# Explicating *mathematical concept* and *mathematicalconception* as theoretical constructs for mathematics education research

## Abstract

Mathematical understanding continues to be one of the major goals of mathematics education. However, what is meant by “mathematical understanding” is underspecified. How can we operationalize the idea of mathematical understanding in research? In this article, I propose particular specifications of the terms *mathematical concept* and *mathematical conception* so that they may serve as useful constructs for mathematics education research. I discuss the theoretical basis of the constructs, and I examine the usefulness of these constructs in research and instruction, challenges involved in their use, and ideas derived from our experience using them in research projects. Finally, I provide several examples of articulated mathematical concepts.

### Keywords

Mathematical concept Mathematical conception Mathematical understanding Mathematics learning Instructional goals“In the field of mathematics education, the ‘quest for understanding’ is akin to the (re)search for the Holy Grail.” (Llewellyn, 2012, P 385)

For more than 25 years, mathematics educators have been stressing the goal of mathematical understanding. The authors of the National Council of Teachers of Mathematics (1989) asserted that an emphasis of the curriculum should be “mathematical concepts and understanding.” Mathematics education researchers and instructional designers take mathematical understanding as a primary emphasis.^{1} However, what is meant by “mathematical understanding?”

In the literature, there are many discussions of mathematical understanding. However, they tend to be of three types: vague specification (e.g., “Conceptual understanding refers to an integrated and functional grasp of mathematical ideas,” Mathematics Learning Study Committee, 2001, p. 118.), descriptions of the state of having understanding (e.g., “rich in relationships,” Hiebert & Lefevre, 1986, p. 3), and identification of what students can do when they understand (e.g., “justify… why a particular mathematical statement is true or where a mathematical rule comes from,” National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 4).

How do we translate the idea of mathematical understanding into usable constructs that can support research and design focused on the learning of particular mathematical topics? How can we translate this emphasis into specific goals for instruction? How can we assess and characterize student learning with respect to these instructional goals? In this article, I present an evolving characterization of the theoretical constructs, *mathematical conception* and *mathematical concept*—constructs that can be used in addressing these questions. After characterizing each construct, I explain both the intersection and the non-intersection of the sets specified by these constructs.

^{2}combined with theoretical work that my colleagues and I have been doing. The expressions

*mathematical concept*and

*mathematical conception*are commonly used in mathematics education. However, the meanings of the terms are often not problematized and as a result are typically underspecified. Balacheff and Gaudin (2010) described this problem:

The word “conception” has been used for years in research on teaching and learning mathematics. It functions as a tool, but its definition remains implicit; it has not yet been taken as an object of study per se. … The expectation is that a better definition will allow us to analyze the differences and commonalties between conceptions in such a way that we will have a better ground to design learning situations. (p. 212)

Some mathematics education researchers distinguish conception from concept by using the former to refer to the understanding of learners and the latter to represent taken-as-shared ideas of the mathematics education community (c.f., Sfard, 1991). This distinction still leaves both terms underspecified. In this article, I go beyond this distinction. I describe the results of our efforts to increasingly specify these constructs and make them useful for mathematics education research and development.

Defining mathematical conception and mathematical concept is problematic in that any definition involves other terms (e.g., understanding, thinking, meaning, knowing) that are themselves not precisely defined. So rather than attempting to generate a precise definition, I treat mathematical concept and mathematical conception as evolving theoretical constructs. As such, I elaborate aspects of the constructs in an attempt to give them greater specificity and power. At points in the article, I highlight in italics important aspects of the constructs just elaborated to make explicit the accumulating characterizations of the constructs.

Multiple interrelated goals are important in mathematics education, including problem posing and solving, justification of mathematical claims, communication of mathematical ideas, and conceptual understanding. In this article, I foreground conceptual understanding and specify how the constructs of mathematical concept and conception can be useful in organizing thinking about promoting and analyzing understanding. I begin by laying a foundation for the discussion.

## 1 Theoretical orientation

In Simon (2009), I expressed the idea that different theories of learning are tools for doing different kinds of work. The work my colleagues and I engage in is informed by multiple theories, particularly constructivist, emergent, and sociocultural. However, the work described here is primarily grounded in constructivism.

The idea of a concept is not historically limited to constructivism. Key to Vygotsky’s (1962) sociocultural theory was a focus on “scientific concepts” as distinct from “spontaneous concepts.” Conceptual development was central to the work of both Vygotsky and Piaget (e.g., Piaget, 2001). The more significant divide is between scholars who take conceptual structures to be the goal of learning and scholars who take social participation to be the goal of learning (c.f., Lave & Wenger, 1991). I have argued (Simon, 2009) that the question is not the validity of these theoretical stances, but rather the affordances of different perspectives on learning. I leave it to others to elaborate the affordances of situated learning and focus here on elaborating the constructs of mathematical concept and mathematical conception based on a constructivist perspective.

Rather than reviewing relevant constructivist ideas ahead of the main thesis of the article, I have elected to discuss these ideas where needed throughout the article. A detailed discussion of our elaboration of reflective abstraction can be found in Simon, Placa, and Avitzur (2016).

In the next three sections, I discuss mathematical conceptions, mathematical concepts, and the relationship between the two constructs.

## 2 Mathematical conceptions

Many of the products of mathematics education research are in the form of models … that attempt to describe and explain the beliefs or behavior of pupils and/or teachers. (Brown, 1998, p. 265)

I use mathematical conception to refer to an explanatory model generated by mathematics educators^{3} to explain behaviors (including verbalizations) of mathematics learners in terms of what they think, know, and understand, what Steffe (1995) called “second order models.”^{4}^{5} As such, it is a theoretical model created by the observer (researcher, teacher) to make sense of her observations and not a claim about what is true for the learner.

Scientists often think in terms, of theoretical explanatory models, such as molecules, waves, fields, and black holes, that are a separate kind of hypothesis from empirical laws. Such models, are not merely condensed summaries of empirical observations, but, rather, are inventions that contribute new mechanisms and concepts that are part of the scientists’ view of the world and that are not “given” in the data. (p. 549):

A conception is a researcher’s invention of a learner’s way of knowing that would yield the observed results. Postulating particular conceptions allows researchers to characterize or model sets of student (or teacher) data and to predict further performance and potential learning. The effectiveness of a postulated mathematical conception is not judged by its fit with “reality,” a comparison that cannot be made, but by its viability, its ability to do the work for which the model was postulated. As Ricco (1993) asserted, “The utility of theories or models, … is a thoroughly empirical question” (p. 142).

*A mathematical conception is an explanatory model used to explain observed abilities and limitations of mathematics learners in terms of their (inferred) ways of knowing.*^{6}

## 3 Mathematical concepts

I now turn to characterizing mathematical concepts. After doing so I will discuss the relationship between mathematical concepts and mathematical conceptions.

### 3.1 Relationship between mathematics and mathematical concepts

There is an important gap between mathematical knowledge and knowledge in other sciences such as astronomy, physics, biology, or botany. We do not have any perceptive or instrumental access to mathematical objects, even the most elementary, as for any object or phenomenon of the external world. We cannot see them, study them through a microscope or take a picture of them. (p. 61)

Sierpinska (1994) wrote,

Mathematical objects are creations of the human mind. But, embedded in a system of logical necessities and consequences of their relations with other mathematical objects, they may have properties that can be hard to discover, or difficult to prove or disprove. (p. 30)

Whereas Sierpinska, referred to mathematical objects as “creations of the human mind,” they can also be understood as socially constituted (Godino, 1996). The logic applied to considering the relations between objects is itself a social construction.

Understanding mathematics is not reducible to mathematical definitions and theorems (Vergnaud, 1997). For example, rational numbers are defined as numbers that can be expressed as *a/b,* where *a* and *b* are integers, and *b* is not equal to 0. This definition can be learned, and students can use it to distinguish between rational numbers and numbers that are not rational numbers. However, the definition does not provide insight into what it means to understand fractions or ratio. It is this distinction that underlies our attention to mathematical concepts. *Mathematical concepts consist of mathematical (mental) objects and the relationships among those objects*.

### 3.2 A classroom scenario

In a fourth-grade class, I asked the students to use a blue rubber band on their geoboards to make a square of a designated size, and then to put a red rubber band around one half of the square. Most of the students divided the square into two congruent rectangles. However, Mary cut the square on the diagonal, making two congruent right triangles. The students were unanimous in asserting that both fit with my request that they show half of the square. Further, they were able to justify that assertion by explaining that each of the parts was 1 of 2 equal parts and that the two parts made up the whole. I then asked, “Is Joe’s (rectangular) half larger; is Mary’s half larger, or are they the same size?” Approximately a third of the class chose each option. In the subsequent discussion, students defended their answers. However, few students changed their answers as a result of the arguments presented.

When I have shared this scenario with groups of mathematics educators and asked them for a teaching intervention they would use, they overwhelmingly responded that they would engage students in cutting up the triangular half and superimposing it on the rectangular half to demonstrate that the areas are the same size. I take these responses to be serious attempts to promote understanding. However, I contend that the proposed intervention does not promote the mathematical understanding that needs to be learned. The proposed activity might help students learn that the two halves in this case *happen* to be the same size (or with additional similar tasks, that there seems to be a regularity in the results), but they are unlikely to learn from that activity that the two halves *must be* the same size. We mathematics educators know that the halves must be the same size, even though most of us have never carried out the empirical exercise of cutting up and superimposing one half on the other. I will discuss this in greater depth in the next section and use this scenario to illustrate several of the points I make in this article.

### 3.3 The nature and origin of mathematical concepts

Piaget distinguished between three types of knowledge: physical knowledge, logico-mathematical knowledge, and social knowledge (Kamii, 1986). Physical knowledge refers to knowledge of the properties of physical objects (e.g., fire is hot). Logico-mathematical knowledge refers to knowledge that logically follows from other aspects of our knowledge (e.g., partitioning a whole into a greater number of equal parts creates smaller parts). Although from a sociocultural perspective, all knowledge is social, Piaget used “social” to indicate knowledge that has been arbitrarily determined by a social group (e.g., vocabulary, symbols). In mathematics education, we are concerned with all three types of knowledge. However, mathematical *concepts* belong to the category of logico-mathematical knowledge.

In Simon (2006), I distinguished between two processes of learning, *reflective abstraction* and *empirical learning processes*. Piaget (2001) contributed the former construct and I modified Piaget’s *empirical abstraction* in specifying the latter. Piaget (2001) used *empirical abstraction* to refer to an abstraction of properties of physical objects or the material aspects of a physical action. In contrast, I use empirical learning processes to include all inductive processes, not only those based on physical objects or actions. Empirical learning processes are conclusions that derive from inductive processes, that is, multiple trials in which students select (or observe) inputs and observe corresponding outputs -- essentially a black-box process. For example, a student could multiply integers by 6 and find that the results are even. This is a result of an empirical learning process, which results in knowing *that* something seems to be true. The process does not generate understanding of the logical necessity of that result. Likewise, students in the scenario in Section 3.1, who superimpose pieces of the triangular half on the rectangular half, come to know *that* the halves are the same size, but lack understanding of the logical necessity of that equivalence.

There are two related claims about mathematical concepts embedded in the last paragraph:

*Mathematical concepts involve logico-mathematical knowledge and, therefore, involve knowing the logical necessity of the relationships involved.*^{7}

*Mathematical concepts are the result of reflective abstraction and not the results of an empirical learning process*. Empirical processes have their place in mathematics, for example as a basis for making a conjecture. The claim here is that a mathematical concept is *not* the direct result of an empirical process.

I now provide a brief discussion of reflective abstraction. I build on Ohlsson and Lehtinen’s (1997) notion that “the cognitive function of abstraction is not to provide wide applicability -- transfer -- but to enable the assembly of existing ideas into more complex ideas” (p.38). This is a critical distinction and avoids conflation of *abstraction* and *generalization*.

Skemp (1986) argued that a mathematical concept is the result of abstracting “an activity by which we become aware of similarities … among our experiences” (p. 21, cited in White & Mitchelmore, 2002). Similarly, Piaget (2001) defined reflective abstraction as a coordination of actions. “According to Piagetian constructivist theories, people’s acts must be considered the genetic source of mathematical conceptualization” (Godino, 1996, p. 3). So, an important contribution of Piaget’s reflective abstraction is that mathematical concepts are not abstractions of one’s observations (as in an empirical learning process), but rather an abstraction from one’s own activity (e.g., numerosity abstracted from rudimentary counting, arithmetic mean abstracted from “balancing” a set of data). Piaget (2001) pointed out that the process of reflective abstraction results in a learned anticipation, so that the learner no longer needs to enact the activity to know its result. *A mathematical concept is the result of reflective abstraction, a learned anticipation based on the learner’s activity*. Note, concepts can be modified by processes other than reflective abstraction, such as generalizing assimilation (Piaget, 1952). My claim is focused on the construction of *new* concepts.

### 3.4 What is a mathematical concept?

If mathematics education theories want to provide suitable accounts of learning they need to clarify what they believe constitutes knowledge and knowing in the first place. … we cannot understand learning if we do not provide a satisfactory explanation of what learning is about. (Radford, 2013, p. 8)

I consolidate the ideas about mathematical concepts into the following statement. *A mathematical concept is a researcher’s articulation of intended or inferred student knowledge of the logical necessity involved in a particular mathematical relationship*. I emphasize four points about this statement. First, my use of “intended or inferred” indicates that mathematical concept can be used to characterize a learner’s understanding (a conception) or to specify a learning goal (an intended conception). Second, researchers may specify different concepts related to the same mathematical relationship. There are diverse ways to characterize understanding in a particular area. Third, the specification of a concept is always in relation to assumptions about prior knowledge of learners involved; coming to know the logical necessity must be based on prior knowledge and available reasoning. (See the example in the next paragraph.) Fourth, the specification of the concept is the researchers’ characterization of the students’ knowledge; it is *not* meant to capture student language. Anticipating student language may be a useful exercise, but specification of a mathematical concept is done by using the researchers’ most accurate language for specifying the learner’s understanding of what might be an elementary idea.

The denominator of a unit fraction gives the number of parts of that size that make up the related whole. Equal partitioning of a whole into a greater number of parts (sharing it more ways) results in each part being smaller. Therefore, the larger the denominator is, the smaller the unit fraction must be.

Note that this specification of the concept represents my attempt to articulate it clearly. Students’ language might not be as clear or well organized. However, in articulating the concept, I have tried to capture that which the student knows, can think about, and use –such as the ideas of the diminishing size of a share as the whole is shared in more ways.

Partitioning a quantity into

nequal parts creates parts that are 1/nthe size of the original quantity. That is, becausenof the parts are equivalent to the quantity, the quantity isntimes as large as the parts. Therefore, 1/n is a particular size relative to the original quantity.

Note, that the concept articulated is the one judged to represent the critical understanding and says nothing about cutting a square in different ways. If a student has the concept articulated, then the comparison of the triangular piece and the rectangular piece is simple, “They are the same, they are both half.”

## 4 Reviewing and relating the two constructs

In this section, I summarize the points made for each construct and then discuss the relationship between the two constructs.

A mathematical conception is an explanatory model used to explain observed abilities and limitations of mathematics learners in terms of what they understand. A mathematical concept is the result of reflective abstraction – a learned anticipation. A mathematical concept is a researcher’s articulation of intended or inferred student knowledge of the logical necessity involved in a particular mathematical relationship.

*A*in Fig. 1) are a particular type of conception. When the knowledge modeled does not meet our criteria for a concept, we classify it as a conception, but not a concept (region B in Fig. 1). Finally, a concept that is constructed to model potential knowledge of learners would be classified as a concept, but not a conception (region C of Fig. 1).

Let us go back to the scenario in Section 3.1 to exemplify a conception that is not a concept. One can ask a question that was probably not considered in the thinking of the mathematics educators who suggested having students cut up the triangular half: How might the students, who said that the triangular half was bigger, conceive of one-half? Whenever I pose this question to a group of mathematics educators, it proves to be challenging. In Simon (2006), I postulated that these students have a conception of a “fraction as an arrangement.” That is, they see *1/n* as a part of a whole that contains *n* identical parts. This notion of identical parts is not a claim about quantity (amount), just sameness and the set that they are part of (the *n* identical parts of that whole).^{8}

Whereas fraction as an arrangement is a mathematical conception that I postulated to account for observed behavior (verbalization) – an explanatory model, it is not a mathematical concept. Specifically, the mathematical conception as described could be explained resulting from an empirical learning process, the perception of a pattern between a set of representations (wholes cut into identical parts) and a set of labels (names and symbols for the unit fractions). In contrast, the *concept* of fraction that we might envision is not about identical parts, but rather about quantity.

## 5 Contrasting mathematical concept and *scheme*

Because development of the construct of mathematical concept is grounded in constructivist theory, it is appropriate to inquire into the relationship between mathematical concept and scheme, a construct that has been an important part of constructivist theory (Steffe, 1983). Much of the work using scheme theory overlaps with and provides a foundation for the discussion of concepts and conceptions presented here. In this section, I highlight a few distinctions.

Piaget developed the idea of a scheme during his early work, which was predicated on producing a biological explanation of knowledge development (Montangero & Maurice-Naville, 1997). The scheme construct was (and is) used to characterize adaptation through equilibration (e.g., Steffe, 1992).

- 1.
Recognition of a certain situation;

- 2.
A specific activity associated with that situation; and

- 3.
The expectation that the activity produces a certain, previously experienced result. (p. 65)

In Piaget’s later work, reflective abstraction became a central idea (Campbell, 2001). This change from a biological (equilibration) to a structural (reflective abstraction) account of learning was a major shift and a different type of explanation of conceptual learning. My colleagues and I found it useful to move away from the three-part scheme concept, because of its association with equilibration theory, and create a construct, mathematical concept, that is more in line with our evolving understanding of reflective abstraction in mathematics learning.

Whereas, both scheme and mathematical concept are ways of characterizing knowing with a focus on learners’ activity, I explain three points of contrast that underlie our use of mathematical concept.

### 5.1 Relationship to reflective abstraction

*not*a claim that the activity has resulted in reflective abstraction. In contrast, I consider a mathematical concept to be the result of reflective abstraction. Some mathematics education researchers, who use the scheme construct, refer to the result of reflective abstraction as an “interiorized scheme” (c.f., Boyce, 2014; Steffe & Wiegel, 1994). Hackenberg (2010) made explicit this relationship between mathematical concept and scheme,

I take a concept to be the results of schemes that people have abstracted from their use of the schemes. … An equivalent way to talk about concepts is as the results of schemes that have been interiorized.

A focus on mathematical concepts, the result of reflective abstraction, provides direction for discussion of understanding as a goal of mathematics education. That is, the growth of understanding can be understood as the building of higher-level concepts from extant ones through reflective abstraction.

### 5.2 Usefulness in explicating reflective abstraction

*a complex of a goal and an action*.

We represent a concept as G

_{n}-A_{n}. G_{n}is the goal, and A_{n}is an action that produces a result that accomplishes the goal. The complex of goal and action, represented by G connected to A by a hyphen, signals an available concept that was constructed through reflective abstraction. We use the subscripts to be able to talk about concepts and their goals and activities at different levels. We use subscript 1 for the concept whose development we are trying to explain and subscript 0 for the concepts that were already established and served as building blocks for the developing concept. (p. 83)

^{9}I summarize our chain of reasoning (See Simon et al., 2016, for details of our elaboration of reflective abstraction.):

Reflective abstraction derives from an activity

An activity is a sequence of actions already available to the learner

Actions that are called on as part of an activity do not exist in isolation; they are part of prior concepts (G

_{0}-A_{0}).^{10}These actions are called on, because the goal of the available concept (G_{0}) is a subgoal of the activity.Reflective abstraction involves a coordination of actions (A

_{0}) to produce a higher level action (A_{1}).That coordination is actually a coordination of the concepts (of which these actions are part) into a higher level concept.

### 5.3 Articulation of a concept

This third point is more of a question about the use of these constructs in research. Articulation of a mathematical concept involves researchers’ best efforts to characterize learners’ understanding by articulating the knowledge of logical necessity. Thompson and Saldanha (2004), asserted that research focused on conceptual schemes aims to answer the question, “Which mental operations do we perform in order to conceive a situation in the way we conceive it?” (p. 99). To what extent do researchers using these constructs produce different models?

To summarize the three differences just discussed, knowledge can be characterized using the scheme construct. However, a claim of a concept is a stronger claim than a claim of a scheme -- not all schemes are concepts. An interiorized scheme, like a mathematical concept, identifies the results of reflective abstraction, however, to what extent do models based on the former specify the knowledge of logical necessity constructed?

## 6 Contrast with concept image and concept definition

*concept image*and

*concept definition*are widely used in mathematics education. In contrasting these constructs and the constructs discussed in this article, we can see differences in how the constructs were built up and used. After quoting the definition of Tall and Vinner’s constructs, I highlight some of these differences.

We shall use the term

concept imageto describe the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes. It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures.We shall regard the

concept definitionto be a form of words used to specify that concept. It may be learnt by an individual in a rote fashion or more meaningfully learnt and related to a greater or lesser degree to the concept as a whole. It may also be a personal reconstruction by the student of a definition. It is then the form of words that the student uses for his own explanation of his (evoked) concept image. Whether the concept definition is given to him or constructed by himself, he may vary it from time to time. In this way apersonalconcept definition can differ from aformalconcept definition, the latter being a concept definition which is accepted by the mathematical community at large. (Tall & Vinner, 1981, p. 152)

I now highlight differences between the two pairs of constructs. Concept definition and concept image are both defined in relation to a “concept.” However, the meaning of *concept* is not explained. Concept seems to be considered as an unproblematic referent indicated by the name of the concept. The researchers refer to “concept of subtraction,” “concept of limit,” and “concept of continuity.” Thus, the researchers are interested in the definition and/or image that is evoked by the student in response to the term (or symbol) representing the “concept.” A conception on the other hand, is a model of the learner’s knowing; there is no preconceived notion of what mathematical objects might be implicated. Postulating a conception for a learner starts with interpretation of the learner’s actions and verbalizations, not with a particular mathematical object.

Concept definition and concept (as I have defined it) are quite different. The former refers to specification of a particular mathematical object. The latter is meant to capture an aspect of knowing. For example, one could specify a definition of rational number without discussing the knowledge of logical necessity related to comparing the size of two fractions.

One final point of contrast: mathematical concept and mathematical conception are researcher-invented models. This notion was not an explicit part of Tall and Vinner (1981).

## 7 The usefulness of the constructs mathematical concept and mathematical conception

### 7.1 Specifying learning goals using mathematical concepts

“Conditions for improved instruction entail an enduring discussion of what the community intends students learn” (Thompson & Saldanha, 2004, p. 110). I start with the assumption that effective mathematics instruction requires mathematics educators to have a clear idea about the mathematical ideas to be promoted. This is not a deterministic position. Instruction does not result in a replication of the goal understanding in students. Rather, the articulation of the (revisable) goal understanding serves to organize pedagogical thinking about instruction. The construct of mathematical concept can be used to articulate goal understandings. As an envisioned mathematical conception, a mathematical concept specifies the nature of student knowledge that is the aim of the instructional intervention.

How have mathematics educators specified their goals for instruction when mathematical concepts, as characterized in this article, are not articulated, but understanding is valued? I have observed two common approaches. The first is a goal of understanding a *topic* (e.g., “understanding fractions” or “understanding limit”). The other approach (which can be explicit or implicit) is to have a goal that students learn to solve a particular type of problem. However, these approaches offer no description of the intended student knowledge and therefore may be inadequate for focusing instruction, assessment, and research.^{11}

The scenario in Section 3.1 can serve to exemplify the disadvantage of specifying goals in terms of tasks to be solved and the advantage of using the construct of mathematical concept for establishing instructional goals. What was the goal (likely implicit) of the educators who advocated having students superimpose the parts of the triangular half on the rectangular half? It is reasonable to assume that their efforts were aimed at the students coming to know *that* the two halves are equal in area, that is to be able do the task correctly. They may have had a more general goal, such as knowing that fractions of different shapes have the same area. This still does not extend much beyond the ability to do these kinds of tasks. It is a natural human response, when faced with students failing to solve a problem, to help them learn to solve the problem successfully. However, an explicit and more useful instructional goal might afford a more effective instructional intervention.

Let us consider the effect of establishing a mathematical concept as the goal and how it might change the set of potential instructional interventions. What is it that the students in the scenario do not seem to understand that we would like them to understand? In Simon (2006, p. 361), I termed the idea “fraction as quantity^{12}” to contrast it with “fraction as arrangement.” I can articulate the concept (including the logical necessity) in the following way. *Partitioning a unit into n equal parts creates parts one of which will iterate n times to make the whole. Iterating a small quantity n times produces a specific large quantity that is n times as large. So partitioning a unit into n equal parts creates parts of a particular size.* I now use this articulation of the mathematical concept as the learning goal to examine how it might change the consideration of potential instructional interventions.

- 1.
Estimate the size of the piece

- 2.
Iterate the estimate 5 times to make a new stick.

- 3.
Compare the lengths of the new stick and the original stick.

- 4.
If the lengths are different, adjust the estimate of the part and repeat the process.

Tzur reported that students using this strategy, for a set of tasks of this type, developed an understanding that a unit fraction is a part of a particular size relative to the whole.

### 7.2 Focusing assessment items using mathematical concepts

An ongoing challenge for mathematics education is the assessment of mathematical understanding. Unlike the assessment of procedural knowledge and rote knowledge, assessment of understanding has some additional challenges. I focus on two of the challenges here. First, it is more difficult to specify the knowledge to be assessed. Second, if one creates assessment tasks similar to the instructional tasks used to promote the understanding, it is difficult to know if a successful solution is due to understanding or just the ability to reproduce a solution strategy learned in class.

Mathematical concepts provide a way of addressing both of these challenges. First articulation of the mathematical concept as the goal of instruction also provides a specification of the focus of assessment. Second, as described in the last section, a mathematical concept is not linked to a particular type of mathematical task. Thus, a diverse set of tasks can be used to assess a concept, providing potential items that are different from the tasks used in instruction.

### 7.3 Accounting for student data using mathematical conceptions

Above, I defined mathematical conception as an explanatory model. Endeavoring to find an explanatory model that accounts for a (frequently puzzling) set of data is a productive form of analysis. It is a way of making sense of the data and of having a holistic description of the student’s understanding. Mathematical conceptions already described in the literature give mathematics education researchers (and teachers) tools for examining student understanding and noticing key aspects of student behavior.

### 7.4 Characterizing differences in understanding using mathematical conceptions

Once individuals’ understandings have been characterized in terms of mathematical conceptions, there is an opportunity to compare individuals’ understandings. Mathematical conceptions offer a way to make distinctions among students, distinctions that go beyond how they perform on a particular task.

Mathematical conceptions also provide a basis for claims of and specification of learning. Research that demonstrates and analyzes learning is one of the significant challenges in mathematics education research (Siegler, 1995). Comparing the mathematical conceptions of a student at two points in time can provide not only a warrant for the claim that learning took place, but also the basis for specifying the nature of the change.

### 7.5 Developing learning trajectories using mathematical conceptions and mathematical concepts

Design of a hypothetical learning trajectory (HLT, Simon, 1995) for promoting particular conceptual learning requires knowledge of students’ extant knowledge and clear articulation of the goal understanding. The juxtaposition of two mathematical concepts or a mathematical conception and a related mathematical concept can provide the “endpoints” for the conjectured learning. Mathematical conceptions can be used to characterize the relevant mathematical thinking that the students are capable of engaging in at the outset. The mathematical concept provides a specific goal for the pedagogical intervention.

An active area of work in mathematics education is the development of learning progressions. A learning progression is a multi-step trajectory through a mathematical area. To date there has been a lack of attention to and problematizing of the unit of a learning progression. How might we specify the knowledge represented by each step in the progression? Certainly, if our priority is student understanding, it is insufficient to characterize those steps in terms of behaviors of students or tasks that they can solve (Tzur et al., 2013). Mathematical concepts can be used to articulate the conjectured steps in the progression.

## 8 The challenge of specifying mathematical conceptions and mathematical concepts

### 8.1 Specifying mathematical conceptions

I have discussed mathematical conceptions as explanatory models (Clement, 2000), inventions of the researcher to account for pertinent aspects of the data. Researchers are challenged to make sense of students’ thinking when students act in ways that are inconsistent with how the researchers would act in the situation and different from expected behaviors. The challenge is for the researchers to specify a mathematical conception that would account for the set of observed behaviors. My research team works from the maxim, “We do not understand the research subjects’ perspective, if we have not postulated a perspective from which everything they do makes sense.”^{13} This is often difficult to achieve.

Researchers’ postulation of a mathematical conception is neither the result of inductive (pattern noticing) nor deductive (logical) thinking, although both are necessary ways of thinking in conducting research and may provide groundwork for postulating a mathematical conception. Rather, the generation of an explanatory model, which is not observable in the data nor deducible from them, can be thought of as an abductive process (Anderson, 1986). This is a somewhat mysterious process. I will use an analogy to describe it.

Readers, for decades, have been fascinated by stories about Sherlock Holmes, the detective (Doyle, 1930). One of Holmes’ abilities was his extremely acute power of observation. He was able to perceive clues that others would not notice. However, the ability that is relevant here was his ability to come up with an explanation for the diverse set of clues that he had amassed. How did he come up with this explanation that we readers were not able to come up with? The mathematics education researcher is challenged to come up with explanations in much the same way.

The mysterious part of the abductive process is that an idea for an explanatory model may come to mind all of a sudden, or it may not come at all. I know of no way to explain or directly teach this abductive process. However, I share three ideas about preparation for coming up with an explanatory model. Each idea is exemplified using the scenario in Section 3.1. First, the researchers consciously *set aside their own perspective* (Steffe & Thompson, 2000). In the scenario, students were considering two different shaped halves. I do not assume that what the students meant by “one half” is what I mean by “one half.” Second, the researchers *immerse themselves in the data*. This is not just a process of knowing the basic story. It is in some ways similar to an actor preparing to portray a real person. The actor watches not only what the subject does and says, but also her expressions, gestures, and tone of voice. The goal is to know the subject well enough, to act *as* that person. Third, the researchers *make small local inferences*, asking themselves what each relevant action or verbalization might indicate. These local inferences also require abductive reasoning, but on a much smaller scale. For example, when the students say that the two rectangular halves are the same size, what might they be claiming? They might to be claiming that one part is superimposable on the other (i.e., they look identical in size and shape). This set of local inferences creates a set of clues that support postulating a mathematical conception (making a larger abduction), a level of support that is not available from the data themselves.

I have given three mental activities that support the coming up with a mathematical conception. However, even more fundamental is having the goal of coming up with a mathematical conception and a sense of what it would mean to do so successfully.

### 8.2 Specifying mathematical concepts

The articulation of goal understandings (concepts) is difficult and remains an ongoing problem for mathematics education. There seems to be consensus that we want students to understand place value, ratio, statistical variation, and a host of other mathematical topics. However, what does it mean to understand these topics? What understandings are involved? How might researchers go about addressing these questions?

It is generally impossible for mathematics educators to remember what it was like to not understand particular topics and to have access to how that understanding developed. Some useful preliminary work can be done by analyzing a particular conceptual area, what Dubinsky and Lewin (1986) called genetic decomposition, or by analyzing the knowledge needed to generate a particular solution to a problem known to be difficult for students. However, I have found the most generative approach to specifying concepts is the opportunity to observe contrasts in individuals’ mathematical functioning. (Of course, the nature of the task is important in what can be observed.) Sometimes the useful contrast is between a student and the mathematics educator who is observing the student. Sometimes the useful contrast is between two students or a single student at two points in time. The contrasts provide a particular focus for the inquiry into understanding. In such situations, the postulation of a mathematical concept becomes a specific explanation of the difference in functioning observed. As in the discussion of mathematical conceptions above, a mathematical concept cannot be deduced or observed. It is an abductive act on the part of the researcher.

Returning again to the scenario in Section 3.1, we can examine how the contrast can frame the abductive process. Consider the contrast between the students who said the triangular half was bigger and those that said the two different shaped halves were the same size. By focusing on that contrast, I realized that the former students did not assume that half of the original square was a part of a particular size relative to the whole. My postulation of the concept fraction as a quantity-- equi-partitioning a unit results in the creation of a new (partial) unit of a particular size relative (multiplicatively related) to the whole –was an attempt to characterize in a useful way the understanding that separated the individuals contrasted. The critical point is not the concept I postulated, but that I might never have focused on that concept, if I did not have the opportunity to observe the students who did not know that the triangular and rectangular halves were the same size. In the same way, Piaget (2001) would likely not have thought about class inclusion or conservation without observing students who acted in ways that we now define as not having those concepts.

One final point: for any articulated mathematical concept, subsequent research and/or pedagogical activity may reveal the inadequacy of the articulation of the concept (i.e., it does not do the work we need it to do) and may require articulating the concept in greater detail, emphasizing a different aspect of understanding, and/or postulating component understandings. Thus, articulation of mathematical concepts is a process that has no end.

## 9 Examples of mathematical concepts

One of my challenges in communicating about the construct of mathematical concept is to communicate what it means to articulate a concept. In our work, we attempt to articulate a concept by saying exactly what we would expect the student to understand. The articulation is meant to capture the logical necessity that the student would come to know. This does not mean that someone who develops the understanding would be able to articulate it in this way (or at all). However, it is helpful to use language that captures the thinking of the student.

Understanding comparing unit fractions: If I have a whole, the more equal parts I cut it into (think, more ways we share a pizza), the smaller the parts will be. Because the denominator represents the number of equal parts that the whole is cut into, the unit fraction with the larger denominator represents the smaller part (assuming that both fractions refer to the same whole).

Understanding ASA (angle-side-angle for demonstrating congruence): I am given the measures of two angles of a triangle and the length of the included side. If I were to draw the given side and extend rays from the end of the line segment (on the same side of the line segment) at the angles given, the rays would intersect at one and only one point.

^{14}This point would be the third vertex of the triangle and define one and only one triangle. If only one triangle can be made from the information given, it must be identical to the triangle from which the information was taken.Understanding cardinality: When I count, I say my counting words in order and touch a different object for each word. When I have no more objects to touch I stop counting. If there are more objects I can say more counting words. So depending on how far I get in my counting, that reflects how many objects there are. The last number tells me how many objects there are. [The reader will note that young children who develop this concept will not be able to articulate any of this. This is a researcher’s attempt to articulate their understanding.]

Understanding which unit fractions are equivalent to terminating decimals. Every terminating decimal can be written as an integer divided by a power of ten (e.g., 469/10,000). The fraction

*1/n*can be rewritten as (10^{p})/(*n*x 10^{p}). The numerator of this new fraction has prime factors of 2 and 5 only. So, if*n*has only 2’s and 5’s as prime factors, they will cancel out (form ones) with factors of the numerator, leaving any uncancelled factors in the numerator and only 10^{p}in the denominator – our definition of a terminating decimal. However, if*n*has any prime factors other than 2 and 5, they will not cancel out. In this case, the fraction is not equivalent to a terminating decimal.

## 10 Summary and implications

In this article, I discussed how two previously underspecified terms in mathematics education, mathematical conception and mathematical concept, can be characterized to serve as useful theoretical constructs. I distinguished between the two constructs and demonstrated how they can be used separately and together. I have described how their use can provide greater specification to what is meant by mathematical understanding. I have argued that the two constructs give us ways to articulate learning goals, characterize students’ extant knowledge, distinguish individual differences, specify assessment targets, organize and make sense of student data, and provide key components of learning trajectories/progressions. The power in the use of these two constructs derives from their potential to represent a level of analysis that is deeper than a description or categorization of students’ behaviors or the tasks that they can solve.

In Simon (2004), I argued that one of the reasons that mathematics education research findings often do not build on each other is that studies frequently do not have sharply defined research problems. The use of the two constructs described here is a way to more precisely define both students’ current and potential knowledge. A better defined learning challenge can provide a specification of a research problem that can be addressed by multiple research teams and, thus, make possible a greater potential for accumulation of research knowledge. Further, the articulation of mathematical concepts allows for productive discussion about what it means to understand particular mathematical ideas.

My observation (and the observation of colleagues with whom I have spoken) is that the mathematical portion of the Common Core State Standards (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) and many recent mathematics textbooks, are uneven across topics in the extent to which they focus on mathematical understandings. Well-articulated mathematical concepts for each topic would provide direction for instructional design that fosters understanding. In addition, clear articulation of mathematical concepts would potentially be helpful to teachers by giving them a clearer image of the goal of their instruction.

## 11 Directions for future work

As discussed, mathematical understanding has been a somewhat vague idea and mathematical conception and mathematical concept have been underspecified terms in mathematics education. This article describes theoretical work to address these issues. However, it is only a beginning and work is needed to take these efforts further. I specify here two important areas of investigation that derive from the ideas presented.

First, whereas specifying meaning for mathematical concept creates greater clarity about what is and what is not an example of a mathematical concept, it also may exclude or ignore other types of mathematical knowing that need to be recognized. Other research teams, armed with the distinctions articulated here, can contribute to important aspects of mathematical understanding that have not been addressed by this formulation of mathematical concept.

Second, the mathematical domain that has been the context for my empirical and theoretical work, and therefore for the development of the constructs discussed in this article, stretches from elementary number concepts through first courses in geometry and algebra in secondary schools. I have not worked on the development of advanced mathematical thinking. It is an interesting and important question as to whether the theoretical development of mathematical concept, discussed here, is applicable as is, in need of modification, or not applicable for research on the learning of advanced mathematics.

## Footnotes

- 1.
Whereas, there is a major focus within mathematics education on understanding, researchers, who employ non-cognitive theories, do not necessarily take understanding as a goal of mathematics instruction (c.f., Dowling, 2013).

- 2.
Because of the synthetic nature of this article, I do not do a literature review in advance. Rather, I bring in foundational and related literature as it fits into the development of the constructs.

- 3.
I use “mathematics educators” as a general term to refer to all who are involved in mathematics education (researchers, teachers, curriculum developers, university mathematicians). Key, as discussed, is their role as observer.

- 4.
Second-order models can be considered as a subset of Marton’s (1981) “second-order perspectives.”

- 5.
Ulrich, Tillema, Hackenberg, and Norton (2014) recently exemplified second-order models.

- 6.
Balacheff and Gaudin (2010) have taken a different approach to modeling a conception within the framework of French Didactical Theory. Rather than focusing specifically on the students’ thinking, Balacheff and Gaudin focused on the state of the “subject/milieu system.”

- 7.
The usefulness of this distinction will become clearer in the Section 7, “The usefulness of the constructs mathematical concept and mathematical conception.”

- 8.
I started with the assumption that as 4th graders, 9–10 years old, they already had conservation of area.

- 9.
One of the reasons for this formulation is that evolution in the learners’ goals is key to our explanation of learners’ progression through the sub-stages and stages of developing a concept. See Simon et al. (2016) for this progression.

- 10.
Boyce (2014) seemed to hold a similar idea, “Only an interiorized scheme can become part of another cognitive scheme.”

- 11.
- 12.
The fact that I gave names to the mathematical conception (fraction as arrangement) and the mathematical concept (fraction as quantity) is neither typical nor important.

- 13.
This maxim is consistent with Steffe’s longstanding push for researchers to focus on the rationality of students’ mathematics (e.g., Steffe & Thompson, 2000).

- 14.
Note that logical necessity and mathematical proof are not the same. In this case, the student can reason about the unique paths of the rays, relative to the line segment, and the single point of their intersection. Unlike mathematical proof, logical necessity does not require the formal building of an axiomatic system.

## Notes

### Acknowledgments

This work is supported by the National Science Foundation under Grant No. DRL-1020154. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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