In the Absence of Meaning…



There are many diagnoses of the bad state of U.S. mathematics education, ranging from incoherent curricula to low-quality teaching. In this chapter I will address a foundational reason for the many manifestations of failure—a systemic, cultural inattention to mathematical meaning and coherence. The result is teachers’ inability to teach for understanding and students’ inability to develop personal mathematical meanings that support interest, curiosity, and future learning. In developing this argument I discuss the subtle ways in which actual meanings with which teachers currently teach and actual meanings students currently develop in interaction with instruction contribute to dysfunctional mathematics education. I end by proposing a long-term strategy to address this situation.


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  1. Bauersfeld, H. (1980). Hidden dimensions in the so-called reality of a mathematics classroom. Educational Studies in Mathematics, 11, 23–42.CrossRefGoogle Scholar
  2. Bauersfeld, H. (1988). Interaction, construction, and knowledge: Alternative perspectives for mathematics education. In T. J. Cooney & D. A. Grouws (Eds.), Effective mathematics teaching. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  3. Branca, N. A. (1980). Communication of mathematical structure and its relationship to achievement. Journal for Research in Mathematics Education, 11, 37–49.CrossRefGoogle Scholar
  4. Bringuier, J. C. (1980). Conversations with Jean Piaget. Chicago, IL: University of Chicago Press.Google Scholar
  5. Cai, J. (2010). Conceptions of effective mathematics teaching within a cultural context: Perspectives of teachers from China and the United States. Journal of Mathematics Teacher Education, 13, 265–287.CrossRefGoogle Scholar
  6. Carlson, M. P., Oehrtman, M. C., & Engelke, N. (2010). The precalculus concept assessment (PCA) instrument: A tool for assessing students’ reasoning patterns and understandings. Cognition and Instruction, 28, 113–145.CrossRefGoogle Scholar
  7. Carpenter, T. P. (1986). Conceptual knowledge as a foundation for procedural understanding. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 113–132). Hillsdale, NJ: Erlbaum.Google Scholar
  8. Castillo-Garsow, C. C. (2010). Teaching the Verhulst model: A teaching experiment in covariational reasoning and exponential growth. Unpublished Ph.D. Dissertation, Arizona State University, Tempe, AZ. Retrieved from
  9. Clark, J. M., Cordero, F., Cottril, J., Czarnocha, B., DeVries, D. J., St. John, D., et al. (1997). Constructing a schema: The case of the chain rule? Journal of Mathematical Behavior, 16, 345–364.CrossRefGoogle Scholar
  10. Cobb, P. (2007). Putting philosophy to work: Coping with multiple theoretical perspectives. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 3–38). Charlotte, NC: Information Age.Google Scholar
  11. Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28, 258–277.CrossRefGoogle Scholar
  12. Cobb, P., & Glasersfeld, E. V. (1983). Piaget’s scheme and constructivism. Genetic Epistemology, 13(2), 9–15.Google Scholar
  13. Dewey, J. (1910). How we think. Boston, MA: D. C. Heath.CrossRefGoogle Scholar
  14. Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. Boston, MA: D. C. Heath.Google Scholar
  15. Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 85–106). Washington, DC: Mathematical Association of America.Google Scholar
  16. Dugdale, S., Wagner, L. J., & Kibbey, D. (1992). Visualizing polynomial functions: New insights from an old method in a new medium. Journal of Computers in Mathematics and Science Teaching, 11(2), 123–142.Google Scholar
  17. Ferrini-Mundy, J., Floden, R. E., McCrory, R., Burrill, G., & Sandow, D. (2005). Knowledge for teaching school algebra: Challenges in developing an analytic framework. East Lansing, MI: Michigan State University, Knowledge of Algebra for Teaching Project.Google Scholar
  18. Ferrini-Mundy, J., & Gauadard, M. (1992). Secondary school calculus: Preparation or pitfall in the study of college calculus. Journal for Research in Mathematics Education, 23, 56–71.CrossRefGoogle Scholar
  19. Ferrini-Mundy, J., & Graham, K. (1994). Research in calculus learning: Understanding of limits, derivatives, and integrals. In J. J. Kaput & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning. Washington, DC: Mathematical Association of America.Google Scholar
  20. Glasersfeld, E. v. (1995). Radical constructivism: A way of knowing and learning (studies in mathematics education). London, England: Falmer.CrossRefGoogle Scholar
  21. Glasersfeld, E. V. (1998, September). Scheme theory as a key to the learning paradox. Paper presented at the 15th Advanced Course, Archives Jean Piaget. Geneva, Switzerland.Google Scholar
  22. Grice, H. P. (1957). Meaning. Philosophical Review, 66, 377–388.CrossRefGoogle Scholar
  23. Hackworth, J. A. (1994). Calculus studentsunderstanding of rate. Unpublished Masters Thesis, San Diego State University, Department of Mathematical Sciences. Retrieved from http://pat-­
  24. Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool. Journal for Research in Mathematics Education, 19, 3–25.CrossRefGoogle Scholar
  25. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 65–97). New York, NY: Macmillan.Google Scholar
  26. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 3–20). Hillsdale, NJ: Erlbaum.Google Scholar
  27. Hiebert, J., Stigler, J. W., Jacobs, J. K., Givvin, K. B., Garnier, H., Smith, M., et al. (2005). Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS 1999 video study. Education Evaluation and Policy Analysis, 27, 111–132.CrossRefGoogle Scholar
  28. Hill, H. C. (2010). The nature and predictors of elementary teachers’ mathematical knowledge for teaching. Journal for Research in Mathematics Education, 41, 513–545.Google Scholar
  29. Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., et al. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26, 430–511.CrossRefGoogle Scholar
  30. Johnckheere, A., Mandelbrot, B. B., & Piaget, J. (1958). La lecture de l’expérience (observation and decoding of reality). Paris, France: P. U. F.Google Scholar
  31. Kaput, J. J. (1993). The urgent need for proleptic research in the graphical representation of quantitative relationships. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Integrating research in the graphical representation of functions (pp. 279–311). Hillsdale, NJ: Erlbaum.Google Scholar
  32. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Charlotte, NC: Information Age.Google Scholar
  33. Kilpatrick, J., Hoyles, C., Skovsmose, O., & Valero, P. (Eds.). (2005). Meaning in mathematics education (mathematics education library vol. 37). New York, NY: Springer.Google Scholar
  34. Lehrer, R., Schauble, L., Carpenter, S., & Penner, D. E. (2000). The interrelated development of inscriptions and conceptual understanding. In P. Cobb & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 325–360). Hillsdale, NJ: Erlbaum.Google Scholar
  35. Lortie, D. C. (1975). Schoolteacher. Chicago, IL: University of Chicago Press.Google Scholar
  36. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ knowledge of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.Google Scholar
  37. Machín, M. C., Rivero, R. D., & Santos-Trigo, M. (2010). Students’ use of Derive software in comprehending and making sense of definite integral and area concepts. In F. Hitt, D. Holton, & P. W. Thompson (Eds.), Research in collegiate mathematics education, 7, issues in ­mathematics education (Vol. 16, pp. 29–62). Providence, RI: American Mathematical Society.Google Scholar
  38. Montangero, J., & Maurice-Naville, D. (1997). Piaget or the advance of knowledge (A. Curnu-­Wells, Trans.). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  39. Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 69–100). Hillsdale, NJ: Erlbaum.Google Scholar
  40. National Mathematics Advisory Panel. (2008). Foundations for success: Final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Retrieved from
  41. Oehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ understandings of function. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and practice in undergraduate mathematics (pp. 27–42). Washington, DC: Mathematical Association of America.CrossRefGoogle Scholar
  42. Ogden, C. K., & Richards, I. A. (1923/1989). The meaning of meaning. Orlando, FL: Harcourt Brace.Google Scholar
  43. Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14, 235–250.CrossRefGoogle Scholar
  44. Pask, G. (1975). Conversation, cognition and learning: A cybernetic theory and methodology. Amsterdam: Elsevier.Google Scholar
  45. Pask, G. (1976). Conversation theory: Applications in education and epistemology. Amsterdam, The Netherlands: Elsevier.Google Scholar
  46. Percy, W. (1975a). The message in the bottle: How queer man is, how queer language is, and what one has to do with the other (Kindleth ed.). New York, NY: Open Road.Google Scholar
  47. Percy, W. (1975b). The delta factor. Sourthern Review, 11, 7–14.Google Scholar
  48. Piaget, J. (1968). Six psychological studies. New York, NY: Vintage Books.Google Scholar
  49. Piaget, J. (1977). Psychology and epistemology: Towards a theory of knowledge. New York, NY: Penguin.CrossRefGoogle Scholar
  50. Piaget, J. (1995). Sociological studies (T. Brown, R. Campbell, N. Emler, M. Ferrari, M. Gribetz, R. Kitchener et al., Trans.). New York, NY: Routledge.Google Scholar
  51. Piaget, J., & Garcia, R. (1991). Toward a logic of meanings. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  52. Piaget, J., & Inhelder, B. (1969). The psychology of the child. New York, NY: Basic Books.Google Scholar
  53. Putnam, H. (1973). Meaning and reference. Journal of Philosophy, 70, 699–711.CrossRefGoogle Scholar
  54. Putnam, H. (1975). The meaning of “meaning.” In Mind, language and reality. Philosophical papers (Vol. 2, pp. 215–271). Cambridge, UK: Cambridge University Press.Google Scholar
  55. Schmidt, W. H., Houang, R., & Cogan, L. S. (2002, Summer). A coherent curriculum: The case of mathematics. American Educator, 1–17. Retrieved from
  56. Schmidt, W. H., Wang, H. C., & McKnight, C. (2005). Curriculum coherence: An examination of US mathematics and science content standards from an international perspective. Journal of Curriculum Studies, 37, 525–559.CrossRefGoogle Scholar
  57. Scott, B. (2009). Conversation, individuals, and concepts: Some key concepts in Gordon Pask’s interaction of actors and conversation theories (electronic version). Constructivist Foundations, 4, 151–158. Retrieved from
  58. Simon, M. A., Tzur, R., Heinz, K., Kinzel, M., & Smith, M. S. (2000). Characterizing a perspective underlying the practice of mathematics teachers in transition. Journal for Research in Mathematics Education, 31, 579–601.CrossRefGoogle Scholar
  59. Skemp, R. (1961). Reflective intelligence and mathematics. The British Journal of Educational Psychology, 31, 44–55.CrossRefGoogle Scholar
  60. Skemp, R. (1962). The need for a schematic learning theory. The British Journal of Educational Psychology, 32, 133–142.CrossRefGoogle Scholar
  61. Skemp, R. (1979). Intelligence, learning, and action. New York, NY: Wiley.Google Scholar
  62. Sofronos, K. S., & DeFranco, T. C. (2010). An examination of the knowledge base for teaching among mathematics faculty teaching calculus in higher education. In F. Hitt, D. Holton, & P. W. Thompson (Eds.), Research in collegiate mathematics education, 7, issues in mathematics education (Vol. 16, pp. 171–206). Providence, RI: American Mathematical Society.Google Scholar
  63. Steffe, L. P., & Thompson, P. W. (2000). Interaction or intersubjectivity? A reply to Lerman. Journal for Research in Mathematics Education, 31, 191–209.CrossRefGoogle Scholar
  64. Stigler, J. W., Gonzales, P., Kawanaka, T., Knoll, S., & Serrano, A. (1999). The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States (National Center for Education Statistics Report, Number NCES 99-0974). Washington, DC: U.S. Government Printing Office.Google Scholar
  65. Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York, NY: Free Press.Google Scholar
  66. Stroud, C. (2010). Students’ understandings of instantaneous rate of change: Report of a pilot study. Tempe, AZ: School of Mathmatics and Statistics, Arizona State University.Google Scholar
  67. Thompson, P. W. (1987). Mathematical microworlds and intelligent computer-assisted instruction. In G. Kearsley (Ed.), Artificial intelligence and education (pp. 83–109). New York, NY: Addison-Wesley.Google Scholar
  68. Thompson, P. W. (1989). Artificial intelligence, advanced technology, and learning and teaching algebra. In C. Kieran & S. Wagner (Eds.), Research issues in the learning and teaching of algebra (pp. 135–161). Hillsdale, NJ: Erlbaum.Google Scholar
  69. Thompson, P. W. (1994a). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229–274.CrossRefGoogle Scholar
  70. Thompson, P. W. (1994b). Students, functions, and the undergraduate mathematics curriculum. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), Research in collegiate mathematics education, 1, issues in mathematics education (Vol. 4, pp. 21–44). Providence, RI: American Mathematical Society.Google Scholar
  71. Thompson, P. W. (1994c). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179–234). Albany, NY: SUNY.Google Scholar
  72. Thompson, P. W. (1996). Imagery and the development of mathematical reasoning. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 267–283). Hillsdale, NJ: Erlbaum.Google Scholar
  73. Thompson, P. W. (2000). Radical constructivism: Reflections and directions. In L. P. Steffe & P. W. Thompson (Eds.), Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld (pp. 412–448). London, England: Falmer.Google Scholar
  74. Thompson, P. W. (2002). Didactic objects and didactic models in radical constructivism. In K. Gravemeijer, R. Lehrer, B. V. Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 197–220). Dordrect, The Netherlands: Kluwer.Google Scholar
  75. Thompson, P. W. (2008a). On professional judgment and the National Mathematics Advisory Panel Report. Educational Researcher, 38, 582–587.CrossRefGoogle Scholar
  76. Thompson, P. W. (2008b). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sépulveda (Eds.), Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 45–64). Morélia, Mexico: PME. Retrieved from
  77. Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education, WISDOMe monographs (Vol. 1, pp. 33–57). Laramie, WY: University of Wyoming.Google Scholar
  78. Thompson, P. W., Carlson, M. P., & Silverman, J. (2007). The design of tasks in support of teachers’ development of coherent mathematical meanings. Journal of Mathematics Teacher Education, 10, 415–432.CrossRefGoogle Scholar
  79. Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), Research companion to the principles and standards for school mathematics (pp. 95–114). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  80. Thompson, P. W., & Thompson, A. G. (1992, April). Images of rate. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA. Retrieved from
  81. Thompson, A. G., & Thompson, P. W. (1996). Talking about rates conceptually, part II: Mathematical knowledge for teaching. Journal for Research in Mathematics Education, 27, 2–24.CrossRefGoogle Scholar
  82. Tucker, M. S. (2011). Standing on the shoulders of giants: An American agenda for education reform. Washington, DC: National Center on Education and the Economy.Google Scholar
  83. White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27, 79–95.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Arizona State UniversityTempeUSA

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