Geometry Students’ Arguments About a 1-Point Perspective Drawing
https://doi.org/10.17583/redimat.2017.2015
Downloads
Abstract
The practice of formulating and justifying claims is a fundamental aspect of doing mathematics, and in geometry, students’ use of diagrams is integral to how they establish arguments. We applied Toulmin’s model to examine 23 geometry students’ arguments about figures included in a 1-point perspective drawing. We asked how students’ arguments drew upon their knowledge of 1-point perspective and their use of the diagram provided with the problem. Students warranted their claims based upon their knowledge of perspective, both in an artistic context as well as from experiences in everyday life. Students engaged in multiple apprehensions of the diagram, including using the given features, adding features, or measuring components, to justify claims about the figures. This study illustrates the importance of students’ prior knowledge of a context for formulating arguments, as well as how that prior knowledge is integrated with students’ use of a geometric diagram.
Downloads
References
Bartolini Bussi, M. G. (1996). Mathematical discussion and perspective drawing in primary school. Educational Studies in Mathematics, 31, 11-41.
Google Scholar CrossrefBartolini Bussi, M. G. (1998). Joint activity in mathematics classrooms: A Vygotskian analysis. In F. Seeger, J. Vogt, & U. Waschescio (Eds.), The culture of the mathematics classroom (pp. 13–49). Cambridge, UK: Cambridge University Press.
Google Scholar CrossrefChazan, D. (1993). High school geometry students’ justifications for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359–387.
Google Scholar CrossrefClements, D. H., Swaminathan, S., Hannibal, M. A. Z., & Sarama, J. (1999). Young children’s concepts of shape. Journal for Research in Mathematics Education, 30(2), 192-212.
Google Scholar CrossrefCox, D. C. (2013). Similarity in middle school mathematics: At the crossroads of geometry and number. Mathematical Thinking and Learning, 15(1), 3–23.
Google Scholar CrossrefDavis, P. J. (2006). Mathematics and common sense: A case of creative tension. Boca Raton, FL: CRC Press.
Google Scholar CrossrefDimmel, J. K., & Herbst, P. G. (2015). The semiotic structure of geometry diagrams: How textbook diagrams convey meaning. Journal for Research in Mathematics Education, 46(2), 147–195.
Google Scholar CrossrefDrake, C., Land, T. J., Bartell, T. G., Aguirre, J. M., Foote, M. Q., McDuffie, A. R., & Turner, E. E. (2015). Three strategies for opening curriculum spaces. Teaching Children Mathematics, 21(6), 346-353.
Google Scholar CrossrefDuval, 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.
Google Scholar CrossrefForman, E. A., Larreamendy-Joerns, J., Stein, M. K., & Brown, C. A. (1998). “You’re going to want to find out which and prove it”: Collective argumentation in a mathematics classroom. Learning and Instruction, 8(6), 527-548.
Google Scholar CrossrefForman, E. A., McCormick, D. E., & Donato, R. (1997). Learning what counts as a mathematical explanation. Linguistics and Education, 9(4), 313-339.
Google Scholar CrossrefHallowell, D. A., Okamoto, Y., Romo, L. F., & La Joy, J. R. (2015). First-graders’ spatial-mathematical reasoning about plane and solid shapes and their representations. ZDM Mathematics Education, 47, 363-375.
Google Scholar CrossrefHerbst, 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(3), 283–312.
Google Scholar CrossrefHerbst, P. (2004). Interaction with diagrams and the making of reasoned conjectures in geometry. Zentralblatt für Didaktik der Mathematik, 36(5), 129-139.
Google Scholar CrossrefHerbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry: What is it that is going on for students? Cognition and Instruction, 24(1), 73–122.
Google Scholar CrossrefHollebrands, K. F., Conner, A., & Smith, R. C. (2010). The nature of arguments provided by college geometry students with access to technology while solving problems. Journal for Research in Mathematics Education, 41(4), 324-350.
Google Scholar CrossrefInglis, M., & Mejía-Ramos, J. P. (2009). On the persuasiveness of visual arguments in mathematics. Foundations of Science, 14(1–2), 97–110.
Google Scholar CrossrefInglis, M., Mejía-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3-21.
Google Scholar CrossrefJackson, K., Garrison, A., Wilson, J., Gibbons, L., & Shahan, E. (2013). Exploring relationships between setting up complex tasks and opportunities to learn in concluding whole-class discussions in middle grades mathematics instruction. Journal for Research in Mathematics Education, 44(4), 646–682.
Google Scholar CrossrefJahnke, H. N. (2008). Theorems that admit exceptions, including a remark on Toulmin. ZDM Mathematics Education, 40(3), 363-371.
Google Scholar CrossrefKrummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229-270). Hillsdale, NJ: Lawrence Erlbaum Associates.
Google Scholar CrossrefLave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.
Google Scholar CrossrefMarrades, R., & Gutiérrez, Á. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44, 87–125.
Google Scholar CrossrefMoore-Russo, D., Conner, A., & Rugg, K. I. (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction. Educational Studies in Mathematics, 76(1), 3-21.
Google Scholar CrossrefMoschkovich, J. (2012). How equity concerns lead to attention to mathematical discourse. In B. Herbel-Eisenmann, J. Choppin, D. Wagner, & D. Pimm (Eds.), Equity in discourse for mathematics education: Theories, practices, and policies (pp. 89-105). New York, NY: Springer.
Google Scholar CrossrefNational Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Google Scholar CrossrefNational Governor’s Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington DC: Author. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
Google Scholar CrossrefNetz, R. (1999). The shaping of deduction in Greek mathematics. Cambridge: Cambridge University Press.
Google Scholar CrossrefParzysz, B. (1988). “Knowing” vs “seeing”: Problems of the plane representation of space geometry figures. Educational Studies in Mathematics, 19(1), 79-92.
Google Scholar CrossrefSchoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334-370). New York: MacMillan.
Google Scholar CrossrefSinclair, N., Pimm, D., Skelin, M., & Zbiek, R. (2012). Developing essential understanding of geometry for teaching mathematics in grades 9-12. Reston, VA: NCTM.
Google Scholar CrossrefStephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428-464.
Google Scholar CrossrefStephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21(4), 459–490.
Google Scholar CrossrefToulmin, S. (1958). The uses of argument. New York, NY: Cambridge University Press.
Google Scholar CrossrefTurner, E. E., Drake, C., McDuffie, A. R., Aguirre, J., Bartell, T. G., & Foote, M. Q. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children’s multiple mathematics knowledge bases. Journal of Mathematics Teacher Education, 15, 67-82.
Google Scholar Crossrefvan den Heuvel-Panhuizen, M., Elia, I., & Robitzsch, A. (2015). Kindergartners’ performance in two types of imaginary perspective-taking. ZDM Mathematics Education, 47, 345-362.
Google Scholar CrossrefYackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. Journal of Mathematical Behavior, 21, 423-440.
Google Scholar CrossrefYackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.
Google Scholar CrossrefZahner, W. (2012). “Nobody can sit there”: Two perspectives on how mathematics problems in context mediate group problem solving discussions. REDIMAT–Journal of Research in Mathematics Education, 1(2), 105-135.
Google Scholar CrossrefDownloads
Published
Almetric
Dimensions
Issue
Section
License
Authors retain copyright and grant the journal the right of first publication but allow anyone to share: (unload, , reprint, distribute and/or copy) and adapt (remix, transform reuse, modify,) for any proposition, even commercial, always quoting the original source.