^{*}

^{a}

^{b}

The overwhelming majority of efforts to cultivate early mathematical thinking rely primarily on counting and associated natural number concepts. Unfortunately, natural numbers and discretized thinking do not align well with a large swath of the mathematical concepts we wish for children to learn. This misalignment presents an important impediment to teaching and learning. We suggest that one way to circumvent these pitfalls is to leverage students’ non-numerical experiences that can provide intuitive access to foundational mathematical concepts. Specifically, we advocate for explicitly leveraging a) students’ perceptually based intuitions about quantity and b) students’ reasoning about change and variation, and we address the affordances offered by this approach. We argue that it can support ways of thinking that may at times align better with to-be-learned mathematical ideas, and thus may serve as a productive alternative for particular mathematical concepts when compared to number. We illustrate this argument using the domain of ratio, and we do so from the distinct disciplinary lenses we employ respectively as a cognitive psychologist and as a mathematics education researcher. Finally, we discuss the potential for productive synthesis given the substantial differences in our preferred methods and general epistemologies.

Mathematical literacy and engagement is essential for participation in modern society, with research showing success in mathematics to be an important determinant of children’s later educational, occupational, and even health prospects (^{i} principles – principles that do not align well with a large swath of the mathematical concepts we wish for children to learn. We argue that this misalignment presents an important impediment to teaching and learning. We further suggest that one way to circumvent these pitfalls is to leverage children’s non-numerical experiences that can provide intuitive access to foundational mathematical concepts.

In this piece, we explicitly critique the current practice of building nearly all of early mathematics out of counting and associated natural number concepts. Although natural numbers provide an undoubtedly flexible and powerful tool for supporting mathematical thinking, we argue that the practice of introducing almost every fundamental concept in terms of natural numbers often comes with a cost. Namely, to the extent that natural number concepts are misaligned with to-be-learned concepts, they are unlikely to serve as adequate foundations for building understanding of such concepts. There are many ways to conceive of these costs, depending upon one’s disciplinary commitments. They can be characterized in terms of negative transfer (

that there are often multiple alternative choices, symbolic and nonsymbolic, available for representing early mathematical concepts,

that each of these alternatives typically comes with conceptual affordances and constraints, and

that number is no exception to points 1 & 2.

Ultimately, we advocate for explicit leveraging of perceptually based intuitions about quantity that may offer different affordances than symbolic numbers. These intuitions can support ways of thinking that may at times align better with to-be-learned mathematical ideas, and thus may serve as a productive alternative for particular mathematical concepts when compared to number.

Natural number abounds in K-12 mathematics in the U.S., with standard curricular and instructional approaches building early mathematics from counting activities. Students build from counting to whole number addition and subtraction, followed by multiplication (initially as repeated addition) and division (initially as partitioning). Students are then introduced to negative numbers and rational numbers, and only reach ideas about the real numbers in advanced algebra courses in high school (

Overreliance on number can lead educators and learners to focus on number symbols themselves as opposed to the referent fields to which they should refer (compare with ^{ii} – our policies are structured so that early numerical comparison is unnecessarily tied to natural number symbols and the discrete quantity logic they represent. In this case, we have privileged number symbols and relegated the core to-be-learned concept to the background. Moreover, this step is wholly unnecessarily given that children’s perceptual abilities provide them intuitive access to spatial relations that easily illustrate comparative magnitude (

This is but one example of what we argue is a more general pattern. In building the majority of early mathematics out of counting and natural number, we have chosen to privilege a particular conceptual basis for much of children’s mathematical thinking, and this choice has a number of drawbacks. For instance, prior research has argued convincingly that teaching the multiplication operation primarily in terms of repeated addition of whole numbers can hamper more nuanced understandings of multiplication later on (e.g.,

Another major deleterious effect of overreliance on counting and natural number is particularly dangerous because it is so counterintuitive: Namely, overreliance on natural number can impoverish the broader concept of number itself. Teaching number as fundamentally tied to counting obscures key continuities shared by all real numbers – for instance, that they can all be ordered and assigned specific locations on number lines (

There is evidence that children can build up mathematical thinking in ways that are not always bound to natural number (e.g.,

Indeed, some researchers have proposed that the ability to engage with a generalized sense of magnitude is psychologically prior to the conception of number, laying the foundation for its emergence (e.g.,

In the sections that follow, we will unpack this argument using the domain of ratio, and we will do so from the distinct disciplinary lenses we employ respectively as a cognitive psychologist (Matthews) and as a mathematics education researcher (Ellis). We chose to focus on ratio for two reasons. First, researchers from mathematics education research and from psychology, despite substantial differences in approach, both agree that understanding ratio – and associated rational number concepts – is critical (

In the section immediately below, one of us (Matthews) will argue 1) that research from multiple fields suggests that humans have a perceptually-based sensitivity to nonsymbolic ratio that emerges prior to formal instruction and 2) that leveraging these intuitions may prove an effective base for building up formal ratio concepts. In the next, (Ellis) will address how ratio can be developed from magnitude comparison, and will propose a way to foster the development of internalized-ratio from children’s perception and mathematization of covarying quantities. In the final section, we will discuss how the divergence in our general methods presents opportunities for synthesis. We ultimately underscore the importance of the convergence in our conclusions given the backdrop of substantial differences in methods and philosophy.

In a recent series of papers, my colleagues and I have argued for a cognitive primitives approach to grounding ratio concepts (

Sample nonsymbolic ratios used in nonsymbolic comparison tasks. Ratios are composed of various type of nonsymbolic quantities, including (a & c) dot arrays, (b) line segments, and (c) circle areas. Note that dot arrays, while countable, are often used in task where the number of dots is high (i.e. > 40) and the time limit is set low (i.e. < 2 seconds) to preclude the possibility of counting. In these instances, dot array numerosity is processed perceptually (for an example, see

For my part, I plead guilty to these charges. Indeed, psychologists elide these distinctions every day. Even taking a most cursory glance through a psychology journal, one can find many charts with a ^{nd} definition is “the relationship of one thing to another in terms of quantity, size, or number; the ratio”? I cite these definitions not because I take the dictionary to be canon, but because I respect

My approach is motivated in part by

Rational numbers as fractions

Rational numbers as decimal fractions

Rational numbers as equivalence classes of fractions

Rational numbers as ratios of integers

Rational numbers as operators

Rational numbers as elements of the quotient field

Rational numbers as measures or points on a number line.

For Kieren, robust understanding of rational number depends upon having adequate experience with their many interpretations. Although the taxonomies vary somewhat, several prominent researchers have endorsed such a mega-concept view (e.g.,

Given the complexity of the rational number mega-concept, it is arguably important to find interpretations that square well with children’s intuitions. To that end, my research has focused on promoting the addition of an eighth interpretation to the list above that essentially combines subconstructs #4 and #7:

This nonsymbolic perspective is consistent with Davydov’s project to ground fractions instruction in intuitions about measurement (

It is noteworthy that ratios of nonsymbolic quantities are actually much more expansive in reach than the typical conception of ratio introduced with natural numbers. When composed of continuous nonsymbolic quantities such as line segment lengths, the ratios formed extend beyond those corresponding to rational numbers and include all real numbers. Pi, the ratio between a circle’s circumference and its diameter, is one such “irrational” ratio that corresponds to a pair of line segments similar to those in

It is arguable that the single greatest shortcoming of typical approaches to teaching rational numbers lies in their dependence upon natural number symbols and count-based logic. Although number symbols are typically seen as relatively abstract and flexible, it is arguable that Arabic numerals are rendered concrete relative to many other symbols due to the information that they automatically communicate to experienced learners (e.g.,

Indeed, much prior research demonstrates that natural number symbols are learned so thoroughly that simply seeing them evokes thoughts about whole number magnitudes. For example, whole number values are encoded so automatically that even children demonstrate Stroop effects whereby they are faster and more accurate in judging that the 7 is physically larger than the 5 in

One consequence alluded to above is the inappropriate application of natural number schemas to rational number contexts. For instance, when dealing with rational numbers, learners commonly struggle with the

Although this problem is well known in both mathematics education research and in psychology, the depth of the problem may be underappreciated. Research shows that even mathematicians and undergraduates at highly selective universities are not completely immune to this bias (e.g.

What shows up as relatively small costs in terms of reaction times for the highly educated adult seems to impose a much higher cost on children acquiring rational number knowledge. The prevalence and severity of the costs associated with introducing ratio and fraction with natural number has led some to hypothesize that engagement with whole number schemas

In this context little hope is to be gained from any of Kieren’s seven interpretations of rational number. Each relies on notations that employ repurposed natural number symbols and are therefore likely to evoke count based logics that are not easily reconciled with the relational logic of rational numbers. However, the eighth subconstruct I have offered – that of the nonsymbolic ratio – is not dependent upon number symbols. If such an interpretation of ratio could be made sensible, it might provide access to important aspects of the rational number mega-concept without falling subject to the pitfalls of symbolic representations.

Above I alluded to

A narrow mathematical conception disregards the role physical quantity plays in the meaning of the concept and consequently obscures the psychological origins of fractions, for there is little doubt that number concepts, including rational number concepts, are developed through acting and reflecting upon physical quantities. (p. 241)

He proposed that nonsymbolic ratio could go beyond the number line or measurement interpretation (#7), which fundamentally involves a project of mapping nonsymbolic ratios to symbolic numbers (see

By suppressing the number line altogether we can work in a non-metric space, with no unit of measure, in which the lengths of line segments are mutually defined according to their relative magnitude…this characteristic makes possible the creation of partially "de-arithmetized" tasks, for which there exist no mere numerical solutions and for which students must reflect upon ratios of quantities. (p. 285)

Adapted from

This conception presaged recent work that cast number lines as proportions between nonsymbolic ratios and pairs of number symbols (e.g.,

A growing number of studies demonstrate that human beings do in fact have a perceptually-based sensitivity to nonsymbolic ratios (

Nonsymbolic ratio stimuli used (a) with nonverbal infants by

Recent research with adults has begun to systematically detail the robust nature of nonsymbolic ratio sensitivity and how it relates to symbolic mathematical abilities (

Additionally, research has found that nonsymbolic ratio processing is sometimes automatic (

Nonsymbolic font ratio was defined as the ratio of the physical area occupied by the numerator to that occupied by the denominator in fractions used in a comparison task. When the fraction with the larger nonsymbolic font ratio was congruent with the numerical comparison task (i.e., on the same side as the numerically larger fraction), participants were faster and more accurate compared to when nonsymbolic ratio was incongruent (adapted from

Participants were faster and more accurate on congruent trials when compared to incongruent trials. Notably, the experimenters never made reference to nonsymbolic ratio for the duration of the experiment. Nevertheless, nonsymbolic ratios were processed automatically, and this nonsymbolic processing influenced symbolic numerical activity. Specifically, larger nonsymbolic ratios were associated with larger symbolic ratio values, and smaller nonsymbolic ratios were associated with smaller symbolic ratios. The research briefly outlined here is but a small swath of the growing body that suggests that humans and other animals (e.g.,

It is perhaps natural to ask, if sensitivity to nonsymbolic ratios is so pervasive and even automatic, why do learners continue to have difficulties with rational number symbols? The answer is a simple: a basic sensitivity does not an elaborated concept make. It is one thing to be perceptually sensitive to nonsymbolic ratio and quite another to have verbally-mediated conceptual knowledge about ratio. It is another thing still to link such conceptual knowledge to repurposed number symbols. On this point, I converge with Ellis in the belief that cultivating formal ratio concepts requires effortful action on the part of the learner. Furthermore, educators will continue to play a crucial role in spurring and supporting this effortful action.

It currently remains the case that most curricula do not attempt to leverage this nonsymbolic capacity. Despite the work cited above to make the case humans can perceive nonsymbolic ratios, very little empirical work in psychology has directly sought to leverage this ability (but see

Finally, how might we best use the nonsymbolic ratio subconstruct to make symbolic ratios, fractions, and percents more meaningful to learners? Here it is especially important to recognize two conspicuous limitations that nonsymbolic ratios have relative to symbolic numbers: First, despite their intuitive leverage, nonsymbolic ratios still only provide only partial coverage of the rational number mega-concept. If adopted as an eighth construct to

I (Ellis) define ratio to be a multiplicative comparison of two quantities. Following

By quantities, I mean mental constructions composed of a person’s conception of an object, such as a piece of string, an attribute of the object, such as its length, an appropriate unit or dimension, such as inches, and a process for assigning a numerical value to the attribute (

In order to form an intensive quantity and conceive of an internalized-ratio, one must form a

Are there ways to better support children’s attention to quantities in ratio development? Above, Matthews thoroughly documented the conceptual drawbacks associated with building ratio from natural number symbols, but is there an alternate instructional route? Indeed there is precedent for developing early mathematical ideas by building on children’s natural predilection for noticing and comparing quantities they encounter in their lives. In particular, consider the Elkonyn-Davydov curricular approach, a first through third grade Russian curriculum derived from the work of Vygotsky and Leontiev (

The quantities children attend to in the Elkonyn-Davydov and

With fractions and ratios, researchers using variants of the Elkonyn-Davydov curriculum have worked with children to develop ratio reasoning from a basis of measuring continuous quantities rather than counting and partitioning (

Within the Elkonyn-Davydov curriculum and its counterparts, the conceptual origin of ratio is a relationship between two magnitudes. This is similar to Matthews’ description of ratio as a quantity that emerges from a comparative relationship, with one important difference: Here, after engaging in activities of noticing and comparing, children eventually quantify magnitudes with numbers. Critically, this quantification is built on a foundation that draws on children’s proclivities to notice and assess magnitudes. In contrast, typical approaches introduce fraction and ratio through partitioning, relying on a model of division that is inherently discrete. When working with a continuous attribute, such as length, children can choose a measure that is any size relative to the attribute being measured. The attribute being measured may not be an exact multiple of the measure, which affords attention to the relationship between the change of numbers and the change of measures. Further, it moves children out of the world of natural numbers and, at least mathematically, situates ratio within the reals in the sense that neither the value of the measure nor the measured must necessarily be rational.

The above approach to ratio builds on children’s abilities to notice and perceive non-symbolic extensive quantities such as length or volume. One could also extend this approach to connect to children’s perception of non-symbolic ratios, or intensive quantities. For instance, as Matthews discussed, it may be fruitful to build instruction on having children assess differences in ratio magnitude based on a perceptual comparison of quantities (e.g.,

Covariation is critical for developing an understanding of internalized-ratio, proportionality, and more broadly the concept of function as a foundation for algebra and calculus.

How can we foster the development of these quantitative images to support the construction of internalized-ratio? A key component of such an endeavor is situating students’ mathematical activity, initially, in contexts that are not reliant on a basis of counting, number, or even measurement. Removing students’ abilities to count and therefore discretize continuous, changing, or unspecified quantities could potentially encourage the construction of multiplicative objects. In fact, I propose that it may be helpful to introduce a variety of tasks that encourage covariational reasoning in order to foster such a Way of Thinking (

The first task is the triangle/square task (^{iii}

Four screen captures of a movie of a point sweeping out a triangle.

In a teaching experiment (^{th}-grade participants, Olivia and Wesley^{iv}, I provided the students with the triangle/square movie and asked them to think about how the area of the triangle changes compared to the distance P accumulates as it travels around the perimeter of the square. It is possible for one to invent measures for the dimensions of the square and actually calculate areas and distances, which would eliminate the need to simultaneously attend to the two varying quantities. However, given my knowledge of Olivia and Wesley’s mathematical backgrounds and our prior experiences, I was confident that they would not do so. Indeed, both students attended to the changing area and the distance traveled without making any measurements. Olivia and Wesley produced similar graphs comparing area with distance traveled (see

So on the first increase, I thought that this, it gets from smaller to bigger [pointing from A to B], and then here was sort of this [pointing along segment BC] because I thought that it would stay about the same area. That’s just kind of how I saw it. And then it gets smaller [pointing from C to D]. And then it stays the same at [points along segment DA], I see nothing.

Olivia’s graph comparing the triangle’s area to distance traveled

Explaining why she made the angled portions of her graph straight rather than curved, Olivia said:

I think because like here [points to A], it has no area, and here [drags finger towards B] it has a little bit more. Like if you went up by increments here, maybe it would be, like, doubled, and so it kind of goes up. Like kind of what we were talking about before. Here it would be a whole bunch of little lines because it just kind of keeps going, but if it stopped, I imagine it going steadily up, like slowly getting to half.

Olivia’s construction of the graph relied on a non-numerical image of comparisons as she thought about the triangle’s area value

Olivia’s graph comparing the triangle’s area to the distance between P and A.

The triangle/square task encouraged the students to attend to two quantities that changed together. It is one of a class of tasks that can potentially foster covariation by asking students to think about the changing values of two quantities simultaneously, particularly when it is difficult or impossible to measure (and therefore discretize and calculate with) the quantities’ magnitudes. The second task is the Gainesville Task, which we have borrowed from

Wesley’s graph (

So I started by going like this (points to the line departing from the origin), but I had to get closer to Gainesville so I’d have to be going backwards so I decided to start here (points to the non-origin point on the

Wesley’s graph comparing the distance from Athens to the distance from Gainesville

Building on a foundation of covariational reasoning, one can then shift to tasks that foster building internalized-ratio as an invariant multiplicative comparison in which the related quantities co-vary. For instance, in the same teaching experiment (

Paint roller task.

By this point, the students were accustomed to attending to how two quantities varied together, and both students graphed the area of the rectangle compared to the length swept as a straight line. Olivia explained, “I sort of pictured it in my head…for every length that you’ve pulled, it should be the same amount of area.” Olivia could imagine the change in area for an unspecified length (the “length that you’ve pulled”), but unitized the length pulled to imagine how much area would be added for each same-increment amount of length. Wesley, in contrast, invented specific values: “I decided I would think the height would be 1 meter…so like if you drag it out 1 meter, so that the length is 1 meter and the height is 1 meter, and then to find the area you times the length by the height which is one times one, is 1 square meter actually.” Before introducing specific lengths (which the students called heights) for the paint rollers, I asked them to think about the rate of change of the area of a rectangle made by a taller paint roller versus a shorter paint roller. Both students graphed steeper lines for the taller paint roller (see

Olivia’s graph for the area painted versus the length swept for a shorter and taller paint roller.

Wesley compared the second unspecified height to his original 1-meter paint roller, and explained, “Because since it’s, the height is bigger than 1 meter now, then for every length that you pull it out one meter, it gets more area.” Similar to Olivia’s unspecified length increments, Wesley thought about 1-meter increments and then could compare the amount of area being added for each rectangle while holding the length increment steady. Olivia compared the area swept (which she referred to as the height of the wall) to the slope, stating, “The steeper it is, the larger height of the wall it is.”

The ratio of area to length swept is a multiplicative invariant that depends on the height of the paint roller. Thanks to multiple experiences with covariation tasks before encountering the paint roller task, Wesley and Olivia could not only observe the quantities area and length varying together, but could also explicitly think about both quantities changing. It was only then that they were introduced to tasks with specific numbers. When considering paint rollers of specific heights, such as 10 inches or 16.25 inches, the students could construct ratios as representations of various snapshots in the paint-rolling journey. For instance, consider a paint roller that has swept a length of 20 inches to produce an area of 160 square inches. One could imagine the ratio of 160 square inches: 20 inches as having swept out 8 inches in area for each inch in length swept, but one could also conceive of the 160: 20 ratio as one of an equivalence class of ratios, each representing a different point in the paint-rolling journey for an 8-inch paint roller. Further, the height of the paint roller does not have to be a natural number, nor does the amount of inches swept. When one’s reasoning is grounded in images of continuously changing quantities rather than discrete collections of sets, non-natural numbers do not pose a problem. Students also have a conceptual foundation for making sense of how changing the magnitude of one quantity or the other can change the value of the ratio.

Another example in which students can reason about how changing initial quantities affects the value of a ratio is from a different covariation context, this time with two spinning gears (

In the gear scenario, the students constructed a ratio as an

Matthews asked how one might use nonsymbolic ratios as didactic objects to support mathematical discourse in the learning of symbolic ratio. One potentially fruitful avenue is to leverage students’ perception and mathematization of co-varying quantities. In the above examples, symbolic number was introduced late, after students had constructed an invariant relationship between two quantities as they varied. Significant time and attention was devoted to supporting students’ construction of multiplicative objects, and students had opportunities to learn how to make sense of two quantities varying together before they measured anything. Numbers then became expressions of magnitudes for the students, and thus were imbued with quantitative meaning as representations of their intended referents. In contrast, beginning ratio instruction with natural numbers situates ratio in the discrete world, as a static comparison of sets. It is difficult to then bootstrap up to an equivalence class conception of ratio in which each component value can take on any real number. Situating ratio instruction instead in contexts of variation and change can leverage students’ experiences with quantity and variation in order to foster a more robust understanding.

Building early mathematics concepts exclusively on natural number and count-based logic comes at a cost. We have presented arguments that converge on this conclusion, despite substantial differences in our preferred methods, contrasts in the types of warrants we choose for our claims, and disagreements in our general epistemologies. Specifically, we have each argued that natural number concepts do not neatly align with the structure of many foundational mathematical concepts, using ratio as a case for illustration. This misalignment can impoverish mathematical thinking, standing as an obstacle to conceptual development. Consistent with this critique, we have also sought to highlight the importance of embracing alternative methods of introducing early mathematics concepts. Despite the present curricular primacy of natural number, other methods of representation often exist for building early mathematics concepts. In particular, we have focused on nonsymbolic representations, arguing that they a) may better leverage children’s intuitions built from everyday experiences and b) may better align with the structure of some to-be-learned concepts.

At this point, however, we have yet to give an extensive treatment to the sometimes dramatic differences in our theoretical perspectives and how a theoretical synthesis or interdisciplinary research project might be possible. In what follows, we first attend explicitly to our points of divergence and the implications of this divergence for our research questions and preferred methods. Next, we offer a detailed exposition of the potential for synthesis in the face of this divergence. Finally, we conclude with a general consideration of several ways in which collaborations among researchers with divergent theoretical commitments can help enrich inquiry.

In the test domain of ratio, we have pressed our arguments from significantly different epistemological stances. In fact, our perspectives differ in fundamental ways that we have not fully reconciled. These differences have implications for the definition of ratio itself: In particular, are ratios ‘constructed’, or are they ‘things in the world’?

Matthews’ argument is built on the notion that humans come equipped with a ratio processing system – a primitive perceptual apparatus that is sensitive to ratio magnitudes. For Matthews, a full accounting of mathematical knowledge involves both metacognitive and associative mechanisms (e.g.,

From this perspective, human brains did not evolve to do mathematics. Instead, cultural inventions such as mathematics and reading co-opt pre-existing brain systems to support new competencies (see

For Ellis, mathematical knowledge develops as part of a process in which children gradually construct and then experience a reality as external to themselves (

Where the realist believes mental constructs to be a replica of independently existing structures, Ellis takes these structures to be constituted by people’s activity of coordination (

These differences in perspective naturally shape the research questions we each pose and the methods we use to investigate them. Matthews begins by asking, how do human brains – these computers composed of neurons – come to fluently understand concepts that they clearly did not evolve to support? From this perspective, it is natural to focus on primitive sensitivity to nonsymbolic perceptual ratios that extends across species and to ask how this sensitivity can be

On this view, the first step in theorizing about potential for learning is sometimes to identify and detail these basic primitive abilities (e.g., sensitivity to nonsymbolic perceptual ratios). Methods for exploring these abilities include using various tasks to measure the acuity with which people at various points in development can discriminate among nonsymbolic ratios that serve as analogs to specific numerical ratios. These tasks may involve timed magnitude comparisons (e.g.,

In contrast, rather than taking ratio as given and studying how well students perceive, leverage, or manipulate ratios and their associated concepts (or how they can be trained to improve in these tasks), Ellis investigates a different set of questions: What do students construct as ratio? What is the structure of the epistemic student’s (

The methods appropriate for addressing the above questions include the design and analysis of a) written assessments (e.g.,

In order to investigate the nature of student learning and the development of concepts over time, some mathematics educators rely on the teaching experiment (

These differences noted, we argue that the very divergence in our questions and methods holds much productive potential. Multi-method techniques have long been theorized to be good ways to build richer understandings of complex constructs (e.g.,

This appears to be a classic case of what

For example, consider the dynamic perceptual ratio matching task shown in

Sequential screenshots from a dynamic perceptual ratio matching task. (a) Participants are initially presented with a target ratio composed of line segments on the left side of the screen along with an incomplete ratio to the right. Participants are asked to adjust the components of the incomplete ratio so that the two match. In this case participants are asked to adjust the length of the orange bar to try to make the orange:blue line ratios match. (b) Once the participant submits a response, the computer gives feedback on what the correct answer would have been (farthest right). In this case, the red box lets the participant know that their submission (center) is quite far from the correct value. Note that for these ratios, the jitter between orange and blue lines is an irrelevant dimension that ensures participants cannot make a match by simple scaling of an identical figure.

The above-described matching tasks would enable us to not only identify how well students perceive equivalent ratios, but also potentially to foster better attention to and conscious appreciation of perceptual intuitions over time. Interviewing a subset of the participants would provide additional data about the particular relational features students consciously attend to, the manner in which they attend to the simultaneous changes in the lengths of the provided lines, and the concepts on which they draw when constructing ratio matches. The combined data from the dynamic perceptual ratio matching tasks and clinical interviews would then inform the development of a hypothetical learning trajectory (

This is but one example of many questions of mutual interest in this domain that might be more profitably investigated by the cooperation between our disciplines than if we continued to conduct research in our respective silos. Others include: Are some quantities more optimal than others for helping learners develop mathematical ratio through observing covariaton? How do students’ perceptual intuitions about ratio affect the quantities they attend to when constructing mathematical ratio? Can we measure how integrating measures from mathematics education researchers with those from psychologists might provide more resolution for measuring rational number knowledge than either used in isolation?

This is not to suggest that integrating methods will completely bridge the theoretical and methodological divides between us. In the final analysis, some of our prior theoretical commitments may be irreconcilable, and the key is accepting this fact and moving on in spite of such differences. Our fields stand to profit in several ways from embracing methodological pluralism and forging on with such integrative projects. By incorporating different methods, we generate enriched data sets that we can each use within our preferred analytics. That is, independent of deep theoretical shifts, the presence of more abundant data obtained by more varied measures can present a larger space for inquiry for our preferred analytics. Second, in sharing data and perspective, we create real reflexive opportunities that can lead to genuine shifts in theoretical perspective. Surely the reach of such shifts will be limited somewhat by the depths of our commitments, but the potential remains significant despite these constraints. Third, although our methodological orthodoxies may constrain our abilities to push for a deep synthesis, those steering conventions need not limit our students in the same ways. By offering emerging researchers the opportunities to analyze phenomena of interest through different lenses, we create the possibility that our trainees may produce a more profound synthesis.

With this project, we have not resolved all of our disagreements, nor do we think this would have been a realistic goal. Our epistemological orientations are far enough apart that hope of substantial reconciliation is probably chimerical – and we feel this is not unusual (but also not inevitable) for cognitive psychologists and mathematics education researchers. Still, we have found substantial common ground on a topic that is of great importance for each of us and for our respective fields. In the process, we have each gained considerably by engaging with the other’s mode of thought. We have grown, and our work is richer as a result. It is our sincere hope that this can be a model for collaborations between mathematics education researchers and psychologists more generally. Each has its unique insights, and each has its blind spots. By working across boundaries, and learning to bracket our differences, it may be that focusing on our points of convergence can lead to new productive points of inquiry. In this fashion, we might hope to more exhaustively investigate constructs of mutual interest and to push knowledge further.

The authors would like to thank Nicole Fonger, José Francisco Gutiérrez, Edward Hubbard, Mark Lewis, Brandon Singleton, and Ryan Ziols for their assistance with task development, data collection, and the fruitful discussions that followed.

The natural numbers are typically defined either as the set of non-negative integers (e.g., 0, 1, 2, 3…) or the set of positive integers (e.g., 1, 2, 3,…). For the purposes of this paper, we mean the set of positive integers when we refer to natural numbers.

Thanks to Brandon Singleton for developing the triangle/square task.

Olivia and Wesley are gender-preserving pseudonyms.

Support for this research was provided in part by National Institutes of Health, project 1R03HD081087-01 and National Science Foundation grant no. DRL-1419973.

The authors have declared that no competing interests exist.