“I’ll just try to mimic that”: an exploration of students’ analogical structure creation in abstract algebra

Abstract

Despite the prominence of analogies in mathematics, little attention has been given to exploring students’ processes of analogical reasoning, and even less research exists on revealing how students might be empowered to independently and productively reason by analogy to establish new (to them) mathematics. I argue that the lack of a cohesive framework for interpreting students’ approaches to analogical reasoning in mathematics contributes to this issue. To address this, I introduce the Analogical Reasoning in Mathematics (ARM) framework. Constructed from an analysis of interviews with four abstract algebra students, ARM identifies several analogical activities that serve to analyze students’ analogical reasoning with a finer grain size than was previously possible with existing frameworks. Using this framework, I present an analysis of the students’ constructions of a ring-theoretic analogy to subgroup, thus revealing that even constructing simple analogies can elicit diverse pathways of analogical reasoning across students. Implications for further research related to analogies and analogical reasoning in mathematics education are discussed.

Analogical reasoning offers a way to develop insight into new situations and produce rich connections between mathematical contexts. Reasoning analogically appears to have a symbiotic relationship with the development of productive mathematical reasoning (Buehl & Alexander, 2004), possesses ties to other processes such as abstraction (English & Sharry, 1996; Gentner & Hoyos, 2017; Hicks et al., 2023), and even serves as catalyst for mathematics research (Ouvrier-Buffet, 2015) and creative problem-solving (Pólya, 1954). Analogies are leveraged in curriculum design and instruction, sometimes for the purpose of quickly and efficiently conveying information about a new concept by referring back to related concepts. Such is the case within many abstract algebra texts when introducing ring theory after group theory (e.g., Gallian, 2013) or vice versa (e.g., Hungerford, 2012). Despite the observed importance of analogies, little is known about how mathematics students themselves reason by analogy, and what exists is typically focused on children’s reasoning (e.g., English, 1997; 2004). While many prominent frameworks exist outside of the mathematics education literature (e.g., Gentner, 1983; Holyoak & Thagard, 1989), they do not capture the intricacies of students’ reasoning about mathematics. Indeed, the typical framework for analogical reasoning describes only that reasoning that is considered legitimate and correct, and analysis is constrained to describing the structures themselves that are being compared analogically. Such frameworks may fail to identify productive reasoning, whether correct or not.

The lack of a cohesive framework for interpreting students’ mathematical analogical reasoning has contributed to a negative view of the pedagogical potential of analogies, especially for scenarios where students are expected to generate mathematics by analogy. Cobb et al. (1992) argued that introducing mathematics with pre-designed analogies is a flawed method, claiming that students may not be able to access the deep connections readily available to instructors. Greer and Harel (1998) suggested that analogies are best used for reinforcing pre-existing concepts rather than building new ones. However, these perspectives on analogy may limit the potential for designing fruitful tasks or lessons in mathematics that leverage analogical reasoning for concept creation. I contend that with the appropriate tools for interpreting students’ ways of reasoning by analogy, it becomes possible to make informed pedagogical choices about supporting students to invent new (to them) mathematics by analogy, and ultimately develop stronger pedagogical sense for productive analogical reasoning. As such, this paper advocates a positive perspective of analogical reasoning as a tool for constructing mathematics. To this end, I propose an initial analytic framework that analyzes and interprets students’ processes of reasoning by analogy: The Analogical Reasoning in Mathematics (ARM) framework.

While analogies are often associated with comparisons between largely different contexts (e.g., understanding the Rutherford model of an atom by analogy with the solar system), it is not uncommon for analogies in advanced mathematics to be made between rather similar contexts, such as those found between group and ring-theoretic structures. Such strong similarity is not accidental given the purposeful standardization of mathematical structures in modern mathematics (see Hausberger, 2018). Unfortunately, the fact that several similar structures exist across the advanced mathematics spectrum has led to some structures (or their properties) being presented as immediately accessible by analogy with a previous structure. For example, Gallian (2013) suggests, “Just as was the case for subgroups, there is a simple test for subrings.” (p. 240). While these analogies may be construed as fairly evident from the perspective of those who already possess knowledge of the mathematical objects of subgroup and subring, statements such as these conceal how students’ might understand the analogy. Thus, I argue that abstract algebra provides a suitable context for building an initial analytic framing while simultaneously showcasing that students’ analogical reasoning can be complex even in similar contexts. In this study, I attended to the following research questions:

• RQ1. What are four students’ analogical activities as they reason by analogy in abstract algebra?

• RQ 2. How do four students’ processes of analogical reasoning differ as they reason about a ring-theoretic analogy for subgroups?

1 Literature pertaining to analogy in mathematics education

Before describing the analytic framework, I first provide a brief overview of analogy in mathematics education. English (2004) described three types of analogy found in mathematics education: classical analogy, problem analogy, and pedagogical analogy. This useful categorization provides a way to discuss research orientations toward analogies in mathematics education. First, proportional reasoning describes the common orientation toward analogies within the category of classical analogy (Modestou & Gagatsis, 2010). Classical analogies are typically presented as a proportion of the form A is related to B as C is related to D, expressed symbolically as:

A:B::C:D

Standardized instruments for measuring intelligence often leverage classical analogies as a multiple-choice testing item by omitting one of the components and requesting to fill in the blank with the appropriate component (English, 2004). Although some theoretical developments have been made by investigating classical analogies (e.g., Piaget, 1952), little research has focused on classical analogies in recent years, perhaps due in part to their roots in assessment rather than focusing on other qualitative aspects, such as student thinking. In contrast, there has been greater interest overall in students’ reasoning with the second analogy type: problem analogies. In general, research on problem analogies focuses on the creation of isomorphisms between problem types and determining students’ ability to solve problems in new contexts by making analogies to previous problems (Carbonell, 1983). For example, researchers have explored children’s ability to formulate solutions to word problems by identifying relational structure to previous ones (English, 1998), and Lockwood (2011) explored student-generated connections between structurally related counting problems.

Third, pedagogical analogies are those that are related to teaching. Research on pedagogical analogies is often concerned with the efficacy of a particular instructional method, such as leveraging a pre-determined analogy to reduce the abstraction of a concept (e.g., Dawkins & Roh, 2016). Examples of research focused on teacher education exist as well, such as developing pre-service teachers’ awareness of student reasoning processes (Peled, 2007). However, negative perspectives on pedagogical analogies are present in the literature. Except for cases where students’ analogical reasoning is guided, pedagogical analogies may be counteractive (Sidney & Alibali, 2015); Cobb et al. (1992) argued that pedagogical analogies are mostly ineffective due to difficulty in conveying intended connections to students who may lack the sophisticated insight possessed by the teacher. Exacerbating this issue further, evidence suggests that teachers create the majority of analogies in the classroom (Richland et al., 2004).

A prominent theme of these categories in mathematics education research is that analogies are often pre-determined or are restrictive in terms of allowing students to explore or create analogies on their own. In contrast, as discussed below, only a handful of studies have focused on the process of students generating analogies and creating mathematics by analogy. To allow for a more precise conversation of these studies, I introduce a fourth category for orienting teachers and researchers toward analogy: constructive analogy. I define the category of constructive analogies to include analogies that are intended to be constructed by students for the purpose of investigation or concept creation. Constructive analogies are not pre-determined (even if the intended concept is) and are potentially idiosyncratic.

Pushing back on the constrained nature of classical analogies, Lee and Sriraman (2011) proposed the Open Classical Analogy (OCA), an expansion of classical analogies that introduced a constructive component to classical analogies and allows for students to generate their own analogy-based conjectures. The OCA constitutes a marriage of classical and constructive analogy types, although the constructive category allows for even further possibilities beyond classical analogies. For example, Hejný (2002) recognized the possibility for students to construct internal mathematical structures through analogical reasoning, detailing the process as a transfer of structure from one context to another. Along the same vein, Stehlikova and Jirotkova (2002) investigated the processes of reasoning analogically to construct a finite arithmetic structure by analogy with integers. Significant results of this research were the identification of several phenomena that were related to the building of a structure by analogy as well as an existence proof that students could indeed develop mathematical structures by analogy in undergraduate mathematics. Building on this work, Hicks (2020) identified pathways of analogical reasoning¹ as a way to distinguish between students’ processes of analogical reasoning and argued for a tighter connection between pedagogical analogies and constructive analogies in advanced mathematics by introducing two ways of thinking associated with productive analogical reasoning in advanced mathematics: (1) a purpose-oriented way of thinking, and (2) a relation-oriented way of thinking (Hicks, 2022).

Even with the introduction of the constructive category to parse the literature on analogies in mathematics education, much remains to be learned about the nature of student-generated mathematical analogies and how to promote productive analogical reasoning associated with constructive analogies. The present study contributes to research on this phenomenon.

2 Conceptual framing

In this paper, the conceptualization of analogy and analogical reasoning rests upon three foundations: (1) an operationalization of analogical reasoning that builds upon key characteristics from Structure-Mapping Theory (Gentner, 1983), (2) the adoption of the Actor-Oriented perspective (Lobato, 2012) to place students’ activity in the foreground, and (3) the assertion that analogical reasoning can be parsed into individual activities.

I borrow the following three features of structure mapping:

• Analogies compare across domains.

• Analogies consist of mappings between domains.

• Mappings consist of linking content between the source domain and target domain.

Analogical reasoning is defined to be reasoning about the underlying structural or relational similarity between domains. Like the notion of concept image (Tall & Vinner, 1981), a domain is a collection of knowledge that one possesses about a given topic or idea. Domains may vary in size and scope depending on the context in which one is reasoning by analogy. For instance, one may hold the singular concept of a subgroup as constituting a domain, or one may reason with the domain as being the entirety of group theory including anything relevant to groups within the domain. To operationalize the creation of an analogy, Structure-Mapping Theory introduced a formal description of mapping content between domains: the generation of connections from one domain to another. The original domain being mapped from is typically already known to the thinker and is called the source domain, while the domain that is being investigated is the target domain.

Because observers tend to be well-versed in the contexts which they are studying, they bring along biases concerning the analogies expected to be generated by the participant. This is especially pertinent to analogies in mathematics in general, where several concepts are indeed pre-determined to adhere to convention. If a participant does not generate the expected concepts, then the participant is often thought to have failed in creating any analogy at all. The actor-oriented perspective mitigates this bias by orienting the examination of student activity from the perspective of the student and situates the role of the researcher as an interpreter of students’ activity without holding expectations of correct concepts or analogies. Rather, the activities enacted by the students while engaging with analogical reasoning are faithfully interpreted from the perspective of the student. In this way, there are no “incorrect” activities or connections that may be generated by a participant while reasoning by analogy. This orientation is key to developing a framework for how students reason by analogy in mathematics, rather than modeling perfect representations of analogical reasoning.

Finally, as my goal is to describe analogical reasoning during concept/structure creation, I focus primarily on the constructive category of analogy. I view analogical reasoning as a process that can be dissected into individual activities, thus allowing for a closer inspection of a student’s reasoning beyond attending to the resulting concept in the target domain and the connections made. In other words, even if two students generate the same concept and make the same connections, the constructed analogy may differ between them.

3 Research methods

In this section, I outline the research methods implemented to access students’ analogical reasoning in abstract algebra and the process for developing the ARM framework.

3.1 Participants and setting

The setting for this study was a large public research university in which only one course of undergraduate abstract algebra is offered. I acquired rosters from two sections of undergraduate abstract algebra courses at the university and emailed an invitation to all students previously enrolled, totaling approximately 40 students. Four students accepted the invitation to be interviewed, three of whom were undergraduates in mathematics and one of whom was graduate student in mathematics education. The graduate student had taken the course as a leveling course (i.e., they had not taken the course as an undergraduate, but were required to take the course to fulfill requirements for their graduate degree).

Of the four participants interviewed, two of them had previously seen the definition of a ring within their class, and one had seen the definition of ring, integral domain, and field. However, the definitions were only covered at the end of their course along with basic examples, and ring theory was not studied in any further depth, meaning that none of the students had seen subrings, ring homomorphisms, or quotient rings. This light introduction to rings was typical of most undergraduate abstract algebra courses at the university.

Although three students in the study had seen the definition of ring prior to being interviewed, all participants were provided the definition of ring in their second interview (see the discussion of the interview tasks below.) This assisted with establishing a more balanced origin point for investigating other structures beyond the definition of ring.

3.2 Interview tasks and procedures

With each participant, I conducted five 60–90-min semi-structured interviews designed to elicit analogical reasoning. The first three students were interviewed in person and tasks were administered on paper. One camera was positioned to record the students’ writing and a lapel microphone recorded audio. Due to complications with COVID-19, the fourth student was interviewed using a video-conferencing tool for each interview. Tasks and questions were administered verbally and through a chat interface within the tool, and the student wrote and recorded their work with the use of an electronic tablet. The tablet screen was visible to myself during the interviews and a real-time recording of the tablet screen and synchronous audio was made available from the video-conference tool.

Because the goal was to understand and interpret each student’s reasoning and construction of analogical structures, follow-up questions were often posed in the moment that were based in the student’s ideas. Thus, the interviews were not standardized, and not all tasks were posed to every student. The purpose of the initial interview was to determine what students recalled about group theory as well as ascertain what students knew about ring theory before further interviews were conducted. The second interview provided participants with the definition of ring and various tasks designed to acclimate the student to working with rings. This interview focused primarily on getting students acquainted with basic properties of rings, but also included opportunities for the students to compare rings to groups analogically. The tasks used for the initial group theory interview and the introduction to rings interview can be found in Appendices 1 and 2 respectively.

Each of the third, fourth, and fifth interviews focused on constructing a ring-theoretic analogy with a concept in group theory. The first task of each of these interviews was the analogizing task: Make a conjecture for a structure in ring theory analogous to X in group theory. In this study, X was taken to be one of subgroup, homomorphism, or quotient group. In each interview, this initial task lasted between approximately 10 and 25 min per student, providing deep insight into the students’ thinking and reasoning about their analogical constructions. Once the student finalized their structure, further tasks were provided that were constructed around three basic types: (1) elicitation of comparisons between structures (i.e., “In what other ways are the structures of group and ring the same?”), (2) example generation and checking (i.e., “Give an example of a subring.”), and (3) conjecturing and proof-writing (i.e., “Is the homomorphic image of a commutative ring commutative?”) As the interviews progressed, I allowed students to refer to their work from previous interviews. An example of the type of tasks provided for the constructive interviews can be seen in Appendix 3.

3.3 Data analysis

Before describing the creation of the ARM framework in more detail, I first provide an overview of the general process. The development of the ARM framework relied on open coding and the constant comparative method (Corbin & Strauss, 2015) to identify emergent themes distinguishing the participant’s analogical reasoning. Although some early codes were informed by the literature (e.g., mapping content across domains), the codes were not applied a priori and instead were identified within students’ analogical reasoning.

As a part of ongoing analysis, I routinely leveraged three techniques: microanalysis, diagramming, and memoing (Corbin & Strauss, 2015). Microanalysis involves close examination of portions of data with the goal of deeply understanding phenomenon contained within. I intermittently performed microanalysis when the nature of the analogical reasoning was especially unclear. In addition, diagramming was incorporated to make sense of how concepts arising from the coding fit together in new (or otherwise difficult to detect) ways. Diagrams consisted of (a) constructing visual representations of my participants’ analogical activity, making heavy use of the naturally visual nature of mapping between domains, or (b) creating tables that connected and categorized codes with one another. Finally, I regularly wrote memos that explicated my thinking about concepts. These memos acted as a record of my thinking over time and allowed older thoughts and observations to be rediscovered and reintegrated when appropriate. The results of microanalysis, diagramming, and memoing were regularly shared with colleagues through conversation and presentation to assist in ensuring the viability of my interpretations.

The developed codes were iteratively refined over a period of two years with the specific goal of searching for differences in describing students’ analogical reasoning. The result was a set of codes, called analogical activities, that provided greater nuance and explanatory power for describing analogical reasoning extending beyond a final constructed analogy alone. In the sections that follows, I describe the evolution of the unit of analysis and the key observations that led to the development of these codes.

3.3.1 Instances of analogical reasoning

Classical approaches to analyzing analogies parse the finalized structure itself, meaning that the primary unit of analysis is the final set of content mapped from one domain to another. In this way, two students’ construction of a concept or a structure by analogy may appear to have the same final result, such as producing essentially exact definitions for subring by analogy with subgroup. In this research, I attend to examples of longer, sustained reasoning about a concept or structure by analogy as an episode of analogical reasoning.

However, a closer inspection of the episodes that make up the construction of an analogy can reveal quite different understandings of the final concept, even if the structures are indistinguishable on the surface. To this end, the additional unit of analysis developed for ARM was an instance of analogical reasoning; that is, the creation or acknowledgment of mathematical content within a domain, or the creation or acknowledgment of a relation between content over the course of an episode. Instances of analogical reasoning are characterized by localized analogical activity that relate to the three features of structure mapping, meaning that instances describe when fine-grained activity is occurring that is relevant to comparing across domains, and forming mappings that link content between a source and target domain. Instances are not necessarily required to include explicit activity of comparing or mapping but may also include activity that lends itself to future analogical reasoning, or activity that leverages previously formed analogical comparisons. Thus, while content is still essential to understanding the process of reasoning by analogy in this research, the focus has been shifted to the localized activity associated with the content formed during an episode as opposed to the content alone. Examples of content relevant to the setting of abstract algebra can be found in Appendix 4.

To analyze the transcripts, episodes of analogical reasoning were segmented in the transcript indicating when a student engaged in sustained reasoning about a particular concept or structure, and instances within an episode were denoted by [] surrounding the text. An example of this is shown in the following brief excerpt of an episode:

• (1) [So, now we’re going to call this normal subring. We give this one a name.] (2) [First condition is that S is a subring of R.] (3) [Second condition, I don’t know. Maybe we say rSr⁻¹ is in S, just to copy it.]

Within all three instances above, evidence of content or relations was determined by language such as “we give this one a name,” “conditions,” and “just to copy it.” In the first instance, the student is attending to naming conventions as the mathematical content. In the second and third instances, the student is attending to definitional properties of normal subrings. In this manner, an episode of analogical reasoning was parsed as a series of instances of analogical reasoning, the sum of which constituted the process of analogical reasoning within the episode.

At times, a student might engage in a brief bout of analogical reasoning that did not constitute an entire episode (i.e., the analogical reasoning was not sustained for a longer length of time or was not focused primarily on comparing or constructing a focal concept.) These instances that were not contained within episodes were also identified. In these cases, the instances could be a one-off statement made while responding to a task, or a brief series of instances related to a smaller concept.

3.3.2 An expansion of aspects of analogical reasoning

During the initial coding process, two ubiquitous aspects of analogical reasoning remained apparent: mapping across domains and identifying similarity between domains. However, by attending to students’ analogical reasoning, other aspects quickly became observable that are less prominent in the literature but are still greatly important for appropriately describing how students reason analogically. Thus, ARM expands on the aspects of mapping across domains and similarity. The aspects that were developed through the iterative analysis are found in Table 1.

Table 1 Aspects of analogical reasoning

The first aspect is the presence of activity that did not necessarily involve making connections across two or more domains. I refer to this as intra-domain activity, in contrast to the inter-domain nature of mapping across domains. Although inter-domain activities are a hallmark of analogical reasoning, intra-domain activities exemplify the temporally complex nature of analogical reasoning; it is often beneficial to reason about an individual domain either before or after reasoning across domains. The second aspect is the attention given to differences between domains, rather than just pure attention to similarities. As is the case with mapping across domains, attention to similarity is an essential component to reasoning by analogy, but the inclusion of differences allows for a deeper analysis of what one is explicitly attending to while reasoning by analogy. The final aspect I introduce is an observation that accounts for where the students’ attention is focused as they reason analogically: foregrounding a domain. Foregrounding is crucial for identifying shifts in attention from one domain to another, thus allowing for meaningful distinctions between activities that may otherwise appear to be the same.

Instances of analogical reasoning were initially coded based on the aspects of analogical reasoning that were present. Through axial coding, the interrelated nature of the aspects was identified in each coded instance leading to the development of new codes that reflected the analogical activity contained within the instance. These analogical activities are discussed in the next section.

4 Findings

The resulting codes describing the students’ analogical activities as they developed ring-theoretic structure by analogy with group theoretic structures provides an answer for the first research question. The complete list of activities found in this study can be seen in Table 2.

Table 2 Overview of analogical activities for parsing analogical reasoning

To answer the second research question, I focus on the episodes of analogical reasoning in which the students developed a ring-theoretic analogy for subgroups, a structure that would appear to many to be a straightforward structural analogy. However, by parsing analogical activities within the episodes, differences become visible across the students’ reasoning by analogy. To exhibit this, I present an analysis of the four students, Nathan, Andrew, Brandon, and Ellen (all pseudonyms), as they reasoned about and constructed an initial ring-theoretic analogy for subgroups. As I will show, their pathways to creating the structure were distinct and possessed implications for how they understood their structure. In describing these pathways, I do not aim to compare and contrast the quality or correctness of the students’ constructions; rather, my goal is only to document their analogical activity while creating their initial structure.

4.1 Andrew’s attention to language and general adaptations

Andrew’s pathway was what many might expect to be the obvious construction of the subring concept by analogy with subgroups. Andrew began the analogizing task for subgroups by silently writing out a general overall definition (see Fig. 1). As he was writing, I asked him to think aloud. He responded:

Yeah, so, I’m gonna let there be a ring and let there be some subset of the set that is included in the ring… And just say if that’s a ring <referring to the set H>, then it’s a ‘sub ring’, I guess.

Fig. 1

Andrew’s initial attempt at a definition for a “sub ring”

Although not explicit at first, Andrew appeared to be copying the general definition of subgroup as described in the first interview and made modifications to (1) the number of operations involved (possessing two binary operations) and (2) the language within the definition (replacing the word group with ring). Thus, Andrew’s construction of the subring concept was a general definition that was lightly modified to fit the new context of rings. Further evidence of this interpretation came about when Andrew was asked to discuss what he was thinking about when creating the analogue to subgroups. He explained:

Well, that’s pretty much the definition for groups. If we have some group G and some operation, and if H is a subset of G and H is a group under that same operation, then we say H is a subgroup of G. So, I literally just took the exact same thing from group theory and applied it to rings.

The fact that Andrew described the construction as taking “the exact same thing from group theory” and applying it to rings suggests that Andrew’s initial construction was focused on the exporting the entire definition of subgroup and making general adaptations to fit the context of rings. While Andrew would continue later in his interview to expand his initial construction to include specific properties related to being a subring, his original construction only described the general definition of a subring.

4.2 Nathan and Brandon’s attention to structural properties and specific adaptations

As was the case with Andrew, Nathan and Brandon also appeared to begin by leveraging their own general definition for subgroup. However, in contrast to Andrew, Nathan and Brandon also inquired into the specific properties needed to form a subring as well. Here, I primarily discuss Nathan’s construction as an exemplar of this pathway. Consider Nathan’s reaction to the analogizing task below:

Interviewer:So, the first question here is that I want you to make a conjecture for a structure in ring theory that is analogous to subgroups in group theory.

Nathan:So I’m just making it up? Like a subring? I guess we’ll call it that. So, subgroup, subring… I’m going to say has to exhibit closure under… I don’t know if it has to be both of them <referring to the two binary operations defined on rings>, I don’t know if it is at all, but we’ll just say and it’s always what are plus times. So let’s just say under plus and then... What were the two steps again?

From his initial statement, Nathan began his construction by recalling his understanding of a subgroup, and wrote down criteria associated with a standard theorem in group theory known as the two-step subgroup test.² That is, given a nonempty subset of a group, only the properties of closure and inverses must be checked (see Fig. 2). Based on this recollection, Nathan continued his construction by making specific adaptations to the new structure as follows:

Okay, so I’ll just try to mimic that. We’ll say the inverse under multiplication. I guess we’ll call subring, so this is… Let’s say an S. Subring, let’s say denoted [sic]. Closure under plus, inverse under times or multiplication is in S. But then it’s got to have something else, it’s got to be more stuff. What else could we have? Maybe it has one of the identities. I mean, it has to have one of them. Maybe it has one. I don’t remember what it was under.

Fig. 2

Nathan’s recalling of his understanding of subgroup

At this point, Nathan reasoned about which specific properties would be needed for the new subring structure, and Nathan was careful to avoid assuming that the exact same properties would hold true given that he knew groups and rings were different structures as well. This is further evidenced by Nathan’s summary of his construction below:

So, I just compared it to subgroup. So, we know that subgroup has a closure under its binary operation and it contains the inverse. So, I said maybe it has closure under just one of them, one of the binary operations and then maybe the inverse is under the other operation. And then I said, well since ring had more properties than the group, then maybe a subring has more properties than the subgroup. So, then I just added a unity or what do you call it, identity, under one of the operations. Because it’s a subring, maybe it doesn’t exhibit all the properties, just like subgroup doesn’t. So, I said maybe it just has one of them.

Thus, Nathan’s pathway began by explicitly associating subgroups to a new analogous object in ring theory. In particular, he exported the specific properties of closure and inverse from the source domain of subgroups, but then made a small modification about where those properties are needed for the subring structure (assuming closure for one operation, and inverses for the other). Nathan adapted again to form a structural property: an identity is needed for one of the operations in the subring because rings have more properties than groups. Finally, Nathan argued that the criteria to be a subring do not necessarily require all properties of a ring, which is a property exported from his understanding of subgroup. Nathan’s initial writing while developing a ring-theoretic analogy to subgroup can be seen in Fig. 3.

Fig. 3

Nathan’s initial attempt at creating a ring-theoretic analogy to subgroup

4.3 Ellen’s extension of the subgroup structure

Ellen began her creation of a ring-theoretic analogy for subgroup by requesting to see the formal definitions of group and subgroup that were discussed in the first interview. After studying the definition of group and subgroup, Ellen began to think aloud: “I’m trying to see how I can say that if something is a subgroup [sic] and there’s gonna be some qualifying sentence that relates to how it could be a ring.” Thus, Ellen’s approach to constructing a ring-theoretic analogy for subgroup began with using properties as a base. However, unlike Nathan, Ellen investigated the properties that a group does not have with the goal of identifying the totality of differences between groups and rings. As shown in Fig. 4, Ellen observed that groups are not necessarily commutative, and do not possess multiple operations.

Fig. 4

Ellen’s list of properties possessed by rings, but absent from groups

Ellen’s initial investigation resulted in her first description of a ring-theoretic analogy to subgroups. In particular, Ellen adorned the definition of subgroup to include properties relevant to being a ring which were previously absent from the subgroup definition (e.g., additive commutativity and the distributive properties.) When asked to reflect on her reasoning process, Ellen summarized her construction as follows:

All right, so, we don’t ever deal with both, like more than one operation interacting for groups of ... We also don’t have an inverse for multiplication, which you need an inverse for a group, therefore you need an inverse for the sub-group. So, I think that if we have a subgroup, then we have all of these <points to group definition on the desk>. We have the identity, the inverse. We have associativity and closure, but what we don't have is commutativity, necessarily, or this use of multiple properties at once. So, we could say that if we have a subgroup, then if this subgroup is also ... so this was just kind of me brainstorming up here, but if a subgroup H is commutative and distributes... Right-hand side and left-hand side, then H is a ring… Then H is a ... yeah. Yeah, that’s what I’m doing, trying to like form this H that meets all the properties.

Thus, Ellen’s pathway to creating a ring-theoretic analogy to subgroup was to make several observations of the differences between groups and rings by distinguishing between structural properties. In order to create the analogical structure itself, Ellen extended the mathematical structure of subgroup by adding on the missing properties of ring she identified in the previous instances, thereby forming a ring using subgroup as a base. Ellen’s initial description of the analogical structure can be seen in Fig. 5.

Fig. 5

Ellen’s initial attempt at creating a ring-theoretic analogy to subgroup

4.4 Cross-examining the pathways

This analysis has revealed three pathways that students might take to begin constructing a ring-theoretic analogy to subgroups in group theory. Having presented each of the students’ constructions, it is useful to consider not only how their constructions differed, but also how their constructions influenced their understanding of the concept of subring for the remainder of the interview.

Each of Andrew, Nathan, and Brandon readily developed the notion of “subring” and proceeded to create their structure by comparing directly to subgroups. The way in which their pathways primarily differed was on the type of content that they were focused upon. In Andrew’s case, the notion of a subring was formed by taking “the exact same thing from group theory” and applying it to rings, placing much of the emphasis on the language of the overall definition itself. Indeed, Andrew’s attention lied primarily on generalities of the structure of subgroup. Because Andrew’s construction was focused on generalities when constructing the definition of subring, Andrew proceeded to grapple with the specifics of the definition when confronted with further tasks, such as example generation. When posed the task of constructing an example of a subring, Andrew responded: “Oh man… < long pause > So real numbers with the standard addition and multiplication. Subring would be the same thing but with integers.” Upon being asked to prove that the integers under standard addition and multiplication are a subring of the real numbers, Andrew began exporting the structural property of inverses for group operations to the multiplicative operation for integers. However, he paused as he was writing, erased his work, and began reasoning about inverses only under addition, and only attended to closure for multiplication (see Fig. 6).

Fig. 6

Andrew’s generation of an example of a subring

In contrast, both Nathan and Brandon had attended to specific, individual properties of subgroup when constructing the ring-theoretic analogy. Nathan’s approach involved attending to the subgroup test as though it were the definition for subgroup. The properties entailed in this definition were exported and adapted within the context of rings to create the analogous structure.³ This proved to be an affordance for them as they were posed new tasks that furthered their reasoning about the subring structure. For example, when asked to consider an example of a subring, Nathan readily exhibited his knowledge of the properties that he believed were required to form a subring:

Nathan:So [is Z] a subring for anyone? If I had to guess, I would just say R.

Interviewer:You talked about that one earlier.

Nathan:Then if I had to check I would check... Is it closed? Yeah, I know that it’s closed. Does it have negative? Yeah, cause it’s positive/negative. Does it have... But yeah. Yeah, it’s closed. Oh, this one. Does it have one <points to multiplicative identity>? Yeah, it has one. And then boom, we’re a subring.

Thus, because Nathan’s initial construction of a ring-theoretic analogy for subgroups involved the exportation and adaptation of specific properties known about subgroups to the ring context, Nathan was able to fluidly reason about proving a subset was a subring.

Unlike the other students, Ellen’s initial construction was centered around an eventual extension of the subgroup structure to form a ring. In other words, Ellen’s activity relied on thinking about what properties of rings were missing from her definition of subgroup, and then added properties to the subgroup structure based on her analysis of what was different. Of particular note in Ellen’s initial construction of a ring-theoretic analogy to subgroup is that she never explicitly identified a structure that was separate from subgroup. Instead, the structure of a subgroup was transformed into a ring, and the name “subring” only made an appearance after the initial construction of the analogy itself. This had a profound impact on Ellen’s reasoning for the remainder of the interview in contrast to the other students, who all possessed conceptions of subgroups and subrings as completely separate structures from the beginning. For example, when Ellen reasoned about examples of the subring concept, she found herself unable to articulate that her constructed analogy for subgroups would need to be a subset of a parent ring. This is in line with her definition, in which the subring concept is built out of subgroups that are given extra properties to become rings without ever identifying an explicit parent ring itself.

5 Discussion

This study contributes two major findings relevant to understanding students’ analogical reasoning in mathematics. First, through an exploratory analysis using student interview data, I developed an initial analytic framework for analyzing students’ reasoning by analogy: the ARM framework. ARM parses individual analogical activities within students’ reasoning rather than focusing solely on the finalized set of content mapped across domains. In developing ARM, several aspects of analogical reasoning were identified as important for understanding how students reason by analogy, namely the aspects of intra-domain activity, attention to differences, and the aspect of foregrounding a domain (see Table 1). These aspects allowed for a wider range of interpretations of students’ analogical activity, thus providing insight into understanding distinctions between the ways that different students might reason by analogy (see Table 2).

Second, this research exhibits evidence that even in cases where analogies are perceived as basic, students’ analogical reasoning is diverse, multifaceted, and complex. Although the structural analogies between concepts in group theory and ring theory might appear obvious on the surface, I found that productive opportunities for students to reason by analogy in distinctive and robust ways still existed. Indeed, the analysis of the three students’ initial constructions of a ring-theoretic analogy to subgroups revealed this diversity by exemplifying a variety of pathways, including Andrew’s focus on the similarity of the general written definition and minor differences in the language used for each, Nathan and Brandon’s focus on specific structural properties and subsequent adaptations to account for the context of rings, and Ellen’s extension of the subgroup concept to form a ring. Thus, echoing the warning of using instructional analogies posed by Cobb et al. (1992), as students encounter similar structures across mathematics, we must be careful to notoverlook the complexity involved with establishing analogical connections from the students’ perspective.

5.1 Connections to literature

The results of this study are intended to complement studies that I identified as being within the category of research on constructive analogy as well as studies related to pedagogical analogies. In contrast to the argument that analogies are best for reinforcement of previous ideas (Greer & Harel, 1998), this research has focused on inventive, student-driven analogy and provides further evidence to work by Stehlikova and Jirotkova (2002) that students can in fact generate mathematics through analogical reasoning. Furthermore, the present study expanded on the scope of mathematical activity that can be expected from implementing analogical tasks: students not only conjectured by analogy (i.e., Lee & Sriraman, 2011), but also engaged in the act of defining mathematical concepts by analogy.

Given that the results in this paper have only analyzed four students’ analogical reasoning in an interview setting, there is much to be learned about how students might productively leverage analogical reasoning over an extended period. By achieving this goal, we can better support students’ analogical reasoning by designing more structured curriculum (e.g., Hicks, 2024). A unique characteristic of the present research was attention to how students construct mathematical structures by analogy with known structures in abstract algebra. In many ways, this was similar to guided reinvention (Gravemeijer & Doorman, 1999). The task design implemented for data collection in this research suggests the potential for a new heuristic: analogical reinvention. While inroads have been made into the guided reinvention of ring, integral domain, and field (e.g., Cook, 2014, 2018), there has yet to be an established curriculum developing further concepts, such as ideals. The structure of ideal arose naturally during my participants’ development of the quotient ring concept by spontaneously making a connection to normal subgroup. Thus, one approach to achieving reinvention of ideal could include investigating how students might leverage analogical reinvention to reason about the structure of ideal by analogy with normal subgroups. Further research can explicate what such a reinvention process would entail by exploring the usefulness of analogical reinvention and generating models of the analogical reinvention process in various content areas.

While this study has primarily focused on identifying differences of students’ analogical reasoning processes by attending to analogical activity at a smaller grainsize, it is also important to consider how and in what ways students’ pathways of analogical reasoning might possess common features across different students as well as different structures for the same student. For example, each of the students began their construction of a ring-theoretic analogy to subgroups by recalling aspects of the subgroup concept, although they did so in distinct ways. Situating this observation within the broader literature on analogical reasoning, a general name for this phenomenon of initiating analogical reasoning through recall is analogical access (Hummel & Holyoak, 1997). Following access, the students each engaged in a different form of analogy generation, followed by the establishment of new content (see Fig. 7.) I hypothesize that general overarching templates for pathways of analogical reasoning can be identified as further cases of students’ creation of structures by analogy are investigated in future research.

Fig. 7

A general pathway of analogical reasoning

5.2 Limitations

Acknowledgements of limitations are warranted to properly make sense of the analytic framing proposed in this paper. First, I note that my participants’ analogical reasoning may have been influenced by their varied exposures to topics in abstract algebra. Of the interviewed students, one claimed to have never seen any topics in ring theory, two had seen only the definition of ring, and one had previously seen the definition of ring, integral domain, and field. Thus, although none of the students had been exposed to the concepts of subring, ring homomorphism, or quotient ring, I note that great variability may have been present among the participants’ responses to certain tasks (such as example generation) corresponding to the depth of their previous exposure to abstract algebra, as well as their personal memory of the subject.

Second, I was the sole researcher interpreting the range of my participants’ analogical activity. As such, individual bias may be present in the ARM framework in terms of the selected aspects, activities, and mathematical content. As discussed in the methods, steps were taken to curb this potential bias by sharing my interpretations with colleagues at various stages in the data collection and analysis. I also acknowledge that the analogical reasoning elicited from students may be context dependent. While abstract algebra proved to be fertile ground for eliciting analogical reasoning, the analogies between concepts in group theory and ring theory may not have captured the full range of possible aspects within other cases of analogical reasoning. Thus, before the usefulness of ARM can be fully substantiated, further work is necessary in more diverse mathematics settings.

5.3 Implications

As was the case in this research, the analogical creations given by students can appear very similar when looking only at the end result of an episode of analogical reasoning. Attention to analogical activity provides the groundwork for researchers of mathematics education and instructors of mathematics to pivot away from focusing on final products as the basis for evaluating a student’s analogical construction, thus allowing us instead to focus on the holistic process of how the analogy was created. As shown in the findings, even with similar analogical creations, the students’ future reasoning about the subring concept was influenced by their particular pathway to construct the analogical structure.

Building on the finding that students exhibit diverse pathways of analogical reasoning, an immediate implication of this study is that structural analogies should not be taken for granted in undergraduate classrooms by instructors of advanced mathematics. Even when structural analogies may seem obvious, careful attention must be given to the variety of pathways that students might employ for developing analogical structures. In addition, instructors should carefully consider how they might most effectively employ analogical reasoning in their classrooms. The diverse nature of the students’ reasoning suggests that there is value in allowing students to generate and openly discuss their analogical creations. Doing so could lead to potentially fruitful discussions that would enrich students’ understanding of the underlying structures being constructed, while also allowing students to engage in an open forum for sharing a variety of perspectives.

In summary, this research has contributed an analytic framework for parsing students’ analogical reasoning at a smaller grainsize than was previously available, and also shifts attention away from finalized analogical concepts and structures as the focal point for understanding students’ analogical reasoning. Instead, the ARM framework favors focusing on the parsing the developmental process of constructing analogies, thereby allowing for the identification in variation across students’ reasoning even in cases where their analogies appear to be the same concept or structure.

Appendices

Appendix 1. Group Theory Interview Protocol

Participant’s History with Abstract Algebra

• 1. How many courses in modern algebra have you had? When did you last take a course in modern algebra?

• 2. Was there any coverage of ring theory in any of your classes? If so, what did you learn?

Definition of Group

• 3. What is the definition of group?

Student Generated Examples

• 4. Provide an example of a group.

  • If no answer, ask them to recall examples from class.

  • If no answer, ask specifically about integers under addition

• 5. Provide a nonexample of a group.

  • If no answer, ask them if they can recall basic example from class.

  • If no answer, ask about integers under subtraction.

Determining if given sets are Groups

• 6. Determine if the following sets are groups:

  • i. The set of integers modulo $n$ under addition, ${\mathbb{Z}}_{n}.$

  • ii. The set of natural numbers under addition, ${\mathbb{N}}$.

• 7. Can the set {0,2,4,6,8} form a group? Why or why not?

• 8. Can the set {a,b,c} form a group? Why or why not?

Subgroups

• 9. What is the definition of subgroup?

• 10. Is Z₃ a subgroup of Z₉?

  • If no details given: How do you determine if a H is a subgroup of G?

Group Homomorphisms

• 11. What is the definition of group homomorphism?

• 12. Is the function defined by $f\left(x\right)= \frac{1}{2}x$ a homomorphism from 2Z to Z?

Definition of Normal Subgroups and Quotient Groups

• 13. What is the definition of a normal subgroup?

• 14. What is the definition of a quotient group?

Asking about Other Topics

• 15. Do any other topics from group theory come to mind that were not on this list? If so, please list them.

Appendix 2. Ring Theory Interview Protocol

Introducing Definition of Ring

• Present the participant with the definition of ring. Allow them time to study the definition. Ask them to think aloud as they study the definition.

Working with Examples of Rings

• 1. Determine if the following sets are rings under their usual addition and multiplication:

  • iii. The set of integers modulo $n,$ ${\mathbb{Z}}_{n}.$

  • iv. The set of natural numbers, ${\mathbb{N}}$.

    • If the student does not recognize the structures as groups, ask if and how knowledge of groups could have helped assess these examples.

• 2. Can the set {0,2,4,6,8} form a ring? Why or why not?

  • If answer is same/different from that given for groups, ask them to compare.

Student Generated Examples of Ring

• 3. Construct an example of a ring.

  • If no answer, ask them to recall examples from groups.

  • If no answer, ask specifically about integers under addition.

• 4. Construct a nonexample of a ring.

  • If no answer, ask them if they can recall examples from groups.

  • If no answer, ask about integers under subtraction.

Basic Proofs Involving Rings

• 5. Prove that the additive identity of a ring is unique.

  • If spontaneous analogy is not present, ask about the analogous theorem in group theory.

• 6. Let $(R,+,*)$ be a ring. Prove that ${a}^{2}- {b}^{2}=\left(a+b\right)\left(a-b\right)$ for all $a,b \in R$ if and only if $R$ is commutative.

Properties of Rings

• 7. Determine if the following properties are true for the ring ${\mathbb{Z}}_{6}$.

  • a ${a}^{2}=a$ implies that $a=0$ or $a=1$.

  • b $ab=0$ implies $a=0$ or $b=0$.

  • c $ab=ac$ and $a \ne 0$ imply $b=c$

Connections to Group Theory

• 8. What connections (if any) have you made to group theory during this session? How might you expect the study of rings to proceed?

Appendix 3. Example of Constructive Analogy Interview

Constructing the Definition of Subring

• 1. Make a conjecture for a structure in ring theory that is analogous to subgroups in group theory.

  • a If the participant is unsure of the meaning of “analogous,” assist them with an example (i.e. analogy between atom and solar system) or a diagram.

  • b When finished with spontaneous comparisons: Ask what similarities there are to subgroups.

  • c Ask what differences there are to subgroups.

Student Generated Examples of Subrings (given a Ring)

• 2. Is ${\mathbb{Z}}$ (with usual addition and multiplication) a subring of any rings?

  • If no answer, ask if Z is a subgroup of any groups.

• 3. Does ${\mathbb{Z}}$ (with usual addition and multiplication) contain any proper subrings?

Conjecturing the Subring Test

• 4. If test did not show up during structure creation, present the following task: Conjecture a way to test whether $S$ is a subring of a ring $R$.

  • If no answer, prompt them to recall the subgroup test.

Determining Examples of Subrings

• 5. Is ${\mathbb{Z}}_{3}$ a subring of ${\mathbb{Z}}_{9}$?

  • If no answer, ask to compare to response given for groups.

• 6. Suppose that R is a ring and a is an element of $R$ such that ${a}^{2}=1.$

Let $S = \{ara|r\in R\}$. Prove that $S$ is a subring of $R$.

Proofs Involving Subrings.

• 7. Suppose $R$ is a ring and $S$ and $T$ are subrings of $R$. Prove that $S\cap T$ is a subring of $R$.

Connections to Group Theory

• 8. What connections (if any) have you made to group theory during this session?

• 9. How might you expect the study of rings to proceed?

Appendix 4. Types of content

Name	Attends to:	Example
Structural Property	A specific property of a structure	Recognizing that groups and rings both require closure
Definition or Definitional Property	A mathematical definition, either taken as a whole or just one part	Looking back at the definition of ring when developing subrings
Naming Convention	The name of a structure or property in the source or target	Defining a structure in ring theory analogous to normal subgroups in group theory as “normal subrings.”
Relation	A connection within a domain between two or more other forms of content	Recognizing that subrings are a subset of rings (whether or not the same relation is explicated for groups and subgroups.)
Example	An example, usually of a mathematical structure	Calling on the example of integers from groups to initiate exploration of an example in rings
Process	An algorithm or step-by-step rules	The subgroup test
Theorem	Statements of theorems, lemmas, etc	The subgroup lemma
Proof	A mathematical proof, either taken as a whole or just one part	Recalling the proof of the subgroup lemma to assist with conjecturing a subring lemma