STUDENT-GENERATED CONNECTIONS IN LEARNING ABOUT COMPOUND PROBABILITY AND THEIR EMERGENCE DURING INSTRUCTION
DOI:
https://doi.org/10.52041/serj.v22i1.51Keywords:
Statistics education research, compound probability, connections, qualitative researchAbstract
In this report, we analyze students’ learning of compound probability by describing connections they generated during individual interviews and group lessons. Several of their connections were compatible with the development of expertise, such as recognizing the need to determine sample spaces across a variety of situations and noting structural similarities among tasks, even when their task solutions were incomplete from a normative standpoint. Students reasoned about dimensions of context, variation, mathematical structure, sample space, and probability quantification. We describe the extent to which they coordinated these dimensions. We also describe teaching moves, such as posing idealized situations and shifting to structurally similar tasks, which prompted students to attend to multiple relevant task dimensions.
References
Alston, A., & Maher, C. A. (2003). Modeling outcomes from probability tasks: Sixth graders reasoning together. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education, Honolulu, HI (Vol. 2, pp. 25–32).
Bakker, A., & van Eerde, D. (2015). An introduction to design-based research with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Doing qualitative research: Methodology and methods in mathematics education (pp. 429–466). Springer.
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), 612–637. https://doi.org/10.1037/0033-2909.128.4.612
Batanero, C., Henry, M., & Parzysz, B. (2005). The nature of chance and probability. In G. A. Jones
(Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 15–37). Springer. https://doi.org/10.1007/0-387-24530-8_2
Bransford, J. D., & Schwartz, D. L. (1999). Rethinking transfer: A simple proposal with multiple implications. Review of Research in Education, 24(1), 61–100. https://doi.org/10.3102/0091732X024001061
Chen, Z., & Daehler, M. W. (1989). Positive and negative transfer in analogical problem solving by 6-year-old children. Cognitive Development, 4(4), 327–344. https://doi.org/10.1016/S0885-2014(89)90031-2
Chow, A. F., & Van Haneghan, J. P. (2016). Transfer of solutions to conditional probability problems: Effects of example problem format, solution format, and problem context. Educational Studies in Mathematics, 93, 67–85. https://doi.org/10.1007/s10649-016-9691-x
Cobb, P., Jackson, K., & Sharpe, C. (2017). Conducting design studies to investigate and support mathematics students’ and teachers’ learning. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 208–233). National Council of Teachers of Mathematics.
Common Core State Standards Initiative. (2010). Common core state standards for mathematics. http://www.corestandards.org/
Corbin, J. & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory (3rd ed.). SAGE Publications.
DeCuir-Gunby, J. T., Marshall, P. L., & McColloch, A. W. (2011). Developing and using a codebook for the analysis of interview data: An example from a professional development project. Field Methods, 23(2), 136–155. https://doi.org/10.1177/1525822X10388468
English, L. D., & Watson, J. M. (2016). Development of probabilistic understanding in fourth grade. Journal for Research in Mathematics Education, 47(1), 28–62. https://doi.org/10.5951/jresematheduc.47.1.0028
Fischbein, E., & Snarch, D. (1997). The evolution with age of probabilistic, intuitively based misconceptions. Journal for Research in Mathematics Education, 28(1), 96–105.
https://doi.org/10.2307/749665
Friel, S. N., Curcio, F. R., & Bright, G. W. (2001). Making sense of graphs: Critical factors influencing comprehension and instructional implications. Journal for Research in Mathematics Education, 32(2), 124–158. https://doi.org/10.2307/749671
Halcyon Software. (2006). Inspiration software (Version 8) [Computer Software].
Iversen, K., & Nilsson, P. (2019). Lower secondary school students’ reasoning about compound probability in spinner tasks. Journal of Mathematical Behavior, 56, Article 100723. https://doi.org/10.1016/j.jmathb.2019.100723
Kimball, D. R., & Holyoak, K. J. (2000). Transfer and expertise. In E. Tulving & F. Craik (Eds.), The Oxford handbook of memory (pp. 109-122). Oxford University Press.
Konold, C. (1995). Issues in assessing conceptual understanding in probability and statistics. Journal of Statistics Education, 3(1). https://doi.org/10.1080/10691898.1995.11910479
Konold, C. (1996). Representing probabilities with pipe diagrams. Mathematics Teacher, 89(5), 378–382. https://www.jstor.org/stable/27969794
Konold, C., & Miller, C. (2011). TinkerPlots (Version 2.3 [Computer Software]. Learn Troop.
Langrall, C., Nisbet, S., Mooney, E., & Jansem, S. (2011). The role of context expertise when comparing data. Mathematical Thinking and Learning, 13(1&2), 47–67. https://doi.org/10.1080/10986065.2011.538620
Lineback, J. E. (2015). The redirection: An indicator of how teachers respond to student thinking. Journal of the Learning Sciences, 24(3), 419–460. https://doi.org/10.1080/10508406.2014.930707
Lobato, J. (2003). How design experiments can inform a rethinking of transfer and vice versa. Educational Researcher, 32(1), 17–20. https://www.jstor.org/stable/3699930
Lobato, J. (2012). The actor-oriented transfer perspective and its contributions to educational research and practice. Educational Psychologist, 47(3), 232–247. https://doi.org/10.1080/00461520.2012.693353
Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. Journal of Mathematical Behavior, 21(1), 87–116. https://doi.org/10.1016/S0732-3123(02)00105-0
Lockwood, E. (2011). Student connections among counting problems: An exploration using actor-oriented transfer. Educational Studies in Mathematics, 78(3), 307–322. https://doi.org/10.1007/s10649-011-9320-7
Lysoe, K. O. (2008). Strengths and limitations of informal conceptions in introductory probability courses for future lower secondary teachers. In M. Borovcnik, D. Pratt, Y. Wu, & C. Batanero (Eds.), Research and development in the teaching and learning of probability (pp. 1–14). International Association for Statistical Education.
Makar, K., & Ben-Zvi, D. (2011). The role of context in developing reasoning about informal statistical inference. Mathematical Thinking and Learning, 13(1&2), 1–4. https://doi.org/10.1080/10986065.2011.538291
Miles, M. B., Huberman, A. M., & Saldaña, J. (2020). Qualitative data analysis: A methods sourcebook (4th ed.). SAGE Publications.
Moyer, P. S., & Milewicz, E. (2002). Learning to question: Categories of questioning used by preservice teachers during diagnostic mathematics interviews. Journal of Mathematics Teacher Education, 5, 293–315. https://doi.org/10.1023/A:1021251912775
Nelson, C., & Williams, N. (2008). A fair game? The case of rock, paper, scissors. Mathematics Teaching in the Middle School, 14(5), 311-314. https://www.jstor.org/stable/41183143
Nesbit, J. C., & Adesope, O. O. (2006). Learning with concept and knowledge maps: A meta-analysis. Review of Educational Research, 76, 413–448. https://doi.org/10.3102/00346543076003413
Pratt, D. (2000). Making sense of the total of two dice. Journal for Research in Mathematics Education, 31(5), 602–625. https://doi.org/10.2307/749889
Ron, G., Dreyfus, T., & Hershkowitz, R. (2017). Looking back to the roots of partially correct constructs: The case of the area model in probability. Journal of Mathematical Behavior, 45, 15–34. https://doi.org/10.1016/j.jmathb.2016.10.004
Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 957–1009). Information Age Publishing.
Shaughnessy, J. M., & Ciancetta, M. (2002). Students’ understanding of variability in a probability environment. In B. Phillips (Ed.), Proceedings of the Sixth International Conference on the Teaching of Statistics, Cape Town. https://iase-web.org/documents/papers/icots6/6a6_shau.pdf
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340. https://doi.org/10.1080/10986060802229675
Vidakovic, D., Berenson, S., & Brandsma, J. (1998). Children’s intuition of probabilistic concepts emerging from fair play. In L. Pereira-Mendoza, T. Wee Kee, W.-K. Wong (Eds.), Proceedings of the Fifth International Conference on Teaching Statistics, Vol. 1, Singapore (pp. 67–73). International Statistical Institute.
Wagner, J. F. (2010). A transfer-in-pieces consideration of the perception of structure in the transfer of learning. The Journal of the Learning Sciences, 19(4), 443–479. https://doi.org/10.1080/10508406.2010.505138
Watson, J. M., & Kelly, B. (2004). Expectation versus variation: Students’ decision-making in a chance environment. Canadian Journal of Science, Mathematics, and Technology Education, 4(3), 371–396. https://doi.org/10.1080/14926150409556620
Watson, J. M., & Moritz, J. B. (2003). Fairness of dice: A longitudinal study of students’ beliefs and strategies for making judgments. Journal for Research in Mathematics Education, 34(4), 270–304. https://doi.org/10.2307/30034785
Webb, P., Whitlow, J. W., Jr., & Venter, D. (2017). From exploratory talk to abstract reasoning: A case for far transfer? Educational Psychology Review, 29, 565–581. https://doi.org/10.1007/s10648-016-9369-z
Wheeldon, J., & Åhlberg, M. K. (2011). Visualizing social science research: Maps, methods, and meaning. SAGE Publications.
Zawojewski, J. S., & Shaughnessy, J. M. (2000). Data and chance. In E. A. Silver & P. A. Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 235–268). National Council of Teachers of Mathematics.
