Students’ solution of arrangement problems and their connection to Cartesian product problems

References

• Australian Curriculum, Assessment and Reporting Authority [ACARA]. (n.d.). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/
   Google Scholar
• Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32(2), 181–199. doi:10.1023/A:1002954428327
   Google Scholar
• Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 843–908). Charlotte, NC: Information Age Publishing.
   Google Scholar
• Behr, M., Harel, G., Post, T., & Lesh, R. (1994). Units of quantity. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 121–176). Albany: State University of New York Press.
   Google Scholar
• Chini, J. J., Carmichael, A., Rebello, N. S., & Puntambekar, S. (2009, November). Does the teaching/learning interview provide an accurate snapshot of classroom learning?. In: Sabella, M., Henderson, C., & Singh, C. (Eds.). AIP conference proceedings (Vol. 1179, No. 1, pp. 113–116). AIP. Ann Arbor, MI: University of Michigan.
   Google Scholar
• Confrey, J., & Lachance, A. (2000). Transformative teaching experiments through conjecture-driven research design. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 231–265). Mahwah, NJ: Lawrence Erlbaum Associates.
   Google Scholar
• Eizenberg, M. M., & Zaslavsky, O. (2004). Students’ verification strategies for combinatorial problems. Mathematical Thinking and Learning, 6(1), 15–36. doi:10.1207/s15327833mtl0601_2
   Google Scholar
• English, L. D. (1991). Young children‘s combinatoric strategies. Educational Studies in Mathematics, 22, 451–474. doi:10.1007/BF00367908
   Google Scholar
• English, L. D. (1993). Children‘s strategies for solving two- and three-dimensional combinatorial problems. Journal for Research in Mathematics Education, 24, 255–273. doi:10.2307/749347
   Web of Science ®Google Scholar
• English, L. D. (1996). Children‘s construction of mathematical knowledge in solving novel isomorphic problems in concrete and written form. The Journal of Mathematical Behavior, 15, 81–112. doi:10.1016/S0732-3123(96)90042-5
   Google Scholar
• English, L. D. (1999). Assessing for structural understanding in children‘s combinatorial problem solving. Focus on Learning Problems in Mathematics, 21(4), 63–83.
   Google Scholar
• Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. Zentralblatt fur Didaktik der Mathematik, 5, 193–197.
   Google Scholar
• Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). New York, NY: Macmillan.
   Google Scholar
• Greer, B. (2010). Overview of papers: Why is linear thinking so dominant? Mathematical Thinking and Learning, 12(1), 109–115. doi:10.1080/10986060903465996
   Web of Science ®Google Scholar
• Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. The Journal of Mathematical Behavior, 26, 27–47. doi:10.1016/j.jmathb.2007.03.002
   Google Scholar
• Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 1–50. doi:10.1080/07370008.2010.511565
   Web of Science ®Google Scholar
• Hackenberg, A. J., Creager, M., Eker, A., & Lee, M. L. (2016, April). Understanding how to differentiate instruction for middle school students. Paper presentation at the Research Conference of the NCTM, San Francisco, CA.
   Google Scholar
• Hackenberg, A. J., & Tillema, E. S. (2009). Students’ fraction composition schemes. Journal of Mathematical Behavior, 28(1), 1–18. doi:10.1016/j.jmathb.2009.04.004
   Google Scholar
• Halani, A. (2012). Students’ ways of thinking about enumerative combinatorics solution sets: The odometer category. In: Stacy Brown, Sean Larsen, Karen Marrongelle, (Eds). The electronic proceedings for the fifteenth special interest group of the MAA on research on undergraduate mathematics education (pp. 59–68). Portland, OR: Portland State University.
   Google Scholar
• Jones, G. A., Langrall, C. W., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 909–955). Charlotte, NC: Information Age Publishing.
   Google Scholar
• Kapur, J. N. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 3(1), 111–127.
   Google Scholar
• Kobiela, M., & Lehrer, R. (2019). Supporting dynamic conceptions of area and its measure. Mathematical Thinking and Learning, 21(1), 1–29. doi:10.1080/10986065.2019.1576000
   Google Scholar
• Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2002). Clever counting. Upper Saddle River, NJ: Prentice Hill.
   Google Scholar
• Lobato, J. (2008). Research methods for alternative approaches to transfer: Implications for design experiments. Handbook of Design Research Methods in Education: Innovations in Science, Technology, Engineering, and Mathematics Learning and Teaching, 1, 167–194.
   Google Scholar
• Lockwood, E. (2011). Student connections among counting problems: An exploration using actor-oriented transfer. Educational Studies in Mathematics, 78(3), 307–322. doi:10.1007/s10649-011-9320-7
   Web of Science ®Google Scholar
• Lockwood, E., & Gibson, B. (2016). Combinatorial tasks and outcome listing: Examining productive listing among undergraduate students. Education Studies in Mathematics, 91(2), 247–270. doi:10.1007/s10649-015-9664-5
   Web of Science ®Google Scholar
• Lockwood, E., & Swinyard, C. A. (2016). An introductory set of activities designed to facilitate successful combinatorial enumeration for undergraduate students. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 26(10), 889–904.
   Google Scholar
• Maher, C. A., & Martino, A. (1996). The development of the idea of mathematical proof. Journal for Research in Mathematics Education, 27(2), 194–214. doi:10.2307/749600
   Web of Science ®Google Scholar
• Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.). (2010). Combinatorics and reasoning: Representing, justifying and building isomorphisms (Vol. 47). New York, NY: Springer Science & Business Media.
   Google Scholar
• Maher, C. A., & Yankelewitz, D. (2010). Representations as tools for building arguments. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and reasoning: Representing, justifying, and building isomorphisms (pp. 17–25). New York, NY: Springer.
   Google Scholar
• Mathison, S. (1988). Why triangulate? Educational Researcher, 17(2), 13–17. doi:10.3102/0013189X017002013
   Google Scholar
• Mcgillivray, H., Petosz, P., Evans, M., & Jones, P. (2013). Statistics and probability for the Australian curriculum. Sydney, Australia: Cambridge University Press.
   Google Scholar
• Ministry of Education, Science, and Technology, Korea. (2007). The school of curriculum of the Republic of Korea: General guidelines (Notification No. 2008-160). Seoul, Gyeonggi : Author.
   Google Scholar
• Morsanyi, K., Handley, S. J., & Serpell, S. (2013). Making heads or tails of probability: An experiment with random generators. British Journal of Educational Psychological Society, 83, 379–395.
   PubMed Web of Science ®Google Scholar
• Mulligan, J., & Mitchelmore, M. (1997). Young children‘s intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28(3), 309–330. doi:10.2307/749783
   Web of Science ®Google Scholar
• National Council of Teachers of Mathematics. (2000). Principles and standards of school mathematics. Reston, VA: Author.
   Google Scholar
• National Governors Association for Best Practices, Council of Chief State School Officers. (2010). Common core state standards mathematics. Washington, DC: Author.
   Google Scholar
• Nunes, T., & Bryant, P. (1996). The progress to multiplication and division. In Children doing mathematics (pp. 142–201). Oxford, UK: Blackwell Publishers.
   Google Scholar
• Nunes, T., Light, P., & Mason, J. (1993). Tools for thought: The measurement of length and area. Learning and Instruction, 3, 39–54. doi:10.1016/S0959-4752(09)80004-2
   Google Scholar
• Outhred, L. (1996). Children‘s drawings of multiplicative structures: Cartesian products and area. In J. Mulligan & M. Mitchelmore (Eds.), Children‘s number learning (pp. 185–204). Adelaide, Australia: Australian Association of Mathematics Teachers for the Mathematics Education Research Group of Australasia.
   Google Scholar
• Outhred, L., & Mitchelmore, M. (2000). Young children’s intuitive understanding of rectangular area measurement. Journal of Research in Mathematics Education, 31(2), 144–167. doi:10.2307/749749
   Web of Science ®Google Scholar
• Piaget, J. (1970). Genetic epistemology (E. Duckworth, Trans.). New York: W.W. Norton & Company Inc.
   Google Scholar
• Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children (L. J. Leake, P. Burrell & H. D. Fishbein, Trans.). New York, NY: W.W. Norton & Company.
   Google Scholar
• Rubel, L. H. (2007). Middle school and high school students’ probabilistic reasoning on coin tasks. Journal for Research in Mathematics Education, 38(5), 531–556.
   Web of Science ®Google Scholar
• Saldana, J. (2013). The coding manual for qualitative researchers. London, UK: Sage Publications Ltd.
   Google Scholar
• Simon, M. A., & Blume, G. W. (1994). Building and understanding multiplicative relationships: A study of prospective elementary teachers. Journal for Research in Mathematics Education, 25(5), 472–494. doi:10.2307/749486
   Web of Science ®Google Scholar
• Smith, J. P., III, & Barrett, J. E. (2018). Learning and teaching measurement: Coordinating quantity and number. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 355–385). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Difference, 4(3), 259–309. doi:10.1016/1041-6080(92)90005-Y
   Google Scholar
• Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 1–41). Albany: State University of New York Press.
   Google Scholar
• Steffe, L. P. (2007, April). Problems in mathematics education. Paper presented for the Senior Scholar Award of the Special Interest Group for Research in Mathematics Education (SIG-RME) at the annual conference of the American Educational Research Association in Chicago, Illinois.
   Google Scholar
• Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267–307). Hillsdale, NJ: Erlbaum.
   Google Scholar
• Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children‘s counting types: Philosophy, theory, and application. New York City, NY: Praeger Publishers.
   Google Scholar
• Tillema, E. S. (2013). A power meaning of multiplication: Three eighth graders’ solutions of Cartesian product problems. Journal of Mathematical Behavior, 32, 331–352. doi:10.1016/j.jmathb.2013.03.006
   Google Scholar
• Tillema, E. S. (2014). Students’ coordination of lower and higher dimensional units in the context of evaluating sums of consecutive whole numbers. Journal of Mathematical Behavior, 36, 51–72. doi:10.1016/j.jmathb.2014.07.005
   Google Scholar
• Tillema, E. S. (2018). An investigation of 6th graders’ solutions of combinatorics problems and representation of these problems using arrays. Journal of Mathematical Behavior, 52, 1–20. doi:10.1016/j.jmathb.2018.03.009
   Web of Science ®Google Scholar
• Tillema, E. S., & Gatza, A. (2016). A quantitative and combinatorial approach to non-linear meanings of multiplication. For the Learning of Mathematics, 36(2), 26–33.
   Google Scholar
• Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (Part 1). For the Learning of Mathematics, 35(3), 2–7.
   Google Scholar
• Ulrich, C. (2016). Stages in constructing and coordinating units additively and multiplicatively (Part 2). For the Learning of Mathematics, 36(1), 34–39.
   Google Scholar
• Ulrich, K., Tillema, E. S., Hackenberg, A. J., & Norton, A. N. (2014). Units coordination: An example of the utility of radical constructivist thought in education. Constructivist Foundations, 9(3), 328–339.
   Web of Science ®Google Scholar
• Van Dooren, W., De Bock, D., Janssens, D., & Verschaffel, L. (2008). The linear imperative: An inventory and conceptual analysis of students‘ overuse of linearity. Journal for Research in Mathematics Education, 39(3), 311–342.
   Web of Science ®Google Scholar
• Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquistion of mathematics concepts and processes (pp. 127–174). New York, NY: Academic Press.
   Google Scholar
• Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 557–628). Charlotte, NC: Information Age Publishing.
   Google Scholar
• Von Glasersfeld, E. (1982). Subitizing: The role of figural patterns in the development of numerical concepts. Archives De Psychologie, 50, 191–218.
   Google Scholar
• Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning (Vol. 6). Washington, DC: The Falmer Press.
   Google Scholar
• Watson, J., & Neal, D. (2012). Preparing students for decision-making in the 21st Century—Statistics and probability in the Australian curriculum: Mathematics. In B. Atweh, M. Goos, R. Jorgensen, & D. Siemon (Eds.), Engaging the Australian national curriculum: Mathematics – Perspectives from the field (pp. 89–113). Online Publication: Mathematics Education Research Group of Australasia. Available from: https://tatagyes.files.wordpress.com/2007/10/book_curriculum-math_australia.pdf
   Google Scholar