An integrated theory of whole number and fractions development (Siegler, Thompson, & Schneider, 2011)

Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273-296. doi:10.1016/j.cogpsych.2011.03.001

Advocates of privileged domains theories argue that specialized learning mechanisms make it easier to learn about whole numbers than about fractions or other types of numbers (Gelman & Williams, 1998; Wynn, 2002) and that constraints that facilitate learning about whole numbers interfere with learning about fractions. Fractions learning is hindered (and whole number learning helped) by children being predisposed to assume that each number has a unique successor, that sets can be counted by assigning numbers to objects in a 1:1 fashion, and that the final number in a count can be used to represent the cardinality of the set that was counted (also see Geary, 2006; Leslie, Gelman, & Gallistel, 2008; Wynn, 2002). Evolutionary theories propose that whole numbers are biologically primary and that fractions and other types of numbers are biologically secondary. Conceptual change theories(Ni & Zhou, 2005; Vosniadou, Vamvakoussi, & Skopeiliti, 2008; Vamakoussi & Vosniadou, 2010) are similar in emphasizing differences between learning about whole numbers and fractions and in emphasizing how the ‘‘whole number bias’’ interferes with fractions learning.All posit qualitative differences between an early developing, ‘‘natural’’ understanding of whole numbers and a later developing, flawed or hard-won, understanding of fractions. The earlier developing understanding of whole numbers is said to interfere with the later developing understanding of rational numbers. The theory of numerical development proposed in this article differs in emphasizing a crucial continuity between acquisition of understanding of whole numbers and fractions, as well as differences between the acquisitions. The authors propose that numerical development represents a process of 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. (Similar to Case and Okamoto’s (1996) proposal that the central conceptual structure for whole numbers, a mental number line, is eventually extended to other types of numbers, including rational numbers.)

“One way of thinking about this pair of developments (learning the properties shared and not shared by different types of numbers) is as a gradual change from initially conceptualizing numbers in terms of characteristic features (salient properties of whole numbers that are not necessarily properties of other types of numbers) to later distinguishing between defining features (properties of all real numbers, in particular their magnitudes) and features that apply to some but not all classes of numbers. This change is analogous to the shift from characteristic to defining features in semantic development described by Keil and Batterman (1984).”

“One implication of the present theory of numerical development is that acquisition of knowledge about fractions emerges as a crucial process in numerical development, rather than being of secondary importance. Learning about fractions provides the first major opportunity for children to learn that a variety of salient and invariant properties of whole numbers are not definitional for numbers in general.”

If fractions are crucial for overall mathematical understanding, and if understanding magnitudes is crucial for understanding fractions, then (1) Understanding of fraction magnitudes should be strongly related to proficiency at fractions arithmetic; and (2) Understanding of fraction magnitudes should be strongly related to overall mathematical knowledge.

“Children could memorize fraction arithmetic algorithms without understanding the magnitudes of the fractions being manipulated. Indeed, many mathematics educators have lamented that this is exactly what most students do (e.g., Cramer, Post, & del Mas, 2002; Hiebert & Wearne, 1986; Mack, 1995; Sowder et al., 1998). However, rote memorization without understanding tends to be inaccurate over short time periods and becomes even more inaccurate over longer periods (Reyna & Brainerd, 1991). Thus, fraction arithmetic procedures seemed likely to be more accurately remembered by children who understand the magnitudes of the fractions used in the computation than by children who do not understand the fraction magnitudes. One reason is that accurate fraction magnitude representations make it possible to estimate the results of fraction arithmetic operations and to reject implausible solutions. This, in turn, might lead children to reject flawed arithmetic procedures that produced implausible solutions and to continue trying to learn a procedure that generates reasonable answers. Consistent with this perspective, Hecht (Hecht, 1998; Hecht, Close, & Santisi, 2003; Hecht & Vagi, 2010) has found repeatedly that mea- sures of conceptual understanding of fractions correlate positively with fraction addition skill. Another reason for the prediction is that knowledge of whole number arithmetic correlates positively with knowledge of whole number magnitudes (Booth & Siegler, 2008; Siegler & Ramani, 2009). If knowledge of magnitudes play the same role with fractions as it does with whole numbers, as envisioned by the present theory, similar relations between knowledge of numerical magnitudes and arithmetic would be expected with both types of numbers.”

Understanding of numerical magnitudes has been shown to be related to many other aspects of mathematical development, including counting (Ramani & Siegler, 2008; Whyte & Bull, 2008) arithmetic (Booth & Siegler, 2008; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Geary, Hoard, Nugent, & Byrd-Craven, 2008; Siegler & Ramani, 2009), memory for and categorization of numbers (Laski & Siegler, 2007; Opfer & Thompson, 2008; Thompson & Siegler, 2010), and mathematics achievement test scores (Booth & Siegler, 2006; Geary et al., 2007, 2008; Halberda, Mazzocco, & Feigenson, 2008; Siegler & Booth, 2004). Manipulations that improve numerical magnitude knowledge have been shown to be causally related to increased proficiency in arithmetic (Booth & Siegler, 2008; Siegler & Ramani, 2009) and counting (Ramani & Siegler, 2008; Whyte & Bull, 2008).

The number line task has several important advantages for measuring representations of numerical magnitudes:

  • It can be used with any real number (large/small, positive/negative, integer/fraction, rational/irrational).
  • It transparently reflects the ratio characteristics of the number system.
  • The task is practiced infrequently compared to skills such as counting and arithmetic, so estimates reflect people’s sense of the magnitudes of the numbers rather than memorization of procedures.

Two other estimation tasks: numerosity estimation (‘‘There is 1 dot in this beaker and 1000 in this one; hold down the mouse until there are N dots in this empty beaker.’’) and measurement estimation (‘‘This short line is 1 zip long; this long line is 1000 zips long; draw a line N zips long.’’) (Booth & Siegler, 2006).

Individual differences in numerical magnitude representations have been found to be positively related to other individual differences in mathematical knowledge — arithmetic competence (Booth & Siegler, 2006; Gilmore, McCarthy, & Spelke, 2007; Halberda et al., 2008; Holloway & Ansari, 2008; Mundy & Gilmore, 2009; Schneider et al., 2008) and scores on standardized math achievement tests (Booth & Siegler, 2006; Halberda et al., 2008; Laski & Siegler, 2007; Siegler & Booth, 2004).

Link btw numerical magnitude representations and arithmetic: If learning answers to arithmetic problems is a meaningful process, accurate magnitude representations might indicate the implausibility of many answers and the plausibility of a few, producing more peaked distributions of activation around the correct answer, thus facilitating correct retrieval. Thus, experiences that improve numerical magnitude representations not only increase subsequent learning of correct answers to arithmetic problems but also lead to errors being closer to the correct answer on trials where children err (Booth & Siegler, 2008; Siegler & Ramani, 2009).

“Another important way in which development of knowledge of fractions magnitudes seems likely to differ from development of whole number magnitudes is in the role of strategies. Although strategic influences have been recognized in a wide range of problem solving and reasoning contexts (Siegler, 1996), reviews of the literature on whole number magnitude representations (e.g., Ansari, 2008; Dehaene, Dehaene-Lambertz & Cohen, 1998; Fias & Fischer, 2005; Hubbard et al., 2005) typically do not even mention strategies or strategy choices (for exceptions, see Geary et al., 2007, 2008). The implicit assumption is that people invariably use a particular representation of numerical magnitudes and that the research task is to determine the characteristics of that representation. Sometimes, the assumption is explicit, as when Dehaene (1997, p. 78) described logarithmic representations of whole number magnitudes as occurring ‘‘like a reflex’’ that cannot be inhibited. Consistent with the assumption that magnitude representation is an automatic, non-strategic process, number line estimation with whole numbers is no less accurate under time pressure than without time pressure (Siegler & Opfer, 2003).”

Study design:

  • 6th and 8th graders (11- and 12-year-olds and 13- and 14-year- olds)
  • Assessments of fraction magnitude knowledge: 0–1/ 0–5 number line estimation,  0–1 magnitude comparison, fraction arithmetic problems.
  • Verbal reports of strategy use were obtained immediately after each number line estimation and arithmetic problem, to allow examination of relations between strategy use and speed and accuracy on each task.
  • Obtained students’ mathematics achievement test scores, to examine their relation to the three measures of fraction magnitude knowledge and to fraction arithmetic proficiency.

We made six predictions:

  1. Even after years of fractions instruction, fraction magnitude representations, whether measured by magnitude comparison or number line estimation, will be quite inaccurate in both 6th and 8th grade.
  2. Despite information about fractions magnitudes being taught explicitly in 3rd and 4th grade (NCTM, 2007), this understanding should still be increasing between 6th and 8th grade, due to students learning about fraction magnitudes from solving problems involving proportions and percentages in those grades.
  3. No logarithmic to linear transition should be present with fractions, because frequency of encountering fractions (and therefore knowledge of specific fractions) is correlated minimally if at all with fraction magnitudes, at least in the 0–1 range.
  4. Students should use a variety of strategies to solve fraction number line estimation and arithmetic problems, and the quality of these strategies should be related to students’ accuracy and speed in solving problems, as with whole number arithmetic.
  5. Individual differences in knowledge of fractions magnitudes should correlate highly with success at solving fraction arithmetic problems.
  6. Individual differences in knowledge of fractions magnitudes should correlate highly with individual differences in overall mathematics achievement test scores.

Number line estimation task:

  • participants were sequentially presented 10 number lines on a computer screen.
  • 0-1 number line items: 1/19, 1/7, 1/4, 3/8, 1/2, 4/7, 2/3, 7/9, 5/6, and 12/13.
  • 0-5 number line items: 1/19, 4/7, 7/5, 13/9, 8/3, 11/4, 10/3, 7/2, 17/4, and 9/2.
  • One fraction was drawn from each tenth of the number line, and presentation order was random.
  • Participants responded on each trial by moving the cursor to the desired position on the number line and clicking the mouse.
  • Accuracy of number line estimation was indexed by percent absolute error (PAE). PAE = (|Child’s Answer – Correct Answer|) / Numerical Range. For example, if a child was asked to locate 5/2 on a 0–5 number line, and marked the location corresponding to 3/2, the PAE would be 20% ((|1.5 – 2.5|) / 5).

Magnitude comparison task:

  • Participants were asked to compare to 3/5 a fraction shown on the computer screen: 3/8, 5/8, 2/9, 4/5, 4/7, 5/9, 8/9, or 2/3.
  • If the fraction was smaller than 3/5, the participant was to press the ‘‘a’’ key; if the fraction was larger than 3/5, the participant was to press the ‘‘l’’ key.”

Fraction arithmetic assessment:

  • Participants were presented 8 problems, 2 for each of the 4 arithmetic operations: 3/5 + 1/2, 3/5 + 2/5, 3/5 – 1/2, 3/5 – 2/5, 3/5 * 1/2, 3/5 * 2/5, 3/5 / 1/2, and 3/5 / 2/5.
  • One of the two problems for each arithmetic operation had operands with equal denominators. Problems appeared one at a time on the computer screen, and participants typed their answers.

Classification of strategies:

  • Strategies were classified on the basis of overt behavior and immediately retrospective self-reports.
  • When overt behavior clearly indicated the child’s approach, that behavior was the basis of the strategy assessment; otherwise, the child’s self-report was used.
  • The two main types of number line estimation strategies were numerical transformation strategies, in which participants transformed the presented fraction to a more convenient number, and number line segmentation strategies, in which participants generated subjective landmarks on the number line.
  • Both types of strategies could be, and often were, used on a single trial; children could transform the fraction to a more convenient numerical form and segment the number line in a way that helped them locate the fraction.
  • In coding numerical transformation strategies, we distinguished only between using a numerical transformation and not using one.
  • The most common numerical transformations were rounding the fraction (‘‘5/9 is a bit more than ½’’), simplifying it (‘‘9/5 = 1 and 4/5, which is a little less than 2’’), or translating it into a different form (‘‘12/13 is about 90%). The reason for not distinguishing among these numerical transformations was that they overlapped and could not be reliably distinguished.
  • The main number line segmentation strategies were division into halves; division into fifths or whole number units (e.g., placing marks on a 0–5 number line at the estimated positions of 1, 2, 3, and 4); division into units corresponding to the denominator (e. g., dividing a 0–1 number line into sevenths to locate 4/7); flawed approaches (e.g., on a 0–1 number line, reporting, ‘‘I put 3/7 near 0 because 3 rounds down’’); and none unknown (e.g., saying, ‘‘I don’t know’’).
  • “The lengthy solution times on none unknown trials, roughly 10 s, suggested that children might have used some strategy, but neither self- reports nor overt behavior indicated what it was.”
  • The mean of the median solution times across number lines and age groups was 9.5 s. On 0–1 number lines, the means of the medians of individual children’s solution times were 8.4 s for 6th graders and 7.3 s for 8th graders; on 0–5 number lines, they were 11.9 s for 6th graders and 10.3 s for 8th graders.
  • The response times suggest that fractions number line estimation is far from automatic; rather, it appears to be a controlled, strategic process.”

“The present findings indicate that understanding of fraction magnitudes and fractions arithmetic are closely related. If learning fraction arithmetic algorithms reflected rote memorization, as has often been claimed (e.g., Cramer & Bezuk, 1991; Hiebert, 1986; Kerslake, 1986), there would be no reason to expect such a relation. However, the strong correlations between fractions arithmetic and all three measures of magnitude knowledge in both 6th and 8th grades indicate that conceptual and procedural knowledge of fractions are intertwined. (See Hecht (1998), Hecht, Close, and Santisi (2003), Hecht and Vagi (2010), and Schneider and Stern (2010) for similar findings.) One plausible interpretation of these results is that magnitude knowledge makes it easier to learn and remember fraction arithmetic algorithms. This might occur through children with good magnitude knowledge of fractions rejecting procedures that produce unreasonable answers, such as operating independently on numerator and denominator often does, and searching longer for procedures that produce reasonable answers. For example, children might reject the procedure that produces arithmetic errors of the form 3/5 – 1/2 = 2/3 if they recognized that subtracting a positive number can- not lead to an answer larger than the number being subtracted from. This could lead them to try other procedures and test whether they yielded plausible answers.”

“Another implication for understanding fraction arithmetic is methodological: The same strategy assessment techniques that have proved useful with whole number arithmetic also are useful for investigating fractions arithmetic. As with whole number arithmetic, individual children used a variety of fractions arithmetic strategies. Even on a single fractions arithmetic operation, strategy use varied with problem characteristics, notably with the equality or inequality of denominators. The quality of fraction arithmetic strategy use was related to both knowledge of numerical magnitudes and to overall mathematics achievement test scores. In whole number arithmetic, these strategy assessment techniques have provided a base for computer simulation models of arithmetic learning that accounted for numerous findings regarding variations in accuracy, solution times, and strategy use across problems; discovery of useful new strategies; individual differences in arithmetic proficiency; and changes in speed, accuracy, and strategy use with problem-solving experience (Shrager & Siegler, 1998; Siegler & Shipley, 1995). The prominence of strategy use in fractions arithmetic suggests that similar models might be applicable to that area, and that it might be possible to formulate a common model of development of whole number and fractions arithmetic.”

“A third implication for understanding fractions arithmetic is that fraction arithmetic errors often reflect confusion about the right strategy, together with a lack of constraints on the magnitudes of answers, rather than a consistent whole number bias or other systematic misunderstanding.”

“[whole number bias] has been said to lead to children treating numerators and denominators as independent whole numbers and operating on them independently, for example by subtracting numerator from numerator and denominator from denominator (Carpenter et al., 1981; Gelman, 1991; Kilpatrick, Swafford, & Findell, 2001). However, the present findings revealed greater variability in fraction procedures than implied by this attribution of errors to a systematic misconception.”

“Roughly half of arithmetic errors stemmed from applying whole number algorithms independently to numerators and denominators, but a similar percentage reflected using parts of algorithms that would have been correct for a different fraction arithmetic operation or trying other erroneous procedures. The inconsistency of strategies even within a single arithmetic operation was striking; 40% of children correctly solved one of the pair of problems for a single arithmetic operation and erred on the other. This variability suggests that the whole number bias is only part of the problem in understanding fractions arithmetic. Rather than reflecting a systematic misconception, fractions arithmetic knowledge seems piecemeal; understanding of whole numbers is one source of ideas about how to solve fractions arithmetic problems, but other types of numerical knowledge are also incorporated. It also is unclear whether children who use erroneous fractions arithmetic procedures believe that those procedures are correct. They might well be skeptical about their correctness but have learned that saying ‘‘I don’t know’’ is not an acceptable alternative in school mathematics. This issue could be addressed through studies that examine children’s confidence in their fraction arithmetic answers.”

“In the United States, instruction in fractions emphasizes part-whole interpretations far more than other interpretations of fractions (Ni & Zhou, 2005; Sophian, 2007; Thompson & Saldanha, 2003). For example, students are taught to interpret 1/5 as one of five slices of pizza, but less often to think of 1/5 as one fifth of the distance from zero to one on a number line (Moseley, Okamoto, & Ishida, 2007). This is quite different than the approach to teaching fractions in Japan, China, and other countries where students understand fractions better. Indeed, many teachers in the US can only explain fractions in terms of the part-whole interpretation, unlike teachers in China and Japan who also emphasize num- ber line and other interpretations (Ma, 1999; Moseley et al., 2007). Part-whole interpretations have the advantages of concreteness and accessibility. When numerators and denominators are small and positive and the numerator is less than the denominator, it is easy to think about N parts of a whole that includes M parts. For example, children have little difficulty understanding that if a pizza is cut into four pieces, then each piece is 1/4 of the pizza (Mix, Levine, & Huttenlocher, 1999). However, the part-whole interpretation of fractions also has some serious limitations. Negative fractions cannot be represented in this way, it is very difficult to imagine fractions with large numerators and denominators (e.g., 734/878), and improper fractions can be confusing within the part-whole interpretation, as illustrated by one learner’s reaction to being presented 4/3, ‘‘You cannot have four parts of an object that is divided into three parts’’ (Mack, 1993). Moreover, there is nothing in the operation of dividing an object into N parts that says that the size of the parts must be equal; many students fail to understand that the parts must be equal for fractions to have any consistent meaning (Sophian, 2007).”

“emphasizing that fractions are measurements of quantity might improve learning about fractions. Indeed, a common feature of instructional studies that have yielded especially promising results in teaching rational numbers, such as work by Robbie Case and his associates, is that they emphasize that fractions are measures of quantity (e.g., Cramer et al., 2002; Fujimura, 2001; Keijzer & Terwel, 2003; Moss & Case, 1999; Rittle-Johnson & Koedinger, 2002, 2009). The present integrated theory of numerical development helps to explain the prevalence of this common feature of successful instruction: If magnitudes are central to understanding fractions as well as whole numbers, then instruction that emphasizes magnitude understanding is more likely to succeed than instruction that does not emphasize magnitude understanding.”

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