learning binary operations, groups, and subgroups

JMB JOURNAL OF MATHEMATICAL BEHAVIOR, 16 (3), 187-239 ISSN 0364-0213. Copyright 0 1997 Ablex Publishing Corp. All rights of reproduction in any form reserved.

Learning Binary Operations, Groups, and Subgroups

ANNE BROWN

Indiana University South Bend

DAVID J. DEVRIES

Georgia College & State University

ED DUBINSKY

Georgia State University

KAREN THOMAS

University of Wisconsin-Platteville

This paper is one in a series of studies by members of the “Research in Undergraduate Mathemat-

ics Education Community,” or RUMEC, concerning the nature and development of college stu-

dents’ mathematical knowledge. The present paper examines how abstract algebra students might

come to understand binary operations, groups, and subgroups. We give preliminary theoretical

analyses of what it could mean to understand these topics, expressed in terms of the Action-Pro-

cess-object-schema epistemological framework. We describe an instructional treatment designed

to help foster the formation of mental constructions postulated by the theoretical analysis, and dis-

cuss the results of interviews and performance on examinations. These results suggest that our

pedagogical approach was reasonably effective in helping students to develop strong conceptions

of binary operations, groups, and subgroups. Based on the data collected as part of this study, we

propose revised epistemological analyses of these topics, and give some pedagogical suggestions related to these topics.

1. INTRODUCTION

This paper reports on a study of the nature of abstract algebra students’ understanding of binary operations, groups, and subgroups. The study was carried out according to a very specific research methodology that is being developed by the members of the Research in Undergraduate Mathematics Education Community, or RUMEC, for the purpose of studying the learning of collegiate mathematics. Our framework for conducting research has three components: an initial theoretical analysis, an instructional treatment, and empirical data. We will begin by describing our approach briefly; the reader is referred to Asiala, Brown, DeVries, Dubinsky, Matthews and Thomas (1996) for a complete discussion of each of the three components.

Direct all correspondence to; Anne Brown, Department of Mathematics and Computer Science, Indiana Univer-

sity South Bend, South Bend, IN 46634.

187

188 BROWN, DEVRIES, DUBINSKY, THOMAS

The first step is to make an initial theoretical analysis of the epistemology of the concept

of interest. The purpose of the theoretical analysis of the concept is to propose a genetic decomposition which is a model of cognition: that is, a description of specific mental constructions that a learner might make in order to develop her or his understanding of the con-

cept. These constructions are called actions, processes, objects, and schemas, so that the

theory is sometimes called the APOS Theory.

We begin with an explication of the general APOS theoretical perspective, and describe how it was applied in Breidenbach, Dubinsky, Hawks, and Nichols (1992) to characterize

student understandings of the concept of function. An action is any transformation of

(mental or physical) objects to obtain other objects. It is perceived by the individual as being at least somewhat externally directed, as it has the characteristic that at each step, the

next step is triggered by what has come before. For example, an individual is limited to an action conception of function if he or she requires that the function be given by an expression containing complete and explicit instructions on what steps to take in order to evaluate

the function at a point. Someone with a deeper understanding of a concept may well perform actions when appropriate; if the individual is not limited to performing actions, he or she is said to have moved beyond an action conception of the concept.

When an action is repeated, and the individual reflects upon it, it may be interiorized

into aprocess. In contrast to actions, processes are perceived as being internal to, and under the control of, the individual. We say that an individual has aprocess conception of a given concept if the individual can think of the concept as a process. Thus, for example, if one is

capable of thinking of a function as receiving one or more inputs, performing one or more operations on the inputs, and returning the results as outputs without needing to actually

calculate them, then he or she is considered to have a process conception of function. That is, the individual has interiorized the actions which define a function by constructing a mental process that is under her or his own control, rather than simply responding to exter- nal cues. Further indications of a process conception include the ability to transform processes through reversal and the ability to coordinate processes.

If an individual reflects on operations applied to a particular process, becomes aware of

the process as a totality, realizes that transformations can act on it, and is able to actually construct such transformations, then we say the individual has encapsulated the process as a cognitive object, and the individual has an object conception of the concept. One who has

an object conception is also able to de-encapsulate the object back to the process from which it came in order to work with it. In the case of functions, de-encapsulation is often required when one wishes to perform actions or processes on functions, such as adding or

multiplying functions, or forming sets of functions.

In attempting to understand a given mathematical concept, an individual may find it necessary to coordinate several previously constructed concepts. Such a coordination is referred to in our theoretical analysis as an individual’s schema for the concept. A schema

may also be thematized to become an object.

A discussion of the stages through which a schema might develop can be found in Clark, Cordero, Cottrill, Czarnocha, DeVries, St. John, Tolias, and Vidakovic (1997). Until now, very little has been done with regard to developing criteria for deciding whether or not thematization of a schema has occurred. Certain criteria that appear to be useful have emerged in our analyses; we report on these in Section 8.1.

LEARNING BINARY OPERATIONS 189

Returning to our overall framework, we note that the initial analysis is based primarily on the researchers’ understanding of the concept, on their experiences as learners and teachers of the concept, and on other published research. The resulting genetic decomposition for the concept forms the basis for the design of instruction that is intended to help students make the proposed constructions. The pedagogical method that drives this instructional treatment is referred to as the ACE teaching cycle (Activities, Class discussion, and Exercises); the main strategies of this method include having students construct mathematical ideas on the computer using a mathematical programming language, and having them work in cooperative learning groups for problem solving and discussion of the results of the computer activities. Implementing the instruction provides an opportunity for gathering data. There are two ways in which the data are related to the theoretical analysis. First, the theoretical analysis directs the analysis of data by asking the question: did the proposed mental constructions appear to be made by the students? Second, the analysis of the data is used to reconsider the genetic decomposition. According to our method of analysis, student responses are compared to find very fine mathematical points which some students seem to understand (or operations that some can perform) but others cannot. Then we try to find some explanation for the difference in terms of some construction of actions, processes, objects and/or schemas. If we can find an explanation that seems to work, then it is used to revise the genetic decomposition. The entire cycle of theoretical analysis, instruction, data collection, and analysis may be and often is repeated in subsequent studies as new insights are gained which lead to a revised theoretical analysis or to changes in the design of instruction. At the same time, data is also gathered to report on the performance of students on mathematical tasks related to the concept in question. Our analysis of this type of data is expressed in mathematical terms, rather than in terms of what mental constructions might, or might not have been made.

Thus the outcome of this approach is, by nature, two-fold. One result of the research is the deepening of the researcher’s understanding of the epistemology of the concept. The second result is the creation of pedagogical strategies which are better aligned with the way we believe that students come to understand the concept; these improved strategies should thus lead to increased learning by the students. As a consequence, many of the studies conducted by RUMEC contain findings both about epistemological issues and about pedagogical issues, as well as the relations between them.

The reader should note that our framework does not involve trying to establish the truth of the theory, or even to compare the APOS theory with other theories of mathematical knowledge. Our aim is only to develop a theory that provides one possible explanation of the observations that have been made and can thus be used to design instruction that results in improved learning.

The goals of the current study are: to determine to what extent the APOS epistemological framework is useful for understanding the mental constructions made by students leam- ing about binary operations, groups, and subgroups; to increase our understanding of how learning about these topics might take place; to evaluate the extent to which our instructional treatment leads to students successfully performing mathematical tasks that require an understanding of these concepts; and to develop a base of information which sheds light on the epistemology and pedagogy associated with these topics.

The structure of this paper mirrors the framework of the research methodology. That is, we start by describing our initial genetic decompositions for binary operations, groups, and


subgroups. This is followed by a description of the instructional treatment that was designed to help students make the mental constructions proposed in the genetic decompositions. Next, we discuss the instruments of assessment, and the results of those assess- ments. Finally, we provide revised theoretical analyses for binary operations, groups, and subgroups, and conclude by making some suggestions concerning pedagogy for these topics and stating some questions for further study.

In addition to the contents described above, we also include a strand throughout the paper concerning one of the problems that students were asked to solve during the interviews-proving that the center of a group is a subgroup. Initially, this problem was viewed simply as an application of the students’ knowledge about subgroups but, during the data analysis, it became clear that the students’ construction of the center as a subset is also an issue. The students’ work on the construction of the subset and showing it is a subgroup is particularly interesting because it involves enhancing and coordinating virtually all of the other schemas discussed in this paper. Hence, our study of the center tends to unify the various epistemological and mathematical components of the paper. The strand on the center follows the structure described above throughout the paper, with the exception that there was no instructional treatment that was designed to help students make the constructions needed, beyond that which focused on subgroups in general.

2. LITERATURE

In this section, we give a brief overview of the literature regarding the teaching and learning of binary operations, groups and subgroups. We have not attempted a comprehensive review; rather, our goal is to give the reader a sense of what is available.

The literature contains many examples of papers which offer suggestions on the teaching of abstract algebra. We provide a sampling of this type of paper, but we do not review them in depth because they do not, in general, report on research into how students learn the topics in question. Some give an innovative approach to a particular topic or theorem (see, for instance, Johnson, 1983) while others present alternative ways to structure the class as a whole (Barbut, 1987; Freedman, 1983; Hirsch, 1981; Leron and Dubinsky, 1995). A few, such as Maruszewski (1991), Simmonds (1982), and Kraines, Kraines and Smith (1990), outline approaches to integrating computer activities into the class.

There are a number of studies of children’s learning of these and related topics which

appeared during the time of the “New Math’ movement. (Branca, 1974; Lant, 1980; Dienes and Jeeves, 1965; Dienes, 1959). In general, these studies dealt with younger students (in elementary or middle grades) and involved experiments in which students were presented with concrete instantiations of groups and subgroups. The results of these experiments give information about the students’ abilities to deal with the specific examples, but do not address the learning of the underlying conceptual mathematics.

Still other studies exist (Halford, 1982; Leskow and Smock, 1970; Bum, 1996) in which the mathematical understandings of very young children are identified and described as being instances of advanced concepts such as binary operation, group, or subgroup. Such studies frequently attribute to the children a deep understanding of the underlying mathematical concept on the basis of the observed use of a concrete example (see, for example, the comment by Bum, 1996, who asserts that quotient groups are not difficult because the


additive properties of even and odd “provides a description of a quotient group of the integers which is familiar to many school children.“).

Recently, a few studies have been done which investigate the ways in which students learn these concepts in various settings. Hart (1994) studied the proof-writing behavior of mathematics majors in two levels of undergraduate abstract algebra courses and in a beginning graduate course. While the goal of the study was to investigate proof-writing rather than the learning of topics in elementary group theory, the proofs were chosen with the intent of reflecting the level of understanding of elementary group theory possessed by the student. Students were provided with a page of relevant facts from group theory in order to reduce the amount of variation in proof-writing performance due to ability to recall the necessary facts. On the basis of performance on certain proofs, the students were classified according to levels of conceptual understanding of the concepts of elementary group theory. It was found that “the amount of academic experience with abstract algebra does not necessarily reflect the level of understanding” (p. 56).

Dubinsky, Dautermann, Leron, and Zazkis (1994) studied the learning of topics in group theory in the context of a six-week summer workshop for high school teachers. They found that understanding of groups and subgroups may progress somewhat simultaneously. It was observed that “understanding may move from seeing groups and subgroups as primarily sets of discrete elements, to a stage where the operations as well as the group elements are incorporated into the necessary definition” (p. 273).

3. PRELIMINARY THEORETICAL ANALYSES

This paper reports on the second iteration of the research and curriculum development cycle described in the introduction, following up on Dubinsky et al. (1994) and, as such, the preliminary genetic decompositions used here are based on the results of that study. In the following sections, we describe the preliminary genetic decompositions for group and subgroup that directed the instructional design and data collection of this study, and then compare it with the results of the analysis in the previous study.

In keeping with its status as a separate, unifying strand, we will consider the center separately from the other schemas.

3.1. Binary Operation

The essence of our initial theoretical analysis is that a binary operation is a function (of two variables), and hence our genetic decomposition will be very close to a genetic decomposition for the function concept. (See Breidenbach et al., 1992). Following is a description of various conceptions of binary operation in terms of the APOS framework.

l Action. The student can perform a binary operation only when given an explicit formula as in modular arithmetic.

l Process. The student can think in terms of a process for a generic binary operation in which two objects come in, something is done to them, and a new object comes out.

l Object. The student is able to distinguish between different binary operations, and/or consider more than one binary operation defined on the same set. The student is able to de-encapsulate a binary operation in order to work with it as a process.


l Schema. The student has a schema for binary operation that can be invoked and used in dealing with mathematical problem situations.

3.2. Group

The concept of group can be understood as a schema that consists of three schemas: set, binary operation, and axiom. The schemas of set and binary operation have been thematized to form objects and they are coordinated through the axiom schema.

The axiom schema includes the general notion that a binary operation on a set may or may not satisfy a property, which is essentially the process of checking the property. It also includes four specific objects obtained by encapsulating the four processes corresponding to the four group axioms. Checking an axiom consists of coordinating the general notion of satisfying a property with the specific process for the axiom (de-encapsulated from the object) and applying it to a particular binary operation and set. In doing this, the binary operation and the set are de-encapsulated to their processes and the three processes (axiom, binary operation, set) are coordinated to establish that the axiom is satisfied. The four instances of this operation are coordinated into the total process of satisfying the axioms.

The group schema is thematized to form an object to which actions can be applied. Examples of such actions include determining that a particular set and binary operation form a group, checking various properties a group might have, and considering whether two given groups are isomorphic.

An important component of one’s group schema is the ability to consider a generic group as well as particular examples of groups.

3.3. Subgroup

The concept of subgroup can be understood as a coordination of three schemas: group, subset, and function. The function and subset schemas are coordinated to obtain the process of restriction of a function to a subset of its domain. This process is then coordinated with the binary operation in the group schema to obtain the restriction of the binary operation to a subset. Finally, the axiom schema in the group schema is applied to the pair consisting of the subset and the restriction of the binary operation to that subset.

3.4 Center of a Group

In constructing the center of a group, the group schema is maintained as a generic group, producing a set and a function. The defining condition of the center of a group (as the set of all elements of the group which commute with every element of the group) is applied to construct the required subset of the generic group. The subgroup schema, as described above, is applied to the specified subset to establish the center as a subgroup of the group.

3.5 Comparison with Epistemological Analyses in Previous Study

As we have indicated, this study follows up a part of the study reported in Dubinsky et al. (1994), in the sense that we repeated the research cycle for binary operations, groups, and subgroups. Thus, the preliminary analyses given here are very close to the analyses


arrived at in that study. We take as our starting point (which may or may not be the case for the students) the existence of schemas for set, subset and function and focus on the coordinations of these schemas and other schemas that result from those linkings. The analysis in Dubinsky et al. focuses on how these coordinations arise, beginning, for example, with the concept of group and subgroup initially understood almost entirely in terms of the underlying set with the binary operation brought in later.

This was seen particularly in students thinking that a group whose underlying set was a subset of another group would also be a subgroup. For example, many students in that study thought that Z n is a subgroup of Z .

One difference between the two analyses is that in Dubinsky et al. there is suggested the possibility that the group and subgroup concept may develop simultaneously whereas here we begin with the idea that the group concept is more or less present when the development of the subgroup concept begins. It is possible that this difference may be related to the way in which the subject is made available to the students.

One result of Dubinsky et al. that is not considered here is the idea of a generic group as an equivalence class (under isomorphism) of pairs, each consisting of a set and a binary operation.

4. PARTICIPANTS

The participants in this study were undergraduate students at a large midwestern university who had taken or were taking a first course in abstract algebra which was designed for mathematics majors but was not the honors course. The main group of participants con- sisted of 3 1 students who were taking an experimental version of the course during the fall of 1991; a full description of the instructional treatment used in this course is given in the next section. The students were mostly pre-service secondary mathematics teachers, and the instructor and a graduate assistant are two of the authors of this study.

In addition, there were 20 students who had taken an abstract algebra course taught according to standard methods at various times ranging from the same time as the students in the experimental course, back to two semesters earlier. More specifically, 5 of these students took the course in the fall of 1991, 8 in the spring of 1991, and 3 in the fall of 1990. One student had taken it in the spring of 1990 but was taking a second undergraduate course in abstract algebra at the time the data for this study was collected. There were 3 students who were not asked when they took the course.

It is possible that some of the other students in either group were taking courses that touched on concepts in abstract algebra before or at the time the data was collected. Finally, two of the 3 1 students in the experimental course had previously taken the standard course.

5. INSTRUCTIONAL TREATMENTS

As mentioned earlier, the overall structure of the experimental course was based on the ACE teaching cycle; our implementation utilized both computers and cooperative groups. Students were grouped into permanent teams consisting of three or four members each, and the majority of course work was completed in groups. Material was broken into topical sections, each of which ran for approximately one week. One two-hour class session per week


was spent in a computer laboratory, and two one-hour class sessions per week were held in a classroom with no computers. In the computer laboratory, students completed computer activities using the mathematical programming language ISETL. In order to stimulate

reflection, the computer activities usually dealt with concepts that had not been formally studied in class. These concepts were then discussed in the successive class meetings. To encourage further exploration of the concepts, students were assigned homework to be completed outside of class; both computer exercises and traditional exercises were

included in the assignments. The course uses a textbook that was written explicitly to support this pedagogical approach, Learning Abstract Algebra with ISETL, by Dubinsky and

Leron (1994).

There are several references to ISETL in our description of computer activities. In many cases, the meaning should be clear from the context. Some of the syntax is explained in Dubinsky and Leron (1994) and for others, the reader may wish to consult Dautermann

(1992).

We now describe the specific instructional strategies used in the experimental course to help students develop their understandings of each of the topics considered in this study.

5.1. Binary Operations

Of course, the concept of binary operation is implicit in all aspects of any abstract algebra course, including the one on which this study is based, because of its key role in the content. All of the work with groups would tend to solidify a student’s understanding of the binary operation concept. The binary operation concept is revisited when the subject turns from groups to other algebraic structures such as rings and fields. At this point, the student must think about two binary operations defined on the same set but having different properties, as well as consider conditions that relate the two operations.

In the experimental course, all of the explicit references to binary operations are in Chapter 1, which is covered in the first two weeks of the course. All of the situations occurred in the context of learning ISETL on the computer, and were designed with the initial theoretical analysis in mind. Examples of situations in which students encountered binary operations during this period follow.

l The expression (x+y) mod(6) appears (without explanation) in the following selection of code used as an illustration in the textbook:

x:= 4; y := 2;

if(x+y)mod6=Othen

ans := “Additive Inverses!“; end;

ans;

Working with such code is expected to help students construct actions for binary operations other than ordinary arithmetic.

l Several kinds of binary operations appear as ISETL predefined operations. These include: mod, div, min, max, and, or. The students work with these without explicit mention of the idea of binary operations. This work is intended to lead the student to


augment her or his experiential base of binary operations and, ultimately, could strengthen an individual’s object conception of binary operation.

The students construct ISETL funcs (i.e., functions) to implement various binary operations such as modular arithmetic and composition of permutations. For example:

220 := {0..19};

op := func(x,y);

if (x in 220 and y in 220) then return (x+y) mod 20; end;

end;

op(3s); op(9,16); op(4,20);

The goal of such activities is for the student to interiorize binary operation actions to processes.

The students use ISETL’s infix notation (.op) for binary operations. That is, if op is any function of two variables in ISETL, the expression a .op b may be used as an altema- tive to op(a,b). For example, the student could enter

3 .op 5; 9 .op 16; 4 .op 20;

to obtain the same results as the last line of the previous activity. Our expectation is that this will help the student make the connection between ISETL work and manipulations in ordinary mathematical notation.

The students form ordered pairs of two elements in which the first is a set and the second is a binary operation on that set. This is expected to help the student interpret the set and the function as objects and also to begin to construct a binary operation schema.

the students write funcs which accept a set and a binary operation on it and return true or false depending on whether a certain property is satisfied. They are asked to use certain names for these funcs and the names are suggestive, such as: is_closed, is_associative, etc. Our expectation is that such activities will help the students thema- tize their binary operation schema.

Formal treatment of binary operations in the text is very brief, consisting of just a few lines. Aside from these lines being contained in a reading assignment, the course did not emphasize this formal treatment.

5.2. Groups

The instructional strategy for the group concept is for students to construct all of the ingredients of the group schema, including the coordination through the axioms (see Sec- tion 3.2). In the implementation being discussed here, this work also took place during their study of Chapter 1, concurrent with learning about binary operations and ISETL.

In addition, students are expected to construct a mental concept of set as process by constructing set formers and iterating through forall, exists, and choose statements in ISETL.


For example, the set S, of all permutations of { 1,2,3} is constructed as an example of a set former:

S3 := { [a,b,c] : a,b,c in { 1,2,3} I #{a,b,c} = 3 };

Having constructed the set 220 and addition mod 20 in a previous computer activity, students construct the following process that iterates through the set 220 to check whether an

identity exists:

exists e in 220 I (forall g in 220 I (e+g) mod 20 = g);

They construct sets as objects by working with properties of sets, operations on sets, sets as inputs to a func and sets as components of tuples (i.e., finite sequences.)

Chapter 2 of the text begins with some computer activities aimed at making explicit what the group axioms are and how they apply in specific examples. Emphasis in these activities is on the idea that a particular set and binary operation might satisfy some of the axioms, but not the others. Other properties that a group might have, such as commutativity, are also explored.

The discussion in Chapter 2 begins with a formal definition of group as a set together with a binary operation satisfying the four axioms. Thus, the wording of the definition cor- responds to the genetic decomposition. Over a three week period, the students apply the definition to two classes of examples, modular groups Z, and groups of symmetries S,, and begin to prove properties of groups.

Of course, the group concept in general and understanding a group as an object in particular pervade all of the remaining work in the course.

One very specific strategy was developed with the intention of helping students construct the notion of a group as a generic object. As shown below, code is written that allows students to use generic group notation (such as denoting the group by G, and the identity by e) during their computer work. The idea being explored here is whether using this generic notation helps students move beyond thinking just in terms of a single concrete example, and begin to see particular examples as simply specializations of the generic group concept.

This strategy comes into play after they have studied groups for a week or two. At that point, they are given the following computer program which incorporates several computer functions they have written themselves.

name-group := proc(set, operation);

G := set; o := operation;

e := identity(G,o);

i := Ig -> inverse(G,o,g)l;

writeln “Group objects defined: G, o, e, i .“;

end;

To see how this program is used, suppose the student has constructed, on the computer, the set 212 and a func al2 which implements the binary operation of addition mod 12.


Then they run name-group on the pair (212, a12), by writing and executing the following line:

name_group(Z12,al2);

During the remainder of the session (or until the command is run again with another group and operation), the computer will recognize the symbol G as standing for 212, o for addition mod 12. It will have found the identity (0, of course) and assigned it to the variable e. Finally, it will have constructed a func i which selects the inverse of an element of the group.

While the students are asked to apply name_group to other examples and are advised to use it throughout the course in working with specific groups, it is not clear how many of the students in this implementation of the experimental course actually took this advice.

5.3. Subgroups

Towards the end of the study of the group concept, students are asked to think about subsets of a group, restrictions of the binary operation and the group concept. Then, at the end of the chapter on groups there is a very brief discussion of the subgroup concept including the formal definition of subgroup and examples of subsets of the group D, of symmetries of the square.

After approximately 5 weeks of the course, the students begin a chapter in which they spend three weeks studying subgroups. The first week is about the concept of subgroup, the second is about cyclic groups and their subgroups, and the third is about cosets and Lagrange’s theorem.

The main approach to the notion of subgroup is to replace the set with one of its subsets, but to keep the “same” binary operation. The notion of restricting a function (of two variables) to a subset is barely mentioned and the formal definition of subgroup is repeated after the students have worked with the concept for some time.

5.4. Center of a Group

There was no formal instructional treatment of this topic in the experimental course. However, as part of their homework, students were assigned to show that the center of a group is a subgroup. They were also asked to find the center of each of four groups: Z,, S,, and two different subgroups of the group of 2 x 2 invertible matrices with real entries.

6. INSTRUMENTS

This paper is reporting on a portion of a large-scale study of students learning abstract algebra. In the full study, there were a total of five instruments used to gather data: three written examinations in the course and two sets of interviews. All but the second interview were administered only to students in the experimental course. The interviews were conducted by a team consisting of two of the authors of this paper and four research assistants. In the appendix, we list those examination and interview questions that are related to the topics of


binary operations, groups, and subgroups. We also give, for each question, an indication of what we expected to learn from the responses.

The first two instruments were two examinations given during the semester as part of the course. The students took the first exam in their permanently assigned cooperative

groups. They were given unlimited time for this closed book examination and each group turned in one exam for which each member of the group received the same grade. The second exam was also closed book and unlimited time, but the students took it individually and did not communicate with each other during the exam. Each student received two grades for this exam: one was the score on her or his own paper and the other was the average of the scores received by all the members of the student’s group.

The final examination for the experimental course was given in a traditional way in a two hour period with each student taking the exam individually and receiving only one grade. The exam had three parts: (1) definitions; (2) true/false questions; and (3) a set of 11 propositions from which the student selected any two to prove.

Our data also includes two sets of interviews covering topics from abstract algebra. Audio-taped interviews with 24 of the 31 students in the experimental course were conducted during the last week of the Fall 1991 semester. The second set of audio-taped interviews were conducted during the following semester with 17 of the 31 students from that course, together with 20 students who had taken a standard course in abstract algebra. Transcripts of all sessions were produced to complement the record of written work which the student completed during the interview. The transcripts were carefully read and analyzed in order to produce a list of mathematical issues that arose during the interviews. Focusing on these issues, we obtained results about the mental constructions that students appear to have made, as well as a general statement on performance, all of which is reported, along with the results of the examinations, in the next section.

7. RESULTS

We present two kinds of results regarding student work on the instruments described in the previous section and in the appendix. First, we consider the nature of their responses in terms of the mental constructions proposed in the preliminary genetic decompositions given in Section 3. That is, we are using our data to see whether the students appeared to be making these mental constructions and to see what other constructions they might appear to be making.

The second kind of result is simply a summary of the performance of the students in the mathematical sense. That is, did they answer the questions and solve the problems reasonably well, what typical errors did they make, and of what mathematical concepts did they demonstrate their knowledge?

In each case, we select the relevant interview questions and/or test items listed in the appendix. In discussing each type of result, we consider separately the topics of binary operation, group, subgroup, and the center of a group.

When we report the results of work submitted by teams, we recognize that the work may reflect the knowledge of one member, or it may reflect the knowledge of more than one member. Some of the knowledge may be shared, and some may not be. Therefore, when we refer to a schema constructed by a team, we recognize that it is possible that not all indi-


viduals in the team have the same components and linkages in their schemas. In particular, the work submitted by a team may or may not be an application of any single individual’s

schema for the concept.

On the other hand, for the topic of inverse functions, Vidakovic (unpublished) compared how the concept developed in calculus students who studied the concept individually

with how it developed in students who were assigned to work in cooperative learning groups. She concluded that there were no differences in the constructions of this concept

for the two types of students. Thus, we might expect to learn something about the individual constructions for the topics under study in this paper by considering the work done by teams.

7.1. Mental Constructions

The main contribution to what we can say about the students’ mental constructions comes from the interviews. We also include in this section results from any test item that

informed us about the student’s possible mental constructions. We omit the true/false questions on the final which are, at most, performance indicators.

For each of the three main topics, we will start with what we learned from the interviews, and support this, when possible, with further results from the written examinations,

including the team exam.

7.1.1. BINARY OPERATIONS

First Interview, Question 1: The responses of the students to the questions about the binary operation in D4 are related to both their understanding of binary operation in general and their understanding of symmetries of the square. There is a focus on the latter in

another report (see Asiala, Brown, Kleiman and Mathews, in press), but the two cannot be completely separated since a student’s difficulty with understanding a symmetry as an object or process will affect her or his response to these questions about the binary opera-

tion in D,. In the excerpts of student reponses to the interview questions, we have changed the

names of the interviewees to protect their identities. Also, while the interviews were con-

ducted by several different people, we refer to each with the same designation, I. We dis- tinguished five different kinds of responses, three of which contained variations having to do with how the student viewed the product of two symmetries.

1. The lowest level of response, in which the student shows no general understanding of

the notion of binary operation, and even has difficulty performing an actual computation, is exemplified by the response of Eli. He started by explaining how to manipulate an actual square in order to perform a single rotation of 90 degrees and then, the following exchange takes place.

I: Ok. All right. Can you think of another way to do it?

Eli: Um,...(brief pause)... yes, actually, urn, as far as what? As far as product combining? Or.. . ?


I: No, I mean another way to take these elements and multiply two of the

elements to get the outcome? Besides taking the piece of paper?

Eli: Well, actually, urn, well, the product of rotations is urn, just, urn, the

addition of, you know, Rgu plus, urn, combined with R27u would be Ru.

Urn, the . . you can just use the degrees of rotation mod 360.

I: M-hmmm.

Eli: Urn, you can add them that way...

We consider this to indicate an action conception of binary operation in general and

composition of symmetries in particular, because Eli appears able to refer to only very

specific examples, and he describes composition of symmetries only in terms of the

explicit procedures of manipulating an actual square and adding angles mod 360.

2. The first progress that was made was to think in terms of two symmetries to which

something had to be done, but without reference to the result either in general, or in the

context of a particular example. Some students at this level appeared to be using a pro-

cess conception of symmetries as functions and attempted to compose them, as in the

following excerpt.

Arlene: We’d make a little square, R,J is, we’d number all of our corners and

we’d say look we’re going to do R,, and then H. We’ll make a 90 degree

turn and then a horizontal flip and see where we end up, that would be,

that was the easiest way to see exactly what it was going to be .

Other students tried to combine two symmetries by thinking of one as acting on the

other. This is similar to a situation involving composition of functions reported by

Vidakovic (1993) in which, when asked to define the composition of functions, f o g,

one student considered g to be an object to which the processfcould be applied.

Lorrie: We used tables where we wrote out each element and we did what like

R,, does to like RI,,, you know?

In these responses, we do not see a completed process of taking two symmetries and

doing something to produce another symmetry. It is possible that these students lack

strong object conceptions of symmetries of a square, which would make it difficult for

them to perform actions on symmetries.

3. Some students did indicate an interpretation of two symmetries going in and a symme-

try coming out, but only in the context of a specific example, so their conception might

be no stronger than action. It is interesting that all of these students seemed more com-

fortable working with the permutation representation of the symmetry. Although it is

hard to follow her calculations, in the following example Kathy does correctly com-

pute R,u followed by a horizontal flip to be a flip along the diagonal from the right.

Kathy: Ok, let’s try Rgu and . .if I do R9u which is 4,1,2,3 and I do a horizontal

flip, which is equal to 3,2,1,4 you get 4,1,2,3 times 3,2,1,4 . . .


Kathy: This equals 3,2,2,3,3,1,3,1,1,4, which, do we have 2,1,3,4? No. How would we get 2134? Let’s see. 2,1-it’d be 270. Hmmm. If we do 2,l whoops! 2,1,4,4,2 is that what I had? Oh, yes, I did! Which would have been, how did I do that? Would have been Rz7u with an, urn Dr. So, it would have been diagonal from the right? Would that have been it? Yeah.

4. At the next level, we see responses in which the student appears to think of a binary operation as a process that takes two inputs, and produces an output, that is, as a function of two variables. In the following excerpt, Darrel does this when he reports the results of the operation without having to refer explicitly to the details of the calculation.

Darrel: My H and V, okay, so when you multiply those by themselves you get the identity element. Uh, when you multiply them together, you get the Rtsu, which is the cycle 1,4, cycle 2,3.

I: Can you show commutativity? Darrel: Yeah, and then yeah, if you take HV you get Rlsu and in this case it

works out that you get the same thing when you take VH.

In the following excerpt, Ted refers to the binary operation as a process of interaction between two symmetries, a description that appeared only once in our data.

Ted: Just use these two, urn, I just sort of look at it and picture, you know, what happens to each of them as they interact with each other, so this is where you started with and it ends up, let’s see (pause) urn, this here is just a, ok, this here has just been turned 180 degrees so then what I do if I was taking this one with this one is just turn this one 180 degrees.

5. Finally, there were some students who just gave correct answers and indicated a strong knowledge without going into details about the nature of that knowledge during the interview.

Next, we turn to the results from the tests.

Test 1, Question 1: Each team was successful in verifying the axiom of closure, and all but one was successful in verifying associativity, with respect to the new operation. At a minimum, this indicates that each team was able to distinguish between two operations defined on the same set, and that they had constructed a thematized schema for each binary operation.

Moreover, in order to verify these group axioms, the teams had to use a link between their schemas for each of the two binary operations. This indicates that each team had constructed a general binary operation schema that contained, in particular, a process for switching back and forth between two operations defined on the same set. Finally, when a team is able to think about different binary operations on a set and decide to work with one instead of the other, we have an indication that the team is reflecting on this schema and


seeing it as a total entity, that is, they have thematized their general binary operation

schema. We saw indications of differences in the adequacy of the teams’ general binary opera-

tion schemas in their work on verifying the identity and inverse axioms for the new operation. Some teams’ responses indicated confusion regarding which of the two operations should be used in the calculation of formulas for the identity and inverses for the new oper-

ation.

Test 2, Question 3(a): This item is considered both here for binary operations and in the next section for groups. We will focus here on the operation (which is inherited from the original ring), and on the subset specified by the unit condition, which serves as the domain of the restricted binary operation. In the section on groups we will consider the results in relation to the group axioms.

In this context, all responses showed the existence of a binary operation schema including the property of an element being a unit, but there was considerable variation in what an individual student was able to do with this construction. The following categories of responses do not necessarily lie in order of a developmental sequence.

.

.

At the lowest level, a student was able to focus on the operation of “multiplication” in a ring and the idea of a unit and was aware of the properties which this operation could have (in terms of the group axioms), but was unable to check any properties. In some responses, students could check associativity, but not closure. They were aware that the identity and inverses existed, but they did not show that that they were contained in R*. Some students dealt correctly with all axioms except for closure. At the highest level, students solved the problem correctly, in some cases making minor computational errors.

The responses on this question suggest that after constructing a set and a binary operation, the student still must work to coordinate the various ways in which the binary operation can relate to the set or sets on which it is defined. There appears to be a tendency to construct an environment consisting of a set and an operation and to work as if this is the entire, and only, universe in which things are taking place. In a sense, the student “lives” entirely in this one environment while working on a problem. This can be problematic when a shift in environment is required in order to consider the algebraic structure of a subset, as it is in this problem.

Progress is made when the student recognizes that there is more than one environment, game, or set of rules and, under her or his control, the rules to be used can vary from time to time within the solution of a given problem. We can interpret these conscious choices of environment as actions on binary operation schemas.

Test 2, Question 6: Our interest in this question relates to student understanding of binary operations on quotient groups, in particular, whether the student understands that both the set of cosets and the binary operation are important. Ignoring the binary operation, as some students did, could signal a difficulty in understanding binary operations, or groups, or quotient groups.


The responses to this question fell into three mutually exclusive sets:

a. students who made little progress on the problem,

b. students who found a normal subgroup, but did not mention the binary operation, and

C. students who found a normal subgroup, described the operation on the quotient

group, and identified the quotient.

Because none of those who described the binary operation had difficulty calculating and identifying the quotient, it appears that the crux of the problem may consist in constructing the binary operation.

Test 2, Question 7(a): Again the responses to this question showed some students

paying attention to the binary operation and working with it correctly, and some never mentioning it. There were also students who considered the operation, but either used an inappropriate one or used the right one incorrectly.

Test 2, Question 7(b): In checking the isomorphism, the question of the binary oper-

ation arises in a manner similar to the previous questions. Here the point is that some stu-

dents compared both the set and the binary operation in thinking about isomorphism, while others consider only the set. There were also some responses that had little in them that

indicated progress on the problem.

7.1.2. GROUPS

First Interview, Question 7: In the first interview, the lowest level of response to the question of finding an element of order 6 in a commutative group that contains elements of

order 2 and order 3, was for the student to be totally confused and not say anything that can be recognized as progress. One reaction that occurred was that, in struggling to make some sense out of the situation, a student would try to use Lagrange’s theorem inappropriately (for example, suggesting that since 6 divided the order of the group, the group must have an element of order 6, a type of response that has been discussed by Hazzan and Leron, 1994). For the most part, however, students at this level just fumbled around with incoher-

ent phrases.

Some students appeared able to think about this problem only in the context of a specific example, such as Z 1X or S,, indicating that they may have had difficulty with the concept of a generic group. In the latter case it is, of course, necessary to ignore the fact that the group is supposed to be commutative. In some cases, these students were led to think of a counterexample and they could find one in S,. A reminder from the interviewer about commutativity sometimes led to the student reconsidering and then solving the problem with a

good explanation of why ab has order 6.

An important first step in solving the problem is to think of a candidate for an element of order 6. The most natural choice was ab, but some students discovered that ab2 must have order 6. In some cases, prompting was necessary before a candidate could be found.

The task of showing that ab is an element of order 6 can be analyzed in terms of actions, processes, and objects.


Action. The first action is to form various products of powers of the given elements a,h without a clear strategy or organization of the data. The written work of the following student was a collection of symbols such as k, 111, k21 without any apparent coherence. This together with the following comments, suggests to us that for this student, performing the group operation successively on one or more elements is an action which he is not really able to reflect on at this point.

Nathan: Urn. I wasn’t really sure. Well, you know, if you have an element 2 times 3 is 6, right, okay, so you’re gonna have. . . You see how I can make that connection, right? How can I make this connection 2 times 3 is 6, right So you got k times k, you got 111. And if you multiply you get an elemeni that’s in there, but not necessarily in order. Well, no, okay. Okay, e times e is e. So k times k times 1 times 1 times 1, that’s gotta be e. That’s an element of order 5 isn’t it? And this isn’t the same element though. So

I: You’ve got k squared times 1 cubed.

Nathan: Yeah. k squared times 1 cubed equals e. So, urn, if, okay if this were tc equal some element, say f to the 6th, if that equals could that . Does that necessarily have to be the case? Well, we know it’s commutative, SC it’s the same thing as 1 cubed times k squared equals e. I don’t know if that makes any difference. Let’s see. Urn, does that equalf to the 6th. another element? Well, when you multiply numbers.... I don’t know. I’m forgetting using properties of exponents here probably. This basic crap. For the same number you add the exponents. So it’s staring me in the

face and I’m not seeing it.

A more effective approach to the problem might include organizing the actions intc sequences of computations such as a, a2 = e, h, h2, h3 = e and uh, u2b2 = b2, ah3 = a, a2E

= b, ab2, a2b3 = e. The following student indicated such an organization in what he wrote down as well as what he says here as he struggles to obtain the desired conclusion. Initially. his calculations are undirected, but the interviewer brings him back to task by repeating the problem and then he proceeds to the desired conclusion.

Ted:

I: Ted:

I: Ted:

I: Ted:

I: Ted:

I: Ted:

I: Ted:

I’m trying to think, there has to be more elements . . . ah times at squared, because it’s commutative, a squared b cubed which is e

Okay. Because a squared is e and b cubed is e. Okay, so those two are inverses, I’m with you so far. Urn, these two are inverses multiplied by each other. Uh-huh. sure.

a is its own inverse.

Okay. Urn, you multiply, a, u, anything you multiply together . Uh-huh. You’re not going to get anything new. Okay, so you are saying that, your thought right now is what? Is that it does not have to have any more numbers.

I:

Ted:

I:

Ted:

I:

Ted:

I:

Ted:

I:

Ted:

I:

Ted:

I:

Ted:


Okay, so those are the only ones you can definitely say it has to have.

Yeah.

Okay, I’m with you so far. So, the question is, must it have an element

of order six?

Well, we already know the orders of these.

Do we know the orders of. . .

b squared, b squared times b squared is b to the fourth times, which is b

cubed times, 2 b’s cubed times b, which is e times b, and times b

squared, so you multiply that by another b squared . . . Uh-huh.

And that’s going to be b cubed, which is e. So then that order is three.

Okay.

Urn, ah times ah is equal to a squared b squared and we know that a

squared is e so it would be b squared. Times ah . . . is . . .times ab is ab

cubed, which is the same as a, multiplied by ab is a squared b which is b.

Okay.

There’s probably a faster way of doing this but . . urn, multiplied by ah

is ab squared, multiplied by ab . . . (??) . . a squared b cubed which is e

so that’s an order six element.

So your conclusion is what?

It does have to have an element of order six.

Another feature of Ted’s thinking that was shared by some students is that, although he

is clearly forming powers until the first occurrence of e, this is not explicit in anything he

says or writes.

Process. The student thinks about the computation and expresses it, for example, as

the powers cycling back, or as computing powers until there is nothing left over. The fol-

lowing student took the latter view in explaining “why it works.” Note that Mitch does

seem to be paying attention to the importance of not reaching the identity earlier.

Mitch: Obviously it going to be, the reason why it works is because, urn, the

least common multiple, is that right, yeah, urn, is going to have to,

they’re going to have to be, you’re going to have to go that far, I guess,

like, so that they’re both equal to e and you don’t have either g or h left

over. And that’s why it has to exist, or that, yeah, that is has to exist

because otherwise you won’t, g times h, urn, and that won’t be equal to

the identity until, until they’re, until g, urn, is equal to the identity and h, or not g itself, but, urn h times itself is equal to the identity.

Object. The processes become objects of thought. The student thinks about a cycling

back in two steps and b cycling back in three steps and realizes that these two processes

must be coordinated. The question then is how many steps must be taken before the two

processes both cycle back to the identity.


Joelyn: OK so we’ve got a having a certain order meaning a certain number of times you have to multiply it by itself to get back to the identity. And b having a certain order, the number of times you have to multiply it by itself to get back to the other, So if you have ab together and you take powers of ab . . . It’s like has a certain no, a has a certain order and b has a certain order, so you are taking ab times itself. So you are going to get to a point where a is going to cycle back to the identity and, you know, b might have a bigger order so it is going to keep getting a bigger order until it cycles back to the identity, while it’s doing that a is like going through its cycle and . . .

For the general case (orders 2,3 replaced by n,k), some students said that the group has to have an element whose order was the least common multiple of n,k and some of these were able to use thinking as just described to explain why.

Joelyn: so there is like going to be a point when they are going to match up. And that will be the least common multiple.

Test 1, Questions 1,2. The responses to Test 1, Questions 1,2 uniformly indicated that the teams had constructed a schema for group consisting of a set and a binary operation, that they could use in working with problems such as these.

Regarding the thematization of this schema there were the following three kinds of responses.

Although each team had apparently constructed a group schema, it was possible for a team to not progress very far towards thematizing it. In the response to Question 1, there was confusion between the identity under the original operation and a possible identity in the new operation and, when it was necessary to distinguish between two groups, there were some cases in which there seemed to be an inability to argue coher- ently. In some cases, the team appeared to be making progress in thematizing the schema. They were able to compare two situations when the settings were clearly distinct as in Question 2, but when the two binary operations were defined on the same set, as they were in Question 1, some confusion appeared. The work of some teams indicated that they had thematized this schema in that they gave correct answers to Question 1 and all parts of Question 2. They were able to coordinate the use of the schema for two different situations in comparing groups to determine whether they were the same, and also could move easily from one situation to the other.

Test 2, Question 3(a). This item was considered in the section on binary operations. Here, we will focus on the group axioms.

Several responses indicated confusion on the connection between the set and the binary operation as it related to group properties. In various situations students took the position that once a property was present in a situation, then it was present in all aspects of the situation. Thus some students wrote that a subset of a ring is closed because the ring is closed.

207

Other students wrote that the subset has an identity because the given ring is assumed to

have an identity.

Interestingly, although this type of thinking appeared to occur with most students and

with all three group axioms, very few students displayed it on all three. In fact, most of the

students gave an essentially correct argument on at least one of the three. This inconsis-

tency suggests that we should be very careful about drawing conclusions from student

responses on this question.

A second difficulty with interpreting student responses occurred in our reading of stu-

dent work on verifying the inverse and identity axioms. At times, we simply cannot tell,

when a point is omitted from a response, whether the student does not understand the point,

or just did not think it necessary to mention.

Specifically, several students did not check that the inverse of an element in R* not only

existed in R, but was in R*. But this point may have been obvious to an individual, since

the inverse of the inverse of an element is the element, so that some may just have

neglected to mention it explicitly. A similar situation occurred with the identity for R* in

that many students explained why the multiplicative identity of R is in R*, but neglected to

mention explicitly the fact that it is an identity in R*. There is an additional point to mention here, regarding student understanding of a subset

S determined by a defining property P. Two related issues arise in student work. First, a

student needs to be able to conclude that an element is in S whenever it is known to have

property P and, second, the student needs to be able to conclude that the element has prop-

erty P whenever it is known to be in S. In the responses to this question, and in the inter-

views about the center of a group (reported in a later section), we found evidence that

indicated students had more difficulty with the second situation than the first.

7.1.3 SUBGROUPS

We focus in this section on what we learned from student responses to one question in

the second interview, since none of the responses to the test items provided useful informa-

tion about students’ possible mental constructions regarding the concept of subgroup.

Second Interview, Question 5. Students were first asked to give examples of sub-

groups of Z and to make a general statement about their form, which gave them an oppor-

tunity to apply their subgroup schema in a particular situation. For those whose schema had

been constructed and thematized, we can describe their progress in using it. Some students

may not have constructed a very useful subgroup schema and others appeared to have such

a schema but may not have thematized it.

Their responses fell into four broad categories which we discuss in some detail. All but

the first category indicate a thematized subgroup schema, and analyses of the responses

have to do with the examples they gave and the general statement they made.

1. No Useful Thematized Subgroup Schema. The first excerpt is from the inter-

view of a student whose responses indicate that he apparently does not have a useful sub-

group schema.

208

I:

Peter:

I: Peter:

I: Peter:

BROWN, DEVRIES, DUBINSKY, THOMAS

Urn, let’s let capital Z denote the group of integers with addition. Okay, so all positive and negative integers with addition. Can you give an example of a subgroup of that group? (Pause.) I know there’s a little chart that helps me on this one, in the book. What does the chart look like? Oh there’s a, in the current book, that just has all the groups listed . . all the way up to like R. Isn’t R the one with 360. . . or is that D? It’s D, isn’t it. It just tells you what each of them are. Okay. And then . . . (mumbling) . . and the last one just tells you whether or not it’s abelian, it tells you whether or not, yes or no if it’s commutative or . . you know it just gives a bunch of different things.

After some discussion of the group axioms and much prompting, Peter suggests that the set (2, 3) is a subgroup. When the interviewer points out that closure fails, Peter suggest the even integers, but then also suggests the odd integers. No further progress is made.

In the next excerpt, we see a student who has a subgroup schema, but it appears not to be thematized. First, Hillary focuses on the fact that a subgroup is a subset, and considers that Z, might be a subgroup of Z (an error also noted by others who have studied the learning of abstract algebra, such as Dubinsky et al., 1994; Hazzan and Leron, 1994). Next, she begins to talk about the group properties for the subset, but does not connect the operation on the original group Z with the operation on the subset. When the interviewer reminds her about this, she acts as if it is knowledge that she has but did not use.

I: Hillary:

I: Hillary:

I: Hillary:

I: Hillary:

I: Hillary:

I: Hillary:

Ok, let’s move on to the next question about subgroups and ideals in Z . [silently reading the question, which states that 2 is the group of integers under addition and asks the student to give an example of a subgroup of Z.] What is a group of Z? I don’t know. I’m not sure. Z, maybe, all these elements are in Z. [She is referring to what she has written: Z, = (0, 1, 2, 3)] How would you check? What do you need to check? Ammm, those, an inverse, identity, and then closed under the operation. If you add all these elements they are also there. Is it closed then? It is closed under the addition. Inverses? Inverse of zero is zero, inverse of one would be three, the inverse . . . amm, when you take an number and its inverse is equals the identity. So, zero’s inverse would be zero, one’s inverse would be three, two’s inverse is itself two, three inverse is one. So each element has the inverse; then the identity would be zero. What is your operation in ZJ? It’s addition mod four . . . What is the operation in the given group? It’s addition. Are those operations same? No, so 55, is not a subgroup.


To summarize, the fact that she is able to recognize and correct her errors in response to questioning by the interviewer suggests that she does have a subgroup schema, but initially she did not have ready access to it. We interpret this as an indication that she has not thematized her subgroup schema.

The remaining categories of responses are those of students who show clear evidence of having a thematized subgroup schema. They are organized in terms of an APOS analysis of the task of finding the subgroups of Z . At the action level, we see students forming the set of all multiples of a single integer II to produce a subgroup of Z , and possibly repeating this action for certain other values of 12. Then, we see progressively more sophisticated examples of responses of students having constructed this action for certain classes of values of IZ. This culminates in the situation in which a student constructs a process that produces a subgroup of Z for any IZ, the encapsulation of which is the set of all subgroups of Z .

2. nZ for a Single Value of IZ and Then Repeat for Other Values of n. In this excerpt we see an example of a student who can repeat the action for specific examples, but is unable to make any general statement. After some errors, he responds as follows.

Jeff:

I: Jeff:

I: Jeff:

I: Jeff:

I:

Jeff: I:

Jeff? I:

Jeff: I:

Jeff: I:

Jeff:

Integers. So, that would be the other way around. No. OK. So, it would be . OH! I got it, I got it, I got it. What have you got? 22. What does 2 Z represent? The set of all even integers. Could you write that out for me just so we can discuss this .? Like this? Is that what you mean? Yeah. Whatever 22 represents to you. I just want to see that actual set. OK. So, it’s got zero and positive even numbers in it? . . . Oh. Positive and negative. OK. You’re going to throw the negatives in there. I appreciate that. OK. And you’re confident that’s a subgroup? Can you name another subgroup? Uh...3,uh,4Z. OK. Which again . . . Is the same idea, multiples of 4. So, you say it looks like what? Plus or minus 4, Plus or minus 8, . . OK. Do you remember any general statement from class about the subgroups of Z Uh.... . .I remember . . .Hm . . . Oh, I remember, no, yeah. It has something to do with ideals . . .

The interviewer does considerable prompting, but Jeff remains unable to make a general statement. He does not seem to have interiorized his action of forming a subgroup with a single integer to a process. We did see some students progress towards constructing a process in that they moved from seeing that nZ is a subgroup for individual values of R to considering nZ for values of IZ in various classes such as all the even numbers, or all the primes.

210 BROWN, DEVRIES, DUBINSKY. THOMAS

3. nZ For a Single Value of n, Possibly Repeat For Other Values of n, and Then Extend to All Values of n. In this case, we saw evidence of students having constructed a satisfactory process by being able to state that nZ is a subgroup of Z for all values of n. For example, here we see Joelyn moving from the action of a single value of n to the general case with ease.

Joelyn: OK, ammm, the subgroup of Z could have ammm the set of all even integers, that would work, ammm anything in the form nZ where n in Z would work.

We also saw cases in which the student went from a single value to considering classes of values of n, other cases in which the student went from considering several individual values to considering all values, and still other cases in which students started with classes of values and then moved to all values of n.

4. nZ For All Values of n. Finally, some students just displayed their process by going directly to all values.

The interview question did not ask for more than a general statement so not all of the interviewers pushed to see if the student realized that nZ , n an integer (or positive integer) gave all of the subgroups of Z However, some did and here we see Diane thinking about her process and, after some difficulty, coming to a reasonable position. We can see in Diane’s responses, an interesting visualization of her process for specifying the subgroups of Z

I:

Diane:

I:

Diane:

I: Diane:

I:

Diane: I:

Diane:

I:

Diane:

How about back to the subgroups. Can you make a statement about all the possible subgroups of Z ?

OK. All possible subgroups of G-of Z.

Of Z . Uh-huh.

They’re all the groups where elements are equal distance from the number 1. We see a number line now. Like 1, 1 Z obviously. The Z itself. 2 Z ,3 Z ,42. Another subgroup of addition is 0 itself.

O.K...So...

All . . All the possible subgroups I could come up with could be written in what, what way?

xZ, where x is any positive, or actually any integer.

OK.

But there’s probably more.

But there’s probably more subgroups?

No, there couldn’t be because then you’d start getting into repeating. If you had a subgroup with, let’s say that it wasn’t evenly--or even.

There was a misunderstanding on the part of the some of the interviewers about what to ask the students to prove. In some cases, the student was asked to prove that every nZ , is a subgroup of Z and in other cases, the student was asked to prove that every subgroup of


Z is of the form nZ for some positive integer n. A few students were not asked for any proof at all at this point.

All of the students who were asked to prove that nZ is a subgroup of Z proceeded to check the group axioms. Thus they indicated that they had a subgroup schema which they called upon and used in this situation. Some students omitted associativity, but since this was never pursued by the interviewer it may be that the student did not think to state explicitly that it was inherited. More serious was the omission of checking the existence of inverses. The students’ calculations were generally correct although some stumbled with some details.

Among students who were asked to prove that every subgroup of Z is of the form rzZ , most had little or no success, but a few made reasonable progress. However, none was able to give a completely acceptable proof, even with prompting from the interviewer.

While we were not able to learn very much about the mental constructions students might make in proving this result, we can provide two excerpts that give some indication of how students might think about two main issues involved in arriving at the proof. The issues are

1. With an arbitrary subgroup of Z in mind, construct a process for selecting a candidate for a generator of the subgroup.

2. Construct a process for showing that every element of the subgroup is a multiple of the proposed generator.

We consider the work of two students. James was able to deal with the first issue, but not the second, while Diane made progress with the second issue after the interviewer suggested an appropriate choice of generator.

With regard to the first issue, it appears that the process that yields a generator for each arbitrarily chosen subgroup can arise as a sort of reversal of the process of generating a subgroup by multiples. In the following excerpt, James appears to construct an action corresponding to the generation of the subgroup of multiples of 3, but then with the phrases “etc. . . . , On up into infinity” he indicates that it is becoming a process for him. He then thinks about reversing this process to get some II. After a prompt to think about the order- ing, he finds n and indicates how the generation process with n might be carried out.

James: I:

James: I:

James:

I: James:

I: James:

I: James:

I:

So that if n is, uh, let’s say n is three. Mmm-hmmm. And then I know I’m going to generate zero, three, six, nine, etc. Mmm-hmmm. On up into infinity. So that, what, the way I’d show it? Show H equals nZ , show that there exists an n, so, show that there is such an n . . . Okay. . . . possible.

Okay. That generates . . . that would generate, uh, H.

So if you, if you took H and its elements were all in numeric order. Mmm-hmmm. How could you show that?


James: If I took H and they were all in, in numerical order I’d take the, the first positive integer and I would say that that would be iz.

I: Okay. James: And I would say that, okay, you know, n, uh, n one, n two, n three, etc.

generates all of H.

James made no further progress on the proof beyond this point. With regard to the second issue, we consider the work of Diane. In the following

excerpt, she is prompted to consider the smallest positive element x in a subgroup as a generator, and immediately relates this idea to the set xZ . She shows that she understands, at the level of an action on specific numbers, what occurs when x is repeatedly subtracted from a number that is not a multiple of x. Then, she is able to interiorize this action to a pro-

cess and describes what could be done in general.

I:

Diane: I:

Diane: I:

Diane:

I: Diane:

I: Diane:

I: Diane:

I: Diane:

What about working with the subgroups themselves? Instead of going with this contradiction, if you worked with the form of these subgroups themselves could you prove it? Have you ever heard anything about smallest positive elements? You know the smallest element is your x, like the 2 here.

Right. Right. Or the . . Takes us back to what you said before, right? Or, if you add another element into, let’s say the smallest one is 5, if you add another element 12 in there, then 5 would not be your smallest number. Why is that? Because you can take 12 - 10 and you’d end up with 2. And then you could take-ultimately it would end up being the entire group, after you play around with the numbers for a little while. Urn, why? Why? Because! Urn, I know it’s in the form xZ. O.K... I know it’s in the form xZ . If we add another element in there, that’s not in xZ , it’s either going to be less than the smallest element . . . Uh-huh. Or greater than the smallest element. And you can keep subtracting the smallest element from it until you get it down . . O.K.

. . to either it equal to or less than, in which case you’re-you’d be able to generate the whole group or another subgroup.

The difficulty with Diane’s work on this proof is that she does not seem to hold in mind the fact that she is working within the environment of an arbitrarily chosen subgroup of Z . In particular, she applies the “smallest positive element” property to a previously constructed xZ , rather than to the arbitrarily chosen subgroup. Thus, while she is able to construct a process that could be used to complete the proof of the result, her lack of awareness of the environment in which she is working prevents her from seeing the immediate signif-


icance of being able to construct a positive number in the subgroup that is smaller than its designated smallest positive element.

7.1.4. CENTER OF A GROUP

The responses of the students to the interview questions on the center of a group bring together the notions of binary operation, group and subgroup in a context that has the added difficulty of a non-trivial construction of a subset of a group. Our discussion of the group of units of a ring has a similar character, but the data related to that includes only a test question which does not give us nearly as much information as do the interviews.

We present here evidence of two categories of mental constructions that students might, or might not have made, while trying to prove that the center of a group is a subgroup: those associated with the construction of the center as a subset, and those associated with showing that the center is a subgroup. In the first category, we see a student who has difficulty instantiating her group schema to a generic group, as well as students who appeared to have difficulty formulating or encapsulating a process for selecting group elements for the center. In the second category, we see a variety of examples in which students have difficulty coordinating the de-encapsulation of the center with the de-encapsulation of one of the group axioms to provide a proof that the axiom holds for the center.

Second Interview, Question 4. Definition of the Center. Following is an example of

a student who had some difficulty instantiating her group schema to a generic group. To produce a set, Ada found it necessary to list specific elements. It is possible that this also has to do with the adequacy of her set schema for this task. The fact that she first has to list elements, writing

G = Group { GI, .GN}

but then describes (and writes) a reasonable set former suggests that she may be in transition from action to process in her conception of the underlying set in a group.

Ada:

I: Ada:

I: Ada:

I: Ada:

I: Ada:

Ok. Let G be a group. The center of G is defined to be the set of those elements of G which commute with every element of G. Now, could you attempt to write that in some set notation for me? The set of all elements . . We have a lot of paper here, so feel free. Ok. G is a group. Ok. Ok. Let’s call it gl to g, Ok. Ok. It’s the set of g, of little g primes, g whatever, such that . . . a, oh. Thesetg... such that . . , a operation g equals g operation a, where for all a in the group G.

In the next example, we see that Peg appears to have a general schema for subsets that includes the process of applying some criteria to select elements from the original set, but


it is not a very strong conception and it appears to break down when she is asked to apply it to this situation.

I:

Peg:

I:

Peg: I:

Peg: I:

Peg: I:

Peg:

I:

Peg:

And it’s the set of all those elements of G which commute with every element of G. So now using set notation, urn, how would you write that set? If, like, C was the set . and it has to be like, containing . . the elements, or something, or to . . Ok, if you were going to use, like, that set former notation, are you familiar with set former. notation . . where you have, you know, the curly brackets, and you say, it describes what the set is . . Oh, yeah. Ok, that’s the type of thing I’m looking for here. So it’s like C is this, here. M-hm I still don’t . I don’t understand what you’re looking for

Ok, what I want is, like, you know, something like this, but then some statement inside that describes, urn, what the elements are. Just like what we had before, but (can’t understand) like this? (Mum- bles, draws pair of set brackets on the paper) Yeah, like that. I’m not sure what commute means.

With prompting from the interviewer, Peg is able to write set notation for the center:

{XE G:XE G,~E G,xy=yx}

We see here that Peg does appear to understand that the subset is constructed by selecting elements in G, and that this is done with a commutativity condition. Her difficulty seems to lie in an inability to construct a coordinated set of processes in which first x ranges through G, but then, for each fixed x, it is subjected to a test consisting of iterating y throughout G and checking the condition. Instead, she writes down the generic element x, but then mixes the two iteration processes, and states the condition without quantification. In the remaining portion of this question, Peg is unable to work at all with the set she has constructed and it could be that her conception of the set is largely syntactic.

In contrast to the preceding examples, Carla is an example of a student who does appear to be able to work with a generic group and she apparently has a process conception for constructing the center subset. Moreover, in moving from the construction of the center as a subset to showing that it is a subgroup, it is necessary to encapsulate the process used to construct this subset to an object which may or may not have certain properties. Carla apparently is able to do this as indicated by her references to “it” in thinking about the

group axioms as properties of an object.

Carla: OK, center is element g in G such that, ummm I mean is a set of g in G such that g times an element x is the same as x times an element g for all x in G. So you gonna take an element in G and you gonna see if you take


that element and perform an operation with another element in G if that

equals the other element with the operation in G and you’ll get all g that work, that make this statement true.

I: Can you show that the center is a subgroup of G?

Carla: Ok, well you know that it’s gonna contain elements of G. You need to show that it’s closed, and has an identity, and it has inverses.

In showing a few minutes later that the center has an identity, Carla is de-encapsulating this object.

Carla: For an element to be in the set this statement has to be true. So if g is the identity it’ll be true because you have the identity, identity operated on x and x operated on the identity which both of those will give you x.

However, in considering closure, Carla seems to lose the ability to de-encapsulate and is unable to express the meaning of an element being in the center, and the general process she had before seems to be breaking down.

I: What if you take two elements, say a and b, from the center how would you show that a operation b is also in the center?

Carla: So if you have a and b from the center and you want to show that a operation b is in there and that b operation b is in there and that a operation b and take the operation a again and to show that all of those in there and just keep on going. I don’t know how to prove to be in there, that’s my problem.

A possible explanation is that she does not have a fully formed object conception of the center, so that, as it breaks down, she is reduced to trying to perform an action on an object that she is still in the midst of constructing.

Following this exchange, the interviewer begins to prompt her with questions, and she is able to complete the proof, but only after some help in reconstructing the processes that must be used..

I: Carla:

I: Carla:

I:

Carla: I:

Carla: I:

Carla:

I:

What does it mean for a to be in there? It means that a operation x equals to x operation a. Can you write that down? Yeah, for all x in G, so you go through all of G. What does it mean for b to be in there? You just change the a. It’ll be b operation x equals to x operation b. So, what does it mean for ab to be in there? It means a operation b operation x equals x operation a operation 6. So you want to show that, the last statement. Ok, since this (b operation x) is in there than you are taking a times an element in there and x operation a is in there. Let’s stop for a moment. You have here a operation b operation x and you said that b operation x is what?


Carla: x operation b, OK, and then this could be written as a operation x operation b and this is the same as a operation x operation b. OK, that’s it, that make sense

Since she is successful in carrying out the proof with the prompting provided by the interviewer, it might be that one benefit of having an object conception of the center is the knowledge (or impulse) that the interviewer is providing here, that is, thinking to ask one- self what it means for an element to be in the center.

Unlike Carla, who seems to have constructed a rich process for defining the center of a group but has not been able to fully encapsulate it to an object which she can always de- encapsulate, here is an example of a student who has constructed and encapsulated a vaguely conceived process for determining membership in the center. When the process is shown to be inadequate for proving that the center is a subgroup, she drops it and resorts to treating the set notation for the center as a syntactic object.

When the interviewer notices that Lynette has written the set notation for the center as C = {ab = ba : a, b E G}, Lynette is asked to give her interpretation of the center:

I: What’s the idea, what type of elements will I find inside this center? If I take one out, what can you tell me about it?

Lynette: If you take two elements together no matter what operation you perform on them, no matter what order you perform on them it’s not going to make a difference, it will be the same element.

Since she does not differentiate the roles of the two elements, the possibility that she has not constructed a correct process for determining membership in the center is raised. The discussion with the interviewer about her notation for and interpretation of the center does not clarify the matter, and her difficulty continues when she considers closure. She has taken a and b in the center, and is trying to establish that a operation b is in the center. Without a process for set membership to apply to the element ah, she seems, in this excerpt, to try to match it with the syntax of her set notation for the center.

Lynette: Umm, basically it’s going to be closed, because it’s going to be in the same form, so if you know that a and b are in G then you know that a operation b is also going to be in G because they’re the same form. umm. so that proves closure, there.

The interviewer again probes her understanding of the meaning of an element being in

the center:

I: But I want to know if this element [a operation b] you picked is in the center. Now, what do I have to check to see if it’s in the center?

Lynette: I guess you, uh, I don’t know if I’m understanding the question but I guess you, you have to check that a operation b is equal to b operation a.

While she is eventually able, with prompting from the interviewer, to prove that closure holds, the difficulty she is having with placing elements in the center returns when she tries


to show the inverse axiom holds. In particular, when she runs into trouble in trying to verify

that the inverse of an element in the center is again in the center, she reverts once again to a syntactic rule, as opposed to a process, for determining that something is in the center:

Lynette: . . . I guess the reason why I started out using these letters here is because when we talked about a operation b, to me, that signifies that

it’s in the center.

Second Interview, Question 4-Showing the Center is A Subgroup. The first issue that arises in showing that the subset that has been constructed is a subgroup has to do with the binary operation. In the following excerpt, Tammy may not be able to think about a generic operation. She thinks of inverse as reciprocal as if the operation were multiplication and writes expressions like l/c for the inverse of c. When the interviewer points out that the operation may not be ordinary multiplication, Tammy accepts this, but then refers to it as “plus.”

I:

Tammy:

I:

Tammy:

I:

Tammy:

I:

Tammy:

I:

Tammy:

I:

Tammy: (Mumbling) . . . this gives me c+c.

Okay. If we pick some particular a in there, then, urn, we know a has an inverse in the group, right, and what we want to show is that it’s actually in the center, so this property holds for a inverse. So, how could you

attack that?

(Long pause.) Just. . if I had a, a is in the center, and a is in G.

Mmm-hmmm.

So if, if one over a is in G . . . Mmm-hmmm.

. . . then it’s still going to . . . commute.

Okay. Can you write something down to show that?

Yes.

Okay, why would that work necessarily? (Pause.) I mean you know that’s true for a, but this may not be ordinary multiplication, so . . . we need to show for a inverse that it still works. (Pause.) Okay, can I suggest a trick? Okay, you start with this statement, so ac=cu.

Okay.

And then, uh, if you multiply in strategic places by one over a, uh, it should work out.

Some students, such as Jeff, did appear to have some conception of a generic binary operation, but did not seem to be able to restrict it to the subset. Here, he appears to focus only on g and h being elements of G instead of considering their roles as elements of a specific subset.

Jeff: I can attempt to. OK. Well, closure’s fairly, I mean, I don’t know if it’s like an actual, real, hard, definite proof. But, if you use this set notation, here, g little h equal little h little g, where h is also in G, gh will be in G because g is in G and h is in G so, therefore, the product will be in G,


because the original group is closed. So, therefore, this has to be in the original group, as well. Right?

Another way to see students’ difficulties with restricting the binary operation to the subset is to look at responses related to associativity where this restriction is the issue. In the following excerpt, we see that Ada is apparently aware of this by name, but she is unable to give it any meaning in her discussion.

Ada:

I: Ada:

I:

Ada:

I: Ada:

I:

. Well first of all I’d try to say associative is inherited from the group. Because . . it’s a pain to prove.

Ok. Because it’s a pain to prove, huh. It’s very long. And normally it is sometimes, assumed. The ones I always worried about were closed, inverses, and identity. Ok. Why would it make any sense to be able to assume that it’s associative? That’s what we always did in class. It was so long. It inherited from the group. What does it mean to you for it to be inherited from the group?

It’s like brought down. If it . If it’s associative in the group to be able to be in the subgroup, it will also be associative. It doesn’t make much sense, but I’m trying to . . uhhh. ‘Cause if it’s not associative, then it’s not a group. So you’re saying that . . the idea it’s inherited from the original group makes no sense to you?

And Ada has no response to the last question of the interviewer. Once the subset has been equipped with the restriction of the original binary operation,

there is the question of verifying the group axioms. This requires first that the student is aware that the group axioms must be checked for this set and operation and remembers what these axioms are. Awareness here indicates that the center and the axioms are objects for the individual.

There were students who did not seem to understand anything about subgroups, even with prompting. One example was Peg, whose difficulty constructing the center as a subset was discussed earlier. The following excerpt took place just after she wrote down the set notation for the center.

I: Ok, yeah, that would do it. Ok, and so that’s our set notation. Urn, now, given that set, can you show that that’s a subgroup of the larger group?

Peg: I don’t know how you . . . I don’t know how to do that, either. I: Ok. For, for a subset of a group to be a subgroup, remember the subset

has to be a group, itself. So we have this, and it’s a subset of G, and so what we need to show is that this is a group, with the same operation as G.

Peg: Oh.. I: So it needs to be closed, it needs to have the identity in it, it needs to

have, urn, inverses for every element in there.


Peg: I remember doing that . . . but I don’t remember the notation, or any-

thing.

Some students needed considerable prompting and it is hard to tell how much, if any-

thing they understood and how much was just agreeing with the interviewer. In one case,

there was a long and laborious effort of the interviewer to get the student to think of the axi-

oms, but no trace of understanding in his responses. But then, at some point, he simply

wrote down the four group axioms, apparently without any prompting.

Another student, Kristen, responded without difficulty in stating that axioms had to be

verified, but left one out, associativity. Her use of the word “things” in referring to the axi-

oms suggests that she may be interpreting them as objects.

I: How would you show that the center of G is a subgroup of G?

Kristen: There are three things I need to show: identity, inverse and . . ammm . . .

closed under the operation.

Other students omitted closure and some omitted both closure and associativity. For

each of the axioms, the students had to de-encapsulate the axiom object and the center

object to obtain the underlying processes which were then coordinated, perhaps with some

manipulations of group elements and the operation. We have discussed associativity

above, and here we see Eli struggling with the de-encapsulation and coordination for the

axiom of closure. Just prior to the excerpt quoted below, he states that in order to show the

center is closed, he has “to show that ab will be commutative with everything.” But then,

he carries out a calculation that shows only that abG = Gab. (Equality here is understood,

apparently, as sets .)

I: So, if a is in the center what do you know about a?

Eli: That it commutes with everything else, and b is in the center. . . (pause-writing : a in Center, b in Center, aG = Ga, bG = Gb; show

abG = Gab.) Oh, OK. Since both sides (aG = Ga, bG = Gb) are equal

let me try this. Let’s see how I do this. . . ammm . . let me try to see if

this works. Let me see. What property can I use here? Since it is assoc

. . . Ok, since b is commutative we can change that, so we’ll have ab and

then G operation G is just G anyway, so it will be abG on that side would

be the same over here only opposite way, so that’s commutative, equals

Gab. As long as a and b are there then ab will be an element of the center

regardless, so it will be closed.

In the following excerpt, we see that Mel appears to de-encapsulate his center object and

states its defining condition as a process. In what follows, however, he is unable to see how

to correctly apply the process to the inverse of an element in the center.

I: But, say g is in the center, then is its inverse also in the center?

Mel: Oh . . . hmmm . . . I don’t know. Not sure.

220

I:

Mel: I:

Mel:

I: Mel:

I: Mel:

I: Mel:

I:

Mel:

I: Mel:

BROWN, DEVRIES, DUBINSKY, THOMAS

Okay, if you’re given, say any element of the group and you wanted to check whether it was in the center or not, how would you go about doing

that? Any element of the group?

Yeah. Urn, I would test it with every other element of the group and see if it would have commuted. Okay. So let’s try that with f. (Pause.) You might . Oh.

. like to writefas g-l. Sometimes the notation is suggestive.

Okay, okay. Urn, so okay, so that would be g-‘. Mmm-hmmm. Oh, I see. So g, that little g . equals e, I think we already said that. Also there exists, also . . . there exists . . . the inverse g and g equals e . . . so, urn, g g-’ is equal to the inverse of g, which is an element of my

G center. (has written: g o g-’ = g-’ o g E G center) Right. Now let’s check it with any, say, you know, not necessarily with g but let’s check g-’ against some other h in the group. Well we know g . . well now I know g-’ is an . . element of, uh, G cen-

ter. Is it? I don’t think we’ve quite shown that yet. We haven’t? Well . .

From his comments and his written work, we can see that Mel has interpreted the problem of showing g-’ is in the center as consisting of showing that 8-l commutes with g.

Jeff makes more progress, in that he can state clearly what he needs to show, but he is not able to succeed with the calculations necessary to complete the proof.

I:

Jeff:

I: Jeff:

I: Jeff:

I: Jeff:

I:

Jeff: I:

Jeff:

Urn . . . Can you show me-wait, we’ve got inverses left. How about inverses? OK. Well, inverses has to do with the identity. Like, ggel will equal the identity. OK. So Can you assume Can you assume that it has inverses if it has an identity? Or is it the other way around? I can’t remember how that works. Well, does g-’ exist, at all? Oh, yeah, it exists in the group. OK. So, we know that it exists. What you’re wanting to show is what? Is that it commutes. That it’s, right, that it’s in the center. So, it exists in the set. Right. I want to show that the inverse commutes. So, how do I do that? Can you do it the same idea with g and h, just using g-l? Talk to me. OK. If you have gg-‘, element j again, where it is something different,

that has to equal jgg-’ inverse.

I: Jeff:

I: Jeff:

I: Jeff:

I:

Jeff:

I:


OK. Now, that’s not actually what I want to show. No? Oh, you want to show that it’s in the group.

I just want to show that g-’ commutes. so, g-1 . . . So, you want to show s-‘j = jg-‘? OK. OK. That’s what I have to accomplish, somehow . . . You want to take one side and work to the other. Right. (pause) You can’t-you’re trying to show that this works, so you can’t just say you’re going to switch these 2 around? Right, you’ve got to prove it.

Similar difficulties arose in connection with the identity although this was somewhat easier for the students, as we saw with Carla above.

We conclude this discussion with an example of a student who seems to have overcome all of the difficulties connected with showing that the center is a subgroup. Not only is the mathematics correct but Joelyn’s discussion suggests that all of the appropriate mental constructions have been made.

I: Ok, can you show that the center of a group is a subgroup? Joelyn: OK, to be a subgroup has to be closed. So, if a and b are in the center

then ah is in the center. If a is in the center that means that ax equal to xu for all X, b is in the center means that bx equals xb for all X, so we have to show that (ab)x equals x(ub) for all x. OK, ammm we will take the fact that bx equals xb for all X, and we multiply, we operate a on the left hand side so we get ubx is uxb, and then knowing that a is in the center we know that ax is xu for any x, so we can switch this around. So we get ubx is xub for all x. So, that proves that the center is closed. Ammm, associativity follows from the fact that G is a group and the center is just contained of elements in G so they have to be associative. And then for the identity, well you know that e times any element x is the same as x times e, just because it’s the identity. Ammm, inverses, so if a is in the center u inverse has to be in the center. OK, a being in the center means ax is xu for all x and we wonna show that u-lx = xu-‘. So, a is in the group so a does have an inverse, so we operate both sides of this by a-‘. So you get u-‘ux is uM1xu. This two cancel and it’s just x = u-‘XLZ and then we operate on the right by u-‘u-~ au -I

again so you get x&l = u-lx, and then would cancel. So we get what we wanted. So if a is in the center

then its inverse is also. So that means that the center is a subgroup.

7.2. Performance

7.2.1. BINARY OPERATIONS

First interview, Question 1. Following are the numbers of students who gave responses of the five kinds described in the previous section.


1. A total of 4 students did not indicate any general understanding of binary operation in

D,. 2. A total of 6 students indicated they were able to think in terms of two symmetries to

which something had to be done, but without reference to the result. 3. A total of 4 students did indicate an interpretation of two symmetries going in and a

symmetry coming out, but only in the context of a specific example. 4. A total of 8 students responded in terms of a general function of two variables and con-

nected it with specific examples. 5. The remaining 2 students gave correct answers and seemed to have strong knowledge,

but the interview did not probe its nature.

Test 1, Question 1. Of the seven groups, three got full marks. All seven groups dealt correctly with using two different binary operations on the same set with respect to closure and six did so with respect to associativity.

Five groups understood that there could be different identities relative to these two operations and found the new identity. A sixth group found the new identity but did not use it when considering the inverse axiom.

Three groups found the formula for inverses with respect to the new operation and a fourth came close. Two groups did not keep the two operations distinct when working with inverses.

The seventh group seemed unclear about identities and inverses with respect to the new

operation. This problem was also part of a study of proof-writing performance reported by Hart

(1994). Hart rated it as the second most difficult of the six proofs of standard results from elementary group theory that were part of this study. It is difficult to make a meaningful comparison of our more positive results with his because the situations were quite different. On the one hand, the students in Hart’s study worked on the problem individually, rather than in groups, and had only 15 minutes to complete it. On the other hand, Hart studied a set of 29 students, only 10 of whom were taking an elementary undergraduate course in abstract algebra, as were the students in the present study. The other 19 students in Hart’s study were taking either an advanced undergraduate or a graduate course in abstract algebra.

Test 2, Question 1. The average score of the individuals was 77% and the errors did not appear to have anything to do with coordinating the two binary operations in the ring.

Test 2, Question 3(a). All 3 1 students worked correctly with the operation on R*, distinguishing it from addition.

Seven students got full marks and another 5 were very close with only minor errors. Fourteen others appeared to understand about the identity and inverse in R*, but were confused about closure; and three of these did not seem to be fully clear about the need for the identity and inverse to be in R* as opposed to just being in R. The five remaining students did not show much more understanding than knowing the axioms for a group.

Test 2, Question 6. Twenty six students found a normal subgroup of SJ, and two others found subgroups that are not normal. All 28 of the students computed the cosets cor-


rectly. Two of these students chose the subgroup to be the trivial subgroup (this had not

been ruled out in the question) and the product of two cosets was not an issue. Of the other 26, 12 computed the product of cosets correctly. One other referred to the operation but did not compute it explicitly and 3 identified the quotient group correctly, but did not show any

computations with cosets. Seven ignored the operation completely and the remaining 3 seemed to be confused about the operation.

The issue appeared to be whether or not they paid attention to the binary operation. About half did and all of these performed the computations correctly.

Test 2, Question 7(a). Twenty eight students worked correctly with the two binary

operations in the set of cosets. Of the other three, one recovered and used the operations correctly in the second part of the question, one used an incorrect operation and one never

mentioned any operations.

Test 2, Question 7(b). Fourteen students mentioned the operation explicitly in con-

sidering the isomorphism and fifteen just wrote down tables and compared them. The remaining two seemed very confused.

7.2.2. GROUPS

First Interview, Question 7. Of the 24 students who were interviewed, 11 had no difficulty in solving this problem correctly. Two more had some difficulty but eventually got it on their own. Three students did not know the definition of order of an element, confusing it with the order of the group or the cardinality of some set other than the cyclic sub-

group the element generates. The interviewer eventually provided the definition for these three students and two of them were able to go on to solve the problem.

Five students needed some serious prompting and four of these eventually solved the problem with some degree of autonomy. In two cases, the difficulty had to do with being able to think about a generic group and not just specific examples.

The remaining three students were not able to do very much that was productive on this problem.

On the general question of two elements of order y1 and k in an abelian group, 8 students saw that the order of the product is the least common multiple of n and k and at least two of them were able to explain why (not all of the interviewers pressed for an explanation).

Another six students seemed close to considering the idea of the least common multiple, but did not get it. Four students thought it would be the product of the orders. The remaining six students did not have any idea of what the result would be in general.

Test 1, Question 1. The overall average score for the seven groups on this question

was 89%.

Test 1, Question 2. Six of the seven groups got full marks on all parts of this question. The seventh group got full marks on the first part but less than half scores on the other two.

Test 1, Question 3. All seven groups got full marks on this question.


Test 2, Question 3(a). The average score of the 3 1 individuals who took this test wa 68%.

There were five issues on part (a) of this question: knowing the four group axioms an’ applying each of them to the set of units in a ring. A total of 29 students knew the axiom and 28 realized that associativity is inherited. Nine students checked closure correctly anI another three were close. Only five students were able to verify the identity axiom COI rectly, but another 16 were close and may have only neglected to write down somethin they knew. Ten students correctly verified the axiom of inverses, and another 6 were clos and, again, may have known more than they wrote.

7.2.3. SUBGROUPS

Second Interview, Question 5. We use the convention n(a,b) where a+b = n to indi cate n students, a from one of the standard courses and b from the experimental tours described in this study. Recall that, in this case, we are reporting on the reponses o 37(20,17) students.

On the question of finding the subgroups of Z, 2(2,0) students were not able to find any l(l,O) student was able to find only a single example; 10(7,3) students were able to fin several examples and/or classes of examples, but were not able to find all subgroups; an 24( 10,14) were able to state that every nZ is a subgroup of Z.

Of the latter group of 24, 12(4,8) students made the general statement without prompt ing and without preliminaries.

A total of 4( 1,3) students offered the observation that this gave all of the subgroups. L total of 17( 12,5) students gave examples of Z, for particular values of n as subgroups o Z (in some cases these were given in addition to correct examples). Of these, 2(1,1 changed their minds after some prompting.

Turning to the proofs, we note that a total of 11(5,6) students were asked to prove tha nZ is a subgroup of z, 22( 12,10) were asked to prove that every subgroup of Z is equal tc some nZ, and the remaining 4(3,1) were not asked about either.

All of the 11 students who were asked to prove that nZ is a subgroup of Z went directl: to checking the group axioms. In general, they succeeded except l(l,O) student who wa unable prove closure. The others all proved what they stated although 5(3,2) did not men tion associativity and l( 1,O) omitted associativity and closure.

Of the 22 students who were asked to prove the characterization of subgroups of Z 3( 1,2) of them made reasonable progress and the rest had little in the way of success. Non of the 3 who made reasonable progress was among those who thought that some Z, were subgroups of Z. On the other hand, of those who did not get anywhere with the proof 11(8,3) students were among those who had thought that some Z, were subgroups of 2 and did not change their minds. There was no discussion in any interview about the contra diction between the statement that Z, is a subgroup of Z and that every subgroup wa, some nZ.

Test 1, Questions 3, 4(b), 5. The scores on these problems were very high. On the first and third, all seven teams received perfect scores, and on the second, four team; received perfect scores. The other three teams received more than half credit for Question

4(b).


In Question 3, six teams chose either S3 or S4 as an example of a non-abelian group, and

one team also considered S,, in general. Only the latter proved that their chosen group was not commutative. All but one team chose a two element subgroup and they checked that it was commutative. One team chose a four element subgroup and showed only that two particular elements commuted. They did not lose any points because proofs were not specifically requested on this problem.

On Question 4(b), three teams missed one or two of the three subgroups with four ele-

ments.

On Question 5, four teams merely listed the elements of the subgroup and the other three specified the subgroup as those elements for which one of the objects to be permuted is fixed. Two of these teams explained it in words and one used set former notation with the condition x(4) = 4.

True/False Questions on the Final. Of the 31 students, 28 got the first one correct,

28 got the second one correct, 22 got the third one correct and 24 got the fourth one correct. No student had more than two marked incorrect.

Proofs on the Final. A total of seven students selected one of the proofs related to subgroups and an eighth student selected two. Their choices and scores were as follows.

3d. No students selected this proof.

3e. Three students selected this proof and the scores in percentages were: 100, 76, 32.

3f. Three students selected this proof and the scores in percentages were: 96,92,40.

3g. Two students selected this proof and the scores in percentages were: 60, 56.

3i. One student selected this proof and the score in percentage was: 96.

The overall average score was 71%

On this problem, students received 20% of their score for stating the theorem correctly and the remaining 80% for the proof. The percentage correctness of the statements was 89% If this is removed from the above calculations, then the overall average score for proof alone was 67%.

7.2.4. CENTER OF A GROUP

Second Interview, Question 4. A total of 16(8,8) students constructed the center as a subset without difficulty. An additional 9(6,3) students succeeded in this construction with some difficulty. The remaining 12(6,6) students made little progress in dealing with this

task.

A total of 26( 11,15) of the students seemed to understand the idea of applying the group axioms to the center. An additional 2( 1,l) appeared to understand this after some prompting. Of these 28 students, 2(1,1) mentioned explicitly that a subgroup is a subset with the same operation as the group, and an additional 5(3,2) students used the term “subset.”

Regarding the axioms, 14( 1,13) students considered all four, 7(5,2) omitted associativity, and 10(8,2) students left out other axioms and/or needed prompting. The remaining 6(6,0) did not appear to know any of the axioms.


A total of 2( 1,l) students established without difficulty that the center satisfies all four

axioms. In addition, 9(3,6) gave arguments for two or three of the axioms with little or no

prompting. The number of students who managed to give one or more proofs, but only with

a great deal of prompting was 19(12,7). Of the remaining students, 2(0,2) had only the

slightest idea of how one might verify that the axioms hold, and 5(4,1) seemed to have no

idea at all.

8. DISCUSSION

8.1. Theoretical Analyses Reconsidered

Before considering the specific genetic decompositions of the main topics we are inves-

tigating in this study, we make some comments about the thematization of a schema. We

are still very much at the beginning of our understanding of the construction of a schema,

and the results of this study are helpful in clarifying our ideas. Following these comments,

we reconsider our genetic decompositions of binary operation, group and subgroup.

Finally we analyze the construction of the concept of the center of a group.

8.1.1. THEMATIZING A SCHEMA

We consider a schema to be thematized if the individual can think of it as a total entity

and perform actions on it. These actions can be as simple as the act of distinguishing two

or more instantiations of the schema. This study provided a number of examples in which

students were either successful or unsuccessful in thematizing a schema.

A first example of thematization was mentioned in Section 7.1.1 on mental construc-

tions for binary operations, in connection with responses to Test 1, Question 1. When a stu-

dent, or group of students is able to coordinate the use of two different binary operations on

the same set, and make a choice about which one to use in a given situation, then they may

be interpreting their binary operation schema as an object.

Another example occurred in Section 7.1.2 on mental constructions for groups, in con-

nection with responses to both Questions 1 and 2 of Test 1. When a student, or a group of

students is able to think about a generic group situation and distinguish among several

instantiations-especially when these instantiations have something in common, such as

the underlying set-and coordinate the application of these instantiations in order to com-

pare them, then they may be interpreting their group schema as an object.

Examples arose in Section 7.1.3 on mental constructions for subgroups, in connection

with responses on the Second interview, Question 5. Successful performance on this ques-

tion requires not only the application of the subgroup schema more than once, but to think

about all possible applications of the schema in order to reason about the set of all sub-

groups of Z .

We also saw in this interview question, an example of a student, Hillary, who may have

constructed a schema for subgroup but did not appear to have ready access to it and there-

fore the schema may not, for her, be an object of thought.


8.1.2. BINARY OPERATION

The results of this paper generally tend to support the genetic decomposition given in Section 3.1. Here is our refinement of that analysis based on the results in this report.

Action. The student can perform a binary operation only when given an explicit formula as in modular arithmetic.

In order to make the transition to thinking of a generic binary operation as a process, the student’s function schema must be reconstructed so that he or she can reflect on the domain set and the function’s operation separately, and in coordination. This requires that both the set concept and the function concept are understood by the student as objects. These two objects are coordinated to form a pair that, in the context of a function of two variables, can become a binary operation schema.

Process. The student can think in terms of a process for a generic binary operation in which two objects come in, something is done to them, and a new object comes out. At the same time, the student is aware that an underlying domain set is specified for this process, and that the domain set chosen may vary (particularly in the special case of restricting the binary operation to a subset). The student must be clear that both the function and its domain must be considered in a binary operation and that focusing on just one of them is not adequate.

Object. The student can think about various properties that a binary operation might have. He or she is able to consider more than one binary operation on a given set. The student is able to de-encapsulate a binary operation in order to work with the process of its function together with its set (for example, to verify properties).

Schema. As the binary operation object is coordinated with other concepts, a schema is formed. The student’s schema for binary operation can be invoked and used in dealing with mathematical problem situations. In invoking the schema, the student sets up an environment in which calculations using only the specified binary operation are made. The student must be able to operate entirely within this environment but, when necessary, consciously switch to a different environment (binary operation) in which to calculate. The latter action can take place only when the student has a thematized binary operation schema.

8.1.3 GROUP

Our preliminary analysis of the construction of the group concept as a coordination of schemas for set, binary operation and axiom system, with the axiom schema used to coordinate the first two, seems to be supported by the results of this study. We are able to add some detail about specific issues that can arise in the development of the construction of this coordination.

1. At the early stages of coordinating these schemas, there is a tendency to assume that a feature that appears in one part of an environment applies throughout the entire envi-


ronment. Thus, for example, a student might express the opinion that a given subset of a group is closed because closure is a property that a group has. Or, if a new operation on a given group is considered, it will be assumed that the identity of the original group is still the identity.

2. Another observation, really concerned with the student’s set schema, is that in constructing a set based on a certain condition on elements, it seems to be easier to see that satisfying the condition implies that an element is in the set than to see that being in the set implies that the condition is satisfied.

3. The student must construct an understanding of the notion of a generic group and be able to perform calculations therein. One example of this occurred in the First interview, Question 7, in which students showed that if a and b are elements of an abelian group with orders 2 and 3, respectively, then ab has order 6. A second example occurred in the Second interview, Question 4, in which students were to prove that the center of a group is a subgroup. A prerequisite for success in tasks like these is the ability to calculate in an environment established by a generic group.

8.1.4 SUBGROUP

Again, the preliminary analysis of construction of the subgroup concept as a coordination of schemas for group, subset and function seems to hold up when considered in light of the data obtained in this study. As with the group concept, we can add some specific details about this coordination.

1. It may be that the student does not have a very strong conception of set and so might be restricted to thinking about subgroups in which the underlying set can be listed explicitly.

2. A critical component of the construction is the ability to think about the restriction of the binary operation function to the subset.

Finally, as a result of this study, we can provide an analysis of the construction of two concepts related to subgroup: finding subgroups of Z ; and proving that every subgroup of Z is an nZ for some positive integer n. In the first case, the analysis is very detailed, but in the second, our comments are more speculative and point to issues for further study.

Finding Subgroups of Z. In our discussion of mental constructions in connection with the First Interview, Question 5, we laid out four categories of responses to this task viewed as an application of a student’s thematized schema for subgroup, and related them to the APOS framework.

No Thematized Subgroup Schema. In the first category are students who have not constructed a subgroup schema, or have constructed it, but have not thematized it.

Action. This category consists of students who pick a value for II and construct a single, specific, subgroup of Z by constructing the multiples of this value of II, and then are able to repeat the action to obtain other sets of multiples, but not in a systematic way.

Transition to Process. This category consists of students who tried to systematize in some way or other the repeated action of constructing a single subgroup. In these cases,


they tended to look for a rule or condition for n to satisfy and then began to realize that all values of n could be used.

Process. Finally the student sees, without the help of any conditions, that a subgroup can be constructed using any value of n and expresses this process as taking any value of n and seeing that its multiples form a subgroup.

Proving that Every Subgroup of Z is an n Z . As mentioned earlier, very few of the students made any progress at all with this proof, and none gave a completely correct proof even with prompting from the interviewer. The question of what makes this such a difficult task naturally arises. The limited amount of data we have on this question precludes formulating any final conclusions, but we can make some tentative statements about how the complex development of this proof might relate to the APOS framework. Analysis of the data also suggests that the framework needs further development, particularly in explaining how students might use their schemas for binary operations, groups, and subgroups in proof-making situations.

First, the student chooses an overall proof strategy, for example, the strategy of direct proof. To see direct proof as a possible strategy, the student must have constructed some understanding of the set of all subgroups of Z . Although we have no data on this point, it seems reasonable to assume that to construct the set of all subgroups, one coordinates an aspect of one’s set schema (the set of all subsets of a set) with one’s subgroup schema to produce a mental process of considering each subset of Z and checking whether it is a subgroup.

To progress, the student must be able to think about picking a generic subgroup of 55 and consider how to verify that its underlying subset is the set of all multiples of a single element n. The first step is to construct a process that identifies a certain element of the subgroup as a candidate for a generator. Specifically, the construction of the generator n involves a process of iterating through the elements of the (nontrivial) subgroup to find the smallest positive element. As we saw in the data, this construction can arise as the reversal of the process of forming the set of multiples of a given positive integer. Completing the proof requires the selection of another, arbitrary, element of the subgroup, and the subsequent construction of a process involving division that shows that this element is necessarily a multiple of K As we saw in the data, this construction may appear in the form of division as repeated subtraction.

In reflecting on the proof, we see that it involves a kind of mental construction that is common and very important in mathematics. That is, in many situations, an individual must construct in her or his mind a certain collection of objects, any one of which may or may not satisfy a certain property. Then, a particular object that does satisfy the property is selected using a selection process that includes a component that verifies that an object with the property exists in the set. In practice, the degree of specificity of the object that comes from this process can vary considerably. In the case of the proof under discussion here, the desired II is somewhat specific in that it is the smallest positive element of the subset.

Finally, it is not enough to have constructed the generator. The student must reflect on the act of construction to realize its significance to the proof. That is, it is not the actual value of n that is of importance here, but rather the fact that such an n can be produced for


each subgroup of Z . This type of mental construction, which is based on reasoning about

generic objects, is often important in proof-making but it is not explicitly included in the

existing APOS framework, other than as a consequence of being able to construct objects by encapsulating processes that are not explicitly performed, but only imagined.

We close this section with one further comment about the process used to select n in this proof which will allow us to highlight a difference between APOS theory and another theory that describes mathematical knowledge in terms of processes and objects, Gray and

Tall’s theory of proceprs. In Gray and Tall (1993), the authors write:

The idea of a process giving a product, or output, represented by the same symbol is seen to

occur at all levels in mathematics. It is therefore worth giving this idea a name: We define a

precept to be a combined mental object consisting of a process, a concept produced by that process, and a symbol which may be used to denote either or both. We do not maintain that all mathematical concepts are precepts. But they do occur widely throughout mathematics. .

We interpret this description of precept to say, expressed in the terminology of APOS, that precepts can occur only when the encapsulation of a process is the object that the pro-

cess produces. Thus, the integer symbolized by n in the proof is not a precept because the process there does not create the n (it has been created earlier). Rather, the process simply selects the integer from a collection of integers (and possibly other objects), all of which pre-exist for that individual. Such processes that do not encapsulate to the object they pro-

duce are often of interest in APOS theory, but are not considered in the theory of precepts.

8.1.5. CENTER

In our initial theoretical analysis, we considered the construction of the center of a group

only as an application of the subgroup schema. However, the data that we obtained suggests that developing a schema for the concept of the center of a generic group raises issues for the student that relate to the more general schemas for binary operation, group, and subgroup.

Based on the data considered here, we see that it is not just a matter of coordinating existing schemas; there are several points at which a student must change or enhance an existing schema in order to deal with this new situation. In addition, the underlying schema for subset needs to be strong enough to handle a subset defined by the center condition, and

the function schema adequate for the restriction of a binary operation to a subset.

We suggest that the construction of the concept of center consists of two main parts: the construction of the center as a subset and the determination that this subset is a subgroup.

Construction of the Center as a Subset. The issue in this first part is the construction

of a generic subset of a generic group. Thus the student must be able to apply her or his group schema to think about processes and objects in a group without any concern for what group it is. Some students, for example, felt the need to list specific elements of the group in order to think about which of them might be in the subset. Others had to think about the operation as multiplication or addition. These difficulties indicate weaknesses in the group schema which must be strengthened in order to make progress towards a general concept of a center.


The coordination of the subset schema and the group schema becomes an issue because

the condition defining the subset in the case of the center is relatively complicated (as com-

pared, for example, to the criteria that specifies the even integers as a subset of the set of

integers). We interpret the students’ difficulties in coordinating these notions to indicate

that an individual’s subset schema may be adequate to assimilate a newly defined subset in

some cases, but it may not be adequate in other cases. It has been noted in the literature that

this is true of functions, that is, that the way in which a particular function is defined has an

effect on the way in which the concept develops in a student’s mind (Thomas, 1995;

Asiala, Cottrill, Dubinsky, and Schwingendorf, 1997).

In the case of the center of a group, the defining condition is that a group element x belongs

to the center of a group G if, for all g E G, xg = gx. Our analysis of the data leads to the

following descriptions of the construction of this subset. First, the individual constructs a

process for determining whether or not each element in the group is a member of the set,

and then the individual encapsulates that process to an object which is the set all of those

elements for which the process returned a value of true. Finally, this object may be named.

In the case of the center of the group, the set membership process actually consists of the

coordination of three processes: the first is the process of iterating through each element x

of the group G; the second is a set of processes, one for each value ofx, in which each element

g of G is considered and the two values xg, gx are compared; and finally the latter set of

processes is coordinated via a universal quantification and, for those cases in which xg =

gx for all g, there is the process of placing the element x in the set being constructed.

There appear to be three possibilities in relation with this construction.

1. The student constructs and encapsulates this process of determining membership in the

center.

2. The student constructs this process of determining membership in the center, but does

not encapsulate it.

3. The student constructs (and possibly encapsulates) a process for determining member-

ship in the center which is much less rich than the one described above. In some cases,

the student might treat the set notation for the center as a syntactic object. Such a stu-

dent has a conception of the center that is pseudostructural (see Sfard, 1991; Sfard and

Linchevski, 1994), in that he or she is confusing the object with one of its representa-

tions.

Determination That the Center is a Subgroup. The second part of constructing the

concept of center consists in applying the subgroup schema to this subset. This application

requires two things: first, the student’s conception of the binary operation on the original

group must be adequate for restricting this function to the subset, and second, the student

must either apply theorems for determining that a subset is a subgroup, or apply her or his

group schema to this subset and restricted binary operation.

We did not see any students in this study who used a theorem to determine that a subset

is a subgroup and so we cannot say anything about how this notion fits into the subgroup

schema. As far as this data is concerned, the only method is to apply the group schema by

checking the definition of group.

232 BROWN. DEVRIES, DUBINSKY, THOMAS

The first step in applying the group schema to the center is to unpack this schema to extract the axioms. This amounts to realizing that there are axioms which must be checked for this set/binary operation pair and remembering what the axioms actually are.

Once this is done, then each axiom must be de-encapsulated to its process(es), the set involved in the center must be de-encapsulated to the process of set membership used in defining the center, and these processes must be coordinated with the use of the binary operation to check the condition of the axiom. Each of the axioms present different difficulties. For closure it is this coordination; for associativity it is a matter of constructing the concept of preserving properties when restricting a binary operation to a subset; for identity and inverse, it is the idea that the desired object exists in the original group and it must be checked that it exists in the subset.

In this last step of the development of the concept of center of a group, an issue arose that has not been dealt with yet by our theoretical framework. We saw that some students reached this point, and understood what they were to prove, but then had difficulty completing the proof because of a lack of skill in performing manipulations in a generic group and/or reasoning about these manipulations.

8.2. Student Understandings

8.2.1. OVERALL LEARNING

The data that we have suggests that the students from the experimental course had a reasonable degree of success in learning about binary operations, groups and subgroups. On the first test, which was taken in groups, it is perhaps not surprising that the overall scores of the seven groups on questions related to these topics were very high-ranging from about 80% to full marks on the five items. However, this level of performance was maintained on three and a half of the four relevant items of the second exam, which the students took individually.

The question on the second exam that showed a spread of understanding concerned showing that the set of units in a ring forms a group. The students in the experimental course had no difficulty distinguishing between the two binary operations in a ring and selecting the appropriate one to work with, and more than a third succeeded with little or no difficulty in checking the group axioms. Most of the rest seemed to know what they were doing to some extent, but got into trouble with one or two of the axioms. Only about 15% appeared to be totally confused about the units of a ring forming a group.

In this situation, closure was problematic for almost half of the students who had trouble checking the axioms, and others did not understand the subtleties distinguishing the existence (of an identity or inverse) from its being in the appropriate subset. These difficulties continued to appear in the work of some students for the remainder of the course and in the interviews that came later.

Perhaps we can summarize the situation by saying that the overall student understanding of using the axioms to determine that a subset is a subgroup was, for this experimental class, at an acceptable but not satisfactory level. In some cases, students might have had only a tenuous understanding of the idea of checking axioms to determine whether a subset is a group, and that understanding fell apart when the situation was unfamiliar. Neverthe-


less all of these difficulties occurred in only a small portion of the instruments that were

used.

There appeared to be a similar level of success, perhaps slightly better, on the interview

question concerning the order of the product of two elements in a group.

The students also showed, on the final exam, some ability to make proofs.

In summary, we can say that at least one-third of the students in the experimental class

had fully succeeded in understanding the concepts investigated in this report and most of

the rest had made significant progress, but still had some way to go. There did not appear

to be any students in the experimental class who were totally unable to deal with this mate-

rial.

8.2.2. COMPARISONSBETWEENSTUDENTSFROMSTANDARDAND

EXPERIMENTAL COURSES

Although we have some data that can be used to compare the performance of students

who took a standard abstract algebra course with those who took the experimental course,

there are a number of reasons why one should treat these comparisons with the greatest of

caution and, in fact, drawing strong conclusions from this data should be avoided.

We made no attempt to control the two groups for any factors other than the course that

was taken. Moreover, the time elapsed between when the material was originally studied

and the time the data was collected varies a great deal, with a larger time lag for some of

the students who took standard courses. Half of the comparative data was based on the con-

cept of center of group which was barely mentioned in the experimental course but may

have been studied in the standard courses. The interviews were conducted by people known to be connected with the experimental course. Finally, some potential interviewees

from standard courses declined to participate because of an expressed strong dislike for the

subject matter (see Clark, Hemenway, St. John, Tolias, and Vakil, in press, for a complete

discussion of this issue).

Therefore, we will only summarize the comparative data very briefly. All of the data

comes from the Second interview, Questions 4 and 5. We identified several performance

indicators in the data, and categorized them as either “successful” or “unsuccessful.” The

list of indicators follows:

A. Indicators of Successful Performance

l Knowing that every subset of the form nZ , n a positive integer, is a subgroup of Z

l Knowing that all subgroups of Z have the form nZ

l Being able to construct the center as a subset without difficulty

l Understanding the idea of applying the group axioms to the center to determine that it is a subgroup

l Realizing that all four axioms needed to be considered in order to prove that the center is a subgroup

l Verifying with little or no difficulty that the axioms of a group are satisfied by the center

of a group.


B. Indicators of Unsuccessful Performance

l Making essentially no progress in finding subgroups of 25 l Thinking that some Z, are subgroups of Z l Making little progress in constructing the center as a subset l Not knowing a single group axiom

l Knowing some axioms, but having absolutely no idea of how to verify that the center of a group satisfies them.

Forty-six percent of the participants of the study were from the experimental course. What we found was that on each indicator of successful performance, more than forty-six percent of those exhibiting that indicator were from the experimental course. Furthermore on all but one indicator of unsuccessful performance, less than forty-six percent of the students exhibiting that indicator were from the experimental course. The only exception was the indicator of making little progress in constructing the center as a set, in which case 50% were from the experimental course.

It is interesting to note that almost all of the figures are in a direction which favors the students from the experimental course. We may not, however, conclude from this anything other than that there is nothing in this data to suggest that the students taking the experimental course learned any less than students who took a standard course. The evidence suggesting that, in fact, the students taking the experimental course may have learned more than those who took the standard course is sufficient to warrant a careful study comparing our pedagogical strategy with one or more standard approaches.

8.3. Pedagogical Suggestions

One conclusion that does appear to be warranted by the analysis of the data that we have presented is that our pedagogical approach is a reasonable way to help students construct an understanding of the concepts of binary operation, group and subgroup. Therefore, our pedagogical suggestion is to continue more or less in the direction of the experimental course with a view toward finding ways of improving in certain areas.

One area in which improvement is needed is for students to make a better coordination of their schemas for subset and subgroup. Perhaps some students come to an abstract algebra course with a set schema that is sufficient for simple situations but breaks down when considering a set such as the center of a group or the set of units of a ring whose definition is more problematic for them. More attention should be focused on issues such as the necessity of showing that the identity or inverse is in a subset that is a candidate for a subgroup, as opposed to existing in the original group. The question of closure is, of course, precisely a matter of coordinating construction of a subset with binary operation.

We leave for future studies the question of whether the difficulty here lies in the subset schema or in coordinating it with the group schema.

Another issue that needs more attention is the restriction of a binary operation to a subset, for example, in understanding why certain properties, such as associativity, are auto- matically inherited by the subset.

Finally we note that not many students appeared to take advantage of the name_group feature. Increased emphasis on using this in the course might help students construct the concept of a generic group.


REFERENCES

Asiala, Mark, Brown, Anne, DeVries, David J., Dubinsky, Ed, Mathews, David & Thomas, Karen. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education, 2, l-32.

Asiala, Mark, Brown, Anne, Kleiman, Jennifer, & Mathews, David. (in press). The development of students’ understanding of permutations and symmetries. International Journal of Computers for Mathematical Learning.

Asiala, Mark, Cottrill, Jim, Dubinsky, Ed, & Schwingendorf, Keith. (1997). The development of students’graphical understanding of the derivative. Journal of Mathematical Behavior, 16.

Barbut, Erol. (1987). Abstract algebra for high school teachers: An experiment with learning groups. Journal of Mathematical Education in Science and Technology, 18(2), 199-203.

Branca, Nicholas A. (1974). Learning mathematical structures. Paper presented at the annual meet- ing of the American Educational Research Association, Chicago, 1974. (ERIC document ED092384).

Breidenbach, Danny, Dubinsky, Ed, Hawks, Julie, & Nichols, Devilyna. (1992). Development of the process conception of function, Educational Studies in Mathematics, 23,247-285.

Bum, Bob. (1996). What are the fundamental concepts of group theory? Educational Studies in Mathematics, 31, 37 l-378.

Clark, Julie, Cordero, Francisco, Cottrill, Jim, Czamocha, Broni, DeVries, David, St. John, Dennis, Tolias, Georgia, & Vidakovic, Draga (1997). Constructing a schema: The case of the chain rule. Journal of Mathematical Behavior, 16.

Clark, Julie, Hemenway, Clare, St. John, Dennis, Tolias, Georgia, & Vakil, Roozbeh. (in press). Stu- dent attitudes about abstract algebra: a comparative research study. PRZMUS.

Dutermann, Jennie. (1992). Using ZSETL 3.0: A Languagefor Learning Mathematics. St. Paul: West.

Dienes, Zoltan P. (1959). Concept formation and personality. Leicester, U.K.: Leicester University Press.

Dienes, Zoltan P., & Jeeves, Malcolm A. (I 965). Thinking in Structures. London: Hutchinson Edu- cational.

Dubinsky, Ed, Dautermann, Jennie, Leron, Uri, & Zazkis, Rina. (1994). On learning fundamental concepts of group theory, Educational Studies in Mathematics, 27, 267-305.

Dubinsky, Ed, & Leron, Uri. (1994). Learning Abstract Algebra with ZSETL. New York: Springer- Verlag.

Freedman, Haya. (1983). The teaching of mathematics: A way of teaching abstract algebra. American Mathematical Monthly, 90(9), 64 l-644.

Gray, Eddie, & Tall, David. (1993). Success and failure in mathematics: the flexible meaning of symbols as process and concept, Mathematics Teaching, 142,6-10.

Halford, Graeme. (1982). How do we define the limits to children’s understanding? Paper presented at the 20th International Congress of Applied Psychology, Edinburgh, Scotland, 1982 (ERIC document 258726).

Hart, Eric. (1994). A conceptual analysis of the proof-writing performances of expert and novice students in elementary group theory. In E. Dubinsky and J. Kaput (Eds.), Research issues in mathematics learning: Preliminary analyses and results (pp. 49-62), MAA Notes, E. Wash- ington: The Mathematical Association of America.

Hazzan, Orit, & Leron, Uri. (1994). Students’ use and misuse of mathematical theorems: the case of Lagrange’s Theorem . For the Learning of Mathematics, 16(l), 23-26.

Hirsch, Christian R. (198 1). An exploratory study of the effectiveness of a “Didactical Shadow” sem- inar in abstract algebra. School Science and Mathematics, 81(6), 459-466.

Johnson, Wells. (1983). A note on Lagrange’s Theorem. American Mathematical Monthly, 90(2), 132-133.

236 BROWN, DEVRIES. DUBINSKY. THOMAS

Kraines, David P., Kraines, Vivian Y., & Smith, David A. (1990). Classroom computer capsules Binary operations. College Mathematics Journal, 2/(3), 240-241.

Lant, Rainier M.C. (1980). A comparative study of the learning and understanding of a mathematics group structure in different representations. ERIC document ED190405.

Leskow, Sonia, & Smock, Charles D. (1970). Developmental changes in problem-solving strategies Permutation. Developmental Psychology, 2(3), 412-422.

Leron, Uri, Hazzan, Otit, & Zazkis, Rina. (1995). Learning group isomorphism: A crossroad of many concepts. Educational Studies in Mathematics, 29, 153-174.

Leron, Uri, & Dubinsky, Ed. (1995). An abstract algebra story. American Mathematical Monthly 102(3), 227-242.

Maruszewski, Richard F., Jr. (1991). ADA in a theoretical mathematics course. Mathematics am Computer Education, 25(2), 131-143.

Sfard, Anna. (1991). On the dual nature of mathematical conceptions. Educational Studies in Math, ematics, 22, l-36.

Sfard, Anna, & Linchevski, Liora (1994). The gains and pitfalls of reification-the case of algebra Educational Studies in Mathematics, 26, 191-228.

Simmonds, Gail. (1982). Computer discovery of a theorem of modem algebra. Mathematics am

Computer Education, 16(l), 58-61.

Thomas, Karen. (1995). The fundamental theorem of calculus: An investigation into students’ con

structions. Unpublished doctoral dissertation, Purdue University, West Lafayette, IN.

Vidakovic, Draga. (1993). Cooperative learning: Differences between group and individual processes of construction of the concept of inverse function. Unpublished doctoral dissertation Purdue University, West Lafayette, IN.

Vidakovic, Draga. (unpublished manuscript). Learning the concept of inverse function in a group

versus individual environment.

APPENDIX

Instruments

We list in this section the interview questions and the test items relating to binary operations, groups, and subgroups. The interview questions are listed first, and are followed by the test items in the order in which they were given.

Because the present study is part of a large-scale study of the learning of abstract algebra, there were items relating to group theory that are not included here. In particular, there were several questions involving subgroups that were mainly about Lagrange’s theorem and normality. These will be considered in a separate report on learning the concepts of cosets, normality, and quotient groups.

First Interview, Question 1. Students were questioned about how they would gc about computing the product of two elements in D,.

This is the main item on which we base our considerations of our genetic decomposition for binary operation.

First interview, Question 7. Suppose that G is a commutative group with an element of order 2 and an element of order 3. Must G have an element of order 6? Give a proof OI

counterexample. What would happen if 2 and 3 were replaced by other numbers?


The way in which students talk about this problem should indicate how they think about groups and, in particular, about a generic group. We should also learn something about their ability to solve a problem about groups that is not completely on the surface. This

problem was not considered in the course.

Second Interview, Question 4. The student was given the definition of the center of

a group, expressed in words. The student was asked to express the definition in set notation, and then show that the center of a group is a subgroup.

In the experimental course, this problem was given as part of a homework assignment, but it was not emphasized in class. In some of the interviews, there were indications that

this topic was specifically covered in some of the standard courses which the students took.

We expect to learn about the quality of a student’s subgroup schema from her or his work on this question, as well as something about the set schema with which the student is

operating.

Second Interview, Question 5. The student was asked to provide examples of sub-

groups of Z and then was asked for a general statement. Once the characterization of sub-

groups of Z was on the table, the student was asked about a proof.

Unfortunately, there was a misunderstanding on this question and some students were not asked about the proof of the characterization of subgroups. In most of these cases, the students were asked to prove that nZ is a subgroup for an arbitrary n. We will consider the results from both of these questions.

The question of the subgroups of Z was on the official syllabus for the course, which was covered in both the experimental and the standard courses, and it appeared promi- nently in the texts that were used in these courses. Because the students took the course in different semesters and from a variety of instructors, the emphasis on this topic may have varied. In the experimental course, the analysis of the subgroups of Z was dealt with at length. Hence, for the students in the experimental course, we are seeing what students learned from that experience as opposed to her or his ability to deal with a new situation.

Test 1, Question 1. Let (G,*) be an abelian group, t a fixed element of G, and define

the binary operation 0 by

X 0 y = x * y * t-l, x,y E G

Prove or disprove that (G,O) is a group.

Student reaction to this problem has been studied by Hart (1994). It was expected to tell us about the students’ ability to distinguish between different binary operations on a given set. With regard to groups, the issue is whether the students are thinking about a group as a totality or schema which consists of a set together with an operation, and whether they can take that schema apart to look at different operations on the same set.

Test 1, Question 2. G,, G, be groups and let

G=G,xGz= {[a,b]:aE Gl,bE G2}

Define an operation on G by


[a,bl[c,dl = [ac,bdl

a Show that GI x G2 is a group.

b. Find a familiar group that is the same as 22 x Z3 and explain why it is the same.

c. Is Z, x 24 the same as Zs? Explain.

This is a topic that was not considered in the course so it was expected to give us information about how the students deal with groups in an unfamiliar situation. In particular, it was again expected to tell us if students are thinking about a group as a schema which consists of a set together with an operation, and if they can put these together in a specific example to form a new group. The second and third part should indicate whether students are thinking about a group as an object.

Test 1, Question 3. Give a proof of, or counterexample to, the following statement. Every subgroup of a non-abelian group is non-abelian.

For groups, this problem should tell us about students understanding of the concept of commutativity. This item also should give an indication of the quality of the subgroup schema through using it in a problem situation. It was also expected to tell us if the students appear to have a number of examples of subgroups available to them, especially in the non- commutative case, and if they can think about these examples.

Test 1, Question 4(b). Find all subgroups of D,.

Again, this question tells about students’ ability to use the subgroup concept in a problem situation.

Test 1, Question 5. Find a subgroup of S4 that is the same as S3.

Our expectations are the same as for the previous two questions.

Test 2, Question 1. Let R be a ring and 0 the additive identity. Show that for all x E R it is the case that x . 0 = 0.

The results on this question should show the extent to which students can coordinate two different binary operations on the same set. It should also show how students work with properties of a binary operation.

Test 2, Question 3. If R is a ring with identity, denote by R* the set of units in R with

the multiplication operation from the ring.

a Show that R* is a group.

b. Prove or disprove that the group (Z 9)* is isomorphic to the group (Z rg)*. Here, Z n refers to the ring Z n, +,,, ‘n.

Regarding binary operations, part (a) is concerned with the same issues as Test 2, Ques- tion 1, but in a different context, where different issues might be raised. For groups, one purpose of the question was to see how students think about groups that arise in a more complicated context.

Test 2, Question 6. Let S, be the group of permutations of three objects. Find a normal subgroup N of S3 and identify the quotient group S&W.


Here the question is whether the binary operation process can be constructed for cosets, assuming that they are objects for the students. Thus this is also a question about coset formation and coset products, but they are topics for a separate report.

Test 2, Question 7(a). Let Z be the ring of integers. Describe the ring 2 Z /6Z. This is about the same issues as the previous question but in a completely different con-

text.

Test 2, Question 7(b). Prove or disprove that the ring 22/62 is isomorphic to

(Z 3,+3> .3h The question here is whether the students are satisfied with matching the sets or if they

understand the necessity of comparing both binary operations in the two systems.

True/False Questions on the Final

2m. In a finite group, the set of all positive powers of a single element is a subgroup.

2n. If you have a subset of a group and you know that the closure axiom is satisfied, then you can be sure that the subset is a subgroup.

20. If n is a positive integer, then S, is isomorphic to a subgroup of S,,+z.

2P. A coset of a subgroup is always a subgroup.

These questions have meaning for student learning only if we consider the total performance on all of them. Even this gives us only a minor piece of information.

Proofs on the Final

The following five propositions had to do with subgroups.

3d. If S is a non-empty subset of a group with the property. that for every a,b E S it follows that ab-’ E S, then S is a subgroup.

3e. If G = (Z,+,,) and k is a factor of n, then G has a subgroup isomorphic to (Zk, +k). 3f. If G is a group and a E G, then the conjugate of any subgroup by a is again a sub-

group. 3g. Every subgroup of a cyclic group is cyclic. 3i. The image under an onto homomorphism of a normal subgroup is again a normal

subgroup.

In cases where these problems are chosen, we should get a strong idea of the students’

conception of subgroup.

learning binary operations, groups, and subgroups

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