Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process.
01 Jun 2001-Journal of Educational Psychology (American Psychological Association)-Vol. 93, Iss: 2, pp 346-362
TL;DR: In this paper, the authors propose that conceptual and procedural knowledge develop in an iterative fashion and that improved problem representation is 1 mechanism underlying the relations between them, and demonstrate that children's initial conceptual knowledge predicted gains in procedural knowledge.
Abstract: The authors propose that conceptual and procedural knowledge develop in an iterative fashion and that improved problem representation is 1 mechanism underlying the relations between them. Two experiments were conducted with 5th- and 6th-grade students learning about decimal fractions. In Experiment 1, children's initial conceptual knowledge predicted gains in procedural knowledge, and gains in procedural knowledge predicted improvements in conceptual knowledge. Correct problem representations mediated the relation between initial conceptual knowledge and improved procedural knowledge. In Experiment 2, amount of support for correct problem representation was experimentally manipulated, and the manipulations led to gains in procedural knowledge. Thus, conceptual and procedural knowledge develop iteratively, and improved problem representation is 1 mechanism in this process.
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TL;DR: Although low-income children performed more poorly than their middle-income counterparts in mathematics achievement and progressed at a slower rate, their performance and growth were mediated through relatively weak kindergarten number competence.
Abstract: Children's number competencies over 6 time points, from the beginning of kindergarten to the middle of 1st grade, were examined in relation to their mathematics achievement over 5 later time points, from the end of 1st grade to the end of 3rd grade. The relation between early number competence and mathematics achievement was strong and significant throughout the study period. A sequential process growth curve model showed that kindergarten number competence predicted rate of growth in mathematics achievement between 1st and 3rd grades as well as achievement level through 3rd grade. Further, rate of growth in early number competence predicted mathematics performance level in 3rd grade. Although low-income children performed more poorly than their middle-income counterparts in mathematics achievement and progressed at a slower rate, their performance and growth were mediated through relatively weak kindergarten number competence. Similarly, the better performance and faster growth of children who entered kindergarten at an older age were explained by kindergarten number competence. The findings show the importance of early number competence for setting children's learning trajectories in elementary school mathematics.
900 citations
Cites result from "Developing Conceptual Understanding..."
...This is in keeping with arguments that conceptual knowledge and procedural skill are iterative, each prompting the learning of the other (National Mathematics Advisory Panel, 2008; Rittle-Johnson, Siegler, & Alibali, 2001 )....
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TL;DR: Key findings from the small body of research on mathematics difficulties (MD) relevant to early identification and early intervention are highlighted and the presence of reading difficulties seems related to slower progress in many aspects of mathematics.
Abstract: This article highlights key findings from the small body of research on mathematics difficulties (MD) relevant to early identification and early intervention. The research demonstrates that (a) for many children, mathematics difficulties are not stable over time; (b) the presence of reading difficulties seems related to slower progress in many aspects of mathematics; (c) almost all students with MD demonstrate problems with accurate and automatic retrieval of basic arithmetic combinations, such as 6 + 3. The following measures appear to be valid and reliable indicators of potential MD in kindergartners: (a) magnitude comparison (i.e., knowing which digit in a pair is larger), (b) sophistication of counting strategies, (c) fluent identification of numbers, and (d) working memory (as evidenced by reverse digit span). These are discussed in terms of the components of number sense. Implications for early intervention strategies are explored.
775 citations
Cites background from "Developing Conceptual Understanding..."
...Interventions should follow the finding of Rittle-Johnson, Siegler, and Alibali (2001) , who noted that procedural and conceptual abilities in mathematics “lie on a continuum and cannot always be separated” (p. 346)....
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...Interventions should follow the finding of Rittle-Johnson, Siegler, and Alibali (2001), who noted that procedural and conceptual abilities in mathematics " lie on a continuum and cannot always be separated " (p. 346)....
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TL;DR: The present findings with 11- and 13-year-olds indicate that, as with whole numbers, accuracy of fraction magnitude representations is closely related to both fractions arithmetic proficiency and overall mathematics achievement test scores.
560 citations
TL;DR: The Knowledge-Learning-Instruction framework is described, which promotes the emergence of instructional principles of high potential for generality, while explicitly identifying constraints of and opportunities for detailed analysis of the knowledge students may acquire in courses.
511 citations
TL;DR: In this paper, the authors randomly assigned 70 seventh grade students to learn about algebra equation solving by either comparing and contrasting alternative solution methods or reflecting on the same solution methods one at a time, and found that students in the compare group had greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge.
Abstract: Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics, and comparison is emerging as a fundamental learning mechanism. To experimentally evaluate the effects of comparison for mathematics learning, the authors randomly assigned 70 seventhgrade students to learn about algebra equation solving by either (a) comparing and contrasting alternative solution methods or (b) reflecting on the same solution methods one at a time. At posttest, students in the compare group had made greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom.
473 citations
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TL;DR: This article seeks to make theorists and researchers aware of the importance of not using the terms moderator and mediator interchangeably by carefully elaborating the many ways in which moderators and mediators differ, and delineates the conceptual and strategic implications of making use of such distinctions with regard to a wide range of phenomena.
Abstract: In this article, we attempt to distinguish between the properties of moderator and mediator variables at a number of levels. First, we seek to make theorists and researchers aware of the importance of not using the terms moderator and mediator interchangeably by carefully elaborating, both conceptually and strategically, the many ways in which moderators and mediators differ. We then go beyond this largely pedagogical function and delineate the conceptual and strategic implications of making use of such distinctions with regard to a wide range of phenomena, including control and stress, attitudes, and personality traits. We also provide a specific compendium of analytic procedures appropriate for making the most effective use of the moderator and mediator distinction, both separately and in terms of a broader causal system that includes both moderators and mediators.
80,095 citations
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01 Jan 1997
TL;DR: In this article, the authors discuss the importance of biology for human development and the role of the human brain in the development of human cognition and behavior, and propose a model of human development based on the Bioecological Model of Human Development.
Abstract: VOLUME 1. 1. Developmental Science, Developmental Systems, and Contemporary Theories of Human Development (Richard M. Lerner). 2. Developmental Psychology: Philosophy, Concepts, Methodology (Willis F. Overton). 3. The Making of Developmental Psychology (Robert B. Cairns and Beverley D. Cairns). 4. Developmental Epistemology and Implications for Methodology (Jaan Valsiner). 5. The Significance of Biology for Human Development: A Developmental Psychobiological Systems Views (Gilbert Gottlieb, Douglas Wahlsten and Robert Lickliter). 6. Dynamic Systems Theories (Esther Thelen and Linda B. Smith). 7. Dynamic Development of Action and Thought (Kurt W Fischer and Thomas R. Bidell). 8. The Person in Context: A Holistic-Interactionistic Approach (David Magnusson and Hakan Stattin). 9. The Developing Person: An Experiential Perspective (Kevin Rathunde and Mihaly Csikszentmihalyi). 10. Action Perspectives on Human Development (J. Brandstadter). 11. Life Span Theory in Developmental Psychology (Paul B. Baltes, Ulman Lindenberger and Ursula M. Staudinger). 12. The Life Course and Human Development (Glen H. Elder and Michael J. Shanahan). 13. The Cultural Psychology of Development: One Mind, Many Mentalities (Richard A. Shweder, Jacqueline J. Goodnow, Giyoo Hatano, Robert A. Levine, Hazel R. Markus and Peggy J. Miller). 14. The Bioecological Model of Human Development (Urie Bronfenbrenner and Pamela A. Morris). 15. Phenomenologitcal and Ecological Systems Theory: Development of Diverse Groups (Margaret Beale Spencer). 16. Positive Youth Development: Theory, Research, and Applications (Peter L. Benson, Peter C. Scales, Stephen F. Hamilton and Arturo Sesma). 17. Religious and Spiritual Development Throughout the Life Span (Fritz K. Oser, W. George Scarlett and Anton Bucher). Author Index. Subject Index. VOLUME 2. SECTION ONE: FOUNDATIONS. 1. Neural Bases of Cognitive Development (Charles A. Nelson, Kathleen M. Thomas and Michelle de Haan). 2. The Infant's Auditory World: Hearing, Speech, and the Beginnings of Language (Jenny R. Saffran, Janet F. Werker and Lynne A. Werner). 3. Infant Visual Perception (Philip J. Kellman and Martha E. Arterberry). 4. Motor Development (Karen E. Adolph and& Sarah E. Berger). 5. Infant Cognition (Leslie B. Cohen and Cara H. Cashon). SECTION TWO: COGNITION AND COMMUNICATION 6. Acquiring Linguistic Constructions (Michael Tomasello). 7. Early Word Learning (Sandra R. Waxman and Jeffrey L. Lidz). 8. Nonverbal Communication: The Hand's Role in Talking and Thinking (Susan Goldin-Meadow). SECTION THREE: COGNITIVE PROCESSES. 9. Event Memory (Patricia J. Bauer). 10. Information Processing Approaches to Development (Yuko Munakata). 11. Microgenetic Analysis of Learning (Robert S. Siegler). 12. Cognitive Strategies Michael Pressley and Katherine Hilden). 13. Reasoning and Problem Solving (Graeme S. Halford and Glenda Andrews). 14. Cognitive Science and Cognitive Development (Frank Keil). 15. Culture and Cognitive Development in Phylogenetic, Historical, and Ontogenetic Perspective (Michael Cole). SECTION FOUR: CONCEPTUAL UNDERSTANDING AND ACHIEVMENTS. 16. Conceptual Development (Susan A. Gelman and Charles W. Kalish). 17. Development of Spatial Cognition (Nora S. Newcombe and Janellen Huttenlocher). 18. Development of Mathematical Understanding (David C. Geary). 19. Social Cognition (Paul L. Harris). 20. Development in the Arts: Drawing and Music (Ellen Winner). 21. Extraordinary Achievements: A Developmental and Systems Analysis (Seana Moran and Howard Gardner). SECTION FIVE: THE PERSPECTIVE BEYOND CHILDHOOD. 22. The Second Decade: What Develops (and how) (Deanna Kuhn and Sam Franklin). Author Index. Subject Index. VOLUME 3. 1. Introduction (Nancy Eisenberg). 2. The Development of the Person: Social Understanding, Relationships, Conscience, Self (Rosa A. Thompson). 3. Temperament (Mary K. Rothbart and John E. Bates). 4. Biology, Culture, and Temperamental Biases (Jerome Kagan and Nathan A. Fox). 5. Emotional Development: Action, Communication, and Understanding (Carolyn Saarni, Joseph J. Campos, Linda A. Camras and David Witherington). 6. Personality Development (Avshalom Caspi) and Rebecca L. Shiner. 7. Socialization Processes (Daphne Blunt Bugental and Joan E. Grusec). 8. Socialization in the Family: Etnnic and Ecological Perspectives (Ross D. Parke and Raymond Buriel). 9. The Self (Susan Harter). 10. Peer Interactions, Relationships, and Groups (Kenneth H. Rubin, William M. Bulkowski and Jeffrey G. Parker). 11. Prosocial Development (Nany Eisenberg, Richard A. Fabes and Tracy L. Spinrad). 12. Aggression and Antisocial Behavior in Youth (Kenneth A. Dodge, John D. Coie and Donald Lynam). 13. The Development of Morality (Elliot Turiel). 14. Gender Development (Diane N. Ruble, Carol Lynn martin and Sheri A. Berebaum). 15. Development of Achievement Motivation (Allan Wigfield, Jacquelynne S. Eccles, Ulrich Schiefele, Robert W. Rosser and Pamela Davis-Kean). 16. Adolescent Development in Interpersonal Context (W. Andrew Collins and Laurence Steinberg). Author Index. Subject Index. VOLUME 4. PART I: INTRODUCTION. Applying Research to Practice (K. Renninger & I. Sigel). PART II: RESEARCH ADVANCED AND IMPLICATIONS FOR PRACTICE IN EDUCATION. 1. Early Childhood Development and Education (M. Hyson, et al.). 2. Assessments of Early Reading (S. Paris & A. Paris). 3. Becoming Bilingual, Biliterate, and Bicultural (C. Snow & J. Kang). 4. Mathematical Thinking and Learning (E. De Corte & L. Verschaffel). 5. Scientific Thinking and Science Literacy (R. Lehrer & L. Schauble). 6. Character Education (D. Lapsley & D. Narvaez). 7. Learning Environments (P. Blumenfeld, et al.). PART III: RESEARCH ADVANCED AND IMPLICATIONS FOR CLINICAL APPLICATIONS. 8. Self-REgulations and Effort Investment (M. Boekaerts). 9. Risk and Prevention (R. Selman & A. Dray). 10. Learning Disabilities (V. Berninger). 11. Mental Retardation (R. Hodapp & E. Dykens). 12. Developmental Psychopathology and Preventive Intervention (D. Cicchetti & S. Toth). 13. Families and Early Childhood Interventions (D. Powell). 14. School-based Social and Emotional Learning Programs (J. Kress & M. Elias). PART IV: RESEARCH ADVANCED AND IMPLICATIONS FOR SOCIAL POLICY AND SOCIAL ACTION. 15. Cultural Pathways Through Human Development (P. Greenfield, et al.). 16. Children and War Trauma (A. Klingman). 17. The Child and them Law (M. Bruck, et al.). 18. Media and Popular Culture (G. Comstock & E. Scharrer). 19. Children's Health and Education (C. Ramey, et al.). 20. Parenting Science and Practice (M. Bornstein). 21. Nonparental Child Care (M. Lamb & L. Ahnert). 22. Research to Practice Redefined (I. Sigel). Afterword.
9,880 citations
TL;DR: Results from sorting tasks and protocols reveal that experts and novices begin their problem representations with specifiably different problem categories, and completion of the representations depends on the knowledge associated with the categories.
5,091 citations