows that the pre-test mean scores obtained by the remaining 91 (56 + 35) students (Expt. 83.2, Cont. 82.0) closely matched the pre-test mean scores of the original groups (Expt. 83.1, Cont. 82.6). It is therefore being assumed that the remaining groups are reasonably representative of the original groups.

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Table 8.4 Test Z: Group mean scores and standard deviations Group Integer operations (/10) Mean SD Expt. 8.8 2.0 (N = 56) Cont. 8.3 2.2 (N = 35) t-test (t, d.f., p)

Total score (%) Mean 76.4

SD 19.9

70.3

21.6

1.35, 89, 0.180 (n.s.)

Four-test comparison The experimental and control group performances across the four tests are compared in Figure 8.1. 2 90 80

Mean % score

70 60 50

Expt.

40

Cont.

30 20 10 0 Pre-test

Post-test

Ret.test

L.T.test

Figure 8.1 Comparison of experimental and control group mean percentage scores across the four tests

Summing up, the experimental and control groups were practically level at the beginning of Study E. The experimental group performed significantly better than the control group immediately following and several months after the teaching of negative number. No significant difference between the groups was found two years after the initial teaching.

Detailed analysis of selected test items and questions Consideration will now be given to finer details and features behind the broad results outlined above. The tabulated statistics of selected questions and items are used as indicators of the relative strengths and weaknesses of

2

The lines joining points are to distinguish the groups and do not show score trends. The content of the four tests differed. The pre-test score was relatively high because the test

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the groups at each test stage (W, X, Y, Z) of the study. The analysis of Tests X, Y and Z will include, where possible, evidence of group performance changes on similar items that seem to have occurred in the interval between tests. The quantitative data are supported, where possible, by the researcher’s observational data and recollections of classroom incidents and behaviour. There were also conversations with students and teachers.

Pre-test analysis A majority of the students (Experimental 72.6%, Control 70.7%) were able to provide correct answers for at least 7 of the 14 integer operations items. Two School B students managed perfect scores. Five students (one experimental., four control) who scored zero did not attempt the items. Some students who attempted either none or few items, wrote comments on their papers such as “I haven’t done these yet and I don’t know them”. Test W integer operations items have been sorted into operational categories. Percentages of students correct on each item in the experimental and control groups are compared.

Addition Table 8.5 shows the group results to be quite uniform. A good proportion of the students entering the study displayed either prior knowledge or the ability to make a good guess with regard to integer addition operations. Table 8.5 Test W: Percentage of students correct on addition items 7 + -3 4 + -5 2 + -6 3 + -3 Expt. (n = 84) 75.9 79.3 58.6 72.4 Cont. (n = 58) 72.4 75.9 58.6 65.5

4+ 0 87.4 86.2

0 + -4 71.3 75.9

Subtraction The students were less successful with subtraction. Table 8.6 shows that over half the students in both the experimental and control groups were correct for 0 - 3 but achieved lower results on the other items. Very few

included unsigned number items (tables and number facts) in which most students showed good mastery levels.

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managed 8 - -4 the most intuitively difficult of the items. The performance patterns for the groups were again very similar. Table 8.6 Test W: Percentage of students correct on subtraction items 5-5 6 - -2 2 - -2 Expt. (n = 84) 36.8 49.4 36.8 Cont. (n = 58) 46.5 34.5 36.2

8 - -4 3.5 13.8

0-3 71.3 60.4

Multiplication Table 8.7 reveals that the experimental and control groups produced similar results with over half of the students correct for each of the multiplication items involving only one negative integer. Table 8.7 Test W: Percentage of students correct on multiplication items 2 × -3 4×2 Expt. (n = 84) 56.3 52.8 Cont. (n = 58) 60.3 56.9

-

5 × -2 21.8 12.1

The percentage of students correct for the product of two negative integers was small although almost half of the students in the School B experimental class (46.4%) gave the correct answer. The proportions in other classes ranged from 20% (School B control) to none (School C control). The overall percentage correct for the product of two negatives was 17.9%.

Summary of pre-test findings The pre-test revealed that the majority of students appeared to have a good knowledge of positive number addition and multiplication facts. No student was identified as being unready to continue to the topic of integers and negative numbers. Many students displayed ability to perform some integer operations prior to the commencement of teaching of negative numbers. At the commencement of the study the ability levels of the experimental and control groups, on the number skills tested, appeared to be practically equal.

Post-test analysis Test X was used to probe the following questions;

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•

How do the experimental and control group performances differ across the range of skills tested?

•

What are the relative group performance changes on particular integer operation items since Test W?

•

Can performance differences be related to the teaching strategies used?

•

What are the relative strengths and weaknesses of the experimental and control groups?

For each test question the overall experimental and control group mean scores are provided first. This is followed by tabulation of group performances on particular items. For some items the tables contain examples of common incorrect responses (CIRs) to assist the analysis of student misunderstandings and errors.

Integer Operations Q.1 contained 30 integer operations items. Table 8.8 shows the mean group scores. The mean score difference in favour of the experimental group was statistically significant. (p < 0.05). Table 8.8 Test X: Q.1 Integer operations mean scores and standard deviations Group Mean SD t-test (t, d.f., p) Expt. (n = 82) 23.1 5.4 2.61, 137, 0.010 Cont. (n = 57) 20.0 7.2 (sig.)

Comparison of integer operation skills The items have been sorted into categories (addition, subtraction, multiplication and division, equations, and brackets). Where similar items were included in Test W the percentages for the latter test are shown in brackets.

Addition Q1. contained three addition items. Table 8.9 shows the percentages correct in each group.

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Table 8.9 Test X: Percentage of students correct on addition items (Test W percentages in brackets) Group 5+2 8 + -8 4 + -2 Expt. (n = 82) 88.0 96.4 (72.4) 85.5 (58.6) Cont. (n = 57) 80.7 75.4 (65.5) 75.4 (58.6) Answer 3 0 6 CIRs 7, 3 16, 16 6, -2

The experimental group performed better and also produced higher proportions of students correct on the items common to both tests. Within particular schools there were only two instances of control classes showing higher proportions correct; for -5 + 2 (School A) and -

4 + -2 (School C). Eleven control students (19.3%) and six experimental

students (7.3%) provided the incorrect response, -7, for -5 + 2 by interpreting the item as -(5 + 2). On the addition items some students tended to add the numbers regardless of the signs; if there was one negative sign the answer was given as negative, whilst if there were two negative signs the answer became positive. Seven control students (12.3%) and six experimental students (7.3%) showed this behaviour for -4 + -2. Students of both groups spoken to justified it with reasons such as “Because two minuses make a plus”. It had been anticipated that students who had used the integer tiles would not make this type of error. For a small minority this ideal was not achieved. The other common error (-2) by some students in both groups was substituting ‘−’ for ‘+ -’ and then evaluating -4 − 2 as -(4 − 2). There were two particular experimental group students who continually baulked at using the tiles and both frequently resisted teacher and researcher offers of help with the tiles. They thus missed ‘seeing’ -4 + 2 as a ‘collection’ of two negative ‘sets’ and resorted instead to relying on rules. Control students also tended to rely on rules and were reluctant or unable to explain working in terms of the number-line model used during their teaching. The process of moving backwards for the purpose of adding a negative integer (as explained in the textbooks used) did not appear to be generally understood. Explanation of addition by drawing tail to head directed line segments on the number line was not used by any of the control class teachers. The text book is also inconsistent with regard to the

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numbers in the operations. The left-hand number is interpreted as a point on the line. The right-hand number is a directed distance. The answer is then read as a point. This may be one of the causes of confusion. There may also be a language factor. The work booklets encouraged students to verbalise expressions; i.e. to read -4 + -2 as ‘negative four plus negative two’ as part of the tile modelling (combining heaps) process. The textbooks do not encourage to students to read and verbalise mathematical expressions. As a consequence the control students appeared to be less able to show understanding of the meaning of expressions. As mentioned in Chapter 5, Mukhopadhyay 3 reports research relating the need for language skills for the development of mathematical understanding and problem solving ability. Observations in this study appear to support her findings.

Subtraction Table 8.10 shows the percentages of the groups correct on the subtraction items. Table 8.10 Test X: Percentage of students correct on subtraction items (Test W percentages in brackets) Group 2 - -5 6 - -4 3 - -3 8 - -6 0 - -4 Expt. (n = 82) 77.1 91.6 (49.4) 96.4 (36.8) 69.9 (3.5) 65.1 Cont. (n = 57) 70.2 63.2 (34.5) 75.4 (36.2) 59.6 (13.8) 57.9 Answer 3 2 0 14 4 CIRs 7, 7, -3 10, 10, 2 6, 6 2, -2, -14 4, 0

Higher proportions of experimental group students were correct on each of the items. The experimental group also showed much greater gains on the similar pre-test items with from 42.2% to 66.4% more of the groups learning compared with 28.7% to 45.8% for the control group. The lowest results were on the item, 0 - -4, for which 24.4% of the experimental group and 28.1% of the control group gave -4 as the answer. The other incorrect response given for the item was 0 (3.7%, 7.0%). No student was questioned about this response but it may be due to the ‘intuitive impossibility’ of

3

Mukhopadhyay (1995).

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taking ‘a negative number from nothing’. 4 The item 8 - -6 produced similar incorrect response patterns, 2 (14.8%, 14.0%), -2 (9.8%, 15.8%) and -14 (1.2%, 3.5%). For other items the error patterns of experimental and control class students differed somewhat. This was particularly evident for the item -

2 - -5 for which the common errors were -3 (13.4%, 7.0%), -7 (4.9%, 14%)

and 7 (2.4% and 8.8%). Classroom observations suggest the following possible reasons for some student errors. Understanding subtraction of negatives as explained in the textbook (i.e. ‘facing and pacing’) is confusing and difficult for many students so the process was generally performed using sign rules and strategies for applying the rules were taught. Emphasis was placed on like and unlike signs. In control classes students often circled and replaced double ‘-’ signs with a single ‘+’ signs or unlike sign pairs with single ‘-’ signs (a chalkboard and correction strategy often used by teachers). Thus -2 - -5 would become -2 + 5. However 19.5% of control class students were shown to get this type of item wrong on an earlier question. Errors accumulate if each teaching step is not thoroughly understood and learned. The experimental method interpreted subtraction operations in terms of tile manipulations first adding and then replacing the zero tile with a zero equivalent to evaluate examples such as -2 - -5. +0 0 = -3 + 3 − -5 [ 1] [ 1] → [ 1] [ 1] [0] → [ 1] [ 1] [ 1] [ 1] [ 1] [1] [1] [1] → 2 -2 + 0 2 + -3 + 3 -

-

[1] [1] [1] 3

However during the teaching activities some students were observed to start with five ‘-1’ tiles (the bigger set) and then remove two of them, thus obtaining the answer -3. As indicated above this was the most common experimental group student error. Such subtraction misconceptions are common. 5 Weaker students may be tempted to make the task look easier. Almost all experimental group students coped with -6 - -4 but some appeared to reverse -2 - -5. This indicates the need to ensure that students

4

Recall that some great mathematicians were also deeply troubled by such ‘absurdities’. (see Chapter 2). Some secondary level students may also think such expressions are a bit strange.

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can correctly read expressions (i.e. ‘negative two, minus negative five’). Also the process of adding zero to facilitate the subtraction process appears to require a lot of practice and frequent revision for weaker students in the early stages. It is also possible that, for some students, the tiles experience may have impeded learning. Boulton-Lewis, Wilss and Mutch 6 have researched conceptual difficulties associated with the use of structured materials in the context of subtraction (e.g. counters, blocks, ...). Sometimes another teaching and learning difficulty is introduced. In this case students may have visualised five [-1] tiles and imagined taking away two of them. To use the tiles students must be able to interpret the given expression correctly and then apply a correct sequence of learned actions. For subtraction (take away) using the tiles the students need to learn that the subtrahend must always be represented (actually or imaginary) first and the operation follows. The subtraction results show that some students in both groups displayed weaknesses following teaching. In the experimental classes there were students did not fully master the tile strategy for some operations. The assumption was that the students would use tiles to discover and justify the sign rules and thus move beyond the need for tiles. The minority for whom this did not happen may have ended up with neither a valid model interpretation nor a rule. However overall the experimental group appeared to perform better than the control group across the five items.

Multiplication and Division The group outcomes for the multiplication and division items, shown in Table 8.11, indicated little overall performance difference between the experimental and control groups with the exception of the product of two 5

English and Halford, pp.149-163. G. M. Boulton-Lewis, L. A. Wilss, & S. J. Mutch (1996). Representations and Strategies for Subtraction Used by Primary School Children. Mathematics Education Research Journal, 8, 2, pp.137-152. 6

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negatives on which the experimental group showed a substantially higher success rate. The performance gains on the multiplication items since Test W are approximately equal for both groups. 7 Table 8.11 Test X: Percentage of students correct on multiplication & division items (Test W percentages in brackets) Group 3 × -2 2×5 2 × -6 12 ÷ 4 6 ÷ -2 Expt. (n = 82) 92.8 (56.3) 86.7 (52.8) 77.1(21.8) 81.9 74.7 Cont. (n = 57) 94.7 (60.3) 82.5 (56.9) 64.9 (12.1) 82.5 77.2 Answer 6 10 12 3 3 CIRs 6 10 12 3 3

It may have been expected that the simple like sign rule would be easy for the number line and rule-based teaching used with the control group. However this was not so for over a third of the group members. Errors (apart from those due to carelessness mis-reading signs, e.g. reading × as +) consisted of giving the incorrect sign. In the experimental group tile strategies were used to interpret two types of multiplication, i.e. (+ × +) and (+ × −) and three types of division, i.e. (+ ÷ +), (− ÷ −) and (− ÷ +). The multiplications, (− × +) and (− × −) were taught by first using the distributive law and then by referring back to earlier tile-based results. The students were led to the rules by the sequential activities (see Chapter 6). The control classes were taught two types of multiplication, (+ × +) and (+ × −) either as number-line hops or repeated addition, whilst (− × +) was justified using the commutative rule. Consistency pattern or tables were then used to justify the rule for (− × −). Division was treated as the inverse of multiplication to justify the rules. By the time of the post-test both groups were using rules for their calculations. The experimental group did better than the control group on the multiplication of two negatives but apart from that item there was little to separate the groups. After the teaching and testing students from both groups who were interviewed only offered rules as justifications for answers and none tendered reasons relating to either experimental or control class teaching strategies. 7

Integer division was not tested in Test W.

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Summing up, the results on four of the multiplication and division items were fairly uniform. The experimental group did better than the control group on the product of two negatives. This may be due to the more careful build up to, development and justification of the rule given in the experimental materials. A greater proportion of experimental group students may have ended up believing the rule. According to theory the rule is therefore more likely to be permanently remembered. 8

Equations Tables 8.12A and 8.12B provide the outcomes for the equations items. Table 8.12A Test X: Percentage of students correct on equation items Group 4+[ ]=-9 3+[ ]=-2 3+[ ]=4 3-[ ]=8 Expt. (n = 82) 92.8 91.6 84.3 75.9 Cont. (n = 57) 77.2 75.4 73.7 71.9 Answer 5 5 7 5 CIRs 5, 13 1, 1, 5 7, 1, 1 11, 11, 5 Table 8.12B Test X: Percentage of students correct on equation items Group 3-[ ]=-9 8-[ ]=2 2× [ ]=8 Expt. (n = 82) 59.0 51.8 69.9 Cont. (n = 57) 54.4 43.9 70.2 Answer 6 10 4 CIRs 12, 12, 6 10, 6, 6 4

-

-

[ ]--2=5 65.1 54.4 3 7, 7, -3

5-[ ]=2 67.5 70.2 7 7, 3,-3

20÷[ ]= -5 78.3 77.2 4 4

-

30÷[ ]=10 66.3 70.2 3 3

On the equations involving addition or subtraction the experimental group performed better whilst on equations requiring multiplication or division the group performances were almost the same. The easiest item overall was -4 + [ ] = -9 with the experimental group doing better than control group. On balance the experimental class students performed better on the subtraction equations although many students in both groups made a variety of errors. The most difficult of the subtraction equations was -8 - [ ] = 2. The incorrect responses for the equations involving addition and subtraction show a consistent pattern. Students performing the correct operation gave the wrong sign whilst many of those using the wrong operation seem to decide on the sign by the number of minus and negative 8

Sfard (1994).

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signs in the equation. In general this appears to indicate either an inability to interpret the equations and confusion with rules or a tendency for some students to just make a quick guess. Talking with some of the students, from both experimental and control classes later, revealed that the method of solution used tended to be informal. For example for the equation, -8 - [ ] = 2 a successful experimental class student said, “What (number) taken from negative eight gives two? It has to be negative ten!” A control class student said, “I first thought ten but that wasn’t right so it must be minus ten.” Another control class student incorrectly reasoned, “Two minuses make a plus so eight plus something equals two. It’s got to be minus six.” Some, from both experimental and control classes, ignored the negative sign in front of the 8 and converted the item into 8 - 6 = 2. None of the students spoken to mentioned making practical or imaginary use of a tile, number-line or any other model strategy to help them obtain answers to the equation items. Students from both groups seem to have relied on rules or (good or bad) number sense.

Brackets The experimental group performed considerably better than the control group in all of the six integer operations items containing brackets. The results are given in Table 8.13. Table 8.13 Test W: Percentage of students correct on items containing brackets Group 5× (3-5) 4× (2--5) -12÷(1-4) -2× (6-9) (-3+-6)÷3 Expt. (n = 82) 74.7 65.1 69.9 69.9 69.9 Cont. (n = 57) 63.2 56.1 49.1 47.4 38.6 Answer 10 28 4 6 3 CIRs 10, 75 12, -3, 28 -4, 36, -3 -6, -21, -30 3, 1, -1

(3-5)÷-2 77.1 50.9 1 1, 0, -4,

Considerable emphasis was placed on interpretation of the distributive law using the tiles in the experimental material. However many errors were made by students in all classes on each of the six items. The most common errors in each of the items was giving the correct numeral but the wrong sign. A few students attempted to expand the brackets first but failed to

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correctly apply the distributive rule. For example 5 × (3 - 5) = 15 - 5 = 10 was the reason for error by two control students questioned rather than 5 × -2 = 10. The latter students were the two who produced -21 for 2 × (6 - 9).

-

Conclusion On the Q.1 integer items, the experimental group had a greater percentage correct on 24 of the 30 items. The two largest differences in favour of the experimental group were for the items, (-3 + -6) ÷ 3, (31.3% difference), and -6 - -4, (28.4% difference). In comparison, for the six items on which the control group performed better, the group differences were very small. The two largest were for -30 ÷ [ ] = 10 (3.9% difference) and -5 - [ ] = 2 (2.7%). The item by item analysis of Q.1 above indicates that the experimental group developed substantially higher levels of overall integer operation skills during the three weeks of teaching.

Further integer operations Integer operations were also tested in a variety of other forms of question. The results follow below. Scramble tables Questions 2, 3, 4 & 5 consisting of scramble tables (see Figures 8.2, 8.3, 8.4 and 8.5) provided a further comprehensive measure of integer operation skills.

+ 1 2

2 3

-

4 3

0

3 4 1

0 -

1

Figure 8.2 Test X: Addition Scramble Table

× 1 2 3

2 2

-

4

0

3

4 Figure 8.3 Test X: Multiplication Scramble Table

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3 2 0 − 5 2 0 1 Figure 8.4 Test X: Subtraction Scramble Table

2 -3 1 ÷ 12 6 18 0 Figure 8.5 Test X: Division Scramble Table

A total of 43 missing integers were required across the four tables. The class and group results are shown in Table 8.14. Table 8.14 Test X: Scramble table mean scores and standard deviations Group Addition Multiplicn. Subtraction Division (/12) (/15) (/8) (/8) Mean SD Mean SD Mean SD Mean SD Expt. 9.9 2.4 13.0 2.6 5.7 1.8 5.7 2.3 Cont. 8.4 3.3 11.6 4.1 4.6 2.4 5.2 2.7

Total (/43) Mean SD 34.3 6.7 29.7 10.2

The experimental group registered higher mean scores than the control group for each of the four operations and also in most within school situations. The group mean total scores were significantly different (t = 2.84, d.f. = 137, p = 0.005). This supports the evidence of Q.1 indicating the superior performance of the experimental group on integer operations.

Mixed Operations The two items in question six required students to evaluate expressions containing mixed integer operations. The group outcomes are shown in Table 8.15. Table 8.15 Test X: Percentage of students correct on mixed operations items (sign errors highlighted) Group 6 + -5 × -2 3 + 4 × -2 - -12 ÷ 4 Expt. (n = 82) 72.3 39.8 Cont. (n = 57) 35.1 26.3 Answer 16 2 CIRs 4, 2, 16 8, 8, -11

The experimental group performed better on both items. For 6 + -5 × -2 a total of eleven different incorrect values were given due to incorrect order of operation and sign errors. The most common was a sign error; 6 - 5 × -2 = 6 - 10 = -4 (8.5% Experimental, 14.0% Control). Ignoring the order of operation convention produced 1 × -2 = -2 by ten students (2.4%, 14.0%).

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The complicated item 3 + 4 × -2 - -12 ÷ 4 produced a total of 22 different incorrect values. The proportions correct were low for both groups. However the experimental group again performed better. The most common incorrect answers were due to sign errors resulting in 3 - 8 - 3 = -8 (7.3%, 15.8%) and 3 + 8 - 3 = 8 (7.3%, 7.0%). Another error was due to incorrect order of operations producing 7 × -2 + 3 = -14 + 3 = -11 (signs correct but order wrong) by a few students in each group.

Total integer operations and calculations scores Considering all of the post-test questions testing integer operations together shows that the experimental group performed better than the control group on all questions except Q.9 (the recognising zero question) which produced almost identical scores for both groups. The total number of marks on the test for integer calculations was 79. Table 8.16 shows the group mean scores. The experimental group performed significantly better (p < 0.01). Table 8.16 Text X: Total integer calculations mean scores and standard deviations Group Mean SD Expt. (n = 82) 61.3 14.3 Cont. (n = 57) 53.1 19.1 t-test (t, d.f., p) 2.88, 137, 0.005 (sig.)

Conclusion The integer operations and calculations component of the post-test showed the experimental group to perform better than the control group on eight of the nine questions and, as the foregoing analysis shows, almost all of the test items within the questions. The experimental group performed clearly better on integer addition and subtraction. On integer multiplication and division their appeared to be less separation between the groups, however the experimental group were more successful on the multiplication of two negatives.

Other questions

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The remaining post-test questions were intended to test knowledge of ordering signed numbers and the ability to use integers in answering simple word problems. Order Question eight required the students to place the following list of ten numbers in order with the lowest number on the left and the highest number on the right; -

3, 2, -2, 0, -2.5, 2.3, 0.4, -0.5, -5, -0.2.

Students received 2 marks toward their test score if they were completely correct and one mark if they had two numbers out of order. The group outcomes are shown in Table 8.17. The experimental group performed clearly better than the control group. Table 8.17 Test X: Percentage of students at performance levels shown in ordering signed numbers Group Correct Two out of order More than 2 out of order Expt. (n = 82) 73.5 3.6 22.9 Cont. (n = 57) 56.1 8.8 35.1

Initially this appeared to be an unexpected result because the integer tiles only demonstrates ordering of integers as absolute values. However students are usually familiar with counting activities including the use of negative numbers in this context. The number-line was introduced late in the experimental work booklets to define and demonstrate real number order for the purpose of graphing and extension to negative rational numbers prior to teaching of division. Some ordering activities were included. 9 In contrast the control groups were taught from the outset using the number-line and order definitions and extensive exercises were given before proceeding to operations. 10 Some students made careless errors (e.g. omitting signs, omitting numbers, arranging numbers in the opposite order). A common error by control students, but infrequent in experimental students was mis-ordering of the negative decimals, for example; ..., -2, -2.5, -0.2, -0.5, 0, ..., and -0.2, -0.4, -0.5, -5, -2, -2.5, -3, ...

9

Work Booklet Two, pp. 5-6 (Appendix 10). see Daly et al., pp. 5-8, Lynch et al., pp. 51-53.

10

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In general the experimental students were superior because they could order decimals better. Although the textbooks used in the control classes entitled the chapters as ‘Directed Numbers’ their treatment ignored negative decimals. The experimental class work booklets included exercises involving the ordering of negative decimals. This probably explains the better performance of the experimental group on these items.

Word Problems The final two post-test questions were word problems. Q.9 required finding the altitude difference between locations above and below sea-level. The top of Mt Ecks is 5400 metres above sea level. Close by down in Devils Canyon there is a cavern which is 250 metres below sea level. What is the difference in altitude between the two places?

Q.10 used a temperature theme. During a particular week in winter at Mt Buller the maximum daily temperature (in degrees Celsius) is shown in the following table: Day 0 C a) b) c)

Sun -

Mon 9

Tues 5

Wed 3

Thur 0

Fri 3

Sat 6

6 How many degrees difference was there between the highest and lowest maximum temperatures for the week? Between which two successive days was there the greatest change in maximum temperature? What was the average maximum daily temperature for the week?

Table 8.18 shows the group outcomes. Table 8.18 Test X: Percentage of students correct on word problems Group Altitude Temperature Successive day difference difference greatest change (Q.9) (Q.10a) (Q.10b) Expt. (n = 82) 57.8 80.7 78.3 Cont. (n = 57) 47.4 64.9 66.7 Answer 5650 15 Mon-Tues CIRs 5150 12 Sat-Sun

Average max. temperature (Q.10c) 54.2 43.9 2 2, -3

Overall a higher proportion of experimental group students obtained the correct answers on each of the four items. Apart from a few arithmetic slips, practically all students who made an error in the altitude difference item showed the calculation 5400 - 250 = 5150. The majority of students, in both experimental and control classes, who gave the correct answer, showed 5400 + 250 = 5650. Only a small

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number started by interpreting the problem as 5400 - -250. Each student who did this provided the correct answer. In solving word problems many students avoid the use of negative numbers and regard them as unnecessary. Student behaviour in tackling problems such as this (which are common in textbooks) raises the issue of their usefulness. They are generally included for teaching purposes as simple applications. Realistic applications in contexts where the use of negative numbers is unavoidable are needed. Students could not avoid the use of negative numbers in tackling the temperature difference item. A total of thirteen different incorrect values were tendered due to the fact that students were required to interpret the problem and decide on the appropriate pair of temperature values (highest and lowest) from the week-long table before calculating. Several students were unable to manage this. The most common error in the successive day change comparison was to give Saturday-Sunday as successive days although the table started with Sunday and finished with Saturday. The greatest successive days change was between two negative (minus) temperatures, -90C (Monday) and -50C (Tuesday). The sum of the temperatures for the week was -14 producing an average of -14 ÷ 7 = -2. The technique of finding the average seemed to be generally well known. However 23 incorrect answer values were generated with the most common being 2 by omitting the minus sign (7.3% Expt., 11.5% Cont.). The value -3 (by three students) was puzzling until a student, when asked why , said, “Oh, that was the middle (Wednesday) temperature!”. Three control group students summed the temperatures and obtained 32 (because, according to one student, “A minus plus a minus equals a plus and there are four minuses!” ) giving respective average values of 4.57,

4 4/7, and 4 r 4.

Overall the experimental group performed better than the control group on the word problems due to their apparently superior negative and general number skills following the three week teaching programs.

Summary of post-test findings

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The detailed analysis of the test questions and items revealed that the experimental group, •

produced significantly better results than the control group in eight of the ten question categories tested,

•

showed a higher proportion of students giving correct answers on 33 of the 41 items analysed in detail,

•

was shown to be superior on integer addition and subtraction,

•

performed better on items (equations, brackets, mixed operations) requiring the use of integer operations in conjunction with other mathematical knowledge (e.g. order of operation),

•

performed better on the tasks of ordering negative numbers and answering word problems.

By the time of the post-test students in both groups were using sign rules for calculations. The experimental group students were expected to discover and articulate the sign rules for themselves through the neutralisation model-based integer tile activities and associated exercises in the work booklets. The control group students were taught addition and subtraction using the number line however the sign rules tended to be given rather than thoroughly explored. A consequence of the contrasting approaches was that a larger proportion of experimental class students seemed to have developed a more confident and secure knowledge of the rules and showed less tendency to confuse sign rules across operations. This appeared to be the main reason for the difference in performance levels observed on the post-test.

Retention test analysis Text Y was administered to each group of students involved five to eight months after formal completion of the topic and the post-test. Seventy four of the experimental group students were available for testing and 58 of the control group. The data will be used to answer the following questions;

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•

Were the performance differences between the groups observed on the post-test for integer operations sustained or were there any changes?

•

Do the groups differ with regard to positive number sense and ability to use the sign rules?

•

Have the relative strengths and weaknesses of the experimental and control groups changed?

Integer Operations Q.1 contained 30 integer operation items similar to those contained in Test X. Only the numerical values were changed. The mean score for the experimental group was 24.4 (sd = 6.3) and for the control group 19.8 (sd = 7.7). Again the items have been sorted into the categories of addition and subtraction, multiplication and division, equations and brackets. Group performances are compared with those for the corresponding items on Test X.

Addition and Subtraction The group performances on the addition and subtraction items, are shown in Table 8.19. The percentages have been rounded off to save table space. Table 8.19 Test Y: Percentage of students correct on addition and subtraction items(Test X in brackets) Group Expt.(n = 74) Cont.(n = 58)

-6 + 3 94 (88) 80 (81)

-4+-3

87 (86) 66 (75)

1 - -5 74 (70) 53 (60)

-3--8

-5--2

-4--4

80 (77) 62 (70)

77 (92) 63 (63)

82 (96) 75 (75)

0--2 75 (65) 51 (58)

The experimental group continued to perform clearly above the control group on each of the items. Comparison with Test X results shows that the experimental group improved on some items and regressed on others. The items they regressed on should have been easy for integer tile experienced students. The incorrect responses (in order of frequency) for -5 - -2 (-7, 6, 7) and 4 - -4 (8, -8) indicate rule misapplication and/or misinterpretation of expressions of the type discussed earlier for Test X. Paradoxically the improvement was on the more difficult items. The control group showed no improvement and regressed on four of the items.

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Multiplication and Division The results on the multiplication and division items are given in Table 8.20. Table 8.20 Test Y: Percentage of students correct on multiplication and division items (Test X in brackets). Group 4 × -3 3× 6 4 × -2 15 ÷ 3 Expt.(n = 74) 87.0 (92.8) 85.8 (86.7) 90.6 (77.1) 87.9 (81.9) Cont.(n = 58) 77.0 (94.7) 69.9 (82.5) 64.7 (64.7) 79.2 (82.5)

6 ÷ -6 69.6 (74.7) 50.6 (77.2)

The experimental group performed better than the control group on all five items and also improved on post-test performance on the product of two negatives. Control group performance regressed on four of the items with no change on the product of two negatives.

Equations Table 8.21 shows that the experimental group performed better on each of the eleven items. Table 8.21 Test Y: Percentage of students correct on equations items (Test X in brackets) Group Expt.(n = 74) Cont.(n = 58) Group Expt.(n = 74) Cont.(n = 58)

-4

+[ ]= -8 86.4 (92.8) 78.4 (77.2) -4-[ ] = -8 76.2 (59.0) 59.5 (54.4)

3+[ ] = -3 94.1 (91.6) 79.8 (75.4) - 6-[ ] = 3 69.8 (51.8) 48.2 (43.9)

-2+[

]= 6 86.8 (84.3) 62.9 (73.7) -2×[ ] = 6 79.5 (69.6) 64.9 (70.2)

[ ] - -3=6 76.4 (65.1) 52.9 (54.4) -27÷[ ] = -9 90.6 (78.3) 71.6 (77.2)

-6-[ ] = 2 5-[ ] = 7 75.4 (75.9) 67.1 (67.5) 68.7 (71.9) 58.4 (70.2) -20÷[ ] = 20 65.1 (66.3) 50.0 (70.2)

The experimental group showed improvement on five of the items and stayed about the same on the others. The control group appeared to improve slightly on three items but showed regression on six items.

Brackets On the brackets items the experimental groups performed substantially better than the control group on all six items. Table 8.22 shows the details. Table 8.22 Test Y: Percentage of students correct on brackets items (Test X shown in brackets) Group 2 × (7-8) -3 × (2--4) -3 × (6-8) -15 ÷ (1-4) (-4+-8) ÷ 3 (4-8) ÷ -2 Expt.(n = 74) 77.4 (74.7) 68.1 (65.1) 85.0 (69.9) 81.5 (69.9) 66.0 (69.9) 77.9 (77.1) Cont.(n = 58) 63.7 (63.7) 48.6 (56.1) 56.8 (47.4) 62.8 (49.1) 40.8 (38.6) 57.9 (50.9)

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Compared with Test X the experimental group improved on two items and stayed about the same on the other four. The control group also showed improvement on some items but regressed on one.

Findings On the integer operation items in Test Y the experimental group showed higher proportions of students correct on all 30 items. The differences were substantial on many of the items. The experimental group also showed improvement on several of the items following the interval between the previous test. The control group showed small improvements on some items but also regressed on several. Overall the experimental group showed much higher levels of skill than the control group on the types of integer operations tested. The teachers claimed that in the intervening months between Test X and Test Y no specific class teaching of negative number had been done apart from where they were used in later topics (e.g. algebra, graphs, indices). Knowledge of negative number and related skills tended to be assumed. Incidental individual assistance, normally in the form of reminding students of the rules (oral or as corrections in workbooks, assignments and tests) are claimed to have taken place in all classes. The type of assistance given to both groups following the initial teaching of negative number seems to have been generally similar. It appears that the skill level in negative number operations shown earlier by the experimental group tended (apart from a few types of items) to increase, presumably through practice and application in later topics. In the absence of deliberate further teaching of the topic no other explanation can be provided. Similar improvement did not occur in the control group for which some evidence of regression was apparent. The latter findings seem to concur with the findings of Kuchemann 11 and the interpretation offered by Sfard. 12 Kuchemann found that negative number skills tended to decrease following teaching. Sfard 11 12

Kuchemann (1981). Sfard (1994).

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postulated that the decrease was probably due to not gaining understanding at the time of teaching. It is the author’s view (supported by observations, talking with students and the evidence of the test results described above) that the integer tiles method provided more students with understanding and meaning at the time of teaching resulting in, according to Sfard, reification of the concepts. For larger proportion of control group students such reification did not occur thus the concepts tended to be “... fragile and quick to disappear.” 13

Knowledge of sign rules Q.2 of Test Y was intended to provide a novel, compact and comprehensive indicator of students’ number sense and ability to use the sign rules. The twenty items contained large integers and the students were required to indicate only the sign of the number resulting from the calculation using the letters P (positive), N (negative) or Z (zero). The items have been categorised as ‘addition and subtraction’ or ‘multiplication and division’.

Addition and Subtraction Table 8.23 gives the results for the addition and subtraction items. It shows that the experimental students displayed higher proportions correct for all of the ten items. Table 8.23 Test Y: Percentage of students giving the correct sign on addition and subtraction items Group 567+764 654+-432 897+-675 891+594 686-753 Expt.(n = 74) 96.6 85.6 80.1 89.1 90.6 Cont.(n = 58) 86.5 53.2 66.6 80.8 82.2 Group 345--231 675--675 643--643 973+973 765+-765 Expt.(n = 74) 73.5 80.8 73.1 84.3 68.9 Cont.(n = 58) 48.0 66.3 46.9 55.1 48.1

The poorest performance on these items, for both groups, was on the item, 765 + -765, for which uncertainty was seemingly caused by the equality of the two numbers. Students did better on the similar item -897 + -675. Items involving subtraction of negative numbers again showed out as being more difficult for control students. 13

Ibid., p.268.

-

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Why did the experimental group perform better? The following explanation is offered as the reason. The experimental approach emphasised modelling and visualising outcomes of integer addition and subtraction operations in terms of the relative sizes and manipulations of positive and negative tile collections. As well as manipulating the tiles for small valued calculations the students were encouraged to imagine and write descriptions for calculations with large numerical values. An intended outcome of the tile activities was that students would both find out and articulate the rules for themselves. The teachers were requested to avoid statements such as “a plus and a minus makes a minus” and “two minuses makes a plus”. In contrast the control students experienced less exploration and no clear use of the number line model to justify the rules, particularly for subtraction of negatives. Some ‘complete the sentence’ type exercises were included in the textbooks but often the answer(s) was (were) given in an adjacent highlighted box. 14 As a consequence the control group seemed to show more uncertainty with the rules and markedly less ability to apply them to the large values used.

Multiplication and Division The outcomes for the multiplication and division items are given in Table 8.24. Table 8.24 Test Y: Percentage of students giving the correct sign on multiplication and division items Group 878×878 564×543 876×-642 764×0 0×-234 Expt.(n = 74) 77.8 82.7 82.7 82.7 84.0 Cont.(n = 58) 72.4 73.6 67.0 76.7 71.1 Group 987×-876 786÷987 554÷554 459÷-459 765÷127 Expt.(n = 74) 84.7 53.4 84.5 82.4 86.8 Cont.(n = 58) 66.3 36.9 67.3 64.0 76.4

The experimental group again had higher proportions correct for all items. By far the lowest scoring item was the all positive division, 786 ÷ 987. In the experimental group 40.5% responded ‘N’ compared with 54.2% in the 14

e.g. Daly et al. p.14.

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control group. When a sample of students was questioned on this response some could not give a reason, whilst others thought it was negative because the value was less than one. An experimental class student responded ‘Z’ and when questioned said, “ Because 987 into 786 won’t go it’s nought.” This item provides an indication of how poorly students understand the number line for positive numbers. Thirteen students (7 experimental, 6 control) did not attempt the item. In general the results of question two indicate that although the experimental classes performed better on all twenty items, at least 20% of such students had not developed full mastery of the operational rules and understanding of how integer magnitudes affect calculations. More than 30% of control class students indicate similar characteristics. The appearance of the large numbers and the reluctance of weaker and perhaps less motivated to think sufficiently deeply about the meaning of the given expressions and operations affected performance. Some students, able to successfully give the correct signs when small numbers were used for similar operations, were seemingly confused by the large numbers. All students questioned seemed to know that multiplying a number by zero produces zero and that zero times any number is zero. However around 20% said that 764 × 0 was positive and that 0 × -234 was negative. In written classwork occasional instances of ‘-0’ were observed. It appears that some students consider ‘0’ to be a positive number and ‘-0’ to be negative number. Emphasis that zero is neither positive nor negative is needed.

Substitutions and Evaluations Q.3 of Test Y required students to make integer substitutions in and evaluations of twelve algebraic expressions. The purpose was to test skill in using integers and integer operations in an algebraic context. The class and group outcomes are given in Table 8.25. The results of matching Q.1 items on the same test are shown in brackets. The experimental group performed substantially better than the control group on all twelve items. Mean scores

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were 7.9 (sd = 4.0) for the experimental group and 5.9 (sd = 3.8) for the control group. Table 8.25 Test Y: Percentage of students correct on substitution and evaluation items (Matching Q.1 item performances in brackets) Find the value of each expression if a = 4, b = -3, c = -2. Group b+c bc a-b 0-a 2-ab Expt.(n = 74) 77.8(86.9) 77.8(90.6) 69.1(74.1) 81.5 49.4 Cont.(n = 58) 52.8(66.4) 54.5(64.2) 42.4(53.2) 70.3 27.0 Group -c b2 a-(b+c) a-bc a÷c Expt.(n = 74) 61.7 56.8 51.9 64.2 72.8(69.6) Cont.(n = 58) 42.4 38.3 30.6 50.3 52.4(50.6)

0-b 66.7(75.4) 50.8(51.0) ab÷c 64.2 45.5

The performance of both groups was affected by algebra skill deficiencies, as the comparison with the matching integer operations items in Q.1 show. Several students were unable to correctly interpret the algebraic symbolism. For example bc was interpreted as b + c and evaluated as -

3 + -2 by some students. Four control class students generated 24 from

2 - ab by interpreting it as 2 × -a × b leading to 2 × -4 × -3 = -8 × -3 = 24. A total of 21 incorrect response values were given for the latter item. Similar faulty algebraic behaviour was apparent across all items. 15 Such short comings (also extremely common in the Years 9, 10 and 11 classes tested earlier in Study P) are well documented in algebra teaching and learning research. 16

Retention test findings On the quantitative part of the retention test the experimental group had substantially higher proportions correct on every one of the 62 items. In the five to eight month period between the post-test and the retention test the skill gap on integer operations appeared to widen between the groups. The only mathematical program difference between the groups was during the three-week negative number teaching experiment. The final worksheet in Work booklet 2 contained a short set of substitution and 15

Common examples of faulty interpretations included, 2 - ab = (2 - a) × b; a - (b - c) = a - b - c; b2 = b + b. 16 C. Kieran (1992). The Learning and Teaching of School Algebra, in Grouws, D. A. (Ed.) (1992). Handbook of Research on Mathematics Teaching and Learning. New York: Macmillan.

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evaluation exercises. The experimental class teachers treated it as an optional extra because the topic of symbolic algebra 17 (linear algebra 18) had not been taught. By the time of Test Y all classes had been taught the latter topics based on the approaches contained in the chosen textbooks.

Long-term retention test analysis In 1997, two years after the initial teaching of negative numbers a long-term retention test (Test Z) was given to as many as possible of the 1995 cohort. In Schools A and B the students had progressed to Year 10, whilst in School C the students were in Year 9. Fifty six of the experimental group and 35 of the control group were available for testing. The students were not spoken to after the test. However there were conversations with some teachers when test results were taken to schools.

Substitution in and evaluation of algebraic expressions Q.1 contained ten single operation algebraic expression for students to evaluate using the given integer values for the pronumerals. The mean scores were almost identical (Expt. 8.5 (sd = 2.4), Cont. 8.3 (sd = 2.2). Table 8.26 shows the group outcomes in terms of the percentage of students correct in each item. Performances on matching Test Y items are shown in brackets. Table 8.26 Test Z: Percentage of students correct on substitution and evaluation items (Matching Test Y items shown in brackets) Evaluate the given expressions if a = -1, b = 2, c = -3 and d = 0. Group d-b a+c c+b c×b a×d Expt. (n = 56) 93.0 93.0 93.0 84.2 (77.8) 87.7 Cont. (n = 35) 94.4 91.7 88.9 100.0 (52.8) 83.3 2 Group c-a c d-c b-a a×c Expt. (n = 56) 86.0 78.9 (77.8) 80.7 (56.8) 78.9 (66.7) 75.4 (69.1) Cont. (n = 35) 86.1 86.1 (54.5) 77.8 (38.3) 63.9 (50.8) 69.4 (42.4)

The results show that the substantial overall gap between the experimental group and control group evident two years earlier had virtually closed On the matching items from Test Y both groups improved however the control 17 18

Lynch et al. Chapter 4. Daly et al. Unit 12.

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group’s improvement was substantial. The incorrect responses for each item were similar to those observed and discussed on the earlier tests.

Expansions and simplifications Q.2 and Q.3 were intended to test positive and negative number operational skills in the context of algebraic manipulations. The results are contained in Table 8.27 which also shows the most common incorrect response(s) for each item. The experimental group performed slightly better on each of the four items. However the results for both groups are poor. Table 8.27 Test Z: Percentage of students correct on expansion and simplification items (sign rule errors highlighted) Question 2 (expand) Question 3 (simplify) Item u+3(v-w) p-2(q-r) 2x-3y-x-4y 3v-4w-5v+7w Expt. (n = 56) 50.9 24.6 50.9 52.6 Cont. (n = 35) 47.2 19.4 41.7 38.9 Answer u + 3v - 3w p - 2q + 2r x - 7y 3w-2v or -2v+3w (most) CIRs u + 3v - w p - 2q - 2r x + 7y, 3x - 7y -2v+11w, -2v-11w

The attempt to study the use of negative number skills in this context was affected by widespread student lack of understanding of the meaning of the given expressions and inability to correctly interpret them. For example a minority treated u + 3(v - w) as (u+3)(v-w) and p - 2(q - r) as (p-2)(q-r). This caused some of the variety of incorrect responses produced for each item. However the evidence of the most commonly occurring incorrect responses indicates that most mistakes were caused by sign rule errors (multiplication errors in Q.2, addition and subtraction errors in Q.3).

Completing tables of values Q.4 and Q.5 required students to display their integer operational skills by completing tables of values for simple linear functions. The two items tested integer multiplication and subtraction in an algebraic context. The given values were scrambled in order to force students to calculate each dependant value required. The results in Table 8.28 show the experimental group having higher proportions correct on seven of the eight items. For

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each item a variety of incorrect responses were given. The most common are shown. Table 8.28 Test Z: Percentage of students correct on completing tables of values items Question 4 (q = -2p)

Question 5 (m = 3 - n) -3

Group

p=1

p=0

p=

Expt.(n=56)

73.7 61.1

77.2 61.1

64.9 58.3

Cont.(n=35)

n=2

n=0

n=4

n = -1

n = -3

75.4 66.7

71.9 75.0

77.2 72.2

75.4 69.4

71.9 66.7

Answer

-2

0

6

1

3

-1

4

6

CIRs

-1, 2, -1p

-2, 2, -2p

-6,-5,-5p, 6p

-1, 5, 6-2n

-3, 0

1, 7

-2, 2, -4, 7

0, -6

The lowest overall performances were for q = -2p when p = -3. The curious effect of presenting examples of simple integer calculations in an algebraic context is clearly evident in Q.5 when some students in both groups failed to correctly evaluate m = 3 - n when n = 2 and n = 0. A total of 26 students failed to evaluate 3 - 2 and 25 students made a mistake on 3 - 0. In contrast similar items in Q.1 were handled comfortably. Teachers also found the result to be puzzling and could not offer an explanation.

Total Score on Algebra Items (Q2 - Q5) Questions two to five contained a total of 12 algebra items. The experimental group obtained a mean score of 7.6 (sd = 3.4) whilst the control group mean was 6.7 (sd = 3.7). The difference is not statistically significant. (t = 1.19, d.f. = 89, p = 0.208) Word Problem Q.6 was a four item word problem with a quiz show positive and negative points scoring scheme. In the ‘Get Rich Quick’ Quiz Show contestants score 5 points for each correct answer and -5 for each wrong answer. a) Bill answered two questions correctly and got four questions wrong. What was his score? b) Mick’s score was twice Bill’s score. What did Mick score? c) Jane answered six questions correctly and got two wrong. What did she score? d) What was the difference between Bill’s and Jane’s scores?

The class and group item by item results are given in Table 8.29. (Items have been written as formulae in the table). As well as the numerical answers given, the strategies used to obtain solutions (working shown, use of formulae, positive and negative numbers, operations, ...) were analysed.

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Table 8.29 Test Z: Percentage of students correct on quiz show problem items Question 6 (quiz show, correct +5 points, wrong -5 points) Group B=2c+4w M=2B J=6c+2w Expt. (n = 56) 87.7 71.9 91.2 Cont. (n = 35) 94.4 77.8 88.9 CIRs -15, -30, -2 10, 0, -30, 60, 20 25, 40, 4, 10 Answer -10 -20 20

d = J-B 68.4 77.8 40, -40, 0 30 or -30

The control group did a little better than the experimental group on three of the four items. Mick scoring twice Bill’s score (when Bill’s score was negative) confused some students, hence the responses 10, 0, 20 and 60 appearing as incorrect answers for the second item. Finding the difference between Bill’s score (negative) and Jane’ score (positive) was the item found most difficult. Students were not penalised if their answer for the latter item was consistent (and correct) with earlier responses on which it was dependant. Few students, in either the experimental and control groups, showed any working. The majority worked mentally and simply wrote down their answers as positive or negative numbers.

The following are examples from the few students who did show some working; ‘Peter’ (experimental group) a) b) c) d)

5 × 2 + (-5 × 4) = -10 10 × 2 = -20 5 × 6 + (-5 × 2) = 20 20 + 10 = 30

(Bill’s score) (Mick’s score) (Jane’s score) (Diff. Bill’s & Jane’s scores)

Peter’s working is all correct. He avoided use of -10 in (d) and understood that difference can be found by adding when the numbers have opposite signs.

‘Mary’ (control group) a) b) c) d)

2 × 5 = 10 10 × 2 = -20 5 × 6 + -5 × 2 20 - (-10)

4 × -5 = -20

10 + -20 = -10

30 + -10 30

20

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Mary is also correct. She used different setting out for similar items (a) and (c) and switched the order of the multiplications in (c). She knows that subtracting negative is equivalent to adding positive in (d).

Both Peter and Mary made use of signed numbers in their working. However another common strategy (typical of students’ approaches to word problems) was to avoid using signed numbers during the working stage and just use a negative sign in the answer if necessary as in the following case; ‘Paul’ (experimental) a) b) c) d)

5 + 5 - 5 × 4 = 10 - 20 = -10 10 (incorrect, no working shown) 5 × 6 - 5 × 2 = 30 - 10 = 20 (correct) 30 points (no working shown)

Paul was correct on (a), (c) and (d). The left hand side in (a) is interesting. He considered 10 to be ‘twice -10’.

The latter example illustrates the tendency shown by many students to avoid the use of negatives unless it is absolutely necessary or forced (as in item 6(a). The mean scores on the four-item quiz show problem were; Experimental, 3.2 (sd = 1.2) and Control, 3.4 (sd = 1.1). The question did not reveal any skill difference between the groups.

Recollections of negative number teaching methods The final question on the test asking students to describe how they would teach a friend to understand and perform a certain integer operation. Five different examples were used and each student was given one example to explain. The examples were randomly distributed through the test papers distributed to classes. Imagine you have a friend who hasn’t learned about negative numbers. Show (or explain) how you would teach your friend to do and understand the following problem: 0 − −3 (The four other examples used were; -5 - -3, 3 - -5, -5 + -3, 4 + -6.)

No marks were awarded for this question. The purpose of the question was to assess student understanding of the operations and seek evidence of

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residual effects of the initial experimental and control class teaching methods experienced.

Outcomes The types of explanations written were divided into four categories. Some students offered no response. Rule based explanations were by far the most common. Table 8.30 shows the group distributions of the explanation categories used. Table 8.30 Test Z: Group distributions of numbers of students using types of operation explanation Group Tiles, counters, ... Number-line Rule ‘Logic’ No attempt Expt. (n = 56) 3 9 27 6 11 Cont. (n = 35) 1 5 20 1 8

The table shows that the numbers of students in either the experimental group or control group referring to the initial teaching models experienced were small. The number of experimental group students citing the number line to explain either addition or subtraction was unexpected. This may indicate that either that some students may have been given number line explanations by someone subsequent to the initial teaching and post-testing or that they may have read their textbooks. 19 Some explanations categorised under the heading of ‘Logic’ seem to treat signed numbers as ‘objects’. For example, experimental group student explanations for -5 + -3 included; “Adding -3 is the same as making it more negative so -5 and 3 more negative is -8.” “Two negative numbers added together equal a negative number so the answer would be read as -8.” The above explanations are similar to some written in work booklets following integer tiles activities when experimental students were asked to ‘recap’ their findings. 20The tile-based method encouraged students to visualise addition as combining heaps. Adding two negative heaps produces

19 All experimental class students possessed a copy of the textbook used in the control class within their schools. 20 Work Booklet 2, p.1. (Appendix 10)

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a larger (more) negative heap. Elements of these ideas may underpin the explanations quoted above but by Year 10 the students may have considered it to be unnecessary to think of numbers as ‘heaps of tiles’. 21 A control student explained -5 + -3 thus; “Let’s say you lost 5 lollies and your friend lost 3 lollies that would mean you lost 8 lollies together.” The latter imaginative explanation is similar to some used in the textbook. 22 The interesting feature is the linking of ‘lost’ with ‘negative’. Tile or counter-based explanations appeared only for the explanation of 4 + -6. An experimental student drew four [+] counters and six [-] counters and followed with the comment; “Each counter is the same value in positive and negative. 4 negatives & 4 positives would be 0, but as there are 2 more negatives than positives the answer is -2.” Another experimental student in a different school provided; “You have 4 positive cubes and 6 negative cubes (drawn as squares). One positive cube & one negative cube when added equals zero. Therefore +4 + -4 (drawn) equals zero, but you still have 2 negative cubes, therefore answer is -2.” Number-line explanations for, 4 + -6, -5 + -3, and, -5 - -3, were attempted by approximately equal proportions of experimental and control students. Examples included; “... use a number line to explain ... start at -5 and count backwards by 3 to -8 ... this is the answer.” (-5 + -3, Expt.) Drew a number line from -6 to +6. Showed six (-1) hops to left starting at +4 finishing at -2 with an overarching ‘-6’. No written explanation was provided. (4 + -6, Cont.) Drew a number line from -5 to +5. “Start at -5. When you minus a negative it becomes positive. Move three spaces towards the zero. Your answer is negative two.” 23 (-5 - -3, Cont.)

21 In retrospect it may have been better to ask the students to provide the explanation to a Year 7 or Year 8 student rather than a ‘friend’. 22 Daly et al. p.11. (The textbook does not use the ‘lost lolly’ story.) 23 This explanation also includes use of a rule.

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No students attempted to use tiles (counters) or the number line to explain either 3 - -5 or 0 - -3. The students given these tasks quoted rules and neglected the instruction to provide understanding. Examples included; “2 negative signs equal a positive 0 - -3 = 0 + 3 = 3.” (0 - -3, Cont.)

“Any equation that has a minus sign followed by a negative number translates to addition ... = 3 + 5 = 8.” (3 - -5, Cont.)

The following novel rule interpretation for -5 - -3 was provided by a control student; “... because it is take away and a negative number follows this number must be added. To remember this rule pretend that the two dashes (minus signs) will form a (+) plus signal if they are beside one another.” Rule explanations for addition items included; “Well a plus and a minus is equal to a minus so it juts (sic) 4 - 6 which equals -2.” (4 + -6, Cont.) Drew a ring around ‘+ -’ and wrote, “4 - 6 = -2.” 24 (4 + -6, Expt.)

“... the minus takes precendence (sic) over the plus. The question is +4 - 6 = -2.” (4 + -6, Expt.) The confusion that some students have with rules is shown by the following explanations given for, -5 + -3; “I would say that when you add an (sic) negative and a negative you get a positive so therefore -5 + -3 = 8.” (Cont.) “I would say 2 neagatives (sic) make a positive so then you add the numerals 5 + 3 so it would be 8 positive.” (Cont.)

Many students made no response. One did not regard teaching ‘a friend’ as a responsibility judging by the following remark; “I wouldn’t say anything. I’d tell them their (sic) stupid and send them to special school.” (Expt.)

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Findings Application of rules dominated the thinking and method of handling negative numbers for a major proportion of the students who were content to explain ‘how’ but not ‘why’. A wide variety of rule interpretations were offered, most of which were valid. As found often throughout the study, misapplication of ‘two minuses make a plus’ is a recurring error for some students. For the relatively few students who offered model-based explanations, number line interpretations were more common for both experimental and control students. The control students were initially taught using this method so their use of it could be expected. The number line may also have been used occasionally for revision purposes by some teachers in the following years. The number line is also used by all students for the purpose of illustrating order and graphing whereas the integer tiles have no essential application following the teaching of the operations. For the experimental students the tiles (or tile-based ideas) were not used for revision purposes because teachers in following years were not familiar with the method. However some experimental students obviously did become familiar with number-line methods, possibly in class or perhaps from their textbooks. Three students managed to recall the experimental method but the majority had either forgotten about it or perhaps considered they had gone beyond the use of tiles or counters by Year 9 or Year 10 for the purpose of explanation. No student actually mentioned the integer tiles. The three who used the neutralisation model for their explanations explained it by referring to ‘counters’ or ‘cubes’. No student attempted model-based explanations for the difficult subtractions (0 - -3 or 3 - -5). Experimental and control students were able to either give versions of the required sign rule (e.g. “When you have to (sic) negatives facing each other it becomes positive.”) or describe a procedure (e.g. “Circle the two ‘-’ signs and write ‘+’.”) but no student produced an explanation that indicated an understanding of the process. This supports the author’s view (mentioned previously) that the type of 24 A textbook strategy and common quick correction strategy observed in one of the schools. (see Lynch et al.)

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number line explanations used in the chosen textbooks for both addition and subtraction (i.e. ‘facing and pacing’) is providing little understanding for many students. Number line diagrams using directed line segments (head to tail for addition, tail to tail for subtraction) are considerably easier to illustrate and explain. 25

Conclusion On the evidence of the Test Z results it appears that, during the two year period that elapsed since the initial formal teaching of signed numbers, the performance gap found around 18 months earlier in favour of the experimental group, had almost closed. In general the former experimental class students seem to have continued at about the same measured knowledge and skill levels, whilst the former control students have markedly improved. It is possible (perhaps highly likely) that the teaching given to the former experimental class members did not build sufficiently on and more effectively develop their superior early skills. Most experimental class students became competent at integer calculations although some showed subtraction weaknesses. The evidence suggests that subtraction is the most difficult negative number calculating skill to understand and an effective teaching model is essential. Compared with the number line method used, the integer tile method was instrumental in developing much higher levels of subtraction skill in the early part of the study. However some students did not master it at the time of teaching and weaknesses have been allowed to persist. Since the initial teaching in 1995, the classes in each school have been regrouped at least twice, they have been taught by different mathematics teachers and no attempt was made to influence teaching methods. Regardless of the negative number teaching method used initially, the responses to Q.7 on the long-term retention test, suggest that students usually end up relying on the operational sign rules. The tile-based activities encouraged experimental class students to find, understand and articulate the rules themselves during the initial three-week teaching 25

The neutralisation model can also be used on the number line.

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program. However the impact of earlier teaching diminishes and some students forget. Mathematics teachers thus find it necessary to re-teach or revise topics to classes or individuals. With regard to negative number, the author’s observations of classroom events over many years indicate that many teachers shortcut the teaching process and merely re-state the sign rules (often in carelessly abbreviated form) without appropriate justification and/or reference to a suitable model. As a consequence uncertainty and confusion with rules can result for weaker students. This is almost certainly a causal factor in the lack of mastery observed in some students. When difficulties, weaknesses and forgetfulness became apparent, particularly for weaker students, the integer tiles (or more effective use of the number line) could (should) have been used for revision purposes rather than the giving of trite reminders of the sign rules as appears to have been the case. 26 Weakness in basic algebra prevented several students (both experimental and control) from demonstrating signed number skills in the context of algebra. The suggestion offered is that algebra should be taught using only the numbers of arithmetic over a reasonable period (perhaps a year) before complicating it with negative number. Algebra can then be used to assist the introduction of negative number. Chapter summary and findings This chapter contained the quantitative outcomes of the four tests used as indicators of negative skill development and retention resulting from the two teaching methods investigated. Statistical analysis of test scores and detailed comparative examination of test items indicated that; •

the experimental and control groups appeared to start level with regard to arithmetic skills and prior knowledge of integer operations,

•

the experimental group developed significantly different and substantially higher levels of negative number skills than the control group during the period of teaching,

26 The fact that some former experimental class members referred to the number line for their explanations may indicate that some teachers may do a little more than just restate sign rules when revising or giving help to students.

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•

the observed difference between the groups appeared to widen for a period of several months subsequent to teaching,

•

after two years the experimental group maintained the same level of skill whilst the control group improved to a comparable skill level,

•

subtraction was the weakest operational skill and there were some students in both groups who did not master it during the period of teaching and still showed weaknesses two years later,

•

only a minority of students appeared to be able to write explanations of integer operations using reference to a model used during the teaching process - the norm was to give sign rule explanations.

The overall evidence suggests that neutralisation model, embodied in the integer tiles, can be the basis of a superior initial negative number teaching method to the number line model-based method popularly used. However more needs to be done in subsequent teaching to take fully exploit the initial learning benefits provided by the neutralisation model. By Year 10 practically all students should be expected to attain complete mastery of the types of skills tested in Test Z. The latter aspect needs further study and indepth research investigation.

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Chapter 9

Outcomes within Schools

In the previous chapter the results were presented as a quantitative comparison between the experimental and control groups comprising the experimental and control classes across the three schools involved. The discussion in this chapter will concentrate mostly on events and outcomes within particular schools and classes, as a basis for discussing how the respective teaching approaches affected and influenced learning. Apart from the researcher explaining the neutralisation model, and the integer tiles and work booklets prior to beginning the teaching of the topic, the experimental class teachers were free to organise and conduct lessons in their own way. The teachers were requested to work carefully through the work booklets themselves before starting teaching. Control class teachers received no teaching method instructions and the expectation was that they would teach their lessons using their normal approaches. The researcher’s observations and conversations with teachers and coordinators suggest that this seemed to have occurred. The intention was to keep the classrooms as natural as possible. On the basis of school-based events the questions to be considered are; •

How did the teachers organise and teach their classes and handle the use of the tiles and work booklets?

•

How did teachers’ use of the experimental materials vary?

•

What difficulties did the teachers encounter with the materials?

•

What approach did the teachers use for correcting work booklets and providing feedback to students?

•

How did the students use and react to the tiles and work booklets?

•

To what extent did the experimental materials appear to assist learning?

•

What difficulties did students have in using the materials?

•

What did the teachers think of the tiles and work booklets and what changes and improvements did they suggest?

•

How did the control class teachers teach and manage their classes?

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•

Within each school how did experimental and control class performances compare?

•

What evidence emerged to support the integer tile embodiment of the neutralisation model as superior to the number line for the purpose of teaching integer concepts and operations?

The above questions are answered on the basis of the researcher’s classroom observations, conversations with teachers and students, videotaped interviews with samples of students in each school and examination of a sample of students’ completed or partially completed work booklets. The performance on six basic integer operations items of the seven classes involved in Study E and the two classes tested at corresponding stages of Year 8 and Year 10 for Study P in 1994 will also be compared.

School A School A provided two year 8 classes, each with 21 students. ‘Helen’ taught the experimental class and ‘Irene’ the control class. Experimental class The experimental class worked with the tiles and work booklets in selfpaced fashion with Helen (who was also the mathematics coordinator) providing a minimal amount of up-front class teaching. Helen became firmly convinced of the value of the neutralisation model approach as a result of the pilot study in the previous year. After the first lesson the students were organised to distribute and collect the work booklets and tiles. 1 Students worked around tables mostly in small groups or pairs. One excellent student, who worked quickly and meticulously, preferred to work mostly alone but occasionally help others if both Helen and the researcher were already engaged. Helen collected the work booklets for correction following each lesson and did not permit them to be taken away by the students for homework. She provided detailed individual written feedback

1

The tiles were kept in bulk in two icecream containers, the [+1]/[-1] tiles in one and the [0]/[0] tiles in the other between lessons. Monitors placed sufficient tiles on each table at the beginning of each lesson.

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comments and corrections and a numerical score (on the basis of one mark for each small task) was given for each worksheet in Work Booklet One. Pressure of work lessened the opportunity for the same standard of detailed correction for Work Booklet Two. During the first lesson (Work Booklet One, Worksheet 1) in which the tiles were introduced and used for only positive integers in simple additions and subtractions, some students questioned the necessity of the tiles for doing exercises which merely revised primary level number facts and operations. Sensing imminent rebellion threatening to derail the experiment there and then (mutterings included, “This is stupid.” “It’s boring.” “Why are we doing this?” “You don’t need these silly things.”) Helen stopped the class and explained that actually using the tiles was not necessary but the ideas behind the tiles and the tile activities were needed for understanding how to use the tiles. “I don’t care if you’d rather not use the tiles when you think they’re not needed, but I want you to at least think about and imagine them (the tiles) while you are working.” This seemed to satisfy most. A few took little notice and merely worked the ‘sums’ and skipped over many of the other assigned activities requiring reading, thinking, writing or drawing tile representations. As a result the latter students generally found the next stages of the topic more difficult. Convincing some students of the need to do the activities apart from the numerical exercises, particularly tasks leading to finding patterns and being able to articulate the sign rules for themselves, was an ongoing teaching problem throughout the unit. There was less resistance to using the tiles after calculations involving both positive and negative integers and the need to use zero pairs were introduced in Worksheet 3. From then on the benefits of the tiles for modelling operations became more obvious and consequently more acceptable. Most students appeared to have little difficulty in using the tiles to show the neutralisation process when necessary in performing addition. They readily accepted that balanced quantities of positive and negative tiles had value zero and could be replaced by a zero tile. No student appeared to question this assumption. Nor did students have difficulty in accepting that

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a zero tile could be replaced by a balanced quantity of positives and negatives, when necessary, in performing difficult (more advanced) subtractions. 2 By the completion of Worksheet 6, students were expected to have discovered, as a consequence of tile activities, the sign patterns for adding and subtracting integers. At this stage no students in the experimental class were adding two negatives to get a positive 3 and most appeared to have found that subtracting an integer was equivalent to adding its opposite. The remainder of Work Booklet One dealt with integer multiplication. Positive by positive and positive by negative were treated as repeated addition of equal sized tile collections. To find the value of 3 × -4 we can use three lots of four [-1] tiles. In this case 3 × -4 = -4 + -4 + -4 = -12. 4

The students appeared to accept this interpretation and as a consequence saw that a positive by a negative produced a negative. Also introduced were similar multiplications using different tile representations as a lead in to multiplying by zero and expanding brackets. Make a collection containing three [+1] tiles and three [-1] tiles. Make another identical collection. Combine the two collections. What is the value of the total collection? You have shown that: 2 × (3 + -3) = (3 + -3) + (3 + -3) = 0 + 0 = 0 5 Let's work out 2 x (4 + -3) using the tiles. If we first consider the part in the brackets we need four [+1] tiles and three [-1] tiles. Make two of these collections and combine them. How many [+1] tiles and how many [-1] tiles are there altogether? What is the final value? We have used the tiles to show that 2 × (4 + -3) = (2 × 4) + (2 × -3) = 8 + -6 = 2 6

Most students coped with the set of similar examples that followed. A few students omitted the full working (although instructed to show it) but still found the correct answers. An intention of this section was to show that such evaluations could be done in two ways; either by simplifying the

Using integer tiles 8 − 6, -9 − -5, and -500 − -270 are ‘easy’ subtractions whilst 6 − 8, 5 − -9, and 270 − -500 are difficult subtractions (insertion of zero (neutral pairs) needed). 3 Of course at this stage they had not met the product of two negatives. 4 Work Booklet One, p.14. 5 Ibid., p.15. 6 Ibid., p.17. 2 -

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brackets before multiplying or by expanding the brackets and then simplifying. The development then extended to examples of right-hand distribution in preparation for evaluating negative by positive and negative by negative multiplications. Now we'll try (4 − 2) × 3. You can probably see that the answer is 6 but let's work it out using the tiles. There are two ways we can do it. Doing inside the brackets first we can start with four [+1] tiles and then subtract two [+1] tiles. (4 − 2 = 2) Our example has 'collapsed' to 2 × 3 and you know that two lots of three [+1] tiles gives six [+1] tiles. Written out in full it looks like this. (4 − 2) × 3 = 2 × 3 = 6. In words we could say, "Starting with four, take away two and then multiply what's left by three". Instead we could start by expanding the brackets first. Let's try it! Make four lots of three [+1] tiles. This gives twelve [+1] tiles. From this collection take away two lots of three [+1] tiles. How many [+1] tiles are left? (There should be six!) In full it can be written out as; (4 − 2) × 3 = (4 × 3) − (2 × 3) = 12 − 6 = 6

7

The work continued with the following exercises; Complete the following expansions by filling in the missing numbers. a)

(6 − 3) × 2 = (6 × ) − (3 × ) =

−

= 6

b)

(2 + 4) × 3 = (2 × ) + (4 × ) =

+

= 18

c)

(5 − 5) × 4 = (5 × ) − (5 × ) =

−

=

d)

(2 − 5) × 3 = (2 × ) − (5 × ) =

−

=

e)

(0 − 4) × 2 = (0 × ) − (4 × ) =

−

=

f)

(5 − 15) × 4 = ( × ) − ( × ) =

−

=

Multiplying negative numbers by positive numbers Look back at your answers for exercises 3d, 3e and 3f above. You should have got negative number answers. Why? For exercise 3d by working out the brackets first we would get: 7

Work Booklet One, pp.17-18.

158 (2 − 5) x 3 = -3 x 3. From your working above you can see that the value must be -9. So -3 x 3 = -9. Do 3e and 3f again by working out the brackets first. 8

Most students correctly completed the above exercises and the negative by positive examples that followed. The few who had difficulties were usually students who felt it unnecessary to include the full working and wrote only what they thought would be the answer. Some (usually weaker) students appear to be inhibited or intimidated by long chains of mathematical reasoning. On the other hand some brighter students preferred to skip steps and do it in their heads. By not engaging in thinking through the process fully, some gave (or guessed) incorrect positive values. Work Booklet One concluded at this point with a note and a small exercise on the commutative rules for integer addition and multiplication. Some students completed the first booklet (containing nine worksheets) by the end of the first week and most had finished it by the middle of the second week. Work Booklet Two commenced with a review of the key findings from the learning activities and exercises in the first booklet (Worksheet 10). It was suggested that this could be handled as a class discussion. However by this stage the class was too spread out for this to be done. The teacher checked findings either with individuals or table groups. All but one student provided at least rough but substantially correct descriptive responses to each question as required. The other did not bother to respond to the descriptive questions but did get all of the mixed set of numerical examples at the end of the worksheet correct. Thus it appeared that all class members knew the sign patterns for the operations covered to that point. Worksheet 11 dealt with multiplying negative numbers by negative numbers. The approach used earlier subtraction findings and right hand distribution as was the case for negative by positive in Worksheet 9. Earlier work with the tiles showed that 0 − 3 = -3.

8

Work Booklet One, p.18.

159 (Recall that we started with zero in the form of three [+1] tiles and three [-1] tiles and then took away (subtracted) the three [+1] tiles. Do this again for yourself with the tiles if you still aren't sure about it. You can always use zero to 'create' missing numbers.) Activities with the tiles have shown that when any positive number is subtracted from zero the result is the negative of that number. (eg. 0 − 5 = -5, 0 − 1000 = -1000, 0 − a = -a) Tile activities also showed that when any negative number is subtracted from zero the result is the positive of that number. (eg. 0 − -5 = 5, 0 − -1000 = 1000, 0 − -a = a) We will now use our findings to explore negative by negative multiplications. As an example let's look at -3 × -2. Using the tiles we found that 0 − 3 = -3. We'll begin by replacing -3 in the expression by (0 − 3), so -3 × -2 becomes (0 − 3) × -2. Next we can expand out the brackets. (0 − 3) × -2 = (0 × -2) − (3 × -2) = 0 − -6 = 6 -3 × -2 = 6 So What is the most 'curious' feature of this result? 9 1. Work through this one by filling in the missing numbers yourself. -5 × -4 = ( − ) × -4 = ( × -4) − ( × -4) = − = -5 × -4 = 2. Write this one out in full yourself. -100 × -6 = By now you have probably realised that you do not need to write the working out in full each time. What is the sign of the number that will always result from multiplying a negative integer by a negative integer? 10

Four of the students did not fully complete the exercises shown above and correctly respond ‘positive’ to the question regarding the resulting ‘sign of the number’ for the product of two negative numbers. The rest of the class

9

“It’s very interesting that when you multiply a negative by a negative, you get a positive.” was one student’s comment. Most students neglected to respond to this question, but did complete the exercises fully and correctly and seemed to have discovered that the sign is always positive. When the work booklets are revamped for more general use care will be taken to number all questions and tasks for which students are expected to provide written responses.

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did as required. “It’s very interesting that when you multiply a negative by a negative, you get a positive.” was one student’s comment. The four students who had not bothered to attempt Q.1 and Q.2 also did not (or were unable to) satisfactorily complete the rest of Worksheet 11. Errors in the mixed set of operations exercises and in completing the integer multiplication table for values from -4 to 4 show that the four students had not mastered the multiplication sign rules. The other students all completed the exercises successfully and appeared to have achieved a good grasp of the integer sign rules for multiplication. Three of the four students mentioned previously did not get beyond this stage of Work Booklet Two. Worksheet 12, introduced decimals, number lines and graphs into the topic. The same sign 'rules' we have discovered for integers also apply to positive and negative decimal numbers. 11

The students who had shown mastery of the sign rules in performing integer operations were generally able to use them for decimals. Errors tended to be calculation slips (e.g. -3.1 × -6 = 19.6). The numerals were simple and the students did not use calculators. 12 The number line was then introduced and used for the purpose of ordering, defining order properties and the use of greater than (>) and less than (