Modelling in Primary School: Constructing Conceptual Models and Making Sense of Fractions

Article

Abstract

This article describes sixth-grade students’ engagement in two model-eliciting activities offering students the opportunity to construct mathematical models. The findings show that students utilized their knowledge of fractions including conceptual and procedural knowledge in constructing mathematical models for the given situations. Some students were also able to generalize the fraction model and transfer it to a new situation. Analysis of the students’ work demonstrates that they made use of four fraction constructs—part-whole, operator, quotients, and ratio. The activities also revealed difficulties in the students’ knowledge of fractions, some of which were overcome in the process of organizing and mathematizing the problem.

Keywords

Fractions Modelling Model-eliciting activity Operator Part-whole 

References

  1. Barlow, A. & Drake, J. (2008). Division by a fraction: Assessing understanding through problem writing. Mathematics Teaching in the Middle School, 13(6), 326–332.Google Scholar
  2. Behr, M. J., Harel, G., Post, T,. & Lesh, R. (1993). Rational numbers: Toward a semantic analysis—emphasis on the operator construct. In T. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 13–47). Hillsdale, MI: Erlbaum.Google Scholar
  3. Behr, M. J., Lesh, R., Post, T. R. & Silver, E. A. (1983). Rational numbers concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–125). New York, NY : Academic.Google Scholar
  4. Bill, D. (2002). Student teachers’ understanding of rational numbers: Part-whole and numerical constructs. Research in Mathematics Education, 4(1), 53–67.CrossRefGoogle Scholar
  5. Blum, W. & Leiss, D. (2005). “Filling up”: The problem of independence-preserving teacher interventions in lessons with demanding modelling tasks. In CERME 4–Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (pp. 1623–1633).Barcelona, Spanish: Universitat Ramon Llull.Google Scholar
  6. Bonotto, C. (2010). Realistic mathematical modeling and problem posing. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies: ICTMA13 (pp. 399–408). New York, NY: Springer.CrossRefGoogle Scholar
  7. Brown, G. & Quinn, R. J. (2006). Algebra students’ difficulty with fractions: An error analysis. Australian Mathematics Teacher, 62(4), 28–40.Google Scholar
  8. Byrnes, J. P. (1992). The conceptual basis of procedural learning. Cognitive Development, 7, 235–237.CrossRefGoogle Scholar
  9. Charalambos, Y. C. & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64, 293–316.CrossRefGoogle Scholar
  10. Clarke, D. M. & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72(1), 127–138.CrossRefGoogle Scholar
  11. Doerr, H. & English, L. (2003). A modeling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 34(2), 110–136.CrossRefGoogle Scholar
  12. Doerr, H. M. & English, L. D. (2006). Middle grade teachers learning through students’ engagement with modeling tasks. Journal for Mathematics Teacher Education, 9, 5–32.CrossRefGoogle Scholar
  13. English, L. (2003). Mathematical modelling with young learners. In S. J. Lamon, W. A. Parker, & S. K. Houston (Eds.), Mathematical modelling: A way of life (pp. 3–17). Chichester, England: Horwood.CrossRefGoogle Scholar
  14. English, L. (2006). Mathematical modeling in the primary school: Children’s construction of a consumer guide. Educational Studies in Mathematics, 63(3), 303–323.CrossRefGoogle Scholar
  15. English, L. (2010). Young children’s early modelling with data. Mathematics Education Research Journal, 22(2), 24–47.CrossRefGoogle Scholar
  16. English, L. D. & Fox, J. L. (2005). Seventh-graders’ mathematical modelling on completion of a three-year program. In P. Clarkson et al. (Eds.), Building connections: Theory, research and practice (Vol. 1, pp. 321–328). Melbourne, Australia: Deakin University Press.Google Scholar
  17. English, L. D. & Walters, J. (2004). Mathematical modeling in the early school years. Journal for Research in Mathematics Education, 16(3), 59–80.Google Scholar
  18. Eric, C. C. M. (2008). Using model-eliciting activities for primary mathematics classrooms. The Mathematics Educator, 11(1), 47–66.Google Scholar
  19. Hallett, D., Nunes, T., Bryant, P., & Thorpe, C. M. (2012). Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience. Journal of Experimental Child Psychology, 113(4), 469–486.CrossRefGoogle Scholar
  20. Hart, K. (1981). Fractions. In K. Hart (Ed.), Children’s understanding of mathematics (pp. 66–81). London, England: John Murray.Google Scholar
  21. Hecht, S. A. & Vagi, K. J. (2012). Patterns of strengths and weaknesses in children’s knowledge about fractions. Journal of Experimental Child Psychology, 111, 212–229.CrossRefGoogle Scholar
  22. Hiebert, J. & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York, NY: Macmillan.Google Scholar
  23. Julie, C. (2002). Making relevance in mathematics teacher education. In I. Vakalis, D. Hughes Hallett, D. Quinney, & C. Kourouniotis (Eds.), Proceedings of the Second International Conference on the Teaching of Mathematics. New York, NY: Wiley.Google Scholar
  24. Julie, C. & Mudaly, V. (2007). Mathematical modelling of social issues in school mathematics in South Africa. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 503–510). New York, NY: Springer.CrossRefGoogle Scholar
  25. Kaiser, G. & Schwarz, B. (2010). Authentic modelling problems in mathematics education—examples and experiences. Journal für Mathematik-Didaktik, 30, 51–76.CrossRefGoogle Scholar
  26. Kieren, T. E. (1976). On the mathematical, cognitive and instructional foundations of rational numbers. In R. Lesh (Ed.), Number and measurement (pp. 101–144). Columbus: ERIC/SMEAC.Google Scholar
  27. Kieren, T. E. (1980). The rational number construct: Its elements and mechanisms. In T. E. Kieren (Ed.), Recent research on number learning (pp. 125–149). Columbus, OH: ERIC/SMEAC.Google Scholar
  28. Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 49–84). Hillsdale, MI: Erlbaum.Google Scholar
  29. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.Google Scholar
  30. Lesh, R. & Doerr, H. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 3–34). Mahwah, NJ: Erlbaum.Google Scholar
  31. Lesh, R. & Harel, G. (2003). Problem solving, modeling and local conceptual development. Mathematical Thinking and Learning, 5(2–3), 157–189.CrossRefGoogle Scholar
  32. Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In R. Lesh & A. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 591–644). Mahwah, NJ: Erlbaum.Google Scholar
  33. Lesh, R. & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5(2–3), 109–129.CrossRefGoogle Scholar
  34. Marshall, S. P. (1993). Assessment of rational number understanding: A schema-based approach. In T. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 261–288). Mahwah, NJ: Erlbaum.Google Scholar
  35. Mousoulides, N., Sriraman, B., & Lesh, R. (2008). The philosophy and practicality of modeling involving complex systems. The Philosophy of Mathematics Education Journal, 23, 134–157.Google Scholar
  36. Naiser, E. A., Wright, W. E., & Capraro, R. M. (2004). Teaching fractions: Strategies used for teaching fractions to middle grades students. Journal of Research in Childhood Education, 18(3), 193–198.CrossRefGoogle Scholar
  37. Organization for Economic Cooperation and Development (OECD). (2004). Learning for tomorrow’s world: First results from PISA 2003. Paris, France: Author.Google Scholar
  38. Rittle-Johnson, B., Siegler, R. S., & Wagner Alibali, M. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346–362.CrossRefGoogle Scholar
  39. Shahbari, J. A. & Peled, I. (2015). Using modelling tasks to facilitate the development of percentages. Canadian Journal of Science, Mathematics and Technology Education. Advance online publish. doi:10.1080/14926156.2015.1093201.
  40. Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences, 17(1), 13–19.CrossRefGoogle Scholar
  41. Siegler, S. R., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296.CrossRefGoogle Scholar
  42. Silver, E. A. & Kenney, P. A. (2000). Results from the seventh mathematics assessment of the national assessment of educational progress. Reston, VA: National Council for Teachers of Mathematics.Google Scholar
  43. Smith, C. L., Solomon, G. E. A., & Carey, S. (2005). Never getting to zero: Elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology, 51, 101–140.CrossRefGoogle Scholar
  44. Stafylidou, S. & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 508–518.CrossRefGoogle Scholar
  45. Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for Research in Mathematics Education, 30, 390–417.CrossRefGoogle Scholar
  46. Zawojewski, J. & Lesh, R. (2003). A models and modeling perspective on productive problem solving strategies. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: a models and modeling perspective on problem solving, learning and instruction in mathematics and science education (pp. 317–336). Mahwah, NJ: Erlbaum.Google Scholar

Copyright information

© Ministry of Science and Technology, Taiwan 2015

Authors and Affiliations

  1. 1.University of HaifaHaifaIsrael
  2. 2.Al-Qasemi Academy - Academic College of EducationBaqa al-GharbiyyeIsrael
  3. 3.The College of SakhninSakhninIsrael

Personalised recommendations