GUIDED REINVENTION IN RING THEORY:

STUDENTS FORMALIZE INTUITIVE NOTIONS OF EQUATION SOLVING

John Paul Cook

University of Oklahoma

The literature is replete with evidence of student difficulty in abstract algebra. In response,

innovative approaches for teaching group theory have been developed, yet no corresponding

methods exist for ring theory. In an effort to simultaneously fill this void and build upon

Larsen’s (2009) guided reinvention efforts in group theory, I conducted a study to investigate

how students might be able to reinvent fundamental notions from introductory ring theory.

Rooted in the theory of Realistic Mathematics Education, this paper reports on a teaching

experiment conducted in nine sessions (up to 90 minutes each) with two students, neither of

whom had prior exposure to abstract algebra. Using the construct of an emergent model, I

show how these students formalized their intuitive understandings of linear equation solving

and used them to reinvent the definitions of ring, integral domain, and field. In particular,

the milestones of the reinvention process are identified and explicated.

Key words: Abstract algebra, ring, guided reinvention, Realistic Mathematics Education

Introduction and Research Questions

Rings are central structures in mathematics and enjoy an important place in the

undergraduate mathematics curriculum. For the typical mathematics major, ring theory not

only serves as the culmination of their mathematics careers but also lays a foundation for

future study of advanced mathematics. Indeed, a solid understanding of the fundamental

notions of ring theory is crucial for those students who wish to continue their study of

mathematics in graduate school. Additionally, future mathematics teachers have much to

gain from ring theory as it provides an underlying context for the techniques and axioms used

in school algebra.

Despite the importance of rings, both in mathematics in general and in the undergraduate

curriculum, there is reasonable evidence which suggests that students struggle mightily with

the subject. While there are no studies which examine student difficulty with specific ring

theoretic concepts, the literature is replete with evidence of students failing to understand

even the most basic concepts in group theory (Dubinsky, Dautermann, Leron, & Zazkis,

1994; Hazzan & Leron, 1996). As rings are similar to, yet arguably more complex than,

groups, it is quite reasonable to suggest that students experience rings with a comparable

amount of difficulty. Compounding this issue is the fact that research addressing student

learning of rings is almost nonexistent. In fact, only one study can be found in the literature

(Simpson & Stehlikova, 2006). Thus, there is a considerable disparity between the

significance of rings and the amount of information available to address student troubles with

them.

In response to their own assertion that “the teaching of abstract algebra is a disaster, and

this remains true almost independently of the quality of the lectures” (p. 227), Leron and

Dubinsky (1995) suggested developing discovery-based methods for teaching the subject as

an alternative to the traditional lecture. They proposed an investigative approach to

instruction using the computer programming language ISETL. Additionally, using the theory

of Realistic Mathematics Education (Freudenthal, 1991), Larsen (2004, 2009) developed an

instructional theory which supports the guided reinvention of the concepts of group and

group isomorphism. These efforts have since been expanded to create a complete

reinvention-based curriculum for group theory (Larsen, Johnson, Rutherford, & Bartlo, 2009;

Larsen, Johnson, & Scholl, 2011). However, there are still no corresponding innovative

instructional methods in the literature for ring theory. This study aims to begin filling this

void by building upon Larsen’s reinvention efforts in the arena of ring theory. In particular,

this paper reports on a teaching experiment with two students designed to investigate how

they might be able to reinvent the definitions of ring, integral domain, and field. The

teaching experiment and its corresponding results are part of my larger dissertation project,

wherein the ultimate goal is to constitute an instructional theory supporting the guided

reinvention of these definitions. My research questions are as follows:

How might students reinvent the definitions of ring, integral domain, and field?

What models and activities are involved in developing these concepts when the students

start with their own reasoning and intuition?

What models and activities enable students to see the need for, define, and differentiate

between additional ring structures like integral domain and field?

Literature

Of particular interest to this project is Larsen’s (2004) dissertation wherein he produced

an instructional theory supporting the guided reinvention of the definition of group. Using a

developmental research design (Gravemeijer, 1998), he conducted three iterations of the

constructivist teaching experiment (Cobb, 2000; Steffe, 1991) with two students apiece as a

means of testing and revising his instructional theory. His instructional tasks centered on

student manipulation of the symmetries of a triangle (and eventually other polygons). His

students gradually formalized their intuitive notions with these symmetries and used them to

write a precise mathematical definition of group. Larsen’s dissertation and subsequent work

established that the methods of guided reinvention are able to be used quite effectively in

abstract algebra. Seeking a similar goal in the arena of ring theory, I adopted a similar

theoretical perspective and research design (detailed in the methods section).

Like Larsen’s work, nearly all of the literature concerning abstract algebra involves only

group theory. Fortunately, the group theory literature does prove somewhat helpful, as rings

and groups are structurally similar (in fact, a ring is an additive group with an additional

multiplicative structure). Several features and learning mechanisms for groups which have

been explored in the literature have direct analogs in ring theory. Those involving the

definition of ring or the ring structure include, for example, binary operation (Brown,

DeVries, Dubinsky, & Thomas, 1997; Iannone & Nardi, 2002), student proficiency (or lack

thereof) with the group axioms (Dubinsky et al., 1994), confusion of the associative and

commutative properties (Findell, 2000; Larsen, 2010), and the use of operation tables

(Findell, 2000). Despite any possible application of this knowledge to student learning of

rings, however, even introductory ring theory possesses several key, nontrivial features for

which there is no analog in ring theory: zero divisors, an additional binary operation, and the

distributive property (to name a few). Information regarding these concepts can only be

obtained by research which directly examines student learning of rings.

The lone article found in the literature which directly addresses student learning of ring

theory is Simpson and Stehlikova’s (2006) case study of how one student came to understand

the commutative ring Z99. This case study was used to draw conclusions regarding how

students “apprehend” mathematical structure, defined as the shift of attention from the

objects and the operations to the interrelationships between the objects as a result of the

operations. The study examined the process by which a female student, Molly, apprehended

a ring isomorphic to Z99 for her undergraduate thesis over a period of three years. It is worth

noting that Molly had previously taken courses in abstract algebra, and consequently the

researchers used a ring isomorphic to Z99 so that Molly would not immediately connect it

with her prior, formal knowledge. Molly’s primary self-guided method of apprehending this

structure involved solving basic linear and quadratic equations. In addition to elucidating the

need for the traditional ring axioms, this activity illuminated several key aspects of the ring

structure: the existence of inverse operations, zero divisors, and units. These features arose

as she attended to “the sense of interrelationships between the objects caused by the

operations.” Despite her three years of work with this structure, Molly never identified it as

Z99, nor did she exhibit any signs of accessing any of her formal knowledge from abstract

algebra. Thus, it is reasonable to conclude that she used equation solving as method of

discovering this ring structure (instead of a method of affirming what she already knew to be

true about the structure).

In relation to my project, this case study suggests that equation solving can be used quite

effectively by students in order to explore and apprehend an unfamiliar algebraic structure.

In fact, this conclusion agreed with Kleiner’s (1999) commentary on the historical role of

equation solving in the rise of the axiomatic definition of a field: “In the solving of the linear

equation ax+b=0, the four algebraic operations come into play and hence implicitly so does

the notion of a field” (p. 677).

Theoretical Perspective

I adopted Realistic Mathematics Education (RME) as a theoretical perspective which

guided both the instructional design and the data analysis. Two RME heuristics, in particular,

were of critical significance to this study. First, the principle of guided reinvention

(Freudenthal, 1991) served as the overarching guide for the study. The reinvention principle

seeks “to allow learners to come to regard the knowledge they acquire as their own private

knowledge, knowledge for which they themselves are responsible” (Gravemeijer &

Doorman, 1999, p. 116). Secondly, the notion of an emergent model (Gravemeijer, 1998)

was integral to the design of the instructional tasks and was used to identify milestones of the

reinvention process. The purpose of an emergent model is to mediate a shift between

informal mathematical activity to a new, more formal mathematical reality. The model is

said to emerge as a model of the student’s informal mathematical activity while gradually

developing into a model for more formal mathematics. This process is known as the model-

of to model-for transition. Gravemeijer (1999) delineated this transition into four phases of

mathematical activity:

1. The situational phase involves working to achieve mathematical goals in an

experientially real context.

2. The referential phase includes models-of that refer to previous activity in the original

task setting.

3. The general phase is characterized by models-for that support interpretations

independent of the original task setting.

4. The formal phase entails student activity that reflects the emergence of a new

mathematical reality.

I viewed these phases as a continuous progression wherein activity within one phase would

gradually progress toward the next. Because of the tendency for informal procedures to

“anticipate” the emergence of more formal mathematical reasoning (Streefland, 1991), I

argue that the progressive formalization within each phase anticipates the next. This

expansion of Gravemeijer’s four phases, then, can be expanded (if needed) to accommodate

more detail by inserting three sub-phases. Namely, I introduce and define the following:

The situational anticipating referential phase involves activity still firmly rooted in

the original situational setting that lays the groundwork for future referential activity.

The referential anticipating general phase is characterized by models-of that provide

an overview of previous work in preparation for abstract or general activity.

The general anticipating formal phase includes models-for which promote more

efficient or concise use of the mathematics at hand in preparation for formal use.

I used these seven phases as a lens through which I present the results of the teaching

experiment and identify the significant milestones of the reinvention process. This, in turn,

provided a means by which I can begin to answer my research questions. Furthermore, it

informs the creation of the emerging instructional theory being developed to support the

reinvention of ring, integral domain, and field.

For the purposes of this project, I am viewing equation solving as an emergent model.

Specifically, I anticipated that solving equations would initially serve as a model-of the

students’ informal activity with the ring structure, and that this would gradually transform

into a model-for defining the desired ring structures.

Methods

I employed a developmental research design (Gravemeijer, 1998), which was compatible

with and followed from my theoretical perspective because the primary goal is “the

constitution of a domain specific instructional theory for realistic mathematics education” (p.

278). Following Gravemeijer’s (1995) suggestion that the teaching experiment methodology

is useful for such a purpose, I adopted the guidelines of the constructivist teaching

experiment (Cobb, 2000; Steffe, 1991; Steffe & Thompson, 2000). In the constructivist

teaching experiment, the researcher serves as the teacher and interacts with the students

individually or in small groups (Cobb, 2000). I worked together with two students in the

teaching experiment, which consisted of 9 sessions of up to 2 hours each.

Participants

The participant pool included students who had recently completed a course in discrete

mathematics at a large comprehensive research university. Potential participants were

recruited on a volunteer basis. At this university, the discrete mathematics course doubled as

an introduction to advanced mathematics course and, aside from the course content, focuses

on proof construction. To ensure the validity of the reinvention process, I wanted the

participants to have had no direct prior exposure to abstract algebra, including group theory.

I did require, though, that they had a working knowledge of modular arithmetic, polynomials,

and matrices. Their familiarity with these concepts was assessed in a pre-survey

administered after they had volunteered for participation. In addition to meeting the stated

requirements, they were chosen on the basis of perceived compatibility with me and each

other. I wanted two above average students and, ideally, one male and one female. The

following table includes information on the two selected participants, Jack and Carey

(pseudonyms):

Participants Age Major(s) Discrete Math Grade

Jack 21 Mathematics B

Carey 19 Mathematics & Physics B

Instructional Tasks

Due to its potential for explaining the ring structure (Kleiner, 1999; Simpson &

Stehlikova, 2006), solving linear equations became the focal point of the instructional tasks.

Specifically, activities were designed that would culminate in solutions to additive and

multiplicative “cancellation” equations x+a=a+b and ax=ab (a nonzero), respectively.

Throughout the rest of the paper, I suppress the “a nonzero” qualifier so as not to detract

focus from the two equations. I used the equation x+a=a+b instead of the traditional

x+a=b+a to eliminate any ambiguity regarding the necessity of the additive commutativity

axiom, which can be derived from the other ring axioms in a ring with identity (Dummit &

Foote, 2004). These equations were chosen for their potential to both justify the ring axioms

and enable students to differentiate between ring, integral domain, and field. For example,

x+a=a+b can be solved on an algebraic structure if and only if its additive structure forms an

abelian group. The different methods of solving ax=ab make use of all of the multiplicative

ring axioms aside from commutativity (including multiplicative inverses). Additionally,

ax=ab serves to distinguish rings from integral domains, and integral domains from fields: it

has a unique solution (x=b) if and only if the structure is an integral domain. In fields, this

may be shown using multiplicative inverses or the zero-product property. On the other hand,

in integral domains that are not fields it may only be proved by the zero-product property.

The structures upon which the specific linear equations and the cancellation equations

would be solved were selected to incorporate examples of rings (that are not integral

domains), integral domains (that are not fields), and fields so that each set of examples would

be distinct in a meaningful way from the others. The structures I chose for the instructional

tasks are the integers modulo 12, integers modulo 5, integers, polynomials in one

indeterminate over the integers, and 2x2 matrices over the integers (throughout this paper,

assume that these structures are accompanied by their usual operations):

Structure 12Z 5Z Z xZ ZM 2

Rationale finite,

includes

zero divisors

example of a

finite field

prototypical

ring

structure;

integral

domain that

is not a field

prototypical

ring

structure;

integral

domain that

is not a field

prototypical

noncommutative

ring, includes

zero divisors

Notice that I only included one example of a field, and one that is likely to be unfamiliar to

students, at that. I additionally neglected to include the more familiar examples of fields,

such as the real or rational numbers, in this initial set of examples, opting instead for an

example of a finite field with five elements. This was done purposefully, in line with Zazkis’

(1999) recommendation that “working with non-conventional structures helps students in

constructing richer and more abstract schemas, in which new knowledge will be assimilated”

(p. 651). Additionally, I planned for the students to generate their own examples after

solving equations on the structures I provided, anticipating that they would introduce the

more conventional examples of fields themselves.

Results

Recall that I am using equation solving as an emergent model to support the guided

reinvention of the definitions of ring, integral domain, and field. The results of the teaching

experiment are revealed, then, through the lens of Gravemeijer’s (1998) phases of the

emergent model transition, along with the three intermediate phases I introduced in the

theoretical perspective section. In order, these phases are: situational, situational anticipating

referential, referential, referential anticipating general, general, general anticipating formal,

and formal. It is worth noting that, due to the gradual process of formalization I attempted to

foster during the sessions, many of the students’ initial solutions or responses to instructional

tasks were not necessarily complete (or even correct). Instead of only presenting the

students’ finished products, I have included snapshots from the various stages to provide the

reader with some context and feel for the reinvention process.

1. Situational: solving specific linear equations on Z12, Z5 , Z ,Z[x] , and ZM 2

In addition to being designed as the original task setting, I classified the solving of

specific linear equations on these structures as situational because it involves the students

working towards a mathematical goal in an experientially real context. The students were

initially directed to solve specific equations on the given structures, both to familiarize

themselves with the features of each structure and with the activity of equation solving. The

following was presented as a solution to the equation x+3=9 on the integers modulo 12:

As this example was taken from one of their initial responses, the solution is not yet complete

and ignores, for example, associativity of addition. At first, the left hand side of the equation

on second line of the solution read as x+3-3. I inquired about what was meant by -3, since it

was not yet a defined element of the set. The students responded by defining -3 to be +9, and

wrote this above their solution. When I asked them how this might be done for all

“negatives” in Z12, Carey responded by constructing a “negative number line” (the “as seen

on a clock” addendum refers to a previous instructional task designed to increase their

familiarity with modular arithmetic by likening addition modulo 12 to clock arithmetic):

Thus, the solving of x+3=9 enabled students to recognize the need for additive inverses.

Additionally, examining the solution above makes it clear that the students recognized on

some level, if not formally, that 12 is the additive identity of Z12. Next, the students were

prompted to solve multiplicative equations on Z12. In particular, I gave them the equations

5x=10 and 4x=8 with the idea that they would recognize that x=2 is a solution for both but is

only unique for 5x=10. A near-complete solution to 5x=10 is on the left, and an attempt to

solve 4x=8 is on the right:

Interestingly, regarding 5x=10, the students opted for multiplication on the right (which

necessitates the use of commutativity of multiplication) instead of the simpler multiplication

on the left. The solving of these equations brought several other ring axioms to the fore as

well: multiplicative inverse, multiplicative identity, distributivity, and, even though it was not

yet recognized at this point by the students, associativity of multiplication. Additionally, in

their attempts to solve 4x=8, the students recognized a conceptual difference between the

elements 4 and 5. While Jack and Carey were struggling to find an algebraic way to solve

4x=8, I asked them about their need for a different technique:

JP: So why didn’t you use the same method for 4x=8?

Jack: It only works for numbers that are not a factor of our base.

JP: Right. So what is it that doesn’t work in this other case?

Jack: 4 times any number does not make it 1.

While a correct solution to 4x=8 was not produced until later, it is significant in that the

students noticed that not all multiplicative equations can be solved in the same fashion.

2. Situational anticipating referential: solving x+a=a+b and ax=ab on each of the given

structures

This activity could be easily be classified as simply “situational,” because these equations

could have been the focus of the original task setting on their own. In other words, the

students’ ability to solve these equations could have been independent of solving specific

equations beforehand. On the other hand, however, the specific equations were used as a

paradigm upon which the students could reference to solve x+a=a+b and ax=ab. Solving

these general equations was also designed to promote the summarization of their previous

activity, thus anticipating the need for referring to these results at a later stage. Consequently,

I classified this activity in the intermediate stage of situational anticipating referential. The

following excerpt was their near-completed solution to the equation on Z[x] (they used

capital letters to denote polynomials in x):

Setting aside the fact that additive associativity was omitted between lines 2 and 3

(though is used correctly throughout the rest of the solution), all of the additive ring axioms

are in play here. By this point, they had written out a solution nearly identical to this one for

the preceding three structures as well, prompting them to remark:

Jack: Adding [polynomials] is basically adding integers.

Carey: So you do the same thing that you did before.

A similar exchange occurred when they started to solve x+a=a+b on ZM 2 :

Jack: It is commutative, A + B = B + A.

Carey: You can do the same thing that you did for the addition, because you just add the

complements.

JP: And they are [matrices] over the integers. So the components are commutative. So

you guys are saying that it’s exactly the same thing as the others?

Carey: Yeah.

Jack: Yes.

This dialogue suggests that, in addition to successfully motivating the need for all of the

additive ring axioms, the equation solving model effectively highlighted the identical additive

structure present in all rings. However, the multiplicative structure is a different story, and

this was recognized at once by the students when they wrote up their solutions to ax=ab for

Z12 (left) and 5Z (right). Recall that each unit is its own inverse in Z12.

The critical difference the students noticed here was that the solution in Z5 was valid for all

nonzero elements, whereas the solution in Z12 only held for a small subset. And while the

above methods are similar, Jack and Carey noticed that they were not able to use this method

in general when they were faced with solving ax=ab over the integers,

Carey: did we define division?

JP: What would happen if you did that?

Carey: Like x over a equals x times 1 over a.

Jack: The problem is what is this? 1 over a. It doesn’t exist. It is not necessarily the

inverse.

They opted to use distributivity with the zero-product property instead:

This solution, in addition to identifying the necessity of the distributive and zero-product

property, also helped the students to mentally differentiate the integers (and then, eventually,

polynomials) from the modular rings with which they had worked previously. Thus, the

students’ solving of x+a=a+b and ax=ab on each of the five structures:

1. reinforced the need for the axioms used to solve the specific equations,

2. enabled them to see that all of the examples had identical additive structures, and

3. enabled them to notice the differences in multiplicative structure.

3. Referential: summarizing the results from solving x+a=a+b and ax=ab

After the equation solving activities were completed, I gave the students a task prompting

the students to organize their solving of the equations x+a=a+b and ax=ab. Specifically,

they were asked to identify the different methods they used to solve the equations, and

whether the given method could be solved always, sometimes, or never on each of the

structures. I classified this task as referential because it was distinct from the original task

setting yet referenced the previous activity in the original task setting. Additionally, at this

point, the model is still a model-of their equation solving activity and had not yet transitioned

into a model-for (which takes place in the general phase).

Once Jack and Carey had discussed the different methods for solving the equations on

each of the examples, I had them organize their results in a chart by writing “A” for “always

works”, “S” for “sometimes works”, and “N” for “never works” (across the top row:

x+a=a+b; ax=ab, 0a using mult. inverses; ax=ab, 0a using distributivity and the zero-

product property):

In addition to summarizing their previous work, the students were also required to build

off of it. For example, the students had not yet considering whether ax=ab could be solved

on polynomials by using multiplicative inverses:

Jack: Polynomials over integers. [Multiplicative inverses] held, didn’t it?

Carey: We didn’t do it that way.

JP: What would happen if you tried to construct a multiplicative inverse for a

polynomial? 1/x2. Is that a polynomial, based on how we defined it?

Carey: Basically, we are starting with n = 0, which we are.

Jack: So we can’t do things with polynomials.

The discussion continued until they realized that 1 and -1 were the only polynomials which

had multiplicative inverses, earning Z[x] a rating of “sometimes” in that column. Similar

discussions were held for methods which had not yet been applied to other structures.

A number of interesting patterns emerged in the chart, both from my perspective and the

students’. First, the students recognized that there is essentially only one way to solve the

additive equation. Jack noticed this during the activity by referencing their previous work

solving x+a=a+b, remarking, “I think that this method works in all of the cases.” Second,

notice that the sets with “identical ratings” do indeed have substantial features in common.

The always-sometimes-sometimes rating appears for Z12 and ZM 2 , which are the structures

containing zero-divisors. The always-sometimes-always rating appears for Z and Z[x], which

are the integral domains that are not fields. Lastly, 5Z and its always-always-always rating

stands alone as the only field under consideration (at this time).

At this point, I encouraged the students to generate their own examples of structures upon

which the given equations could be solved (in other words, sets endowed with addition and

multiplication). Then I prompted them to fill out a similar chart for their new examples:

As I had previously anticipated, the students’ own examples were dominated by fields. In

fact, four of the six student generated examples were fields (specifically, the real numbers,

complex numbers, rational numbers, and integers modulo a prime). Notice also that they

differentiated between Zn for n prime and composite (this occurred before the chart activity as

a result of generalizing their reasoning about Z5 and Z12). As expected, the fields and their

always-always-always ratings agree with the ratings for Z5 on the previous chart. The only

example I had not anticipated was {0}, the trivial ring. Because this example is markedly

different from the other examples and does not lend any insight into the ring structure, I

intervened and removed it from further consideration (though, in the interests of using

student-generated ideas as much as possible, I did re-introduce it after the definitions had

been reinvented).

4. Referential anticipating general: the sorting activity

Now that Jack and Carey had organized the results of their equation solving, I gave them

a sorting task to encourage them to sort the structures based on what they felt were common

characteristics using their charts. I classified this as referential anticipating general because it

involved referring to previous activity (the chart activity and, to a lesser extent, the actual

equation solving activity). In this way, this task was not yet “independent of the original task

setting”, a characteristic of the general phase. Sorting based on common features, however,

does anticipate the mathematical activity of abstraction, which certainly qualifies as general

activity. Since the students performed the bulk of the mathematical activity for this task by

filling out the charts, this activity proved to be quite simple. Jack commented “If we are not

categorizing them by the first column, which is trivial, we are categorizing them by the

second column and the third column,” suggesting that the equation solving chart is now a

model-of the identical additive structure for all rings (ratings in the x+a=a+b column are all

“always”) as well as the differing multiplicative structure (differing ratings for the ax=ab

columns). This realization enabled the students to sort based on the ratings for ax=ab:

Thus, at this point, the students have used equation solving as a means to sort these

structures. Jack’s comment above emphasizes that the primary criteria for sorting included

how ax=ab can be solved on each of the structures. The underlying ring-theoretic concepts

which govern how this equation be solved, of course, are the existence (or lack) of zero-

divisors and multiplicative inverses. Whether the students were formally aware of these

features at the time of the sorting is unclear. It is clear, however, that the ratings for different

methods of solving ax=ab on each structure served as a model-of these ideas for the students.

5. General: defining by abstracting common features

At this point, I asked the students to define a list of criteria for inclusion in each of the

three sets. This required them to identify the common characteristics of each collection. This

activity was classified as general because, finally, the equation solving model had emerged as

a model-for the formal activity of defining the different ring structures, independent of the

original situational task setting. Again, I asked them to display their results in a chart by

listing the rules they had used to solve the equations and determining if the given rule holds

for each group. The chart is reproduced here:

Group 1

R, C, Q, 5Z , Zn

for n prime

Group 2

Z, Z[x]

Group 3

Zn for n composite,

Z12, ZM 2

Additive identity X X X

Multiplicative identity X X X

Associativity of addition X X X

Commutativity of addition X X X

Distributivity X X X

Zero-product property X X

Associativity of multiplication X X X

Commutativity of multiplication X X

Multiplicative inverse X

Additive inverse X X X

This served as a springboard to being the process of defining. I followed Larsen’s (2004)

guidelines for supporting a cyclic process of presenting and revising a definition:

1. The students prepared a definition.

2. I read and interpreted the definition, calling attention to particular choices made by

the students.

3. The students revised their definitions as necessary and restarted the process.

I suggested that they start with group 3, the idea being that starting with what I knew to be the

most general structure would provide them with the possibility of defining subsequent

structures in terms of this one. Here is one of Jack and Carey’s initial attempts to write out

the criteria for a structure to be included in group 3:

This, of course, is a preliminary definition of a ring with identity. At this stage of the

defining process, the students still need to address the existence of the binary operations and

issues with quantifiers, among other things. After this definition was completed, the students

wrote a similarly rough definition for group 2. It was at this point that I gave them the names

for the structures in each of the three groups so that they could finalize their formal

definitions. In writing their initial definition of an integral domain, Jack and Carey did not

immediately see the potential for defining integral domain in terms of a ring with identity.

6. General anticipating formal: writing “nested” definitions

When the students were repeating their definition of integral domain to incorporate

revisions, the students notified me of what a mundane process rewriting the same axioms

would be. I used this as an opportunity to engage them in a conversation about how they

could shorten the process:

JP: So as you guys have correctly noted writing all of these out is a huge [inconvenience],

so if we wanted to write out, say the next one, knowing that we have this definition down

now, what’s a way that we could shorten, shorten the next one.

Jack: We just say, if it’s A ring, and has the following properties.

JP: Okay. So, how would you do that? …

Jack: Uh, oh if you wrote the main rings then the difference between a ring with identity

is that [an integral domain] has a few more properties.

JP: Okay. Yeah.

As a result, Jack and Carey wrote a definition of an integral domain in terms of a ring with

identity. Shown is their finalized version of this definition:

They used the same technique to define a field in terms of an integral domain:

I categorized this activity as general anticipating formal because it still involves the defining

of mathematical structure (which I previously argued is general), while the “nesting” of these

definitions served as a tool for classifying other ring structures and emphasizes the

interrelationships between the three definitions. Thus, nesting the definitions prepared the

definitions for their use in a more formal setting.

7. Formal: using the reinvented definitions to classify other examples of rings

Upon the reinvention of the definitions of ring with identity, integral domain, and field, I

turned the students’ attention to tasks in which they would use the definitions to classify other

examples of rings. These tasks qualify as formal as they reflect the emergence of a new

mathematical reality. Specifically, one of the tasks asked the students to classify 33 ZZ (with

the usual component-wise operations modulo 3). I anticipated that they would initially

conjecture that it is a field (since 3Z is a field), and that they would find this to not be the

case. Indeed, after verifying that all of the axioms for a ring with identity (plus multiplicative

commutativity) held, they turned their attention to the zero product property and

multiplicative inverses. When examining the zero-product property, I named and defined the

term “zero divisor”, a concept with which they were familiar at this point due to their

experience with Z12 and ZM 2 . I then asked them if there were any zero divisors present in

33 ZZ . In a similar fashion, I asked Jack and Carey which elements had multiplicative

inverses (and named these, accordingly, as units). These were the results (zero-divisors are

on the left and units are on the right; note that ZZ should be 33 ZZ ):

Thus, they concluded that 33 ZZ

is a ring with identity that also has a commutative

multiplication (I use this opportunity to introduce the notion of a commutative ring). Then I

asked the students to classify the infinite ring ZZ . A conversation ensued regarding the

zero product property:

Jack: Z cross Z would be…

Carey: Don’t we have a similar problem?

Jack: It would still be a ring.

Carey: Yeah.

Jack: It would have the exact same problem with zero-product property ‘cause there’s

going to be…you can just take pairs of zeros out of it.

They concluded that, since the zero-product property did not hold, that ZZ could not be an

integral domain or a field. This excerpt, in addition to displaying the students’ activity in a

new mathematical reality, demonstrates that the students having a functional, working

knowledge of the definitions they reinvented.

Conclusions

In addition to providing information about how students come to understand fundamental

concepts in ring theory, this paper supports two primary conclusions which contribute to the

knowledge of the field. First, I introduced an expansion of Gravemeijer’s (1999) phases of

the emergent model transition based on the idea of anticipation and progressive

formalization. Using the results of a teaching experiment designed to investigate how

students might able to reinvent the definitions of ring, integral domain, and field, I presented

evidence that demonstrates how such an expansion can be useful when explaining and

interpreting the emergence of a model.

Second, the results of the teaching experiment demonstrate how students might be able to

capitalize on their informal knowledge of solving equations in order to reinvent the

fundamental ring structures. Additionally, the adaptation of Gravemeijer’s model to seven

phases highlighted the significant milestones of the reinvention process, laying the

groundwork for a domain-specific instructional theory to support the reinvention of these

definitions:

1. Solving specific linear equations on a variety of ring structures

2. Solving the equations x+a=a+b and ax=ab on each of the structures

3. Summarizing the different methods used to solve x+a=a+b and ax=ab

4. Sorting the structures based on similar methods used to solve x+a=a+b and ax=ab

5. Defining by abstracting the common features of each set of sorted structures

6. Writing “nested” definitions, i.e. writing specific definitions in terms of general ones

7. Using the definitions for more formal activity (such as classifying other rings)

This emerging instructional theory framework will be tested, refined, and elaborated through

another iteration of the teaching experiment as a part of my dissertation research project.

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