The Korea Society Of Educational Studies In Mathematics
[ Special ]
Journal of Educational Research in Mathematics - Vol. 30, No. SP1, pp.91-114
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 31 Aug 2020
DOI: https://doi.org/10.29275/jerm.2020.08.sp.1.91

Learning to Solve Counting Problems: Challenges and Opportunities for Non-Math Majors

Orit Zaslavsky*, ; Katherine V. Pauletti** ; Victoria Krupnik***
*Professor, New York University, USA
**Graduate Student, New York University, USA
***Graduate Student, Rutgers University, USA

Correspondence to: Email: orit.zaslavsky@nyu.edu, kpauletti@nyu.edu, victoria.krupnik@gse.rutgers.edu

Please cite this article as: Zaslavsky, O., Pauletti, K. V., & Krupnik, V. Learning to solve counting problems: challenges and opportunities for non-math majors.

Abstract

Our study stems from the belief that meaningful and challenging mathematics can be made accessible to all students. To this end, we designed and implemented a special course for undergraduate non-mathematics and non-mathematics education majors. The course entitled Counting and Chance satisfied the requirement of a core course in quantitative reasoning. The content chosen as the focus of this course was basic combinatorics (i.e., counting problems) – a topic that is non-procedural in nature and could be made accessible to students with limited mathematical background. The design of the course was inspired by inclusive pedagogical principles that motivate and support students’ conceptual learning. We present the characteristics of the learning environment that was developed within the framework of this course, point to the rich and sophisticated kinds of reasoning that students developed throughout the course, and discuss how students’ collaborative engagement in the learning process seemed to have contributed to their views of themselves as math learners and their conceptions of what math is. Our findings point to the feasibility of such a course for non-math students, and its potential merit in helping students: (i) develop appreciation of mathematics as a topic that requires thinking, reasoning, and convincing; (ii) become more confident in their ability to do math.

Keywords:

Counting problems, Collaborative problem solving, Reasoning, Inclusion, Dispositions, Equity

Acknowledgments

We would like to thank Nick Wasserman for his thoughtful input during discussions we held at early stages of this study.

References

  • Bass, H. (2017). Designing opportunities to learn mathematics theory-building practices. Educational Studies in Mathematics, 95(3), 229-244. [https://doi.org/10.1007/s10649-016-9747-y]
  • Batanero, C., Navarro-Pelayo, V., & Godino, J. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32, 181-199. [https://doi.org/10.1023/A:1002954428327]
  • Boaler, J. (2002). The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms. For the Learning of Mathematics, 22(1), 42-47.
  • Boaler, J. (2015). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. John Wiley & Sons.
  • Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171-200). Greenwood Publishing Group, Incorporated.
  • Common Core State Standards in Mathematics (CCSSM) (2010). Mathematics standards. Retrieved from: http://www.corestandards.org/Math/
  • Cobb, P., & Hodge, L. L. (2007). Culture, identity, and equity in the mathematics classroom. In N. S. Nasir and P. Cobb (Eds.), Improving access to mathematics: Diversity and equity in the classroom (pp. 159–171). New York: Teachers College Press.
  • Davidson, N., & Kroll, D. L. (1991). An overview of research on cooperative learning related to mathematics. Journal for Research in Mathematics Education, 22(5), 362-365. [https://doi.org/10.5951/jresematheduc.22.5.0362]
  • Davis, B., Towers, J., Karpe, R., Drefs, M., Chapman, O., & Friesen, S. (2018). Steps toward a more inclusive mathematics pedagogy. In A. Kajander, J. L. Holm, & E. J. Chernoff (Eds.), Teaching and learning secondary school mathematics: Canadian perspectives in an international context (pp. 89-99). Char, Switzerland: Springer. [https://doi.org/10.1007/978-3-319-92390-1_11]
  • English, L. D. (1991). Young children’s combinatorics strategies. Educational Studies in Mathematics, 22(5), 451-474. [https://doi.org/10.1007/BF00367908]
  • English, L. D. (1993). Children’s strategies for solving two-and three-dimensional combinatorial problems. Journal for Research in Mathematics Education, 24, 255-273. [https://doi.org/10.5951/jresematheduc.24.3.0255]
  • English, L. D. (2011). Complex Learning through cognitively demanding tasks. The Mathematics Enthusiast, 8(3), 483-506.
  • Fischbein, E. (1987). Intuition in Science and Mathematics: An educational approach. Dordrecht, The Netherlands: Kluwer Academic Publisher.
  • Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. ZDM – Zentralblatt für Didaktik der Mathematik, 5, 193-198.
  • Forman, E. (1989). The role of peer interaction in the social construction of mathematical knowledge. International Journal of Educational Research, 13, 55-70. [https://doi.org/10.1016/0883-0355(89)90016-5]
  • Gentner, D., & Maravilla, F. (2018). Analogical reasoning. In L. J. Ball & V. A. Thompson (Eds.), International handbook of thinking and reasoning (pp. 186-203). New York: Psychological Press.
  • Gresalfi, M. S., & Cobb, P. (2006). Cultivating discipline-specific dispositions as a critical goal for pedagogy and equity. Pedagogies: An International Journal, 1(1), 49-57. [https://doi.org/10.1207/s15544818ped0101_8]
  • Kondratieva, M. (2011). The promise of interconnecting problems for enriching students’ experiences in mathematics. The Montana Mathematics Enthusiast, 8, 335-382.
  • Krupnik, V. (2020). Early development and application of proof-like reasoning: Longitudinal case studies. Unpublished Rutgers Dissertation.
  • Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29-63. [https://doi.org/10.3102/00028312027001029]
  • Leikin, R. (2011). Multiple-solution tasks: From a teacher education course to teacher practice. ZDM – Zentralblatt für Didaktik der Mathematik, 43, 993-1006. [https://doi.org/10.1007/s11858-011-0342-5]
  • Leikin, R., & Zaslavsky, O. (1997). Facilitating students’ interactions in mathematics in a cooperative learning setting. Journal for Research in Mathematics Education, 28(3), 331-354. [https://doi.org/10.5951/jresematheduc.28.3.0331]
  • Lockwood, E. (2013). A model of students’ combinatorial thinking. The Journal of Mathematical Behavior, 32(2), 251-265. [https://doi.org/10.1016/j.jmathb.2013.02.008]
  • Lockwood, E. (2014). A model of students’ combinatorial thinking. For the Learning of Mathematics, 34(2), 31-37.
  • Lockwood, E., Swinyard, C. A., & Caughman, J. S. (2015). Patterns, sets of outcomes, and combinatorial justification: Two students’ reinvention of counting formulas. International Journal of Research in Undergraduate Mathematics Education, 1(1), 27-62. [https://doi.org/10.1007/s40753-015-0001-2]
  • Lockwood, E., Wasserman, N., & McGuffey, W. (2018). Classifying combinations: Investigating undergraduate students’ responses to different categories of combination problems. International Journal of Research in Undergraduate Mathematics Education, 4(2), 305-322. [https://doi.org/10.1007/s40753-018-0073-x]
  • Maher, C. A., & Martino, A. M. (1996). The Development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194-214. [https://doi.org/10.5951/jresematheduc.27.2.0194]
  • Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.) (2011a). Combinatorics and reasoning: Representing, Justifying, and building isomorphisms. New York: Springer. [https://doi.org/10.1007/978-0-387-98132-1]
  • Maher, C. A., Sran, M. K., & Yankelewitz, D. (2011b). Making pizzas: Reasoning by cases and by recursion. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.). Combinatorics and reasoning: Representing, Justifying, and building isomorphisms (pp. 59-72). New York: Springer. [https://doi.org/10.1007/978-94-007-0615-6_6]
  • Mashiach-Eizenberg, M. & Zaslavsky, O. (2003). Cooperative problem solving in combinatorics – the inter-relations between control processes and successful solutions. Journal of Mathematical Behavior, 22(4), 389-403. [https://doi.org/10.1016/j.jmathb.2003.09.001]
  • Mashiach-Eizenberg, M. & Zaslavsky, O. (2004). Students’ verification strategies for combinatorial problems. Mathematical Thinking and Learning, 6(1), 15-36. [https://doi.org/10.1207/s15327833mtl0601_2]
  • National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. Reston, VA: Author.
  • Nasir, N. S. (2002). Identity, goals, and learning: Mathematics in cultural practice. Mathematical Thinking and Learning, 4, 213-247. [https://doi.org/10.1207/S15327833MTL04023_6]
  • Pólya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton University Press.
  • Ruef, J. (2017). Changes in student perspectives: What it means to be “good at math.” In E. Galindo & J. Newton (Eds.). Proceedings of the 39th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1083-1086). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators.
  • Schulman, S. M. (2013). According to Davis: Connecting principles and practices. The Journal of Mathematical Behavior, 32, 230-242. [https://doi.org/10.1016/j.jmathb.2013.02.002]
  • Silver, E. A. (1979). Student perceptions of relatedness among mathematical verbal problems. Journal for Research in Mathematics Education, 10, 195-210. [https://doi.org/10.5951/jresematheduc.10.3.0195]
  • Solomon, Y. (2007). Not belonging? What makes a functional learner identity in undergraduate mathematics?. Studies in Higher Education, 32(1), 79-96. [https://doi.org/10.1080/03075070601099473]
  • Stein, M. K., Grover, B., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33, 455-488. [https://doi.org/10.3102/00028312033002455]
  • Wasserman, N. H. (2019). Duality in combinatorial notation. For the Learning of Mathematics, 39(3), 16-21.
  • Wasserman, N. H., Galarza, P. (2019). Conceptualizing and justifying sets of outcomes with combination problems. Investigations in Mathematics Learning, 11(2), 83-102. [https://doi.org/10.1080/19477503.2017.1392208]
  • Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458-477. [https://doi.org/10.2307/749877]
  • Yackel, E., Cobb, P., & Wood, T. (1991). Small-group interactions as a source of learning opportunities in second-grade mathematics. Journal for Research in Mathematics Education, 22, 390–408. [https://doi.org/10.5951/jresematheduc.22.5.0390]
  • Zaslavsky, O. (2005). Seizing the opportunity to create uncertainty in learning mathematics. Educational Studies in Mathematics, 60, 297-32. [https://doi.org/10.1007/s10649-005-0606-5]
  • Zaslavsky, O., Nickerson, S., Stylianides, A., Kidron, I., & Winicki, G. (2012). The need for proof and proving: Mathematical and pedagogical perspectives. In G. Hanna & M. de Villiers (Eds), Proof and proving in mathematics education (pp. 215-229). New York: Springer. [https://doi.org/10.1007/978-94-007-2129-6_9]