Understanding Qualitative Calculus: A Structural Synthesis of Learning Research

  • Walter M. Stroup


More than a decade of research and innovation in using computer-based graphing and simulation environments has encouraged many of us in the research community to believe important dimensions of calculus-related reasoning can be successfully understood by young learners. This paper attempts to address what kinds of calculus-related insights seem to typify this early form of calculus reasoning. The phrase “qualitative calculus” is introduced to frame the analysis of this “other” calculus. The learning of qualitative calculus is the focus of the synthesis. The central claim is that qualitative calculus is a cognitive structure in its own right and that qualitative calculus develops or evolves in ways that seem to fit with important general features of Piaget's analyses of the development of operational thought. In particular, the intensification of rate and two kinds of reversibility between what are called “how much” (amount) and “how fast” (rate) quantities are what interactively, and collectively,characterize and help to define understanding qualitative calculus. Although sharing a family resemblance with traditional expectations of what it might mean to learn calculus, qualitative calculus does not build from ratio- or proportion-based ideas of slope as they are typically associated with defining rate. The paper does close, however, with a discussion of how understanding qualitative calculus can support and link to the rate-related literature of slope, ratio and proportion. Additionally, curricular connections and implications are discussed throughout to help illustrate and explore the significance of learning qualitative calculus.

calculus computer simulation modeling Piaget qualitative calculus rate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Artigue, M. (1991). Analysis. In D. Tall (Ed.), Advanced Mathematical Thinking. Boston: Kluwer Adademic Publishers.Google Scholar
  2. Barclay, W.L. (1986a). A graph is worth how many words? Classroom Computer Learning (Feb.): 46–50.Google Scholar
  3. Barclay, W.L. (1986b). Graphing Misconceptions and Possible Remedies Using Microcomputer-based Labs. National Educational Computing Conference (NECC): NECC.Google Scholar
  4. Bell, A. and Janvier, C. (1981). The interpretation of graphs representing situations. For the learning of mathematics 2(1): 34–42.Google Scholar
  5. Booth, L. (1981). Graphs in mathematics and science. Mathematics in School 10(4): 205.Google Scholar
  6. Boyer, C. (1969). The history of calculus. Historical Topics for the Mathematics Classroom, 31st Yearbook, National Council of Teachers of Mathematics, 1969.Google Scholar
  7. Brekke, G. (1988). Graphical Interpretation: A Study of Pupils’ Understanding. Shell Centre.Google Scholar
  8. Brasell, H. (1987). The effect of real-time laboratory graphing on learning graphic representations of distance and velocity. Journal of Research in Science Teaching 24(4): 385–395.Google Scholar
  9. Buckingham, D. and Shultz, T.R. (2000). The developmental course of distance, time, and velocity concepts: A generative connectionist model. Journal of Cognition and Development 1: 305–345.CrossRefGoogle Scholar
  10. Chen, D. and Stroup, W. (1993). General systems theory: Toward a conceptual framework for science and technology education for all. Journal of Science Education and Technology 2(3): 447–459.CrossRefGoogle Scholar
  11. Clement, J. (1985). Misconceptions in Graphing. Noordwijkerhout, The Netherlands: IGPME.Google Scholar
  12. Clement, J., Mokros, J.R. and Schultz, K. (1985). Adolescents’ Graphing Skills: a descriptive analysis. AERA.Google Scholar
  13. Collins, J.D. (1991). Communicating with graphs: are you sure? Mathematics in School 20(2): 18–19.Google Scholar
  14. Confrey, J. and Smith, E. (1989). Alternative representations of ratio: The Greek concept of anthyphairesis and modern decimal notation. In D.E. Herget (Ed.), The History and Philosophy of Science in Science Teaching.Google Scholar
  15. Confrey, J. and Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. In P. Cobb (Ed.), Learning Mathematics: Constructivist and Interactionist Theories of Mathematical Development. Boston: Kluwer Academic Publishers.Google Scholar
  16. Dennis, D. (1995). Historical Perspectives for the Reform of Mathematics Curriculum: Geometric Curve Drawing Devices and Their Role in the Transition to an Algebraic Description of Functions. Ph.D. Thesis (Math Ed), Cornell University 1995.Google Scholar
  17. Dennis, D. and Confrey, J. (1997). Drawing Logarithmic and Exponential Curves with the Computer Software Geometer's Sketchpad: A Method Inspired by Historical Sources. In J. King and D. Schatschneider (Eds.), Geometry Turned On: Dynamic Software in Learning, Teaching and Research (pp. 147–156). Washington D.C.: Mathematical Association of America.Google Scholar
  18. Dienes, Z.P. (1971). The Elements of Mathematics. New York: Herder and Herder.Google Scholar
  19. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical Thinking. Boston: Kluwer Adademic Publishers.Google Scholar
  20. Dugdale, S. (1993). Functions and graphs ‐ Perspectives on student thinking. In T.A. Romberg, E. Fennema and T.P. Carpenter (Eds), Integrating Research on the Graphical Representation of Functions (pp. 101–130). Hillsdale, New Jersey: Lawrence Erlabaum Associates.Google Scholar
  21. Dunham, P.H. and Osborne, A. (1991). Learning how to see: Students graphing difficulties. Focus on Learning Problems in Mathematics 13(4): 35–49.Google Scholar
  22. Forbus, K. (1984). Qualitative process theory. Artificial Intelligence 24: 85–168.CrossRefGoogle Scholar
  23. Forbus, K. (1996). Qualitative Reasoning. CRC Handbook of Computer Science and Engineering. CRC Press.Google Scholar
  24. Goldenberg, E.P. (1987). Believing is Seeing: How Preconceptions Influence the Perception of Graphs. Montreal, Canada.Google Scholar
  25. Goldenberg, E.P., Harvey, W., Lewis, P.G., Umiker, R.J., West, J. and Zodhiates (1988). Mathematical, Technical, and Pedagogical Challenges in the Graphical Representations of Functions (88-4). Educational Technology Center, Harvard Graduate School of Education.Google Scholar
  26. Goldberg, F.M. and Anderson, J.H. (1989). Student difficulties with graphical representations of negative values of velocity. The Physics Teacher: 254–260.Google Scholar
  27. Goldenberg, F.M. and Ferrini-Mundy, J. (1989). Student difficulties with graphical representations of negative values of velocity. The Physics Teacher (April): 254–260.Google Scholar
  28. Harel, G. and Confrey, J. (1994). The Development of Multiplicative Reasoning the Learning of Mathematics. Albany NY: State Univ. of New York.Google Scholar
  29. Huetinck, L. (1992). Laboratory connections: Understanding graphing through microcomputer-based laboratories. Journal of Computers in Mathematics and Science Teaching 11(1): 95–100.Google Scholar
  30. Hughes-Hallett, D. (1992). Calculus. New York: John Wiley & Sons, Inc.Google Scholar
  31. Kaput, J. (1986). Information technology and mathematics: Opening new representational windows. Journal for Research in Mathematical Behavior 5: 187–207.Google Scholar
  32. Kaput, J.J. (1988). Supporting Concrete Visual Thinking in Multiplicative Reasoning: Difficulties and Opportunities (Technical Report No. 88-16). Educational Technology Center.Google Scholar
  33. Kaput, J. (1994). Democratizing access to calculus. In A. Schoenfeld (Ed.), Mathematical Thinking and Problem Solving (pp. 77–156). Hillsdale: Erlbaum.Google Scholar
  34. Kaput, J. and Roschelle, J. (1996). Math Worlds. Dartmouth MA: SimCalc. Project, University of MassachusettsGoogle Scholar
  35. Kaput, J.J., Schwartz, Judah L., Poholsky and Joel S. (1985). Extensive and intensive quantities in multiplication and division word problems: A preliminary report and a software response. In S. K. D. a. M. Selton (Ed.), Seventh Annual Meeting of the PME-NA (pp. 139–144). Columbus, Ohio.Google Scholar
  36. Kaput, J.J., West, M.M., Luke, C. and Pattison-Gordon, L. (1988). Concrete representations for ratio reasoning. In Tenth Annual Meeting of PME-NA (pp. 93–99). Northern Illinois University.Google Scholar
  37. Kieran, C. (1993). Functions, graphing and technology: Integrating research on learning and instruction. In T.A. Romberg, E. Fennema and T.P. Carpenter (Eds), Integrating Research on the Graphical Representation of Functions (pp. 189–278). Hillsdale NJ: Lawrence Erlbaum Associates.Google Scholar
  38. Kuipers, B. (1994). Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge. Cambridge MA: MIT Press.Google Scholar
  39. Krabbendum, H. (1982). The Non-Quantitative Way of Describing Relations and the Role of Graphs. Enschede, The Netherlands: National Institute for Curriculum Development.Google Scholar
  40. Lapp, D.A. and Cyrus, V.F. (2000). Using Data Collection Devices to Enhance Student Understanding. Google Scholar
  41. Leinhardt, G., Zaslavsky, O. and Stein, M.K. (1990). Functions, graphs, and graphing: Tasks, learning and teaching. Review of Educational Research 60(1): 1–64.CrossRefGoogle Scholar
  42. Linn, M., Layman, J. and Nachmias, R. (1987). Cognitive consequences of microcomputerbased laboratories: graphing skills development. Contemporary Educational Psychology 12: 244–253.CrossRefGoogle Scholar
  43. McCloskey, M. (1983). Naive theories of motion. In D. Gentner and A.L. Stevens (Eds), Mental Models. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  44. McDermott, L., Rosenquist, M. and vanZee, E. (1987). Student difficulties in connecting graphs and physics: Example from kinematics. American Journal of Physics 55: 503–513.CrossRefGoogle Scholar
  45. Michael, V. (1989). Graphical analysis. Science Teacher 56(7): 37–40.Google Scholar
  46. Mokros, J. (1985). The Impact of Microcomputer-based Science Labs on Children's Graphing skills (85-2). TERC.Google Scholar
  47. Mokros, J. & Tinker, R. (1987). The impact of microcomputer-based science labs on children's ability to interpret graphs. Journal of Research in Science Teaching 24(4): 369–383.Google Scholar
  48. Monk, G. (1989). Student understanding of function as a foundation for calculus curriculum development. In AERA. San Francisco.Google Scholar
  49. Monk, G. (1992). A study of calculus students’ constructions of functional situations: the case of the shadow problem. In American Educational Research Association. San Francisco.Google Scholar
  50. Moschkovich, J., Schoenfeld, A.J. and Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T.A. Romberg, E. Fennema and T. P. Carpenter (Eds), Integrating Research on the Graphical Representation of Functions (pp. 69–100). Hillsdale, New Jersey: Lawrence Erlabaum Associates.Google Scholar
  51. Nachmias, R. and Linn, M. (1987). Evaluations of science laboratory data: The role of computer-presented information. Journal of Research in Science Teaching 24(5): 491–506.Google Scholar
  52. Nemirovsky, R. (1992). On the Basic Types of Variation. TERC, 2067 Massachusetts Avenue, Cambridge, MA 02140.Google Scholar
  53. Nemirovsky, R. (1993). Symbolizing Motion, Flow and Contours: The Experience of Continuous Change. Doctorate: Harvard Graduate School of Education.Google Scholar
  54. Nemirovsky, R. and Rubin, A. (1992). Students’ Tendency to Assume Resemblances Between a Function and its Derivative (2-92). TERC, 2067 Massachusetts Avenue, Cambridge, MA 02140.Google Scholar
  55. Nemirovsky, R., Tierney, C. and Wright, T. (1998). Body motion and graphing. Cognition and Instruction 16(2): 119–172.CrossRefGoogle Scholar
  56. Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics 14(1983): 235–250.CrossRefGoogle Scholar
  57. Papert, S. (1980). Mindstorms: Children, Computers, and Powerful Ideas. NewYork: Basic Books.Google Scholar
  58. Peterson, S. and Richmond, B. (1992). STELLA II. Hanover NH: High Performance Systems, Inc.Google Scholar
  59. Phillips, R (1990). Everyday Graphs. Published by the Shell Centre.Google Scholar
  60. Piaget, J. (1946:1970). The Child's Conception of Movement and Speed. New York: Basic Books, Inc.Google Scholar
  61. Piaget, J. (1968:1970). Structuralism. New York: Basic Books, Inc.Google Scholar
  62. Rubin, A. and Nemirovsky, R. (1991). Cars, Computers, and Air Pumps: Thoughts on the Roles of Physical and Computer Models in Learning the Central Concepts of Calculus. Virginia USA.Google Scholar
  63. Sawyer, W. W. (1999). What is Calculus About? The Mathematical Association of America.Google Scholar
  64. Schultz, K., Clement, J. and Mokros, J. (1986). Adolescent Graphing Skills: A Descriptive Analysis. San Francisco, CA: AERA.Google Scholar
  65. Schwartz, J.L. (1988). Intensive quantity and referent-transforming operations. In J. Hiebert and M. Behr (Eds), Number Concepts and Operations in the Middle Grades (pp. 41–52). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  66. Schwartz, J.L. (1991). Personal communication.Google Scholar
  67. Schwartz, J.L. (1994). Personal communication.Google Scholar
  68. Schwartz, J.L. (1995). Semantic Aspects of Quantity (Unpublished).Google Scholar
  69. Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics 22: 1–36.CrossRefGoogle Scholar
  70. Silberstein, E.P. (1986). Graphically speaking. Science Teacher 53(5): 41–45.Google Scholar
  71. Stein, J.S. (1986). Evaluating the Classroom Effectiveness of Microcomputer-based Labs. San Francisco, CA: AERA.Google Scholar
  72. Stroup, W. (1991, 1994). Kine-Mouse Software (Unpublished).Google Scholar
  73. Stroup, W. (1991, 1994). Vis Viva: A Conceptual Investigation of Energy and Entropy (Unpublished).Google Scholar
  74. Stroup, W. (1993). Kine-Calc Software (Unpublished).Google Scholar
  75. Stroup, W. (1994). What the Development of Non-universal Understanding Looks Like: An Investigation of Results from a Series of Qualitative Calculus Assessments. Havard Graduate School of Education.Google Scholar
  76. Stroup, W. (1995). Dynamics and calculus for the young learner. Connect 8(5): 16–18.Google Scholar
  77. Stroup, W. (1996). Embodying a Nominalist Constructivism: Making Graphical Sense of Learning the Calculus of How Much and How Fast. Dissertation, Harvard Graduate School of Education.Google Scholar
  78. Stroup, W. (2000). Learning Entropy and Energy Project (Proposal). CAREER Award 0093093 (National Science Foundation, Washington, D.C.).Google Scholar
  79. Swan, M. and the Shell Centre Team (1985). The Language of Functions and Graphs (The Red Box Materials). Published by Shell Centre & Joint Matriculation Board 1985.Google Scholar
  80. Tall, D.O. (1986a). Building and Testing a Cognitive Approach to the Calculus Using Computer. Mathematics Education Research Centre, University of Warwick.Google Scholar
  81. Tall, D.O. (1986b). Using the Computer to Represent Calculus Concepts (pp. 238–264). Orleans: IMAG Grenoble.Google Scholar
  82. Thompson, P. (1994a). The development of the concept of speed and its relationship to concepts of rate. In G.H. & J. Confrey (Eds), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 179–234). Albany, NY: SUNY.Google Scholar
  83. Thompson, P. (1994b). Images of rate and operational understanding of the fundamental theorem of calculus. In P. Cobb (Ed.), Learning Mathematics Constructivist and Interactionist Theories of Mathematical Development (pp. 125–170). Boston: Kluwer Academic Publishers.Google Scholar
  84. Thompson, P. and Thompson, A. (1996). Talking about rates conceptually, part II: Mathematical knowledge for teaching. Journal for Research in Mathematics 27: 2–24.CrossRefGoogle Scholar
  85. Thornton, R. (1987). Tools for scientific thinking: Microcomputer-based laboratories for physics teaching. Physics Education 22(4): 230–238.CrossRefGoogle Scholar
  86. Thornton, R.K. and Sokoloff, D.R. (1990). Learning motion concepts using real-time micorcomputer-based laboratory tools. American Journal of Physics 58(September 1990): 858–867.CrossRefGoogle Scholar
  87. Tinker, R. (1984). Microcomputers in the Lab: Techniques and Applications. Cambridge MA: Technical Education Research Center.Google Scholar
  88. Tierney, C. Nemirovsky, R. and Noble, T. (1995). Patterns of change: Walks, tables and graphs. Curricular Unit for Grade 5. California: Dale Seymour Publications.Google Scholar
  89. Trowbridge, D.E. and McDermott, L.C. (1980). Investigation of student understanding of the concept of velocity in one dimension. American Journal of Physics 48; 1020–1028.CrossRefGoogle Scholar
  90. Wilensky, U. et al. (2000). NetLogo. Scholar
  91. Wittgenstein, L. (1958). Philosophical Investigations. New York: Macmillan Publishing Co., Inc.Google Scholar
  92. Yerushalmy, M. and Schwartz, J.L. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T.A. Romberg, E. Fennema and T.P. Carpenter (Eds), Integrating Research on the Graphical Representation of Functions (pp. 41–68). Hillsdale NJ: Lawrence Erlbaum Erlbaum Associates.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Walter M. Stroup
    • 1
  1. 1.The Center for Generative LearningUniversity of UtahSalt Lake CityUSA

Personalised recommendations