Almeida D.F. (1997) Proof and the eye: a means for understanding Group theory. La lettre de la Preuve, juillet/août 1997.
	Abstract Students meet mathematics structured as in formal mathematics texts so their learning takes place within this structure. These texts present mathematics as a summary of the knowledge. Mathematical knowledge, on the other hand, may be better understood by a process that is faithful to the historical psychogenesis (Burn, 1996). Such a psychogenesis includes generic language (Kline, 1980) and the use of the eye (Davis, 1993). This article argues that mathematics students of all abilities will benefit by being encouraged to utilise generic language and visual aids in their learning. Some evidence from teaching Group theory and from a empirical study are presented.
© Dennis Almeida

Introduction Mathematics is in the main presented in textbooks and lectures as a summary. There are axioms, definitions, theorems and proofs concisely expressed. But that is not the way that mathematics is discovered. Mathematical results may be discovered through investigative enquiry, intuition, speculation, etc. with proof being the validator (Mac Lane 1994) If mathematicians understand mathematics in an organic rather than in the rigorously structured way suggested by textbooks and formal lectures then this may have some implications for the classroom. I suggest that the flow diagram for understanding mathematics in the classroom may be expressed as: Structured investigation or problem solving -> formation of conjectures -> Hunger for proof. This has some support in the mathematics education field (for example, Burn, 1996; Dreyfus, 1990). In this article we describe how particular results in Group Theory/Geometry were received by a sample of undergraduates following a traditional mathematics courses and compare this with the treatment during lectures of the same results for students following mathematics courses in a teacher training degree programme. Those following the traditional mathematics courses were taught formally whereas the latter were taught in a way that, in some way, reflected the origins of the Groups theory - that is, by a study of symmetry. The rationale being that "...attention to historical origins provided the strongest pointer available to an authentic genesis of this subject, a pointer which lends some support to the Action, Process, Object, Schema theory of learning." (Burn, 1996, p 376) The undergraduates following the teacher training degree in mathematics were third year students taught by the author. A sample of 15 second year undergraduates, all following a three year BSc. degree course in mathematics in an established UK university, were involved in an empirical about proof (reported elsewhere in Almeida, 1996). In the next section the teaching styles for the two groups of students are described. In the following section we describe how each group of students approached similar problems in Group theory. And finally we conclude with some suggestions for teaching. Teaching styles The group theory course for the teacher trainees consisted of 50 hours of a study of geometric symmetry before axioms are offered. The course was developed by Burn (1996) and allows for work with subgroups, cosets, normal and quotient groups before they are formally defined. The mathematics undergraduates, on the other hand, learn about groups in a traditional but formal way. The lecturer in the Groups course, however, had expressed concern about his students general lack of experience in proving and with formal abstract mathematics. However the use of informal and generic arguments in proofs and exposition by the lecturer was very limited. These is an intersection of the mathematics curricula of the two groups of students; in particular there is an intersection in the Group theory course with several theorems and results being studied by both groups. Whilst the teacher trainees were guided to some of these results by their lecturer using a generic approach (that attempts to replicate the historical genesis), the mathematics undergraduates had to prove the same results themselves. In the next section we compare the way that the two the teacher trainees were guided to these results with the way the mathematics undergraduates attempted to verify the same results. The teacher trainees have, on average, lower mathematics 'A' (high school) level grades than their counterparts in the mathematics department. Thus we will not compare the relative abilities of the two groups but merely ask whether the undergraduates' responses might benefit from a teaching style that includes generic language and the use of the eye. Student responses i) The theorem of two reflections and applications. The theorem of two reflections in a plane is introduced to teacher trainees as follows - if A and B are reflections in lines through the origin O inclined an angle q as shown then the composite AB of the reflections is an anticlockwise rotation of angle 2q. On the other hand, mathematics undergraduates use 2x2 matrix representations of rotations and reflections and express the theorem in symbols without recourse to a diagram. When teacher trainees attempt the following coursework project:

 

they have a good understanding of this problem and its possible proof: PROBLEM: Show that there are matrices A and B in O(2) such that A2 = B2 = I and AB has infinite order. For example, by taking q = p/sqr(2), say, AB is an anticlockwise rotation of angle sqr(2p) which has infinite order. However, of the seven mathematics students in the study who attempted this task none were successful in proving the existence of A and B. They attempted a direct approach by investigating the two matrix equations A2 = B2 = I. None applied the theorem of two reflections from which the result may be deduced directly. Because the theorem of two reflections are stated for them in formal terms (using matrices) without recourse to visual illustration, we could surmise that these students were not able to make progress in this task. ii) Generic understanding of inverse. With the image of symmetry groups in mind teacher trainees view inverse generically. A reflection is self-inverse - your reflection reflected is you. As for rotations inverse is viewed as follows: Each such g is paired uniquely with its inverse g-1 unless q = 0 or 360. And so the argument for this task is none too difficult. PROBLEM: Show that the number of elements g in a finite group G with g ¼ g-1 is even. Mathematics students in the project had some difficulty with this problem. Most appear not to think of generic examples preferring to rely on the rules of mathematical logic which they misapply often. This is exemplified by Nat's formal argument: "We can list the elements of the group as (e, e), (g1, g1-1),......,(gn, gn-1). Obviously the inverse of an element could equal itself. However with the stipulation that g ¼ g-1 we only drop the identity. This shows that the set of these elements is even." His formal 'proof' is less than convincing. Clearly the essential argument (for which a hint was given) : 'g ¼ g-1 Þ (g-1 ) ¼ g Þ (g-1 ) ¼ (g-1)-1 = g and hence such elements occur in pairs' is missing. They way some students appear to think and work (ineffectively) in mathematics may be induced by the dichotomy between the way professional mathematicians teach and the way they do mathematics. We have seen that it is widely held that the way professional mathematician do mathematics involves a particular flow: intuition -> trial and error -> speculation -> conjecture -> proof (MacLane, 1994, p 191): but the way they teach often focuses only on the end point of the flow thereby marginalising the inherent sense making activity in mathematics. Alibert and Thomas (1991) put it this way: "...(the) apparent conflict between the practice of mathematicians on the one hand, and their teaching methods on the other creates problems for students. They exhibit a lack of concern for meaning, a lack of appreciation of proof as a functional tool and an inadequate epistemology." (p.215) iii) Cauchy's theorem. The apparent reluctance to use generic examples to illuminate concepts and structure at the fundamental level has implications for understanding at a higher level. For example the generic example of inverses has a direct bearing on students appreciation of the following special case of Cauchy's theorem: PROBLEM: G is a non-cyclic group with 6 elements. Prove that G must contain an element of order 2 and an element of order 3. This statement is proved in class for the teacher trainees in a way that forces the students to focus on the properties of symmetry groups. It involves the use of the following three arguments: (a) Lagrange's theorem implies that the orders of the elements in G are 1, 2 or 3. (b) The statement in ii) above: 'Let G be a finite group. Show that the number of elements gŒG with g ¼ g-1 is even' implies that not all elements (apart from the identity) can be of order 3. Hence there must be an element of order 2. (c) The statement 'A group with identity e has the property that g2 = e for all group elements. Show that the group is abelian' is given credibility by considering the symmetry group D2 which has all elements satisfying this criteria - it is abelian which students appreciate because the two reflections a and b in D2 are at right angles and the theorem of two reflection tells them that these two elements commute; their product being a half turn. By a further consideration of the same theorem they see that the half turn itself commutes with the two reflections. A formal proof is given by considering the identity a2b2 = (ab)2 = e in such groups. Hence if all elements of G have order 1 or 2 then this result implies G is abelian. This allows the construction of a dihedral subgroup {e, a, b, ab} of order 4 in contradiction to Lagrange's theorem. The contradiction implies that there must be an element of order 3. The special case of Cauchy's theorem was given as a special optional assignment for mathematics students to prove. The six who responded did so out of personal interest and motivation. Two students presented a convincing proof which contained the arguments (a) and (c); the simple counting argument (b) was however replaced by a proof by exhaustion (which, arguably, was inferior in that it required no structural insights). The other four students did not use b) or c). They all began their 'proofs' by using (a) and then proceeded to exhaustively examine using the structure of the Cayley table the two impossible situations - the group elements, apart from the identity e, either all have order 2 or all have order 3. That they failed is understandable given the very high probability of 'clerical' error. This is part of Simone's 'proof':   'Let G = {A, B, C, D, E, F} and let E be the identity. Suppose A, B, C, D, F have order 2.   Square brackets contain responses from Simone during interview (what Simone may have meant in the last two steps was: AB=C Þ BC=A and so DC=D or F. DC=D and DC=F are untenable- which is the required contradiction). We may argue that Simone explored the construction of a proof in purely algebraic terms because she did not have an alternative. Some suggestions for the teaching of Group theory We argue that students' understanding of mathematics can be improved if they were to focus some of their attention on the meaning and understanding of the concepts that they have to deal with. We argue that this can be accomplished by the following strategies: i) Lecturers ought to help develop students' concept images so that it is synonymous with the concept definition. The concept image of a concept is the ".....the total cognitive structure associated with the concept, which includes all the mental pictures and associated properties and processes". (Tall and Vinner, 1981, p152). For example, the concept image of groups can be properly developed by focusing on the symmetry groups of regular polygons. An evocative and proper concept image can help students to be better prepared to prove the results. ii) Informal, generic and visual proofs ought to be used by lecturers in lectures. Ideally lecturers ought to demonstrate that these proofs can be easily be transformed into formal language (see also Van Asch, 1993). It must be said that not all students will have the same the same concept image of a particular concept. What will work well with one student may not work as well with another and what may be needed is the presentation in lectures of as many possible additions to concept images as can be afforded. And providing the concept image is free of potential conflict with the concept definition (the definition of the concept: see Tall and Vinner, 1981, p153) then informal or generic arguments, such as the ones described above, may help the transition to formal proof by lecturers using such proofs in lectures and demonstrating that such proofs can easily be translated into formal language (Blum and Kirsch, 1991; Hersh, 1993). By way of an example consider the statement PROBLEM: a) In O(2) the reflections form a single conjugacy class. b) If R1 and R2 are rotations in O(2), then R1 and R2 are conjugate in O(2) if and only if R1 = R2 or R1 = R2-1.   A proof for the first part a) 'the reflections form a single conjugacy class' might involve using matrices (favoured by all the mathematics students in the study) but a generic proof using diagrams may be more meaningful and may add to students concept images (of congujacy and conjugates). This is described below and brief details of the transformation into formal language given. For an argument that verifies a) we let Land M be any two reflections in O(2). Let X be another reflection in O(2) whose axis bisects the angle between the axes of L and M: The composite X-1MX = XMX can be shown to be the same as L merely by tracking any geometrical object O as it is transformed by XMX. So any two reflections belong to the same conjugacy class. And the demonstration above can be easily be transformed in to formal language using matrices or mappings of complex numbers. That rotations can never belong to this class can be demonstrated informally by noting that reflections reverse orientation while rotations preserve orientation (if X is a reflection then the conjugate XMX being the product of three reflections will reverse orientation and hence cannot be a rotation; if X is a rotation then by considering the composite effect on orientation of the conjugate it can be seen that XMX will reverse orientation and hence also cannot be a rotation). Then this argument can be easily be transformed into formal language using properties of determinants or mappings of complex numbers. A similar argument may be used to verify b) References Alibert D and Thomas M, 1991, 'Research on Mathematical Proof', in Tall D (ed) Advanced Mathematical Thinking. Kluwer Academic Publishers, Dordrecht, 215-230 Almeida D, 1996, 'Mathematics Undergraduates' Perceptions of Proof', Teaching Mathematics and its Applications, 14, 171-176 van Asch A G, 1993, 'To Prove, Why and How? International Journal of Mathematics Education in Science and Technology, 24(2), 301-313. Burn R P, 1985, Groups: A Path to Geometry, Cambridge University Press, Cambridge. Dreyfus T, 1990, 'Advanced Mathematical Thinking' in Nesher P and Kilpatrick J (editors) Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education, Cambridge University Press, Cambridge, 113-134. Hersh R, 1993, 'Proving is Convincing and Explaining', Educational Studies in Mathematics, 24, 389-399 MacLane S, 1994, Responses to "Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics", Bulletin of the American Mathematical Society, 30(2), (178-207) Tall D, 1991, 'The Psychology of Advanced Mathematical Thinking', in Tall D (ed), Advanced Mathematical Thinking, Kluwer Academic Publishers, Dordrecht, 3-21 Tall D and Vinner S, 1981, 'Concept Image and Concept Definition in Mathematics with Particular Reference to Limits and Continuity, Educational Studies in Mathematics ,12, 151-169