Naturally biased? In search for reaction time evidence for a natural number bias in adults

Abstract

A major source of errors in rational number tasks is the inappropriate application of natural number rules. We hypothesized that this is an instance of intuitive reasoning and thus can persist in adults, even when they respond correctly. This was tested by means of a reaction time method, relying on a dual process perspective that differentiates between intuitive and analytic reasoning. We measured fifty-eight educated adults’ accuracies and reaction times in a variety of rational number tasks. In half of the items (congruent), the correct response was compatible with natural number properties (thus intuitive reasoning led to a correct answer). In contrast, in the incongruent items, intuitive reasoning would lead to an incorrect answer. In comparing two numbers, there were hardly any natural-number-based errors but correct responses to incongruent items took longer. Regarding the effect of operations, more mistakes were made in incongruent items, and correct responses required longer reaction time. Incongruent items about density elicited considerably more errors than congruent items. These findings can be considered as evidence that the natural number bias is an instance of intuitive reasoning.

Introduction

Research in cognitive-developmental psychology and mathematics education has repeatedly shown that a major source of difficulty in the learning of rational numbers¹ is the inappropriate application of natural number rules. This phenomenon is associated with the so-called “whole number bias”² (Ni & Zhou, 2005). Systematic errors arise when the “behavior” of rational numbers differs from that of natural numbers (Lamon, 1999, Moss, 2005, Ni and Zhou, 2005, Smith et al., 2005, Resnick et al., 1989, Vamvakoussi and Vosniadou, 2010). Evidence for this phenomenon typically comes from paper-and-pencil tests, interviews, and classroom observations. The majority of studies in this area—and all of the above-mentioned ones—have been conducted with primary and secondary school students. Some, however, also targeted adults (e.g., Post et al., 1988, Stacey et al., 2001, Tirosh et al., 1999). It appears that some of these systematic errors diminish or even disappear with age and level of instruction, whereas others are remarkably persistent.

The central hypothesis in the present study is that natural number reasoning still interferes with educated adults’³ reasoning when dealing with rational numbers tasks, even when they give correct responses and thus appear unaffected by a natural number bias. To test this hypothesis, we applied, besides error analysis, a new method to investigate the natural number bias, namely reaction-time methodology. This way, the interference of natural number reasoning could not only manifest itself through erroneous responses, but also through the increased time that subjects need in order to provide a correct response. The use of reaction time data and the underlying assumptions are inspired by dual-process accounts of reasoning (Epstein, 1994, Evans and Over, 1996, Kahneman, 2000, Sloman, 1996, Stanovich, 1999).

Mastery of the rational numbers is considered an important aspect of mathematical literacy. It is, however, widely documented that learning about rational numbers presents students with many difficulties. In particular, students are found to make systematic errors when a task requires reasoning that is not in line with their prior knowledge and experience about natural numbers (Moss, 2005, Ni and Zhou, 2005, Smith et al., 2005, Vamvakoussi and Vosniadou, 2010). On the other hand, students deal effectively with rational number tasks when these are compatible with natural number strategies (e.g., Nunes and Bryant, 2008, Stafylidou and Vosniadou, 2004). New information about rational numbers that is presented in a context that allows for natural number reasoning is deemed easier for students. One such example is the part-whole aspect of fraction, which is typically used in instruction as an introduction to fractions. Students are asked to count the number of parts in a shape that represents the whole (e.g., a pie). Then they count the number of parts shaded. Subsequently, they use these counts as the basis for naming and symbolically representing fractions. This practice allows students to treat the parts of the whole as discrete objects and apply familiar counting strategies, which in turn helps them perform simple operations on fractions (e.g., to add two fractions with the same denominator). Interestingly, it is documented that over-reliance on the part whole aspect of fraction eventually becomes an obstacle to further understanding of rational number concepts (Mamede et al., 2005, Moss, 2005). Another example is that presentation of information in terms of frequencies instead of rates facilitates probabilistic reasoning. Butterworth (2007) explains this finding as preference for natural numbers. Thus students’ reliance on natural number reasoning when dealing with rational number tasks is pervasive, facilitates reasoning when it is appropriate, but has an adverse effect when it is not. Hence the term bias, first introduced by Ni and Zhou (2005) in relation to this phenomenon, seems justified.

There is no consensus regarding the origins of the natural number bias. Although some researchers argue that early quantitative representation is limited to discrete quantities—and thus the natural number concept is privileged—this issue is still controversial (Ni and Zhou, 2005, Rips et al., 2008). It appears that some pre-instructed ideas pertaining to rational number are also present in young children, even pre-schoolers (e.g., Moss & Case, 1999). However, it is clear that the externalization and systematization of such intuitions about rational numbers is typically much less socially supported in the first years of a child's life (Greer, 2004). In contrast, the development of the natural number concept is mediated from early on by cultural representational tools [e.g., language (Carey, 2004)] and practices such as finger counting (Andres, Di Luca, & Pesenti, 2008) as well as counting songs, rhymes, and cardboard games. In addition, early instruction focuses on natural number arithmetic, thus supporting the systematization and validation of children's initial understandings of number as natural numbers.

It thus appears that, before students are introduced to rational numbers through instruction, they have already constructed rich understandings of number, which are mainly tied around their formal as well as informal knowledge of natural numbers. These understandings sustain students’ beliefs about what counts as a number and how numbers are supposed to “behave” (Gelman, 2000, Smith et al., 2005, Vamvakoussi and Vosniadou, 2010). Several expectations related to the behavior of number are violated when rational numbers—in the form of decimals and fractions—are introduced in the curriculum. To make sense of these new mathematical constructs that are presented as numbers, it appears that students draw heavily on natural number knowledge. This explains why systematic errors occur precisely where the behavior of rational numbers deviates from that of the natural numbers.

In the following we discuss three kinds of rational number tasks that are found to elicit such systematic errors. These tasks will lie at the basis of our research instrument.

Comparison tasks have been widely used to assess understanding of rational numbers. In the comparison of decimals, the literature reports a salient intrusion of prior knowledge about natural numbers underlying the judgment that “longer decimals are larger”, which may be correct in cases such as 2.15 > 2.1, but not in cases such as 2.12 > 2.2 (Resnick et al., 1989). This judgment may be grounded in the observation that one of the numbers has more digits than the other—which for natural numbers characterizes a larger number. Another plausible explanation is that students compare the fractional parts of the two numbers as if they were natural numbers (i.e., 2.12 is deemed larger than 2.2, because 12 is larger than 2). It appears that this type of errors is typical at the early stages of students’ encounter with decimal numbers, but tends to decrease with age (Desmet et al., 2010, Stacey and Steinle, 1998, Stacey and Steinle, 1999) and is not common in educated adults (Stacey et al., 2001). An error that is more likely to be found even in educated adults is the “shorter is larger” one (Stacey et al., 2001). This error is usually explained as an intrusion of knowledge about fractions. For example, 2.3 is deemed larger than 2.32, because the fractional parts of the two numbers refer to “tenths” and “hundredths’, respectively, and “tenths are always larger than hundredths” (see also Peled and Awawdy-Shahbari, 2009, Resnick et al., 1989).

In the comparison of fractions, a crucial factor is students’ difficulty to understand that the magnitude of a fraction depends on the relation between its terms (Moss, 2005, Ni and Zhou, 2005, Smith et al., 2005). Instead, students initially tend to interpret the symbol a/b as two independent natural numbers, separated by a bar (e.g., Stafylidou & Vosniadou, 2004). This leads them to conclude that a fraction increases when its numerator, its denominator, or both increase. Focusing on each term of the fraction separately can result in correct judgments in cases such as 2/5 < 3/5, but also in incorrect ones in cases such as 2/5 < 2/7. Recently, there has been an interest in educated adults’ processing of fractions. Neuro-psychological studies indicate that when comparing fractions, adults rely mainly on componential strategies, in the sense that they access the terms of the fractions separately (Bonato et al., 2007, Kallai and Tzelgov, 2009). Meert et al., 2009, Meert et al., 2010 suggested that even when students (10 and 12 year olds) and also adults process fractions holistically (i.e., by accessing the magnitude of the whole fraction), there are still indications of componential processing. For instance, in the case of fractions with the same numerators (such as 2/5 and 2/7), there was interference of the relative magnitude of the denominators. Componential processing in the comparison of fractions may arguably trigger natural-number based reasoning, which has to be inhibited for making a correct comparison.

Within the natural numbers set, whenever two numbers are added or multiplied, the outcome is always bigger than the initial numbers. Similarly, when two numbers are subtracted or divided, the outcome is smaller than the minuend and the dividend, respectively. None of the above is necessarily true within the rational numbers set: The effect of operations depends on the numbers involved. For example, 3 + (−5) is smaller than 3; and 8 ÷ 0.5 is larger than 8.

In a seminal paper, Fischbein, Deri, Nello, & Marino (1985) argued that there are intuitive models of the four operations, associating addition with putting together, subtraction with taking away, multiplication with repeated addition, and division with equal sharing. These intuitive models are assumed to be implicit and to shape students’ expectations about the effect of operations, as well as the role of the numbers involved. Later research challenged some of Fischbein et al.’s claims (see for example De Corte and Verschaffel, 1996, De Corte et al., 1988, Mulligan and Mitchelmore, 1997). Nevertheless, there is plenty of evidence showing that in extending the meaning of operations from natural to non-natural numbers, the idea that “multiplication makes bigger” and “division makes smaller” is difficult to overcome (Greer, 1994); and that it might be present in adults as well (Graeber, Tirosh, & Glover, 1989). There is also evidence showing that students associate “more” and “less” with addition and subtraction, respectively, when solving word problems (De Corte, Verschaffel, & Pauwels, 1990), which is compatible with the idea that “addition makes bigger” and “subtraction makes smaller”. Some researchers point out the possibility that, similarly to multiplication and division, students also hold intuitive beliefs about the effect of addition and subtraction (e.g., Tirosh, Tsamir, & Hershkovitz, 2008). Nevertheless, so far this phenomenon has—to the best of our knowledge—never been explicitly and systematically investigated.

Between any two non-equal natural numbers there is a finite number of numbers. On the contrary, between any two non-equal rational numbers there are infinitely many intermediates. It is amply documented that the density property of rational numbers is difficult to grasp for elementary and secondary students (Hannula et al., 2006, Hartnett and Gelman, 1998, Merenluoto and Lehtinen, 2002, Smith et al., 2005, Vamvakoussi and Vosniadou, 2004, Vamvakoussi and Vosniadou, 2010, Vamvakoussi et al., 2011). In younger ages, the response that there are no other numbers between two given numbers that are pseudosuccessive, such as 0.005 and 0.006 or 1/2 and 1/3, is quite frequent, but it decreases with age. Older students usually refer to some intermediate numbers between such pseudosuccessive numbers, but limit their answers to numbers such as 0.0051, 0.0052, …, 0.0059, and therefore do not accept that there are infinitely many. There is evidence that university students also face difficulties with the notion of density (Giannakoulias et al., 2007, Tirosh et al., 1999).

To sum up, erroneous judgments that can be attributed to the natural number bias occur with respect to the comparison, the operations, and the dense ordering of rational numbers. There is evidence that some of the errors decrease substantially with age, while others are more persistent and present even in educated adults. Thus it appears that the manifestation of the natural number bias in terms of errors might not occur across different mathematical tasks. Comparison tasks, in particular the comparison of decimals, appear to be less challenging for educated adults. On the other hand, tasks related to the density property are more likely to elicit erroneous responses even in educated adults. This is an indication that the over-reliance on natural number reasoning is not a one-dimensional bias that either manifests itself through erroneous responses across all kinds of rational number tasks, or does not exert any influence at all. There is a vast literature on the acquisition of rational number concepts. Some studies investigate rational number competence using various tasks simultaneously (e.g., Tirosh et al., 1999), but they do not focus explicitly on natural number interference. There is also quite some work on the way in which the natural number bias affects rational number reasoning, but these studies focus only on specific tasks such as the comparison of fractions (Stafylidou & Vosniadou, 2004) or the density property (Hartnett and Gelman, 1998, Smith et al., 2005, Vamvakoussi and Vosniadou, 2010, Vamvakoussi et al., 2011). To the best of our knowledge, there are no studies investigating the natural number bias across different rational number tasks, in particular at the individual level. Moreover, we are not aware of any study investigating whether the natural number bias can still manifest itself in the reasoning process even when errors are no longer committed. To this end, we looked at the natural number bias from a different perspective.

It is arguably unavoidable—and to some extent also recommendable—for students to build on their prior knowledge of natural numbers in their early attempts to make sense of rational numbers (Steffe & Olive, 2010). Several authors, however, stress that learning about rational numbers eventually calls for substantial reorganization of students’ prior knowledge about number, namely conceptual change (Ni and Zhou, 2005, Smith et al., 2005, Vamvakoussi and Vosniadou, 2010). This is a difficult and time consuming process because—as explained above—the conception of number as counting number is firmly established on the basis of in and out of school experiences. In addition, this process is typically impeded by the fact that students remain unaware of the discrepancy between what they already know about numbers and what is to be learned. Thus they tend to regard their beliefs as necessarily correct and do not challenge them in view of the new information regarding rational numbers presented at school. In the words of Greer (2009), students do not recognize that this is a situation where they need to “stop and think”.

We suggest that this problem can be examined from the perspective of the distinction between intuitive and analytical reasoning. In fact, reasoning that leads to erroneous responses based on natural numbers bears many similarities to intuitive reasoning as described in science and mathematics in the influential work of Fischbein (1987) (see Merenluoto and Lehtinen (2004) for a similar observation). Fischbein described intuitions as (cognitive) beliefs characterized by self evidence, intrinsic certainty, and coerciveness. The latter means that intuitions are taken to be necessarily true beyond the need for any further justification while any possible alternatives are readily discarded as unacceptable. They are also characterized by globality (i.e., they allow for an immediate and integrated grasp of a situation, via the selection of features that are deemed relevant). Furthermore, in contrast to analytical reasoning, intuitions are implicit (i.e., they are not under the conscious control of the individual). Intuitions have a (mini) theory status, in the sense that they are not isolated, unitary perceptions, skills, or beliefs and that are characterized by extrapolativeness (i.e., they provide the basis upon which inferences are made that go beyond the information at hand). Finally, they are characterized by perseverance (i.e., once established they are robust and therefore not easily eradicated by instruction). Fischbein made the rather strong claim that some intuitions are never completely abandoned, but survive—and may coexist with scientific accounts—throughout a person's life.

This analysis of mathematical intuitions is obviously relevant to the natural number bias. Fischbein (1987) himself offered examples related to the number concept throughout his book, in particular the above-mentioned successor principle, as well as the conceptualization of multiplication as repeated addition. The question arises: Is there empirical evidence that the natural number bias is indeed an instance of intuitive reasoning?

This question can be placed into the more general frame of intuitive reasoning in the domain of mathematics. Recently, it has been argued that the dual-process theories in cognitive psychology and their accompanying methodologies could be a valuable tool in establishing the intuitive nature of erroneous reasoning in various mathematical domains (Gillard et al., 2009a, Leron and Hazzan, 2006, Leron and Hazzan, 2009). Dual-process accounts of reasoning (e.g., Epstein, 1994, Evans and Over, 1996, Kahneman, 2000, Sloman, 1996, Stanovich, 1999) were originally developed to account for poor performance in reasoning and decision-making by individuals who otherwise possessed the knowledge and skills necessary to deal with the tasks at hand. In these theories, it is assumed that humans have an intuitive/heuristic (S1) and an analytic processing system (S2). S1 is deemed fast, automatic, associative and undemanding of working memory capacity, whereas S2 is deemed slow, controlled, deliberate and effortful. Fast S1-heuristics often lead to correct responses, but sometimes they do not. In these latter cases, either an incorrect response is provided, or S2 needs to intervene and override the initial response. Hence, errors may be attributed to S1's pervasiveness and S2's failure to intervene. There are at least two processing claims within the dual-process framework that may serve as a basis to empirically identify whether a response is the result of a heuristic or an analytic process. The first is the differential processing speed—S1 is faster than S2—and the second is the differential involvement of resources—S1 is less demanding in working memory resources than S2.

The dual-process accounts of reasoning are not without criticism (see for example Osman, 2004, Osman and Stavy, 2006). Nevertheless, reaction time data and manipulation of working memory capacity are less contested, and have been extensively used to study the underlying reasoning processes in cases of interference, such as in the Stroop Color-Word Task, namely the well-known Stroop effect (see McLeod, 1991 for a comprehensive review). Recently, such methods have been fruitfully employed in the case of mathematical tasks that are known to systematically elicit erroneous responses (Gillard et al., 2009a). For example, Babai, Levyadun, Stavy, and Tirosh (2006) studied a widely documented error that students make when comparing two polygons with respect to their perimeter, consistent with the assumption that “the larger the area, the larger the perimeter”, which these researchers take to be an instance of a more general intuitive rule termed more A–more B (Stavy & Tirosh, 2000). They showed that incorrect responses in line with the more A–more B rule were provided faster than correct answers. Similar results were obtained in a probability task (Babai, Brecher, Stavy, & Tirosh, 2006). Gillard, Van Dooren, Schaeken, & Verschaffel (2009b) studied in two experiments the overuse of proportional solution methods in arithmetic word problems from a dual-process perspective. They (a) restricted the solution time and (b) manipulated participants’ available working memory capacity by increasing working memory load using a dual-task method. In both experimental conditions, there was an increase of the errors based on inappropriate proportional reasoning when compared to a control condition without time pressure or working memory load, respectively. On the other hand, appropriate proportional reasoning remained unaffected under time pressure or working memory load.

From a dual-process framework perspective, these findings support the hypothesis that systematic errors in a variety of mathematical tasks are the result of intuitive, heuristic reasoning. Applying this framework in the case of the natural number bias can further provide methods to test for its intuitive character that go beyond the mere observation of errors.

The central hypothesis underlying this study is that errors due to the natural number bias can be characterized as intuitive and thus are persistent and present even in educated adults, who otherwise have the knowledge and skills necessary to respond correctly. We assumed that when an individual, even an educated adult, is confronted with a rational number task—such as comparing fractions—an intuitive response grounded in natural number reasoning comes to mind first. When an item is compatible with natural number properties—hereafter called a congruent item—(e.g., “1/3 < 2/3, True or false?”), the intuitive response leads to a correct answer. For incongruent items (e.g., “1/5 < 1/9, True or false?”) this intuitive response needs to be inhibited for a correct answer to be given. We thus predicted that incongruent items would trigger more incorrect responses (because the intuitive response was not inhibited) than congruent ones; and also that correct responses for incongruent items would have a longer reaction time than congruent items (because inhibiting the intuitive response required time).

We also hypothesized that there would be differences across the different kinds of tasks involved in this study (comparison, operations, and density tasks). Given the above literature review, we predicted that the comparison tasks would elicit fewer errors, and the density tasks would elicit more errors, than the other tasks, while even in the absence of errors the bias would still be manifested in terms of reaction times. Moreover, we expected the difference in accuracy between the tasks to be present not only at the group but also at the individual level.