This paper presents an analytic approach for understanding the interplay through time between “scientific” and “everyday concepts” in a mathematics classroom community. To illustrate the approach, we focus on an elementary classroom implementing an integers and fractions lesson sequence that makes use of the number line as a principal representational context. In our analysis of the community's emerging collective practices (recurring structures of joint activity), we trace the interplay between children's sensorimotor actions (displacing, counting, and splitting) and the mathematical definitions supported in the classroom, like definitions of unit interval or equivalent fractions. In our illustrative analysis, we find that the teacher orchestrated collective practices to support the use of actions to make sense of the formal definitions, and the use of definitions to regulate actions. Though we illustrate the analytic approach for a particular classroom community, the approach illuminates teaching-learning dynamics that transcend any particular classroom or subject matter domain.

Keywords:
Classroom practices, Embodied cognition, Everyday concepts, Longitudinal analysis, Mathematical cognition, Mathematics education, Number line, Scientific concepts
1.
Barrett, J.E., Sarama, J., Clements, D.H., Cullen, C., McCool, J., Witkowski-Rumsey, C., & Klanderman, D. (2012). Evaluating and improving a learning trajectory for linear measurement in elementary grades 2 and 3: A longitudinal study. Mathematical Thinking and Learning, 14, 28-54.
2.
Beth, E.W., & Piaget, J. (1974). Mathematical epistemology and psychology (W. Mays, Transl.). New York, NY: Springer.
3.
Bofferding, L. (2014). Negative integer understanding: Characterizing first graders' mental models. Journal for Research in Mathematics Education, 45, 194-245.
4.
Booth, J.L., & Siegler, R.S. (2008). Numerical magnitude estimations influence children's learning. Child Development, 79, 1016-1031.
5.
Chrisomalis, S. (2010). Numerical notation: A comparative history. New York, NY: Cambridge University Press.
6.
Clements, D.H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge.
7.
Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26, 66-86.
8.
Daro, P., Mosher, F.A., & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. CPRE Research Report #RR-68. Philadelphia, PA: Consortium for Policy Research in Education.
9.
Davis, G.E., & Pitkethly, A. (1990). Cognitive aspects of sharing. Journal for Research in Mathematics Education, 21, 145-153.
10.
Davydov, V.V. (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula (J. Teller, Transl., Vol. 2). Reston, VA: National Council of Teachers of Mathematics.
11.
Earnest, D.S. (2012). Supporting generative thinking about number lines, the Cartesian plane, and graphs of linear functions (unpublished doctoral dissertation). Berkeley, CA: University of California.
12.
Empson, S.B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom. Cognition & Instruction, 17, 283-342.
13.
Engle, R.A., & Conant, F.R. (2002). Guiding principles for fostering productive disciplinary engagement: Explaining an emergent argument in a community of learners classroom. Cognition and Instruction, 20, 399-483.
14.
Ford, M.J., & Forman, E.A. (2006). Redefining disciplinary learning in classroom contexts. Review of Research in Education, 30, 1-32.
15.
Forman, E.A., McCormick, D.E., & Donato, R. (1997). Learning what counts as a mathematical explanation. Linguistics and Education, 9, 313-339.
16.
Fosnot, C.T., & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division. Portsmouth, NH: Heinemann.
17.
Fuson, K.C. (1988). Children's counting and concepts of number. New York, NY: Springer.
18.
Gearhart, M., & Saxe, G.B. (2014). Differentiated instruction in shared mathematical contexts. Teaching Children Mathematics, 20, 426-435.
19.
Gelman, R., & Gallistel, C.R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.
20.
Gutiérrez, K.D., & Rogoff, B. (2003). Cultural ways of learning: Individual traits or repertoires of practice. Educational Researcher, 32, 19-25.
21.
Halliday, M.A.K. (1978). Language as social semiotic. London, England: Arnold.
22.
Hedegaard, M. (1990). The zone of proximal development as basis for instruction. In L.C. Moll (Ed.), Vygotsky and education: Instructional implications and applications of sociohistorical psychology (pp. 349-371). New York, NY: Cambridge University Press.
23.
Hedegaard, M. (2007). The development of children's conceptual relation. In H. Daniels, M. Cole, & J.V. Wertsch (Eds.), The Cambridge companion to Vygotsky. New York, NY: Cambridge University Press.
24.
Hershkowitz, R., Schwarz, B.B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32, 195-222.
25.
Hunting, R.P., & Sharpley, C.F. (1988). Fraction knowledge in preschool children. Journal for Research in Mathematics Education, 19, 175-180.
26.
Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence (A. Parsons & S. Milgram, Transl.). New York, NY: Routledge & Kegan Paul.
27.
Inhelder, B., & Piaget, J. (1969). The early growth of logic in the child. New York, NY: Norton.
28.
Karpov, Y.V. (2003). Vygotsky's doctrine of scientific concepts. In A. Kozulin, B. Gindis, V.S. Ageyev, & S.M. Miller (Eds.), Vygotsky's educational theory in cultural context (pp. 65-82). Cambridge, England: Cambridge University Press.
29.
Lakoff, G., & Núñez, R.E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.
30.
Langer, J. (1980). Origins of thought. New York, NY: Academic Press.
31.
Langer, J. (1986). The origins of logic: One to two years. Orlando, FL: Academic Press.
32.
Lean, G.A. (1992). Counting systems of Papua New Guinea and Oceania (unpublished doctoral dissertation). Lae, Papua New Guinea: Papua New Guinea University of Technology.
33.
Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W.G. Martin, & D.E. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 179-192). Reston, VA: National Council of Teachers of Mathematics.
34.
Menninger, K. (1969). Number words and number symbols: A cultural history of numbers. Cambridge, MA: MIT Press.
35.
Moll, L.C. (1990). Introduction. In L.C. Moll (Ed.), Vygotsky and education: Instructional implications and applications of sociohistorical psychology (pp. 1-30). New York, NY: Cambridge University Press.
36.
Nelson, K. (1995). From spontaneous to scientific concepts: Continuities and discontinuities from childhood to adulthood. In L.M.W. Martin, K. Nelson, & E. Tobach (Eds.), Sociocultural psychology: Theory and practice of doing and knowing (pp. 229-249). Cambridge, England: Cambridge University Press.
37.
Núñez, R. (2011). No innate number line in the human brain. Journal of Cross-Cultural Psychology, 42, 651-668.
38.
Núñez, R., & Marghetis, T. (in press). Cognitive linguistics and the concept(s) of number. In R. Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition. Oxford, England: Oxford University Press.
39.
Panofsky, C.P., John-Steiner, V., & Blackwell, P.J. (1990). The development of scientific concepts and discourse. In L.C. Moll (Ed.), Vygotsky and education: Instructional implications and applications of sociohistorical psychology (pp. 251-268). New York, NY: Cambridge University Press.
40.
Penuel, W.R., Confrey, J., Maloney, A., & Rupp, A.A. (2013). Design decisions in developing learning trajectories-based assessments in mathematics: A case study. Journal of the Learning Sciences, 23, 47-95.
41.
Petitto, A.L. (1990). Development of number line and measurement concepts. Cognition and Instruction, 7, 55-78.
42.
Piaget, J. (1952). The child's conception of number. New York, NY: Norton.
43.
Piaget, J. (1963). The origins of intelligence in children (M. Cook, Transl.). New York, NY: Norton.
44.
Piaget, J. (1970a). Genetic epistemology. New York, NY: Columbia University Press.
45.
Piaget, J. (1970b). Piaget's theory. In P.H. Mussen (Ed.), Carmichael's manual of child psychology (pp. 703-731). New York, NY: Wiley.
46.
Piaget, J., & Inhelder, B. (1956). The child's conception of space. London, England: Routledge & K. Paul.
47.
Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child's conception of geometry. New York, NY: Harper and Row.
48.
Pothier, Y., & Sawada, D. (1983). Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14, 307-317.
49.
Radford, L. (2014). Towards an embodied, cultural, and material conception of mathematics cognition. ZDM, 46, 349-361.
50.
Sarnecka, B.W., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108, 662-674.
51.
Saxe, G.B. (1977). A developmental analysis of notational counting. Child Development, 48, 1512-1520.
52.
Saxe, G.B. (2012). Cultural development of mathematical ideas: Papua New Guinea studies. New York, NY: Cambridge University Press.
53.
Saxe, G.B., de Kirby, K., Le, M., Sitabkhan, Y., & Kang, B. (2014). Understanding learning across lessons in classroom communities: A multi-leveled analytic approach. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Doing (qualitative) research: Methodology and methods in mathematics education. New York, NY: Springer.
54.
Saxe, G.B., Diakow, R., & Gearhart, M. (2013a). Towards curricular coherence in integers and fractions: A study of the efficacy of a lesson sequence that uses the number line as the principal representational context. ZDM, 45, 343-364.
55.
Saxe, G.B., Earnest, D., Sitabkhan, Y., Haldar, L.C., Lewis, K.E., & Zheng, Y. (2010). Supporting generative thinking about the integer number line in elementary mathematics. Cognition & Instruction, 28, 433-474.
56.
Saxe, G.B., Guberman, S.R., & Gearhart, M. (1987). Social processes in early number development. Monographs of the Society for Research in Child Development, 52, Serial No. 162.
57.
Saxe, G.B., & Posner, J. (1982). The development of numerical cognition: Cross-cultural perspectives. In H.P. Ginsburg (Ed.), The development of mathematical thinking (pp. 291-317). New York, NY: Academic Press.
58.
Saxe, G.B., Shaughnessy, M.M., Gearhart, M., & Haldar, L.C. (2013b). Coordinating numeric and linear units: Elementary students' strategies for locating whole numbers on the number line. Mathematical Thinking and Learning, 15, 235-258.
59.
Schleppegrell, M.J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly, 23, 139-159.
60.
Schmid-Schönbein, C., & Thiel, T. (Writers) (2010). From direct encounter with objects to reflection on actions - A cinematic reconstruction of an individual cognitive process [film, 90 minutes]. Presented at the 40th Annual Meeting of the Jean Piaget Society, St. Louis, MO.
61.
Siegler, R.S., & Opfer, J.E. (2003). The development of numerical estimation: evidence for multiple representations of numerical quantity. Psychological Science, 14, 237-243.
62.
Simon, M.A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145.
63.
Smith, J.P., Males, L.M., Dietiker, L.C., Lee, K., & Mosier, A. (2013). Curricular treatments of length measurement in the United States: Do they address known learning challenges? Cognition and Instruction, 31, 388-433.
64.
Steffe, L.P., & Olive, J. (2010). Children's fraction knowledge. New York, NY: Springer.
65.
Steffe, L.P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children's counting types: Philosophy, theory, and applications. New York, NY: Praeger.
66.
Tabach, M., Hershkowitz, R., Rasmussen, C., & Dreyfus, T. (2014). Knowledge shifts and knowledge agents in the classroom. The Journal of Mathematical Behavior, 1, 192-208.
67.
Temina-Kingsolver, V. (2008). Psycholinguistics - language and thought. Retrieved from http://languageavenue.com/psycholinguistics/language-and-thought/scientific-vs-spontaneous-concepts-vygotskian-perspective/all-pages.
68.
Thiel, T. (2012). Micro-genesis and micro-analysis of cognitive processes within the framework of Piaget's theory of reflecting abstraction - A pleading for re-constructing real empirical processes. Paper presented at the 42nd Annual Meeting of the Jean Piaget Society, Toronto, Canada.
69.
Vygotsky, L.S. (1986). Thought and language. Cambridge, MA: MIT Press.
70.
Wells, G. (1994). Learning and teaching “scientific concepts”: Vygotsky's ideas revisited. Paper presented at the conference, ‘‘Vygotsky and the Human Sciences'', Moscow, Russia.
71.
Wertsch, J.V. (1991). Voices of the mind: A sociocultural approach to mediated action. Cambridge, MA: Harvard University Press.
72.
Whitehead, A.N. (1959). The aims of education. Daedalus, 88, 192-205.
73.
Wittgenstein, L. (1953). Philosophical investigations. New York, NY: Macmillan.
74.
Yoshida, K. (2004). Understanding how the concept of fractions develops: A Vygotskian perspective. Paper presented at the International Group for the Psychology of Mathematics Education, Bergen, Norway.
© 2015 S. Karger AG, Basel
2015
Copyright / Drug Dosage / Disclaimer
Copyright: All rights reserved. No part of this publication may be translated into other languages, reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, microcopying, or by any information storage and retrieval system, without permission in writing from the publisher.
Drug Dosage: The authors and the publisher have exerted every effort to ensure that drug selection and dosage set forth in this text are in accord with current recommendations and practice at the time of publication. However, in view of ongoing research, changes in government regulations, and the constant flow of information relating to drug therapy and drug reactions, the reader is urged to check the package insert for each drug for any changes in indications and dosage and for added warnings and precautions. This is particularly important when the recommended agent is a new and/or infrequently employed drug.
Disclaimer: The statements, opinions and data contained in this publication are solely those of the individual authors and contributors and not of the publishers and the editor(s). The appearance of advertisements or/and product references in the publication is not a warranty, endorsement, or approval of the products or services advertised or of their effectiveness, quality or safety. The publisher and the editor(s) disclaim responsibility for any injury to persons or property resulting from any ideas, methods, instructions or products referred to in the content or advertisements.
You do not currently have access to this content.