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An Evaluation of ULTRA; an Experimental Real Analysis Course Built on a Transformative Theoretical Model

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Abstract

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.

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Notes

  1. http://ultra.gse.rutgers.edu/

  2. The research team consisted of six mathematics educators: four faculty and two doctoral students.

  3. We note that in this third iteration, one student at Teachers College, Columbia University was a “math coach” and not technically a teacher and at Rutgers University, two of the students had not enrolled in a teacher certification program, so there were three students in total who, technically speaking, may not have been PISTs. However, all participants had some degree of interest in mathematics education. We refer to all study participants as PISTs for rhetorical convenience.

References

  • Abbott, S. (2015). Understanding analysis. New York, NY: Springer-Verlag.

    Book  Google Scholar 

  • 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.

  • 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.

    Article  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 

  • Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.

    Article  Google Scholar 

  • Common Core State Standards in Mathematics (CCSSM). (2010). Retrieved from: http://www.corestandards.org/the-standards/mathematics. Last accessed May 30 2019.

  • Conference Board of Mathematical Sciences (CBMS). (2001). The mathematical education of teachers. Providence, RI and Washington, DC: American Mathematical Society and Mathematical Association of America.

    Book  Google 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.

    Book  Google Scholar 

  • Darling-Hammond, L. (2000). Teacher quality and student achievement: A review of state policy evidence. Educational Policy Analysis Archives, 8, 1.

    Article  Google Scholar 

  • Edelson, D. C. (2002). Design research: What we learn when we engage in design. The Journal of the Learning Sciences, 11(1), 105–121.

    Article  Google Scholar 

  • Even, R. (2011). The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: Practitioner’s views. ZDM—The International Journal of Mathematics Education, 43(6–7), 941–950.

  • 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.

  • Fitzpatrick, P. M. (2006). Advanced Calculus (2nd ed.). Providence, RI: American Mathematical Society.

    Google Scholar 

  • Goulding, M., Hatch, G., & Rodd, M. (2003). Undergraduate mathematics experience: Its significance in secondary mathematics teacher preparation. Journal of Mathematics Teacher Education, 6, 361–393.

    Article  Google Scholar 

  • Herbst, P., Chazan, D., Chen, C., Chieu, V. M., & Weiss, M. (2011). Using comics-based representations of teaching, and technology, to bring practice to teacher education courses. ZDM—The International Journal of Mathematics Education, 43(1), 91–103.

    Article  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 

  • Ko, Y. Y., & Knuth, E. (2009). Undergraduate mathematics majors’ writing performance producing proofs and counterexamples about continuous functions. The Journal of Mathematical Behavior, 28(1), 68–77.

    Article  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.

    Article  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.

  • 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.

  • Monk, D. H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125–145.

    Article  Google 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.

    Article  Google Scholar 

  • Sandoval, W. (2014). Conjecture mapping: An approach to systematic educational design research. Journal of the Learning Sciences, 23(1), 18–36.

    Article  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.

  • TeachingWorks (2018). High leverage teaching practices. Downloaded from: http://www.teachingworks.org/work-of-teaching/high-leverage-practices. Accessed 31 May 2018.

  • Ticknor, C. S. (2012). Situated learning in an abstract algebra classroom. Educational Studies in Mathematics, 81(3), 307–323.

    Article  Google Scholar 

  • Wasserman, N., Villanueva, M., Fukawa-Connelly, T., Mejía-Ramos, J. P., & Weber, K. (2017). Making real analysis relevant to secondary teachers: Building up to and stepping down from practice. PRIMUS, 27, 559–578.

    Article  Google Scholar 

  • Wasserman, N., Weber, K., Villanueva, M., & Mejía-Ramos, J. P. (2018). Mathematics teachers’ views about the limited utility of real analysis: A transport model hypothesis. Journal of Mathematical Behavior, 50, 74–89.

    Article  Google Scholar 

  • 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.

  • Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.

    Article  Google 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.

  • 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 

  • Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. Educational Studies in Mathematics, 69(2), 131–148.

    Article  Google Scholar 

  • Zazkis, R., & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice: Perceptions of secondary mathematics teachers. Mathematical Thinking and Learning, 12, 263–281.

    Article  Google Scholar 

Download references

Acknowledgements

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.

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Correspondence to Timothy Fukawa-Connelly.

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Fukawa-Connelly, T., Mejía-Ramos, J.P., Wasserman, N.H. et al. An Evaluation of ULTRA; an Experimental Real Analysis Course Built on a Transformative Theoretical Model. Int. J. Res. Undergrad. Math. Ed. 6, 159–185 (2020). https://doi.org/10.1007/s40753-019-00102-8

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