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In this paper we focus on the development of rational number knowledge and present three research programs that illustrate the possibility of bridging research between the fields of cognitive developmental psychology and mathematics education. The first is a research program theoretically grounded in the framework theory approach to conceptual change. This program focuses on the interference of prior natural number knowledge in the development of rational number learning. The other two are the research program by Moss and colleagues that uses Case’s theory of cognitive development to develop and test a curriculum for learning fractions, and the research program by Siegler and colleagues, who attempt to formulate an integrated theory of numerical development. We will discuss the similarities and differences between these approaches as a means of identifying potential meeting points between psychological and educational research on numerical cognition and in an effort to bridge research between the two fields for the benefit of rational number instruction.

Numerical cognition is a research area that appeals to mathematics education researchers, to cognitive-developmental psychologists and to neuroscientists. However, the researchers coming from these different fields approach numerical cognition in different ways in terms of theoretical perspectives, questions asked, methodologies used, and most importantly, of end goals (

With this article we contribute to this effort, arguing that some of the research that lies in the intersection of cognitive-developmental psychology and mathematics education can be fruitful for both fields and very relevant for instruction. We focus on the development of rational number knowledge and we present our research program that is theoretically grounded in the framework theory approach to conceptual change (

Rational numbers are notoriously difficult for primary and secondary students and even for educated adults. Drawing on empirical evidence as well as on conceptual analyses coming from numerous studies,

The natural number bias has attracted the interest of researchers from diverse fields, such as mathematics education, educational psychology, cognitive and developmental psychology and neuroscience. It has been studied using different methodologies (i.e, paper-and-pencil tests, interviews, reaction time studies, brain imaging studies etc.), and different populations (i.e., spanning from infants to adults, and from novices to expert mathematicians). As could be expected, research on the natural number bias is multifaceted. One research strand documents systematic errors in incongruent rational number tasks (i.e., tasks that target aspects of rational numbers which are not compatible with natural number knowledge). Numerous misconceptions have been identified in various contexts, such as in the comparison of decimals (e.g., “longer decimals are bigger”,

Another strand of research looks into the reasoning processes that underlie the manifestation of the bias. Reaction time studies have been used to investigate the observation that natural numbers “come to mind first”, showing that—besides making more errors in incongruent tasks— people need more time to answer correctly in incongruent than congruent tasks (

Research related to the natural number bias provides valuable insights into the effects of the bias on rational number learning, the reasoning processes that underlie it, its connections to other cognitive functions, its neurological underpinnings and in a more general fashion, into numerical cognition. In this article we focus only on one aspect of this bias that we believe could be of value to mathematics instruction. Specifically, we discuss a research program that examines the natural number bias from the perspective of the framework theory approach to conceptual change, and we contrast it with two research programs that have clear roots in psychological theory but are also relevant to mathematics education research. Unlike the framework theory approach to conceptual change, which emphasizes differences between natural and rational numbers, the two other research programs emphasize similarities between natural and rational number knowledge. We will discuss these three approaches with a view to synthesizing ideas and suggestions for rational number instruction, taking into consideration relevant insights stemming from mathematics education research.

Joan Moss in collaboration with Robie Case designed and evaluated a rational number curriculum grounded in Case’s theory of cognitive development. This theory has been applied to the domain of mathematics accounting for the development of natural number concepts (

The rational number curriculum developed by Moss and Case was based on a number of well-elaborated ideas. First, they aimed at capitalizing on students’ initial understandings and experiences. True to their analysis, they targeted 4^{th}, 5^{th}, and 6^{th} graders; they started with the construct of ratio; they worked with situations rich with visual props (e.g., full, half full, and empty containers, number ribbons) building on children’s assumed schema for proportional evaluation; and they presented ratio as percent, so that students could use their natural number strategies to deal with the numerical aspects of the tasks (e.g., halving, doubling). Then they introduced other symbolic representations for rational numbers (first decimal and then fraction notation) emphasizing the connection with the ones that were already familiar to students. Finally, they presented children with exercises in which all types of representations were to be used interchangeably, using number lines extensively.

This curriculum was evaluated in experimental, pre/post test interventions (^{th} graders. The control group was introduced to fractions with the customarily used part-whole aspect of fraction. The experimental intervention consisted of twenty 40-minute instructional sessions spread over a period of 5 months. The results were very encouraging: The experimental group outperformed the control group in all tasks targeting conceptual understanding of rational numbers, and did not lack behind in tasks on procedural fluency. Moreover, the experimental group students showed qualitative differences in their reasoning about rational numbers, indicating that they engaged in multiplicative reasoning. On the contrary, the control group students still relied heavily on additive reasoning, resulting in errors.

Similarly to

[...] progressively broadening the class of numbers that are understood to possess magnitudes and of learning the functions that connect that increasingly broad and varied set of numbers to their magnitudes. In other words, numerical development involves coming to understand that all real numbers have magnitudes that can be ordered and assigned specific locations on number lines (p. 274).

This progression is thought to reflect the gradual expansion of the mental number line to the right to include larger natural numbers, leftward to encompass negatives, and interstitially to include fractions and decimals (

The main tenet of the integrated theory of numerical development is that understanding the magnitude of numbers is instrumental for the development of numerical competence. This view is supported by many studies of natural as well as rational numbers (see ^{th} graders (9-10 years old). The students were pre-and post-tested on several outcome measures of conceptual as well as procedural aspects of fraction knowledge. In all studies, the results showed that the experimental group outperformed the control group with respect to all outcome measures. In addition, the gap between high and low achievers decreased for the experimental group, whereas it increased for the control group.

The framework theory approach to conceptual change (

When non-natural numbers are introduced in the curriculum, although they are also called “numbers”, they clearly violate practically all the background assumptions underlying students’ initial framework theory of number. The question arises, how do students assign meaning to these new mathematical objects?

The framework theory approach to conceptual change is a constructivist approach. We argue that students draw heavily on their prior knowledge of natural numbers to make sense of non-natural numbers (e.g., reasoning by analogy). They typically employ additive mechanisms of learning to gradually enrich their initial framework theory of number with new information about non-natural numbers. When the incoming information, however, is not compatible with their knowledge base, the use of additive mechanisms destroys the coherence of the original structure and may result in fragmentation, internal inconsistencies, and the formation of misconceptions. From our theoretical perspective, this is a slow and gradual process during which a special type of misconception, namely synthetic conceptions, is formed. Synthetic conceptions reflect the assimilation of new information while retaining many of the background assumptions of the initial framework theory of number (see

Applying the framework theory approach to conceptual change in mathematics (

^{th} to 10^{th} graders’ understanding of the numerical value of fractions. They stressed that, unlike what students know about natural numbers, fractions cannot be ordered in terms of their position on the counting list, they are not bounded by a “smallest” fraction, and are not associated with a unitary symbol. In line with the framework theory approach to conceptual change, Stafylidou and Vosniadou predicted that students would make systematic errors related to these differences, which could be interpreted as synthetic conceptions of the fraction concept. Two hundred students from 5^{th} to 10^{th} grade were asked to a) write the smallest and biggest fraction that they could think of and explain their answers, and b) to compare and order fractions. The great majority of these students (89%) were placed in three categories corresponding to an initial and two intermediate states of fraction understanding. In the first category, the students did not take into consideration the multiplicative relation between the numerator and the denominator and considered fractions to consist of two independent natural numbers. In the second, fractions were considered to be always smaller than the unit, again ignoring the relation between the terms of the fraction. Only in the third category were students able to take into consideration this relation. Students in the intermediate categories exhibited synthetic conceptions of fractions. One such example comes from the majority of students in the third category who—although apt to consider the relation between the terms of the fraction—still believed that fractions are bounded by a smallest and a biggest fraction.

In a series of studies, ^{th} to 11^{th} graders) understanding of the dense order of rational numbers. The distinction between discrete and dense order is a fundamental difference between natural and rational numbers: Within the natural numbers set, there is a finite number of intermediates between any two natural numbers. In other words, given a natural number, one can always find its successor. On the contrary, within the rational numbers set, there are always infinitely many numbers between any two numbers. In other words, the successor principle is no longer valid.

The above findings indicate that students’ conceptualizations of rational numbers are far from the view of the rational numbers set as a unified system of numbers that are invariant under different symbolic representational forms. Rather, students seem to view (the positive) rational numbers as a collection of unrelated sets (natural numbers, decimals, fractions) that are allowed to behave differently with respect to order (discrete/dense). For the point of view of the framework theory approach this is a synthetic conception of the rational numbers set.

In a second line of research,

Guided by the framework theory approach to conceptual change,

A series of studies with secondary students (^{th} up to 10^{th} grade tend to think that only natural numbers can be substituted for variables. For example, students answered that 2x stands only for multiples of 2, that a/b stands only for positive fractions, and that -b stands for negative integers. In agreement with our hypothesis, we also found that even the students who would accept non-natural numbers as substitutes for the variables, were not ready to accept any number as a possible substitute. In certain cases students accepted decimal numbers—but not fractions—as possible substitutes for variables, and in other cases they accepted fractions but not decimals. The finding that a literal symbol, denoting a (real) variable, stands only for specific types of numbers points to a synthetic conception of this mathematical notion. This finding has been replicated by a study with Flemish secondary students, who tended to assign only natural numbers far less frequently than their Greek peers but still had great difficulty to consistently assign any type of numbers to literal symbols (

An interesting finding of the above research was that the students were particularly reluctant to accept that an algebraic expression that appeared to be negative (such as –b or -2x-1) could take on positive values, and vice versa (

To sum up, the framework theory has provided a new perspective on rational number learning and has generated novel predictions regarding the interference of natural number knowledge in the context of rational numbers as well as in the context of algebra. A series of empirical studies have confirmed these predictions and showed that there are intermediate states in the transition from initial to more sophisticated understandings of complex mathematical concepts, as students’ initial ideas interact with the new counter-intuitive information coming from instruction forming synthetic conceptions. The insights into students’ difficulties gained through this research were employed in the design of instructional interventions that addressed them.

There is a number of principles for the design of instruction stemming from the conceptual change perspective on learning that are relevant for mathematics learning (

^{th} graders) worked in Synergeia, a software designed to support collaborative knowledge building that provides a structured, web-based work space in which documents and ideas can be shared and discussions can be stored. The experimental group had online and offline access to their peers’ answers and could write comments or respond to their peers’ comments on their own answers. The control group (14 9^{th} graders) worked in pairs in their classroom, with paper and pencil, and the results were presented orally and then written on the blackboard by the researcher. One 45-minute session was devoted to each task. Both groups received the same pre- and post-test with tasks regarding the density of numbers. They were also interviewed after the intervention. The experimental group showed improved performance in the density tasks after the intervention and outperformed significantly the control group. In addition, the experimental students appeared to be more aware of the changes in their ideas about numbers before and after the intervention. It appears that the opportunity to express and exchange ideas using specific models, such as the number line, in an environment that allows for structured discussion was profitable for students.

We designed a text that provided explicit information about the infinity of numbers in an interval, made explicit reference to the numbers-to-points correspondence and used the “rubber line” bridging analogy to convey the idea that points (and numbers) can never be found one immediately after the other. The “rubber-line” text was experimentally tested against two other texts that contained the same explicit information. In addition it presented examples of intermediate numbers or figures illustrating the examples. Six classes of 8^{th} and 11^{th} graders (one experimental class per grade), 149 students in total participated in the study. They received a pre-test with density tasks in a arithmetical and a geometrical context, were administered the corresponding text, and then received a post-test containing all the pre-test tasks as well as 5 additional tasks that examined students’ abilities to deal with the no-successor aspect of density. All groups profited from the explicit information about the infinity of numbers presented in the text. However, the experimental group (8^{th} as well as 11^{th} graders) outperformed the other groups in the “no successor” items of the post-test and was more consistent in providing correct answers and justifications for them.

In another experimental intervention,

^{th} graders in total). Both groups were pre- and post-tested with paper and pencil tests that used tasks regarding the sign of algebraic expressions in various mathematical contexts familiar to students (e.g., square root functions, algebraic inequalities, and absolute values). The refutational lecture used the main principles of direct instruction. More specifically, the students were given the definition of the real variable in algebra, and were explicitly told that literal symbols are used in algebra to stand for any real number, unless otherwise specified. Students’ attention was called on to the differences between the (actual) sign of numbers in arithmetic, where the presence or absence of a minus sign indeed denotes a negative or positive value, and the (phenomenal) sign of algebraic expressions which does not necessarily denote an actual sign. Cognitive conflict was induced by providing examples and counter examples of natural and non-natural number substitutions to literal symbols in algebraic expressions, which either retained or changed their phenomenal sign. Students were also offered the strategy to always test the sign of an algebraic expression by substituting at least one negative number to the literal symbols. At the end of the intervention all the above points were discussed with the students. The whole intervention lasted about 25 minutes.

The results showed that the students who attended the refutational lecture made significantly fewer phenomenal sign errors compared to the students who did not attend the lecture. In addition, the learning profits were retained for at least one month after the intervention.

The intervention studies presented in this section addressed certain difficulties of secondary school students due to the natural number bias, by implementing principles for instruction and teaching strategies stemming from conceptual change research. These instructional interventions supported students to use, evaluate, compare, build, and discuss representations of mathematical constructs; promoted understanding via the use of analogies and bridging analogies; and explicitly addressed students’ misconceptions using refutational arguments. The results showed that these teaching strategies can greatly foster the understanding of counter-intuitive mathematical ideas (see also

The interventions discussed above targeted specific aspects of the natural number bias and were short-termed. For the problem of the adverse effects of natural number knowledge in rational number learning to be tackled more effectively, a long-term perspective on the planning and design of instruction is necessary. We argue that instruction should target the problem at the earliest possible time, before the discrepancy between natural and rational number knowledge and experience gets too large. In what follows we will ground this claim on a synthesis of psychological and educational research with reference to the research programs discussed above.

The research programs presented above have similarities as well as differences in terms of their theoretical framing and the implications for rational number instruction. All research groups emphasize the importance of natural number knowledge in rational number learning.

Finally, both ^{rd} grade—as is true for the great majority of experimental intervention programs.

One overarching principle for instruction stemming from conceptual change perspectives on learning is that instruction should be designed not only on the basis of what is easy for children to understand at a given point in their cognitive development, but also by taking a long-term perspective, and by anticipating later expansions of the meaning of mathematical ideas and symbols as much as possible (

In a more general fashion, building rational numbers on the idea that are numbers that have magnitudes and can be placed on number lines, has certain limitations: It addresses the most abstract aspect of rational numbers that students will meet in their school career and it does not address the fundamental question of why rational numbers are considered numbers in a way that is meaningful for students. One should keep in mind that this is a question that has puzzled mathematicians for centuries and it entailed tremendous changes in the meaning of number (see

We value

At this point it is worth turning to the more general discussion on rational number teaching, focusing on suggestions for instruction that tackle, implicitly or explicitly, the problem of the natural number bias. First, it is important to note here that many researchers agree that overemphasis on the part-whole aspect when students first encounter fractions in instruction creates many problems in the long run. This is because the part-whole aspect of fraction, typically represented by the area model, actually evokes students’ natural number knowledge and elicits additive rather than multiplicative reasoning (

Second, it is also important to take into consideration that much effort has been put by mathematics education researchers in identifying deep similarities between natural and rational numbers, so that natural number knowledge can be used productively in rational number learning. To this end, a great deal of attention has been paid to the notion of the unit and to the operation of unitizing. These can serve as basic elements of reasoning with understanding in additive as well as in multiplicative situations and can be applied in the case of natural as well as rational numbers (e.g.,

Based on the above, we argue that measurement is worth-investing on (see also

We are not the first to argue for the importance of starting rational number instruction earlier. Although this suggestion is not as popular as other approaches, there are still some researches that have taken this position (

Finally, there is a strong trend in research on mathematics education to take a developmental approach to the design of curricula and instruction, which puts forward the notion of learning trajectories (

To summarize and conclude: In this paper we argued that some of the research that lies in the intersection of cognitive-developmental psychology and mathematics education can be fruitful for both fields and very relevant for instruction. We illustrated this point with reference to three research programs stemming from psychology that focus on rational number development and learning. We placed the discussion in the more general context of research on rational numbers, synthesizing findings and principles for instruction coming from educational and psychological research. Although we narrowed our synthesis to include research that tackles the issue of the adverse effects of prior knowledge on further learning, we acknowledge that this work is by no means exhaustive. Nevertheless, we hope to have contributed to the effort to establish more intense dialogue between the fields of psychology and mathematics education.

Τhe authors are members of the Centre for Innovative Research "Conceptual Change", supported by the European Association for Research on Learning and Instruction (EARLI).

The authors have no funding to report.

The authors have declared that no competing interests exist.