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Towards a Didactic Model for Assessment Design in Mathematics Education

  • Marja van den Heuvel-Panhuizen
  • Jerry Becker
Part of the Springer International Handbooks of Education book series (SIHE, volume 10)

Abstract

In this chapter we compare and contrast the didactic and psychometric models of assessment design. We begin with background information and an introduction to assessment, and then the relationship between the goals of mathematics education and assessment is presented. This relationship is made explicit by exhibiting two assessment problems, each reflecting a different way of thinking about mathematics education. The primary objections to the current way of assessing students, and alternatives that have been developed for it, are next presented. We then present a theoretical perspective on the psychometric approach, followed by some recent developments regarding this assessment model Here we arrive at the crucial issue in this chapter, namely, the need to extend assessment design in mathematics education via a didactic model. The consequences of the psychometric requirements and assumptions for mathematics education are presented and discussed. Importantly, we discuss how we can overcome certain misconceptions that inhibit an approach to assessment which is aligned with the new way of thinking about mathematics education. Finally, we present examples of assessment problems that represent a didactic model of assessment design. The chapter concludes with a reference to Freudenthal and his concept of mathematics as a human activity. The approach to assessment in the chapter, and the examples that exemplify it, are identified as consistent with Freudenthal ’s thinking.

Keywords

Mathematics Education Test Item Classroom Teacher Local Independence Classroom Assessment 
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.

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References

  1. American Educational Research Association/American Psychological Association/National Council on Measurement in Education (1999). Standards for educational and psychological testing. Washington, D. C.: American Educational Research Association.Google Scholar
  2. Apple, M. (1995). Taking power seriously: New directions in equity in mathematics and beyond. In W. G. Secada, E. Fennema & L. B. Adajian (Eds.), New directions for equity in mathematics education (pp. 329–348). Cambridge, UK: Cambridge University Press.Google Scholar
  3. Baker, E. L., O’Neil Jr, H. F., & Linn, R. L. (1993). Policy and validity prospects for performance-based assessment. American Psychologist, 48(12), 1210–1218.CrossRefGoogle Scholar
  4. Baxter, G. P., & Junker, B. (2001, August). Designing cognitive-developmental assessments: A case study in proportional reasoning. Paper presented at the annual meeting of the National Council for Measurement in Education, April, Seattle, Washington.Google Scholar
  5. Becker, J. P., & Seiter, C. (1996). Elementary school practices. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International handbook of mathematics education (pp. 511–564). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  6. Becker, J. P., & Shimada, S. (1997). The open-ended approach — A new proposal for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  7. Berlak, H., Newmann, F. M., Adams, E., Archbald, D. A., Burgess, T., Raven, J., & Romberg, T. A. (1992). Toward a new science of educational testing and assessment. Albany, NJ: SUNY Press.Google Scholar
  8. Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan, 80(2), 139–148.Google Scholar
  9. Buros, O. (1977). Fifty years in testing: Some reminiscences, criticisms, and suggestions. Educational Researcher, 6(1), 9–15.Google Scholar
  10. Casas, F. R., & Meaghan, D. E. (2001). Renewing the debate over the use of standardized testing in the evaluation of learning and teaching. Interchange, 32(2), 147–181.CrossRefGoogle Scholar
  11. Cizek, G. J. (1993a). Rethinking psychometricians’ beliefs about learning. Educational Researcher, 22(4), 4–9.Google Scholar
  12. Cizek, G. J. (1993b). The place of psychometricians’ beliefs in educational reform: A rejoinder to Shepard. Educational Researcher, 22(4), 14–15.Google Scholar
  13. Clarke, D. (1996). Assessment. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International handbook of mathematics education (pp. 327–370). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  14. Clements, M. A. (1999). Education policy, education research, and the development of mathematics teachers. In N. F. Eilerton (Ed.), Mathematics teacher development: International perspectives (pp. 217–246). Perth, Western Australia: Meridian Press.Google Scholar
  15. Clements, M. A., & Eilerton, N. F. (1996). Mathematics education research: Past, present and future. Bangkok: UNESCO.Google Scholar
  16. Collis, K. F. (1992). Curriculum and assessment: A basic cognitive model. In G. C. Leder (Ed.), Assessment and learning of mathematics (pp. 24–45). Hawthorn, Victoria: Australian Council for Educational Research.Google Scholar
  17. Collison, J. (1992). Using performance assessment to determine mathematical dispositions. Arithmetic Teacher, 39(6), 40–47.Google Scholar
  18. De Corte, E., Greer, B., & Verschaffel, L. (1996). Mathematics teaching and learning. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology. New York: Simon & Schuster/Macmillan.Google Scholar
  19. De Lange, J. (1987). Mathematics, insight and meaning. Utrecht, The Netherlands: OW&OC, Utrecht University.Google Scholar
  20. De Lange, J. (1992). Critical factors for real changes in mathematics learning. In G. C. Leder (Ed.), Assessment and learning of mathematics (pp. 305–329). Hawthorn, Victoria: Australian Council for Educational Research.Google Scholar
  21. De Lange, J. (1999). Framework for classroom assessment in mathematics. Madison, WI: NICLA/WCER.Google Scholar
  22. Elmore, R. F. (2002). Testing trap: The single largest — and possibly most destructive — Federal intrusion into America’s public schools. Harvard Magazine (Alumni), 105(1), 35.Google Scholar
  23. Fiske, E. B. (1997, May 1). America’s test mania. New York Times (Education Supplement), 19.Google Scholar
  24. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, The Netherlands: Reidel Publishing Company.Google Scholar
  25. Froese-Germain, B. (2001). Standardized testing + High-stakes decisions = Educational inequity. Interchange, 32(2), 111–130.CrossRefGoogle Scholar
  26. Glaser, R. (1986). The integration of instruction and testing. In The redesigning of testing for the 21st century. Proceedings from the 6th Annual Invitational Conference of the Educational Testing Service, 26 October 1985.Google Scholar
  27. Grouws, D. A., & Meier, S. L. (1992). Teaching and assessment relationships in mathematics instruction. In G. C. Leder (Ed.), Assessment and learning of mathematics (pp. 83–106). Hawthorn, Victoria: Australian Council for Educational Research.Google Scholar
  28. Haney, W., & Madaus, G. (1989). Searching for alternatives to standardized tests: Whys, whats, and whithers. Phi Delta Kappan, 70, 683–687.Google Scholar
  29. Hashimoto, Y., & Becker, J. P. (1999). The open approach to teaching mathematics — Creating a culture of mathematics in the classroom: Japan. In L. J. Sheffield (Ed.), Developing mathematically promising students (pp. 101–119). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  30. Herman, J. L., & Winters, L. (1994). Portfolio research: A slim collection. Educational Leadership, 52(2), 48–55.Google Scholar
  31. Keitel, C, & Kilpatrick, J. (1999). The rationality and irrationality of international comparative studies. In G. Kaiser, E. Luna & I. Huntley (Eds.), International comparisons in mathematics education (pp. 241–256). London: Falmer Press.Google Scholar
  32. Kilpatrick, J. (1993). The chain and the arrow: From the history of mathematics assessment. In M. Niss (Ed.), Investigations into assessment in mathematics education: An ICMI study (pp. 31–46). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  33. Kohn, A. (2000). The case against standardized testing. Portsmouth, NH: Heinemann.Google Scholar
  34. Kohn, A. (2001). Learning is threatened by specific, measurable, uniform mandates. Education Week, 21(4), 38 and 52.Google Scholar
  35. Krutetskii, V. A. (1976). The psychology of mathematics abilities in schoolchildren (translated by J. Teller and edited by J. Kilpatrick). Chicago: University of Chicago Press.Google Scholar
  36. Lajoie, S. P. (1995). A framework for authentic assessment in mathematics. In T. A. Romberg (Ed.), Reform in school mathematics (pp. 19–37). Albany: NY: SUNY Press.Google Scholar
  37. Leder, G. C. (1992). Curriculum planning + assessment = learning? In G. C. Leder (Ed.), Assessment and learning of mathematics (pp. 330–344). Hawthorn, Victoria: Australian Council for Educational Research.Google Scholar
  38. Linn, R. L., Baker, E., & Dunbar, S. B. (1991). Complex, performance-based assessment: Expectations and validation criteria. Educational Researcher, 20(8), 15–21.Google Scholar
  39. McLean, L. (1982). Achievement testing — Yes! Achievement tests — No!, e + m Newsletter, Ontario Institute of Education, Fall, p. 1.Google Scholar
  40. Mummé, J. (1990). Portfolio assessment in mathematics. Santa Barbara, CA: University of California.Google Scholar
  41. Nagasaki, E., & Becker, J. P. (1993). Classroom assessment in Japanese mathematics education. In N. Webb (Ed.), Assessment in the mathematics classroom (pp. 40–53). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  42. National Center for Fair and Open Testing (2001). How standardized testing damages education. Published on the website of the National Center for Fair and Open Testing.Google Scholar
  43. National Council of Teachers of Mathematics (NCTM) (1997). Student catches SAT Math problem error. NCTM News Bulletin, March, 4.Google Scholar
  44. National Research Council (NRC) (1989). Everybody counts. Washington, D. C.: National Academy Press.Google Scholar
  45. Niss, M. (Ed.) (1993). Cases of assessment in mathematics education. An ICMI study. Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  46. Osterlind, S. J. (1998). Constructing test items. Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  47. Pellegrino, J. W., Chudowsky, N., & Glaser, R. (Eds.) (2001). Knowing what students know: The science and design of educational assessment. Washington, D. C.: National Academy Press.Google Scholar
  48. Pipho, C. (1985). Tracking the reform, part 5: Testing — Can it measure the success of the reform movement? Education Week, 22 May, 19.Google Scholar
  49. Popham, W. J. (1978). Criterion-referenced measurement. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  50. Popham, W. J. (1987). The merits of measurement-driven instruction. Phi Delta Kappan, 68, 679–682.Google Scholar
  51. Popham, W. J. (2001). Standardized achievement tests: misnamed and misleading. Education Week, 21(3), 46.Google Scholar
  52. Resnick, L. B. (1982). History of educational testing. In A. K. Wigdor & W. R. Garner (Eds.), Ability testing: Uses, consequences, and controversies, part 2: Documentation section (pp. 173–194). Washington, D. C.: National Academy Press.Google Scholar
  53. Resnick, L. B., & Resnick, D. P. (1992). Assessing the thinking curriculum: New tools for educational reform. In B. R. Gifford & M. C. O’Connor (Eds.), Changing assessments: Alternative views of attitude, achievement and instruction (pp. 37–75). Dordrecht, The Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
  54. Romberg, T. A. (1992). Evaluation: A coat of many colors. In T. A. Romberg (Ed.), Mathematics assessment and evaluation: Imperatives for mathematics educators (pp. 10–36). Albany, NY: SUNY Press.Google Scholar
  55. Romberg, T. A. (1995). Reform in school mathematics and authentic assessment. Albany, NY: SUNY Press.Google Scholar
  56. Romberg, T. A., Zarinnia, E. A., & Collis, K. F. (1990). A new world view of assessment in mathematics. In G. Kulm, Assessing higher order thinking in mathematics (pp. 21–38). Washington, D. C.: AAAS.Google Scholar
  57. Rotberg, I. (2001). A self-fulfilling prophecy. Phi Delta Kappan, 83(2), 170–171.Google Scholar
  58. Salmon-Cox, L. (1981). Teachers and standardized achievement tests: What’s really happening? Phi Delta Kappan, 62(9), 631–634.Google Scholar
  59. Scheerens, J., & Bosker, R. (1997). The foundations of educational effectiveness. Oxford, UK: Elsevier Science Ltd.Google Scholar
  60. Shafer, M. C, & Romberg, T. A. (1999). Assessment in classrooms that promote understanding. In E. Fennema & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 159–183). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  61. Shepard, L. A. (1988, April). Should instruction be measurement-driven? A debate. Paper presented at the meeting of the American Educational Research Association, April, New Orleans.Google Scholar
  62. Shepard, L. A. (1991). Pychometricians’ beliefs about learning. Educational Researcher, 20(1), 2–16.Google Scholar
  63. Shepard, L. A. (1993). The place of testing reform in educational reform: A reply to Cizek. Educational Researcher, 22(4), 10–13.Google Scholar
  64. Shepard, L. A. (2000a). The role of assessment in teaching and learning. Santa Cruz, CA: CRESST/CREDE, University of California, Los Angeles.Google Scholar
  65. Shepard, L. A. (2000b). The role of assessment in a learning culture. Educational Researcher, 29(1), 4–14.Google Scholar
  66. Shimada, S. (1977) (Ed.). The open-ended approach in arithmetic and mathematics — A new proposal toward teaching mathematics, Mizuumishobo, Tokyo, Japan (in Japanese).Google Scholar
  67. Sproull, L., & Zubrow, D. (1981). Standardized testing from the administrative perspective. Phi Delta Kappan, 62(9), 628–631.Google Scholar
  68. Stake, R. E. (1995). The invalidity of standardised testing for measuring mathematics achievement. In T. A. Romberg (Ed.), Reform in school mathematics — and authentic assessment (pp. 173–235). State University of New York, New York.Google Scholar
  69. Stake, R. E., Cole, C, Sloane, F, Migotsky, C, Flores, C, Merchant, M., Miron, M., & Medley, C. (1994). The Burden: Teacher professional development in Chicago school reform. Urbana-Champaign: University of Illinois.Google Scholar
  70. Stephens M., Clarke, D. J., & Pavlou, M. (1994). Policy to practice: High stakes assessment as a catalyst for classroom change. In G. Bell, B. Wright, N. Leeson & J. Geake (Eds.), Challenges in mathematics education: Seventeenth Annual Conference of the Mathematics Education Research Group of Australasia (pp. 571–580). Lismore, Australia: Mathematics Education Research Group of Australasia.Google Scholar
  71. Streefland, L., & Van den Heuvel-Panhuizen, M. (1999). Uncertainty: A metaphor for mathematics education. Journal of Mathematical Behavior, 27(4), 393–397.Google Scholar
  72. Van den Heuvel-Panhuizen, M. (1995). Toetsen bij reken-wiskundeonderwijs. In L. Verschaffel & E. de Corte (Eds.), Naar een nieuwe reken/wiskundedidactiek voor de basisschool en de basiseducatie (pp. 196–246). Brussels, Deel 1, StOHO.Google Scholar
  73. Van den Heuvel-Panhuizen, M. (1996). Assessment and realistic mathematics education. Utrecht, The Netherlands: Freudenthal Institute, Utrecht University, CD-β Press.Google Scholar
  74. Van den Heuvel-Panhuizen, M. (1997). Bananentoets. een onderzoek naar het schattend vermenigvuldigen met kommagetallen (internal publication). Utrecht, The Netherlands: Freudenthal Institute, Utrecht University.Google Scholar
  75. Van den Heuvel-Panhuizen, M., & Fosnot, C. T. (2001). Assessment of mathematics achievements: Not only the answers count. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 335–342). Utrecht, The Netherlands: Freudenthal Institute, Utrecht University.Google Scholar
  76. Wassermann, S. (2001). Quantum theory, the uncertainty principle, and the alchemy of standardized testing. Phi Delta Kappan, 83(1), 28–40.Google Scholar
  77. Watson, A. (1999). Paradigmatic conflicts in informal mathematics assessment as sources of social inequity. Educational Review, 51(2), 105–115.CrossRefGoogle Scholar
  78. Watson, A. (2000). Mathematics teachers acting as informal assessors: Practices, problems and recommendations. Educational Studies in Mathematics, 41, 69–91.CrossRefGoogle Scholar
  79. Webb, D. C. (2001). Instructionally embedded assessment practices of two middle grades mathematics teachers. Unpublished dissertation, University of Wisconsin, Madison, WI.Google Scholar
  80. Webb, N. L. (1992). Assessment of students’ knowledge of mathematics: Steps toward a theory. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 661–683). NCTM/Macmillan, New York.Google Scholar
  81. Webb, N. L. (1993). Visualizing a theory of the assessment of students’ knowledge of mathematics. In M. Niss (Ed.), Investigations into assessment in mathematics education: An ICMI study. Dordrecht (pp. 31–46). The Netherlands: Kluwer Academic Publishers.Google Scholar
  82. Wiggins, G. (1989a). Teaching to the (authentic) test. Educational Leadership, 46(1), 41–47.Google Scholar
  83. Wiggins, G. (1989b). A true test: Towards more authentic and equitable assessment. Phi Delta Kappan, 70(9), 703–713.Google Scholar
  84. Wiliam, D., & Black, P. (1996). Meanings and consequences: A basis for distinguishing formative and summative functions of assessment? British Educational Research Journal, 22, 537–548.CrossRefGoogle Scholar
  85. Wilson, M., & Sloane, K. (2000). From principles to practice: An embedded assessment system. Applied Measurement in Education, 13(2), 181–208.CrossRefGoogle Scholar
  86. Wolf, D. P. (1989). Portfolio assessment: Sampling student work. Educational Leadership, 46(1), 35–39.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Marja van den Heuvel-Panhuizen
    • 1
  • Jerry Becker
    • 2
  1. 1.Freudenthal InstituteUtrecht UniversityThe Netherlands
  2. 2.Southern Illinois UniversityCarbondaleUSA

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