Mathematical Question Spaces - The Faculty of Mathematics ...

Mathematical Question Spaces C J Sangwin LTSN Maths, Stats and OR Network, School of Mathematics and Statistics, University of Birmingham, Birmingham, B15 2TT, UK Telephone 0121 414 6197, Fax 0121 414 3389 Email: [email protected]

http://web.mat.bham.ac.uk/C.J.Sangwin/

Abstract: This paper considers a particular aspect of computer aided assessment (CAA) in mathematics: randomly generating questions. Adopting a methodology of textual analysis, a consideration of how textbook authors have structured their questions on quadratics leads to drawing a distinction between question space and question instance. While a student will only see an instance of a particular question, underlying this is a question space of pedagogically equivalent questions. Identifying such a space has practical application in randomly generating question sequences in CAA. Keywords:

1

Introduction

This paper considers a particular aspect of computer aided assessment (CAA) in mathematics: randomly generating questions. In doing this we examine not only a single mathematical question, but how such questions are linked together into coherent structured schemes. By considering what can be varied in a particular question, without significantly changing its purpose or effect, the analysis results in a distinction between a “question space” and a “question instance”. That is to say, we consider variation which maintains structure. To support this distinction, the treatment in school text books of the topic of quadratics is examined in detail. The methodology comprises a documentary analysis of existing teaching materials, with a focus on those pertaining to the following ubiquitous question. Solve ax2 + bx + c = 0,

(a 6= 0).

(1)

It is further argued that the question space/question instance distinction is applicable, and potentially useful, more generally when designing schemes of mathematical questions, not only those associated with the algebraic activities connected with quadratics, nor solely with computer aided assessment. This is another example of computer aided assessment concerns acting as a Trojan horse for more general educational discussion. It may be argued that very few teachers will need to design schemes of work from scratch, since question design is the purview of the specialists writing text books, or CAA materials. Actually, as we shall see, structure in problem sequences is fragile. It is hoped that the framework developed will allow teachers to perceive the structure and to be mindful of, and hence preserve, it when using the material. 1

2

Background

2.1

Computer aided assessment in mathematics

Computer aided assessment in mathematics is a large field. However, such computer aided assessment has predominantly used multiple choice questions, or modifications of these. Such modifications provide a significant range of variations, and include multiple response, matching questions and ‘hot spot’ graphical input. What all these have in common is that they are provided response questions (PRQ). That is to say, questions in which a student is provided with a list of potential answers and asked to select one or more, or to match pairs. The use of PRQ is almost always a constraint dictated by the software, and not the choice of the user. Indeed there are well-aired fundamental problems with provided response questions which are (i) question distortion, (ii) assessment only of lower order skills, (iii) strategic learning and possibly (iv) gender bias Hassm´en and Hunt (1994). There is research evidence that (i)–(iii) can be ameliorated to some extent by careful design of questions. However, it is quite uncontroversial to assert that for the majority of mathematical questions, asking the student to provide an answer more honestly reflects the intentions of the teacher. CAA implementations usually include numerical entry and free text entry type questions. The former has only limited use, and until recently tools to mark the latter were primitive. In CAA, the response processing includes not only the assignment of a numerical mark to a student’s response, but also the generation of specific feedback and storage of the outcomes in the system. Contemporary practice in mathematical CAA increasingly employs computer algebra systems (CAS) to process responses, as implementations such as AiM, (Klai et al. (2000); Strickland (2002)), CABLE (Naismith and Sangwin (2004)) and others demonstrate successfully. The prototype response processing establishes the algebraic equivalence of a student’s answer with that of the teacher. However, much more sophisticated response process is possible which establishes the syntactic form, or other mathematical properties, of students’ answers. See, for example, Sangwin (2003). These are precisely the sorts of tasks for which a CAS was designed. Response processing, from the technical, semiotic and pedagogic points of view, are important fields. However, this paper concentrates on the design of schemes of work which are randomly1 generated. While we have in mind CAA system using CAS, in which automatic response processing takes place, there is every reason to suppose that students’ responses to randomly generated questions could also be marked by a teacher in the traditional way. There are two important pragmatic reasons for wishing to generate a random sequence of questions. 1. Randomly generated questions may reduce plagiarism. 1 While the term “random” is used throughout this paper it should be taken to be synonymous with “pseudo-random”. In practice, when generating sequences of problems for students, the statistical differences between random and pseudorandom methods are inconsequential. Furthermore, it is imperative to track which values have been given to each student and an expedient method for doing this is to store a seed, from which the pseudo-random sequence can be recreated at will. Even very simple pseudo-random generators, such as linear recurrence relations, have proved to be sufficient. For example, the system described in Naismith and Sangwin (2004) takes, for each student, r0 as a seed value, and generates a sequence

rn+1 := arn 261 − 1)

mod p

where a is fixed, and p (= is prime. The sequence (rn ) is then used as the pseudo-random sequence, from which random values can be calculated. For example, if the kth random number required is to be an integer between 0 and 9, taking rk mod 10 suffices. The seed values can be generated from a combination of student identity number and system clock time. Full details may be found in, for example Knuth (1969).

2

2. Distinct but equivalent questions may be used for practice. Even if giving each student a distinct problem sequence reduces plagiarism, professional experience unfortunately demonstrates it is not eliminated. Neither is impersonation in CAA. However, in this paper we address only the second question, by examining the notion of “equivalent problems” in some detail. The majority of CAA systems have some facilities to automatically and randomly generate different questions for each student. Just as computer algebra may be used for response processing, so it makes the automatic manipulation of mathematical terms and equations very simple to achieve, when randomly generating problems. As motivation, to generate examples of question (1), one might start by taking two random numbers α and β , which are almost certainly small integers. These are substituted into the term (x−α)(x−β ) which is manipulated symbolically and expanded by the computer algebra system before the resulting unfactored term is given to the student, as part of (1) above. The ability of the CAS to manipulate terms symbolically gives great flexibility to easily generate questions, and a survey of the capabilities of modern CAS implementations may be found in Wester (1999). One might argue that in the case of (1), the use of computer algebra is not necessary. For example, in the case a = 1, one calculates −b = α + β and c = αβ . The first and most significant objection is that such a piecemeal approach to CAA severely limits what is possible. However even in the case of the apparently tractable (1) there are problems. If α + β > 0, then in setting the question one has the irritating technical problem of unary −, interacting poorly with binary +. For example, if α = 1 and β = 2 then we have the real possibly of writing x2 + −3 + 2. Worse examples occur when α = −β , in which case the system would display x2 − 0x − α 2 . Such issues are not confined to the display only, especially when the mathematically well-posed questions are generated from a proper subset of the possible parameter values. These problems need not occur if computer algebra is used sensibly to support CAA. More importantly, if α = ±β or α = 0 or β = 0 then pedagogically we could argue we have generated very different problems. Hence, generation of such random numbers may destroy or alter some structure that the teacher wishes to preserve. It is precisely the issue of such structure within problems sets, in the context of CAA and more general assessment, that this research attempts to address.

2.2

Theoretical background

It is uncontroversial to assert that learning mathematics is only effective when it is an active process on the part of the learner. A ubiquitous technique for attempting to engage students is the setting of questions, both written and oral. Indeed, asking students questions is a central part of all theories of learning. Linguistically, a question is a sentence worded or expressed so as to elicit information. We shall use the term “question” in such a way, when in practice many words are used in text books. For example, “exercise”, “problem”, “task” and so on are often used by text book authors in a way we include under the broad category of question. “Examples” may also be taken to mean question. In the analysis here, a question is also taken to include an instruction, such as “solve”, “factor”, “sketch” and so on. For the purposes of question and test interpretability (QTI) the IMS define an Item to be “the smallest exchangeable assessment object”. In their sense it is significantly more than a question, since it contains details of response processing instructions, and feedback, both hints and solutions, to be given. This specification includes the notion of Item Clone, which are equivalent items created from 3

an Item Template by the substitution of Item Variables. However, the specification operates only at the level of individual items, and takes no account of the sequence of items. It is precisely the sequence of questions and in particular how they are linked together, which is considered in this paper. As evidence in this study we consider questions in mathematical text books, rather than existing CAA material, the majority of which is constrained artificially by the use of provided response question types. Using such schemes of questions is one of the major techniques used for self-study, home work or in the classroom. In one sense, working through such pre-structured exercises is akin to taking part in a dialogue, and such dialogues are an important part of learning. Indeed, the central tenant of the Conversational Theory of Pask, and others, Scott (2001), may be paraphrased as “a conversation is the minimum necessary structure to enable learning.” However, in interpreting the use of a scheme of fixed questions as a dialogue, care is needed. As Richards (1991) comments: “the mere saying of words does not make a conversation, nor does it make for communication. [...] For communication to occur, both participants must have the potential for change.” The dialogue, if it can indeed be described as such, between the textbook author and student is predicted by the book author, and realized as an internal process by the student. In predicting this, the author assumes that the learner is able to comprehend the knowledge in the form in which it is expressed. That is to say, that there does not need to be a negotiation of meaning. Indeed, it may also be argued such a dialogue is a transmission of knowledge which the author asserts as representing some universal truth. Hence, pre-storing dialogues in this way is not a mutual process of coming to know. Therefore, it is not a conversation in the sense of Conversation Theory. How then do such schemes of questions enable learning? Although it is usual for a dialogue to take place between two interlocutors, an internal conversation occurs when one engages in “thinking aloud”. On the nature of this internal conversation Skemp (1971) says, “the mere act of communicating our ideas seems to help clarify them, for, in so doing, we have to attach them to words (or other symbols), which makes them more conscious”. Hence, while one does not have a conversation with the textbook, the textbook may provoke internal enquiry and dialogue. They may also play a part in the learning process by providing mutual ground, or a shared sequences of experiences, about which subsequent conversations can take place. There may be other legitimate uses, such as providing “finger exercises” to promote rather mindless mechanical fluency. Contemporary practice seeks to encourage the use of computer aided assessment, or computer based learning. Here the dialogue is between the machine and a student. Such a dialogue is synchronous, as opposed to the asynchronous traditional paper based approach in which a week, or more, may elapse between answering a scheme of work and receiving any feedback. Such systems increasingly involve an expert system, or other forms of artificial intelligence, to generate feedback which is responsive and appears to be intelligent. While such systems may be able to provide feedback to point out technical slips and algebraic infelicities, it is notoriously difficult to design systems which identify and reveal misconceptions. The problem is most acute when a misconceived method results in a correct answer. There are intelligent tutoring systems, an early example of which is that of O’Shea (1982), which do adapt the scheme of questions given to students, based upon students’ previous responses. However, each such system is restricted to a very limited domain of learning. What CAA, textbooks, and written materials produced by teachers for their individual classes, have in common is their use in assessment. There are at least four types of assessment: diagnostic, formative, summative and evaluative (the latter concerns institutions and curricula). Following Wiliam and Black (1996), these terms “are not descriptions of kinds of assessment but rather of the use to which information arising from the assessments is put”. Accepting this, the function of an individual 4

assessment should therefore have less influence on its design than other considerations. These include whether the schemes of work connected with the topic are conceptually underpinned.

3

Methodology

To focus the study, the teaching of “quadratics” was considered in detail. This topic is ubiquitous in curricula and important mathematically. Indeed, this topic is sufficiently important to have inspired a House of Commons debate on general education in the United Kingdom, see Hansard (2003). The mathematics associated with quadratics is simple, but as we shall see is sufficiently rich to present the course designer with a number of real options in the design of schemes of work. In addition, associated with quadratics are a variety of graphical activities. While it is artificial to take one topic in isolation, the subject of quadratics is sufficiently broad for our purposes. The methodology adopted was that of textual analysis. The research question is the following. How are sequences of mathematical questions structured by text book authors? This is from an overall sense and at the fine grained detail of how individual questions are linked together into sequences of exercises. Of course, the perception of structure in a problem sequence does not imply its presence by the intention of the author. Equally the absence of the perception of structure does not imply thoughtlessness on the part of the author, and may well indicate a blindness by the reader. The key feature here is the search for structure. The methodology of document analysis was chosen since book authors must pay particular attention to the design of their teaching materials, including the question sequences. Essentially one is seeking inspiration by examining closely the work of previous generations of mathematicians and educators. In some cases, contact with authors was possible, and some authors were contacted with specific queries as to their intentions in setting particular problem sequence. The age of much of the materials studied precluded a systematic interview of textbook authors. Each text was read closely, to examine the order, emphasis and implementation of this topic. The focus of the reading was the exercise sequences, rather than any text. Professional experience suggests that students often ignore text and move straight to the questions which constitute the assessment. That is to say, the assessment drives the learning. This reading includes both the strategic planning of work as a whole, the structure of exercises and the fine grained detail of particular exercise design. The evidence base was a large selection of mathematics texts published in the United Kingdom over the last two hundred and fifty years. Information on arithmetic books prior to 1850 may be found in, De Morgan (1847). A historical perspective was considered necessary, as the approach of contemporary texts is dictated to a large extent by National Curricula. Indeed some books, such as Jennings and Dunne (2003), follow a prescribed lesson structure, such as that of the Key Stage 3, National Numeracy Strategy, explicitly. In this book, the work is presented in what could be interpreted as lesson plans with clearly differentiated goals, starter questions, demonstrations, worked examples, plenary activities, exercises and homework. Older texts have the potential for greater variety in approaches. Furthermore, these works also follow mathematical topics, such as algebra and geometry. Hence, each individual book was used by students for a number of years. Since current United Kingdom text book 5

schemes are designed to follow the National Curriculum (see HMSO (1999)), topics are presented in the order prescribed. In general, a different book is used in each year2 . As we shall see, this approach was vindicated inasmuch as providing a variety of approaches and contrasting schemes of work. The purpose was not a comprehensive examination of how quadratics have been taught, although the inspection here indicates that such a study would be very interesting and perhaps valuable. Note that only the sections on quadratics, which constitute a small sample of any particular work, have been examined in any great detail. Hence, it is not the intention to criticize the quality of particular authors’ works cited here. In presenting the results, a few key texts have been selected. The backgrounds to these are given, for completeness, in Appendix A. There are three reasons for including a text here. Firstly, because it is representative of a general approach, secondly because it takes a radically different, but interesting or challenging approach, and thirdly because some exercise sequence illustrates a point with particular clarity.

4

Results: overall strategies

In examining the topic of quadratics it is soon apparent that there are at least four aspects, which include (i) symbolic manipulation, eg expansion and factorization of terms, (ii) extraction of roots, and solution of equations, (iii) graphical activities, including sketching, (iv) application to word problems. What was most surprising was the unanimity of all modern texts as to how the first two of these aspects, manipulation and solution, were treated. This order was as follows. 1. Removal of brackets. 2. Factorization by inspection, variously subdivided into cases. Monic (ie a = 1), and non-monic quadratics, perfect square, or the difference of two squares. 3. Solution of equations by factorization. 4. Completing the square. 5. Use of the formula for solutions. 6. Symmetries of roots, ie (x − α)(x − β ) = x2 − (α + β )x + αβ . [Not always present]. In the vast majority of algebra texts other topics, specifically for example the manipulation of algebraic fractions, are interspersed between the topics above. This is in close agreement with the summary of how algebra is usually developed, as presented by Barnard (1999). How much emphasis is placed on graphical methods differed, as did how graphical considerations interacted with algebraic. By way of contrast, we compare the ubiquitous treatment of the algebra associated with quadratics with the ordering of topics given in Euler (1822). Quadratics first occur in §270 when compound 2 This horizontal, rather than vertical stratification of material may have profound implications on the way students learn, and their perceptions of the interconnections between mathematical topics. Furthermore, horizontal stratification exposes students only to that material prescribed by the teacher. It makes “reading ahead” and “looking back” difficult, and may stifle the opportunities for enrichment material to be included within text books. These issues deserve closer scrutiny.

6

quantities are multiplied together, although they are not singled out and are treated together with other kinds of terms. It is only beginning in §623 that the “Resolution of Pure Quadratic Equations” is first studied. These are quadratic equations without a linear term (ie b = 0), which can be solved immediately by taking a square root, and accepting where necessary imaginary solutions. To quote, “Equations of the second degree, in which all the three kinds of terms are found, are called complete, and the resolution of them is attended with greater difficulties.” These he approaches not by writing quadratics in the current standard form ax2 + bx + c but as an equation x2 + px = q, 2

and noting that the left hand side of the latter is a perfect square if and only if we add p4 to both sides. Essentially he completes the square to solve the equation. In §641 he gives the usual quadratic formula. This formula contains the rule by which all quadratics equations may be resolved; and it will be proper to commit it to memory, that it may not be necessary, every time, to repeat the whole operation which we have gone through. After giving some examples, Euler provides another method which is to remove the linear term by a change of variables. Quoting §645 extensively: If we make this substitution of y + 21 p instead of x, we have x2 = y2 + py + 14 y2 , and px = py + 12 p2 ; consequently, our expression [ie x2 = px + q] will become y2 + py + 14 p2 = py + 21 p2 + q; which is reduced [...] to y2 = 14 p2 + q. This is a pure quadratic equation, which immediately gives √ y = ± ( 14 p2 + q). Now, since x = y + 21 p, we have √ x = 12 p ± ( 14 p2 + q), as we found it before. What is conspicuous is the total absence of the factorization of quadratics by inspection. Indeed factorization of quadratics, is only discussed later in §689 since equations of the second degree admit of two solutions; and this property ought to be examined in every point of view, because the nature of equations of a higher degree will be very much illustrated by such an examination. This detailed discussion includes the cases when the root is repeated, and when two imaginary roots result in a quadratic with real coefficients. Then follows the solution of cubic equations, using Cardan’s technique, and solutions of 4th degree polynomials3 . This technique relies on a similar co-ordinate transformation to that given in §645 (and quoted above) to remove one term. 3 Interestingly,

in §780, Euler discusses the lack of formulae for the solution of higher degree polynomial equations. He gives no sense that such a solution may be impossible, commenting only that “This is the greatest length to which we have yet arrived in the resolution of algebraic equations”. A modern discussion of these issues can be found in, for example Stewart (1989)

7

We reiterate these findings. Modern authors concentrate on the manipulations of terms, to render them in a particular syntactic form. This usually begins with the removal of brackets and collecting terms, to write quadratics in an unfactored canonical form, such as that in (1). It progresses to re-writing terms in completed square or factored forms. Once these techniques have been mastered, they are applied to the solution of equations. Older authors, such as Euler (1822) concentrate on the solving of equations from the outset. Techniques for manipulating the terms in these equations are given. It is rare to find the manipulation of terms, not contained in equations. It was possible to find authors who took an intermediate approach. For example, Chrystal (1893), Chapter 17, begins with quadratic equations of the form (1). Since a 6= 0 (otherwise we would not have a quadratic), we may render (1) in the form x2 + px = q, as does Euler (1822). Without loss of generality, quadratic equations may be considered monic, whereas a quadratic term on its own may not. Then, he gives two methods, the second of which is via a change of variables to remove the linear term, as before. The first, however, involves completing the square and then taking the difference of two squares to derive the factored form of the equation. This technique is very similar to that shown in Figure 2. (However, Hall (1929) is of course using = to denote equality of terms in his train of thought, not an algebraic equation which he is solving.) Hence, in Chrystal (1893), the factored form is now a key step in solving the equation, unlike in Euler (1822). How the overall pedagogic strategy is laid out has a profound effect on the order and overall structure of the individual problem sequences. The two cannot be separated meaningfully. The fine-grained detail of one particular sequence of questions is the next consideration. Since the modern approach to the problem (1) is an attempt to solve by factorizing, the student must first learn how to factor. This is either by inspection, or by a number of systematic methods involving the integer factors of the constants a and c, and comparing arithmetic combinations of these with b. If other strategic approaches, such as in Euler (1822), constitute the main method for solving (1), then the place of the factored form is elsewhere. Hence practice in this technique is unnecessary. The problem sequences reflect this.

5

Results: purposes of questions

A crucial distinction, when considering a mathematical question, is whether or not one cares about the answer. With many questions, no one cares about the actual answer. The purpose of the question, is either to (i) practise some technique, or (ii) help build or reinforce some concept by prompting reflective activity. In other cases the purpose of the question is to obtain the answer. The question itself is a prototype of a practical problem which may be encountered, and hence this result may be useful. This often occurs with word problems, and we shall see examples of this later.

5.1

Practice of techniques

In looking at questions which practice techniques for mechanical fluency, we focus on the detail of two small manipulative tasks, to examine how these have been treated by respective authors. These topics are 1. Factorization by inspection. 2. Completing the square. 8

Ex.

No

type

eg.

a

36

a common factor

23. 2a3 − 4a2 − 2a

b

32

groups with compound factor

12. x2 − ax + 5x − 5a

c

42

quadratic, roots same sign, including perfect squares

See Figure 1

d

48

quadratic, roots opposite sign

14. c2 − 4x − 12

e

39

quadratic, mixture

12. k2 − 14k + 48

f

45

difference of two squares, symbolic and numerical

43. (75)2 − (25)2

g

36

sum or difference of two cubes

13. 64 − p3 q3

h

47

miscellaneous problems

21. k4 − 25l 2

Table 1: Problem sequences in Hall (1929), Chapter XIV Factorization by inspection is seen as an important topic by many authors. For example, this is treated in Hall (1929) in two parts. Initially this is by eight sequences of questions in Chapter XIV. In a moment the differences between sequences will be examined in detail. In these sequences the quadratics are all monic. Questions in which the quadratic terms are not monic are treated separately in later chapters, which are considered to be more challenging. The different question sequences of Hall (1929) Chapter XIV are summerized in Table 1. Note the large number, some 325, of repetitive exercises on this topic alone. All the sequences simply instruct the student to “Resolve into factors”. This large quantity of repetitive practice is typical of many algebra books, including modern ones. The sequences have a number of features in common, beyond the overt type shown in Table 1. For example, all coefficients are integers and all terms are written in descending order of the powers. Looking at one problem set, such as c. which is reproduced in Figure 1, we notice a variety of variables including x. Sometimes the term is a quadratic in a power of a variable, eg problems 37–42. Also quadratics in more than one variable are included, such as 34–36. and 42. Such variety is used consistently across all sequences of exercises in Hall (1929) Chapter XIV. Turning attention to the fine grained level, we consider the internal structure of the problem set shown in Figure 1. Clear structure is evident within the problems when the factored forms are examined. Beginning with Exercises 1–6, we have what are perhaps the simplest quadratics in the variable x which factor over the integers, with distinct factors. Note, 4–6 are problems 1–3 with roots of opposite signs. Exercises 7-12 include increasing integers, a new variable, and the integer factorizations do not follow the pattern of 1–6. Note, the signs of the roots in 1–12 follow a pattern, which is broken in subsequent problems. From exercise 22 onwards, the problems may involve more than one variable, and powers of one variable. While it is unlikely that the earlier problems test whether a student can accurately perform integer factorization, later problems such as 38 or 41 may do. Questions involving perfect squares, ie question numbers 18,25,32,34 and 39, do occur without a discernable pattern.

9

1. 4. 7. 10. 13. 16. 19. 22. 25. 28. 31. 34. 37. 40.

(x + 1)(x + 2) (x − 1)(x − 2) (y + 1)(y + 4) (y − 4)(y − 5) (z + 3)(z + 5) (z − 1)(z − 15) (a − 1)(a − 8) (a + 2b)(a + 7b) (b − 3)(b − 3) (b − c)(b − 9c) (x + 7y)(x + 9y) (ab − 2)(ab − 2) (n2 + 5)(n2 + 13) (p − q)(p − 17q)

2. 5. 8. 11. 14. 17. 20. 23. 26. 29. 32. 35. 38. 41.

(x + 2)(x + 3) (x − 2)(x − 3) (y + 2)(y + 4) (y − 1)(y − 7) (z − 2)(z − 5) (z + 6)(z + 7) (a + 3)(a + 7) (a − 2)(a − 6) (b − 1)(b − 13) (b + c)(b + 8c) (x + 5y)(x + 5y) (ab + 2)(ab + 8) (n2 + 8)(n2 + 17) (p2 + 3)(p2 + 23)

3. 6. 9. 12. 15. 18. 21. 24. 27. 30. 33. 36. 39. 42.

(x + 1)(x + 3) (x − 1)(x − 3) (y + 3)(y + 4) (y − 2)(y − 5) (z + 3)(z + 6) (z + 4)(z + 4) (a + 4)(a + 6) (a + 3b)(a + 8b) (b + 4)(b + 7) (b + 1)(b + 12) (x − 2y)(x − 12y) (ab + 5)(ab + 7) (n3 − 5)(n3 − 5) (pq − 4)(pg − 11)

Figure 1: Hall (1929), Examples XIV. c., pg 143, and solutions

Figure 2: Hall (1929), Example 2, pg 174

10

Figure 3: Tuckey (1904), Quadratic Graphs, pg 62 The completed square is first approached by Hall (1929) in §198, as a general method to write ax2 + bx + c as the difference of two squares and hence as a general method for factorizing quadratics. For example, the worked example reproduced in Figure 2 demonstrates this. This theme is taken up much later in §275, in which the general problem of solving ax2 + bx + c = 0 is addressed, first by factorization, then by completing the square (§278–§282). This is treatment is accompanied by 56 exercises in Examples XXV. b. Interestingly, Hall (1929) treats solution by the formula only on one page (pg 255). This is derived by completing the square in the general case (§283), and giving two short examples (§284). There are no questions involving the solution of quadratic equations by the formula alone, which is exceptional. Hence the inference one draws from Hall (1929), is that completing the square is the general method of factorizing, and hence solving quadratic equations. It is unclear whether the student is expected to remember the general formula, although which method to adopt is addressed in §287, and in Examples XXV. c “the method of solution in each case is left to the pupil’s discretion”. The work on the formula is immediately followed by an extensive graphical interpretation of quadratic graphs in §285–§290.

5.2

Building concepts

Another purpose of mathematical questions is to build concepts. Although the exact purpose may not be clear, and indeed the purpose may be mixed, some sequences are clearly more conceptual than mechanical. A good example of this is the sequence of questions taken from Tuckey (1904) shown in Figure 3. These appear to be a sequence of questions, designed to lead a student through the variety of possible shapes which quadratic graphs can assume. This careful sequence of questions is brought to an end in the synoptic question (9). This brings explicit attention to the features which may have been completed un-noticed in previous parts. We shall comment further on the use of a synoptic question below. Note however, that including such a question may make little sense for a student who has struggled with questions (1)–(8), and has little work of merit from which to form a coherent synopsis. Notice here that precisely which numbers are used in a question is essentially irrelevant. For example, the 2 in question (3), could in reality be substituted for any small natural number, greater than one. Similar statements hold true for most of the other numbers in this sequence of problems. It is identifying such possible variation within the overall structure which has practical application. In questions (7) which has completed square (x + 2)2 + 2, and question (8) which has an integer factorization the structure only becomes apparent when the solutions are considered. The structure in this case would 11

be altered by naively changing a coefficient. Within these criteria, the numbers themselves simply do not matter.

5.3

Word problems

The majority of books examined contain word problems. Part of the process of answering such a question is setting up the equations themselves. Modelling, in its broadest sense. Some of these problems are practical, others mathematical. What they have in common, is that the answer appears to be important. They do not appear to be conceptual, nor for practice, rather they might be termed utilitarian. The following two are taken from Hall (1929), however similar (if not identical) examples may be found in many modern books. Examples XXVII. b. 10. If 6 fewer bottles of wine can be bought for £5 when the price is raised ten shillings per dozen, what is the original price? Other common word problems involve two numbers with their properties given in terms of their sum/different and product. For example Examples XXVII. b.14. The product of two numbers added to their sum is 23, and 5 times their sum taken from their squares leaves 8: find the numbers. Interestingly, Euler (1822) has comparatively few questions, and all of these are word problems, with the solution adjacent. There is no repetitive practice. §651 is a typical question. Question 6. A person buys a certain number of pieces of cloth: he pays for the first 2 crowns, for the second 4 crowns, for the third 6 crowns, and in the same manner always 2 crowns more for each following piece. Now, all the pieces together cost him 110: how many pieces had he?

6

Analysis: question instance vs question space

Just as there are a variety of purposes for asking a question, the form of the question can be of many types. The style of an isolated question can be described using a question taxonomy, such as that of Pointon and Sangwin (2003) or Smith et al. (1996). The former classifies question styles under headings such as, factual recall, use of a routine algorithm (eg, a calculation), classification of some mathematical object, proof, construct an example, criticize a fallacy. The latter three, it is claimed, typically require the use of higher level skills. Such a question taxonomy only describes the external form of an individual question, which is regarded in isolation from a coherent scheme of work. This does not integrate well with the theoretical framework which suggests that learning only takes place as part of a dialogue. What follows from this latter is, that if mathematical questions are to aid learning, they too must form part of this dialogue. Hence, isolated questions are of less intrinsic value than carefully constructed schemes. In turn, these schemes must form part of a coherent pedagogic strategy. It is contended that the sample and order of questions encountered by a student constitutes a large part of their active learning experience, and will certainly affect their conception of a particular topic. 12

Hence, there is a need to describe the intended purpose of the question within the dialogue. Again, this could be done by developing an additional, and orthogonal, taxonomy. The work of Michener (1978), for example, identified four classes of mathematical examples (startup examples, reference examples, model examples, and counter examples). A counterpart to this scheme would consist of classes of questions. These might be Startup questions: a straight forward and reassuring entry. Reference questions: which somehow typify a question in this topic, and allow practice of some technique. Model questions: bring out the generality in the mathematical space, perhaps overtly. These could be synoptic. There appears to be no direct analogy to counter examples, but questions which cause cognitive conflict, perhaps by taking advantage of common misconceptions and exposing them, could act as a counterpart. We take a slightly different approach here, while still drawing parallels with the nature and purpose of mathematical examples, which have received much attention in the literature. One important concept is that of an example space, which in some papers, such as Watson and Mason (2002), is taken to be the cognitive domain possed by the student, rather than some intrinsic mathematical space. As a result of the textual analysis, we seek to develop a dual notion: that of mathematical question space. Since mathematics is an ideal tool for classifying patterns, and making distinctions, it is tempting to consider a “question space” much as one would another mathematical space. As a concrete example of this, we consider indexing the individual instances of (1), by using coordinates (a, b, c) ∈ R3 . Immediately this allows the examination of some mathematical aspects. Clearly, there are some subspaces, with perhaps the most important of these being the subspaces of mathematically possible questions. Although we have required that a 6= 0, so that (1) is a genuine quadratic, any (a, b, c) with a 6= 0, gives a mathematically possible question, provided one is prepared for complex solutions. The subspace satisfying b2 ≥ 4ac characterizes the question subspace with real solutions. One could also consider the degenerate subspaces, with for example c = b = 0, or a subspace of questions with trivial solutions, ie c = 0. There are other subspaces which are very particular to the context of (1), with characteristics such as those where b = 0, a 6= 0 and those with only one real solution, which in this case is easily characterized by the relation b2 = 4ac. The associated notion of “subspace” appears to be useful in discourse associated with question spaces. In many situations, such mathematical aspects of a question space may be far from obvious4 . Even discerning which instances of a question are possible might be a challenging problem. Indeed, setting problems with particular mathematical characteristics is a difficult task, especially at university level. In order to undertake a dialogue about such spaces the notions of the dimensions of possible variation and ranges of permissable change in any question space appear to be very useful. These could be used in a way that is analogous to the work of Watson and Mason (2002) in their consideration of spaces of mathematical examples. Each dimension of possible variation corresponds to an aspect of the question which can be varied to generate a collection different question instances. The range of permissible change is more problematic. “Permissible” may of course be taken to indicate the strict mathematical criteria of well-posedness, or may be used in a pedagogic sense. This is a distinction we examine next. A mathematical approach is certainly important. This may set out to include the largest possible space of well-posed questions, or to identify other mathematical subspaces. However, it seems to miss the point in an educational context, which is to consider a question instance as representative 4 For example, when does Question 1 have rational solutions? For polynomials of higher degree such questions constitute an important mathematical topic in their own right. See, for example, Stewart (1989).

13

Figure 4: Tuckey (1904), Expanding Brackets, pg 50 of the question as part of the dialogue. We reformulate the concept of question space along more pedagogic lines, and pay less attention to an indexing of question instances. Accordingly, a question space is considered to be the collection of instances which are educationally equivalent. That is to say, two instances in a space differ in ways which do not alter the purpose or effect of a question within that particular dialogue. Furthermore, we identify the mathematical question with this pedagogic question space. In essence a student does not answer a question, but the instance of a question. While the student is likely to be aware only of the task in hand: the question instance, to the teacher this instance actually represents the question space and hence the underlying generality. Clearly, the question space is more complex than simply varying a coefficient in a term. For example, in question 7 of the problem set shown in Figure 3, the question is an instance of a quadratic with no real roots, for which the completed square form is tractable. An instance of such a question would probably be given as an expansion of (x − a)2 + b, where a is a small integer, and b > 0 is a small integer. Hence, a particular dimension of variation certainly does not correspond to the direct variation of a coefficient in a question instance. Clearly here it is easy to identify how the dimensions of variation affect the question instances, but it is unlikely that such an algebraic clarity will be evident in many situations. Equally, there is nothing to suppose that a dimension of variation will be algebraic at all. Variation could include which variable is used, the dimensions and orientation of geometric shapes, or the adjectives used in a word problem. It is also possible in some circumstances that a question space will only contain one instance. For example, in Figure 3, question (1), there may be no reasonable alternatives, and the question space consists only of the instance “Draw the graph of x2 ”. In setting questions for students to undertake, it is possible to indicate that the particular numbers are of little importance, and that the questions are “just for practice”, while at the same time preserving structure. One example to this approach may be found in the exercises of Tuckey (1904), part of which are shown in Figure 4. The student is instructed to multiply every term on the left of the large } with every term to the right of {. This gives question combinations 1a, 1b etc. Therefore, each product is an instance from the question space. Here, a student is being asked to complete many question instances from a particular question (space). Despite the apparently mindlessness of this task, there is structure. There are actually six sequences of such exercises, which follow the following pattern. 1 2 3 4 5 5

monic quadratics in x, positive integers (left of Figure 4) monic quadratics in x, negative integers (right Figure 4) monic quadratics in x, positive and negative integers non-monic quadratics in x, positive and negative integers non-monic quadratics in a, positive and negative integers non-monic quadratics in y, positive and negative integers 14

Practice of some technique can therefore be seen to be the repeated completion of question instances from a particular question space. However, practice with the level of repetition shown in Figure 4 is unusual, and exercise sequences more often show some progression through a sequence of slightly different cases. Each of these will be consciously different, and so will be instances from different question spaces. This progression is not to be confused with the notion of subordination, which Hewitt (1996) claims to be so important to the process of learning. Here subordination is taken in the following sense. Skill A is subordinated to a task B, only if [...] (a) I require A in order to do B. [...] (b) I can see the consequences of my actions of A on B, at the same time as making those actions. (c) I do not need to be knowledgeable about, or be able to do, A in order to understand the task B. Hewitt (1996) [emphasis as in original] For example, as it is currently taught, A might be “factoring the quadratic”, and B might be “finding the roots”. If we accept the position of Hewitt (1996), then for a sequence of questions to provide “effective practice”, in a particular task, then the task itself should be subordinated into something larger. However, most question sequences examined in the context of quadratics do not provide the opportunity for this subordination, rather they offer only repetitive, albeit progressively more difficult, tasks. In identifying the question space, there is a delicate balance between the granularity of thinking in terms of repeated instantiation of a single question, and number of distinctions between different cases. The teaching context will dictate the maturity of learners in appreciating what can be generalized and what cannot. For example, students unfamiliar with algebra will need to be exposed to a variety of variables, such as is done in exercises shown in Figure 1. More mature learners can be expected to appreciate this. The issue of distinctions is something we now turn to.

6.1

From pedagogic strategy to individual question: distinctions

In producing a sequence of questions, by which we mean a sequence of question spaces, the staff member has to negotiate his/her way from the pedagogic strategy to the level of the individual question. In order to do this a number of distinctions need to be made. For example, Euler (1822) considered separately the subspaces of (1) where b = 0 (“pure quadratics”) and b 6= 0 (“complete quadratics”), whereas Hall (1929) first separates the cases a = 1, and a 6= 1. Once made, a decision needs to be made as to which are overt, and which remain covert. Overt distinctions are ones to which the students’ attention is consciously drawn. Possibly through the text, possibly by grouping questions in common together. Covert distinctions are made by the staff member in choosing the selection of questions, but the student may not perceive these differences. Other distinctions may be identified as possible but are ignored, inasmuch as this dimension of variation is not used. To be concrete, let us return to the question sequences of Hall (1929) involving factoring by inspection. The first and most major distinction is made between monic quadratic (ie a = 1), and more general quadratic. This is a distinction of sufficient significance to warrant separate book chapters. Within the monic quadratics, further clear distinctions are apparent concerning the roots α and β . The first distinction is the algebraic sign of the roots, and there are three possibilities: that they are both positive, 15

both negative or of opposite sign. The second distinction is whether |α| = |β |. The choices made between these possibilities result in the various sequences of problems c–f shown in Table 1, together with text overtly explaining which distinctions have been made. Exercises XIV c. contain only factors of the same algebraic sign. No overt distinction is made here between cases when the roots are both positive or both negative. Repeated roots, ie the second distinction with |α| = |β |, are included without comment, again a covert distinction. See for example Figure 1, and notice that problems 18, 25, 32, 34, and 39 are included without comment. Exercises XIV d. contain roots of opposite sign, an overt distinction, but do not contain the difference of two squares. An overt distinction is made here with the case |α| = |β |, and these problems are treated separately in Exercises XIV f. Note also in Hall (1929), that in each exercise set the variable is changed, which is the manifestation of a covert distinction. The order of terms in each question is identical. That is to say all problems are of the form ax2 + bx + c, rather than, perhaps bx + ax2 + c. This is a distinction which is ignored. Actually, the majority of texts ignore this distinction, despite advice that “prejudice in favour of the x2 term coming first should not be allowed to grow up; forms like 2 + 5x − 12x2 should be freely used, and without alteration of the order, or else there will be a fruitful source of mistakes.” Tuckey (1934). The problem set given in Figure 3 shows a number of cases which are clearly distinguished. This problem set is included since, unusually, the end of the exercise contains a synoptic question which is clearly designed to draw students’ attention to these distinctions. While a student who automatically takes a deep approach to their learning might ponder the sequence of questions, as a whole, it is more likely that the student will focus only on the individual problems to hand, and not consider the overall structure. Unless prompted to do so. The role of a synoptic question in drawing attention to distinctions, in an attempt to alert a student to the structure which has been carefully incorporated, appears to be key. Synoptic question(s), as part of a scheme of work allow the task to be subordinated, in the sense of Hewitt (1996) above, into something more significant. However, very few of the texts examined contained an overt synoptic question, either sporadically or systematically. Of course, a teacher may well be expected to ask this in a classroom setting. With students who are engaged in self-study or using CAA (as are many in Higher Education), this cannot occur. Furthermore, in the classroom, there may be a significant delay between a student completing the question instances and answering the teacher-provided synoptic question. Not taking the opportunity to include synoptic question(s) routinely in text book schemes, or with CAA, seems to be an opportunity missed. Note that the concept of “question as question space” is applicable more widely than to textbook questions or school algebra. This expanded concept may well apply to any mathematical question, used in learning and teaching. In support of this, consider Pledger et al. (2001) and Newton (1998), which comprise a comprehensive text together with an alternative set of problems of the same structure as those in the text book. In the terminology above, Newton (1998) contains different question instances and clearly here the concept, and implementation, of question as question space is implicit. The ability to produce such alternative question sequences across the curriculum supports the assertion that the concept of question space is applicable beyond the topic of quadratics, or high-school algebra in general.

16

7

Conclusion

This paper reports an analysis of the treatment by school text books of the topic of quadratics. This analysis suggested that the concept of mathematical question should be refined to distinguish between a question instance, and a question space. This concept may well be usefully applicable to any mathematical question, not only those associated with the algebraic activities connected with quadratics. This distinction has clear practical application in the random generation of questions in mathematical CAA. In isolating and examining this topic, it became clear that it is impossible to consider individual questions, or schemes of questions, in isolation from the overall pedagogic strategy. In general, it is to be expected that the role of the pedagogic strategy will certainly be key. Next, we must consider what important distinctions are to be drawn between the mathematical objects, their properties and associated processes. From these considerations, we tentatively propose the following framework for assessment design. The form of this framework is a summary comprising the following prompt questions. 1. What is the mathematical space. What are the objects, properties and processes? 2. What is the pedagogic strategy? That is the overall flow of teaching and learning of this topic? What are the alternatives? 3. What emphasis will be placed on the different properties and processes? 4. What are the possible distinctions? 5. What are the most important distinctions? 6. Which distinctions should be overt, which covert, and which ignored? One clear implication for practice, is that despite care taken by authors, structure in problem sequences is extremely fragile. It is easy to destroy any structure in practical application by using only odd numbered questions, for example, or working down the first column of a page of exercises.

Acknowledgements The author would like to thank Martin Brown and Anne Watson for valuable discussions and encouragement.

References Barnard, T.: 1999, A Pocket Map of Algebraic Manipulation, The Mathematical Association. Chrystal, G.: 1893, Algebra, an elementary text-book for the higher classes of secondary schools and for colleges, Vol. 1, third edn, Adam and Charles Black, London and Edinburgh. De Morgan, A.: 1847, Arithmetical books from the invention of printing to the present time being brief notes of a large number of works drawn up from actual inspection, Taylor and Walton, 28 Upper Gower Street, London. 17

Euler, L.: 1822, Elements of algebra, Longman, Hurst, Rees, Orme and Co. Translated from the French, with the notes of M. Bernoullil and the Additions of M. de La Grange by Hewlett, J. Hall, H. S.: 1929, A School Algebra, 11th printing edn, MacMillian, London. First published 1912. Hansard: 2003, United Kingdom House of Commons. 26 June 2003, Columns 1259–1269. Hassm´en, P. and Hunt, D. P.: 1994, Human self-assessment in multiple choice, Journal of Educational Measurement 31(2), 149–160. Hewitt, D.: 1996, Mathematical fluency: the nature of practice and the role of subordination, For the learning of mathematics 16(2), 28–35. HMSO: 1999, Mathematics: The national curriculum for England. Jennings, S. and Dunne, R.: 2003, I see maths, Vol. Book 2, Letts Education. Klai, S., Kolokolnikov, T. and Van den Bergh, N.: 2000, Using Maple and the web to grade mathematics tests, Proceedings of the International Workshop on Advanced Learning Technologies, Palmerston North, New Zealand, 4–6 December. Knuth, D. E.: 1969, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading, Massachusetts. Michener, E. R.: 1978, Understanding understanding mathematics, Cognitive Science 2, 361–381. Naismith, L. and Sangwin, C.: 2004, Computer algebra based assessment of mathematics online, Proceedings of the 8th CAA Conference 2004, 6th and 7th July, The University of Loughborough, Birmingham, UK. Newton, D.: 1998, London General Certificate of Secondary Education Mathematics: Higher Practice Book (Edexcel GCSE Mathematics), Heinemann Educational Secondary Division. O’Shea, T.: 1982, A self inproving quadratic tutor, in D. Sleeman and J. S. Brown (eds), Intelligent Tutoring Systems, Kluwer Academic Publishers, chapter 13, pp. 309–336. Pledger, K., Sylvester, J. and C., M.: 2001, Edexcel GCSE Mathematics Higher Course (Edexcel GCSE Mathematics), Heinemann Educational Secondary Division. Pointon, A. and Sangwin, C. J.: 2003, An analysis of undergraduate core material in the light of hand held computer algebra systems, International Journal for Mathematical Education in Science and Technology 34(5), 671–686. Richards, J.: 1991, Mathematical discussions, in von Glasersfeld (ed.), Radical constructivism in mathematics education, Kluwer Academic Publishers, chapter 2, pp. 13–51. Sangwin, C. J.: 2003, New opportunities for encouraging higher level mathematical learning by creative use of emerging computer aided assessment, International Journal for Mathematical Education in Science and Technology 34(6), 813–829. Scott, B.: 2001, Gordon Pask’s conversation theory: A domain independent constructivist model of human knowing, Foundations of Science 6(4), 343–360.

18

Skemp, R. R.: 1971, The psychology of learning mathematics, Penguin. Smith, G., Wood, L., Coupland, M. and Stephenson, B.: 1996, Constructing mathematical examinations to assess a range of knowledge and skills, International Journal of Mathematics Education in Science and Technology 27(1), 65–77. Stewart, I.: 1989, Galois Theory, second edn, Chapman and Hall. Strickland, N.: 2002, Alice interactive mathematics, MSOR Connections 2(1), 27–30. http://ltsn.mathstore.ac.uk/newsletter/feb2002/pdf/aim.pdf (viewed December 2002). Tuckey, C. O.: 1904, Examples in Algebra, Bell & Sons, London. Tuckey, C. O.: 1934, The teaching of algebra in schools, A Report for the Mathematical Association, G. Bell & Sons. Watson, A. and Mason, J.: 2002, Extending example space as a learning/teaching strategy in mathematics, in A. D. Cockburn and E. Nardi (eds), Proceedings of the Annual Conference of the International Group for the Psychology of Mathematics Education (PME26, Norwich, United Kingdom), Vol. 4, pp. 378–385. Wester, M.: 1999, Computer Algebra Systems: a Practical Guide, Wiley. Wiliam, D. and Black, P. J.: 1996, Meanings and consequences: a basis for distinguishing formative and summative functions of assessment?, British Educational Research Journal 22(5), 537–548.

A

Background to key texts cited

Hall (1929) A typical book is the now dated, Hall (1929), which was a very popular early twentieth century school algebra text. The “aim has been to provide all that is essential in a school course of Elementary Algebra”. In 550 pages, the book opens with simple use of letters in place of numbers, and continues to cover topics at advanced school level. These include Induction (Chapt XL), The Binomial Theorem (Chapt XLI), Partial Fractions (Chapt XLII) and The Use of Exponential and Logarithmic Series (Chapt XLIII). The book contains both text and theory, worked examples and sequences of exercises, and is a comprehensive text book, rather than an exercise resource for the teacher. However, the writing style and typography are condensed with emphasis very much on “Examples”, that is to say exercises, which appear to occupy over half the text. These are usually set out in multiple columns, see Figure 1. These exercises are predominantly mechanical tasks, in the form of structured sequences. However, there are also many graphical, numerical and word problems, such as the whole of Chapter XXVII, Problems Leading to Quadratics. There were very few general proofs, counter examples or questions of an open ended exploratory nature. Quadratics are covered by Hall (1929) extensively, as one would expect in a comprehensive text.

19

Euler (1822) This work was written as a comprehensive algebra text. Indeed, Euler states that Algebra has been defined, The science which teaches how to determine unknown quantities by means of those that are known. It differs significantly from modern treatments, and may be difficult to obtain, so that we sketch the contents here, outlining significant differences. Euler’s work is in two Parts, half of Part II comprises the additions of LaGrange, with Euler’s actual text comprising approximately 450 A5 pages. This is subdivided into 5 Sections, the modern term for which would be chapters. Section I consists of methods of calculating simple quantities. That is addition of numbers, with a discussion of the signs + and −, multiplication and the nature of integers, fractions, and decimals. Immediately there are some significant differences from the modern approach to algebra. For example, the nature of infinity is discussed in §83 and imaginary numbers are treated in Chapter XIII (beginning with §139), and then used throughout. Square and cube roots are discussed in detail, as are higher powers and there is work on logarithms. Section II contains methods of calculating compound quantities, which contain terms, known and unknown added together. This section consists of various algebraic techniques, such as addition, subtraction, multiplication and division of such compound quantities ending in polynomial long division, ending in 1/(1 − a) which results in infinite series. Section III contains work on ratio and proportion. Arithmetic and geometric series are covered in detail, together with polygonal numbers. Infinite decimals are covered and applications to compound interest are also given. Section IV contains material on resolution of equations, starting with linear equations of a single variable and continuing through quadratic, cubic and quartic equations. These latter are solved by the exact methods of Cardan and Bombelli. Part II, Section V, contains methods of solving various equations, including those with more than one unknown. The subsequent Section contains La Grange’s Additions, which essentially contains material on Diophantine Equations.

Tuckey (1904) The small, unassuming volume of Tuckey (1904), consists of 178 pages. There is no text, and the work consists only of sequences of problems. “These examples are intended to provide a complete course of elementary algebra for classes in which the bookwork is supplied by the teacher”. This volume is almost certainly rare, however this is a great pity since from the large number of volumes examined by the author this contained some of the most clearly and consistently structured exercise sequences.

20