# Fraction Learners: Assessing Understanding Through Language Acquisition

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## Abstract

This study applies the Mathematics Acquisition Framework (MAF) (Authors, 2018) to investigate fifth- through seventh-grade students’ comprehension of fractions and decimals and examines how students build understandin of mathematical principles and concepts regarding fractions and decimals. Based on case studies and discourse analysis, the results revealed students in different stages of the MAF with some students employing informal/social language in their mathematical communication and others using formal/academic language. Additionally, mathematically erroneous student communication was more positively interpreted through the lens of the MAF, which interpreted student errors, misunderstandings, and miscommunications as natural components to learning.

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## Article Details

**International Electronic Journal of Elementary Education**, [S.l.], v. 11, n. 2, p. 113-124, jan. 2019. ISSN 1307-9298. Available at: <https://iejee.com/index.php/IEJEE/article/view/649>. Date accessed: 26 mar. 2019.

## References

Bailey, N., Madden, C., & Krashen, S. (1974). Is there a ‘natural sequence’ in adult second language learning? Language Learning, 24, 235-243.

Barnett, C., Goldenstein, D., & Jackson, B. (Eds.) (1994). Fractions, decimals, ratios, & percents: Hard to teach and hard to learn? Mathematics teaching cases. Portsmouth, NH.: Heinemann.

Bartelet, D., Ansari, D., Vaessen A., & Blomert, A. (2014) Cognitive subtypes of mathematics learning difficulties in primary education. Research in Developmental Disabilities 35 (2014) 657–670.

Biggs, J. B., & Collis, K. F. (1982). Evaluating the quality of learning - The SOLO taxonomy. New York: 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.

Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17, 31–48.

Bruner, J. S. (1966). Toward a Theory of İnstruction, Cambridge, Mass.: Belkapp Press

Chomsky, N. (1957). Syntactic Structures. The Hague: Mouton.

Chomsky, N. (1959). A review of B. F. Skinner’s Verbal Behavior. Language, 35(1), 26-58.

Chomsky, N. (1964). Current İssues in Linguistic Theory. The Hague: Mouton.

Christiansen, I. (1997). When negotiation of meaning is also negotiation of task: Analysis of the communication in an applied mathematics high school course. Educational Studies in Mathematics 34(1), 1-25.

Clements, D. H., Battista, M. T., & Sarama, J. (2001). Logo and geometry. Journal for Research in Mathematics Education, Monograph Series, 10.

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

Cummins, J. (1979). Cognitive/academic language proficiency, linguistic interdependence, the optimum age question and some other matters. Working Papers on Bilingualism, 19, 121-129.

Cummins, J. (1984). Bilingualism and Special Education: Issues in Assessment and Pedagogy. Clevedon, England: Multilingual Matters.

Cummins, J. (1991). Language development and academic learning. In L. Malavé & G. Duquette (Eds.) Language, culture and cognition. (pp. 161-175). Clevedon, England: Multilingual Matters.

DeWolf, M., Rapp, M., Bassok, M., & 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.

Dienes, Z. P. (1960). Building up Mathematics. London: Hutchinson Education Ltd.

Dienes, Z. P. (1971). An example of the passage from the concrete to the manipulation of formal systems. Educational Studies in Mathematics, 3 (3/4), 337-352.

Dienes, Z. P., & Golding, E. W. (1971). Approach to Modern Mathematics. New York: Herder and Herder Co.

Dulay, H., & Burt, M. (1973). Should we teach children syntax? Language Learning, 23, 245-258.

Dulay, H., & Burt, M. (1974). Natural sequences in child second language acquisition. Language Learning, 24, 37-53.

Fuys, D. (1985). Van Hiele levels of thinking in geometry. Education and Urban Society, 17(4), 447-462.

Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education. Monograph 3.

Garfinkel, H. (1967). Studies in Ethnomethodology. Englewood Cliffs, NJ: Prentice- Hall.

uculano, 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.

Kara, F., & Incikabı, L. (2018). Sixth Grade Students’ Skills of Using Multiple Representations in Addition and Substraction perations in Fractions. International Electronic Journal of Elementary Education, 10(4), 463-474.

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.

Krashen, S. (1977). Some issues relating to the monitor model. In H. D. Brown, C. Yorio & R. Crymes (Eds.) Teaching and learning english as a second language: Trends in research and practice. (pp. 144-158) Washington: TESOL.

Krashen, S. D. (1982). Principles and Practice in Second Language Acquisition. Oxford: Pergamon.

Krashen, S. D., & Terrell, T. (1983). The Natural Approach: Language Acquisition in the Classroom. London: Prentice Hall Europe.

Lightbrown, P. M., & Spada, N. (1999). How languages are learned [2nded.]. Oxford: Oxford University Press.

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.

Muzheve, M. T., & Capraro, R. M. (2012). An exploration of the role natural language and idiosyncratic

representations in teaching how to convert among fractions, decimals, and percents. Journal of

Mathematical Behavior, 31(1), 1-14

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.

Pica, T. (1987). Second language acquisition, social interaction, and the classroom, Applied Linguistics, 8, 3–21.

Pica, T. (1996). The essential role of negotiation in the communicative classroom. Japan Association of Language Teaching Journal,18 (2), 241-268

Piaget, J (1979). “Comments on Vygotsky’s critical remarks”. Archives de Psychologie. 47 (183): 237–249.

Schneider, M., & Siegler, R. S. (2010). Representations of the magnitudes of fractions. Journal of Experimental

Psychology: Human Perception and Performance, 36, 1227–1238.

Selinker, L . (1992). Rediscovering İnterlanguage. London:Longman.

Selinker, L. (1972). Interlanguage. International Review of Applied Linguistics, 10, 209–231.

Swain, M. (1985). Communicative competence: some roles of comprehensible input and comprehensible output

in its development. In S. Gass, S. and C. Madden, (Eds.), Input in second language acquisition. Rowley, MA: Newbury House.

Swain, M., & Lapkin, S. (1995). Problems in output and the cognitive processes they generate: a step towards

second language learning. Applied Linguistics, 16(3), 371-391.

Usiskin, Z. (1996). Mathematics as a language. In Communication in Mathematics: K-12 and Beyond 1996 Yearbook. Edited by Elliott, P.C., & Kenney, M.J. Reston, VA: NCTM.

Van Hiele, P. M. (1986). Structure and İnsight: A theory of Mathematics Education. Orlando, FL: Academic Press.

Vygotsky, L. S. (1978). Mind in society. Cambridge, MA: Harvard University Press.

Wodak, R. (2009). Aspects of Critical Discourse Analysis. London. Sage Publications.

Wodak, R., & Meyer, M. (2009). Methods of Critical Discourse Analysis. London and New York. Longman.