# Mathematically and practically-based explanations: individual preferences and sociomathematical norms

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## Abstract

This paper is an initial investigation of teachers’ and students’ preferences for mathematically-based (MB) and practically-based (PB) explanations and the relationship between those preferences and sociomathematical norms. The study focuses on one fifth grade teacher and two of her students and discusses three issues. The first issue concerns students’ abilities to understand and accept MB explanations. The second issue concerns the choices teachers make regarding the types of explanations they introduce to their classes and the basis for these choices. The third issue concerns the place of the individuals’ preferences within the sociomathematical norms of the class. The findings indicate that elementary school students are capable of understanding MB explanations and some might even prefer them. We also found that although a teacher might personally prefer MB explanations, this preference may be set aside for didactical considerations. Finally, we discuss the complex relationship between individual preferences for MB and PB explanations and sociomathematical norms.

## Key Words

elementary school explanations mathematically-based and practically-based explanations preferences sociomathematical norms## Preview

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## References

- Achinstein, P. (1983).
*The nature of explanation*. New York: Oxford Univesity Press.Google Scholar - Ball, D. & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. Phillips (Ed.),
*Yearbook of the National Society for the Study of Education, Constructivism in Education*. Chicago, IL: University of Chicago Press.Google Scholar - Ben-Yehuda, M., Lavy, I., Linchevski, L. & Sfard, A. (2005). Doing wrong with words: What bars students’ access to arithmetical discourses.
*Journal for Research in Mathematics Education*,*36*(3), 176–247.CrossRefGoogle Scholar - Borasi, R. (1992).
*Learning mathematics through inquiry*. Portsmouth, NH: Heinemann.Google Scholar - Bowers, J. & Doerr, H. (2001). An analysis of prospective teachers’ dual roles in understanding the mathematics of change: Eliciting growth with technology.
*Journal of Mathematics Teacher Education*,*4*, 115–137.CrossRefGoogle Scholar - Cobb, P., Wood, T. & Yackel, E. (1991). A constructivist approach to second grade mathematics. In E. von Glaserfeld (Ed.),
*Radical constructivism in mathematics*(pp. 157–176). Dordrecht, The Netherlands: Kluwer Academic.Google Scholar - Cobb, P., McLain, K. & Gravemeijer, K. (2003). Learning about statistical covariation.
*Cognition and Instruction*,*21*(1), 1–78.CrossRefGoogle Scholar - Cramer, K., Post, T. & del Mas, R. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum.
*Journal for Research in Mathematics Education*,*33*, 111–144.Google Scholar - Dreyfus, T. (1999). Why Johnny can’t prove?
*Educational Studies in Mathematics*,*38*, 85–109.CrossRefGoogle Scholar - Fischbein, E. (1987).
*Intuition in science and mathematics*. Dordrecht, The Netherlands: Reidel.Google Scholar - Freudenthal, H. (1973).
*Mathematics as an educational task*. Dordrecht, The Netherlands: Reidel.Google Scholar - Fuchs, L., Fuchs, D., Hamlett, C., Phillips, N., Karns, K. & Dutka, S. (1997). Enhancing students’ helping behavior during peer-mediated instruction with conceptual mathematical explanations.
*Elementary School Journal*,*97*(3), 223–249.CrossRefGoogle Scholar - Ginsburg, H. & Seo, K. (1999). Mathematics in children’s thinking.
*Mathematical Thinking and Learning*,*1*(2), 113–129.CrossRefGoogle Scholar - Hershkowitz, R. & Schwarz, B. (1999). The emergent perspective in rich learning environments: Some roles of tools and activities in the construction of sociomathematical norms.
*Educational Studies in Mathematics*,*39*, 149–166.CrossRefGoogle Scholar - Hiebert, J. & Carpenter, T. (1992). Learning and teaching with understanding. In D.A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 65–97). New York: Macmillan.Google Scholar - Kazemi, E. & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms.
*Elementary School Journal*,*102*(1), 59–80.CrossRefGoogle Scholar - Knifong, J. & Burton, G. (1980). Intuitive definitions for division with zero.
*Arithmetic Teacher*,*73*, 179–186.Google Scholar - Koirala, H. (1999). Teaching mathematics using everyday contexts: What if academic mathematics is lost? In O. Zaslavsky (Ed.),
*Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education*, III (pp. 161–168). Haifa, Israel.Google Scholar - Lampert, M. (1990). When the problem is not the question and the solution is not the answer.
*American Educational Research Journal*,*27*(1), 29–63.CrossRefGoogle Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States*. Mahwah, NJ: Erlbaum.Google Scholar - Mack, K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge.
*Journal for Research in Mathematics Education*,*26*(5), 422–441.CrossRefGoogle Scholar - McClain, K. & Cobb, P. (2001). An approach for supporting teachers’ learning in social context. In F. Lin & T. Cooney (Eds.),
*Making sense of mathematics teacher education*(pp. 207–232). Dordrecht, The Netherlands: Kluwer.Google Scholar - McNeal, B. & Simon, M. (2000). Mathematics culture clash: Negotiating new classroom norms with prospective teachers.
*Journal of Mathematical Behavior*,*18*(4), 475–509.CrossRefGoogle Scholar - National Council of Teachers of Mathematics (2000).
*Principles and standards for school mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Nyabanyaba, T. (1999). Whither relevance? Mathematics teachers’ discussion of the use of ‘real-life’ contexts in school mathematics.
*For the Learning of Mathematics*,*19*(3), 10–14.Google Scholar - Perry, M. (2000). Explanations of mathematical concepts in Japanese, Chinese, and U.S. first- and fifth-grade classrooms.
*Cognition and Instruction*,*18*(2), 181–207.CrossRefGoogle Scholar - Piaget, J. & Inhelder, B. (1969).
*The psychology of the child*. New York: Basic Books.Google Scholar - Raman, M. (2002). Coordinating informal and formal aspects of mathematics: Student behavior and textbook messages.
*Journal of Mathematical Behavior*,*21*, 135–150.CrossRefGoogle Scholar - Schoenfeld, A. (1994). What do we know about mathematics curricula?
*Journal of Mathematical Behavior*,*13*, 55–80.CrossRefGoogle Scholar - Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one.
*Educational Researcher*,*27*(2), 4–13.CrossRefGoogle Scholar - Sfard, A. (2000). On reform movement and the limits of mathematical discourse.
*Mathematical Thinking and Learning*,*2*(3), 157–189.CrossRefGoogle Scholar - Sfard, A. (2001). Learning mathematics as developing a discourse. In R. Speiser, C. Maher & C. Walter (Eds.),
*Proceedings of 21st Conference of PME-NA*(pp. 23–44). Columbus, Ohio: Clearing House for Science, Mathematics, and Environmental Education.Google Scholar - Sfard, A. & Lavie, I. (2005). Why cannot children see as the same what grownups cannot see as different? – early numerical thinking revisited.
*Cognition and Instruction*,*23*(2), 237–309.CrossRefGoogle Scholar - Streefland, L. (1991).
*Fractions in realistic mathematics education: A paradigm of developmental research*. Dordrecht, the Netherlands: Kluwer Academic Publishers.Google Scholar - Szendrei, J. (1996). Concrete materials in the classroom. In A.J. Bishop (Ed.),
*International Handbook of Mathematics Education*(pp. 411–434). Dordrecht the Netherlands: Kluwer Academic Publishers.Google Scholar - Thompson, A., Philipp, R., Thompson, P. & Boyd, B. (1994). Calculational and conceptual orientation in teaching mathematics. In A. Coxford (Ed.),
*1994 Yearbook of the National Council of Teachers of Mathematics*(pp. 79–92). Reston, VA: The National Council of Teachers of Mathematics.Google Scholar - Tsamir, P. & Sheffer, R. (2000). Concrete and formal arguments: The case of division by zero.
*Mathematics Education Research Journal*,*12*(2), 92–106.Google Scholar - Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage.
*Educational Studies in Mathematics*,*54*, 9–35.CrossRefGoogle Scholar - Van Fraassen, B. (1980).
*The scientific image*. Oxford: Clarendon, UK.Google Scholar - Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.)
*Advanced mathematical thinking*(pp. 65–81). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar - Wu, H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy.
*American Educator*,*23*(3), 14–19, 50–52.Google Scholar - Yackel, E. (2001). Explanation, justification and argumentation in mathematics classrooms. In M. van den Heuvel-Panhuizen (Ed.),
*Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education*, Vol. 1 (pp. 9–24). Utrecht, The Netherlands: Freudenthal Institute.Google Scholar - Yackel, E. & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics.
*Journal for Research in Mathematics Education*,*22*, 390–408.CrossRefGoogle Scholar - Yackel, E., Rasmussen, C. & King, K. (2000). Social and sociomathematical norms in an advanced undergraduate mathematics course.
*Journal of Mathematical Behavior*,*19*, 275–287.CrossRefGoogle Scholar