880
Views
3
CrossRef citations to date
0
Altmetric
Articles

Developing and validating proof comprehension tests in undergraduate mathematics

, , &
Pages 130-146 | Received 24 May 2016, Accepted 13 Oct 2016, Published online: 01 Aug 2017

References

  • Alcock, L. J., Hodds, M., Roy, S., & Inglis, M. (2015). Investigating and improving undergraduate proof comprehension. Notices of the American Mathematical Society, 62, 742–752. doi: 10.1090/noti1263
  • Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. The Journal of Mathematical Behavior, 24, 125–134. doi: 10.1016/j.jmathb.2005.03.003
  • Alcock, L., & Wilkinson, H. (2011). e-Proofs: Design of a resource to support proof comprehension in mathematics. Educational Designer, 1(4), Retrieved from http://www.educationaldesigner.org/ed/volume1/issue4/article14/
  • Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment: A tool for assessing students’ reasoning abilities and understandings. Cognition and Instruction, 28(2), 113–145. doi: 10.1080/07370001003676587
  • Conradie, J., & Frith, J. (2000). Comprehension tests in mathematics. Educational Studies in Mathematics, 42, 225–235. doi: 10.1023/A:1017502919000
  • Cowen, C. (1991). Teaching and testing mathematics reading. The American Mathematical Monthly, 98(1), 50–53. doi: 10.2307/2324038
  • Deeds, D., & Callen, B. (Eds.), (2006). Proceedings of the national STEM assessment conference. Jointly sponsored by the National Science Foundation and Drury University, Springfield, MO: Drury University.
  • de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.
  • Dreyfus, T. (1990). Advanced mathematical thinking. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education (pp. 113–134). Great Britain: Cambridge University Press.
  • Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced mathematical thinking (pp. 215–230). Kluwer: The Netherlands.
  • Dubinsky, E. (1986). On teaching mathematical induction I. The Journal of Mathematical Behavior, 5, 305–317.
  • Dubinsky, E. (1989). On teaching mathematical induction II. The Journal of Mathematical Behavior, 8, 285–304.
  • Epstein, J. (2013). The calculus concept inventory–measurement of the effect of teaching methodology in mathematics. Notices of the American Mathematical Society, 60(8), 1018–1026. doi: 10.1090/noti1033
  • Fuller, E., Weber, K., Mejia-Ramos, J. P., Samkoff, A., & Rhoads, K. (2014). Comprehending structured proofs. International Journal for Studies in Mathematics Education, 7(1), 1–32.
  • Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13. doi: 10.1007/BF01809605
  • Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15(3), 42–50.
  • Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zazkis (Eds.), Learning and teaching number theory (pp. 185–212). Norwood, NJ: Ablex.
  • Hazzan, O., & Zazkis, R. (2003). Mimicry of proofs with computers: The case of linear algebra. International Journal of Mathematical Education in Science and Technology, 34(3), 385–402. doi: 10.1080/0020739031000108628
  • Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31, 396–428. doi: 10.2307/749651
  • Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399. doi: 10.1007/BF01273372
  • Hestenes, D., Wells, M., & Swackhamer, G. (1992). Force concept inventory. The Physics Teacher, 30, 141–158. doi: 10.1119/1.2343497
  • Hodds, M., Alcock, L., & Inglis, M. (2014). Self-explanation training improves proof comprehension. Journal for Research in Mathematics Education, 45, 98–137.
  • Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43, 358–390. doi: 10.5951/jresematheduc.43.4.0358
  • Knuth, E. (2000). The rebirth of proof in school mathematics in the United States. International Newsletter on the Teaching and Learning of Mathematical Proof. (Spring 2000). Retrieved from http://www.lettredelapreuve.it/OldPreuve/Newsletter/000506Theme/000506ThemeUK.html. Last downloaded: May 29, 2012.
  • Knuth, E. (2002). Secondary school mathematics teachers conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405. doi: 10.2307/4149959
  • Konior, J. (1993). Research into the construction of mathematical texts. Educational Studies in Mathematics, 24(3), 251–256. doi: 10.1007/BF01275425
  • Lai, Y., Weber, K., & Mejía-Ramos, J. P. (2012). Mathematicians’ perspectives on features of a good pedagogical proof. Cognition and Instruction, 30(2), 146–169. doi: 10.1080/07370008.2012.661814
  • Leron, U. (1983). Structuring mathematical proofs. The American Mathematical Monthly, 90(3), 174–184. doi: 10.2307/2975544
  • Lin, F.-L., & Yang, K.-L. (2007). The reading comprehension of geometric proofs: The contribution of knowledge and reasoning. International Journal of Science and Mathematics Education, 5, 729–754. doi: 10.1007/s10763-007-9095-6
  • Malek, A., & Movshovitz-Hadar, N. (2011). The effect of using transparent pseudo-proofs in linear algebra. Research in Mathematics Education, 13, 33–58. doi: 10.1080/14794802.2011.550719
  • Mamona-Downs, J., & Downs, M. (2005). The identity of problem solving. The Journal of Mathematical Behavior, 24, 385–401. doi: 10.1016/j.jmathb.2005.09.011
  • Mantini, L, Mills, M., & Johnson, J. (2012). Reading comprehension of series convergence proofs in calculus II. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.), Proceedings of the 15th annual conference for research in undergraduate mathematics education (pp. 485–489). Retrieved from http://sigmaa.maa.org/rume/crume2012/RUME_Home/RUME_Conference_Papers_files/RUME_XV_Conference_Papers.pdf
  • Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41–51. doi: 10.2307/749097
  • Mejía-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79(1), 3–18. doi: 10.1007/s10649-011-9349-7
  • Mejía-Ramos, J. P., & Inglis, M. (2009). Argumentative and proving activities in mathematics education research. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proceedings of the ICMI study 19 conference: Proof and proving in mathematics education (Vol. 2, pp. 88–93). Taipei, Taiwan: National Taiwan Normal University.
  • Mejía-Ramos, J. P., Weber, K., de la Torre, J., & Lew, K. (under review). Student performance on proof comprehension tests in transition-to-proof courses.
  • Mills, M. (2011). Mathematicians’ pedagogical thoughts and practices in proof presentation. Presentation at the 14th Conference for Research in Undergraduate Mathematics Education. Available for download from: http://sigmaa.maa.org/rume/crume2011/Preliminary_Reports.html
  • Powers, R., Craviotto, C., & Grassl, R. (2010). Impact of proof validation on proof writing in abstract algebra. International Journal of Mathematical Education in Science and Technology, 41, 501–514. doi: 10.1080/00207390903564603
  • Raman, M. (2004). Epistemological messages conveyed by three high-school and college mathematics textbooks. The Journal of Mathematical Behavior, 23(4), 389–404. doi: 10.1016/j.jmathb.2004.09.002
  • Resnick, L. B., & Resnick, D. P. (1992). Assessing the thinking curriculum: New tools for educational reform. In B. R. Gifford & M. C. O’Connor (Eds.), Changing assessments: Alternative views of aptitude, achievement, and instruction (pp. 37–75). Boston: Kluwer.
  • Rowland, T. (2001). Generic proofs in number theory. In S. Campbell & R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 157–184). Westport: Ablex.
  • Schoenfeld, A. H. (1994). What do we know about mathematics curricula? The Journal of Mathematical Behavior, 13(1), 55–80. doi: 10.1016/0732-3123(94)90035-3
  • Segal, J. (1999). Learning about mathematical proof: Conviction and validity. The Journal of Mathematical Behavior, 18(2), 191–210. doi: 10.1016/S0732-3123(99)00028-0
  • Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4–36. doi: 10.2307/30034698
  • Weber, K. (2001). Student difficulties in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119. doi: 10.1023/A:1015535614355
  • Weber, K. (2004). Traditional instruction in advanced mathematics courses: A case study of one professor’s lectures and proofs in an introductory real analysis course. The Journal of Mathematical Behavior, 23(2), 115–133. doi: 10.1016/j.jmathb.2004.03.001
  • Weber, K. (2010a). Proofs that develop insight. For the Learning of Mathematics, 30(1), 32–36.
  • Weber, K. (2010b). Mathematics majors’ perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12, 306–336. doi: 10.1080/10986065.2010.495468
  • Weber, K. (2012). Mathematicians’ perspectives on their pedagogical practice with respect to proof. International Journal of Mathematical Education in Science and Technology, 43(4), 463–482. doi: 10.1080/0020739X.2011.622803
  • Weber, K., Fuller, E., Mejia-Ramos, J. P., Lew, K., Benjamin, P., & Samkoff, A. (2012, February). Do generic proofs improve proof comprehension? Presented at the 15th Annual Conference on Research in Undergraduate Mathematics Education, Portland, Oregon.
  • Weber, K., & Mejía-Ramos, J. P. (2011). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 76, 329–344. doi: 10.1007/s10649-010-9292-z
  • Weber, K., & Mejía-Ramos, J. P. (2014). Mathematics majors’ beliefs about proof reading. International Journal of Mathematical Education in Science and Technology, 45, 89–103. doi: 10.1080/0020739X.2013.790514
  • Yang, K.-L. (2012). Structures of cognitive and metacognitive reading strategy use for reading comprehension of geometry proof. Educational Studies in Mathematics, 80(3), 307–326. doi: 10.1007/s10649-011-9350-1
  • Yang, K.-L., & Lin, F.-L. (2008). A model of reading comprehension of geometry proof. Educational Studies in Mathematics, 67, 59–76. doi: 10.1007/s10649-007-9080-6

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.