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JUSTIFICATIONS AND EXPLANATIONS IN ISRAELI 7TH GRADE MATH TEXTBOOKS

  • Sarit Dolev
  • Ruhama EvenEmail author
Article

Abstract

This study analyzes six seventh grade Israeli mathematics textbooks, examining (1) the extent to which students are required to justify and explain their mathematical work, and (2) whether students are asked to justify a mathematical claim that is stated by the textbook or a mathematical claim that they themselves generated when solving a problem. Two different units of analysis were used to analyze two central topics in the seventh grade curriculum as follows: (1) equation solving in algebra and (2) triangle properties in geometry. The findings indicate that all six textbooks had considerably larger percentages of geometric tasks than algebraic tasks, which required students to justify or explain their mathematical work. Moreover, considerable differences were found among the six textbooks regarding the percentages of tasks that required students to justify and explain in both topics, but more so with the algebraic topic. Analysis of whether the textbook tasks required students to justify a mathematical claim that is stated by the textbook or a mathematical claim that the students themselves generated also revealed substantial differences among the textbooks. These findings are discussed, as well as the research methods used, in light of relevant literature.

Key words

algebraic tasks curriculum analysis explanations geometric tasks justifications textbook analysis 

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References

  1. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241.CrossRefGoogle Scholar
  2. Ayalon, M. & Even, R. (2009). Are they equivalent? In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 81-88). Thessaloniki, Greece.Google Scholar
  3. Ayalon, M. & Even, R. (2010). Mathematics educators’ views on mathematics learning and the development of deductive reasoning. International Journal of Science and Mathematics Education, 8, 1131–1154.CrossRefGoogle Scholar
  4. Baker, M. (2003). Computer-mediated argumentative interactions for the co-elaboration of scientific notions. In J. Andriessen, M. Baker & D. Suthers (Eds.), Arguing to learn: Confronting cognitions in computer-supported collaborative learning environments (pp. 47–70). Dordrecht: Kluwer.CrossRefGoogle Scholar
  5. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. Mathematics Teachers and Children, 216, 235.Google Scholar
  6. Balacheff, N. (1991). The benefits and limits of social interactions: The case of mathematical proof. In A. Bishop, S. Mellin-Olsen & J. van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 175–192). Dordrecht: Kluwer.Google Scholar
  7. Ball, D. L. & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin & D. Shifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27–44). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  8. Cabassut, R. (2005). Argumentation and proof in examples taken from French and German textbooks. In M. Bosch (Ed.), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (pp. 391–400). Spain: ERME.Google Scholar
  9. da Ponte, J. P. & Marques, S. (2007). Proportion in school mathematics textbooks: A comparative study. In D. Pitta—Pantazi & G. Philippou (Eds.), 5th Congress of ERME, the European Society for Research in Mathematics Education (pp. 2443-2452). Cyprus: LarnacaGoogle Scholar
  10. Davis, P. J. & Hersh, R. (1981). The mathematical experience. Boston: Birkhäuser.Google Scholar
  11. Dreyfus, T. (1999). Why Johnny can’t prove. In Forms of Mathematical Knowledge (pp. 85–109). The Netherlands: Springer.CrossRefGoogle Scholar
  12. Dreyfus, T. & Hadas, N. (1996). Proof as answer to the question why. Zentralblatt für Didaktik der Matematik, 28(1), 1–5.Google Scholar
  13. Eisenmann, T. & Even, R. (2009). Similarities and differences in the types of algebraic activities in two classes taught by the same teacher. In J. T. Remillard, B. A. Herbel-Eisenmann & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 152–170). New York: Routledge.Google Scholar
  14. Eisenmann, T. & Even, R. (2011). Enacted types of algebraic activity in different classes taught by the same teacher. International Journal of Science and Mathematics Education, 9, 867–891.CrossRefGoogle Scholar
  15. Even, R. & Kvatinsky, T. (2010). What mathematics do teachers with contrasting teaching approaches address in probability lessons? Educational Studies in Mathematics, 74(3), 207–222.CrossRefGoogle Scholar
  16. Fujita, T., Jones, K. & Kunimune, S. (2009). The design of textbooks and their influence on students’ understanding of “proof” in lower secondary school. In F. Lin, F. Hsieh, G. Hanna & M. de Villiers (Eds.), Proceedings of the ICMI Study 19 conference: Proof and Proving in Mathematics Education (Vol. 1, pp. 172–177). Taipei, Taiwan: National Taiwan Normal University.Google Scholar
  17. González, G. & Herbst, P. (2006). Competing arguments for the geometry course: Why were American high school students supposed to study geometry in the twentieth century? International Journal for the History of Mathematics Education, 1(1), 7–33.Google Scholar
  18. Grouws, D. A., Smith, M. S. & Sztajn, P. (2004). The preparation and teaching practices of United States mathematics teachers: Grades 4 and 8. In P. Kloosterman & F. K. Lester (Eds.), Results and interpretations of the 1990-2000 mathematics assessments of the National Assessment of Educational Progress (pp. 221–267). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  19. Haggarty, L. & Pepin, B. (2002). An investigation of mathematics textbooks and their use in English, French, and German classrooms: Who gets an opportunity to learn what? British Educational Research Journal, 28(4), 567–590.CrossRefGoogle Scholar
  20. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.CrossRefGoogle Scholar
  21. Hanna, G. (2000). Proof, explanation, and exploration: An overview. Educational Studies in Mathematics, 44(1-2), 5–23.CrossRefGoogle Scholar
  22. Hanna, G. & de Bruyn, Y. (1999). Opportunity to learn proof in Ontario grade twelve mathematics texts. Ontario Mathematics Gazette, 37(4), 23–29.Google Scholar
  23. Harel, G. & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. Research in Collegiate Mathematics Education III, 7, 234–282.Google Scholar
  24. Herbst, P. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49, 283–312.CrossRefGoogle Scholar
  25. Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399.CrossRefGoogle Scholar
  26. Jones, D. L. & Tarr, J. E. (2007). An examination of the levels of cognitive demand required by probability tasks in middle grades mathematics textbooks. Statistics Education Research Journal, 6(2), 4–27.Google Scholar
  27. Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229–269). Hillsdale, NJ: Erlbaum.Google Scholar
  28. Kulm, G., Roseman, J. & Treistman, M. (1999). A Benchmarks-Based approach to textbook evaluation. Retrieved from website of The Project 2061: http://www.project2061.org/publications/articles/textbook/articles/approach.htm
  29. Li, Y., Chen, X. & An, S. (2009). Conceptualizing and organizing content for teaching and learning in selected Chinese, Japanese, and US mathematics textbooks: The case of fraction division. ZDM—The International Journal on Mathematics Education, 41, 809–826.CrossRefGoogle Scholar
  30. Manouchehri, A. & Goodman, T. (1998). Mathematics curriculum reform and teachers: Understanding the connections. The Journal of Educational Research, 92, 27–41.CrossRefGoogle Scholar
  31. Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 173–203). Rotterdam: Sense.Google Scholar
  32. Mejía-Ramos, J. P. & Inglis, M. (2008). What are argumentative activities associated with proof? Proceedings of the British Society for Research into Learning Mathematics, 28(2). Retrieved from http://www.bsrlm.org.uk/IPs/ip28-2/BSRLM-IP-28-2-12.pdf
  33. Ministry of Education (2009). Math curriculum for grades 7-9. Retrieved from http://meyda.education.gov.il/files/Tochniyot_Limudim/Math/Hatab/Mavo.doc (in Hebrew).
  34. Newton, D. & Newton, L. (2007). Could elementary mathematics textbooks help give attention to reasons in the classroom? Educational Studies in Mathematics, 64(1), 69–84.CrossRefGoogle Scholar
  35. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.CrossRefGoogle Scholar
  36. Schwarz, B. B., Hershkowitz, R. & Prusak, N. (2010). Argumentation and mathematics. In C. Howe & K. Littleton (Eds.), Educational dialogues: Understanding and promoting productive interaction (pp. 115–141). London: Routledge.Google Scholar
  37. Senk, S. L., Thompson, D. R. & Johnson, G. (2008). Reasoning and proof in high school textbooks from the USA. Retrieved from http://tsg.icme11.org/document/get/282
  38. Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 14, 15–39.CrossRefGoogle Scholar
  39. Son, J. (2005). A comparison of how textbooks teach multiplication of fractions and division of fractions in Korea and in the US. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 201–208). Melbourne: PME.Google Scholar
  40. Stacey, K. & Vincent, J. (2009). Modes of reasoning in explanations in Australian eighth grade mathematics textbooks. Educational Studies in Mathematics, 72, 271–288.CrossRefGoogle Scholar
  41. Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11, 258–288.CrossRefGoogle Scholar
  42. Tirosh, D., Even, R. & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness and teaching approaches. Educational Studies in Mathematics, 35, 51–64.CrossRefGoogle Scholar
  43. Yackel, E. & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.CrossRefGoogle Scholar
  44. Yackel, E. & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227–236). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  45. Yerushalmy, M. & Chazan, D. (1987). Effective problem posing in an inquiry environment: A case study using the geometric supposer. In J. C. Bergeron, N. Herscovics & C. Kieran (Eds.), Proceedings of the 11th PME Conference (Vol. II, pp. 53–59). Montreal, Canada: PME.Google Scholar
  46. Zhu, Y. & Fan, L. (2006). Focus on the representation of problem types in intended curriculum: A comparison of selected mathematics textbooks from Mainland China and the United States. International Journal of Science and Mathematics Education, 4(4), 609–626.CrossRefGoogle Scholar

Copyright information

© National Science Council, Taiwan 2013

Authors and Affiliations

  1. 1.GivatayimIsrael
  2. 2.Department of Science TeachingWeizmann Institute of ScienceRehovotIsrael

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