Using Geometric Habits of Mind to Connect Geometry from a Transformation Perspective to Graph Transformations and Abstract Algebra

  • Yvonne LaiEmail author
  • Allan Donsig
Part of the Research in Mathematics Education book series (RME)


The Common Core State Standards for Mathematics advocate for geometry from a transformation approach, and we illustrate how this movement opens two important mathematical opportunities. First, geometry from a transformation approach connects to algebra at the secondary level, especially through transformations of graphs; and second, geometry from a transformation approach connects to abstract algebra at the university level, especially through linear algebra and group theory. To make these connections visible, we propose an extension of Driscoll et al.’s (2007) Geometric Habits of Mind (GHOM) framework that can apply to graph transformations and ideas of abstract algebra. We then describe three specific contexts where the extended GHOMs link problem solving and theory building activities (Gowers, The two cultures of mathematics. Retrieved from, 2000) across geometry and algebra at the secondary level and abstract algebra at the university level. These contexts are: (1) reconciling three definitions of congruence, (2) using transformations to understand conventions for defining families of functions and shapes, and (3) analyzing plane transformations of the plane in terms of linear algebra. We conclude with some implications of our work for the preparation of secondary teachers.



We are grateful to the reviewers and the editor for helpful comments, and to Erin Baldinger for pointing us to Driscoll et al.’s (2007) Geometric Habits of Mind framework.


  1. Artin, M. (1991). Algebra. Upper Saddle River, NJ: Prentice Hall.Google Scholar
  2. Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. In G. Sykes & L. Darling-Hammond (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 3–32). San Francisco, CA: Jossey-Bass.Google Scholar
  3. Ball, D. L., & Forzani, F. (2009). The work of teaching and the challenge for teacher education. Journal of Teacher Education, 60(5), 497–511.CrossRefGoogle Scholar
  4. Bass, H. (2017). Designing opportunities to learn mathematics theory-building practices. Educational Studies in Mathematics, 95(3), 229–244.CrossRefGoogle Scholar
  5. Betz, W. (1909). Intuition and logic in geometry. The Mathematics Teacher, 2(1), 3–31.Google Scholar
  6. Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. Providence RI and Washington DC: American Mathematical Society and Mathematical Association of America.CrossRefGoogle Scholar
  7. Coxford, A., Usiskin, Z., & Hirschhorn, D. (1991). The University of Chicago School Mathematics Project: Geometry. Glenview, IL: Scott & Foresman.Google Scholar
  8. Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. The Journal of Mathematical Behavior, 15(4), 375–402.CrossRefGoogle Scholar
  9. Dorier, J. L., & Sierpinska, A. (2001). Research into the teaching and learning of linear algebra. In D. Holton (Ed.), The teaching and learning of mathematics at university level - An ICMI study. New ICMI Study Series (Vol. 7, pp. 255–273). Dordrecht: Kluwer.Google Scholar
  10. Douglas, L., & Picciotto, H. (August, 2017). Transformational proof in high school geometry: A guide for teachers and curriculum developers. Retrieved from Google Scholar
  11. Driscoll, M. J., DiMatteo, R. W., Nikula, J., & Egan, M. (2007). Fostering geometric thinking: A guide for teachers, grades 5–10. Portsmouth, NH: Heinemann.Google Scholar
  12. Dubinsky, E., & Mcdonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton (Ed.), The teaching and learning of mathematics at university level - An ICMI Study. New ICMI Study Series (Vol. 7, pp. 275–282). Dordrecht: Kluwer.Google Scholar
  13. Eureka Math. (2015). Geometry Module 1: Teacher materials. Retrieved from Google Scholar
  14. Gallian, J. (2013). Contemporary abstract algebra. Boston, MA: Brooks/Cole.Google Scholar
  15. Gowers, W.T. (2000). The two cultures of mathematics. Retrieved from Google Scholar
  16. Harel, G. (1998). Two dual assertions: The first on learning and the second on teaching (or vice versa). American Mathematical Monthly, 105(6), 497–507.CrossRefGoogle Scholar
  17. Harel, G., & Sowder, L. (2008). Toward a comprehensive perspective on proof. In F. Lester (Ed.), Handbook of research on teaching and learning mathematics (2nd ed.). Greenwich: Information Age Publishing.Google Scholar
  18. Heid, M. K., Wilson, P. & Blume, G. W. (Eds.). (2015). Mathematical understanding for secondary teaching: A framework and classroom-based situations. Charlotte, NC: Information Age Publishing.Google Scholar
  19. Johnson, H. L. (n.d.) Are transformations of functions giving your students trouble? Try a covariation approach. Retrieved from
  20. Klein, F. (1939/2004). Elementary mathematics from an advanced standpoint: Geometry (Vol. 2). Mineola, NY: Dover Publications.Google Scholar
  21. Mason J. (2018). Combining geometrical transformations: A meta-mathematical narrative. In R. Zazkis & P. Herbst (Eds.), Scripting approaches in mathematics education. Advances in Mathematics Education (pp. 21–51). Dordrecht: Springer.Google Scholar
  22. National Governors Association Center for Best Practices & Council of Chief State School. (2010). Common core state standards for mathematics. Washington, DC: Authors.Google Scholar
  23. Park City Mathematics Institute. (2016). Geometry transformed. Retrieved from Google Scholar
  24. Polya, G. (1954). Mathematics and plausible reasoning (Vol. 1). Princeton, NJ: Princeton University Press.Google Scholar
  25. Schoenfeld, 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, NY: MacMillan.Google Scholar
  26. Smith, M. S., & Stein, M. K. (2011). Five practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  27. Suydam, M. N. (1985). The shape of instruction in geometry: Some highlights from research. The Mathematics Teacher, 78(6), 481–486.Google Scholar
  28. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.CrossRefGoogle Scholar
  29. Tatto, M. T., Schwille, J., Senk, S., Ingvarson, L., Peck, R., & Rowley, G. (2008). Teacher education and development study in mathematics (TEDS-M): Conceptual framework. East Lansing, MI: Teacher Education and Development International Study Center, College of Education, Michigan State University.Google Scholar
  30. Ticknor, C. S. (2012). Situated learning in an abstract algebra classroom. Educational Studies in Mathematics, 81(3), 307–323.CrossRefGoogle Scholar
  31. Usiskin, Z. P., & Coxford, A. F. (1972). A transformation approach to tenth-grade geometry. The Mathematics Teacher, 65(1), 21–30.Google Scholar
  32. Van Hiele-Geldof, D. (1957). De didaktiek van de meetkunde in de eerste klas van het V.H.M.O. Unpublished doctoral dissertation, University of Utrecht.Google Scholar
  33. van der Waerden, B. L. (1931/1949). Modern algebra. New York, NY: Frederick Ungar Publishing Co.Google Scholar
  34. Wasserman, N. (2016). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science, Mathematics and Technology Education, 16(1), 28–47.CrossRefGoogle Scholar
  35. Wasserman, N., Weber, K., & McGuffey, W. (2017). Leveraging real analysis to foster pedagogical practices. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 20th Annual Conference on Research in Undergraduate Mathematics Education (RUME) (pp. 1–15). San Diego, CA: RUME.Google Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Nebraska-LincolnLincolnUSA

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