Cute drawings? The disconnect between students’ pictorial representations and their mathematics responses to fraction questions
Palavras-chave:
Fractions, Modeling, Representations, Elementary SchoolResumo
Third and fourth grade students’ responses to open-ended questions requiring the modeling of fraction concepts were examined in order to determine the types and prevalence of difficulties students exhibit using pictorial representations in the problem-solving process. When developing pictorial representations, students experienced difficulties with model selection, partitioning, and comparison. Four specific difficulties students experienced in using pictorial representations to solve problems were: not answering the problem goal, incorrect model selection, failure to overcome whole number bias, and struggles with part-whole understanding.
Downloads
Referências
Abrams, J. P. (2001). Teaching mathematical modeling and the skills of representation. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, 2001 Yearbook (pp. 269-282). Reston, VA: National Council of Teachers of Mathematics.
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241.
Armstrong, B. E. & Larson, C. N. (1995). Students’ use of part-whole and direct comparison strategies for comparing partitioned rectangles. Journal for Research in Mathematics Education 26(2-19).
Brannon, E. M., Jordan, K. E., & Jones, S. (2010). Behavioral signatures of numerical discrimination. In Primate Neuroethology, Eds., M.L. Platt, A. Ghazanfar. Oxford Press, pp. 144-159.
Bulgar, S. (2009). A longitudinal study of students'representations for division of fractions. Montana Mathematics Enthusiast, 6,165-199.
Charalambous, C.Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions, Educational Studies in Mathematics, 64, 293-316.
Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org.
Confrey, J. (1998). Building mathematical structure within a conjecture driven teaching experiment on splitting. In S. B. Berenson & K. R. Dawkins (Eds.), Proceedings of the Twentieth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 39-50). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Davis, G. E., & Pitkethly, A. (1990). Cognitive aspects of sharing. Journal for Research in Mathematics Education, 21(2), 145-154.
Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17, 283–342.
Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C. L., Cirino, P. T., et al. (2013). Improving at-risk learners’ understanding of fractions. Journal of Educational Psychology, 105, 683-700.
Gabriel, F., Coche, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2012). Developing childrens’ understanding of fractions: an intervention study. Mind, Brain, and Education, 6, 137-146.
Goldin, G. & Shteingold, N. (2001). Systems of representation and the development of mathematical concepts. In A.A. Cuoco & F. R. Curcio (Eds.). The Roles of Representation in School Mathematics: 2001 Yearbook. National Council of Teachers of Mathematics: Reston, VA.
Grobecker, B. (2000). Imagery and fractions in students classified as learning disabled. Learning Disability Quarterly, 23(2), 157-168.
Hunting, R.P., Davis G., & Pearn, C. (1996). Engaging whole-number knowledge for rational-number learning using a computer based tool. Journal for Research in Mathematics Education, 27(3), 354-379.
Jordan, K. E., & Baker, J. (2011). Multisensory information boosts numerical matching abilities in young children. Developmental Science, 14, 205-213.
Jordan, K. E., & Brannon, E. M. (2009). A comparative approach to understanding human numerical cognition. In The Origins of Object Knowledge, Eds., B. Hood, L. Santos. Oxford University Press. pp. 53-84.
Kamii, C., & Clark, F. B. (1995). Equivalent fractions: Their difficulty and educational implications. Journal of Mathematical Behavior, 14, 365-378.
Kiczek, R. D., Maher, C. A., & Speiser, R. (2001). Tracing the origins and extensions of Michael’s representation. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, 2001 Yearbook (pp. 201-214). Reston, VA: National Council of Teachers of Mathematics.
Kieren, T. E. (1980). The rational number construct – its elements and mechanisms. In T. E. Kieren (Ed.), Recent research on number learning (pp. 125-150). Columbus: ERIC/SMEAC.
Kieren, T., Mason, R., & Pirie, S. (1992, June). The growth of mathematical understanding in a constructivist environment. Paper presented at the CSSE Meeting, P.E.I, Charlottetown.
Lamon, S. (1996). The development of unitizing: Its role in children's partitioning trategies. Journal for Research in Mathematics Education, 27, 170-191.
Lamon, S. (2005). Teaching fractions and ratios for understanding (2nd ed.). New York, NY: Routledge.
Lamon, S.J. (2001). Presenting and representing: From fractions to rational numbers. In A.A. Cuoco & F. R. Curcio (Eds.). The Roles of Representation in School Mathematics: 2001 Yearbook. National Council of Teachers of Mathematics: Reston, VA.
Larkin, J., McDermott, J., Simon, D.P., & Simon, H. A. (1980). Expert and novice performance in solving physics problems. Science, 208(20 June), 1335-1342.
Miles, M. B., & Huberman, A. M. (1994). An expanded sourcebook: Qualitative data analysis (2nd ed.). Thousand Oaks, CA: Sage Publications, Inc.
Moyer-Packenham, P. S., Jordan, K, Baker, J., Westenskow, A., Rodzon, K., Anderson, K., & Shumway, J. (2013, April). Hidden Predictors of Achievement: The Equalizing Effect of Virtual Manipulatives for Mathematics Instruction. Paper presented at the Annual Meeting of the American Educational Research Association (AERA), San Francisco, California.
Moyer-Packenham, P., Baker, J., Westenskow, A., Anderson, K., Shumway, J., Rodzon, K., & Jordan, K., The Virtual Manipulatives Research Group at Utah State University. (2013). A study comparing virtual manipulatives with other instructional treatments in third- and fourth-grade classrooms. Journal of Education, 193(2), 25-39.
National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author.
National Council of Teachers of Mathematics (NCTM), (2000) Principles and standards for school mathematics, Reston, VA: NCTM.
Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40 (3), 282-313.
Ni, Y., & Zhou, Y. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52.
Pothier, Y., & Sawada, D. (1983). Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14(5), 307-317.
Reeve, R. A., & Pattison, P. E. (1996). The referential adequacy of students’ visual analogies of fractions. Mathematical Cognition, 2(2), 137-169.
Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing Effective Fractions Instruction for Kindergarten Through 8th Grade: A practice guide (NCEE #2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U. S. Department of Education. Retrieved from whatworks.ed.gov/publication/practice guides.
Smith, J. P. (2002). The development of students’ knowledge of fractions and ratios. In B. Litwiller & G. Bright (Eds), Making Sense of Fractions, Ratios, and Proportions (pp. 3-17). Reston. VA: National Council of Teachers of Mathematics.
Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503-518.
Steffe, L. P. (2004). On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning, 6(2), 129-162.
Steinle, V., & Price, B. (2008). What does three-quarters look like? Students’ representations of three-quarters. In M. Groos, R. Brown, & K.Maker (Eds.), Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp 483-489). Brisbane, Australia: Education Research Group of Australasia Incorporated.
Torbeyns, J., Schneider, M., Xin, Z., & Siegler, R. S. (2014). Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, (2014).
Turner, E. E., Junk, D. L., & Empson, S. D. (2007). The power of paper-folding tasks: Supporting multiplicative thinking and rich mathematical discussion. Teaching Children Mathematics, 13(6), 322-329.
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(4), 390-416.
Van Garderen, D. (2006). Spatial visualization, visual imagery, and mathematical problem solving of students with varying abilities. Journal of Learning Disabilities, 39(6), 496–506.
Whitin, P., & Whitin, D. (2001). Using literature to invite mathematical representations. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, 2001 Yearbook (pp. 228-237). Reston, VA: National Council of Teachers of Mathematics.
Woleck, K. R. (2001). Listen to their pictures; An investigation of children’s mathematical drawings. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, 2001 Yearbook (pp. 215- 227). Reston, VA: National Council of Teachers of Mathematics.
Downloads
Publicado
Edição
Seção
Licença
Este trabalho está licenciado sob uma licença Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
