Advertisement

Exponential Functions, Rates of Change, and the Multiplicative Unit

Chapter

Abstract

Conventional treatments of functions start by building a rule of correspondence between x-values and y-values, typically by creating an equation of the form y = f (x). We call this a correspondence approach to functions. However, in our work with students we have found that a covariational appraoch is often more powerful, where students working in a problem situation first fill down a table column with x-values, typically by adding 1, then fill down a y-column through an operation they construct within the problem context. Such an approach has the benefit of emphasizing rate-of-change. It also raises the question of what it is that we want to cal ‘rate’ across different functional situations. We make two initial conjectures, first that a rate can be initially understood as a unit per unit comparison and second that a unit is the invariant relationship between a successor and its predecessor. Based on these conjectures we describe a variety of multiplicative units, then propose three ways of understanding rate of change in relation to exponential functions. Finally we argue that rate is different than ratio and that an integrated understanding of rate is built from multiple concepts.

Keywords

Exponential Function Function Concept Standard Unit Rate Concept Additive Unit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Afamasaga-Fuata’i, K.: 1991, Students’ Strategies for Solving Contextual Problems on Quadratic Functions, unpublished doctoral dissertation. Cornell University, Ithaca, NY.Google Scholar
  2. Behr, M., Lesh, R., Post, T. and Silver, E.: 1983, ‘Rational-number concepts’, in R. Lesh and M. Landau (eds.), Acquisition of Mathematics Concepts and Processes, New York, pp. 91–126.Google Scholar
  3. Borba, M.: 1993, Students’ Understanding of Transformations of Functions Using Multi- Representational Software, unpublished doctoral dissertation, Cornell University, Ithaca, NY.Google Scholar
  4. Borba, M. and Confrey, J.: 1992, ‘Transformations of functions using multi-representational software: Visualization and discrete points’, a paper presented at the Sixteenth Annual Meeting of Psychology of Mathematics Education-NA, Durham, p. 149.Google Scholar
  5. Boyer, C. B.: 1968, A History of Mathematics,New York.Google Scholar
  6. Carlstrom, K.: 1992, Units, Ratios, and Dimensions: Students’ Constructions of Multiplicative Worlds in a Computer Environment, unpublished thesis, Cornell University, Ithaca, NY.Google Scholar
  7. Confrey, J.: 1990, ‘Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions’, a paper presented at the annual meeting of the American Educational Research Association, Boston.Google Scholar
  8. Confrey, J.: 199la, ‘The concept of exponential functions: A student’s perspective’, in L. Steife (ed.), Epistemological Foundations of Mathematical Experience,New York, pp. 124–159.Google Scholar
  9. Confrey, J.: 1991b, Function Probe©[Computer Program], Santa Barbara.Google Scholar
  10. Confrey, J.: 1991c, ‘Learning to listen: A student’s understanding of powers of ten’, in E. von Glasersfeld (ed.), Radical Constructivism in Mathematics Education, Dordrecht, pp; 111–138.Google Scholar
  11. Confrey, J.: 1992, ‘Using computers to promote students’ inventions on the function concept’, in S. Malcom, L. Roberts and K. Sheingold (eds.), This Year in School Science 1991: Technology for Teaching and Learning,Washington, DC, pp. 141–174.Google Scholar
  12. Confrey, J.: 1993, ‘Learning to see children’s mathematics: Crucial challenges in constructivist reform’, in K. Tobin (ed.), Constructivist Perspectives in Science and Mathematics, Washington, DC: American Association for the Advancement of Science, pp. 299–321.Google Scholar
  13. Confrey, J., in press a, ‘Voice and perspective: Hearing epistemological innovation in students’ words’, in N. Bednarz, M. Larochelle and J. Desautels (eds.), Revue des sciences de l’éducation,Special Issue.Google Scholar
  14. Confrey, J.: in press b, ‘Splitting, similarity, and rate of change: New approaches to multiplication and exponential functions’, in G. Harel and J. Confrey (eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics,Albany.Google Scholar
  15. Confrey, J. and Smith, E.: 1989, ‘Alternative representations of ratio: The Greek concept of anthyphairesis and modem decimal notation’, a paper presented at the First Annual Conference of The History and Philosophy of Science in Science Teaching, Tallahassee, pp. 71–82.Google Scholar
  16. Confrey, J., Smith, E., Piliero, S. and Rizzuti, J.: 1991, ‘The use of contextual problems and multi-representational software to teach the concept of functions’, Final Project Report to the National Science Foundation (MDR-8652160) and Apple Computer, Inc.Google Scholar
  17. Davis, R.: 1988, ‘Is percent a number?, Journal for Mathematical Behavior 7, 299–302.Google Scholar
  18. Dienes, Z. P.: 1967, Fractions: An Operational Approach,New York.Google Scholar
  19. Dubinsky, E. and Harel, G.: 1992, ‘The nature of the process conception of function’, in G. Haret and E. Dubinsky (eds.), The Concept of Function (MAA Notes V. 25 ), Washington, DC.Google Scholar
  20. Greer, B.: 1988, ‘Nonconservation of multiplication and division: Analysis of a symptom’, The Journal of Mathematical Behavior 7 (3), 281–298.Google Scholar
  21. Kaput, J., Luke, C., Poholsky, J. and Sayer, A.: 1986, ‘The role of representation in reasoning with intensive quantities: Preliminary analyses’, Educational Technology Center Tech. Report 869, Cambridge.Google Scholar
  22. Kaput, J. and West, M.: in press, ‘Missing value proportional reasoning problems: Factors affecting informal reasoning patterns’, in G. Harel and J. Confrey (eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics,Albany.Google Scholar
  23. NCTM: 1989, Curriculum and Evaluation Standards for School Mathematics,Reston.Google Scholar
  24. Nemirovsky, R.: 1991, ‘Notes about the relationship between the history and the constructive learning of calculus’, Proceedings of the Segundo Simposia Internacional sobre Investigacion en Educacion Matematica, Universidad Autonoma del Estado va Mexico, Cuernavaca, pp. 37–54.Google Scholar
  25. Nemirovsky, R. and Rubin, A.: 1991, ‘It makes sense if you think about how graphs work, but in reality…’, in F. Furinghetti (ed.), Proceedings of the Fifteenth PME Conference, Assisi, pp. 57–64.Google Scholar
  26. Oresme, N.: 1966, ‘De proportionibus proportionum’, in Edward Grant (ed. and trans.), De proportionibus proportionum and Ad pauca respicientes, Madison.Google Scholar
  27. Pothier, Y. and Sawada, D.: 1983, ‘Partitioning: The emergence of rational number ideas in young children’, Journal for Research in Mathematics Education 14 (4), 307–317.CrossRefGoogle Scholar
  28. Rizzuti, J.: 1991, High School Students’ Uses of Multiple Representations in the Conceptualization of Linear and Exponential Functions, unpublished doctoral dissertation, Cornell University, Ithaca, NY.Google Scholar
  29. Rizzuti, J. and Confrey, J.: 1988, ‘A construction of the concept of exponential functions’, in M. Behr, C. LaCompagne and M. Wheeler (eds.), Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Dekalb, pp. 259–268.Google Scholar
  30. Rubin, A.: 1992, The Patterns of Quilts, unpublished manuscript, Technical Education Research Center, Cambridge.Google Scholar
  31. Schom, A. C.: 1989, Proportional Reasoning by Young Children, unpublished thesis, Cornell University, Ithaca, NY.Google Scholar
  32. Schwartz, J.: 1988, ‘Intensive quantity and referent transforming arithmetic operations’, in M. Behr and J. Hiebert (eds.), Number Concepts and Operations in the Middle Grades, Reston, pp. 41–52.Google Scholar
  33. Smith, E. and Confrey, J.: 1989, ‘Ratio as construction: ratio and proportion in the mathematics of ancient Greece’, a paper presented at the annual meeting of the American Educational Research Association, San Francisco.Google Scholar
  34. Smith, E. and Confrey, J.: 1992, ‘Using a dynamic software tool to teach transformations of functions’, in L. Lum (ed.), a paper presented at the Fifth Annual International Conference on Technology in Collegiate Mathematics, Reading.Google Scholar
  35. Smith, E. and Confrey, J.: in press, ‘Multiplicative structures and the development of logarithms: What was lost by the invention of function?’, in G. Harel and J. Confrey (eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics,Albany.Google Scholar
  36. Steife, L.: 1988, ‘Children’s construction of number sequences and multiplying schemes’, in M. Behr and J. Hiebert (eds.), Number Concepts and Operations in the Middle Grades, Reston, pp. 119–141.Google Scholar
  37. Steffe, L.: 1991, ‘The constructivist teaching experiment: Illustrations and implications’, in E. von Glasersfeld (ed.), Radical Constructivism in Mathematics Education, Dordrecht, pp. 177–194.Google Scholar
  38. Steife, L.: in press, ‘Children’s multiplying and dividing schemes’, in G. Haret and J. Confrey (eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics,Albany.Google Scholar
  39. Thompson, P: 1990, ‘The development of the concept of speed and its relationship to concepts of rate’, a paper presented at the annual meeting of the American Educational Research Association, Boston.Google Scholar
  40. Thompson, P.: in press, ‘The development of the concept of speed and its relationship to concepts of rate’, in G. Haret and J. Confrey (eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics,Albany.Google Scholar
  41. Tierney, C. and Nemirovsky, R.: 1991, ‘Young children’s spontaneous representations of changes in population and speed’, in R. Underhill (ed.), Proceedings of the Thirteenth Annual Meeting of the NA-PME, Blacksburg, pp. 182–188.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  1. 1.Department of Education Mathematics EducationCornell UniversityIthacaUSA

Personalised recommendations