Visualizing Mathematics pp 277-308 | Cite as

# Geometric Modeling of Mesospace Objects: A Task, its Didactical Variables, and the Mathematics at Stake

## Abstract

For decades, mathematics educators have been interested in engaging K-12 students in the practice of creating and using mathematical models. What might this look like in the context of geometry? Inspired by claims that students come to secondary school with knowledge of three-dimensional space that can be leveraged to engage them in modeling, we wondered what it would take to have them do so. We designed a communication task aimed at engaging teenagers in the geometric modeling of mesospace objects—three-dimensional objects of scale comparable to that of the human body. Specifically, we asked a group of teenagers to plan and enact the movement of furniture through a narrow staircase in a residential home. In this paper, we present our original design considerations, an analysis of the teens’ work, and a set of didactical variables that this analysis led us to believe need to be considered to ensure that such an activity engage teenagers in the geometric modeling of mesospace objects. The paper concludes with a discussion of the implications for research on a modeling approach to the teaching and learning of geometry.

## Keywords

Mesospace 3D geometry Modeling Communication Game Diagram Task Didactical variable Design Calculation Sketch Angle Rotation Staircase Instructions Experiential world Multimodal modeling Scale Conception of figure Milieu Devolution Macrospace Microspace Furniture Moving Movers Boxspring Couch Tabletop## References

- Acredolo, L. (1981). Small- and large-scale spatial concepts in infancy and childhood. In L. Liben, A. Patterson, & N. Newcombe (Eds.),
*Spatial representation and behavior across the life span: Theory and application*(pp. 63–81). New York: Academic.Google Scholar - Balacheff, N., & Gaudin, N. (2010). Modeling students’ conceptions: The case of function.
*CBMS Issues in Mathematics Education, 16*, 207–234.CrossRefGoogle Scholar - Battista, M. T. (2007). The development of geometric and spatial thinking. In F. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 843–908). Charlotte, NC: NCTM/Information Age Publishing.Google Scholar - Baxter, J. A., & Williams, S. (2010). Social and analytic scaffolding in middle school mathematics: Managing the dilemma of telling.
*Journal of Mathematics Teacher Education, 13*(1), 7–26.CrossRefGoogle Scholar - Behr, M., & Post, T. (1992). Teaching rational number and decimal concepts. In T. Post (Ed.),
*Teaching mathematics in grades K-8: Research-based methods*(2nd ed., pp. 201–248). Boston: Allyn and Bacon.Google Scholar - Berthelot, R., & Salin, M. H. (1998). The role of pupils’ spatial knowledge in the elementary teaching of geometry. In C. Mammana & V. Villani (Eds.),
*Perspectives on the teaching of geometry for the 21st century: An ICMI study*(pp. 71–77). Dordrecht, The Netherlands: Kluwer Academic.Google Scholar - Blum, W., & Ferri, R. B. (2009). Mathematical modelling: Can it be taught and learnt?
*Journal of Mathematical Modelling and Application, 1*(1), 45–58.Google Scholar - Blum, W., Galbraith, P. L., Henn, H. W., & Niss, M. (2007).
*Modelling and applications in mathematics education*. New York: Springer.CrossRefGoogle Scholar - Boyd, C., Burrill, G., Cummins, J., Kanold, T., & Malloy, C. (1998).
*Geometry: Integration, applications, connections*. New York: Glencoe-McGraw Hill.Google Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics: Didactique des mathématiques 1970–1990 (*N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. and Trans.). Dordrecht, The Netherlands: Kluwer.Google Scholar - Brousseau, G., & Warfield, V. M. (1999). The case of Gaël.
*The Journal of Mathematical Behavior, 18*(1), 7–52.CrossRefGoogle Scholar - Carraher, D., & Schliemann, A. (2007). Early algebra and algebraic reasoning. In F. Lester (Ed.),
*Second handbook for research on mathematics teaching and learning*(pp. 669–705). Charlotte, NC: Information Age.Google Scholar - Chávez, Ó., Papick, I., Ross, D. J., & Grouws, D. A. (2011). Developing fair tests for mathematics curriculum comparison studies: The role of content analyses.
*Mathematics Education Research Journal, 23*(4), 397–416.CrossRefGoogle Scholar - Chazan, D., Herbst, P., & Clark, L. (2016). Research on the teaching of mathematics: A call to theorize the role of society and schooling in mathematics. In D. Gitomer & C. Bell (Eds.),
*Handbook of research on teaching*(5th ed., pp. 1039–1097). Washington, DC: AERA.CrossRefGoogle Scholar - Coliva, A. (2012). Human diagrammatic reasoning and seeing-as.
*Synthese, 186*(1), 121–148.CrossRefGoogle Scholar - Collins, H. (2010).
*Tacit and explicit knowledge*. Chicago: University of Chicago Press.CrossRefGoogle Scholar - d'Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics.
*For the Learning of Mathematics, 5*(1), 44–48.Google Scholar - Dennis, J. (1970). Mondrian and the object art of the sixties.
*Art Journal, 29*(3), 297–302.CrossRefGoogle Scholar - Dimmel, 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.CrossRefGoogle Scholar - Gerdes, P. (1986). How to recognize hidden geometrical thinking: A contribution to the development of anthropological mathematics.
*For the Learning of Mathematics, 6*(2), 10–12.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 - Grugnetti, L., & Jaquet, F. (2005). A mathematical competition as a problem solving and a mathematical education experience.
*Journal of Mathematical Behavior, 24*(3–4), 373–384. https://doi.org/10.1016/j.jmathb.2005.09.012CrossRefGoogle Scholar - Halverscheid, S. (2008). Building a local conceptual framework for epistemic actions in a modelling environment with experiments.
*ZDM Mathematics Education, 40*(2), 225–234.CrossRefGoogle Scholar - Hanna, G., & Jahnke, H. N. (2007). Proving and modelling. In W. Blum et al. (Eds.),
*Modelling and applications in mathematics education*(pp. 145–152). New York: Springer.CrossRefGoogle Scholar - Hegarty, M., Montello, D. R., Richardson, A. E., Ishikawa, T., & Lovelace, K. (2006). Spatial abilities at different scales: Individual differences in aptitude-test performance and spatial-layout learning.
*Intelligence, 34*(2), 151–176.CrossRefGoogle Scholar - Herbst, P. (2006). Teaching geometry with problems: Negotiating instructional situations and mathematical tasks.
*Journal for Research in Mathematics Education, 37*(4), 313–347.Google Scholar - Herbst, P., & Chazan, D. (2012). On the instructional triangle and sources of justification for actions in mathematics teaching.
*ZDM Mathematics Education, 44*(5), 601–612.CrossRefGoogle 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.CrossRefGoogle Scholar - Herbst, P. G., Fujita, T., Halverscheid, S., & Weiss, M. (2017).
*The learning and teaching of geometry in secondary schools: A modeling perspective*. New York: Routledge.CrossRefGoogle Scholar - Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education.
*ZDM Mathematics Education, 38*(3), 302–310.CrossRefGoogle Scholar - Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. von Glasersfeld (Ed.),
*Radical constructivism in mathematics education*(pp. 53–74). Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar - Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema & T. A. Romberg (Eds.),
*Mathematics classrooms that promote understanding*(pp. 133–155). Mahwah, NJ: Erlbaum.Google Scholar - Lochhead, J., & Whimbey, A. (1987). Teaching analytical reasoning through thinking aloud pair problem solving. In J. E. Stice (Ed.),
*New directions in teaching and learning*(pp. 73–92). San Francisco: Jossey-Bass.Google Scholar - Masingila, J. O. (1994). Mathematics practice in carpet laying.
*Anthropology and Education Quarterly, 25*(4), 430–462.CrossRefGoogle Scholar - Meyer, D. (2015). Missing the promise of mathematical modeling.
*The Mathematics Teacher, 108*(8), 578–583.CrossRefGoogle Scholar - Millroy, W. L. (1991). An ethnographic study of the mathematical ideas of a group of carpenters.
*Learning and Individual Differences, 3*(1), 1–25.CrossRefGoogle Scholar - Müller, P., Wonka, P., Haegler, S., Ulmer, A., & Van Gool, L. (2006). Procedural modeling of buildings.
*ACM Transactions on Graphics, 25*(3), 614–623.CrossRefGoogle Scholar - National Governors Association (NGA) Center for Best Practices & Council of Chief State School Officers. (2010).
*Common core state standards for mathematics*. Washington, DC: Authors.Google Scholar - Otte, M. (2006). Mathematical epistemology from a Peircean semiotic point of view.
*Educational Studies in Mathematics, 61*, 11–38.CrossRefGoogle Scholar - Siegel, A. W. (1981). The externalization of cognitive maps by children and adults: In search of ways to ask better questions. In L. S. Liben, A. H. Patterson & N. Newcombe (Eds.),
*Spatial representation and behavior across the lifespan*(pp. 167–194). New York: Academic Press.Google Scholar - Simon, B. (2013). Mondrian's search for geometric purity: Creativity and fixation.
*American Imago, 70*(3), 515–555.CrossRefGoogle Scholar - Skovsmose, O. (2000). Aporism and critical mathematics education.
*For the Learning of Mathematics, 20*(1), 2–8.Google Scholar - Vergnaud, G. (1996). The theory of conceptual fields. In Steffe et al. (Eds.),
*Theories of mathematical learning*(pp. 219–239). Mahwah, NJ: Erlbaum.Google Scholar