How a Measuring Perspective Influences Pre-service Teachers’ Reasoning about Fractions with Discrete and Continuous Models
Keywords:
Fraction Models, Pre-service Teacher Knowledge, Mathematical Reasoning, Measuring PerspectiveAbstract
How teachers interpret and express fractions critically influences their teaching and their students’ fraction knowledge. Internationally, the mathematics education community has been studying ways to enhance pre-service elementary teachers’ rational number knowledge, particularly fractions. To address the challenge of augmenting pre-service teachers’ fraction knowledge warrants theoretical and empirical revisions to standardized practices for teaching fractions. This study investigates how reexamining fractions from a distinctive measuring perspective influences pre-service teachers’ reasoning about fractions. For four 75-minute sessions, 46 pre-service teachers enrolled in a teacher preparation program at a university in the United States revisited fractions from a measuring perspective. They engaged in tasks that focused on comparing continuous quantities and identifying relative magnitudes. The data for this study comprise their pre- and post-tests that assessed how they identify and represent fractions with discrete and continuous models. For each model, we analyzed participants’ reasoning by attending to their written strategies. Findings revealed three main strategies: partition, construction, and symbolic manipulation. In general, participants expressed more strategies on the post-test for all fraction models. Partitioning was the most frequent strategy on the pre- and post-tests. However, the frequencies of strategies changed after the intervention. For example, with all models, there was an increase in partitioning strategy and a decrease in symbolic manipulation strategy. The results highlight affordances of a measuring perspective to support participants to shift from procedural strategies such as symbolic manipulation to more conceptual strategies to identify and represent fractions.
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References
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Behr, M. J., Khoury, H. A., Harel, G., Post, T., & Lesh, R. (1997). Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-Operator Task. Journal for Research in Mathematics Education, 28(1), 48-69. https://doi.org/10.2307/749663
Bobos, G., & Sierpinska, A. (2017). Measurement approach to teaching fractions: A design experiment in a pre-service course for elementary teachers. International Journal for Mathematics Teaching and Learning, 18(2), 203-139.
Bright, G. W., Behr, M. J., Post, T. R., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19(3), 215-232.
Busi, R., Lovin, L., Norton, A., Siegfried, J., Stevens, A., & Wilkins, J. L. (2015). An Investigation of PreK-8 Preservice Teachers' Construction of Fraction Schemes and Operations. In T. G. Bartell, K. N. Bieda, R. T. Putnam, K. Bradfield, & H. Dominguez (Eds.), Proceedings of the 37th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 616-623). Michigan State University.
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Hodges, T. E., Cady, J., & Collins, R. L. (2008). Fraction representation: The not-so-common denominator among textbooks. Mathematics Teaching in the Middle School, 14(2), 78-84.
Hsieh, H.-F., & Shannon, S. F. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277-1288.
Kerslake, D. (1986). Fractions: Children’s strategies and errors. NEER-NELSON.
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Lee, M. Y., & Lee, J.-E. (2019). Pre-service teachers’ perceptions of the use of representations and suggestions for students’ incorrect use. EURASIA Journal of Mathematics, Science and Technology Education, 15(9), 1-21. https://doi.org/doi.org/10.29333/ejmste/103055
Lo, J.-J., & Luo, F. (2012). Prospective elementary teachers’ knowledge of fraction division [journal article]. Journal of Mathematics Teacher Education, 15(6), 481-500. https://doi.org/10.1007/s10857-012-9221-4
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Morano, S., & Riccomini, P. J. (2019). Demonstrating conceptual understanding of fraction arithmetic: An analysis of pre-service special and general educators’ visual representations. Teacher Education and Special Education. https://doi.org/10.1177/0888406419880540
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Behr, M. J., Khoury, H. A., Harel, G., Post, T., & Lesh, R. (1997). Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-Operator Task. Journal for Research in Mathematics Education, 28(1), 48-69. https://doi.org/10.2307/749663
Bobos, G., & Sierpinska, A. (2017). Measurement approach to teaching fractions: A design experiment in a pre-service course for elementary teachers. International Journal for Mathematics Teaching and Learning, 18(2), 203-139.
Bright, G. W., Behr, M. J., Post, T. R., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19(3), 215-232.
Busi, R., Lovin, L., Norton, A., Siegfried, J., Stevens, A., & Wilkins, J. L. (2015). An Investigation of PreK-8 Preservice Teachers' Construction of Fraction Schemes and Operations. In T. G. Bartell, K. N. Bieda, R. T. Putnam, K. Bradfield, & H. Dominguez (Eds.), Proceedings of the 37th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 616-623). Michigan State University.
Carraher, D. W. (1993). Lines of thought: A ratio and operator model of rational number. Educational Studies in Mathematics, 25(4), 281-305. https://doi.org/10.1007/BF01273903
Charalambous, C. Y., Hill, H. C., Chin, M. J., & McGinn, D. (2020). Mathematical content knowledge and knowledge for teaching: Exploring their distinguishability and contribution to student learning. Journal of Mathematics Teacher Education, 23, 579-613. https://doi.org/10.1007/s10857-019-09443-2
Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understanding of fractions. Educational Studies in Mathematics, 64, 293-316.
Cramer, K. A., Post, T. R., & delMas, R. C. (2002). Initial fraction learning by fourth-and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111-144. https://doi.org/10.2307/749646
Cuisenaire, G. (1952). Les nombres en couleurs. Nouveau procédé de calcul par la méthode active, applicable à tous les degrés de l’école primaire. Duculot-Roulin.
Davydov, V. V., & Tsvetkovich, Z. (1991). On the objective origin of the concept of fractions. Focus on Learning Problems in Mathematics, 13(1), 13-64.
Depaepe, F., Torbeyns, J., Vermeersch, N., Janssens, D., Janssen, R., Kelchtermans, G., Verschaffel, L., & Van Dooren, W. (2015). Teachers' content and pedagogical content knowledge on rational numbers: A comparison of prospective elementary and lower secondary school teachers. Teaching and Teacher Education, 47, 82-92. https://doi.org/10.1016/j.tate.2014.12.009
Dienes, Z. P. (1967). Fractions: An operational approach. Herder and Herder.
Erol, M. (2021). Determination of 5th-grade students strategies in comparing fractions. Acta Didactica Napocensia, 14(1), 17-28. https://doi.org/10.24193/adn.14.1.2
Fazio, L., & Siegler, R. (2011). Teaching fractions. International Academy of Education.
Fuchs, L. S., Malone, A. S., Wang, A., Schumacher, R. F., & Namkung, J. (2017). Fraction intervention for students with mathematics difficulties: Lessons learned from five randomized controlled trials [Article]. Journal of Learning Disabilities, 50(6), 631-639. https://doi.org/10.1177/0022219416677249
Gabriel, F. C., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2013). A componential view of children's difficulties in learning fractions. Frontiers in psychology, 4, 1-12. https://doi.org/10.3389/fpsyg.2013.00715
Gattegno, C. (1960/2009). Arithmetic: A teacher’s introduction to the Cuisenaire-Gattegno method of teaching arithmetic. Educational Solutions Worldwide.
Gattegno, C. (1974/2010). The commonsense of teaching mathematics (Second ed.). Educational Solutions Worldwide.
Hamdan, N., & Gunderson, E. A. (2017). The number line is a critical spatial-numerical representation: Evidence from a fraction intervention. Developmental Psychology, 53(3), 587-596. https://doi.org/10.1037/dev0000252
Harvey, R. (2012). Stretching student teachers’ understanding of fractions. Mathematics Education Research Journal, 24(4), 493-511.
Hodges, T. E., Cady, J., & Collins, R. L. (2008). Fraction representation: The not-so-common denominator among textbooks. Mathematics Teaching in the Middle School, 14(2), 78-84.
Hsieh, H.-F., & Shannon, S. F. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277-1288.
Kerslake, D. (1986). Fractions: Children’s strategies and errors. NEER-NELSON.
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). ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 49-84). Erlbaum.
Lamon, S. J. (1996). The development of unitizing: Its role in children’s partitioning strategies. Journal for Research in Mathematics Education, 27(2), 170-193.
Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In J. F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A Project of the National Council of Teachers of Mathematics (pp. 629-667). Information Age.
Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed.). Routledge.
Lee, M. Y., & Lee, J.-E. (2019). Pre-service teachers’ perceptions of the use of representations and suggestions for students’ incorrect use. EURASIA Journal of Mathematics, Science and Technology Education, 15(9), 1-21. https://doi.org/doi.org/10.29333/ejmste/103055
Lo, J.-J., & Luo, F. (2012). Prospective elementary teachers’ knowledge of fraction division [journal article]. Journal of Mathematics Teacher Education, 15(6), 481-500. https://doi.org/10.1007/s10857-012-9221-4
Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21(1), 16-32.
Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26(5), 422-441.
Maher, C. A., & Yankelewitz, D. (2017). Children’s reasoning while building fraction ideas (Vol. 3). Sense Publishers.
Meert, G., Grégoire, J., Seron, X., & Noël, M.-P. (2013). The processing of symbolic and nonsymbolic ratios in school-age children. PloS one, 8(11), 1-11. https://doi.org/10.1371/journal.pone.0082002
Morano, S., & Riccomini, P. J. (2019). Demonstrating conceptual understanding of fraction arithmetic: An analysis of pre-service special and general educators’ visual representations. Teacher Education and Special Education. https://doi.org/10.1177/0888406419880540
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. The National Council of Teachers of Mathematics, Inc.
National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common core state standards for mathematics. National Governors Association Center for Best Practices, Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf
Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52.
Norton, A., & Wilkins, J. L. (2010). Students’ partitive reasoning. The Journal of Mathematical Behavior, 29(4), 181-194. https://doi.org/10.1016/j.jmathb.2010.10.001
Ohlsson, S. (1988). Mathematical meaning and applicational meaning in the semantics of fractions and related concepts. In J. Hiebert & M. J. Behr (Eds.), Number concepts and operations in the middle grades (pp. 53-93). National Council of Teachers of Mathematics.
Olanoff, D., Lo, J.-J., & Tobias, J. M. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, 11(2), 267-310.
Powell, A. B. (2019a). How does a fraction get its name? Revista Brasileira de Educação em Ciências e Educação Matemática, 3(3), 700-713.
Powell, A. B. (2019b). Measuring perspective of fraction knowledge: Integrating historical and neurocognitive findings. Revista Sergipana de Matemática e Educação Matemática, 4(1), 1-19.
Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, XIX(1), 19-43.
Schmittau, J. (2005). The development of algebraic thinking: A Vygotskian perspective. ZDM - International Journal on Mathematics Education, 37(1), 16-22. http://www.scopus.com/inward/record.url?eid=2-s2.0-84855948751&partnerID=40&md5=9a3042942fed27182eb2dfc48797d22e
Sengul, S. (2013). Identification of number sense strategies used by pre-service elementary teachers. Educational Sciences: Theory and Practice, 13(3), 1965-1974. https://doi.org/10.12738/estp.2013.3.1365
Siegler, R. S., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing effective fraction instruction for kindergarten through 8th grade: A practice guide (NCEE #2010-4039). National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education.
Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early Predictors of High School Mathematics Achievement. Psychological Science, 23(7), 691-697. https://doi.org/doi:10.1177/0956797612440101
Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909-918.
Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive psychology, 62(4), 273-296. https://doi.org/10.1016/j.cogpsych.2011.03.001
Steffe, L. P., & Olive, J. (Eds.). (2010). Children’s fractional knowledge. Springer.
Tobias, J. M. (2013). Prospective elementary teachers’ development of fraction language for defining the whole. Journal of Mathematics Teacher Education, 16(2), 85-103.
Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166-175. https://doi.org/10.1016/j.tate.2008.08.003
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Published
2022-03-16
How to Cite
Alqahtani, M., Powell , A., Webster, V. ., & Tirnovan, D. (2022). How a Measuring Perspective Influences Pre-service Teachers’ Reasoning about Fractions with Discrete and Continuous Models. International Electronic Journal of Elementary Education, 14(3), 441–458. Retrieved from https://www.iejee.com/index.php/IEJEE/article/view/1786
