# Semantic and Syntactic Fraction Understanding

## Main Article Content

## Abstract

This study begins by connecting semantic elaboration with conceptual understanding and syntactic elaboration with procedural understanding in the context of fractions. Through case studies and discourse analysis, the work and communication of students in fourth through sixth grade is analyzed to determine the extent of their semantic and syntactic elaboration regarding fractions and fraction operations. Findings are that, while some students emphasized one form of elaboration over the other, some students demonstrated use of both forms of elaboration. Indeed, it is wondered if semantic and syntactic elaboration should be seen as more complementary than adversarial.

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## Article Details

How to Cite

BAYAGA, Anass; BOSSÉ, Michael J.
Semantic and Syntactic Fraction Understanding.

**International Electronic Journal of Elementary Education**, [S.l.], v. 11, n. 2, p. 135-142, jan. 2019. ISSN 1307-9298. Available at: <https://iejee.com/index.php/IEJEE/article/view/653>. Date accessed: 15 oct. 2019.
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Articles

## References

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Behr, M., Lesh, R., Post, T., & Silver, E. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91–125). New York, NY: Academic Press.

Bogdan, R. C., & Biklen, S. K. (2003). Qualitative research for education: An introduction to theories and methods (4th ed.). Boston: Allyn and Bacon.

Braithwaite, D. W., Pyke, A. A., & Siegler, R. S. (2017). A computational model of fraction arithmetic. Psychological Review, 124(5), 603-625.

Bulgar, S. (2003) Children’s’ sense-making of division of fractions. Journal of Mathematical Behavior, 22(3), 319–334.

Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 27(5), 777–786.

Charalambous, C. Y. & Pitta-Pantazi, D. (2007) Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293–316.

Creswell, W. J. (2003). Research design: Qualitative, quantitative and mixed methods approaches (2nd ed.). London: Sage Publications.

D'Ambrosio, B. S., & Spangler Mewborn, D. (1994) Children's constructions of fractions and their implications for classroom instruction. Journal of Research in Childhood Education, 8(2), 150-161.

DeWolf , M., Rapp, M., Bassok, M. and Holyoak, K. J. (2014) Semantic alignment of fractions and decimals with discrete versus continuous entities: A textbook analysis. Proceedings of the Annual Meeting of the Cognitive Science Society, 36, 2133- 2138.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.

Gabriel, F., 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, 715.

Goldin, G. A. (2000). A scientific perspective on structure: task-based interviews in mathematics education research. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517-546). Mahwah, NJ: Lawrence Erlbaum Associates.

Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions. Journal Of Educational Psychology, 102(2), 395-406.

Halliday, M. A. K. (1975). Learning how to mean: Explorations in the development of language. London: Edward Arnold.

Halliday, M. A. K. (1993). Towards a language-based theory of learning. Linguistics and Education, 5(2), 93–116.

Halliday, M. A. K., McIntosh, A., & Strevens, P. (1964). The linguistic sciences and language teaching. London: Longmans.

Hyde, D. C., Khanum, S., & Spelke, E. S. (2014). Brief non-symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children. Cognition, 131, 92–107.

Inglis, M., & Gilmore, C. (2013). Sampling from the mental number line: How are approximate number system representations formed? Cognition, 129, 63–69.

Iuculano, T. & Butterworth, B. (2011). Rapid communication Understanding the real value of fractions and decimals. The Quarterly Journal of Experimental Psychology, 64 (11), 2088–2098.

Jacob, S. N., & Nieder, A. (2009). Notation-independent representation of fractions in the human parietal cortex. The Journal of Neuroscience, 29(14), 4652–4657.

Kallai, A.Y., & Tzelgov, J. (2009). A generalized fraction: An entity smaller than one of the mental number line. Journal of Experimental Psychology: Human Perception and Performance, 35(6), 1845–1864.

Kaput, J. J. (1987a). Representation systems and mathematics. In C. Janvier (Ed.), Problems of representation in teaching and learning mathematics (pp. 19–26). Hillsdale, NJ: Erlbaum.

Kaput, J. J. (1987b). Toward a theory of symbol use in mathematics. In C. Janvier (Ed.), Problems of representation in mathematics learning and problem solving (pp. 159-195). Hillsdale, NJ: Erlbaum.

Kerslake, D. (1986). Fractions: Children’s strategies and errors. A report of the strategies and errors in secondary mathematics project. Windsor, England: NFER-Nelson.

Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. A. Lesh & D. A. Bradbard (Eds.), Number and Measurement. Papers from a Research Workshop (pp. 101–144). Columbus, OH: ERIC

Kolkman, M.E., Kroesbergen, E.H., & Leseman, P.P.M. (2013). Early numerical development and the role of non-symbolic and symbolic skills. Learning and Instruction, 25, 95.

Leibovich, T. and Ansari, D. (2016). The symbol-grounding problem in numerical cognition: A review of theory, evidence, and outstanding questions. Canadian Journal of Experimental Psychology, 1196-1961.

Lyons, I. M., Price, G. R., Vaessen, A, Blomert, L., & Ansari, A. (2014). Numerical predictors of arithmetic success in grades 1–6. Developmental Science, X, pp 1–11.

Meert, G., Grégoire, J., & Noël, M.-P. (2009). Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task. The Quarterly Journal of Experimental Psychology, 62(8), 1598–1616.

Miles, M. B. & Huberman, M. N. (1994). Qualitative data analysis: an expanded sourcebook. Thousand Oaks, CA: Sage.

Murray, H., Olivier, A., & Human, P. (1996). Young students’ informal knowledge of fractions. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the Twentieth Conference of the International Group for the Psychology of Mathematics Education, 4, 43 – 50. Valencia, Spain.

Newstead, K. and Murray, H. (1998). Young students' constructions of fractions. In A. Olivier & K. Newstead (Eds.), Proceedings of the Twenty-second International Conference for the Psychology of Mathematics Education: Vol. 3. (pp. 295-302). Stellenbosch, South Africa.

Opfer, J. E., & DeVries, J. M. (2008). Representational change and magnitude estimation: Why young children can make more accurate salary comparisons than adults. Cognition, 108, 843–849.

Park, J., & Brannon, E. M. (2013). Training the approximate number system improves math proficiency. Psychological Science, 24, 2013– 2019.

Park, J., & Brannon, E. M. (2014). Improving arithmetic performance with number sense training: An investigation of underlying mechanism. Cognition, 133, 188–200.

Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346 –362.

Sasanguie, D., Gobel, S.M., Moll, K., Smets, K., & Reynvoet, B. (2013). Approximate number sense, symbolic number processing, or number-space mappings: what underlies mathematics achievement? Journal of Experimental Child Psychology, 114 (3), 418–431.

Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly, 23(2), 139–159.

Schneider, M., & Siegler, R. S. (2010). Representations of the magnitudes of fractions. Journal of Experimental Psychology: Human Perception and Performance, 36, 1227–1238.

Schulze, J. M. (2016). Understanding the developing persuasive writing practices of an adolescent emergent bilingual through systemic functional linguistics: A case study. International Journal of Learning, Teaching and Educational Research, 15(10), 163–179.

Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M.,... Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(10), 691– 697.

Siegler, R. S., Thompson, C. & Schneider, M. (2011) An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296.

Stafylidou, S. & Vosniadou, S. (2004) The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503–518.

Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Springer Science & Business Media.

Strauss, A., & Corbin, J. (1990). Basics of qualitative research. London: Sage Publications Ltd.

Thwaite, A. (2015). Pre-service teachers linking their metalinguistic knowledge to their practice: A functional approach. Functional Linguistics, 2(1), 1–17.

Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: the case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5–25.

Turkan, S., de Oliveira, L. C., Lee, O., & Phelps, G. (2014). Proposing a knowledge base for teaching academic content to English language learners: Disciplinary linguistic knowledge. Teachers College Record, 116(3), 1–30.

van Lier, L., & Walqui, A. (2012). Language and the Common Core State Standards. Commissioned paper by the Understanding Language Initiative. Stanford, CA: Stanford University. Retrieved from http://ell.stanford.edu/papers/language

Van Steenbrugge, H., Lesage, E., Valcke, M., & Desoete, A. (2014). Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge?. Journal of Curriculum Studies, 46(1), 138-161.

Wodak, R. & Meyer, M. (2009). Methods for critical discourse analysis. SAGE

Baroody, A. J., & Hume, J. (1991). Meaningful mathematics instruction: The case of fractions. Remedial and Special Education, 12(3), 54-68.

Behr, M., Lesh, R., Post, T., & Silver, E. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91–125). New York, NY: Academic Press.

Bogdan, R. C., & Biklen, S. K. (2003). Qualitative research for education: An introduction to theories and methods (4th ed.). Boston: Allyn and Bacon.

Braithwaite, D. W., Pyke, A. A., & Siegler, R. S. (2017). A computational model of fraction arithmetic. Psychological Review, 124(5), 603-625.

Bulgar, S. (2003) Children’s’ sense-making of division of fractions. Journal of Mathematical Behavior, 22(3), 319–334.

Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 27(5), 777–786.

Charalambous, C. Y. & Pitta-Pantazi, D. (2007) Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293–316.

Creswell, W. J. (2003). Research design: Qualitative, quantitative and mixed methods approaches (2nd ed.). London: Sage Publications.

D'Ambrosio, B. S., & Spangler Mewborn, D. (1994) Children's constructions of fractions and their implications for classroom instruction. Journal of Research in Childhood Education, 8(2), 150-161.

DeWolf , M., Rapp, M., Bassok, M. and Holyoak, K. J. (2014) Semantic alignment of fractions and decimals with discrete versus continuous entities: A textbook analysis. Proceedings of the Annual Meeting of the Cognitive Science Society, 36, 2133- 2138.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.

Gabriel, F., 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, 715.

Goldin, G. A. (2000). A scientific perspective on structure: task-based interviews in mathematics education research. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517-546). Mahwah, NJ: Lawrence Erlbaum Associates.

Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions. Journal Of Educational Psychology, 102(2), 395-406.

Halliday, M. A. K. (1975). Learning how to mean: Explorations in the development of language. London: Edward Arnold.

Halliday, M. A. K. (1993). Towards a language-based theory of learning. Linguistics and Education, 5(2), 93–116.

Halliday, M. A. K., McIntosh, A., & Strevens, P. (1964). The linguistic sciences and language teaching. London: Longmans.

Hyde, D. C., Khanum, S., & Spelke, E. S. (2014). Brief non-symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children. Cognition, 131, 92–107.

Inglis, M., & Gilmore, C. (2013). Sampling from the mental number line: How are approximate number system representations formed? Cognition, 129, 63–69.

Iuculano, T. & Butterworth, B. (2011). Rapid communication Understanding the real value of fractions and decimals. The Quarterly Journal of Experimental Psychology, 64 (11), 2088–2098.

Jacob, S. N., & Nieder, A. (2009). Notation-independent representation of fractions in the human parietal cortex. The Journal of Neuroscience, 29(14), 4652–4657.

Kallai, A.Y., & Tzelgov, J. (2009). A generalized fraction: An entity smaller than one of the mental number line. Journal of Experimental Psychology: Human Perception and Performance, 35(6), 1845–1864.

Kaput, J. J. (1987a). Representation systems and mathematics. In C. Janvier (Ed.), Problems of representation in teaching and learning mathematics (pp. 19–26). Hillsdale, NJ: Erlbaum.

Kaput, J. J. (1987b). Toward a theory of symbol use in mathematics. In C. Janvier (Ed.), Problems of representation in mathematics learning and problem solving (pp. 159-195). Hillsdale, NJ: Erlbaum.

Kerslake, D. (1986). Fractions: Children’s strategies and errors. A report of the strategies and errors in secondary mathematics project. Windsor, England: NFER-Nelson.

Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. A. Lesh & D. A. Bradbard (Eds.), Number and Measurement. Papers from a Research Workshop (pp. 101–144). Columbus, OH: ERIC

Kolkman, M.E., Kroesbergen, E.H., & Leseman, P.P.M. (2013). Early numerical development and the role of non-symbolic and symbolic skills. Learning and Instruction, 25, 95.

Leibovich, T. and Ansari, D. (2016). The symbol-grounding problem in numerical cognition: A review of theory, evidence, and outstanding questions. Canadian Journal of Experimental Psychology, 1196-1961.

Lyons, I. M., Price, G. R., Vaessen, A, Blomert, L., & Ansari, A. (2014). Numerical predictors of arithmetic success in grades 1–6. Developmental Science, X, pp 1–11.

Meert, G., Grégoire, J., & Noël, M.-P. (2009). Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task. The Quarterly Journal of Experimental Psychology, 62(8), 1598–1616.

Miles, M. B. & Huberman, M. N. (1994). Qualitative data analysis: an expanded sourcebook. Thousand Oaks, CA: Sage.

Murray, H., Olivier, A., & Human, P. (1996). Young students’ informal knowledge of fractions. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the Twentieth Conference of the International Group for the Psychology of Mathematics Education, 4, 43 – 50. Valencia, Spain.

Newstead, K. and Murray, H. (1998). Young students' constructions of fractions. In A. Olivier & K. Newstead (Eds.), Proceedings of the Twenty-second International Conference for the Psychology of Mathematics Education: Vol. 3. (pp. 295-302). Stellenbosch, South Africa.

Opfer, J. E., & DeVries, J. M. (2008). Representational change and magnitude estimation: Why young children can make more accurate salary comparisons than adults. Cognition, 108, 843–849.

Park, J., & Brannon, E. M. (2013). Training the approximate number system improves math proficiency. Psychological Science, 24, 2013– 2019.

Park, J., & Brannon, E. M. (2014). Improving arithmetic performance with number sense training: An investigation of underlying mechanism. Cognition, 133, 188–200.

Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346 –362.

Sasanguie, D., Gobel, S.M., Moll, K., Smets, K., & Reynvoet, B. (2013). Approximate number sense, symbolic number processing, or number-space mappings: what underlies mathematics achievement? Journal of Experimental Child Psychology, 114 (3), 418–431.

Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly, 23(2), 139–159.

Schneider, M., & Siegler, R. S. (2010). Representations of the magnitudes of fractions. Journal of Experimental Psychology: Human Perception and Performance, 36, 1227–1238.

Schulze, J. M. (2016). Understanding the developing persuasive writing practices of an adolescent emergent bilingual through systemic functional linguistics: A case study. International Journal of Learning, Teaching and Educational Research, 15(10), 163–179.

Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M.,... Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(10), 691– 697.

Siegler, R. S., Thompson, C. & Schneider, M. (2011) An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296.

Stafylidou, S. & Vosniadou, S. (2004) The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503–518.

Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Springer Science & Business Media.

Strauss, A., & Corbin, J. (1990). Basics of qualitative research. London: Sage Publications Ltd.

Thwaite, A. (2015). Pre-service teachers linking their metalinguistic knowledge to their practice: A functional approach. Functional Linguistics, 2(1), 1–17.

Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: the case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5–25.

Turkan, S., de Oliveira, L. C., Lee, O., & Phelps, G. (2014). Proposing a knowledge base for teaching academic content to English language learners: Disciplinary linguistic knowledge. Teachers College Record, 116(3), 1–30.

van Lier, L., & Walqui, A. (2012). Language and the Common Core State Standards. Commissioned paper by the Understanding Language Initiative. Stanford, CA: Stanford University. Retrieved from http://ell.stanford.edu/papers/language

Van Steenbrugge, H., Lesage, E., Valcke, M., & Desoete, A. (2014). Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge?. Journal of Curriculum Studies, 46(1), 138-161.

Wodak, R. & Meyer, M. (2009). Methods for critical discourse analysis. SAGE