Multiplicative transformations 1 RUNNING HEAD : Multiplicative transformations Keywords : Ratio sensitivity , ratios , multiplicative operations , doubling , halving Children ’ s multiplicative transformations of discrete and continuous quantities

Recent studies have documented an evolutionarily primitive, early emerging cognitive system for the mental representation of numerical quantity (the analog magnitude system). Studies with non-human primates, human infants, and preschoolers have shown this system to support computations of numerical ordering, addition, and subtraction involving whole number concepts prior to arithmetic training. Here we report evidence that this system supports children’s predictions about the outcomes of halving and perhaps also doubling transformations. A total of one hundred and thirty-eight kindergarten and first grade children were asked to reason about the quantity resulting from the doubling or halving of an initial numerosity (of a set of dots) or an initial length (of a bar). Controls for dot size, total dot area, and dot density ensured that children were responding to the number of dots in the arrays. Prior to formal instruction in symbolic multiplication, division, or rational number, halving (and perhaps doubling) computations appear to be deployed over discrete and possibly continuous quantities. The ability to apply simple multiplicative transformations to analog magnitude representations of quantity may form a part of the toolkit that children use to construct later concepts of rational number. Multiplicative transformations 3 Children’s multiplicative transformations of discrete and continuous quantities The ability to represent approximate numerical magnitudes without the use of language is common to humans of all ages and to nonhuman animals. Animals, infants, children, and adults prevented from applying their exact verbal counting skills discriminate sets based on their cardinal values (for animals and human adults, see Dehaene, 1997, for a review; for human infants and children, see for example, Xu and Spelke, 2000; Barth, La Mont, Lipton, & Spelke, 2005; Cordes and Brannon, 2008). An “analog magnitude representation” system appears to underlie this ability: the discrete numerosity of the set is internally coded by a mental magnitude, with the magnitude proportional to the number of elements in the set. Like comparative judgments of many kinds of continuous quantities, comparative judgments of discrete number are least accurate when the ratio of compared numerosities is closest to 1:1. Ratio-dependent discrimination in accord with Weber’s Law is a key signature of the analog magnitude system (Gallistel & Gelman, 1992, 2000; Dehaene, 2007). Analog magnitude representations support computations of numerical ordering, addition, and subtraction across species and throughout the course of development. Most relevant here is evidence that nonverbal animals as well as human infants and children use analog magnitude representations to compute the outcomes of additive operations over visually presented sets of elements (McCrink & Wynn, 2004; Barth et al., 2005; Flombaum, Junge, & Hauser, 2005; Slaughter, Kamppi, & Paynter, 2006; Cantlon & Brannon, 2007; Gilmore, McCarthy, & Spelke, 2007; Barth, Beckmann, & Spelke, 2008). More controversial is the question of whether analog magnitude representations of approximate number also support multiplicative operations on sets before young children receive formal training in multiplication and division. Concepts of multiplicative, rather than additive, Multiplicative transformations 4 change are critical to children’s later construction of an understanding of rational number (Smith, Solomon, & Carey, 2005), a famously difficult achievement of middle school math. It is often suggested that children’s early intuitions about quantity transformations may support later learning about fractions, but these intuitions are often thought to rest on protoquantitative, nonnumerical notions of amount (Confrey, 1994; Mix, Levine, & Huttenlocher, 1999; Resnick & Singer, 1993; Resnick, 1992). To our knowledge, the potential role of analog magnitude representations of discrete quantity in children’s intuitive knowledge of multiplicative transformations has not yet been investigated. Gelman and Gallistel (Gallistel & Gelman, 1992, 2000, 2005) hold that mental magnitudes representing number (like those representing non-numerical quantity) do enter into ordering, addition, subtraction, multiplication, and division operations, even in the brains of nonverbal animals (Gallistel, 1990; Leon & Gallistel, 1998; Gallistel, Mark, King, & Latham, 2001; but see Church & Broadbent, 1990; Kakade & Dayan, 2002; Yang & Shadlen, 2007 for alternative views). On this view, analog magnitudes provide a common representational format permitting computations over both continuous and discrete quantities. Some human adult studies also appear consistent with this idea: adults succeed at tasks which may involve multiplying and dividing approximate numerical magnitudes, even when they are prevented from exact counting (Barth, 2002). Because adults have had many years of arithmetic instruction, however, they may have solved these tasks by forming verbal estimates of the quantities involved and then invoking symbolic multiplication or division. Studies of patients with calculation deficits support this latter possibility, because impairments in symbolic multiplication have been linked to impairments in language but not in nonsymbolic number Multiplicative transformations 5 processing (Cohen, Dehaene, Chochon, Lehéricy, & Naccache, 2000; Lemer, Dehaene, Spelke, & Cohen, 2003). Representations of both discrete and continuous quantity do appear to support simpler forms of reasoning about multiplicative relationships. Adults track proportions unconsciously and make use of them when transferring from a discrimination learned for a continuous quantity to a novel discrimination of discrete quantity (Balci & Gallistel, 2006). A recent study has shown that even infants spontaneously represent the ratios between two sets of dots, discriminating new arrays with the same ratios of blue to red dots as those they have seen before from arrays in which the dots are in a different ratio relationship (McCrink & Wynn, 2007). In young children, much previous work on proportional reasoning has focused on continuous quantity. Though some studies have reported earlier competence in proportional reasoning about continuous vs. discrete quantity, these often involve discrete tasks that provide opportunities for exact counting. Children’s apparent lack of competence could stem not from difficulties in reasoning about discrete quantities per se, but from the tendency to count when a task affords the opportunity (Jeong, Levine, & Huttenlocher, 2007; Boyer, Levine, & Huttenlocher, 2008). Here we focus on a simple form of multiplicative reasoning: the ability to apply the multiplicative transformations of halving or doubling to continuous or discrete quantities. We ask children to observe a few examples of such transformations, identify the ratio relationship that holds between the original quantity and the transformed one, and then apply the same transformation to a novel quantity, and then judge whether the transformed quantity would be larger or smaller than a comparison quantity. This task requires more than the detection of a ratio relationship: to succeed, children must apply a transformation that operates on an initial quantity to yield a second quantity that is a fixed ratio of the first. Multiplicative transformations 6 Some evidence suggests that young children may not succeed at these tasks. Children exhibit an understanding of additive relations between quantities before they develop an understanding of multiplicative relations (Resnick & Singer, 1993), and numerous studies from the tradition of information integration theory suggest that younger children apply additive integration rules rather than normative multiplicative rules (e.g. Schlottmann & Anderson, 1994; Wilkening & Anderson, 1991; Wilkening, 1982; Anderson & Cuneo, 1978; but see Gigerenzer & Richter, 1990). These results have led researchers to argue that multiplicative reasoning is not available at all to children under the age of seven or eight. In contrast, other studies have found evidence of intuitive reasoning about multiplicative transformations in younger children, provided that the task situation only required the modification of a single quantity (Schlottman & Tring, 2005; Schlottman, 2001). Also, Confrey and her colleagues have argued that young children possess schemas that form the basis for reasoning about multiplicative operations without relying on repeated addition, proposing that children show intuitive insight into a conceptual primitive called “splitting” that supports later reasoning about ratio, proportion, multiplication, and division (e.g. Confrey, 1994). The present studies address the following questions. First, is there evidence that analog magnitude representations of number can support computations of halving or doubling in young children? We tested kindergarten and first grade children, who have no formal instruction in symbolic multiplication or division, or in symbolic representations of fractions. We used a task in which stimuli are presented rapidly, to discourage attempts at exact counting, and in which a single numerical quantity is transformed, in order to maximize children’s chances of success. Second, are multiplicative computations evident earlier, or more robustly, for continuous Multiplicative transformations 7 quantities than for discrete quantities? To address this question, we adapted the same procedure to a task in which the child mentally doubled or halved the magnitude of a continuous quantity. Experiment 1: Continuous and discrete doubling Kindergarten and first grade children observed a small number of examples of doubling transformations applied to either discrete quantities (blue dot arrays’ numerosities) or continuous quantities (blue bars’ lengths). Animated sequences presented on computer screens showed an initial quantity th