Learning group isomorphism: A crossroads of many concepts · 2016-06-20 · LEARNING GROUP ISOMORPHISM: A CROSSROADS OF MANY CONCEPTS ABSTRACT. This article is concerned with how

URI LERON, ORIT HAZZAN AND RINA ZAZKIS

LEARNING GROUP ISOMORPHISM: A CROSSROADS OF MANY

CONCEPTS

ABSTRACT. This article is concerned with how undergraduate students in their first abstract algebra course learn the concept of group isomorphism. To probe the students' thinking, we interviewed them while they were working on tasks involving various aspects of isomorphism. Here are some of the observations that emerged from analysis of the interviews. First, students show a strong need for "canonical", unique, step-by-step procedures and tend to get stuck of having to deal with some degrees of freedom in their choices. Second, students exhibit various degrees of personification and localization in their language, as in "I can find a function that takes every element of G to every element of G J'' vs. "there exists a function from G to G'". Third, when having to deal with a list of properties, students choose first the properties they perceive as simpler; however, it turns out that their choice depends on the type of the task and the type of complexity involved. That is, in tasks involving groups in general, students mostly prefer to work with properties which are syntactically simple, whereas in tasks involving specific groups, students prefer properties which are computationally simpler.

1. INTRODUCTION

How do we think of the concept of group isomorphism? For an experienced mathematician, it is possible to think of it as a single, unified concept, indeed, as a single object. Students in their first abstract algebra course, however, seldom achieve this mature view. For them, isomorphism is a complex and compound concept, composed of and connected to many other concepts, which in themselves may be only partially understood. For example, understanding group isomorphism involves the understanding of the concepts of group, function and quantifier. Conversely, learnhag about isomorphism may in due time solidify the understanding of these related concepts.

This article is concerned with how undergraduate students in their first abstract algebra course learn the concept of group isomorphism. To approach this question, we interviewed a group of students and identified in their answers several points we judged to be interesting, around which the research results are arranged. The students' interviews reveal connections of the isomorphism tasks to the concepts of orders (both of the group and the group elements), commutativity, etc. In particular, properties of

Educational Studies in Mathematics 29: 153-174, 1995. (~) 1995 Kluwer Academic Publishers. Printed in the Netherlands.

154 URI LERON, ORIT HAZZAN AND RINA ZAZKIS

single elements vs. properties of the whole group (in various degrees of localization) come into play.

Our observations of the students work with the isomorphism tasks invariably lead to larger issues, to which the isomorphism tasks were only a trigger: Issues of similarity and difference, degrees of freedom in constructing an algorithm, proofs and refutations, and constructing mathematical processes and objects. Some of these connections will be discussed further in the concluding section (Section 6).

The main sections of the article describe the research context (Sec- tion 4) and the research results (Section 5). Section 3 contains a literature survey. Section 2 contains some historical notes, some reflections on the concept of isomorphism, the mental processes involved in its understanding, and the various distinctions they involve. These distinctions may be quite simple from the formal point of view, but they are significant in the teaching/learning process. They include the distinction between "naive" vs. formal definition of isomorphism; the relation of two groups being isomorphic vs. the object of isomorphism; the processes involved in proving isomorphism vs. those in proving non-isomorphism; and the processes involved in dealing with isomorphism in the abstract vs. those in specific cases.

2. THE VARIOUS FACES OF ISOMORPHISM

2.1. The Meaning of Isomorphism

Informally, two groups are isomorphic if they are "the same except for notation". Thus, if we take any group and rename its elements and its operation, we get an isomorphic copy of the same group. This is essentially what we shall mean by "naive isomorphism", which will be discussed more broadly in Section 4.

The formal definition is quite far removed from this intuitive notion. An isomorphism from a group [G,o] to a group [G ~,o I] is a one-to-one function f from G onto G I, satisfying f(a o b) = f(a) o I f(b) for all a,b in G. Two groups are called isomorphic if there exists an isomorphism from one to the other. Thus, constructing (mentally) the concept of isomorphic groups according to this formal definition, involves the construction of the concepts of function and of isomorphism-as-object (since we are quantifying over this object). The need for these additional constructions is circum- vented by using naive isomorphism, which may account for the simplicity of the naive version relative to the formal one.

The deeper meaning of isomorphism in either of these definitions is that the implicit correspondence in the naive version, or the explicit function

LEARNING ISOMORPHISM 155

in the formal version, preserve the group structure (i.e., its cardinality and its operation), hence they preserve all its abstract properties.

To illustrate, we bring an example of two isomorphic groups, which plays an important role in the research to be reported here. These two groups are: the symmetric group $3 - the group of all permutations of { 1,2,3 } relative to the operation of function composition; and the dihedral group D3 - the group of all symmetries of the equilateral triangle, again relative to function composition. That the two are isomorphic can be seen by numbering the vertices of the triangle and considering each symmetry to be a permutation of these 3 numbers. This would immediately show that the two are naively isomorphic. To define a formal isomorphism between them, we would need to explicitly construct the function carrying each symmetry to the corresponding permutation. (There are some problems here, however; cf. Zazkis and Dubinsky, in press). This example also demonstrates the importance and usefulness of isomorphism, since it can be used both to work analytically in D3 and to give a geometrical interpretation of $3. For example, now we can see why $3 has three elements of order 2 ("reflections") and two elements of order 3 ("rotations").

2.2. Various Distinctions

The concept of isomorphism is a rich and multi-faceted one. In this discussion we wish to focus on three distinctions. These distinctions may be quite simple from the formal point of view, but they have great importance in the teaching/learning process.

(a) The distinction between the relation of two groups being isomorphic vs. the object of Isomorphism:

Looking at the formal definitions, there is not much to say for this distinction. From the perspective of mental operations, however, there is a great difference. The relation of being isomorphic is symmetric, intuitive (in its naive version) and doesn't require the function concept for its understanding. The object of isomorphism is directional, "functional", and much harder to understand; and we don't have a naive version for it. Inci- dentally, note that homomorphism, too, doesn't admit a naive definition, perhaps for the same reason of directionality, which necessitates the use of function.

(b) Proving isomorphism vs. proving non-isomorphism: The first requires establishing a correspondence or showing the exis-

tence of a function of a certain type. The latter seems to require proof of non-existence- something that is very hard to do. However, students often seem to be able to get around this difficulty by building on what we have called "the deeper meaning" of isomorphism, i.e., they find relatively easy


proofs of non-isomorphism by pointing to a property of one of the groups which doesn't hold in the other.

(c) Isomorphism in the abstract vs. in a specific case: In the abstract context, one can think of all isomorphic groups as being

essentially the same, i.e., one thinks of an abstract group as being essentially an isomorphism class. In dealing with specific examples, however the nature of the specific isomorphism may play a role. Some isomorphisms are highly non-trivial, and it is hard to think of two such groups as being "the same". For example, the reals with addition are isomorphic to the positive reals with multiplication via the exponential function, but no one suggests that they are the same. (There is another reason why these two systems don't "feel" the same: looking at them as groups captures only a small part of their nature since they have a much richer structure than just being groups.)

2.3. Historical Notes

The long and somewhat tortuous history of the group concept is surveyed in Kleiner (1986). The interesting point for our present discussion is the intimate connection between the concept of isomorphism and that of the abstract group. Except for the formal definition, it seems that one can hard- ly think and understand the meaning of "abstract" here, without resorting to some version of isomorphism. This is clearly evidenced in the following quote from von Dyck (1882), appearing in Kleiner (1986):

The aim of the following investigation is to continue the study of the properties of a group in its abstract formulation. In particular, this will pose the question of the extent to which these properties have an invariant character present in all the different realizations of the group, and the question of what leads to the exact determination of their essential group-theoretic content.

It is no accident that von Dyck, who was the first to bring together all the different strands of partial definitions of abstract group, also resorts in the same paragraph to the notion of isomorphism:

in this way all [...] isomorphic groups are included in a single group ]...] the essence of a group is no longer expressed by a particular presentation form of its operations [= generators] but rather by their mutual relations.

And there is of course that quintessential isomorphism of all the groups - the one appearing in the famous Cayley's theorem, though Cayley himself didn't even use the term. In modem terms, Cayley's theorem asserts that every finite group is isomorphic to a permutation group, i.e., a subgroup of some S,~. Cayley's original formulation of the theorem is, of course, not quite the modem one (we are indebted to Kleiner for the following discussion of Cayley's theorem). Cayley:


The general problem of finding all the groups of a given order n, is really identical with the apparently less general problem of finding all the groups of the same order n, which can be formed with the substitutions upon n letters.

As "proof ' he refers to a group of order 6 he gave earlier, and says that each of its elements "may be regarded as a substitution" - essentially the way we do it today, by multiplying the given element by all elements of the group. There is no mention of preservation of operations.

As is always the case, the struggle by some of the best mathematicians to formulate the definitions of abstract groups and isomorphism is an indi- cation of the complexity and sophistication of these notions. No wonder, then, that students find them so difficult and need time and effort to develop them in their minds. Once more we meet the common wisdom (which is not, however, sufficiently applied in the classroom) that it is usually not wise to introduce to students too briskly a definition that has such a history. In fact, our classroom use of naive isomorphism is an attempt in this direction. The following quotes from Cayley (appearing in Kleiner, 1986, pp. 208-209), are also related to our naive isomorphism:

- An abstract group is determined by its multiplication table. - The cyclic group of order n is in every respect analogous to the system of the roots

of the ordinary equation x '~ - 1 = 0. - There exists only one group of a given prime order.

Those references to the multiplication table and to a system being "in every respect analogous" to another system are rather similar to our notion of "naive isomorphism", as described in Section 2.1.

3. LITERATURE SURVEY

Our research is part of a series of studies in advanced mathematical thinking, which was classified by Robert and Schwarzenberger (1991) as "research into students' acquisition of specific concepts". In our literature search, we have found not a single article which is devoted specifically to the topic of our research, namely students' understanding of group isomorphism. We did find, however, a few articles which deal with related topics. These articles fall in three categories, which we survey below.

The concept of isomorphism cannot be studied in isolation. The development (both historical and personal) of the group and isomorphism concepts are, like those of the chicken and the egg, inseparable. On the one hand, group isomorphism is built on the concept of a group; on the other hand, as we have mentioned in Section 2, the concept of isomorphism is essential to the understanding of the abstract group concept. As such, this study forms a part of a recent ongoing research project on groups and


related topics (Dubinsky, Leron, Dautermann and Zazkis, 1994; Leron and Dubinsky, 1995; Hazzan and Leron, 1994).

Another category consists of articles which deal with the teaching of isomorphism, mostly in elementary courses aimed at the high school level. Two articles (Lichtenberg, 1981; Quadling, 1978) describe various non-standard examples of groups and show their equivalence to more standard ones. Usiskin (1975) is aimed at demonstrating how the topics of groups in general and isomorphic groups in particular is applicable to the standard high school curriculum. Thrash and Walls (1991) suggest a teaching strategy based mostly on manipulation of group tables, which is aimed at helping students "having trouble understanding the concept of group isomorphism".

A third category, consisting of articles by Hart (in press) and Selden and Selden (1987) discusses college level proof writing performance, using elementary group theory as a subject domain. Even though none of the proofs discussed is specifically related to the issue of isomorphism, Hart's general conclusion is one of the underlying implicit goals in our study. Hart suggests that "our major goal as teachers should be to promote students' acquisition of stable, powerful, accessible conceptual schemata". Our attempt to break into pieces the complexity of knowledge regarding isomorphism is a contribution towards understanding of what those "conceptual schemata" might be. The study of Selden and Selden (1987) classifies the errors and misconceptions made by students in an introductory abstract algebra course according to their "logical characteristics", that is, according to whether the arise from difficulties in generalization, use of theorems, notation and symbols, nature of proof, or quantification. Their classification directs our attention to the fact that many of the errors and misconceptions in elementary group theory are due not only the complexity of the subject matter, but also to students' weak mathematical background.

Since the very definition of isomorphic groups says that "there exists a function from the group G to the group G ~, such that ...", it follows that understanding the concept of isomorphism is intimately connected with the understanding of at least three other mathematical concepts: group, function and quantifier. We have already surveyed the literature on groups. The literature on functions is vast and we shall not go into any details here (cf. Leinhardt, Zaslavsky and Stein (1990) and Harel and Dubinsky (1992) for extensive survey and bibliography on functions). We shall only mention that most research deals with graphable (i.e., real) functions, while here we are dealing with a much more general (and difficult) type of function.

Dubinsky, Elterman and Gong (1989) suggested a genetic decomposition for students' construction of quantification, and Dubinsky (1993)


followed up with a classroom implementation to support such construction. This genetic decomposition describes a progress from simple declara- tions/single propositions through single-level quantification to multi-level quantification. Isomorphism can be seen as a multi-level quantification: existence of a function between groups G and G I such that for all a,b in G, f(ab) = f(a)f(b). In our study we discuss how the construction of quanfifiers is coordinated with the construction of the other concepts which together make up isomorphism.

4. CONTEXT: DESCRIPTION OF THE RESEARCH

The majority of students interviewed in this research (Group A) come from a one-semester first-year course taught by two of the authors to computer science majors at the Israel Institute of Technology (IIT). A few items of the data are taken from students in a Kent State University summer Institute for inservice high school teachers (Group B), also taught by one of the authors, and from a different, standard IIT course (Group C). We shall indicate the group of each student the first time s/he appears.

The students in groups A and B were taught in a non-traditional method, in which computer activities (programming in ISETL) and team work mostly replace the lecture method. (Cf. Leron and Dubinsky, 1995, for a fuller description of the course and the teaching method). Of special relevance to this research, is the approach used in these classes to introducing isomorphism. It was introduced very early in the course, using the idea of "naive isomorphism" mentioned in subsection 2.1. This intuitive definition is then used extensively throughout the course. For example, students dis- cover that all cyclic groups of order n are (naively) isomorphic to [Z,,, +n], and can see (by constructing the operation table) that the quotient group Z12/{0,4,8} is isomorphic to Z4. Much later in the course, the formal definition of isomorphism makes its appearance, and its relation to the naive version is discussed.

It is quite likely that the research results are influenced both by the non-traditional teaching method and by the early, intuitive introduction of isomorphism. In fact, it is to be expected that some of the "misconceptions" reported here, will apply with even greater force in a traditional class. However, the influence of the teaching method on students' conceptions is not (except for some isolated remarks) under investigation here, and deserves a separate study.

The main body of the research consists of in-depth, semi-structured interviews with 5 out of the 24 students from Group A. The interviews were taken 7 weeks after the final exam, and lasted 60-90 minutes each.

] 60 URI LERON, ORIT HAZZAN AND RINA ZAZKIS

Students from groups B and C were asked about isomorphism only as part of a more general research on understanding group theory. All interviews were audio taped and transcribed. The interview questions for Group A and for Group C were presented to the students verbally, but were based on a pre-prepared questionnaire (cf. Figures 1, 2 and 3). The questionnaire, as well as the data from the interviews were translated into English (from the original Hebrew) for this report. From the questions to Group B students, only the following one was relevant to this research: "Is $3 isomorphic to Z6?"

We elaborate on some of the ideas behind the interview questionnaire. Questions 1 and 2 (cf. Figure 1) are intended to uncover students' understanding of isomorphism in the context of specific examples, ranging from familiar to novel, from tables to abstract presentation. Some group tables, as well as the groups $3 and Z6, were familiar to the students, but they hadn't explicitly seen these particular tables before. Nor have they seen the groups Z2 × Z3 or U9, except for a general discussion of the groups U,~. The task of determining isomorphic and non-isomorphic pairs among the given groups, whether given by tables or names, was unfamiliar to the students. While they had some experience in classifying certain groups into isomorphism classes (e.g., all cyclic groups of a certain order are isomorphic, all groups of order 4 fall into two isomorphism classes, etc.), they didn't specifically have any experience of the required kind.

Questions 3 and 4 (cf. Figure 2) are intended to probe the students' understanding of what isomorphism and non-isomorphism are in the abstract. Question 3 is formulated so as to allow a gradual probing, from the (sup- posedly) easier "being isomorphic" to the more difficult object isomorphism.

Questions 5 and 6 (cf. Figure 3) deal with students' understanding of the variants and invariants of isomorphism. This topic was treated in the classroom at some depth, but the emphasis had always been more on understanding what types of properties were preserved or not and why, and not so much on compiling lists of invariants.

5, THE RESEARCH RESULTS

Because of the combination of a very preliminary research and a very advanced topic, our priorities were first and foremost to uncover some interesting insights and some directions for further research, rather than a coherent conceptual framework or methodological perfection. There- fore, we have arranged this section around some central phenomena that emerged from the interviews. Each subsection relates to one such phe-


Q u e s t i o n i . Y o u h a v e here 3 ope ra t i on - t ab l e s o f g o u p s , a n d y o u r task is to dec ide

w h i c h are i s o m o r p h i c to e a c h o the r and w h i c h are not.

[ A n s w e r : G and G ° are i s o m o r p h i c to $3, and G ' is i s o m o r p h i c to Z6. ]

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b' c ' a ' b' x' y' d '

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Q u e s t i o n 2. Y o u r t a sk n o w is the s ame as in Q u e s t i o n 1, e x c e p t tha t n o w I w i l I j u s t

de sc r ibe the ~ o u p s to you .

[ A n s w e r : U 9 a n d G are i s o m o r p h i c to 76. ]

(i) $3, the s y m m e t r i c ~ o u p on [1 ,2 ,3} .

(ii) U 9, the g o u p o f e l e m e n t s in Z9 w h i c h are r e l a t ive ly p r i m e to 9, w i th the

o p e r a t i o n o f mu l t i p l i ca t i on rood 9.

(iii) G = {[g t ,g2] I g l in Z 2, g2 in Z3} wi th the ope ra t ion :

[g l , g2 ] * [h l ,h2] = [g l +2 h l , g 2 +3 h2]-

Fig. 1. The interview questionnaire, first part.

nomenon, and contains a description, supporting data from the interviews, and suggested mental processes that may account for the phenomenon. These phenomena demonstrate once more the compound nature of isomorphism, and its complex web of relationships with many other concepts.


Question 3.

(i) Suppose you are given two ~oups. How can you tell whether they are isomorphic?

How can you convince someone that they are indeed isomorphic? How can you prove

it?

[In case the student doesn't give a full answer including functions, the interviewer

9roceeds to ask the following:]

ji) What is isomorphism?

[Now repeat question (i) to see if the answer to (ii) induced any change.]

Question 4. Suppose you are given two ~oups. How can you tell that they are not

isomorphic.

Fig. 2. T he in terv iew quest ionnaire , second part.

5.1. Isomorphism and Order Type

One aspect of groups which is salient in students' work is the order of the various elements in the group. When asked whether two given groups are isomorphic, students commonly start by computing the orders of various elements. For example, in a table of a group which is isomorphic to $3 they get one element of order 1 (the identity), three elements of order 2 and two elements of order 3. This information can be condensed into a sequence like [1,2,2,2,3,3], which we call the order type of the group. Formally, the "order type" of a group is an increasing sequence of numbers which are the orders of the group elements. (The term order type is introduced here for the purpose of this report only. It is not assumed that students are familiar with the term, and all references to students' thinking concern only their implicit knowledge of it, as expressed in their work and discussion).

Order type can serve to show that two groups are not isomorphic. Indeed, when asked whether $3 and Z6 are isomorphic, Eric (Group B) answered that the groups are not isomorphic, since $3 has 3 elements of order 2 while Z6 has only one such element.

But what about the converse? If two groups have the same order types, would students tend to conclude that they are isomorphic? Our interviews show that students indeed often make this (unwarranted) assumption. This


Question 5.

(i) Do you know what does it mean for a property to be "preserved" under (or to be an

invariant of) isomorphism?

'[After making sure they know what this means:]

(ii) Can you give 3 examples of such invariants.

(iii) Can you give examples of properties which are not invariant?

Question 6. Suppose you are given two isomorphic goups A and B, and an additional

=,group G. You are a s k e d to decide which of the following statements are mac and

which are false.

(i) If A has an element of order 3, t~'~en B has an element of order 3.

(ii) If A is commutative then so is B.

(iii) If the elements of A are matrices, then the element of B are matrices too.

(iv) If A is a sub~oup of G then So is B.

(v) If G is a subgroup of A, then G is a subgoup of B.

(vi) [optional, for stronger students] If A is a homomorphic image of G, then B too is

a homomorphlc image of G.

(vii) If the operation in A is function composition, then the operation in B is function

COmp osition-

iii) I fA is infinite, then so is B.

Fig. 3. The interview quest ionnaire , third part.

trend is demonstrated by the following excerpts, taken from the work of Tammy and David (Group A) on question 1:

Tammy: [...] G and G O are isomorphic. I'm not sure what all these things mean ...

Interviewer: Why are they isomorphic? Tammy: The orders of the elements. [Doing some calculations with the

table ...] There are 4 of these [elements of order 2]. Since this is a group, [apparently referring to Lagrange's theorem] the other two elements are of order 3. And when I have the same orders for all the elements, I can decide ... It means it will be the same operation table.

I: You checked that all the orders [of elements] in G and G o are the same, the same number of elements of the same order in the two groups, and you have concluded that they are isomorphic?


Tammy: Yes. I: And G ~, why have you excluded it? Tammy: Since it doesn't have ... since it has only one element of order 2.

David seems to believe that equal orders enable us to construct the required map.

David (having calculated some orders): [...] So they must both be of order 3, therefore these two groups are isomorphic. G and G °.

I: Why? David: Because I can now simply change name, simply the names. If

d corresponds to y0, then I can change y0 to d. And so for the other elements.

Later on, while solving question 2, Tammy concludes that Z2 x Z3 and U9 are isomorphic. Since she also referred in the same discussion to the fact (proved in class) that there exists only one commutative group of order 6, it is conceivable that her conclusion might follow from this theorem rather than the equality of order types. To resolve these two possibilities, the following further probe was conducted by the interviewer to confirm that Tammy indeed believes that the order type is sufficient for isomorphism:

I: Tammy, If you had two commutative groups of order 54, with the same orders of elements, respectively, would you conclude ...

Tammy: The same orders? I: I mean, 15 elements of order 3 in this one and 15 elements of order

3 in that one, and so on ... Tammy: If there are the same ... If two groups of the same order,

and all their elements have the same orders [respectively], then they are isomorphic. [...]

I: Because they are commutative, or what? Tammy: [...] As soon as I have the same orders of elements, then the

operation table must be the same.

In what follows we describe some mental processes which may account for this phenomenon.

(a) Perhaps the deepest reason is that any group property having to do with natural numbers automatically assumes high precedence in students' thinking, apparently because numbers are one type of a mathematical object that they have definitely managed to construct. In fact, they so desperately cling to these isolated properties which speak in familiar language, that they tend to neglect all other properties which bear on the problem and


make some of the most spectacular errors especially in those situations involving numbers. If this explanation is correct, we should expect similar trends in other mathematical situations in which natural numbers play a significant role, such as Lagrange's Theorem. Indeed, such trends are reported in "Students' use and misuse of mathematical theorems: The case of Lagrange's theorem" (Hazzan and Leron, 1994).

(b) Another explanation may be connected with the fact that for small finite groups, which is most of the experience these students have had, the order type does indeed often determine isomorphism. For example, two groups of order 6 which have the same order type must be isomorphic. In fact, Ron (Group A) is making use of a version of this fact when, in building the correspondences in Question 1, he cites the fact that "every non-cyclic group of order 6 is isomorphic to $3".

(c) In addition to the above, this observation may result from the usual confusion students tend to have between a statement and its converse. Thus, the true statement that isomorphism implies equal order types, is often confused with the false statement that equal order types imply isomorphism.

5.2. Properties of Elements vs. Properties of Groups

In this section we consider the question of what is easy (or what comes naturally) for students to do when performing a particular task concerning isomorphism. This may be gleaned from the sequence of the various properties which students choose to invoke in doing the given task: Presumably, the easier the property for the student, the earlier it will be invoked. Inter- estingly, it turns out that there are two different (in fact almost opposite) such sequences, depending on whether the task belongs to one or the other of two kinds: either referring to the property in general by using its formal statement, or actually determining whether this property holds in a particular example.

Indeed, when involved in the task of determining whether two given groups are isomorphic, we have found that students often go through the following sequence of checks: what is the order of the group, what is the identity element (in the operation tables), what are the orders of the various elements, what is the "order type" of the given groups, is the group cyclic, is it commutative?

In contrast, when asked the abstract question "what properties are preserved under isomorphism?", students commonly list the following properties in more or less the following sequence: commutativity, cyclicity, order of the group, orders of elements. This is almost the opposite sequence from the previous one.

166 URI LI~RON, ORIT HAZZAN AND RINA ZAZKIS

We demonstrate this phenomenon with excerpts from the interview with Dan (group A). Most of the other students followed a similar pattern. First, here is Dan's work on Question 1:

Dan: OK. First I look at the ... a general impression. I: hmmm ... Dan: I checked that we have the same number of elements, it is the first

thing that I checked. Now I will look for the identity elements, so that I'll have something to compare [...]

[...] Dan: [...] Now I will look for elements of order 2. I mean, the element

times the same element will give me the identity element. [continues looking for elements of order 2 in the various groups.] Dan: First of all, at this stage I can already see that G I is not isomorphic

to any of the groups. I: Why? Dan: Because if it is isomorphic they should have exactly the same

properties, I mean, um, I mean, it is the same group up to the names of the elements, very generally speaking. And if it were isomorphic it should also have 3 element of order 2, actually the same 3 elements, except that here they would have different names.

[continues calculating and comparing additional orders of elements in the various groups.]

Dan: [...] I forgot something very important, I want to check something. y*y is c and c*y is d and y*c is d. Right? [Looking atthe tables in silent.] The group G is also commutative, by the multiplication table. No ... Definitely it is not commutative. No, I see it is not commutative, x*y is not equal to y*x.

I: Hmmm ... Dan: So G is not commutative. So to eliminate, I can check if this, the

second one [G °] is commutative, b°*y °, no ... I ... I'll look ... G O too is not commutative.

The opposite sequence is exhibited by the following excerpts from Dan's work on Question 5.

I: What properties are preserved under isomorphism? [...] Dan: [...] Suppose I have Group B, U9 with multiplication mod 9, and

I am given a group C, which I am told is isomorphic to B. I: Hmmm ...


Dan: Which ... The question is what properties I can ... What do I know about C without seeing the group itself?. I know it has the same number of elements ...

I: Hmmm ... Dan: I know it has the same number of elements, so I will know

the properties ... I will know that if B is commutative then C must be commutative, if B is cyclic then C must be cyclic, if in B there is one element of order 2 or... then C must also have a single element of order 2 [-d

I: Hmmm ... Dan: or any such combination, that is, if I'll know that in B I have 3

elements and 5 ... then I'll know ... [this is his way of referring to what we call the order type.]

We now suggest mental processes which may account for these ten- dencies. To simplify our considerations, we focus our attention on two properties which lie at the two opposite extremes of this spectrum, namely the order of elements vs. commutativity of the group.

There seem to be two kinds of complexity involved here, which operate in different ways, indeed often in opposite ways: syntactical complexity and computational complexity. Our explanation of the observed phenomena is based on the premise that syntactical complexity takes precedence in students' considerations in general settings, while computational complexity takes precedence in specific situations, where the main work required is calculations. We further claim that in comparing students' relative difficulties in dealing with commutativity and with order type, commutativity turns out to be syntactically simpler but computationally more complicated than order type.

To see that commutativity is syntactically simpler than order type is relatively easy - ju s t look at the definitions. We elaborate on the claim that calculating the order type of a given group is computationally simpler than determining whether it is commutative or not. In establishing commutativity there are two difficulties. First, it involves proving (or disproving) a full-fledged proposition, complete with quantifiers and all, on the validity of which the student must venture a conjecture in advance. Second, establishing commutativity involves a global property which holds for all pairs of elements of the group. In contrast, students usually prefer the local, step- by-step nature of order-type calculations, in which at any given moment, all they have to do is calculate a single power of a single element.


5.3. Students' Constructions and Discussions of Isomorphism

By its very definition, the isomorphism concept deals with the existence of a certain function on groups. Thus in analyzing students' interviews about isomorphism, we can distinguish levels of understanding of the various concepts involved: group, function and existential quantifier. The following statement, taken from one of our interviews, is a typical example of partial understanding, in which the three conceptions are intertwined: "... these two groups are isomorphic because I can find a one-to-one function from each element in G to each element in G TM.

Theoretically, it is possible to investigate separately the acquisition of the group, the function and the existential quantifier conceptions. It is also easy to conceive of situations in which students may exhibit various degrees of understanding for each of the concepts involved. In the present research, however, dealing as it does with the notion of isomorphism, we didn't fully make this separation. Instead, the distinction that turned out to be appropriate for analyzing our data, has been between existential quantifier on the one hand and the compound concept "function from G to G I'' on the other hand.

In the remainder of this subsection, we will present three observations: The first one concerning students' craving for "canonical" procedures, the other two dealing with the concept of existential quantifier and the concept of function-from-G-to-G ~, respectively.

Observation a. Students get stuck in constructing an isomorphism between specific groups, when there is more than one way to proceed.

This is seen below in students' answers to Question 1. Attempting to build a correspondence between G and G O directly, some students are stalled in their efforts because of the variety of possibilities to create such a matching. Having followed the usual sequence of checks, most of the students successfully match the identity of G to the identity of G O and then turn to "order type" calculations. Upon encountering three elements of order 2 and two elements of order 3 in these groups, the students get lost. Some of them can't decide which match to choose and therefore fail to construct any.

David: [..i] It's not going to go like that. Must think of the exact definition.

I: What do you mean "it's not going to go like that"? David: Start changing now ... instead of each letter with circle, to find

the corresponding one in G. I: Why? Why shouldn't it go like that?


David: Because I don't know for example what to put instead of a. Because a instead of a ° is of order 2, but I can put it in three ways. So I don't remember any more if it makes a difference. I think it should make a difference what element of order 2 1 choose.

One more example:

Dan: But to find isomorphism like that [by trying to match] is almost impossible, it is many letters and it starts getting triply complicated. How can I know if it is a to b ° or a to d °, and then if a to d °, then b to a °. This is surely not the right method.

This phenomenon may be seen as following from an alternative conception of existential quantifiers: "there exists a function" is taken to mean "there exists a unique function" or, alternatively (and more plausibly), "there is a canonical, algorithmic, way to construct a function". This seems especially plausible when we consider the students' high school experience, where they are always concerned with computational processes (to find a solution, a GCD, etc.) rather than with existence of abstract objects. Proving existence is a more abstract and mature process than performing some computational procedure.

Another explanation of an affective nature may be added: There is much more feeling of security in performing an algorithmic process, where each step is determined by the previous ones, than in trying to construct something from scratch under conditions of vagueness and uncertainty: Making choices (thereby eliminating others) is often a painful process.

In another context, we may cite Ethane's data (Group C), where he doesn't even "know how to begin" to fill in a 4-by-4 group table, since this can be done "any way I want". Having easily filled the first row and column (there was only one way to do that), he complains:

Ethane: Now ... now the problematic part has come. I don't think that there's here any ... Or maybe I don't know. Some exact rule.

The remaining two observations deal with the concepts of existential quantifier and function-from-G-to-G ~, respectively. We will first describe briefly our findings concerning each of these concepts, and then bring data to demonstrate the findings for both concepts together.

Observation b. We have identified three main types of expressions used by students to refer to the existential quantifier:

(i) "... I can find a function...",


(ii) "... it is possible to find a function ...", (iii) "... there exists a function...".

These three phrases express different degrees of students' ability to treat the mathematics as separate from themselves. The first one has a strong component of "me doing something"; it can be characterized as "first person, procedural language". The second expression retains the procedural character, but employs the neutral third person. The last expression employs the fully detached, declarative style of formal mathematics. We conjecture that these three expressions correspond to three levels of development of the existential quantifier conception. Employing the terminology used by researchers who take a developmental perspective on the mental constructions of mathematical concepts, we may conjecture that these three stages correspond to a path from action or process conception of the existential quantifier to its conception as an object (Douady, 1985; Thompson, 1985; Sfard, 1987, 1991; Dubinsky, 1991).

Observa t ion c. This observation parallels the preceding one. Here too we have identified three types of expressions used by students to refer to the compound concept function-from-G-to-G t. The isomorphism function in these expressions maps

(I) "each element of G to each ele~nent of G ~'', (II) "the elements of G to the elements of G ~'',

(III) "G to G ~''.

As in the previous case, using a developmental terminology, these three stages may correspond to the path from action or process conception of function to its conception as a mathematical object.

In the students' data, observations b and c are usually intertwined, and the data for both will be presented together. In fact, we will show data in which students exhibit various combinations of development for the two concepts involved. Using the numbering of the three expressions in the two observations, we will refer to these combinations as (i-H), (ii-I), etc. All the excerpts shown, except when marked differently, were taken from answers to Question 3 by Group A students.

Ron (ii-III): Two groups are isomorphic if it is possible to find a function which maps from one to the other so that the order of the elements is preserved and the operation is preserved.

Dan (ii-I or ii-III): Two groups are isomorphic if [...] it is possible to find a function between them which connects in a one-to-one way every element to every element, every element in group A to every element in group B.


Tammy (i-?): [to show that two groups are isomorphic I need] to show that I can find a one-to-one and onto homomorphism in one direction or to find two homomorphisms [...]

Saul (ii-I or iii-I): [To show that two groups are isomorphic] in principle it is necessary to find some map between every element here to every element there [...] If such a map exists and the two groups behave similarly then [...]

Dan (i-II): [Two isomorphic groups are] a pair of groups which I can match a full map, a one-to-one map between their elements [...]

Ben (Group C) (i-I): I take an element from one group and find its image in ... exactly as with function in mathematics.

The following are taken from answers to Question 6(vi):

Dan (i-I): I can match to each element in G the same homomorphism to B, where instead of matching to a certain element in A, I will match its isomorphic element in B.

Tammy (i-HI): I can form an isomorphism between A and B, and compose the isomorphism on the homomorphism that was from G to A.

6. CONCLUSION

In conclusion, we would like to illuminate the research reported in this article from two additional perspectives. First, the interplay between the specific and the general and, second, the role of "naive" isomorphism in the learning of Abstract Algebra.

Indeed, the very concept of isomorphism is but a formal expression of many general ideas about similarity and difference, most notably, the idea that two things which are different, may be viewed as similar under an appropriate act of abstraction.

Here are some other examples: First, the phenomenon of students getting stuck in constructing a specif-

ic isomorphism may be seen as a special case of the general phenomenon of students' craving for canonical procedures and their fear of loose or uncer- tain procedures, indeed, procedures with any degree of freedom. This in turn can be seen as a special case of the even more general "fear of freedom" phenomenon: Many people find making choices painful, because choosing one option must also mean giving up many others.

Second, difficulties with formulating definitions related to isomorphism may be mere special cases of difficulties with functions and quantifiers;


more specifically, difficulties with quantifying over functions. This in turn may be related to the even more general developmental issues of processes and objects involved in constructing the mathematical conceptions involved.

Next, we would like to transcend the empirical base of this report and offer some more general thoughts about the learning and teaching of isomorphism; more specifically, the important role played by the notion of naive isomorphism in the course. A more comprehensive treatment of this issue, which forms part of an ongoing research, will be given elsewhere.

First, the idea of introducing a complex topic through a naive version is a special case of the deep and general idea of learning via a chain of succes- sive refinements - an alternative way of dealing with complexity without losing the forest for the trees (which is what happens in the traditional approach of learning by mastering little bits in a linear succession).

Second, the idea that "being isomorphic" is a much simpler idea to formulate and learn than that of the object isomorphism itself. The standard sequence of teaching is from homomorphism to isomorphism (a special kind of homomorphism) to the relation of two groups being isomorphic (i.e., when "there exists an isomorphism from one to the other"). We believe that a didactically-appropriate sequence should almost reverse the order of topics. In fact, while a naive version of the relation between two isomorphic groups was easy for us to invent and easy for the students to understand, we are hard put to find similar "naive" versions for the object isomorphism itself and for homomorphism, perhaps because of their inherently directional (rather than symmetrical) nature, which seems to require a substantial use of functions.

ACKNOWLEDGEMENTS

The authors would like thank Ed Dubinsky and Anna Sfard for their help during the various stages of preparing this paper.


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Research in Collegiate Mathematics Education.

URI LERON and ORIT HAZZAN

Department of Science Education, Technion, Israel Institute of Technology, Haifa, Israel

RINA ZAZKIS

School of Education Simon Fraser University Burnaby, BC, Canda

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