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Paths of Justification in Israeli 7th Grade Mathematics Textbooks

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Abstract

When justifying a mathematical statement, textbooks often interweave several instances of justification, creating, what we term, paths of justification. This study aims to characterize the paths of justification of mathematical statements that are offered in eight 7th grade Israeli mathematics textbooks. The study attended to context and compared paths of justification in different mathematical domains (algebra and geometry) and in textbooks intended for students with different achievement levels. Analysis focused on the number of instances of justification in a path, the types of justification of those instances, and their order. Comparing paths of justification between algebra and geometry, and between textbooks intended for students with different levels of achievements, revealed several similarities and differences. The findings portray a complex picture of textbook justifications, underlining the need to be sensitive to context when studying textbook justifications. The study results are discussed in light of relevant literature, highlighting potential factors involved in the way a textbook path of justification of a specific mathematical statement is structured.

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Data Availability

The textbooks analyzed in this study are available to the public. The names of the textbooks are available from the authors upon request.

References   

  • Arcavi, A., Drijvers, P., & Stacey, K. (2017). The Learning and Teaching of Algebra: Ideas. Routledge. https://doi.org/10.4324/9781315545189

    Book  Google Scholar 

  • Ayalon, M., & Even, R. (2010). Mathematics Educators’ views on the role of mathematics learning in developing deductive reasoning. International Journal of Science and Mathematics Education, 8, 1131–1154. https://doi.org/10.1007/s10763-010-9238-z

    Article  ADS  Google Scholar 

  • Ayalon, M., & Even, R. (2016). Factors shaping students’ opportunities to engage in argumentative activity. International Journal of Science and Mathematics Education, 14(3), 575–601. https://doi.org/10.1007/s10763-014-9584-3

    Article  ADS  Google Scholar 

  • Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). Londres.

    Google Scholar 

  • Bell, A. W. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7(1), 23–40. https://doi.org/10.1007/BF00144356

    Article  Google Scholar 

  • Bergwall, A., & Hemmi, K. (2017). The State of Proof in Finnish and Swedish Mathematics Textbooks—Capturing Differences in Approaches to Upper-Secondary Integral Calculus. Mathematical Thinking and Learning, 19(1), 1–18.

    Article  Google Scholar 

  • Bergwall, A. (2021). Proof-related reasoning in upper secondary school: Characteristics of Swedish and Finnish textbooks. International Journal of Mathematical Education in Science and Technology, 52(5), 731–751. https://doi.org/10.1080/0020739X.2019.1704085

    Article  ADS  Google Scholar 

  • Blum, W., & Kirsch, A. (1991). Preformal proving: Examples and reflections. Educational Studies in Mathematics, 22(2), 183–203. https://doi.org/10.1007/BF00555722

    Article  Google Scholar 

  • Branford, B. (1908). A study of mathematical education: Including the teaching of arithmetic. Clarendon Press.

    Google Scholar 

  • Cabassut, R., Conner, A., İşçimen, F. A., Furinghetti, F., Jahnke, H. N., & Morselli, F. (2012). Conceptions of proof – in research and teaching. In G. Hanna & M. de Villiers (Eds.), Proof and Proving in Mathematics Education: The 19th ICMI Study (vol. 15, pp. 169–190). Springer. https://doi.org/10.1007/978-94-007-2129-6

  • Cabassut, R. (2005). Argumentation and proof in examples taken from French and German textbooks. In M. Bosch (Ed.), Proceedings of the 4th congress of the European Society for Research in Mathematics Education (CERME4) (pp. 391–400). ERME.

  • Chazan, D. (2000). Beyond formulas in mathematics and teaching: Dynamics of the high school algebra classroom. Teachers College Press.

    Google Scholar 

  • Choppin, J., Davis, J., McDuffie, A. R., & Drake, C. (2021). Influence of features of curriculum materials on the planned curriculum. ZDM Mathematics Education, 53(6), 1249–1263.

    Article  Google Scholar 

  • Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics (CCSSM). National Governors Association Center for Best Practices and the Council of Chief State School Officers.

  • Davis, P. J., & Hersh, R. (1981). The mathematical experience. Birkhauser. https://doi.org/10.1007/978-0-8176-8295-8

  • Davis, J. D., Smith, D. O., Roy, A. R., & Bilgic, Y. K. (2014). Reasoning-and-proving in algebra: The case of two reform-oriented US textbooks. International Journal of Educational Research, 64, 92–106. https://doi.org/10.1016/j.ijer.2013.06.012

    Article  Google Scholar 

  • Department of Education. (2010). National curriculum for mathematics (England): Key stages 3 and 4 framework document. Department of Education, United Kingdom.

    Google Scholar 

  • Dolev, S. (2011). Justifications and proofs of mathematical statements in 7th grade textbooks. Weizmann Institute of Science.

    Google Scholar 

  • Dolev, S., & Even, R. (2013). Justifications and explanations in Israeli 7th grade math textbooks. International Journal of Science and Mathematics Education, 13(2), 309–327. https://doi.org/10.1007/s10763-013-9488-7

    Article  Google Scholar 

  • 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(4), 867–891. https://doi.org/10.1007/s10763-010-9241-4

    Article  ADS  Google Scholar 

  • Even, R., & Kvatinsky, T. (2009). Approaches to teaching mathematics in lower-achieving classes. International Journal of Science and Mathematics Education, 7(5), 957–985. https://doi.org/10.1007/s10763-008-9141-z

    Article  ADS  Google Scholar 

  • Even, R., & Kvatinsky, T. (2010). What mathematics do teachers with contrasting teaching approaches address in probability lessons? Educational Studies in Mathematics, 74, 207–222. https://doi.org/10.1007/s10649-010-9234-9

    Article  Google Scholar 

  • Fan, L., Xiong, B., Zhao, D., & Niu, W. (2018). How is cultural influence manifested in the formation of mathematics textbooks? A comparative case study of resource book series between Shanghai and England. ZDM Mathematics Education, 50(5), 787–799.

    Article  Google Scholar 

  • Fan, L., Cheng, J., Xie, S., Luo, J., Wang, Y., & Sun, Y. (2021). Are textbooks facilitators or barriers for teachers’ teaching and instructional change? An investigation of secondary mathematics teachers in Shanghai, China. ZDM – Mathematics Education, 53(6), 1313–1330. https://doi.org/10.1007/s11858-021-01306-6

    Article  PubMed  Google Scholar 

  • Finnish National Board of Education. (2003). Lukion opetussuunnitelman perusteet 2003 [National core curriculum for upper secondary schools 2003]. Retrieved from http://www.oph.fi/download/47678_core_curricula_upper_secondary_education.pdf. Accessed 2017-02-03

  • Fu, Y., Qi, C., & Wang, J. (2022). Reasoning and Proof in Algebra: The Case of Three Reform-Oriented Textbooks in China. Canadian Journal of Science, Mathematics and Technology Education, 22(1), 130–149. https://doi.org/10.1007/s42330-022-00199-1

    Article  MathSciNet  ADS  Google Scholar 

  • 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 

  • Gracin, D. G., & Matić, L. J. (2019). Same textbook, different points of view: Students and teachers as textbook users. In S. Rezat, L. Fan, M. Hattermann, J. Schumacher, & H. Wuschke (Eds.), Proceedings of the Third International Conference on Mathematics Textbook Research and Development (pp. 173–178). Universitätsbibliothek Paderborn.

  • 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.

    Article  Google Scholar 

  • Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1–2), 5–23. https://doi.org/10.1023/A:1012737223465

    Article  Google Scholar 

  • Hanna, G., & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge. ZDM - International Journal on Mathematics Education, 40(3), 345–353.

    Article  Google Scholar 

  • Hanna, G., & de Bruyn, Y. (1999). Opportunity to learn proof in Ontario grade twelve mathematics texts. Ontario Mathematics Gazette, 37, 23–29.

    Google Scholar 

  • Hanna, G., & de Villiers, M. (Eds.). (2012). Proof and Proving in Mathematics Education: The 19th ICMI Study (Vol. 15). Springer. https://doi.org/10.1007/978-94-007-2129-6

  • Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Issues in mathematics education: Vol. 7. Research in Collegiate Mathematics Education III (pp. 234–283). Mathematical Association of America. https://doi.org/10.1090/cbmath/007/07

  • Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 805–842). Information Age Publishing.

    Google Scholar 

  • Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399. https://doi.org/10.1007/BF01273372

    Article  Google Scholar 

  • Israel Ministry of Education. (2009). Mathematics curriculum for grades 7–9. Ministry of Education. (Hebrew).

  • Jankvist, U. T., Aguilar, M. S., Misfeldt, M., & Koichu, B. (2021). Launching Implementation and Replication Studies in Mathematics Education (IRME). Implementation and Replication Studies in Mathematics Education, 1(1), 1–19. https://doi.org/10.1163/26670127-01010001

    Article  Google Scholar 

  • Karsenty, R., Arcavi, A., & Hadas, N. (2007). Exploring informal mathematical products of low achievers at the secondary school level. The Journal of Mathematical Behavior, 26(2), 156–177. https://doi.org/10.1016/j.jmathb.2007.05.003

    Article  Google Scholar 

  • Kieran, C. (2018). Algebra Teaching and Learning. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 1–9). Springer International Publishing.

    Google Scholar 

  • Martinez, M. V., Brizuela, B. M., & Superfine, A. C. (2011). Integrating algebra and proof in high school mathematics: An exploratory study. The Journal of Mathematical Behavior, 30(1), 30–47. https://doi.org/10.1016/j.jmathb.2010.11.002

    Article  Google Scholar 

  • Mkhatshwa, T. P. (2022). Quantitative and covariational reasoning opportunities provided by calculus textbooks: The case of the derivative. International Journal of Mathematical Education in Science & Technology, 1–28. https://doi.org/10.1080/0020739X.2022.2129497

  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Author.

  • Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Mathematical Thinking and Learning, 16(1), 51–79. https://doi.org/10.1080/10986065.2014.857802

    Article  Google Scholar 

  • Pedemonte, B. (2008). Argumentation and algebraic proof. ZDM - International Journal on Mathematics Education, 40(3), 385–400. https://doi.org/10.1007/s11858-008-0085-0

    Article  Google Scholar 

  • Pepin, B., Gueudet, G., & Trouche, L. (2013). Investigating textbooks as crucial interfaces between culture, policy and teacher curricular practice: Two contrasted case studies in France and Norway. ZDM Mathematics Education, 45(5), 685–698. https://doi.org/10.1007/s11858-013-0526-2

    Article  Google Scholar 

  • Polya, G. (1973). How to solve it: A new aspect of mathematical method. Princeton university press. (Original work published 1945)

  • Raudenbush, S. W., Rowan, B., & Cheong, Y. F. (1993). Higher order instructional goals in secondary schools: Class, teacher, and school influences. American Educational Research Journal, 30(3), 523–553. https://doi.org/10.3102/00028312030003523

    Article  Google Scholar 

  • Rezat, S., Fan, L., & Pepin, B. (2021). Mathematics textbooks and curriculum resources as instruments for change. ZDM Mathematics Education, 53(6), 1189–1206. https://doi.org/10.1007/s11858-021-013093

    Article  Google Scholar 

  • Senk, S. L., Thompson, D. R., & Johnson, G. J. (2008, July). Reasoning and proof in high school textbooks from the USA. Paper presented at the 11th International Congress on Mathematics Education, Monterey, Mexico.

  • Shield, M., & Dole, S. (2013). Assessing the potential of mathematics textbooks to promote deep learning. Educational Studies in Mathematics, 82(2), 183–199.

    Article  Google Scholar 

  • Sierpinska, A. (1994). Understanding in Mathematics. The Falmer Press.

    Google Scholar 

  • Sriraman, B., & Umland, K. (2014). Argumentation in mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 46–48). Springer.

    Chapter  Google Scholar 

  • Stacey, K., & Chick, H. (2004). Solving the problem with algebra. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI Study (pp. 1–20). Springer Netherlands. https://doi.org/10.1007/1-4020-8131-6_1

    Chapter  Google Scholar 

  • Stacey, K., & Vincent, J. (2009). Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. Educational Studies in Mathematics, 72(3), 271–288.

    Article  Google Scholar 

  • Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258–288.

    Article  Google Scholar 

  • Stylianou, D. A., Blanton, M. L., & Knuth, E. J. (Eds.). (2010). Teaching and learning proof across the grades: A K-16 perspective. Routledge. https://doi.org/10.4324/9780203882009

    Book  Google Scholar 

  • Swedish National Agency for Education. (2021). Ämnesplaner i matematik [Syllabus in mathematics]. Retrieved from https://www.skolverket.se/download/18.4fc05a3f164131a74181063/1535372298267/Mathematics-swedish-school.pdf. Accessed 2023-05-05

  • Tall, D. (2014). Making sense of mathematical reasoning and proof. In M. N. Fried & T. Dreyfus (Eds.), Mathematics & Mathematics Education: Searching for Common Ground (pp. 223–235). Springer.

    Chapter  Google Scholar 

  • Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 43(3), 253–295. https://doi.org/10.5951/jresematheduc.43.3.0253

    Article  Google Scholar 

  • Thompson, D. R. (2014). Reasoning-and-proving in the written curriculum: Lessons and implications for teachers, curriculum designers, and researchers. International Journal of Educational Research, 64, 141–148. https://doi.org/10.1016/j.ijer.2013.09.013

    Article  Google Scholar 

  • Wong, K.-C., & Sutherland, R. (2018). Reasoning-and-proving in algebra in school mathematics textbooks in Hong Kong. In A. J. Stylïanides & G. Harel (Eds.), Advances in mathematics education research on proof and proving: An international perspective (pp. 185–198). Springer. https://doi.org/10.1007/978-3-319-70996-3_13

    Chapter  Google Scholar 

  • 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). NCTM.

    Google Scholar 

  • Zohar, A., & Dori, Y. J. (2003). Higher order thinking skills and low-achieving students: Are they mutually exclusive? The Journal of the Learning Sciences, 12(2), 145–181.

    Article  Google Scholar 

  • Zohar, A., Degani, A., & Vaaknin, E. (2001). Teachers’ beliefs about low-achieving students and higher order thinking. Teaching and Teacher Education, 17(4), 469–485.

    Article  Google Scholar 

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Acknowledgements

This research was supported by the Israel Science Foundation (grant No. 221/12).

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  1. Department of Science Teaching, Weizmann Institute of Science, 76100, Rehovot, Israel

    Ruhama Even & Boaz Silverman

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  1. Ruhama Even
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Correspondence to Ruhama Even.

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Even, R., Silverman, B. Paths of Justification in Israeli 7th Grade Mathematics Textbooks. Int J of Sci and Math Educ 22, 609–631 (2024). https://doi.org/10.1007/s10763-023-10387-6

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