Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers pp 335-361 | Cite as

# Exploring an Instructional Model for Designing Modules for Secondary Mathematics Teachers in an Abstract Algebra Course

## Abstract

In this paper, we elaborate on a theoretically motivated instructional model for designing modules for secondary mathematics teachers in an abstract algebra course. We illustrate this model by elaborating on two modules, Functions and *k*-Product Property, and report some findings from a small-scale study with two secondary mathematics teachers. Findings indicate the potential utility of the instructional model for influencing teaching practice, with various types of instructional changes being identified. We then discuss implications for the teaching of advanced content courses, such as abstract algebra, as they relate to secondary teacher education.

## Keywords

Abstract algebra Instructional model Secondary mathematics teacher education## References

- Ball, D. L., & Bass, H. (2009).
*With an eye on the mathematical horizon: Knowing mathematics for teaching to learners’ mathematical futures*. Paper presented at the 43rd Jahrestagung der Gesellschaft fur Didaktik der Mathematik, Oldenburg, Germany.Google Scholar - Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?
*Journal of Teacher Education, 59*(5), 389–407.CrossRefGoogle Scholar - Barnett, S. M., & Ceci, S. J. (2002). When and where do we apply what we learn?: A taxonomy for far transfer. Psychological Bulletin, 128(4), 612.CrossRefGoogle Scholar
- Bremigan, E. G., Bremigan, R. J., & Lorch, J. D. (2011).
*Mathematics for secondary school teachers*. Washington, DC: Mathematical Association of America.Google Scholar - Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research.
*Educational Researcher, 32*(1), 9–13.CrossRefGoogle Scholar - Conference Board of Mathematical Sciences (CBMS). (2012).
*The mathematical education of teachers II*. Providence, RI and Washington, DC: American Mathematical Society and Mathematical Association of America.CrossRefGoogle Scholar - Creswell, J. W. (2012).
*Educational research: Planning, conducting, and evaluating quantitative and qualitative research*(4th ed.). Upper Saddle River, NJ: Pearson.Google Scholar - Deng, Z. (2007). Knowing the subject matter of a secondary-school science subject.
*Journal of Curriculum Studies, 39*, 503–535.CrossRefGoogle Scholar - Even, R. (2011). The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: Practitioner’s views.
*ZDM, 43*(6–7), 941–950.CrossRefGoogle Scholar - Goulding, M., Hatch, G., & Rodd, M. (2003). Undergraduate mathematics experience: Its significance in secondary mathematics teacher preparation.
*Journal of Mathematics Teacher Education, 6*(4), 361–393.CrossRefGoogle Scholar - Heid, M. K., Blume, G. W., Zbiek, R. M., & Edwards, B. S. (1998). Factors that influence teachers learning to do interviews to understand students’ mathematical understandings.
*Educational Studies in Mathematics, 37*(3), 223–249.CrossRefGoogle Scholar - Heid, M. K., Wilson, P. S., & Blume, G. W. (Eds.). (2015).
*Mathematical understanding for secondary teaching: A framework and classroom-based situations*. Charlotte, NC: Information Age Publishing.Google Scholar - Klein, F. (1932).
*Elementary mathematics from an advanced standpoint: Arithmetic, Algebra, Analysis*(E. R. Hedrick & C. A. Noble, Trans.). Mineola, NY: Macmillan.Google Scholar - McCrory, R., Floden, R., Ferrini-Mundy, J., Reckase, M. D., & Senk, S. L. (2012). Knowledge of algebra for teaching: A framework of knowledge and practices.
*Journal for Research in Mathematics Education, 43*(5), 584–615.CrossRefGoogle Scholar - Monk, D. H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement.
*Economics of Education Review, 13*(2), 125–145.CrossRefGoogle Scholar - Moreira, P. C., & David, M. M. (2008). Academic mathematics and mathematical knowledge needed in school teaching practice: Some conflicting elements.
*Journal of Mathematics Teacher Education, 11*(1), 23–40.CrossRefGoogle Scholar - Perkins, D., & Salomon, G. (1992).
*Transfer of learning*,*International encyclopedia of education*(2nd ed.). Oxford, UK: Pergamon Press.Google Scholar - Rasmussen, C., Wawro, M., & Zandieh, M. (2015). Examining individual and collective level mathematical progress.
*Educational Studies in Mathematics, 88*(2), 259–281.CrossRefGoogle Scholar - Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi.
*Journal of Mathematics Teacher Education, 8*(3), 255–281.CrossRefGoogle Scholar - Sultan, A., & Artzt, A. F. (2011).
*The mathematics that every secondary school math teacher needs to know*. New York, NY: Routledge.Google Scholar - Ticknor, C. S. (2012). Situated learning in an abstract algebra classroom.
*Educational Studies in Mathematics, 81*(3), 307–323.CrossRefGoogle Scholar - Usiskin, Z., Peressini, A., Marchisotto, E. A., & Stanley, D. (2003).
*Mathematics for high school teachers: An advanced perspective*. Upper Saddle River, NJ: Pearson.Google Scholar - 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 - Wasserman, N. (2018a). Nonlocal mathematical knowledge for teaching.
*Journal of Mathematical Behavior, 49*(1), 116–128.CrossRefGoogle Scholar - Wasserman, N. (2018b). Exploring the secondary teaching of functions in relation to the learning of abstract algebra. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.),
*Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics Education (RUME)*(pp. 687–694). San Diego, CA: RUME.Google Scholar - Wasserman, N., Fukawa-Connelly, T., Villanueva, M., Mejia-Ramos, J. P., & Weber, K. (2017). Making real analysis relevant to secondary teachers: Building up from and stepping down to practice.
*PRIMUS, 27*(6), 559–578.CrossRefGoogle Scholar - 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 - Wasserman, N., Weber, K., Villanueva, M., & Mejia-Ramos, J. P. (2018). Mathematics teachers’ views about the limited utility of real analysis: A transport model hypothesis.
*Journal of Mathematical Behavior, 50*(1), 74–89.CrossRefGoogle Scholar - Zazkis, R., & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice: Perceptions of secondary mathematics teachers.
*Mathematical Thinking and Learning, 12*(4), 263–281.CrossRefGoogle Scholar