Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers pp 19-45 | Cite as

# Connecting the Group Theory Concept Assessment to Core Concepts at the Secondary Level

## Abstract

The Group Theory Concept Assessment (GTCA) was developed to meaningfully capture student conceptions around fundamental concepts in introductory group theory. In this chapter, we share results from a large-scale implementation of the GTCA with 375 students across 30 undergraduate institutions in the USA. We include a breakdown of performance based on major. We pair these findings with a detailed look at several GTCA tasks with direct connections to the secondary curriculum. Student conceptions around prior content, including function and operation, often mediated student performance on group theory tasks. Functions play an essential role in student approaches to building isomorphisms, exploring consequences of homomorphisms, and identifying kernels. Binary operations play an essential role in student approaches to exploring properties (such as the associative property), finding identities and inverses, defining groups, and identifying subgroups. We share results from both the multiple-choice inventory and follow-up interviews to illustrate some of these connections. We conclude with a discussion of implications for the abstract algebra classroom, with a focus on opportunities for backward transfer *to* secondary content that can be embedded in conceptual explorations of group theory topics.

## Keywords

Group theory Functions Binary operations Backward transfer## References

- Akkoç, H., & Tall, D. (2002). The simplicity, complexity and complication of the function concept. In A. D. Cockburn & E. Nardi (Eds.),
*Proceedings of the 26th conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 25–32). Norwich: PME.Google Scholar - Blair, R., Kirkman, E. E., & Maxwell, J. W. (2013).
*Statistical abstract of undergraduate programs in the mathematical sciences in the United States: Fall 2010 CBMS survey*. Providence, RI: American Mathematical Society.Google Scholar - Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function.
*Educational Studies in Mathematics, 23*(3), 247–285.Google Scholar - Brown, A., DeVries, D. J., Dubinsky, E., & Thomas, K. (1997). Learning binary operations, groups, and subgroups.
*The Journal of Mathematical Behavior, 16*(3), 187–239.Google Scholar - Clement, L. L. (2001). What do students really know about functions?
*The Mathematics Teacher, 94*(9), 745.Google Scholar - Common Core State Standards Initiative. (2010).
*Common core state standards for mathematics*. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers Retrieved from http://www.corestandards.org Google Scholar - Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory.
*Educational Studies in Mathematics, 27*(3), 267–305.Google Scholar - Elia, I., Panaoura, A., Eracleous, A., & Gagatsis, A. (2007). Relations between secondary pupils’ conceptions about functions and problem solving in different representations.
*International Journal of Science and Mathematics Education, 5*(3), 533–556.Google Scholar - Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept.
*Journal for Research in Mathematics Education, 24*, 94–116.Google Scholar - Hazzan, O. (1999). Reducing abstraction level when learning abstract algebra concepts.
*Educational Studies in Mathematics, 40*(1), 71–90.Google Scholar - Hohensee, C. (2014). Backward transfer: An investigation of the influence of quadratic functions instruction on students’ prior ways of reasoning about linear functions.
*Mathematical Thinking and Learning, 16*(2), 135–174.Google Scholar - Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.),
*Conceptual and procedural knowledge: The case of mathematics*(pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study.
*Journal for Research in Mathematics Education, 31*, 500–507.Google Scholar - Lakoff, G., & Núñez, R. E. (2000).
*Where mathematics comes from: How the embodied mind brings mathematics into being*. New York, NY: Basic Books.Google Scholar - Larsen, S. (2010). Struggling to disentangle the associative and commutative properties.
*For the Learning of Mathematics, 30*(1), 37–42.Google Scholar - Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching.
*Review of Educational Research, 60*(1), 1–64.Google Scholar - Leron, U., Hazzan, O., & Zazkis, R. (1995). Learning group isomorphism: A crossroads of many concepts.
*Educational Studies in Mathematics, 29*(2), 153–174.Google Scholar - Melhuish, K. (2015). The design and validation of a group theory concept inventory (Doctoral dissertation).
*Dissertations and theses*. Retrieved from: http://pdxscholar.library.pdx.edu/open_access_etds/2490 - Melhuish, K. M., & Fagan, J. (2017). Exploring student conceptions of binary operation. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.),
*Proceedings of the 20th annual conference on research in undergraduate mathematics education*(pp. 166–180). San Diego, CA: SIGMAA.Google Scholar - Melhuish, K., Larsen, S., & Cook, S. (2018). When students prove a theorem without explicitly using a necessary condition: Digging into a subtle problem from practice. Manuscript under review.Google Scholar
- Mevarech, Z. R. (1983). A deep structure model of students’ statistical misconceptions.
*Educational Studies in Mathematics, 14*(4), 415–429.Google Scholar - Nardi, E. (2000). Mathematics undergraduates’ responses to semantic abbreviations, ‘geometric’ images and multi-level abstractions in group theory.
*Educational Studies in Mathematics, 43*(2), 169–189.Google Scholar - Novotná, J., & Hoch, M. (2008). How structure sense for algebraic expressions or equations is related to structure sense for abstract algebra.
*Mathematics Education Research Journal, 20*(2), 93–104.Google Scholar - Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. Carlson & C. Rasmussen (Eds.),
*Making the connection: Research and teaching in undergraduate mathematics education*(pp. 27–42). Washington, DC: Mathematical Association of America.Google Scholar - Philipp, R. A. (1992). The many uses of algebraic variables.
*The Mathematics Teacher, 85*(7), 557–561.Google Scholar - Schwarz, B., Dreyfus, T., & Bruckheimer, M. (1990). A model of the function concept in a three-fold representation.
*Computers & Education, 14*(3), 249–262.Google Scholar - Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*(1), 1–36.Google Scholar - Slavit, D. (1997). An alternate route to the reification of function.
*Educational Studies in Mathematics, 33*(3), 259–281.Google Scholar - Slavit, D. (1998). The role of operation sense in transitions from arithmetic to algebraic thought.
*Educational Studies in Mathematics, 37*(3), 251–274.Google Scholar - 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.Google Scholar - Tall, D., McGowen, M., & DeMarois, P. (2000). The function machine as a cognitive root for the function concept. In M. L. Fernandez (Ed.),
*Proceedings of the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 255–261). Tucson, AZ: PME-NA.Google Scholar - Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum.
*Research in Collegiate Mathematics Education, 1*, 21–44.Google Scholar - Ticknor, C. S. (2012). Situated learning in an abstract algebra classroom.
*Educational Studies in Mathematics, 81*(3), 307–323.Google Scholar - Tirosh, D., Hadass, R., & Movshovitz-Hadar, N. (1991). Overcoming overgeneralizations: The case of commutativity and associativity. In F. Furinghetti (Ed.),
*Proceedings of the fifteenth annual conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 310–315). Assisi: International Group for the Psychology of Mathematics Education.Google Scholar - Vinner, S. (1983). Concept definition, concept image and the notion of function.
*International Journal of Mathematical Education in Science and Technology, 14*(3), 293–305.Google Scholar - von Glasersfeld, E. (1995).
*Radical constructivism: A way of knowing and learning*. London: Falmer Press.Google Scholar - Wasserman, N. H. (2017). Making sense of abstract algebra: Exploring secondary teachers’ understandings of inverse functions in relation to its group structure.
*Mathematical Thinking and Learning, 19*(3), 181–201.Google Scholar - Weber, K., & Larsen, S. (2008). Teaching and learning abstract algebra. In M. Carlson & C. Rasmussen (Eds.),
*Making the connection: Research and teaching in undergraduate mathematics*. Washington, DC: Mathematical Association of America.Google Scholar - Warren, E. (2003). The role of arithmetic structure in the transition from arithmetic to algebra.
*Mathematics Education Research Journal, 15*(2), 122–137.Google Scholar - Zandieh, M., Ellis, J., & Rasmussen, C. (2017). A characterization of a unified notion of mathematical function: The case of high school function and linear transformation.
*Educational Studies in Mathematics, 95*(1), 21–38.Google Scholar - Zaslavsky, O., & Peled, I. (1996). Inhibiting factors in generating examples by mathematics teachers and student teachers: The case of binary operation.
*Journal for Research in Mathematics Education, 27*, 67–78.Google Scholar