Abstract
In this paper we present three cases of instructional design that illustrates both horizontal didactising, the activity of using already established principles to design instruction, and vertical didactising the activity of developing new tools and principles for instructional design. The first case illustrates horizontal didactising by elaborating how the constructs chains of signification and models were used to design an instructional sequence involving linear growth. The second and third cases illustrate vertical didactising by developing argumentation analyses and generative listening, respectively, as instructional design tools. In the second case, argumentation analyses emerge as a tool that other designers can use to anticipate the quality of conversations that can occur as students engage in tasks prior to implementing the instructional sequence. The third case develops the notion of generative listening as a conceptual tool within the context of designing differential equations instruction to gain insights into what are, for students, experientially-real starting points that are mathematical in nature and to provide inspirations for revisions to instructional sequences.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Cobb, P.: 2000, 'Conducting teaching experiments in collaboration with teachers', in A.E. Kelly and R.A. Lesh (eds.), Handbook of Research Design in Mathematics and Science Education, Lawrence Erlbaum Associates, Mahwah, NJ, pp. 307–334.
Cobb, P., Gravemeijer, K., Yackel, E., McClain, K. and Whitenack, J.: 1997, 'Symbolizing and mathematizing: The emergence of chains of signification in one first-grade classroom', in D. Kirshner and J. A. Whitson (eds.), Situated Cognition Theory: Social, Semiotic, and Neurological Perspectives, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 151–233.
Davis, B.: 1997, 'Listening for differences: An evolving conception of mathematics teaching', Journal for Research in Mathematics Education 28, 355–376.
Freudenthal, H.: 1991, Revisiting Mathematics Education: China Lectures, Kluwer Academic Publishers, Dordrecht.
Gravemeijer, K.P.E.: 1994, Developing Realistic Mathematics Education, CD-ß Press, Utrecht, The Netherlands.
Gravemeijer, K.: 1999, 'How emergent models may foster the constitution of formal mathematics', Mathematical Thinking and Learning 2, 155–177.
Gravemeijer, K., and Doorman, M.: 1999, 'Context problems in realistic mathematics education: A calculus course as an example', Educational Studies in Mathematics 39, 111–129.
Krummheuer, G.: 1995, 'The ethnography of argumentation,' in P. Cobb and H. Bauersfeld (eds.), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 229–269.
Lacan, J.: 1977, Ecrits: A Selection, Tavistock, London.
Meira, L.: 1995, 'The microevolution of mathematical representations in children's activity', Cognition and Instruction 13, 269–313.
Nemirovsky, R.: 1994, 'On ways of symbolizing: The case of Laura and the velocity sign', Journal of Mathematical Behavior 13, 389–422.
Nemirovsky, R. and Monk, S.: 2000, '“If you look at it the other way...” An exploration into the nature of symbolizing,' in P. Cobb, E. Yackel and K. McClain (eds.), Symbolizing and Communicating in the Mathematics Classroom: Perspectives on Tools and Discourse, Lawrence Erlbaum, Mahwah, NJ 177–221.
Rasmussen, C., Marrongelle, K. and Keynes, M.: 2003, April, Pedagogical Tools: Integrating Student Reasoning into Instruction, Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL. 126 ERNA YACKEL ET AL.
Richards, J.: 1991, 'Mathematical discussions,' in E. von Glasersfeld (ed.), Radical Constructivism in Mathematics Education, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 13–52.
Saussure, F. de: 1986, Course in General Linguistics (C. Bally and A. Sechehaye, with A. Reidlinger, eds.; R. Harris, Trans.), Open Court, La Salle, IL (Original work published in 1916).
Saxe, G.: 1991, Culture and Cognitive Development: Studies in Mathematical Understanding, Lawrence Erlbaum Associates, Hillsdale, NJ.
Seidman, I.E.: 1991, Interviewing as Qualitative Research, Teachers College Press, New York, NY.
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.
Streefland, L.: 1990, Fractions in Realistic Mathematics Education: A Paradigm of Developmental Research, Kluwer Academic Publishers, Dordrecht.
Streefland, L.: 1993, 'The design of a mathematics course: A theoretical reflection', Educational Studies in Mathematics 25, 109–135.
Steffe, L. and Thompson, P.: 2000, 'Teaching experiment methodology: Underlying principles and essential elements,' in A.E. Kelly and R.A. Lesh (eds.), Handbook of Research Design in Mathematics and Science Education, Lawrence Erlbaum Associates, Mahwah, NJ, pp. 267–306.
Stephan, M., Bowers, J. and Cobb, P. (with Gravemeijer, K.) (eds.): in press, 'Supporting students' development of measuring conceptions: Analyzing students' learning in social context', Journal for Research in Mathematics Education Monograph Series.
Stephan, M. and Rasmuseen, C.: 2002, 'Classroom mathematical practices in differential equations', Journal of Mathematical Behavior 21, 459–490.
Thompson, P.: 1994, 'Images of rate and operational understanding of the fundamental theorem of calculus', Educational Studies in Mathematics 26, 229–274.
Toulmin, S.: 1969, The Uses of Argument, Cambridge University Press, Cambridge.
Treffers, A.: 1993, 'Wiskobas and Freudenthal realistic mathematics education', Educational Studies in Mathematics 25, 89–108.
Walkerdine, V.: 1988, The Mastery of Reason: Cognitive Development and the Production of Rationality, Routledge, London.
Wijers, M., Roodhardt, A., van Reewijk, M., Burrill, G., Cole, B. and Pligge, M.: 1997, Building formulas in National Center for Research in Mathematical Sciences Education and the Freudenthal Institute (eds.), Mathematics in Context: A Connected Curriculum for Grades 5–8, Encyclopaedia Brittanica, Chicago.
Yackel, E.: 1997, April, Explanation as an Interactive Accomplishment: A Case Study of one Second-Grade Mathematics Classroom, Paper presented at the annual meeting of the American Educational Research Association, Chicago. Department of Mathematics, Computer Science, and Statistics, Purdue University Calumet, Hammond, U.S.A.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yackel, E., Stephan, M., Rasmussen, C. et al. Didactising: Continuing the work of Leen Streefland. Educational Studies in Mathematics 54, 101–126 (2003). https://doi.org/10.1023/B:EDUC.0000005213.85018.34
Issue Date:
DOI: https://doi.org/10.1023/B:EDUC.0000005213.85018.34