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Opportunities to engage with proof: the nature of proof tasks in two geometry textbooks and its influence on enacted lessons

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Abstract

This paper compares task features and cognitive demand of proof tasks in two US high school geometry textbooks and considers how such differences influence geometry teachers’ facilitation of proof and students’ engagement with proof tasks during enacted lessons. Data were collected via interviews, task cover sheet-before implementation, task reflection sheet-after implementation, samples of students’ work, and classroom observations. Descriptive statistics were used to summarize task features and cognitive demands of proof within textbooks, and a grounded theory approach was used to analyze the enacted lessons. The results revealed variation in the nature of proof tasks within textbooks. Additionally, even though geometry textbooks may have higher-level demand proof tasks, there is no guarantee that such tasks would be assigned, or that the levels of cognitive demand of tasks will be maintained from the written to the enacted curriculum. Factors that can influence how teachers’ use textbooks include: beliefs, students’ disposition, and assessment. Thus, teachers’ actions can limit the extent students engage with proof. This study has implications for unpacking the complexities of students’ engagement with proof.

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Notes

  1. Horizon Research, Inc. developed the artifact packet for the Cases of Reasoning and Proving (CORP) in Secondary Mathematics Project, with funding from the National Science Foundation (Award No. DRL-0732798). CORP seeks to develop curriculum that can be used for professional education that promotes reasoning and proving, and the development of mathematical knowledge needed for teaching.

  2. The observation protocol was adapted from an instrument developed by Horizon Research, Inc. for CORP that documented how teachers use the proof tasks during the lesson.

  3. The dates are read as month, day, and year, so 11-2-11 means November 2, 2011.

  4. This is a standardized exam administered by the state, which assesses the state geometry curriculum.

References

  • Anderson, J. R. (1983). Acquisition of proof skills in geometry. In R. S. Michalski, J. G. Carbonell & T. M. Mitchell (Eds.), Machine learning: An artificial intelligence approach (pp. 191–219).

  • Bass, L. E., Charles, R. I., Johnson, A., & Kennedy, D. (2004). Prentice hall mathematics geometry. Upper Saddle River, NJ: Pearson Education, Inc.

  • Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 843–908). Charlotte: Information Age Publishing.

    Google Scholar 

  • Bieda, K. N. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41(4), 351–382.

    Google Scholar 

  • Boston, M. D., & Smith, M. S. (2009). Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers’ classrooms. Journal for Research in Mathematics Education, 40(2), 119–156.

    Google Scholar 

  • Chang, Y.-P., Lin, F.-L., & Reiss, K. (2012). Learning opportunities for mathematical proof: The presentation of geometry problems in German and Taiwanese textbooks. Paper presented at the 36th Conference of the International Group for the Psychology of Mathematics Education.

  • Cirillo, M. (2008). On becoming a geometry teacher: A longitudinal case study of one teacher learning to teach proof. Unpublished doctoral dissertation. Iowa State University, Ames.

  • Cirillo, M., & Herbst, P. (2010). Moving toward more authentic proof practices in geometry. http://deepblue.lib.umich.edu/handle/2027.42/78169.

  • Creswell, J. W. (2008). Educational research: Planning, conducting, and evaluating quantitative and qualitative research (3rd ed.). Upper Saddle River: New Jersey Merrill.

  • Davis, J. D. (2012). An examination of reasoning and proof opportunities in three differently organized secondary mathematics textbook units. Mathematics Education Research Journal, 24(4), 467–491.

    Article  Google Scholar 

  • Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.), Exploiting mental imagery with computers in mathematics education (pp. 142–157). Berlin: Springer.

    Chapter  Google Scholar 

  • Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.

    Article  Google Scholar 

  • Fujita, T., & Jones, K. (2003). Critical review of geometry in current textbooks in lower secondary schools in Japan and the UK. In N. A. Pateman, B. J. Dougherty & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education. Honolulu, HI, USA.

  • Furinghetti, F., & Morselli, F. (2011). Beliefs and beyond: Hows and whys in the teaching of proof. ZDM—The International Journal on Mathematics, 43(4), 587–599.

    Article  Google Scholar 

  • Glaser, B. G., & Strauss, A. L. (2009). The discovery of grounded theory: Strategies for qualitative research. Piscataway: Transaction Books.

    Google Scholar 

  • Hanna, G., & de Villiers, M. (2012). Proof and proving in mathematics education. The Netherlands: Springer.

    Book  Google Scholar 

  • Harel, G. (2008). DNR perspective on mathematics curriculum and instruction. Part I: Focus on proving. ZDM—The International Journal on Mathematics Education, 40(3), 487–500.

    Article  Google Scholar 

  • Harel, G., & Rabin, J. M. (2010). Teaching practices associated with the authoritative proof scheme. Journal for Research in Mathematics Education, 41(1), 14–19.

    Google Scholar 

  • 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 (Vol. 2, pp. 805–842). Charlotte: Information Age Publishing.

    Google Scholar 

  • Heinze, A., & Reiss, K. (2004). The teaching of proof at the lower secondary level—A video study. ZDM—The International Journal on Mathematics, 36, 98–104.

    Article  Google Scholar 

  • Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549.

    Article  Google Scholar 

  • Herbst, P. G. (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(3), 283–312.

    Article  Google Scholar 

  • Herbst, P. G. (2002). Engaging students in proving: A double bind on the teacher. Journal for Research in Mathematics Education, 33(3), 176–203.

  • Herbst, P. G. (2004). Interactions with diagrams and the making of reasoned conjectures in geometry. ZDM—The International Journal on Mathematics Education, 36(5), 129–139.

    Article  Google Scholar 

  • Herbst, P. G. (2006). Teaching geometry with problems: Negotiating instructional situations and mathematical tasks. Journal for Research in Mathematics Education, 37, 313–347.

  • Jahnke, H. N., & Wambach, R. (2013). Understanding what a proof is: A classroom-based approach. ZDM—The International Journal on Mathematics, 45(3), 469–482.

    Article  Google Scholar 

  • Knuth, E. J. (2002). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5(1), 61–88.

    Article  Google Scholar 

  • Laborde, C. (2005). The hidden role of diagrams in students’ construction of meaning in geometry. In J. Kilpatrick, C. Hoyles, & O. Skovsmose (Eds.), Meaning in mathematics education (pp. 159–179). New York: Springer.

    Chapter  Google Scholar 

  • Larson, R., Boswell, L., Kanold, T. D., & Stiff, L. (2007). Mcdogal littell geometry. Dallas, TX: McDougal Littell.

  • Lin, F. L., Yang, K. L., Lee, K. H., Tabach, M., & Stylianides, G. (2012). Principles of task design for conjecturing and proving. Proof and proving in mathematics education (pp. 305–325). The Netherlands: Springer.

    Google Scholar 

  • Mariotti, M. A. (1995). Images and concepts in geometrical reasoning. In R. Sutherland & J. Mason (Eds.), Exploiting mental imagery with computers in mathematics education (pp. 97–116). Berlin: Springer.

    Chapter  Google Scholar 

  • Mariotti, M. A. (2006). Proof and proving in mathematics education. In Á. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173–204). Rotterdam: Sense Publishers.

    Google Scholar 

  • McCrone, S. M., & Martin, T. S. (2004). Assessing high school students’ understanding of geometric proof. Canadian Journal of Science, Mathematics and Technology Education, 4(2), 223–242.

    Article  Google Scholar 

  • McCrone, S. M., Martin, T. S., Dindyal, J., & Wallace, M. L. (2002). An investigation of classroom factors that influence proof construction ability. Paper presented at the 24th Annual Meeting of Psychology of Mathematics Education, Athens, Georgia.

  • NCTM. (2000). Principles and standards for school mathematics. Reston: Author.

    Google Scholar 

  • Otten, S., Males, L. M., & Gilbertson, N. J. (2013). The introduction of proof in secondary geometry textbooks. International Journal of Educational Research, 64, 107–118.

  • Patton, M. Q. (2001). Qualitative research and evaluation methods. Thousands Oaks: Sage Publications.

    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—The International Journal on Mathematics Education, 45(5), 685–698.

    Article  Google Scholar 

  • Schoenfeld, A. H. (1986). On having and using geometric knowledge. In H. J. (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 225-264). Hillsdale: Erlbaum.

  • Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166.

    Article  Google Scholar 

  • Siu, M. K. (2008). Proof as a practice of mathematical pursuit in a cultural, socio-political and intellectual context. ZDM—The International Journal on Mathematics Education, 40(3), 355–361.

    Article  Google Scholar 

  • Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344–350.

    Google Scholar 

  • Stake, R. E. (2013). Multiple case study analysis. London: Guilford Press.

    Google Scholar 

  • Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.

    Google Scholar 

  • Stylianides, G. J. (2008). Investigating the guidance offered to teachers in curriculum materials: The case of proof in mathematics. International Journal of Science and Mathematics Education, 6(1), 191–215.

    Article  Google Scholar 

  • Stylianou, D. A., Blanton, M. L., & Knuth, E. J. (Eds.). (2009). Teaching and larning proof across the grades: A K-16 perspective. New York: Routledge.

  • Tarr, J. E., Ross, D. J., McNaught, M. D., Chavez, O., Grouws, D. A., Reys, R. E., Sears, R., Taylan, R. D. (2010). Identification of student-and teacher-level variables in modeling variation of mathematics achievement data. Online Submission, 32-32.

  • Thompson, D. R. (2012). Reasoning and justification in the secondary mathematics classroom. In B. Kaur & T. Lam (Eds.), Reasoning, communication and connections in mathematics (pp. 88–106). Danvers: World Scientific Publishing Co., Pte. Ltd.

    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.

    Article  Google Scholar 

  • Weiss, I. R., Banilower, E. R., McMahon, K. C., & Smith, P. S. (2001). Report of the 2000 national survey of science and mathematics education. Chapel Hill: Horizon Research Inc.

    Google Scholar 

  • Wilson, L. (1994). What gets graded is what gets valued. Mathematics Teacher, 87(6), 412–414.

    Google Scholar 

  • Zaslavsky, O., Nickerson, S. D., Stylianides, A. J., Kidron, I., & Winicki-Landman, G. (2012). The need for proof and proving: Mathematical and pedagogical perspectives. In Proof and Proving in Mathematics Education (pp. 215–229). The Netherlands: Springer

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Authors and Affiliations

  1. Department of Secondary Education, College of Education, University of South Florida, 4202 E. Fowler Ave., EDU105, Tampa, FL, 33620, USA

    Ruthmae Sears

  2. Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249-0624, USA

    Óscar Chávez

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  1. Ruthmae Sears
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  2. Óscar Chávez
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Correspondence to Ruthmae Sears.

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Sears, R., Chávez, Ó. Opportunities to engage with proof: the nature of proof tasks in two geometry textbooks and its influence on enacted lessons. ZDM Mathematics Education 46, 767–780 (2014). https://doi.org/10.1007/s11858-014-0596-9

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