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Foreword: Build It and They Will Come
Foreword: Build It and They Will Come Robert S. Siegler Carnegie-Mellon University, Pittsburgh, PA, United States
In 1978, I was a member of the review panel for the first math learning grants competition of the National Institute of Education (NIE). Panel members were told that there was enough money to fund 10 grants; the only problem was that there were only 2 proposals that clearly merited funding and 2 others where opinion was mixed. Once the reviews were presented and discussed, the conversation turned to whether the panel had to recommend 10 grants for funding, because that would mean funding at least 6 proposals for which there was no enthusiasm. Most panel members, me included, argued against funding them. However, Susan Chipman, who was Assistant Director of NIE and in charge of the review competition, stated that giving out all of the money was essential. Although acknowledging that the applications were far from stellar, she argued that once the word got out that there was substantial funding for math learning research, better proposals would soon follow. The panel reluctantly agreed and voted to fund the 10 least dirty shirts in the dirty laundry basket. How things have changed! A similar competition today would elicit at least 20 worthy proposals. The challenge would be to distinguish the really excellent ones from the merely good or very good ones. There would be passionate debates, just as before, but now they would stem from panel members arguing that their favorite proposals just had to be funded no matter what. A web search of the program from the most recent Society for Research in Child Development meeting provides quantitative evidence of the prominence that research in mathematics development has attained. “Math” was a keyword or appeared in the abstract of 216 presentations at the 2015 SRCD Conference. This number of mentions exceeded that for other popular areas of developmental psychology such as “perception” (117), “attention” (166), “memory” (141), “reasoning” (67), “space” or “spatial” (87), “moral” (110), “reading” (91), and “executive function” (180). Susan Chipman was right, and her vision deserves recognition: NIE built it, and they did come. The excellent chapters in the xv
xvi
Foreword: Build It and They Will Come
present volume attest to the theoretical and practical importance of the area that Susan and the NIE helped create; the chapters also say a lot about where the area is today and where it seems to be going. Probably the most striking feature of this volume is the range of topic areas and age groups that are addressed. Before roughly 2010, the overwhelming focus of mathematical development research was on whole numbers and on children from birth to early elementary school. There were many studies of the early development of nonsymbolic numbers, counting, whole number arithmetic, number conservation, and symbolic magnitude representations, but not many on more advanced mathematical topics or with older children and adolescents. Starting around 2010, the focus of math development research widened to include rational numbers: fractions, decimals, percentages, and negatives. The latest and most exciting developments in the study of rational numbers are well represented in this volume, in particular in the chapters of Jordan on improving fraction instruction; of Rittle-Johnson, Star, and Durkin on use of comparison to improve learning of rational numbers; of van Hoof, Van Dooren, Vamvakoussi, and Verschaffel on the developmental transition from natural to rational numbers; and of DeWolf, Bassok, and Holyoak on the importance of relational reasoning with rational numbers for algebra performance. Most of the participants in these studies ranged from late elementary school through the end of middle school, though adults’ knowledge of rational numbers also received a fair amount of attention. Rational numbers have received a great deal of research attention since 2010, so the focus on them in this volume was not too surprising. More surprising was the emphasis on topics beyond rational numbers: on learning of algebra in the chapters of Booth et al.; of Lee, Ng, and Bull; of Rittle-Johnson et al.; and of DeWolf et al.; on geometry in the chapter of Mammarella, Giofre, and Caviola; and on trigonometry in the chapter by Mickey and McClelland. Of course, all of these topics have received some research attention for many years, but the emphasis on them in this volume is striking. I believe that the focus on rational numbers, algebra, geometry, and trigonometry represents an important part of where research on mathematical development is going. Together with prior research on whole numbers, this research will build the database necessary to construct more encompassing theories of mathematical development than have heretofore been possible. The integrated theory of numerical development, which I have formulated in recent years (Siegler & Braithwaite, in press; Siegler, 2016; Siegler, Thompson, & Schneider, 2011), provides a means for bringing together acquisition of whole and rational number knowledge within a single framework. Within that framework, numerical development presents two central challenges. One challenge is to understand that all real numbers share the property of representing magnitudes that can be located and ordered on number lines. The other challenge is to understand that many other properties of natural numbers do not consistently characterize rational numbers. Properties that apply to natural but not rational
Foreword: Build It and They Will Come
xvii
numbers include being represented by a single, unique symbol; having a unique predecessor and a unique successor; never decreasing with multiplication or increasing with division; and so on. It was gratifying for me personally that this theory was used in productive ways by many of the authors in the present volume. A challenge now is to integrate the new work that is emerging on algebra, geometry, and trigonometry with current understanding of whole and rational numbers. We already know that just as precision of whole numbers has proved predictive of precision of representations of rational numbers (e.g., Bailey, Siegler, & Geary, 2014; Mou et al., 2016), so has precision of representations of rational numbers proved predictive of algebraic proficiency (e.g., Booth, Newton, & Twiss-Garrity, 2014; Mou et al., 2016). These relations are present even after many relevant variables are statistically controlled: IQ, working memory, executive functioning, parental income and education, etc. Largely unknown, however, are the causal pathways that lead to these predictive relations. Several chapters in this volume, in particular those of Booth et al., Rittle-Johnson et al., and DeWolf et al., provide promising leads to how the integrative process operates, but a lot of work remains to be done before an integrative theory of mathematical development, as opposed to numerical development, will be possible. Formulating such an integrative theory of mathematical development clearly deserves high priority. Another striking feature of the present volume is the amount of research that focuses on conceptual understanding of mathematical procedures. Although considerable progress has been made in understanding conceptual bases of very early numerical procedures, for example the work of Gelman and Gallistel (1978) on counting, much less research has been devoted to conceptual understanding of whole number arithmetic procedures, much less rational number ones. To be clear, there has been considerable documentation of misconceptions in both whole number arithmetic (e.g., Brown & VanLehn, 1980 and rational number arithmetic (e.g., Resnick & Omanson, 1987), but these studies have left unclear the conceptions on which the flawed procedures are based. To illustrate, consider the well-known long subtraction error of inverting the top and bottom digits in a column when the top digit is smaller (e.g., treating 145-108 as if it were 148-105). More than 35 years after the Brown and VanLehn studies that convincingly documented this error pattern, we still do not know whether this and other “misconceptions” stem from (1) belief that the erroneous procedure is correct, (2) belief that the erroneous procedure is one of a few possibilities that might be correct, or (3) belief that the erroneous procedure is erroneous, but use of it anyway due to not knowing a better alternative. Simply put, we still do not understand the conception on which the “misconception” is based. Thus, it was refreshing to see a great deal of attention in this volume to the conceptual underpinnings of relatively advanced mathematical procedures. Chapters that particularly emphasized these conceptual underpinnings included Robinson’s work on understanding of multiplication and division,
xviii
Foreword: Build It and They Will Come
McNeil et al.’s chapter on understanding of mathematical equivalence, DeWolf et al.’s and Booth et al.’s chapters on algebra, Rittle-Johnson et al.’s chapter on the role of comparison processes, Thevenot’s chapter on arithmetic word problems, and Mammarella et al.’s chapter on geometry. This focus on conceptual understanding of mathematical procedures is welcome for the same reason as the focus on algebra, geometry, and trigonometry. The chapters and related work provide crucial empirical data for identifying the conceptual understanding that influences, and is influenced by, knowledge of mathematical procedures. At the same time, these expansions pose the same type of challenge for formulating an encompassing theory of mathematical development as the expansions of research into more advanced mathematical topics. One sign of the challenge of expanding our theories to include conceptual as well as procedural knowledge is that our descriptions of the development of mathematical procedures are almost invariably far more concrete and specific than our descriptions regarding conceptual understanding of the procedures. Recently, Hugo Lortie-Forgues and I also became intrigued by the conceptual underpinnings of mathematical procedures, in particular, rational number arithmetic procedures. In a pair of studies, one on understanding of fraction arithmetic procedures (Siegler & Lortie-Forgues, 2015) and one on understanding of decimal arithmetic procedures (Lortie-Forgues & Siegler, 2015), we found that large majorities of both middle school students and preservice teachers believed that multiplication of pairs of numbers between 0 and 1 invariably led to answers greater than either multiplicand and that division of pairs of numbers between 0 and 1 invariably led to answers smaller than the number being divided. When asked why they believed that, participants most often answered that multiplication always makes numbers bigger and that division makes them smaller. Such explanations characterized most participants whose multiplication and division of rational numbers was flawless, as well as ones who did not know the procedures as well. This is a striking example of failing to learn from experience, given the hundreds if not thousands of fraction and decimal arithmetic problems that college students especially would have solved during their lives. The findings suggest that studying conceptual understanding of more advanced mathematical procedures is also likely to reveal more misconceptions than correct conceptions, many bits and pieces of knowledge that are only loosely connected to the classes of problems for which they are appropriate, and too often no understanding at all of why mathematical procedures are justified. A third major theme of this volume is the intensifying effort to use findings from cognitive science research to improve math learning. This is most evident in areas where a substantial empirical base is available to guide the instructional efforts, such as mathematical equivalence, arithmetic word problems, and rational numbers, as indicated in the chapters by McNeil et al., Thevenot, Barnes and Raghubar, and Jordan et al. However, it also is evident in areas with a smaller empirical base, in particular in the chapters on algebra by Booth et al. and by Rittle-Johnson et al. These instructional efforts promise to yield practical
Foreword: Build It and They Will Come
xix
benefits for educating students; they also promise to yield theoretical benefits for understanding mathematical development. I’ll close with a few comments about the editors of this volume, Dave Geary, Dan Berch, Rob Ochsendorf, and Kathy Mann Koepke. They were the ones who selected the authors, and they probably anticipated the topics that the authors would emphasize. The extensive coverage in this volume of older children’s mathematical development, of more advanced areas of mathematics, and of conceptual understanding of procedures does not just reflect trends in the field. It also reflects the wisdom of the editors in choosing these authors to contribute chapters to this volume. This wisdom is also evident in their insightful summary chapter. I encourage readers to think hard about the findings reported in this volume, to consider their implications for theory and practice, and to build on them to move toward a fully integrative theory of mathematical development.
REFERENCES Bailey, D. H., Siegler, R. S., & Geary, D. C. (2014). First grade predictors of middle school fraction knowledge. Developmental Science, 17, 775–785. Booth, J. L., Newton, K. J., & Twiss-Garrity, L. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118, 110–118. Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4, 379–426. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Lortie-Forgues, H., & Siegler, R. S. (2015). Conceptual knowledge of decimal arithmetic. Journal of Educational Psychology. Published online ahead of print, doi: http://dx.doi.org/10.1037/ edu0000148 Mou, Y., Li, Y., Hoard, M. K., Nugent, L., Chu, F., Rouder, J., & Geary, D. C. (2016). Developmental foundations of children’s fraction magnitude knowledge. Cognitive Development, 39, 141–153. Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (pp. 41–95). (Vol. 3). Hillsdale, NJ: Erlbaum. Siegler, R. S. (2016). Continuity and change in the field of cognitive development and in the perspectives of one cognitive developmentalist. Child Development Perspectives, 10, 128–133. Siegler, R. S., & Braithwaite, D. W. (in press). Numerical development. Annual Review of Psychology. Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107, 909–918. Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62, 273–296.
In 1978, I was a member of the review panel for the first math learning grants competition of the National Institute of Education (NIE). Panel members were told that there was enough money to fund 10 grants; the only problem was that there were only 2 proposals that clearly merited funding and 2 others where opinion was mixed. Once the reviews were presented and discussed, the conversation turned to whether the panel had to recommend 10 grants for funding, because that would mean funding at least 6 proposals for which there was no enthusiasm. Most panel members, me included, argued against funding them. However, Susan Chipman, who was Assistant Director of NIE and in charge of the review competition, stated that giving out all of the money was essential. Although acknowledging that the applications were far from stellar, she argued that once the word got out that there was substantial funding for math learning research, better proposals would soon follow. The panel reluctantly agreed and voted to fund the 10 least dirty shirts in the dirty laundry basket. How things have changed! A similar competition today would elicit at least 20 worthy proposals. The challenge would be to distinguish the really excellent ones from the merely good or very good ones. There would be passionate debates, just as before, but now they would stem from panel members arguing that their favorite proposals just had to be funded no matter what. A web search of the program from the most recent Society for Research in Child Development meeting provides quantitative evidence of the prominence that research in mathematics development has attained. “Math” was a keyword or appeared in the abstract of 216 presentations at the 2015 SRCD Conference. This number of mentions exceeded that for other popular areas of developmental psychology such as “perception” (117), “attention” (166), “memory” (141), “reasoning” (67), “space” or “spatial” (87), “moral” (110), “reading” (91), and “executive function” (180). Susan Chipman was right, and her vision deserves recognition: NIE built it, and they did come. The excellent chapters in the xv
xvi
Foreword: Build It and They Will Come
present volume attest to the theoretical and practical importance of the area that Susan and the NIE helped create; the chapters also say a lot about where the area is today and where it seems to be going. Probably the most striking feature of this volume is the range of topic areas and age groups that are addressed. Before roughly 2010, the overwhelming focus of mathematical development research was on whole numbers and on children from birth to early elementary school. There were many studies of the early development of nonsymbolic numbers, counting, whole number arithmetic, number conservation, and symbolic magnitude representations, but not many on more advanced mathematical topics or with older children and adolescents. Starting around 2010, the focus of math development research widened to include rational numbers: fractions, decimals, percentages, and negatives. The latest and most exciting developments in the study of rational numbers are well represented in this volume, in particular in the chapters of Jordan on improving fraction instruction; of Rittle-Johnson, Star, and Durkin on use of comparison to improve learning of rational numbers; of van Hoof, Van Dooren, Vamvakoussi, and Verschaffel on the developmental transition from natural to rational numbers; and of DeWolf, Bassok, and Holyoak on the importance of relational reasoning with rational numbers for algebra performance. Most of the participants in these studies ranged from late elementary school through the end of middle school, though adults’ knowledge of rational numbers also received a fair amount of attention. Rational numbers have received a great deal of research attention since 2010, so the focus on them in this volume was not too surprising. More surprising was the emphasis on topics beyond rational numbers: on learning of algebra in the chapters of Booth et al.; of Lee, Ng, and Bull; of Rittle-Johnson et al.; and of DeWolf et al.; on geometry in the chapter of Mammarella, Giofre, and Caviola; and on trigonometry in the chapter by Mickey and McClelland. Of course, all of these topics have received some research attention for many years, but the emphasis on them in this volume is striking. I believe that the focus on rational numbers, algebra, geometry, and trigonometry represents an important part of where research on mathematical development is going. Together with prior research on whole numbers, this research will build the database necessary to construct more encompassing theories of mathematical development than have heretofore been possible. The integrated theory of numerical development, which I have formulated in recent years (Siegler & Braithwaite, in press; Siegler, 2016; Siegler, Thompson, & Schneider, 2011), provides a means for bringing together acquisition of whole and rational number knowledge within a single framework. Within that framework, numerical development presents two central challenges. One challenge is to understand that all real numbers share the property of representing magnitudes that can be located and ordered on number lines. The other challenge is to understand that many other properties of natural numbers do not consistently characterize rational numbers. Properties that apply to natural but not rational
Foreword: Build It and They Will Come
xvii
numbers include being represented by a single, unique symbol; having a unique predecessor and a unique successor; never decreasing with multiplication or increasing with division; and so on. It was gratifying for me personally that this theory was used in productive ways by many of the authors in the present volume. A challenge now is to integrate the new work that is emerging on algebra, geometry, and trigonometry with current understanding of whole and rational numbers. We already know that just as precision of whole numbers has proved predictive of precision of representations of rational numbers (e.g., Bailey, Siegler, & Geary, 2014; Mou et al., 2016), so has precision of representations of rational numbers proved predictive of algebraic proficiency (e.g., Booth, Newton, & Twiss-Garrity, 2014; Mou et al., 2016). These relations are present even after many relevant variables are statistically controlled: IQ, working memory, executive functioning, parental income and education, etc. Largely unknown, however, are the causal pathways that lead to these predictive relations. Several chapters in this volume, in particular those of Booth et al., Rittle-Johnson et al., and DeWolf et al., provide promising leads to how the integrative process operates, but a lot of work remains to be done before an integrative theory of mathematical development, as opposed to numerical development, will be possible. Formulating such an integrative theory of mathematical development clearly deserves high priority. Another striking feature of the present volume is the amount of research that focuses on conceptual understanding of mathematical procedures. Although considerable progress has been made in understanding conceptual bases of very early numerical procedures, for example the work of Gelman and Gallistel (1978) on counting, much less research has been devoted to conceptual understanding of whole number arithmetic procedures, much less rational number ones. To be clear, there has been considerable documentation of misconceptions in both whole number arithmetic (e.g., Brown & VanLehn, 1980 and rational number arithmetic (e.g., Resnick & Omanson, 1987), but these studies have left unclear the conceptions on which the flawed procedures are based. To illustrate, consider the well-known long subtraction error of inverting the top and bottom digits in a column when the top digit is smaller (e.g., treating 145-108 as if it were 148-105). More than 35 years after the Brown and VanLehn studies that convincingly documented this error pattern, we still do not know whether this and other “misconceptions” stem from (1) belief that the erroneous procedure is correct, (2) belief that the erroneous procedure is one of a few possibilities that might be correct, or (3) belief that the erroneous procedure is erroneous, but use of it anyway due to not knowing a better alternative. Simply put, we still do not understand the conception on which the “misconception” is based. Thus, it was refreshing to see a great deal of attention in this volume to the conceptual underpinnings of relatively advanced mathematical procedures. Chapters that particularly emphasized these conceptual underpinnings included Robinson’s work on understanding of multiplication and division,
xviii
Foreword: Build It and They Will Come
McNeil et al.’s chapter on understanding of mathematical equivalence, DeWolf et al.’s and Booth et al.’s chapters on algebra, Rittle-Johnson et al.’s chapter on the role of comparison processes, Thevenot’s chapter on arithmetic word problems, and Mammarella et al.’s chapter on geometry. This focus on conceptual understanding of mathematical procedures is welcome for the same reason as the focus on algebra, geometry, and trigonometry. The chapters and related work provide crucial empirical data for identifying the conceptual understanding that influences, and is influenced by, knowledge of mathematical procedures. At the same time, these expansions pose the same type of challenge for formulating an encompassing theory of mathematical development as the expansions of research into more advanced mathematical topics. One sign of the challenge of expanding our theories to include conceptual as well as procedural knowledge is that our descriptions of the development of mathematical procedures are almost invariably far more concrete and specific than our descriptions regarding conceptual understanding of the procedures. Recently, Hugo Lortie-Forgues and I also became intrigued by the conceptual underpinnings of mathematical procedures, in particular, rational number arithmetic procedures. In a pair of studies, one on understanding of fraction arithmetic procedures (Siegler & Lortie-Forgues, 2015) and one on understanding of decimal arithmetic procedures (Lortie-Forgues & Siegler, 2015), we found that large majorities of both middle school students and preservice teachers believed that multiplication of pairs of numbers between 0 and 1 invariably led to answers greater than either multiplicand and that division of pairs of numbers between 0 and 1 invariably led to answers smaller than the number being divided. When asked why they believed that, participants most often answered that multiplication always makes numbers bigger and that division makes them smaller. Such explanations characterized most participants whose multiplication and division of rational numbers was flawless, as well as ones who did not know the procedures as well. This is a striking example of failing to learn from experience, given the hundreds if not thousands of fraction and decimal arithmetic problems that college students especially would have solved during their lives. The findings suggest that studying conceptual understanding of more advanced mathematical procedures is also likely to reveal more misconceptions than correct conceptions, many bits and pieces of knowledge that are only loosely connected to the classes of problems for which they are appropriate, and too often no understanding at all of why mathematical procedures are justified. A third major theme of this volume is the intensifying effort to use findings from cognitive science research to improve math learning. This is most evident in areas where a substantial empirical base is available to guide the instructional efforts, such as mathematical equivalence, arithmetic word problems, and rational numbers, as indicated in the chapters by McNeil et al., Thevenot, Barnes and Raghubar, and Jordan et al. However, it also is evident in areas with a smaller empirical base, in particular in the chapters on algebra by Booth et al. and by Rittle-Johnson et al. These instructional efforts promise to yield practical
Foreword: Build It and They Will Come
xix
benefits for educating students; they also promise to yield theoretical benefits for understanding mathematical development. I’ll close with a few comments about the editors of this volume, Dave Geary, Dan Berch, Rob Ochsendorf, and Kathy Mann Koepke. They were the ones who selected the authors, and they probably anticipated the topics that the authors would emphasize. The extensive coverage in this volume of older children’s mathematical development, of more advanced areas of mathematics, and of conceptual understanding of procedures does not just reflect trends in the field. It also reflects the wisdom of the editors in choosing these authors to contribute chapters to this volume. This wisdom is also evident in their insightful summary chapter. I encourage readers to think hard about the findings reported in this volume, to consider their implications for theory and practice, and to build on them to move toward a fully integrative theory of mathematical development.
REFERENCES Bailey, D. H., Siegler, R. S., & Geary, D. C. (2014). First grade predictors of middle school fraction knowledge. Developmental Science, 17, 775–785. Booth, J. L., Newton, K. J., & Twiss-Garrity, L. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118, 110–118. Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4, 379–426. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Lortie-Forgues, H., & Siegler, R. S. (2015). Conceptual knowledge of decimal arithmetic. Journal of Educational Psychology. Published online ahead of print, doi: http://dx.doi.org/10.1037/ edu0000148 Mou, Y., Li, Y., Hoard, M. K., Nugent, L., Chu, F., Rouder, J., & Geary, D. C. (2016). Developmental foundations of children’s fraction magnitude knowledge. Cognitive Development, 39, 141–153. Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (pp. 41–95). (Vol. 3). Hillsdale, NJ: Erlbaum. Siegler, R. S. (2016). Continuity and change in the field of cognitive development and in the perspectives of one cognitive developmentalist. Child Development Perspectives, 10, 128–133. Siegler, R. S., & Braithwaite, D. W. (in press). Numerical development. Annual Review of Psychology. Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107, 909–918. Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62, 273–296.