Multiple Representations in Number Line Estimation: A

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Multiple Representations in Number Line Estimation: A Developmental Shift or Classes of Representations? a

Samantha Bouwmeester & Peter P. J. L. Verkoeijen a

a

Erasmus University Rotterdam

Version of record first published: 10 Jul 2012

To cite this article: Samantha Bouwmeester & Peter P. J. L. Verkoeijen (2012): Multiple Representations in Number Line Estimation: A Developmental Shift or Classes of Representations?, Cognition and Instruction, 30:3, 246-260 To link to this article: http://dx.doi.org/10.1080/07370008.2012.689384

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COGNITION AND INSTRUCTION, 30(3), 246–260, 2012 C Taylor & Francis Group, LLC Copyright ISSN: 0737-0008 print / 1532-690X online DOI: 10.1080/07370008.2012.689384

Multiple Representations in Number Line Estimation: A Developmental Shift or Classes of Representations? Samantha Bouwmeester and Peter P. J. L. Verkoeijen

Downloaded by [Erasmus University] at 08:16 16 July 2012

Erasmus University Rotterdam

Children’s estimation patterns on a number line estimation task may provide information about the mental representation of the magnitude of numbers. Siegler and his colleagues concluded that children’s mental representations shift from a logarithmic-ruler representation to a linear-ruler representation. However, there are important methodological issues with respect to their number-line studies that threaten the validity of the conclusions. We discuss these methodological issues and propose an alternative method to analyze estimation data. One hundred nineteen children from kindergarten, first, and second grade performed a number-line estimation task in which they had to estimate the position of 30 numbers on a 0-to-100 number line. The results supported the hypothesis that children show various kinds of estimation patterns. Five classes of children were distinguished, which were characterized by different estimation patterns. A remarkable result was that the logarithmic-ruler representation was not found. Although young children were more likely to show overestimation of small numbers than older children, this developmental trend was small and not significant.

Most people are unaware of how many times they actually estimate in everyday life. We estimate at what time we have to leave home in order to catch the bus, we estimate whether we have enough cash to pay for our groceries, we estimate the amount of salt we add to our meal to make it tasty yet not too salty. For these and many other examples estimation is used because accurate estimates are sufficient for many purposes and because people often lack the knowledge, time, means, or motivation needed to calculate precise values (Siegler & Booth, 2005). However, estimation in everyday life is a very complex process because to be of practical use it may require knowledge of the problem context, knowledge of measurement units like time, distance, quantity (Siegler & Booth, 2004) and a good awareness of the boundaries of an estimate. Despite the fact that people often use estimation in their day-to-day routine, estimation has— until recently—not explicitly been taught in elementary school. However, this may not be problematic, because various studies have demonstrated that estimation performance is related to general arithmetic skills, which do receive a lot of attention in elementary school instruction. For instance, Booth and Siegler (2006, 2008), Siegler and Booth (2004) and Schneider, Grabner, and Paetsch (2009) showed that performance on a number line estimation task was a Correspondence should be addressed to Samantha Bouwmeester, Department of Psychology, Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands. E-mail: [email protected]

MULTIPLE RESPRESENTATIONS IN NUMBER LINE ESTIMATION

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good predictor of math achievement. Furthermore, it should be noted that general estimation ability has become more and more important since problem-based mathematics gained popularity. A central aim of problem-based mathematics is that students think about diverse solution strategies in realistically framed problems (Gravemeijer & Doorman, 1999; Treffers, 1991, 1993). Besides the fact that estimation skills are important in realistic problem solving, they also may play an important role in pure number arithmetic in which a realistic context is not available. In this study, we focus on the estimation performance of children on a number line estimation task. Learning to estimate on a number line is not an educational goal per se but number lines are often used to teach basic arithmetic skills. We will argue in this article that it is very interesting to investigate children’s estimation ability on a number line task because the estimation pattern may tell us a lot about the mental representations children have of magnitudes of numbers and of distances between numbers. NUMBER LINES IN EDUCATION In the United States, not all researchers support the use of number lines in the primary grades to teach arithmetic or for other purposes. According to these researchers the continuous quantity character of the number line does not fit with the discrete quantity perspective children generally have during the first grades of primary school (Ernest, 1985; Fuson, 2009). For this reason, the National Research Council (see Fuson, 2009) has recommended against using number lines as an instructional tool before grade 2. However, in the Netherlands, and also, for example, in Italy (see Di Meo, 2008), children start practicing with number lines from grade 1. Before grade 1, they mainly practice with counting and ordering numbers. During this kindergarten phase, the mental representation of most children is assumed to consist of a list of the counting words. From grade 1, number lines are introduced in the classroom in order to facilitate basic arithmetic skills like addition and subtraction (Beishuizen, 1993; Gravemeijer, 2000). A reason why children in the Netherlands practice with number lines earlier than in the United States may be that until the second half of grade 2 the Dutch arithmetic teaching methods mainly use discrete quantity number lines represented by, for example, ladders on which the numbers are represented by the rungs or on which the numbers are explicitly written. From the second half of grade 2, empty number lines are introduced in which the end poles and the unit lengths are rough or not presented at all. Children are supposed to represent the arithmetic problem by first putting the first number on the line, and then add or subtract a certain distance. When children start using number lines to represent addition and subtraction problems they will gradually learn that numbers can be represented as distances on a number line. Although absolute measurement units may not be introduced yet—the addition and subtraction problems do not require measurement units—a realistic representation of the addends and subtracts ask for an understanding of the underlying scale.

Number Line Estimation and Mental Representation A typical number line estimation task requires children to estimate the position of a number, or the number of a position, on a number line in which the endpoints only are labeled (e.g., 0 and 100). The estimation pattern and the kinds of estimation mistakes children make provide important

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information about the processes involved in representing numbers. Hence, a number line estimation task is a useful tool to obtain insight into the mental representations children have of the distances between numbers on a number line. However, it is also important to realize that children’s estimation patterns—when they show any pattern at all—are not the same as their internal representations. That is, although a certain internal representation is expected to be reflected in an estimation pattern, the reverse may not be true: There may be a multitude of alternative explanations for children’s estimations apart from their internal representations of the magnitude of numbers.


Pure Numerical Estimation and Age A central finding emerging from number line estimation research is that people—but also animals (see, e.g., Dehaene, 1997, 2001)—tend to overestimate the magnitude of small numbers and underestimate the magnitude of large numbers. When the estimated number is presented on the y-axis and the actual number on the x-axis, the resultant estimation pattern of this inappropriate representation of numbers resembles a logarithmic function (Dehaene, 1997, 2001). Dehaene (1997) showed that for young and old people and even for animals the estimations patterns on several tasks can be represented by a logarithmic function. Despite the large amount of empirical evidence for this logarithmic ruler presentation perspective, it may be questioned whether it also applies specifically for estimation patterns and mental representations in a number line task. In number line tasks the end pole is fixed by a number (e.g., 100 or 1000) that determines the scale of the number line. When estimating the position of large numbers, close to the end pole of the line, the end pole may serve as a reference. This means that the underestimation of large numbers may decrease—instead of increase as suggested by the logarithmic ruler account—for numbers close to the end pole. Thus, the estimation of large numbers on a number line task with a reference may be less underestimated than in number estimation tasks—like the estimation of the number of dots on a paper or the number of marbles in a vase—in which no absolute reference is present. Another account that may explain the inaccurate estimations and inappropriate mental representations of numbers is the accumulator model (Gibbon & Church, 1981; Huntley-Fenner, 2000). Indeed, there is a considerable body of empirical evidence consistent with the accumulator model (see, e.g., Huntley-Fenner, 2001). This model assumes that the variability in estimates increases (accumulates) with larger numbers. According to Brannon, Wusthoff, Gallistel, and Gibbon (2001) this occurs because the mental representation of the distance between two numbers becomes fuzzier with increasing number size. Like the logarithmic ruler model of Dehaene (1997), the accumulator model is assumed to be used by people of all ages and even by animals (Brannon et al., 2001). Again, it may be questioned whether this model applies to the number line task. Because the number line task has a fixed end pole, estimation of large numbers is bounded by the endpoint. Therefore, the variation in estimations of large numbers is also bounded by the endpoint. In contrast to the logarithmic ruler representation model and the accumulator model, Case and Okamoto (1996) proposed a model that is not age invariant but shows a developmental shift. According to their multiple representations model young children have a qualitative representation of numbers in which they mainly distinguish between “small” and “large” numbers. When children



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grow up, they shift from a qualitative representation to a linear ruler representation, in which estimates increase linearly with the actual value. Case and Sowder (1990) propose that children, who have not yet acquired the linear ruler representation, are unable to make accurate estimates. On the basis of a number of cross-sectional studies using the number line estimation task, Siegler and his colleagues (Booth & Siegler, 2006; Laski & Siegler, 2007; Opfer & Siegler, 2007; Siegler, 1996; Siegler & Booth, 2004; Siegler & Opfer, 2003; Siegler, Thompson, & Opfer, 2009; Thompson & Opfer, 2008) concluded that children indeed show a developmental shift in their mental representation of numbers on a number line. This shift encompasses a change in the mental representation of numbers from a logarithmic to a linear relationship between the estimated and the real locations of numbers on a number line. According to Siegler and colleagues (2009) younger children’s estimates of numerical magnitude typically follow Fechner’s law (y = k × ln x, where y is the estimated number, x is the actual magnitude, and k is a constant) and increase logarithmically with the actual value. In contrast, estimation patterns of older children tend to reflect a linear ruler representation. Thus, age-group analyses support a developmental shift from logarithmic to linear representations of numbers. These findings were taken as strong evidence for the multiple representations model. However, we will argue that there are some important methodological issues, which undermine the validity of the conclusions with respect to the multiple representations and the developmental trend. As an alternative and to overcome these methodological issues we present in the present study a latent variable model approach to detect estimation patterns of children on a 0 to 100 number line task.

METHODOLOGICAL ISSUES Before we discuss five methodological issues, we will describe the general procedure commonly employed to analyze the estimation patterns. Because the distribution of the estimated numbers is generally skewed, Siegler and Booth (2004, 2006; see also Siegler, 1996; Siegler & Opfer, 2003; Siegler, Thompson, & Opfer, 2009) used for each estimated number the median estimate of an age group as the dependent variable in a regression analysis with the number to be estimated as a predictor variable. Next, they fitted linear, logarithmic, and exponential regression functions on these medians of the estimated numbers. The model with the highest proportion explained variance was chosen to be the best fitting model. In addition, they also fitted the same regression models on the individual estimates of the children. Again, the model that explained the most variance was selected as the best fitting model. The first issue to be discussed is whether the median estimation as a summary score forms an adequate representation of the individual estimation patterns. When children’s individual estimations differ substantively, averaging (or taking the median) over the estimates in a group may result in a summary score that may not be representative of even one of the individuals. By averaging over individuals, one implicitly assumes that individual deviations from the averaged score are error (i.e., unrelated to the actual measurement). However, just because the estimations vary considerably, deviations from the mean, or median, cannot be assumed to be error. Instead, these deviations contain important information about the estimation patterns of the individuals. Suppose a group of same-aged children shows different kinds of estimation patterns. Consider, for example, that some children in this group do not have an appropriate number representation. This


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type of number representation will lead to a random estimation pattern in which no relationship exists between the actual number position and the estimated position. However, other children in this group show a clear linear relationship between the actual number and the estimated number. In this case the average estimation pattern of the complete group of children is a summary of a mixture of estimation patterns. If now, a logarithmic and a linear regression function are compared on the data in this sample, and the logarithmic function provides the best fit then it would be incorrect to conclude that a logarithmic function best describes the estimation patterns of the entire sample. A second issue concerns the use of age groups to investigate the developmental trend. In all studies, summary scores are calculated per grade to investigate a potential developmental shift from a logarithmic to a linear representation of numbers. However, although a developmental shift may be observed in children’s mental representation of numbers, it is a rather strong assumption that this shift is clearly visible between two predefined age groups. Children of the same age may be in different developmental stages dependent on intelligence, gender, experiences with numbers, and so on. As a consequence, a considerable amount of variation in estimation pattern within an age group is treated as unsystematic error, while in fact this variation may be systematically related to development (Wohlwill, 1973). By adding this variation into the error term of the statistical test in which age group is used as between subject factor, the power to detect a developmental trend is suppressed. This issue may be solved by first using a latent class model in which the different latent classes summarize the estimation patterns present in the total sample. Children can be assigned to one of the latent classes on the basis of their estimation patterns. A posteriori it can be investigated how age is related to the estimation patterns distinguished by the latent classes. Of course, when there is a general developmental trend this will be observed roughly in age groups, but it is unfortunate to use a suboptimal design when a more powerful design is available. Using a latent class model, a researcher may conclude that part of the within-age variability may actually be related to systematic differences in estimation ability. A third issue concerns the predefined functions that are used to fit the estimation patterns. Instead of the linear, logarithmic, and exponential functions, Siegler and Booth (2004, 2008) and Siegler and Opfer (2003) fitted, other less restrictive functions may describe the relationship between the estimated and actual numbers better. For example, in a very recent article, Barth and Paladino (2011) proposed that the estimates on a number line task in which the end poles are bounded are likely to be proportional judgments because people base their estimates on one of the known referents. This idea suggests that estimation patterns follow a cyclical power model. According to this model, children overestimate the distance between small numbers and underestimate the numbers to the right of the center of the number line. In addition, the underestimation may decrease for estimated numbers close to the end of the pole. Consistent with their hypothesis, Barth and Paladino found that a cyclical power model could explain the estimations on a number line task quite well. The fourth issue deals with the analysis of the individual estimation patterns. In order to investigate whether individual children within an age group show different kinds of estimation patterns, Siegler and Booth (2004) fitted regression lines on individual estimates and compared the fit of the logarithmic and the linear functions. Beside the questions of whether the two functions fitted the estimation patterns at all (in an absolute sense) and whether a model other than the logarithmic and linear functions might better describe the estimation pattern, the difference in fit was interpreted absolutely and was not statistically tested. As a consequence, it is not clear whether the results of the individual analysis are reliable or just based on chance fluctuation.



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The final issue concerns the fact that all studies on number line estimation in which logarithmic and linear functions are fitted concentrate on the difference between logarithmic and linear functions while not taking into account that the logarithmic and linear functions themselves can be completely different as well. For example, suppose the relationship between the position of the numbers on a 0–100 number line of a certain child is best described by a linear function, say “estimated position = 20+number.” This function indicates a consequent overestimate of the real position of the numbers. Another child’s estimations might best be described by the linear function “estimated position = number,” which indicates a perfectly accurate estimation pattern. Although these two children both show linear representations, this fact is hardly informative when no further information is provided about the differences in intercept and slope. Similarly, a logarithmic function looks completely different when for example “estimated position = 20∗ ln(number)” is used, than when “estimated position = 4∗ ln(number+40)” is used. Thus, when no further details are given about the specific parameters of the function, it is hardly informative to know that a child’s estimation pattern is characterized by a linear or logarithmic function. THE PRESENT STUDY To solve these five issues, we propose a latent variable modeling approach. We use a latent class regression analysis (Wedel & DeSarbo, 1994; Vermunt & Magidson, 2005) to analyze the estimation patterns of the individual children studied (this pertains to issues 1 and 4 above). Different latent classes will be estimated for different kinds of estimation patterns. Because the estimation patterns may be described by widely varying functions, we fitted functions that contain one to three parameters: a linear term, a quadratic term, and a cubic term (issue 3). Adding these higher order terms gives us more freedom to estimate a function, which gives a more realistic reflection of the estimation pattern. That is, when the estimation pattern is best described by a logarithmic curve, the combination of a quadratic and a cubic effect can approach this logarithmic curve. However, the combination of a quadratic and a cubic effect may also describe estimation patterns in which children overestimate small numbers and underestimate large numbers as is proposed by the cyclical power model (Hollands & Dyre, 2000). These latter estimation patterns cannot be approached by a logarithmic curve. To account for individual differences in estimation patterns within a latent class we added a random intercept term to the regression function. This random intercept allows that children have the same kind of estimation pattern but differ with respect to the exact location of the function (issue 5). When different classes of children can be distinguished showing different estimation patterns, age can be related to class membership a posteriori to investigate developmental trends in estimation patterns (issue 2). METHOD Participants The participants were 119 children from two elementary schools in Rotterdam and Spijkenisse, a small town southwest of Rotterdam, the Netherlands [41 kindergarten (Mean age = 6.25, SD = .31), 38 first graders (Mean age = 7.50, SD = .49), and 40 second graders (Mean age = 8.50, SD = .54)]. Informed consent was obtained from the parents of all the children. Further,

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the children were predominantly from a middle socio-economic background. The percentages of females and males were 51.3% versus 48.7%. None of the children had learning problems and it could be assumed that all children had IQ scores above 70. In the Netherlands children with an IQ score below 70 are referred to special education schools.


Materials and Procedure Children met one-on-one with the experimenter for a single 35-min session. The session consisted of three number line estimation tasks. In one task, a plain number line was used; in the other two tasks a context and short story were used. In this manuscript we focus on the first pure, no-context, number line orienting problem. Children were presented with a sheet of paper with a 25-cm line across the middle; the number 0 was printed just below the left end of the line and the number 100 just below the right end. We learned from the pilot study that many young children needed some practice in order to understand the task. Therefore, we followed the procedure of Siegler and Booth (2004) by giving them an introductory task in which they were asked to mark where they thought 50 would go on the line. After they did so, they were shown an identical number line with 50 marked in the correct position. Subsequently they were told that that was where 50 belonged, and they were asked if they knew why 50 went there. All children were then told, “Because 50 is half of 100, it goes directly in the middle, halfway between 0 and 100. So 50 is the middle, and it’s the only number that goes exactly in the middle.” After the orienting problem, children were presented with 30 sheets of paper, each with an identical 25-cm line, and asked to put a single mark on each line to indicate the location of a given number. The lines were identical to the one used in the orienting problem except that a number (different on each trial) was printed above the middle of the line. The 30 numbers that were presented were 3, 4, 6, 8, 10, 12, 14, 17, 18, 20, 21, 24, 25, 29, 33, 39, 42, 48, 52, 57, 61, 64, 72, 75, 79, 80, 81, 84, 90, and 96. We used the same numbers that were used in the study of Siegler and Opfer (2003) in order to replicate this study as closely as possible. Siegler and Opfer (2003) reported that they used more small numbers (below 50) than large numbers in order to increase the ability to detect overestimation of small numbers. The order of the sheets was randomized for each child. There was no time limit for any of the problems. In addition, no feedback was given about specific estimates, though the experimenters frequently offered general encouraging comments to all children. Following completion of all tasks, children were thanked for participating, told they did a good job, and returned to their classroom.

Statistical Analysis A latent class regression analysis was used to investigate whether different representations could be distinguished. In the latent class regression model, latent classes of subjects were distinguished that differed with respect to the regression function. The estimations of the 30 numbers formed the dependent vector with the numbers to be estimated, the quadrate of these numbers, and the cubic of these numbers as predictor vectors. A random intercept was included to the model to allow for individual differences. (See the Appendix for the formal description of the model and the parameters.)



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The program Latent Gold 5.0 (Vermunt & Magidson, 2005) was used to estimate the parameters and calculate the fit of the latent class regression model. The fit of models with one to six latent classes was compared in order to determine the number of estimation patterns to be distinguished. The log-likelihood (LL) is a goodness-of-fit measure that indicates the likelihood of the observed results given the parameter estimates of the model. The Bayesian information coefficient (BIC) and the Akaike information coefficient (AIC) are both indices of the fit of the model relative to the number of parameters. A lower value on these indices signifies a better fit of the model. The percentage of classification errors indicates the percentage of the sample that is wrongly classified. Note that the percentage of classification errors for a one-class model is by definition zero. The R2 indicates the proportion explained variance of the model. Latent class regression analysis is a very useful tool to identify classes of children who show the same estimation patterns. However, a drawback of the analysis is that the number of estimated parameters may become large since a separate regression model has to be estimated for each latent class. To reliably estimate the parameters, a sufficiently large number of observations is required. In our study we had thirty observations per child and a sample of 119 children, therefore we had plenty of observations to estimate reliably the parameters in the five classes.

RESULTS Pure Number Line Estimation Table 1 shows the fit indices of the latent class regression models. The BIC and AIC have the lowest values in the 6-class model, therefore this model can be taken as the best fitting model. However, the class sizes of the 6-class model show that there are two very small classes with class size proportions of .03 and .06. Moreover the percentage of classification errors almost doubled in comparison with the 5-class model. The 5-class model consists of one small class (class size proportion = .03) and four larger classes. Inspection of this small class shows that the estimated curve of the estimation patterns of the children in this small class were horizontal, which indicates that there was no relationship between the estimation of the numbers and the real position on the number line. Figure 1 shows that the children in this class estimated all numbers around 50 (in TABLE 1 Fit Indices of the Latent Class Regression Models Model

LLa

BIC(LL)b

AIC(LL)c

#pard

% Class.Erre

R2

1-class 2-class 3-class 4-class 5-class 6-class

–14589.03 –13726.06 –13465.23 –13371.97 –13288.84 –13239.18

29206.73 27509.47 27016.48 26858.64 26721.05 26650.40

29190.06 27476.12 26966.46 26791.94 26637.67 26550.35

6 12 18 24 30 36

0 0.3 0.8 0.8 1.4 2.7

.73 .80 .82 .83 .83 .84

aLL = Loglikelihood; bBIC = Bayesian information coefficient; cAIC = Akaike information coefficient; d#par = number of parameters in the model; e%Class.Err = percentage of classification errors.



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FIGURE 1 Estimated function (line) of the relationship between actual magnitude and estimated numbers, observed estimates (grey dots), and accurate reference line (dotted line) for the five classes on the number line estimation task.

the center of the number line). Because the 5-class model had lower BIC and AIC values than the 4-class model, we decided to interpret the results of the 5-class model. Figure 1 shows the estimated regression lines of the five classes. The dotted line is a reference line that indicates an accurate estimation pattern. The grey dots are the estimates of the individual children who are assigned to the particular class (note that children are assigned to a particular class when they have the highest probability to be in that class, given their estimation pattern). Class 1 has a sample size proportion of .25 and shows a rather accurate linear relationship between the real position and the estimated position of the numbers. The parameter estimates in Table 2


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TABLE 2 Parameter and Variance Estimates and Significance Level for the Five Latent Classes Class 1

2


3

4

5

Effect

Value

p

Linear Quadratic Cubic Linear Quadratic Cubic Linear Quadratic Cubic Linear Quadratic Cubic Linear Quadratic Cubic

0.0006 0.9062 0.0000 –0.0044 0.0244 –0.0002 1.709 –0.0223 0.0001 2.6620 –0.0485 0.0003 –0.0267 –0.0003 0.0000

.75