Classical versus vector geometry in problem solving. An empirical ...

int. j. math. educ. sci. technol., 2001, vol. 32, no. 1, 105– 125

Classical versus vector geometry in problem solving. An empirical research among Greek secondary pupils ATHANASIOS GAGATSIS Department of Education, University of Cyprus, Cyprus

and HELEN DEMETRIADOU Department of Education, University of Crete, Greece (Received 16 September 1999) An analysis of written solutions to geometry exercises in the last year of Greek Lyceum (15–18 years old) showed relatively low performance on vector methods, which is justiŽ ed on the one hand by pupils’ false preconceptions with regards to the concept of vector, and on the other hand by the strong in uence of classical geometry teaching in the previous years.

1. Introduction 1.1. Some historical elements The science of Geometry can be considered to have been born in Greece and Greece has a long tradition in teaching Geometry. Indeed through the ages, some people have often considered Geometry an entirely Greek aV air. This cannot be seen as an advantage to the Greek educational system since the overemphasis on geometry deters change and make the system in exible to new ideas. The teaching of Geometry in modern Greece started in the1830s when primary and secondary education was institutionalized as part of the newly established state. As part of the educational curriculum in Greece in the period 1830–1884, the teaching of Geometry followed the Bavarian curriculum combined with an attachment to Classical Greek ideas. In 1836, 53.2% of teaching time was devoted to Literature with only 19.2% to Mathematics. The Ž rst syllabus with a detailed description of geometrical items came as follows in 1857: Plane Geometry: deŽ nitions, angles, lines, triangles, congruence, parallelograms, circle – arc – chord, angles and circles, lines and circles, constructions, loci, Thales’ Theorem, area, similar shapes, regular polygons, measurements in the circle. Space Geometry: general features of solid objects, area, volume. In 1935, one century later, the percentage of teaching time devoted to Mathematics was still at 19.2%. Today’ s syllabus in Geometry is very similar to that formulated in 1857. The stability of the system and its resistance to change is remarkable [1]. Tradition in uences decisions concerning Mathematics education in Greece both externally and internally. Externally, educational policy is characterized by International Journal of Mathematical Education in Science and Technology ISSN 0020–739X print/ISSN 1464–5211 online # 2001 Taylor & Francis Ltd http://www.tandf.co.uk/journals

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the reinforcement and promotion of Classical Studies (Classical Greek Literature, Grammar, History and Religion) as is indicated by the excessive proportion of time allocated to these studies. Internally, Mathematics owes its ‘survival’ as a subject of comparatively high esteem basically to its  ourish during antiquity. Thus it feels compelled to serve the image of the glorious past which has in turn haunted textbooks and the teaching of Geometry in Greece for decades. Despite this tremendous in uence of tradition, during this Ž rst period of teaching Geometry in Greece four diV erent translations and several editions of Legendre’ s Elements of Geometry have been used. Thus, on one hand, Euclid is paid lip service as the indisputable leading thinker in Geometry and, on the other hand, Legendre’ s Euclidean adaptation is what is increasingly adopted during this period. Indeed, Legendre’ s adaptation has come to be uniformly used in secondary Geometry in Greece with all modern Greek Mathematics textbooks having adopted this approach [1]. 1.2. The research questions The attachment to tradition discussed above has created some peculiar situations in Greek secondary education regarding the approaches to teaching geometry. Pupils are taught classical (Euclidean) geometry for two years (age 15 to 17). Vector geometry is taught only to pupils of the Ž nal year of Lyceum (age 18). This constitutes part of the content of the entrance examinations to Greek universities. The teaching of the concept of vector in mathematics courses during the previous years is either deŽ cient or non- existent. Thus, in the Ž nal year of Lyceum pupils are forced to use this concept, about which they have either deŽ cient or false conceptions (after its use in physics), in the Ž eld of geometry; a Ž eld where they had worked for three years using classical thought and methodology. Our study concerns the behaviour of pupils of the Ž nal year of Greek Lyceum in solving geometry problems, while using Euclidean or vector methods. We tried to answer the following questions: What are the consequences of this deŽ cient teaching of vectors for pupils in the last year of Lyceum in solving geometry problems? Does their long experience in classical geometry in uence them in solving geometry problems, even if this teaching is neither recent nor particularly emphasized, since it is not examined for the university entrance? Consequently, does the classical approach to geometry con ict with vector methods? 2.

Methodology 2.1. Sample Data were collected from a sample of 361 pupils of the last year of Greek Lyceum (age 18), during 1996. Individual schools were used as the sampling unit. Fifty-one schools were chosen re ecting provision within the major cities of mainland Greece and in some rural areas. 2.2. Instruments For data collection we used: (i) A test of the ability to solve geometric problems concerning mostly metric relations having been taught in the second year of Lyceum.

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(ii) A questionnaire to elicit pupils’ opinions about the appropriateness of methods used to resolve the geometric problems set, and the possible factors in uencing their views. A set of thirteen exercises used in the research project is presented below. From this set only subsets of three or four exercises were given to each school. It needs to be mentioned that these exercises can be solved either by Euclidean or by vector methods. Exercise A1: If the two medians mb and mc of a triangle ABC are perpendicular, then: m2b ‡ m2c ˆ m2a ^ ˆ 1208†, prove that: Exercise A2: Given an obtuse triangle ABC …A 2 2 BC ˆ AC ‡ AB ‡ AC ¢ AB 2

Exercise A3: Given that A and B are the intersections of the two circles (K, R) and (L, r) prove that KL ? AB

Exercise A4: Two circles (K, R) and (L, r) touch each other externally at the point M. Given that AB is the common exterior tangent of the two circles, prove ^ ˆ 908 that AMB Exercise A5: Given that AD is the height of the isosceles triangle ABC with ^ ˆ 908, prove that BC2 ˆ 2AC ¢ CD A

Exercise A6: Given that M is the middle point of the side BC of a triangle ABC, prove that AC2 ‡ AB2 ˆ 2AM2 ‡ 2MB2 Exercise A7: In an isosceles triangle ABC…AB ˆ AC†, let D be a point on BC. Prove that AB2 ¡ AD2 ˆ BD ¢ DC: Exercise A8: p   Given a2 ˆ b2 ‡ c2 ‡ bc 3

a

triangle

ABC

with

^ ˆ 1508, A

prove

that

Exercise A9: Given that AD is the height which corresponds to the hypote^ ˆ 908†, prove that AD2 ˆ BD ¢ DC nuse of a rectangular triangle ABC …A Exercise A10: Given a quadrilateral ABCD with perpendicular diagonals AC and BD, prove that the sums of the squares of the opposite sides are equal, namely: AB2 ‡ DC2 ˆ AD2 ‡ BC2 Exercise A11: Every inscribed angle corresponding to a semi-circle is right

Exercise A12: A rectangle ABCD has AB ˆ 2AD. Given that P is a point of the side DC such as DP ˆ …3=4† DC prove that BP is perpendicular to the diagonal AC Exercise A13: Given a parallelogram ABCD and the points E,Z of its diagonal such as AE ˆ ZC ˆ …1=4†AC, prove that the quadrilateral EBZD is a parallelogram. 2.3. Procedures All tests were administered by regular classroom teachers during the normal school day. The analysis of the results contains two main parts. The Ž rst one concerns success in solving geometric problems, the methods used, and the types of errors made. The second part examines pupils’ opinions with regard to the

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Solver type E V EV

Chosen method Pupils who used only Euclidean methods Pupils who used only vector methods Pupils who used both methods in diV erent problems (in some problems Euclidean and in others vector methods) or a combination of them in the same exercise. Table 1. Categories of solvers.

diV erent geometry methods (advantages, disadvantages) and whether these preferences were in agreement with the methods actually used by them.

3. Geometry methods, success, errors 3.1. Categories of solvers regarding the method used The solvers were divided into three main categories or types, which are presented in table 1. In the next step of analysis we search for characteristic elements in each category. To do this, we attempt: (a) to verify the grade of success in problem solving for each child, and (b) to classify and group children’ s errors. 3.2. ClassiŽ cation model for success in problem solving The work of Malone et al. [2] gives some general criteria for measuring problem-solving ability. Senk [3] has also used this model for secondary school geometry students in the USA. The model uses the following scoring scale: Score 0 Student writes nothing, or writes meaningless deductions. Score 1 Student approaches the problem by at least one valid deduction. Score 2 Student proceeds towards a rational solution by providing a chain of suY cient reasoning, but stops because of major errors or misinterpretations. Score 3 Student has nearly solved the problem, but makes errors in notation, vocabulary or names of theorems. Score 4 Student gives a valid solution. In our research the problem-solving ability was not the main point. Our priority interest was the ability in manipulating one or other method. For this reason a much smaller scale was used for the Ž rst two categories of solvers (i.e. E, V), presented in table 2. Scale 0(n) includes the Ž rst three criteria of Malone et al. model [2], scale 2(n) includes the two last cases of Malone et al., and scale 1(n) is a middle situation between 0(n) and 2(n). This measuring model is slightly modiŽ ed for the last category (EV). In this category the situation was more complicated, since there were for example pupils who succeeded in some exercises while using vectors, but failed in others while using Euclidean methods and inversely. For this reason, two more score types 4(EV) and 5(EV) were added in this category, as indicated in table 3.

Classical versus vector geometry in problem solving

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Score

Solution stage

0(n)

Pupil gives no answer at all, writes wrong deductions to every exercise he deals with, or is not able to reach the end of a solution, although he makes correct steps. Pupil is merely successful; he is not able to solve suY ciently all the exercises he deals with, but solves suY ciently at least one exercise. Pupil provides valid solution to every problem he deals with, or nearly solves the problem with minor errors (in notation, vocabulary or names of theorems) which do not in uence the correct result.

1(n) 2(n)

By the variable n we mean E (Euclidean) or V (vector) Table 2. Scale used for pupils who used only Euclidean or only vector method. Score 0(EV) 1(EV) 2(EV) 3(EV) 4(EV)

Solution stage Failure both in vector and Euclidean methods. Merely success in both methods. Success in both methods. Merely success in Euclidean methods and failure in vector methods. Merely success in vector methods and failure in Euclidean methods. Table 3.

Scale used for pupils who used both methods.

Based on this model, each of the three solver-types was divided into subcategories indicative of pupil success in problem solving. 3.3. ClassiŽ cation model for errors The errors found in both methods were divided into two main categories: general errors concerning the solution procedure and errors related to misconceptions as regards the vector concept. ClassiŽ cation of the general errors which are committed independently from the concept of vector was based on the empirical classiŽ cation model introduced by Movshovitz-Hadar et al. [4]. This model concerns pupils’ errors in high school mathematics (age 17) and is empirical in the sense that ‘the investigation relied solely on data on students’ answer books for a comprehensive examination; the only theoretical assumption was that most student errors in high school mathematics ‘are not accidental and are derived by a quasi-logical process that somehow makes sense to the student’ [4, pp. 3–4]. The model consists of six descriptive categories of errors, which were also identiŽ ed in our research, except for one category, referred to as ‘UnveriŽ ed Solution’ . The other Ž ve categories, with which we deal here, are presented in table 4. Examples for each category of general errors, as found in our research, are provided below: Misused data Incorrect solution: In Exercise A3, the pupil considered that the two circles (K, R) and (L, r), have equal radius r. Analysis of error: The pupil neglected the given information about the radius R and imposed on the Ž rst circle, the radius r.

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Categories of general errors

Analysis of errors

Misused data

Includes errors dealing with a discrepancy between the given data and the way the student treated them. Characteristic elements of this category: neglecting given data and adding extraneous data, stating irrelevant requirements, assigning to some data a meaning that disagrees or is inconsistent with the text, incorrectly copying to the workbook. Misinterpreted language Includes errors related to an incorrect translation of mathematical facts from one language to another. A characteristic element of this category is the designation of a concept by a symbol traditionally designating another concept and operation with the symbol in its conventional use. Logically Invalid Inference Includes erroneous reasoning; e.g. an unjustiŽ ed jump in a logical inference without providing the necessary sequence of arguments. In this category we have also included the following cases: proving that p ˆ p when it is asked to prove that p ˆ q, and concluding that p implies q by providing as argument the validity of q. Distorted theorem Includes errors in applying a theorem outside its conditions, or deŽ nition or an imprecise citation of a recognizable theorem or formula. Technical errors Includes computational errors and errors in mathematical symbols and algorithms. Table 4.

Analysis of general errors for each category.

Misinterpreted language Example 1 Incorrect solution: (Exercise A1) The pupil designated the median AE by the symbol mc . Analysis of error: The pupil considered that the symbol mc designated the median AE, while this symbol traditionally designates the median corresponding to the side AB, and tried without success to prove the given type. Example 2 Incorrect solution: To symbolize a vector with start point A and end point B, some pupils use the symbol AB. Analysis of error: A vector with start point A and end point B is traditionally symbolized by AB. When the arrow is omitted (AB), then the symbol which traditionally designates a line segment, is used to designate a vector. Logically invalid inference Incorrect solution: The pupil provides the relation ^ ‡ L^ ˆ 1808 (Exercise A4). K Analysis of error: The pupil made an unjustiŽ ed jump in his logical inference without providing the necessary arguments which lead to this relation (i.e. KA//LB, since they are both perpendicular to the tangent AB).

Classical versus vector geometry in problem solving

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Distorted theorem or deŽ nition Example 1 In Exercise A8, the law of cosines is written as a2 ˆ b2 ‡ c2 ‡ 2bc cos A instead of a2 ˆ b2 ‡ c 2 ¡ 2bc cos A Example 2 Incorrect solution: ^ ˆL ^ since the two angles correspond to (K the same arc AB" (Exercise A3)). Analysis of error: The pupil applied the theorem about the equality of central ^ and L ^ do not belong to angles, where K the same or to equal circle.

Technical errors Example : When writing : !

!

!

!

!

!

!

!

…AK ‡ KM†:…ML ‡ LB† ˆ AK : ML ‡ KM : LB and omitting the term !

!

!

!

AK LB ‡ KM : ML Besides the general errors made by the three solver-categories, we found another category of errors. We made the assumption that these errors resulted from misconceptions of procedures and concepts related directly to the concept of vector, and therefore we called them ‘vector errors’ . Misconceptions of this type were also found in previous research [5, 6] among Greek pupils of the last year of Gymnasium (aged 15 years) and of the Ž rst two years of Lyceum (aged 16 and 17 years). The classiŽ cation model for vector errors introduced here was based on the diV erent kinds of such misconceptions which were regarded as the main reason for these errors. The misconceptions recognized in this research were Ž ve: the conception of vector as a line segment; misconceptions concerning the sense of a vector; misconceptions regarding the procedure of vector addition, subtraction, and dot product; and Ž nally misconceptions concerning the use of vector coordinates. The classiŽ cation model for vector errors is presented in table 5, and some examples follow. Vector is equivalent to a line segment Example 1 Incorrect solution: In Exercise A4, the pupil replaces ! ! KM by R and LM by r and uses R and r in vector relations. Analysis of error: R and r represent magnitudes but the pupil turns them into vectors.

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Category of vector error

Analysis of errors

Vector is equivalent to a line segment

Includes errors that deal with the misconception of a vector as a concept equivalent or very close to the concept of a line. In this case it is obvious that among the three features of a vector (magnitude, sense, and orientation) magnitude seems to prevail in pupil’ s mind. Characteristic elements of this category: consideration of a line segment as a vector, when a relation with vectors is directly converted to a relation of their magnitudes, when vectors of equal magnitudes are considered as equal, when vectors are considered as equal to their magnitudes, when the convention of parallelism or perpendicularity between vectors is used for line segments. Sense errors Includes errors related to misconceptions about the concept of sense, or where sense is not considered as a feature of a vector. Characteristic elements of this category: errors in vector addition, errors in dot product, opposite vectors are equal. Errors in vector addition Includes errors related to wrong procedure of these and subtraction operations. A characteristic element of this category is the wrong replacement of a vector by the sum of two other vectors which are not its components. Errors in dot product In this case we have erroneous application of the procedure of dot product. Characteristic element of this category: errors as regards the angle between two vectors. Errors in using coordinates Includes errors in expressing a vector by the use of coordinates, and errors in dot product in the case that coordinates are used Table 5. Analysis of vector errors for each category.

Example 2 Incorrect solution: ‘In order to prove that AD2 ˆ BD ¢ DC, it is enough to prove !

!

!

that AD2 ˆ BD ¢ CD since a relation among vectors is also a relation among magnitudes’ (Exercise A4). Analysis of error: In this characteristic comment, the obvious misconception that vectors and their magnitudes are equivalent concepts.

Example 3 Incorrect Solution: ‘In the isosceles triangle ABC …AC ˆ AB† is: ! ! ! ! 2 AC : CD ˆ 2 BA : CD Analysis of error: ! ! The pupil considers that AC ˆ BA since AC ˆ AB. Namely, vectors of equal magnitudes are considered as equal, ignoring their sense and orientation. Example 4. Incorrect Solution: ! ! ! ! 0 BD ¢ DC ˆ j BD j ¢ j DC j 0 (Exercise A9).

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Classical versus vector geometry in problem solving

Analysis of error: ! ! In the dot product, the pupil omits the term cos …BD; DC† considering that the (dot) product of two vectors equals the product of their magnitudes. Sense errors Example 1 In Exercise A9, the pupil uses the relation ! ! ! ! ! AC ‡ AB ˆ CB instead of CA ‡ AB ˆ !

!

!

CB, considering that vectors ACand CA with opposite senses as equal.

Example 2 In Exercise A4, the pupil uses the relation: ! ! ! ! ! BA ¢ AB ˆ BA2 assuming that BA ˆ AB Errors in vector addition and subtraction Example 1 In Exercise A4 the pupil wrongly expresses vector ! ! ! MB as a sum of unrelated terms BA ‡ AK.

Example 2 ! ! ! ! In the relation DA2 ˆ …GD ¡ DA) vector DA is wrongly replaced by the diV erence !

!

GD ¡ DA

Errors in dot product Example 1 In the relation: ! ! OB ¢ AO ˆ R2 cos 1808 the incorrect angle of 1808 is used instead of the correct angle of 908. (Exercise 11)

Example 2 In the same exercise as above, the term cos …2º ¡ ³† is used instead of the correct term cos …º ¡ ³†, in the relation ! ! ! ! OA ¢ OC ˆ j OA j ¢ j OC j cos …2º ¡ ³† Errors in Using Coordinates (Exercise A1) ! OA ˆ ‰X1 ‡ X3 ; Y1 ‡ Y3 Š instead of ‰X3 ¡ X1 ; Y 3 ¡ Y 1 Š.

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4. Pupils’ opinions about the two methods Attempting an investigation concerning pupils’ attitudes towards one or the other geometry method, we asked them to answer a questionnaire, after they had Ž nished the problem-solving procedure. From the initial sample of 497 pupils only 361 were willing to give their responses to the questionnaire, which contained questions as follows: . . . .

Is there another method which could be used to solve these problems? Why have you decided to use this particular method and not another one? Which, according to your opinion, is the best method and why? What diY culties were you faced with, for each method? Namely, what forced you to abandon a method, if this happened, or what has prevented you from reaching a Ž nal solution?

Pupils express positive or negative opinions for one or both methods. These opinions can be classiŽ ed into six general types: positive or negative for Euclidean geometry, positive or negative for vector geometry, and positive for both methods. Each of these six types is presented in tables 6, 7, 8 and 9. There were also pupils who supported both methods. Some of their commends are the following: ‘Both methods are comprehensible’ . ‘We should use both methods, since exercises which are solved easily by Euclidean method can also be solved more easily by vectors, and inversely’ , ‘Vector method is more elegant, while Euclidean is classical, traditional; both are very good’ .

Positive opinions about Euclidean geometry (Simplicity) Quite simple (Long experience) Pupils have a long time experience concerning this method (Aid of theorems) The existence of many theorems make it easier and safer Interesting (imagination, logic, clever thoughts) (Unique in pupils’ mind) It was the only method remembered

Characteristic responses ‘. . . easy’ , . . . brief’ , ‘. . . comprehensible’ ‘. . . accessible’ ‘During the last Ž ve years we used this method in solving exercises’ . ‘It’ s more familiar’ . ‘It’ s more organized and methodical, since it is based on theorems; you know what you want and where you go’ . ‘This method is simpler because it is based on known theorems’ . ‘I think that Thales’ Theorem solves all problems of this kind; reasonably my Ž rst thought concerned this theorem’ . ‘It gives me more mathematical satisfaction, because it excites my imagination’ . ‘It’ s clever’ , ‘. . . enjoyable’ . ‘It’ s based on logical arguments’ . ‘There is no other method’ . ‘I could not Ž nd another method’ . ‘This method came Ž rst in my mind’ .

Table 6. Responses concerning positive opinions about Euclidean geometry.

Classical versus vector geometry in problem solving Negative opinions about Euclidean geometry

115

Characteristic responses

(A lot of theory) ‘Auxiliary lines are needed, as well as a lot of theorems and Large piece of knowledge types’ . and Ž gure diY culties Complicated thought ‘DiY cult thought, . . . imagination is needed ’ . ‘In some exercises you must be very observant. Very often the Ž gure misleads you’ . (Not recent) ‘I cannot remember exactly what we have been taught in the Pupils have forgotten the previous years of Lyceum’ . relative theory ‘I do not remember some theorems’ . Table 7. Responses concerning negative opinions about Euclidean geometry. Positive opinions about vector geometry

Characteristic responses

(Recent) Recent and useful for university entrance

‘Vector Geometry is a new chapter and I Ž nd it interesting’ . ‘It’ s more familiar to me’ . ‘It’ s part of the content for the last year of Lyceum, and also for the examinations for university enrolment’ . (Methodology) ‘Comprehensible for complicated exercises’ . EV ective and ‘General and eV ective’ . standardized ‘Simple relations and operations are used’ . ‘Less knowledge (types and theorems) is needed’ . ‘It does not require imagination’ . ‘Some combinations with vectors are needed’ . ‘No auxiliary lines are needed’ . ‘There is no need to localize something in the Ž gure’ . (Unique in pupils’ mind) ‘There is no other method’ . It was the only method in ‘I could not Ž nd another method’ . the pupils’ mind (More favourable) ‘It’ s better than Euclidean method. Unfortunately I have Pupils feel pleasure in discovered it too late!’ . exploring the new method ‘This method helps us to analyse the problems better than the Euclidean method’ . Table 8. Responses concerning positive opinions about vector geometry.

The above types of response represent Ž ve diV erent trends in pupils’ attitudes towards the two geometry methods (vector and Euclidean). Based on this observation, we discriminated between Ž ve types of pupils concerning attitude towards the two methods: …‡E† …¡E† …‡V† …¡V† …‡EV†

Pupil signiŽ es a preference for the Euclidean method or/and mentions the advantages of this method. Pupil mentions the disadvantages of Euclidean method. Pupil signiŽ es a preference for vector methods or/and mentions the advantages of this method. Pupil mentions the disadvantages of vector methods. Pupil signiŽ es a preference for both methods.

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Negative opinions about vector geometry (Lack of experience) Pupils are not experienced on vector methods

Complicated (Types, operations, vector sense)

Characteristic responses ‘Vectors scare me, since I have not a clear image of them in my mind’ . ‘We do not know vectors well, since these have not been taught during the previous years’ . ‘Vectors are easier (than classical geometry), but we have been taught about them for a shorter period of time’ . ‘The types are more standardized, but I do not know how to continue every time (or they are easily forgotten)’ . ‘Inconvenient types’ . ‘Many and diY cult operations’ . ‘In most cases it is not so brief’ . ‘Vectors need more details’ . ‘Vector senses confuse me, therefore I preferred a safer method’ .

Table 9. Responses concerning negative opinions about vector geometry.

Method supporters Euclidean supporter Vector supporter Both methods supporter

Attitudes towards geometry methods Pupil has a positive attitude towards Euclidean method Pupil has a positive attitude towards vector methods Pupil has a positive attitude towards both methods

Table 10. Pupils’ attitudes towards geometry.

Each response corresponds either to one of the above trends or is a combination of two trends. For example, a pupil was characterized as being of the compound type …‡E†…¡E† if he/she mentioned some disadvantages of the Euclidean method …¡E† although he/she had signiŽ ed a preference for this method …‡E†. Also a pupil was considered of the compound type (1 V)(1 E)(¡E), if he/she had mentioned the advantages of vector methods (1 V) as well as the advantages (1 E) and disadvantages …¡E† of Euclidean methods. In order to simplify the analysis, we considered only three general types of pupils with regards to their attitude towards the two methods. These types are described in table 10. Each of the three types concerns children whose responses show positive opinions for a certain method. Thus, in the type ‘Euclidean supporter’ we included the type …¡V† and also the compound types …‡E†…¡V†, …‡E†…¡E†, and …‡E†…¡E†…¡V†. Similarly in the type ‘vector supporter’ we included also the types …¡E† and …¡E†…‡V†. In the type ‘both methods supporter’ we included pupils who did not oppose any methods, even if in many cases they recognized the advantages and disadvantages of each method. Therefore this type includes also compound types like …‡EV†…‡E†, …‡EV†…‡E†…‡V†, …‡EV†…‡E†…¡V†, …‡EV†…‡V†, …‡EV†…¡E†…¡V†, and …‡E†…‡V†.

Classical versus vector geometry in problem solving Solver types

N

%

Solver type E Solver type V Solver type EV Total

139 51 171 361

39 14 47 100

117

Table 11. Numbers and percentages of pupils using a geometry method. Percentage receiving each score Score

Solver type E

Solver type V

Solver type EV

0(n) 1(n) 2(n) 3(n) 4(n)

16 37 47 — —

25 41 33 — —

14 7 40 33 5

By the variable n we mean E (Euclidean), V (vector), or EV(Euclidean and vector) Table 12. Percentage of pupils receiving each score by solver type.

5. Results 5.1. Success in problem solving Table 11 shows the main classiŽ cation of children based on their use of geometry method. Most pupils decided to use both methods (47%), then follow pupils who dealt exclusively with Euclidean methods (39%), while only a few pupils (14%) chose to deal exclusively with vector methods. The next question was the grade of success for each solver category. Table 12 shows that Euclidean solvers are more successful than vector solvers (entirely succeeded: 47% versus 33%, unsuccessful: 16% versus 25%). Concerning pupils who used both methods, more seem to be successful in Euclidean methods, since 33% of them solved successfully at least one exercise with the classical method and failed when using vector methods, while only 5% solved successfully at least one exercise with the vector method and failed when using classical methods. If we attempt to make a Ž rst comparison between pupils of type E and type V, we could say that there are more pupils using Euclidean methods than vector methods, and that these pupils are also more successful in problem solving than those pupils who deal with vector methods. Furthermore, even pupils who choose to use both methods are more successful in exercises solved by classical methods. 5.2. ClassiŽ cation of errors The next step was to know more about each solver type using the kinds of error made as a criterion to characterize them. As has been mentioned, errors were classiŽ ed into two main categories: general and vector. Altogether we dealt with 407 errors—282 general and 125 vector errors. Table 13 shows that general errors constitute the majority for types E and EV (99% and 65% respectively), while vector errors constitute the majority for type V (61%).

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Solver type V

Solver type EV

Error Categories

n

%

n

%

n

%

General errors Vector errors Total

103 1 104

99 1 100

26 41 67

39 61 100

153 81 234

65 35 100

Table 13. Number and percentage of errors for each solver type.

Percentage of errors Type EV Category Misused data Misinterpreted language Logically invalid inference Distorted theorem or deŽ nition Technical errors

Type E

Type V

E Method

V Method

30 5 28 30 7

8 42 4 15 31

21 7 27 43 1

14 44 9 17 15

Table 14. Percentage of general errors for each solver type.

After this Ž rst analysis, we attempted a further analysis of the two error categories for each solver type. Regarding general errors, we dealt with 282 errors, 103 for type E, 26 for type V, and 153 errors for type EV (67 errors for Euclidean methods and 86 for vector methods). Table 14 presents an analysis of general errors for the three solver types. Errors of type EV were divided into two subcategories, i.e. errors found in problems solved either by classical or vector methods, in order to make a possible comparison with errors of types E and V more convenient. From table 14 it becomes obvious that solvers of any type show in many cases similar behaviour in committing errors, when they deal with the same geometry method. Thus, we observe a correspondence between errors committed by types E and EV when the latter used classical methods, as well as between errors committed by types V and EV when the latter used vector methods. Concerning Euclidean geometry, it seems that errors resulting from insuY cient knowledge of theory (distorted theorem or deŽ nition) predominate among the general errors (30% for E-type and 43% for EV-type). The diV erence of 13% shows that EV-pupils have more diY culty with theory, and this could be a reason why pupils also tried vector methods; they do not feel as safe as E-pupils when using classical methods. In contrast, in vector geometry misinterpreted language errors predominate among general errors (42% for V-type and 44% for EV-type). Then come technical errors (31% for V-type and 15% for EV-type), and then errors resulting from insuY cient knowledge of theory (distorted theorem or deŽ nition: 15% for V-type and 17% for EV-type). Misused data errors and logically invalid inference errors seem to be less prevalent in vector geometry. Classical geometry seems to be the most disliked subject among secondary school pupils. This subject is not only combined with ability in proving theorems,

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but it is also connected with a number of mental skills. Some of these, distinguished by HoV er [7] are verbal, logical, visual, and drawing skills. Verbal skills are useful for pupils in order to reproduce propositions and deŽ nitions and write their own proofs. Deductive reasoning, which characterizes the proving procedure in classical geometry, presupposes that pupils are in a position to make valid arguments in the context of geometric Ž gures. Finally, visual and drawing skills are both connected with the handling of diagrams which are very much used in geometry problems. Based on these comments, we can explain the high percentage of errors concerning knowledge or handling of theory (distorted theorem or deŽ nition and misused data) and also logically invalid inference-errors. Indeed classical geometry demands a good knowledge of a large part of the theory (theorems, deŽ nitions). Here ‘to know’ substantially means ‘to memorize’ and consequently ‘to remember’ . According to Kimball [8] this memorizing obligation has a negative in uence on pupils’ achievement in traditional geometry. Comparing a group of high school pupils who were taught by an ‘open book’ system, with a ‘traditional’ group, he inferred that those who were not required to memorize achieved a superior performance in geometry. In our research we met a number of propositions which had confused pupils. Among them were theorems on congruence and similarity of triangles, trigonometry relations, deŽ nitions and properties of known quadrilaterals, the Pythagorean theorem, theorems on the medians of a triangle, propositions on the perpendicular bisector of a line segment, theorems of acute or obtuse angle of a triangle, and theorems on central or inscribed angles of a circle. The corresponding errors were classiŽ ed as distorted theorem or deŽ nition and as misused data. Such errors have been veriŽ ed by other researchers. [3] refers to diY culties among secondary school geometry students in the USA with congruence and similarity propositions. Carpenter et al. [9] referred to insuY cient knowledge among 17-year-olds, who had studied a year of geometry. Some of the topics mentioned were properties of quadrilaterals, e.g. necessary and suY cient conditions for a quadrilateral to be a rectangle, and properties of congruent and similar Ž gures. The same researchers also veriŽ ed low levels of performance on problem-solving exercises related to simple applications of the Pythagorean theorem and properties of similar triangles. Learning traditional geometry is not only about memorizing propositions, providing precise deŽ nitions, and proofs of theorems. The main feature characterizing this kind of geometry is ‘logical procedure’ . Errors resulting from a logically invalid procedure have been found in our research. For example, many students cite the theorem to be proved in their proofs, an error which has also been found by Senk [3], and has been attributed to the lack of teaching emphasis on the meaning of proof. The same researcher also observed a low level of achievement in using deductive reasoning especially for congruence propositions and similarity propositions, among secondary school geometry students in the USA. Figures are another cause of diY culty for pupils. [3] reported that pupils meet diY culties either when drawing an auxiliary line segment, or with diagrams containing several sets of embedded triangles that appear congruent. Carpenter et al. [9] referred to 17-year-old pupils’ dependence on the appearance of a given diagram, in the sense that they tended to make assumptions about the Ž gure. Such errors were also veriŽ ed among our sample, and were classiŽ ed as misused data. Vector geometry, on the other hand, is very closely related to symbols of vectors and demands attention in using them in vector operations. This particular

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A. Gagatsis and H. Demetriadou Percentage of errors Type EV

Category Vector as a line segment Sense errors Addition and subtraction errors Errors in dot product Errors in using coordinates Table 15.

Type E

Type V

E Method

V Method

100 0 0 0 0

49 34 12 0 5

100 0 0 0 0

57 25 11 4 4

Percentage of vector errors for each solver type.

characteristic of vector geometry helps to explain the relatively high percentage of misinterpreted language errors. The high percentage of technical errors is related to the fact that vector methods include a lot of vector operations. Vector methods are considered rather standardized. Therefore, vector geometry is not dependent on the knowledge of a large number of theorems, or on a strict procedure of logical deductions, as happens with classical methods. This fact also helps to explain the relatively low percentage of distorted theorems or deŽ nitions, misused data, and logically invalid inferences. As regards vector errors, we dealt altogether with 125 errors—1 for solver type E, 41 for solver type V and 83 for type EV (three errors for Euclidean methods and 80 for vector methods). Table 15 presents an analysis of general errors for the three solver types. We can observe a similar behaviour to that identiŽ ed in table 14, in committing errors for solvers of any type, when dealing with the same geometry method. It is obvious that vector errors are substantially connected with vector solvers. Thus, among 125 vector errors only four were committed by pupils who used classical methods, and concern the misconception that a vector is equivalent to a line segment. This misconception predominates among vector errors (49% for Vtype and 57% for EV-type), while sense errors follow (34% for V-type and 25% for EV-type). In third position come errors resulting from vector addition and subtraction procedures. The percentages regarding dot product and coordinates are of less interest. It is to be noted here, however, that dot product plays a more serious role in general and vector errors, since some errors classiŽ ed in other categories (i.e. sense errors, distorted theorem or deŽ nitions, misinterpreted language) were also connected with dot product. It is to be noted here, that teaching and comprehension of the concept of vectors for Greek secondary pupils (aged 15 to 17) has been the object of previous researchers [5, 6]. The results showed relatively low comprehension, which was partly attributed to the peculiarity of teaching vectors in Greek secondary schools. Among errors committed by a sample of 920 pupils aged 16 and 17 years, Ž rst came misconceptions regarding vector features (sense, orientation, and direction), and then followed the misconception of a vector being equivalent to a line segment. In third position came errors in vector operations, while errors concerning vector coordinates were of less interest. Having these previous results in mind helps to

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Classical versus vector geometry in problem solving Method supporters Euclidean …‡E† Vector …‡V† Both methods …‡EV† Total

n

%

157 85 119 361

43 24 33 100

Table 16. Number (n) and percentage of pupils supporting a geometry method.

Percentage of solver types Method supporters Euclidean …‡E† Vector …‡V† Both methods …‡EV† Table 17.

Type E

Type V

Type EV

83 3 14

12 78 10

21 24 55

Percentage of method supporters for each solver type.

explain the errors of Greek pupils (aged 18 years) in vector geometry, which are very closely related to the handling of the concept of vector. Summarizing, we could say that errors resulting from insuY cient knowledge and ‘manipulation’ of theory, as well as errors of logical procedure, characterize Euclidean solvers, while vector solvers make more errors resulting from misconceptions about the concept of vector. 5.3. What pupils believe about the classical or vector approach to geometry So far we have presented our attempts to explore pupils’ attitude towards classical and vector geometry, through their success in problem solving using these two approaches. Finally, we attempted to investigate pupil opinion and disposition regarding the above approaches, and the possible in uence of this disposition in the choice of method for problem solving. For this reason, the pupils of our sample were asked to answer a questionnaire after they had Ž nished the problem-solving procedure. Based on the responses, we distinguished three general types, regarding their critical views about the two methods. Among the 361 pupils who answered the questionnaire, 157 supported classical geometry, 85 supported vector geometry, and 119 supported both approaches. As shown in table 16, more students preferred Euclidean methods (43%) compared to vector methods (24%) or to both approaches (33%). At this point it would be interesting to verify whether pupils’ preferences are in agreement with their choices regarding geometry approaches. In table 17 we present the percentages of supporters of the diV erent methods as they are divided among the three solver types. As we observe, the majority (83%) of pupils who had used classical geometry also expressed a preference for this method. Similarly 78% of vector solvers considered vector methods the most appropriate for geometry exercises. In the same situation pupils who used both methods (55%) expressed the opinion that both methods are equally useful and appropriate for the solution of geometry problems. Consequently, most of the solvers in all three types seem to prefer the method they used. This implies that their choice was not accidental. On the contrary, pupils were aware of the advantages of the chosen method.

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A. Gagatsis and H. Demetriadou Percentage of method supporters Solver type Type E Type V Type EV

Euclidean …‡E†

Vector …‡V†

Both methods …‡EV†

73 4 23

5 47 48

17 4 79

Table 18. Percentage of solver types supporting each method. Percentage of positive features Advantages of Euclidean methods …‡E† Simplicity Long experience Aid of theorems Unique in pupils’ mind Interesting

Type E

Type V

Type EV

47 21 9 21 6

20 40 40 0 0

51 19 12 4 14

Table 19. Percentage of positive features for Euclidean method mentioned by each solver type.

In table 18 we examine the inverse correspondence between chosen and most preferable method; i.e. the percentage of solvers supporting each method. Concerning those pupils who expressed a preference for the Euclidean approach or for both approaches ‡E, and ‡EV), a relatively high percentage of them (73% and 79% respectively) used their favoured method. However, this did not happen in the case of pupils who expressed a preference for vector geometry. Only 47% of them used vector methods, while 48% of them attempted to use both methods. This fact may be attributed to lack of conŽ dence regarding vector methods, as well as the strong in uence of classical methods which have been taught for a longer period of time. This assumption becomes stronger if we bear in mind the relatively low percentage of vector supporters (25%: table 16), combined with the low percentage of pupils who dealt exclusively with vectors (14%: table 11), as well as their poor performance in problem solving (table 12). Concluding, we can say that Euclidean and vector solvers chose the method they declared to be the best. However, with supporters of vector methods, one could observe that their preference was not always strong enough to make them use this method. Almost half of them tried to use classical methods in parallel with vector methods. Does this mean a lack of conŽ dence? Is the in uence of classical geometry so strong that it is used as a refuge, in spite of its disadvantages as recognized by these pupils? The percentages referred in the previous paragraph seem to support this view. Pupils’ responses also gave more information regarding their beliefs about the diV erent characteristics of each approach. We dealt altogether with about 560 comments, 212 of which concern the advantages of Euclidean methods, 36 concern disadvantages of Euclidean methods, 167 concern advantages of vector methods, 20 disadvantages of vector methods, and 125 are positive comments about the two methods. In tables 19, 20, 21 and 22 we present the distribution of some of these

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Classical versus vector geometry in problem solving Percentage of negative features Disadvantages of Euclidean methods …¡E† A lot of theory Not recent Complicated

Type E

Type V

Type EV

75 25 0

50 50 0

45 27 27

Table 20. Percentage of negative features for Euclidean method mentioned by each solver type.

Percentage of positive features Advantages of vector methods …‡V† Methodology More favourable Recent Unique in pupils’ mind Table 21.

Type E

Type V

Type EV

50 44 6 0

44 29 18 9

62 17 20 1

Percentage of positive features for vector methods mentioned by each solver type.

Percentage of negative features Disadvantages of vector methods …¡V† Complicated Lack of experience Table 22.

Type E

Type V

Type EV

50 50

0 0

77 23

Percentage of negative features for vector methods mentioned by each solver type.

types of responses among pupils of the three solver types (E,V, EV), based on the classiŽ cations presented in table 10. As is obvious from table 19, simplicity and the long time experience seem to be the strongest characteristics of Euclidean methods for types E and EV (47%–51%, and 21%–19% respectively). Vector solvers consider equally strong the long experience together with the existence of many theorems (40%). On the other hand, this last characteristic operates as the most important disadvantage (75% of negative comments for type E, 50% for V, and 45% for EV), while the fact that the relative geometry courses are not recent, comes second among the disadvantages of Euclidean geometry, as shown in table 20. As is shown in tables 21 and 22, some of the positive features of one method operate as negative features for the other method. For example, the most popular feature of vector geometry seems to be the comfort of methodology (50% for Esolvers, 44% for V-solvers, and 62% for EV-solvers), which is opposed to the variety in theory, Ž gure diY culties and complicated thoughts characterizing classical geometry. Another advantage of the vector approach is the fact that it is recent, while a part of the relative theory on classical geometry seems to have been

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forgotten by some pupils. Finally, the long time experience with the classical approach competes with the lack of experience of vector methods. 6. Discussion Pupils in the last year of Greek secondary school can be set into three diV erent categories with regard to their attitude to geometry problem solving. The Ž rst category consists of pupils who were strongly in uenced by previous teaching. Thus, in spite of the fact that classical geometry seems to be the most disliked subject among secondary school pupils in many countries, connected as it is with a number of mental skills, and the fact that during the last year only vector geometry is taught (indispensable in the context of university entrance examinations), pupils of this category not only like classical methods, but they also prefer to use them to solve geometry problems, with relatively high performance. In some cases they do not hesitate to express the pleasure they feel when recalling this method: ‘It’ s good to recall some knowledge; unfortunately in the last year we have put Euclidean geometry aside’ . ‘It’ s more scientiŽ c, since it comes from ancient Greeks’ . ‘It’ s more interesting; it’ s a pity that it is not taught in the last year of Lyceum!’ The errors committed by these pupils mostly result from insuY cient knowledge and handling of theory, or from use of a logically invalid procedure. However, pupils of this category consider simplicity as the main feature of classical geometry and recognize this characteristic, together with their long experience, as the main reasons for using classical methods. Pupils in the second category support vector methods and use them in solving geometry problems. Most of these pupils seem to appreciate the eV ectiveness of the standard methods used in vector geometry, which releases them from strong dependence on theory and the strict procedure of logical deductions used in the Euclidean methods. Many of them also express their pleasure in exploring the new method by stating: ‘It’ s better than Euclidean method. Unfortunately I have discovered it too late!’ Some pupils of this category also recognize the fact that vector geometry is taught primarily for use in university entrance examinations. Errors in this category mostly concern the use of vector symbols and operations. Pupils in the third category were in uenced by both approaches. However, the in uence of Euclidean geometry seems to be the strongest. Thus, they did not only use classical methods in most exercises, but they were also more successful when using this method than those who followed the vector methods. Concerning their errors, they have the same attitude as Euclidean or vector solvers when they use classical or vector methods respectively. The simplicity of classical methods at Ž rst, and then their experience, the interest, and the support of the classical theory seem to be the main reasons for using this approach in problem solving. On the other hand, they seem to have the same reasons for using vector methods as the

Classical versus vector geometry in problem solving

125

pupils of the second category. In the third case the ‘concept’ (the representation) of a segment is an important cognitive obstacle for the ‘concept’ (the representation) of a vector. The students very often begin the solution of a problem with segments (vectors) and they continue with vectors (segments). It can be claimed that successful teaching of vectors must be based on the rejection of these cognitive obstacles. In our research we attempted to Ž nd out the level and kind of in uence of each geometry approach on pupils when they solve geometry problems. The strong in uence of classical geometry became obvious. Indeed, pupils’ conceptions surprisingly made them resist change and enrichment by plainer methods. Vectors instil suspicion and distrust, like strange bodies in an already known Ž eld. This is indicated by the relatively low number of pupils who used vector methods, and by the low performance attained when using vector methods to solve problems. In many cases, exercises are solved using the most familiar way, even if this demands diY cult and complicated thought procedures. Of course the percentage of pupils who prefer to use vector geometry should not be underestimated. Many of them refer to the advantages of vector methods. It is disputable however that these advantages make them use this method. In some responses it was obvious that the pressure of time and anxiety for success in national examinations was the motivator. This forces students to use exclusively the methods taught in the Ž nal year, and to reject other diV erent methods. References

[1] Gagatsis, A., 1994, Histoire de l’ enseignement de la Ge´ometrie en Gre`ce: L’ in uence des Geome`tres franc¸ais de 1830 a´ 1884, Repre`res IREM, No 17, 47–69. [2] Malone, J. A., Douglas, G. A., Kissane, B. V., and Mortlock, R. S., 1980, in S. Krulik and R. E. Reys (eds) Problem Solving for School Mathematics (Reston, VA: The National Council of Teachers of Mathematics). [3] Senk, S. L., 1985, Math. Teacher, 78, 448–456. [4] Movshovitz-H ADAR, N., Z ASLAVSKY, O., and Inbar, S., 1987, J. Res. Math. Educ., 18, 3– 14. [5] Demetriadou, H., 1994, The teaching of the concept of vector in Greece—Some remarks on the history of this concept and on the errors to the Greek pupils. M.A. dissertation, University of Surrey. [6] Demetriadou, H., and Gagtsis, A., 1995, in A. Gagatsis (ed.) Didactics and History of Mathematics (Thessaloniki: Erasmys ICP-94-G-2011/11). [7] HoFFer, A., 1981, Math. Teacher, 74, 11–18. [8] Kimball, R. B., 1954, Math. Teacher, XLCII, 13–15. [9] Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., and Reys, R. E., 1981, in M. K. Corbitt (ed.) Results from the Second Mathematics Assessment of NAEP (Reston, VA: The National Concil of Teachers of Mathematics).