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Journal of Educational Psychology 2015, Vol 107, No. 3, 909-918 http://dx.doi.org/10.1037/edu0000025

Conceptual Knowledge of

Robert S. Siegler and Hugues Lortie-Forgues Carnegie Mellon University and The Siegler Center for Innovative Learning, Beijing Normal University

Understanding an arithmetic operation implies, at minimum, knowing the direction of effects that the operation produces. However, many children and adults, even those who execute arithmetic procedures correctly, may lack this knowledge on some operations and types of . To test this hypothesis, we presented pre-service teachers (Study 1), middle school students (Study 2), and math and science majors at a selective university (Study 3) with a novel direction of effects task with . On this task, participants were asked to predict without calculating whether the answer to an inequality would be larger or smaller than the larger fraction in the problem (e.g., “True or false: 31/56 * 17/42 > 31/56”). Both pre- service teachers and middle school students correctly answered less often than chance on problems involving and of fractions below 1, though they were consistently correct on all other types of problems. In contrast, the math and science students from the selective university were consistently correct on all items. Interestingly, the weak understanding of multiplication and division of fractions below 1 was present even among middle school students and preservice teachers who correctly executed the fraction arithmetic procedures and had highly accurate knowledge of the magnitudes of individual fractions, which ruled out several otherwise plausible interpretations of the findings. Theoretical and educational implications of the findings are discussed.

Keywords: fractions arithmetic, conceptual knowledge, mathematical development, mathematical cognition, arithmetic

Mathematical knowledge during schooling of education (Ritchie & Bates, 2013). predicts academic, occupational, and financial Among areas of , fractions success years later. Even after controlling for (including , , , rates, other cognitive and demographic variables, and proportions) seem to be especially important mathematics achievement in high school is for later success. This central role is evident in predictive of college matriculation, college fifth graders’ fraction knowledge predicting their graduation, and early career income (Murnane, algebra knowledge and overall mathematics Willett, & Levy, 1995). Especially striking, achievement in tenth grade, a relation that was mathematics achievement at age 7 predicts present in both the U. K. and the U. S. even after socioeconomic status (SES) at age 42, even after controlling for IQ, reading comprehension, statistically controlling for SES at birth, reading working memory, knowledge of whole achievement, IQ, academic motivation, and years arithmetic, and parental education and income ______(Siegler et al., 2012). Fraction understanding This article was published Online First January 19, 2015. also is essential for a wide range of occupations Robert S. Siegler and Hugues Lortie-Forgues, Department beyond STEM fields (science, technology, of Psychology, Carnegie Mellon University and The Siegler Center for Innovative Learning, Beijing Normal University. engineering and mathematics), including nurse, This article was funded by Grant R342C100004:84.324C pharmacist, automotive technician, stone mason, from the IES Special Education Research & Development and tool and die maker (Davidson, 2012; Centers of the U.S. Department of Education, the Teresa McCloskey, 2007; Sformo, 2008). Heinz Chair at Carnegie Mellon University, the Siegler Center of Innovative Learning at Beijing Normal University, The importance of fractions makes it and a fellowhship from the Fonds de Recherche du Québec – especially unfortunate that many children’s, Nature et Technologies to Hugues Lortie-Forgues. adolescents’, and adults’ fraction understanding Correspondence concerning this article should be addressed to Robert S. Siegler or to Hugues Lortie-Forgues, is poor. On a recent National Assessment of Department of Psychology, Carnegie Mellon University, Educational Progress (NAEP), a test presented to 5000 Forbes Avenue, Pittsburgh, PA 15213. E-mail: a large, nationally representative sample of U.S. [email protected] or [email protected] students, only 50% of 8th graders correctly 2 Siegler and Lortie-Forgues ordered from smallest to largest 2/7, 5/9, and overall mathematics achievement with both 1/12 (Martin, Strutchens, & Elliott, 2007). whole numbers (Geary, et al., 2007; Siegler & Problems in understanding fraction magnitudes Booth, 2004) and fractions (Siegler & Pyke, persist into adulthood; community college 2013; Siegler, Thompson, & Schneider, 2011). students choose the larger of two fractions on Accuracy of numerical magnitude only about 70% of items, where chance is 50% representations also predicts achievement test correct (Schneider & Siegler, 2010; Stigler, scores in later grades for both whole numbers Givvin, & Thompson, 2010). Teachers recognize (Geary, 2011; Watt, Duncan, Siegler, & Davis- the seriousness of the problem; a sample of Kean, 2014) and fractions (Bailey, Hoard, 1,000 U.S. high school algebra teachers rated Nugent, & Geary, 2012; Jordan et al., 2013). knowledge of fractions as one of the two largest Most important, magnitude knowledge is weaknesses in their students’ preparation for causally related to other aspects of mathematical their course, from among 15 topics in knowledge. Randomized controlled trials aimed mathematics (Hoffer, Venkataraman, Hedberg, at improving knowledge of magnitudes produce & Shagle, 2007). Findings like these led the U. gains not only in magnitude knowledge but also S. National Mathematics Advisory Panel in arithmetic with both whole numbers (Booth & (NMAP) to describe fractions as “the most Siegler, 2008; Ramani & Siegler, 2008) and important foundational skill not presently fractions (Fuchs et al., 2013; Fuchs et al., 2014). developed” (NMAP, 2008) and led the National The present study expands the integrated Council of Teachers of Mathematics (2007) to theory of numerical development beyond this emphasize the importance of improving teachers’ emphasis on the central role of the magnitudes of and students’ understanding of them. The individual numbers to include the role of problem is especially serious in the U.S., but it understanding the magnitudes produced by extends to countries that rank high on arithmetic. The two issues show some striking international comparisons of math knowledge, similarities. For example, just as accuracy of including Taiwan and Japan (Chan, Leu, & estimation of individual fractions’ magnitudes is Chen, 2007; Yoshida & Sawano, 2002). related to mathematics achievement, so is In to their importance for estimation of the answers yielded by fraction educational and occupational success, fractions addition (Hecht & Vagi, 2010). However, are also crucial for theories of numerical despite the parallel roles of magnitudes in the development. As noted in the integrated theory two areas, understanding individual fractions of numerical development (Siegler, Thompson, does not imply understanding of the magnitudes & Schneider, 2011), learning fractions requires produced by fraction arithmetic.. differentiating properties of natural numbers Prior research has documented one type of from properties of rational numbers. Students misunderstanding of fraction arithmetic: the need to learn that each has a whole number bias. This well-documented error unique successor but that infinitely many involves treating fraction numerators and fractions fall between any two other fractions; denominators as independent whole numbers, as that, multiplying two natural numbers never when claiming that 1/2 + 1/2 = 2/4 (Gelman, results in a product less than either factor (the 1991; Hecht & Vagi, 2010; Mack, 2005; Ni & numbers being multiplied), but that multiplying Zhou, 2005; Stafylidou & Vosniadou, 2004; Van two proper fractions (fractions less than one) Hoof, Lijnen, Verschaffel, & Van Dooren, always does; and that dividing two natural 2013). Although striking, this error occurs numbers never results in a greater than primarily among low-achieving students; for the number being divided, but dividing by a example, in a study that examined fraction fraction less than one always does. Indeed, one arithmetic of middle school students, children the few major properties uniting natural numbers whose achievement test scores were in the lowest and fractions is that both express magnitudes that one-third of the distribution made such errors on can be located and ordered on number lines. 32% of trials, whereas those whose achievement A variety of types of data are consistent with test scores were in the upper two-thirds made this theoretical emphasis on numerical them on only 10% of trials (Siegler & Pyke, magnitudes. Accuracy of magnitude 2013). The frequency of such errors also representations, as measured by performance on decreases with age, falling from 25% among number line and magnitude comparison tasks, sixth graders to 16% among eighth graders in the correlates consistently and quite strongly with same study. Fraction Arithmetic Understanding 3

In the present study, we examine a different 1986; Rittle-Johnson & Schneider, in press; type of misunderstanding of fraction arithmetic: Rittle-Johnson & Star, 2007). Simply put, direction of effects errors. At minimum, a person students might memorize fraction arithmetic who understands an arithmetic operation should procedures without understanding them. This is know the direction of effects that the operation more than a theoretical possibility. Many U.S. produces. For addition and , the teachers have weak conceptual understanding of direction is the same for all positive numbers: fraction arithmetic (Ma, 1999; Lin, et al., 2013; addition produces answers greater than either Moseley, Okamoto, & Ishida, 2007; Rizvi & addend, and subtraction produces answers less Lawson, 2007), which limits the understanding than the minuend (the number being subtracted that they can convey to students. In principle, from). However, for multiplication and division, students who correctly execute fraction the direction of effects depends on whether the arithmetic procedures could induce some types numbers in the problem are greater or smaller of understanding by themselves. For example, than one. Multiplying numbers between zero and they could infer that multiplying fractions below one always yields an answer less than either one results in answers less than either factor by number being multiplied, and dividing by a connecting the magnitudes of the fractions being number between zero and one always yields an multiplied, the operation being performed, and answer greater than the number being divided the magnitudes of the answers. However, many (the dividend). Such knowledge has been studied students might not make such connections, either with whole numbers (Baroody, 1992; Sophian & due to not attending to the magnitudes of Vong, 1995) and has been claimed to be fractions in the problems or due to forgetting important with fractions, but to the best of our them while executing the procedure. Thus, knowledge, it has not been studied accurately executing fraction arithmetic systematically with fractions. procedures is no guarantee of understanding In principle, the directions of effects of them. arithmetic operations are easy to understand. If The Present Study people think of fraction multiplication as “N of Our main goal was to test the hypothesis that the M’s,” they should be able to predict the even many educated adults with years of whole direction of effects, regardless of whether N and number and fraction arithmetic experience M are natural numbers (e.g., will 6 of the 4’s be nonetheless lack conceptual understanding of more than 6) or proper fractions (e.g., will 1/6 of multiplication and division. This was the 1/4 be more than 1/4). Similarly, if people hypothesized to be true even for people who understand division as the number of times the execute fraction arithmetic procedures flawlessly goes into the dividend, it should not be and who accurately represent fraction hard to realize that a divisor between 0 and 1 will magnitudes. Conceptual understanding of go into the dividend more times than “1” would arithmetic operations was measured by a go into it, so the answer will exceed the direction of effects task (e.g., Is N /M + N /M dividend. 1 1 2 2 > N1/M1). This task, which in its present form is In contrast, if people understand novel to this study, had two advantages for multiplication as an operation that invariably assessing conceptual understanding of increases the size of the numbers involved, and arithmetic: It did not require highly accurate division as an operation that invariably decreases knowledge of the magnitudes of individual their size, as is the pattern with numbers greater numbers (none for addition and subtraction; only than one, then they will perform below chance whether each fraction exceeds one for when both factors are proper fractions (fractions multiplication and division), and application of below one) or the divisor is. the present analysis to it yielded eight Note that mastery of fraction arithmetic predictions, one for each of the eight procedures – the algorithms or rules used to combinations of arithmetic operation and solve mathematical problems – could easily fraction magnitude (factors greater than or less coexist alongside weak or inaccurate conceptual than one). understanding of the procedures – implicit or Our main prediction was that frequency of explicit knowledge of how the procedures work, accurate judgments would be below chance on why they make sense, and how they are related multiplication and division problems with to other procedures and concepts in the domain fractions less than one. This was predicted to be (Byrnes & Wasik, 1991; Hiebert & LeFevre, true even for people who flawlessly multiplied 4 Siegler and Lortie-Forgues and divided fractions and who accurately within countries, Quebec had the highest mean estimated the magnitudes of individual fractions. math achievement score of the 10 Canadian In contrast, we also predicted that performance provinces; its mean PISA math score equalled would be well above chance on addition and that of Japan, and far exceeded that of the U.S. subtraction problems, regardless of the fractions (CMEC, 2013). involved, and on multiplication and division with Tasks. All measures were presented to the fractions greater than one. The reason was that entire class in pencil and paper form. on these problems, either understanding the Conceptual understanding of fraction arithmetic operations or assuming that fraction arithmetic. This was measured by performance arithmetic yields the same patterns as natural on the direction of effects task. On it, number arithmetic produces correct judgments. participants were asked to evaluate the accuracy These predictions were examined with three of mathematical inequalities of the form a/b + populations: pre-service teachers attending a c/d > a/b, where a/b was the larger of the school of education, students attending a middle fractions and the operation was either addition, school, and math and science majors at a highly subtraction, multiplication, or division. The selective university. The pre-service teachers fractions were chosen so that numerators and attended a high quality college of education in denominators were too large for the problem to Canada and provided a test of the main be solved quickly via mental arithmetic. The hypothesis with a sample of adults whose fractions a/b and c/d were either both above one mathematical knowledge was above average for or both below one, and the operator was one of adults in North America. The middle school the four arithmetic operators, resulting in eight students were from a U. S. public school with types of problems. Participants were presented 2 typical achievement levels, according to instances of each of the 8 types of problems, for performance on state tests. They were included a total of 16 items. The same pairs of operands -- to test whether the adults from Study 1 might 31/56 and 17/42, 41/66 and 19/35, 37/19 and have had conceptual understanding of the 58/36, and 51/16 and 47/33 -- were presented for arithmetic operations when they studied them but all four arithmetic operations. forgotten that knowledge in the ensuing years. Each page of the booklet included four Finally, the math and science majors at the problems, which included one instance of each highly selective university were far above arithmetic operation and one of each pair of average in math achievement; including them operands. Participants were instructed not to tested whether the direction of effects task would compute the exact answer, but rather to decide lead even highly knowledgeable adults to without calculating whether the answer would be perform poorly. greater than the answer indicated in the Study 1 inequality. Method Fraction arithmetic computations. Each Participants. Participants were 41 pre- participant was presented four problems for each service teachers (4 men, 37 women, Mean age = of the four arithmetic operations. For each 25.9 years, SD = 8.8) recruited from a French operation, the operand 3/5 was combined with language public university in Montreal, Quebec, 1/5, 1/4, 2/3, and 4/5; for example, the Canada. All were in a mathematical activities subtraction problems were 3/5 - 1/5, 3/5 - 1/4, course in a program aimed at preparing them to 2/3 - 3/5, and 4/5 - 3/5. Thus, all problems teach kindergarten through sixth grade, a period involved positive fractions less than one. Half of in which fractions are a major focus of the problems had operands with equal mathematics instruction. The teacher-training denominators, and half with unequal ones. The program was selective, with 53% of applicants larger operand was always on the left for being admitted. Self-reported R scores, a subtraction and division problems. statistical index used in Quebec to measure Magnitudes of individual whole numbers incoming university students' academic and fractions. Knowledge of whole number performance, were provided by 26 of the 41 magnitudes was measured by a 0-10,000 number participants. Their scores were above the Quebec line estimation task; knowledge of fraction average of 25 (M = 28.87; SD = 2.63; versus M = magnitudes was measured by 0-1 and 0-5 25, SD unknown) (CREPUQ, 2004). On the number line tasks. On each trial, the number 2012 PISA, a test used to compare academic being estimated was printed above the midpoint achievement of different countries and regions of a 16 cm horizontal line with 0 just below the Fraction Arithmetic Understanding 5

left end and 1, 5, or 10,000 just below the right improper fractions and addition and subtraction end; the task was to mark where the target of proper fractions, and on 0% on the 4 problems number belonged on the number line. On the involving multiplication and division of proper whole number task, the target numbers were: fractions. Only 3% (1 participant) answered all 857, 1203, 2589, 3091, 4928, 5762, 6176, 7334, 16 problems correctly. 8645 and 9410. On the 0-1 fractions task, they Fraction arithmetic computations. A were: 1/19, 1/7, 1/4, 3/8, 1/2, 4/7, 2/3, 7/9, 5/6 repeated measures ANOVA with Arithmetic and 12/13. On the 0-5 fractions task, they were: Operation as a within subject variable and 1/19, 4/7, 7/5, 13/9, 8/3, 11/4, 10/3, 7/2, 17/4 and number correct as the dependent variable yielded 9/2. On each magnitude estimation task, one an effect of Arithmetic Operation F(3, 38) = target number was in each 1/10 of the line. 2 12.15, p < .001, ηp = 0.490. Post-hoc Target fractions were also chosen to minimize comparisons with a Bonferroni correction correlations between the magnitude of the showed less accurate performance on fraction numerator and the magnitude of the whole division (M = 51%, SD = 46%) than addition (M fraction, so that basing placements solely on = 87%, SD = 25%; t(40) = 4.76, p < 0.001; numerator magnitude would not yield accurate Hedges' g = 0.95), subtraction (M = 93%, SD = answers. 18%; t(40) = 5.95, p < 0.001; Hedges' g = 1.17) Procedure. All tasks were printed in a and multiplication (M = 87%, SD = 23%; t(40) = booklet and presented in a constant order: first 4.85, p < 0.001; Hedges' g = 0.97). the direction of effects task; then number line The pre-service teachers rarely made the type estimation with whole numbers, 0-1 fractions, of computational error predicted by the well- and 0-5 fractions, in that order; and then fraction documented whole number bias (e.g., a/b + c/d = arithmetic. Two versions were generated by (a+b) / (c+d)). This error appeared on only 4.3% reversing the order of items within each task. of addition and subtraction trials. Thus, the The versions were randomly assigned to inaccurate judgments on the direction of effects participants, who were tested in groups in their task could not be attributed to the whole number classroom. bias. Results and Discussion Number line estimation. Performance on the Conceptual knowledge of fraction number line tasks was measured by percent arithmetic. A repeated measures ANOVA with absolute error (PAE), computed as: (│Estimate - Fraction Size (above or below 1) and Arithmetic Correct answer│) / Numerical Range * 100. Operation (addition, subtraction, multiplication Thus, if a participant estimated the position of and division) as within-subject factors and 1/4 to be 35% of the distance between 0 and 1, number of correct judgments as the dependent PAE on that trial was 10% ((35-25)/100). The variable yielded effects of Fraction Size F(1, 40) higher the PAE, the less accurate the estimate. 2 = 23.17, p < .001, ηp = 0.367, and Arithmetic On the 3.3% of trials on which participants did 2 Operation F(3, 38) = 36.26, p < .001, ηp = not respond, PAE was computed using the value 0.741, both qualified by a Fraction Size x on the number line farthest from the correct Arithmetic Operation interaction, F(3, 38) = answer (e.g., if the problem was to locate 1/19 on 2 6.24, p = .001, ηp = 0.330. As expected, post- a 0-5 fraction number line, the estimate was hoc comparisons with the Bonferroni correction computed as if it were at 5). This was done to showed no differences between number of yield a maximally conservative estimate of correct judgments with proper and improper estimation accuracy. These trials occurred fractions for addition (92% vs. 92%; t(40) = 0, p disproportionately among participants with weak = 1; Hedges' g = 0.00) and subtraction (89% vs. mathematical knowledge and on relatively 92%; t(40) = 0.63, p = 0.53; Hedges' g = 0.00). difficult problems, so deleting the trials would However, large differences between fractions have distorted the data. below and above 1 were observed for The most striking number line estimation multiplication (33% vs. 79%; t(40) = 3.90, p < finding was that the pre-service teachers had 0.001; Hedges' g = 1.11) and division (29% vs. excellent knowledge of the magnitudes of 77%; t(40) = 4.11, p < 0.001; Hedges' g = 1.19). fractions from 0-1, the same range in which they Analysis of individual response patterns showed poor conceptual knowledge of yielded similar results. When considering all multiplication and division. Estimates of fraction problems on the task, 42% of participants were magnitudes in this range were as accurate as correct on 100% of the 12 problems involving those for whole numbers in the 0-10,000 range, 6 Siegler and Lortie-Forgues with both being highly accurate (For the 0-1 r = .09 with number of correct arithmetic fraction task, Mean PAE= 4.41, SD = 3.07; for computations; number of correct direction of the 0-10,000 whole number task, Mean PAE= effects judgments correlated r = -.05 with 3.53, SD = 2.64). Estimates of fractions on the 0- number line PAE; and number line PAE 5 task were less accurate (Mean PAE = 14.94, correlated r = -.04 with number of correct SD =18.67). arithmetic computations. For division, number of Relations of performance across tasks. correct direction of effects judgments and Relations between conceptual and procedural arithmetic computations correlated r = .33; knowledge of arithmetic varied substantially on number of correct direction of effects judgments the four arithmetic operations (Table 1). On and number line PAE correlated r = -.11; and addition and subtraction, performance was strong number of correct arithmetic computations and on measures of both conceptual knowledge (the number line PAE correlated r = -.22. direction of effects task) and procedural In summary, pre-service teachers’ conceptual knowledge (the arithmetic computation task); on understanding of fraction multiplication and multiplication, procedural knowledge was strong division, as indicated by performance on the but conceptual knowledge was weak; on direction of effects task, was weak. This was true division, both were weak. even for participants who flawlessly executed the The discrepancy between conceptual and fraction multiplication and division algorithms. procedural knowledge of multiplication Nor was the problem lack of fraction magnitude illustrated especially clearly that accurate knowledge. Estimation of the magnitudes of execution of procedures does not imply fractions between 0 and 1, the range that elicited understanding of their qualitative effects. The incorrect judgments of the qualitative effects of majority of participants (17 of 29) who correctly fraction multiplication and division, was highly solved all four fraction multiplication problems accurate. Instead, the results indicated weakness erred on both of the direction of effects problems in conceptual understanding of fraction that assessed understanding of multiplication of arithmetic that was not attributable to proper fractions. weaknesses in knowledge of fraction arithmetic computation or magnitudes of individual Although mean levels of performance on fractions. conceptual and procedural measures of division were similar, examination of individual Study 2 participants’ performance revealed a similar In Study 2, we examined whether the Study 1 dissociation. Of participants who correctly results replicated with a different population and answered all four of the fraction division age : U.S. middle school students. This problems, 47% (8 of 17) were incorrect on both population allowed us to test whether students of the direction of effects items that involved possess conceptual understanding of fraction division of fractions below 1. arithmetic at the time they receive instruction in Table 1 it. If so, the difference might reflect Percent Correct on Assessments of Procedural and improvements in fraction instruction in the past Conceptual Knowledge of Fraction Arithmetic: Pre- decade, or it might reflect the pre-service Service Teachers teachers in Study 1 forgetting understanding they Procedural Conceptual once had. The first seemed plausible, because Operation knowledge knowledge mathematics instruction in general, and fraction Addition 87 91 instruction in particular, is focusing increasingly on inculcating conceptual understanding Subtraction 93 89 (NCTM, 2007). The latter interpretation also Multiplication 87 33 seemed plausible because once children learn Division 51 30 correct procedures, they might forget the procedures’ conceptual justifications. Correlations among individual participants’ performance on the direction of effects, Method arithmetic computation, and number line Participants. Participants were 59 6th and 8th estimation tasks with fractions 0-1 were non- graders (28 boys, 31 girls, Mean age = 12.9 significant in 5 of 6 cases, revealing a similar years, SD = 1.19) from middle-income public dissociation. For multiplication, number of schools near Pittsburgh. correct direction of effects judgments correlated Fraction Arithmetic Understanding 7

Tasks. The same tasks were presented as in specific fractions on the direction of effects task the previous study, plus one new task, which were above or below 1. Thus, the poor involved asking whether each fraction on the performance on multiplication and division of direction of effects task was greater or less than fractions below one could not be explained by one. The purpose of this new task was to test students lacking the magnitude knowledge whether failure to discriminate the effects of required to perform well on it. multiplying fractions above and below one might Relations of performance across tasks. As be due to inability to identify whether the in Study 1, there was a clear dissociation fractions were above or below one. This task was between conceptual and procedural knowledge presented after the others, so it could not affect of fraction arithmetic (Table 2). Analysis of performance on them. individual performance showed that among Results and Discussion children who correctly solved all four of the fraction multiplication computation problems, Conceptual understanding of fraction 54% (19 of 35) erred on both direction of effects arithmetic. As in Study 1, a repeated measures items for multiplication of fractions below 1. Fraction Size X Arithmetic Operation ANOVA Similarly, among children who correctly solved on the direction of effects task yielded main all four fraction division computation problems, effects of Fraction Size F(1, 58) = 49.071, p < 39% (16 of 41) erred on both problems .001, η 2 = 0.458, and Arithmetic Operation, F(3, p measuring understanding of division of proper 56) = 28.351, p < .001, η 2 = 0.603, both p fractions. Thus, with children as with adults, qualified by a Fraction Size x Arithmetic successful execution of fraction arithmetic Operation interaction, F(3, 56) = 19.07, p < .001, computations was no guarantee of understanding η 2 = 0.505. Post-hoc comparisons with the p the procedures. Bonferroni correction again showed no differences between fractions below and above Analyses of correlations among individual one for addition (89% vs. 92%; t(58) = 1.36, p = children’s performance on the three tasks with 0.26; Hedges' g =0.09) and subtraction (92% vs. fractions 0-1 yielded non-significant relations in 94%; t(58) = 0.57, p = 0.57; Hedges' g = 0.07), all six cases. For multiplication, number of but large differences between them for correct direction of effects judgments correlated multiplication (31% vs. 92%; t(58) = 8.16, p < r = .16 with number of correct arithmetic 0.001; Hedges' g = 1.74) and division (47% vs. computations; number of correct direction of 70%; t(58) = 2.24, p < 0.05; Hedges' g = 0.51). effects judgments correlated r = -.10 with Also as in Study 1, a fairly substantial number of number line PAE; and number line PAE participants (29%) were correct on 100% of the correlated r = -.20 with number of correct 12 problems involving fractions above one or arithmetic computations. For division, number of addition and subtraction of fractions below one correct direction of effects judgments and and on 0% of the 4 problems involving arithmetic computations correlated r = .13; multiplication and division of fractions below number of correct direction of effects judgments one, and few participants (5 of 59) answered all and number line PAE correlated r = -.17; and problems correctly. number of correct arithmetic computations and number line PAE correlated r = -.13. Fraction arithmetic computations. A repeated measure ANOVA with Arithmetic Operation as the sole variable yielded no Table 2 significant effect. The children’s performance on Percent Correct on Assessments of Procedural and addition (78%), subtraction (83%), Conceptual Knowledge of Fraction Arithmetic: multiplication (81%) and division (82%) did not Middle School Students differ. Procedural Conceptual Number line estimation. Number line Operation knowledge knowledge estimates were very accurate for whole numbers Addition 78 89 in the 0-10,000 range (Mean PAE = 4.08, SD = Subtraction 83 92 1.67), fractions in the 0-1 range (Mean PAE = Multiplication 81 31 3.57, SD = 2.05), and fractions in the 0-5 range Division 82 47 (Mean PAE = 6.28, SD = 3.50).

These middle school students also were correct on 97% of judgments about whether the 8 Siegler and Lortie-Forgues

Study 3 Fraction arithmetic computations. An alternative interpretation to the results of Performance on all four operations was at Studies 1 and 2 was that the direction of effects ceiling, ranging from 97% correct for task could not be understood, even by people multiplication to 100% correct for addition and who had strong mathematics understanding. To subtraction (overall mean = 99%). Thus, as test this interpretation, we presented university shown in Table 3, both conceptual and students with very high mathematics procedural knowledge of arithmetic were achievement scores the same tasks and items as uniformly excellent. were presented in the previous two studies. The Number line estimation. These students’ main prediction was that these students’ deeper estimates were highly accurate on all three understanding of mathematics would allow them ranges: whole numbers from 0-10,000 (Mean to consistently answer correctly all of the PAE = 2.78, SD = 1.30), fractions from 0-1 problems that examined conceptual (Mean PAE = 2.87, SD = 1.87), and fractions understanding of fraction arithmetic. from 0-5 (Mean PAE = 4.17, SD =2.28). Method Thus, results of Study 3 showed that students highly proficient in math exhibited strong Participants. Participants were 17 conceptual understanding of fraction arithmetic. undergraduate students (7 men, 10 women, Mean This ruled out the interpretation that the task age = 19.9 years, SD = 1.3) majoring in precluded accurate judgments. computer science, engineering, physics, chemistry, or biology at a highly selective General Discussion university. Self-reported SAT mathematics Middle school students and pre-service scores for the 16 of 17 students who reported teachers demonstrated excellent understanding of them were in the 99th percentile (M = 778, SD = the magnitudes of individual fractions between 0 23.73), a score consistent with mean SATs in and 1. However, the same middle school these areas at the university. students and pre-service teachers demonstrated Tasks and procedures. The tasks and minimal understanding of the magnitudes procedures were identical to those in Studies 1 produced by multiplication and division of and 2, except that participants were tested fractions in the same range. They consistently individually or in pairs. predicted that multiplying two fractions below “1” would yield an answer greater than either Results and Discussion factor and that dividing by a fraction below “1” Conceptual knowledge of fraction would yield an answer smaller than the number arithmetic. Percent correct on the direction of being divided. effects task was very high on all eight types of Additional results from the present study problems (94% to 100% correct judgments, ruled out a variety of otherwise plausible overall mean = 98%). A Fraction Size X explanations of these findings from the direction Arithmetic Operation ANOVA, parallel to that in of effects task. The incorrect predictions on it the first two studies, yielded no main effect or were not due to the task being impossible; interaction. Most participants (77%) were correct mathematics and science majors at a selective on all 16 problems, none made more than two university performed almost perfectly on the errors, and the few errors were unsystematically same task. The incorrect predictions also were distributed across problems. not due to the previously documented tendency to view fraction arithmetic in terms of Table 3 independent combinations of the numerators and Percent Correct on Assessments of Procedural and of the denominators. Participants showed such Conceptual Knowledge of Fraction Arithmetic: Math independent whole number errors on fewer than and Science Majors at a Selective University 10% of trials when solving fraction arithmetic Procedural Conceptual problems (4.3% of addition and subtraction trials Operation knowledge knowledge in Study 1 and 8.5% in Study 2), but erred on Addition 100 100 more than 70% of trials on the direction of effects task with multiplication and division of Subtraction 100 97 fractions below one. Inaccurate predictions on Multiplication 97 100 the direction of effects task also could not be Division 99 97 attributed to lack of knowledge of individual Fraction Arithmetic Understanding 9

fraction magnitudes or of how to execute fraction recognizing that many properties of natural arithmetic procedures. The pre-service teachers number arithmetic are not properties of in Study 1 generated extremely accurate arithmetic in general. Although understanding estimates of the magnitudes of individual individual fraction magnitudes and fraction fractions and consistently solved fraction arithmetic both require distinguishing properties multiplication problems, yet they judged the of natural numbers from properties of all direction of effects of fraction multiplication less numbers, understanding the magnitudes of accurately than chance. The results also individual fractions does not imply suggested that inaccurate judgments on the understanding how fraction arithmetic operations direction of effects task could not be attributed to transform those magnitudes. forgetting material taught years earlier. Sixth The present findings also shed light on the graders who had been taught fraction division in sources of two types of common errors in the same academic year and fraction fraction arithmetic (Siegler & Pyke, 2013). One multiplication one year earlier also performed involves inappropriately importing procedures below chance on the direction of effects task for from other fraction arithmetic operations into these operations and numbers (30% correct for fraction multiplication. This leads to errors such multiplication and 41% correct for division). as 3/5 * 4/5 = 12/5, in which the numerators of These data from U.S. middle school students do the two fractions are multiplied but the not rule out the possibility that these Canadian denominator is unchanged, as in addition and pre-service teachers had forgotten relevant subtraction of fractions (e.g., 3/5 + 4/5 = 7/5). knowledge that they once had, but the data do The other common error involves dividing make this possibility less likely. fractions by inverting the numerator and One important unanswered question was multiplying (e.g., 2/3 ÷ 1/3 = 3/2 * 1/3 = 3/6 = whether the below chance performance was 1/2) rather than inverting the denominator and specific to common fractions or more general. multiplying (e.g., 2/3 ÷ 1/3 = 2/3 * 3/1 = 6/3 = To address this question, we conducted a pilot 2). The present results suggest that weak study on understanding of multiplication and understanding of the numerical magnitudes division of decimals between 0 and 1. Ten produced by fraction multiplication and division arbitrarily chosen students from the same prevents people from rejecting these incorrect university as the pre-service teachers were procedures on the basis that they yield presented the direction of effects task with implausible answers. Indeed, if learners believe decimals. Each common fraction from the that multiplication yields answers greater than direction of effects task was translated into its either factor and that dividing yields answers nearest 3-digit equivalent (e.g., “31/56 * smaller than the dividend, incorrect answers will 17/42 > 31/56” became “0.554 * 0.405 > 0.554). often seem more plausible than correct ones. A The results with decimals closely paralleled student who believes that multiplication should those with common fractions: High accuracy always yield answers larger than either factor (88% to 100% correct) on the six problem types might well view, “3/5 * 4/5 = 12/5,” as more in which fraction arithmetic produces the same plausible than the correct answer, “3/5 * 4/5 = pattern as natural number arithmetic, and below 12/25.” chance performance on the two problem types The present findings narrowed the range of that showed the opposite pattern as natural potential explanations of weak conceptual numbers (38% correct on multiplication and 25% understanding of fraction multiplication and correct on division of fractions below one). division, but left open at least two interpretations Thus, the inaccurate judgments on the direction of the underlying difficulty. One possibility is of effects task were general to multiplication and that many children and adults have a specific division of numbers from zero to one, rather than belief that arithmetic operations produce the being limited to common fractions in that range. same direction of effects regardless of the Theoretical Implications numbers involved. In particular, they might believe that for all numbers, addition and As noted in previous studies, understanding multiplication yield answers greater than either fractions requires recognizing that many operand, and subtraction and division yield properties of natural numbers are not properties answers smaller than the larger operand. Such a of numbers in general (Siegler et al., 2011). The belief would reflect a conclusion based on the present study makes a related but different point: pattern of answers yielded by natural number Understanding fraction arithmetic requires 10 Siegler and Lortie-Forgues arithmetic. Each natural number operation has (Lemaire & Siegler, 1995) and natural number the same direction of effects regardless of the division in terms of either multiplication (72 ÷ 8 numbers in the problem (excepting = 9 because 8 * 9 = 72) or repeated addition (72 multiplication and division by 1, which are often ÷ 8 = 9 because adding 9 8 times = 72) explicitly described as “exceptions”). Illustrative (Robinson, et al., 2006). However, teachers often of this logic, when asked to “Try to explain what explain multiplication and division in these 1/2 ÷ 1/4 means,” a sixth grader in a pilot study ways, so children might just be repeating what wrote “You’re making 1/2 1/4 times smaller than they have been told and might not have as deep it was”; when asked to explain what 1/2 * 1/4 an understanding of multiplication and division means, the child wrote “You are making 1/2 1/4 as these explanations suggest (Dubé & times bigger.” Robinson, 2010; Robinson & Dubé, 2009). An alternative interpretation is that many Investigating in greater depth people’s people have no strong beliefs about the results understanding of multiplication and division yielded by multiplication and division of with natural numbers as well as fractions, and fractions, and therefore rely on the pattern with analyzing how the two are related, seems a natural numbers as a default option. Within this promising route for future research. interpretation, which resembles one proposed for Educational Implications some pre-service teachers by Simon et al. (2010), Results of Studies 1 and 2 indicated that even participants applied their understanding of people with very accurate knowledge of fraction natural number arithmetic to the direction of arithmetic procedures and magnitudes of effects task with fractions not because they were individual fractions often possess weak convinced that this was correct but because they conceptual understanding of multiplication and did not know what else to do. Relying on default division of fractions below one. These findings knowledge may be quite common in situations are consistent with previous results from where people lack understanding but need to do research (e.g., Heller, et something. For example, if asked how a catalytic al., 1990; Lamon, 2007) as well as the converter works, many people might rely on recommendations of mathematics education knowledge of how air conditioning filters work, organizations (e.g., NCTM, 1989), and point to not because they are convinced that catalytic the need to explicitly teach qualitative reasoning converters work that way, but due to inability to about fraction arithmetic (Behr, et al., 1992; generate a better explanation (see Rozenblit & Huinker, 2002). Keil, 2002, for extensive documentation of such default explanations). Consistent with this A related, straightforward instructional interpretation, Ma (1999) found that most U.S. implication is that teachers and textbooks should teachers in her study could not generate any emphasize that multiplication and division explanation of what 1 3/4 ÷ 1/2 means, or produce different outcomes depending on resorted to explaining a different problem (1 3/4 whether the numbers involved are greater than or ÷ 2). Similarly, an eighth grader in the pilot less than 1, and should discuss why this is true. study mentioned in the previous paragraph Chinese textbooks include such instruction. For explained 1/2 * 1/4 by writing “It means taking instance, in a lesson on multiplication with half a pie, 1/4 of another pie, and showing how decimals, Chinese students are asked to solve much you’d get” and could not generate any and discuss answers to the following three explanation of the meaning of 1/2 ÷ 1/4. problems: 4.9 * 1.01; 4.9 * 1; 4.9 * 0.99 (Sun & Obtaining confidence ratings regarding direction Wang, 2005). The same could be done with of effects judgments could help discriminate triads of fractions, such as 1/2 * 8/7, 1/2 * 7/7, between these possibilities. and 1/2 * 6/7. Such well-chosen problems and accompanying discussion are likely to create Another unresolved issue is whether the weak meaningful and memorable knowledge of the understanding is specific to fraction effects of multiplying by numbers above and multiplication and division or whether it extends below one. to all multiplication and division. Consistent with the view that the difficulty is specific to Another instructional implication is that fractions, children often provide reasonable teachers and textbooks should focus more explanations of natural number multiplication attention on the shifting meaning of wholes in and division. They explain natural number the context of fraction arithmetic. For example, multiplication in terms of repeated addition when asked to use multiplication to find what 1/2 of 1/4 of a pie is, the relevant whole for 1/4 is the Fraction Arithmetic Understanding 11

whole pie, but the relevant whole for 1/2 is the the problem involves division of a smaller by a 1/4 of the pie. Several investigators have noted larger number, such as 1/8 ÷ 5/8, students could that many students fail to gain such be encouraged to think of the problem as “How understanding of wholes in the context of much of N can go into M” (e.g., “How much of fraction arithmetic and have suggested means for 5/8 can go into 1/8”). As in multiplication, these remedying the difficulty (e.g., Ball, 1990; phrasings can be applied to whole numbers as Simon, 1993; Tobias, 2013). Instruction that well as fractions (“How many times can 8 go successfully inculcatesPh such understanding into 48” “How much of 48 can go into 8”), and it would provide a strong foundation for efforts to seems likely to prove useful to proceed from teach children what fraction multiplication using each phrasing first with two whole means. numbers, then with a whole number and a A further instructional implication is that fraction, and then with two fractions instruction should emphasize understanding of (maintaining whether the problems involve a commonalities uniting the processes of whole larger number divided by a smaller number or number and fraction arithmetic. One approach to the opposite). Also as in multiplication, these attaining this goal is to adopt linguistic phrasings phrasings lend themselves to a common spatial of multiplication and division that promote representation of the workings of the operation recognition that these operations have the same on a number line. meaning with whole and with rational numbers. We believe that these alternate phrasings For instance, multiplication of two fractions provide a clearer indication of what (e.g., 1/3 * 1/5) can be expressed as "How much multiplication and division mean than the usual is 1/3 of the 1/5"; multiplication of a whole phrasings. Their applicability to both natural number and a fraction as “How much is 1/5 of numbers and fractions seems a promising way of the 3”; and multiplication of two whole numbers helping students understand that the as “How much is 5 of the 3’s.” Using this mathematical operations have the same meaning phrasing of multiplication first with two whole in both cases, though their effects on numerical numbers, then with a whole number and a magnitudes vary with the numbers involved. We fraction, and then with two fractions might be plan to test the efficacy of these instructional especially effective for promoting analogies from ideas in the near future. better to less understood cases, an instructional strategy that has proved effective in many Limitations and Future Research Directions domains (Gentner & Holyoak, 1997). Illustrating The present study has several limitations, the multiplicative process in all cases in the each of which suggests directions for future common format of a number line might deepen research. Perhaps the most important limitation understanding of multiplication further, by is that the present study examined only one providing spatial as well as verbal support for a aspect of understanding of fraction arithmetic – shared representation. Discussing with students the direction of effects of fraction arithmetic why multiplying two numbers below one always operations. Future research should also examine results in an answer less than one, why other aspects of conceptual understanding of multiplying two numbers above one always fraction arithmetic, including the approximate results in an answer greater than one, and why magnitudes produced by fraction arithmetic multiplying a number less than one by a number operations, the correspondences that can be greater than one sometimes results in one drawn between fraction arithmetic operations outcome and sometimes in the other could and models of their effects, and the varying deepen understanding further. wholes that correspond to the operands in A similar, though somewhat more complex, fraction arithmetic problems. approach could be used with division. Students A second limitation is that the study did not could be encouraged to first judge whether the relate the fraction instruction that students had problem involves dividing a larger by a smaller received to their conceptual and procedural number or dividing a smaller by a larger number. knowledge of fraction arithmetic. Of particular If the problem involves division of a larger by a interest for future research is determining smaller number, for example, 5/8 ÷ 1/8, whether children who receive conceptually students could be encouraged to think of the oriented fraction instruction subsequently show problem as “How many times can N go into M” superior conceptual understanding of fraction (e.g., “How many times can 1/8 go into 5/8”). If 12 Siegler and Lortie-Forgues arithmetic and also superior knowledge and Davidson, A. (2012, January/February). Making it in memory of the fraction arithmetic procedures. America. The Atlantic, 65-83. 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Correction to Siegler and Lortie-Forgues (2015) In the article "Conceptual Knowledge of Fraction Arithmetic" by Robert S. Siegler and Hugues Lortie-Forgues (Journal of Educational Psychology, Advance online publication, January 19, 2015. http://dx.doi.org/10.1037/edu0000025), there were two rounding errors in Table 1. The value 91 in the upper-right corner should be changed to 92, and the value 30 in the bottom-right corner should be changed to 29. Table 1, 2, and 3 also refer only to fractions between 0 and 1, and not fractions above one.

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