An Evaluation of ULTRA; an Experimental Real Analysis Course Built on a Transformative Theoretical Model
- 14 Downloads
Most prospective secondary mathematics teachers in the United States complete a course in real analysis, yet view the content as unrelated to their future teaching. We leveraged a theoretically-motivated instructional model to design modules for a real analysis course that could inform secondary teachers’ actionable content knowledge and pedagogy. The theoretical model and designed curriculum launches the study of advanced mathematics content, in this case, real analysis, via authentic 7–12 classroom situations, abstracts the secondary mathematics, and uses that content to motivate the presentation of the advanced mathematics content. Subsequently, the curriculum then reconnects to practice, asking the teachers to translate ideas from real analysis in ways that are appropriate for teaching high school content to students. This study evaluated the success of the curriculum in terms of the students’ proficiency with real analysis, challenging secondary mathematics content, and the use of that content in teaching situations via both written post-tests and interviews and showed the viability of the model and experimental curriculum.
KeywordsSecondary teacher preparation Advanced mathematics Real analysis Pedagogical practice
This material is based upon work supported by the National Science Foundation under collaborative grants DUE 1524739, DUE 1524681 and DUE 1524619. Any opinions, findings, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Compliance with Ethical Standards
Conflict of Interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
- Alcock, L., & Weber, K. (2010). Referential and syntactic approaches to proving: Case studies from a transition-to-proof course. Research in collegiate mathematics education VII, 93–114.Google Scholar
- Begle, E. (1972). Teacher knowledge and pupil achievement in algebra (NLSMA technical report number 9). Palo Alto, CA: Stanford University, School Mathematics Study Group.Google Scholar
- Common Core State Standards in Mathematics (CCSSM). (2010). Retrieved from: http://www.corestandards.org/the-standards/mathematics. Last accessed May 30 2019.
- Ferrini-Mundy, J., & Findell, B. (2010). The mathematical education of prospective teachers of secondary school mathematics: Old assumptions, new challenges. CUPM discussion papers about mathematics and the mathematical sciences in 2010: What should students know, 31–41.Google Scholar
- Fitzpatrick, P. M. (2006). Advanced Calculus (2nd ed.). Providence, RI: American Mathematical Society.Google Scholar
- Iannone, P., & Inglis, M. (2010). Self efficacy and mathematical proof: Are undergraduate students good at assessing their own proof production ability? In S. Brown, S. Larsen, K. Keene, & K. Marrongelle (Eds.), Proceedings of the 13th annual conference on research in undergraduate mathematics education, 2010. North Carolina: Raliegh.Google Scholar
- McGuffey, W., Quea, R., Weber, K., Wasserman, N., Fukawa-Connelly, T., & Mejía-Ramos, J. P. (in press). Pre- and in-service teachers’ perceived value of an experimental real analysis course for teachers. International Journal of Mathematical Education in Science and Technology, XX(X), XXX. https://doi.org/10.1080/0020739X.2019.1587021.CrossRefGoogle Scholar
- Mejía-Ramos, J.P., & Weber, K. (in press). Mathematics majors’ diagram usage when writing proofs in calculus. Journal for Research in Mathematics Education, XX(X), pp. XXX.Google Scholar
- Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237–266). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
- TeachingWorks (2018). High leverage teaching practices. Downloaded from: http://www.teachingworks.org/work-of-teaching/high-leverage-practices. Accessed 31 May 2018.
- Wasserman, N., Weber, K., Fukawa-Connelly, T., & McGuffey, W. (2019). Designing advanced mathematics courses to influence secondary teaching: Fostering mathematics teachers’ ‘attention to scope’. Journal of Mathematics Teacher Education, 22(4), 379–406. https://doi.org/10.1007/s10857-019-09431-6.CrossRefGoogle Scholar
- Weber, K., Mejía-Ramos, J.P., Fukawa-Connelly, T., Wasserman, N. (submitted). Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions, domain restrictions, and the arcsine function. Journal of Mathematical Behavior.Google Scholar
- Winicki-Landman, G., & Leikin, R. (2000). On equivalent and non-equivalent definitions: Part 1. For the learning of Mathematics, 20(1), 17–21.Google Scholar