Due to the centrality of reasoning and proof (RP) in the practice of mathematics (Schoenfeld, 2009) and the importance of written textbooks in teachers’ instructional practices (Grouws & Smith, 2000; Tarr et al., 2008) researchers have recently turned their attention to analyzing curricular resources for instances of RP (Author, Date, Date; Johnson, Thompson, & Senk, 2010; Stylianides, 2009). Both reform-oriented curricula (programs developed with funds from the National Science Foundation) and conventional mathematics textbooks have been the focus of these studies.

            Stylianides (2009) found that students had differing RP opportunities across grade levels and content areas within the same middle school reform-oriented curriculum. Johnson et al. (2010) reported that only 5.5% of the tasks appearing in high school mathematics textbooks involved RP and as the mathematical content became more difficult, students had more opportunities to engage in RP. Author (Date) examined RP in student tasks and textbook narrative in a reform-oriented textbook unit, a hybrid[1] textbook unit, and a conventional textbook all at the high school level. His research suggests that students had many more opportunities to engage in RP in the reform-oriented textbook unit than in the other two units.

            Despite this work in analyzing the student textbook for RP, less work had been conducted in analyzing the materials that support instruction. Stylianides (2007) is an exception in this area as he examined the nature of support for proof tasks in number, algebra, and geometry units within a reform-oriented middle school mathematics program. Proof tasks were those that provided students with opportunities to construct arguments. He looked at whether resource materials provided students with solutions only or solutions with support. A total of 90% of 227 proof tasks provided teachers with solutions only. The remaining 10% of the tasks contained at least one of three forms of additional guidance.

            The lack of research on RP teacher support at the high school level and the focus by Stylianides (2007) solely on support for proof tasks led to the following research question: What is the nature of teacher support for RP student tasks in the areas of identifying patterns, conjecturing, and proving within three different reform-oriented high school textbook units?

Framework

            This study used a framework adapted from Stylianides (2007) to code RP teacher support, however, instead of examining only those tasks that involved proof, this study investigated the nature of support for the identification of definite and plausible patterns, testing and developing conjectures, as well as proof. The first component of his framework involved coding proof tasks into solution only and solution with support components. Those resource materials that provided support were further categorized using Stylianides (2007) three forms of additional guidance (FAG) (e.g., “explanations about why students’ engagement in a proof task matters” [Stylianides, 2007, p. 197]).  Some resource materials associated with RP tasks did not contain a solution and hence were coded as no solution. Additionally, categories will be developed based upon the support in the materials using a grounded theory approach (Glaser & Strauss, 1967). Thus, a more elaborate framework will be developed based upon the nature of support across the three units.

Methods

            Student textbook questions related to RP in three reform-oriented units were coded as part of a series of earlier studies (Author, Date, Date). The first stage of this study involved the placement of teacher resource materials related to these RP student textbook questions into the previously coded electronic files. The second stage involved coding no solution, solution only, and solution with support categories using HyperResearch (ResearchWare, 2009) by the three study authors. The last stage of the study involves more detailed coding of the support for Stylianides’ (2007) FAG as well as others that emerge from the actual teacher support materials.

Data Sources

            The teacher support appeared in a polynomial functions unit in the teacher’s edition of three reform-oriented textbooks: Core-Plus Mathematics Course 3 (CPM) (Fey et al., 2009), University of Chicago School Mathematics Project Advanced Algebra (UCSMP) (Flanders et al., 2010), and Center for Mathematics Education Algebra 2 (CME) (Education Development Center, 2009). The teacher’s edition for each unit contains both the student textbook as well as the teacher support materials. Each of these three programs has other resources for the teacher such as assessment resources, but these were not analyzed as part of this study.

Results

            This proposal contains preliminary results in the frequency of identifying patterns, developing and testing conjectures, and creating arguments that did not contain solutions, solutions only, and solutions with support. The presentation itself will not only share these findings, but also more detailed results with regard to the similarities and differences in the nature of support among the three curricula as well as between different RP components. The majority of RP tasks (388/569 or 68%) across all three programs contained only solutions. The teacher support for RP tasks varied by program at 15% (43/281) for CPM, 31% (26/83) for UCSMP, and 45% (92/205) for CME. Not only did CPM have the lowest overall teacher support for RP, it also had the highest percentage of tasks that did not contain solutions at 7% (20/281). Among all three textbook units, CME contained the highest percentage of identifying pattern (30/51 or 59%) and constructing argument tasks (61/142 or 43%) that contained teacher support, but UCSMP contained the highest percentage of conjecturing tasks that were supported (2/8 or 25%).

Scientific Importance

            Recently there has been a greater emphasis on examining teacher support materials especially in regard to reform-oriented mathematics programs (Remillard, 2000; Stein & Kim, 2009) and the role of teacher materials in promoting teacher learning (Davis & Krajcik, 2005). This research is in a similar area and builds on the work of Stylianides (2007) and extends that work by examining support in categories of RP other than proof at the high school level. Work of this nature is an important first step in understanding the enacted curriculum. Moreover, this work has benefits for curriculum design around RP and assisting teachers in their RP work with students.

References

Author. (Date).

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Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3-14.

Education Development Center. (2009). Algebra 2. Boston: Pearson Education.

Fey, J. T., Hirsch, C. R., Hart, E. W., Schoen, H. L., Watkins, A. E., Ritsema, B. E., et al. (2009). Core-plus mathematics: Contemporary mathematics in context: Course 3 (2^nd ed.). New York: Glencoe/McGraw-Hill.

Flanders, J., Lassak, M., Sech, J., Eggerding, M., Karafiol, P. J., McMullin, L., et al. (2010). The University of Chicago school mathematics project: Advanced algebra (3^rd ed.). Chicago: Wright Group.

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Grouws, D. A., & Smith, M. S. (2000). Findings from NAEP on the preparation and practices of mathematics teachers. In E. A. Silver & P. A. Kenney (Eds.), Results from the Seventh Mathematics Assessment of the National Assessment of Education Progress (pp. 107-141). Reston, VA: National Council of Teachers of Mathematics.

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Schoenfeld, A. H. (2009). Series editor’s foreword. In D. A. Stylianou, M. L. Blanton, E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. xii-xvi). New York: Routledge.

Stein, M. K., & Kim, G. (2009). The role of mathematics curriculum materials in large-scale urban reform: An analysis of demands and opportunities for teacher learning. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 37-55). New York: Routledge.

Stylianides, G. J. (2007). Investigating the guidance offered to teachers in curriculum materials: The case of proof in mathematics. International Journal of Science and Mathematics Education, 6, 191-215.

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

Tarr, J. E., Reys, R. E., Reys, B. J., Chavez, O., Shih, J., & Osterlind, S. J. (2008). The impact of middle-grades mathematics curricula and the classroom learning environment on student achievement. Journal for Research in Mathematics Education, 39, 247-280.