In the solution of mathematical word problems, problems that are accompanied by text, there is a need to bridge between mathematical language that requires an awareness of the mathematical components, and natural language that requires a literacy approach to the whole text. In this paper we present examples of mathematical word problems whose solutions depend on a transition between a linguistic situation on one side and abstract mathematical structure on the other. These examples demonstrate the need of treating word problems in a literacy approach. For this purpose, a model for teaching and learning is suggested. The model, which was tested successfully, presents an interactive multi-level process that enables deciphering of the mathematical text by means of decoding symbols and graphs. This leads to understanding of the revealed content and the linguistic situation, transfer to a mathematical model, and correspondence between the linguistic situation and the appropriate mathematical model. This model was tested as a case study. The participants were 3 students: a student in the sixth grade, a student in the ninth grade and a college student. All the students demonstrated an impressive improvement in their mathematical comprehension using this model.
KeywordsMathematical LanguageWord Problems
Ball, D. H. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93, 373-397. doi:10.1086/461730
Ben-Chaim, D., Keret, Y., & Ilany, B. (2006). Yahas veproporzia – Mehkar vehoraha behachsharat morim lematematica (Ratio and proportion- research and teaching in mathematics teacher training). Tel-Aviv: Mofet Inst. Press.
Bloedy-Vinner, H. (1998). The understanding of algebraic language in university preacademic students. Ph. D. dissertation, Jerusalem: Hebrew University.
Brown, G., & Yule, G. (1983). Discourse analysis. Cambridge: Cambridge University Press.
Clement, J. (1982). Algebra word problem solution: Thought processes under- lying a common misconception. Journal for Research in Mathemat ics Education, 13, 16-30. doi:10.2307/748434
De Lange, J. 1987 (1987). Mathematics insight and meaning. Utrect, Holland: Rijksuniversiteit.
Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht, Holland: Reidel Pub.
Folman, S. (2000). Hafakat Mashmaut mitext: Hebetim Hakaratiim-tiksortiim shel Heker Hasiah (Decoding meaning from a text: Cognitive and communicational aspects of discourse analysis). Tel-Aviv: Tel- Aviv University.
Freudenthal, H. (1991). Revising mathematics education. Dordrecht, South Holland: Kluw-er.
Gee, J. P. (1996). Social Linguistics and Literacy, Ideology in Discourse. Bristol, PA: Taylor & Francis.
Gravermeijer, K. (1997). Commentary on solving word problems: A case study of modeling?. Learning and Instruction, 7, 389-397. doi:10.1016/S0959-4752(97)00011-X
Greer, B. (1997). Modeling reality in the mathematics classroom: The case of word problems. Learning and Instruction, 7, 293-307. doi:10.1016/S0959-4752(97)00006-6
Halliday, M. A. K., & Hassan, R. (1976). Cohesion in English. London: Long-man.
Hershkovitz, S., & Nesher, P. (1996). The role of schemes in designing computerized environments. Educational Studies in Mathematics, 30, 339-366. doi:10.1007/BF00570829
Hershkovitz, S., & Nesher, P. (2003). The role of schemes in solving word problems. The Mathematics Educator, 7, 1-24.
Hiebert, J., & Carpenter, T.P. (1992). Learning and teaching with under- standing. In: D. A. Grouns (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-92). New York: Macmillan.
Kane, R. B. (1970). The readability of mathematics textbooks revisited. The Mathematics Teacher, 63, 579-581.
Kaput, J. J. (1993). The urgent need for proleptic research in representation of quantitative relationships. In: T. A., Romberg, E. Fennema and T. R. Carpenter (Eds.), Integrating research on graphical representation of functions (pp. 273- 311). London: Lawrence Earlbaum Associates.
Kaput, J. J., & Clement, J. (1979). Letter to the editor of JCMB. Journal of Children’s Mathematical Behavior, 2, pp. 208.
Kintsch, W. (1998). Comprehension: A Paradigm for Cognition. Cambridge, England: Cambridge University Press.
Lester, F. K. (1978). Mathematical problem solving in the elementary school: Some educational and psychological considerations. In: L. L Hatfield and D. A. Bradbard (Eds.), Mathematical problem solving: Papers from a research workshop (ERIC/SMET). Columbus, Ohio: Columbus.
MacGregor, M., & Price, E. (1999). An exploration of aspects of language proficiency and algebra learning. Journal for Research in Mathematics Education, 30, 449-467. doi:10.2307/749709
Margolin, B. (2002). Al defusey lechidut bein tarbutiim [On inter- cultural coherence patterns]. Script – Journal of the Israel Associa- tion for Literacy, 5-6, 81-89.
Nastasi, B. K., & Clements, D. H. (1990). Metacomponential functioning in young children. Intelligence, 14, 109-125.
Nathan, M. J., Kintsch, W., & Young, E. (1992). A theory of algebra-word- problem comprehension and its implications for the design of learning environments. Cognition and Instruction, 9, 329-389. doi:10.1207/s1532690xci0904_2
Nesher, P. (1988). Multiplicative school word problems: Theoretical approaches and empirical findings. In: J. Hiebert and M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 19-41). Mahwah, NJ: L. Erlbaum Associates.
Nesher, P., Greene, J. G., & Riley, M. S. (1982). The development of semantic categories for addition and subtraction. Educational Studies in Mathematics, 13, 373-394. doi:10.1007/BF00366618
Nesher, P., & Katriel, T. (1977). A semantic analysis of addition and subtraction word problem in arithmetic. Educational Studies in Mathematics, 8, 251-269. doi:10.1007/BF00385925
Nir, R. (1989). Semantika hivrit mashmaut vetikshoret (Hebrew semantics meaning and communication. Tel-Aviv: Open University.
Ormell, C. (1991). How ordinary meaning underpins the meaning of mathe- matics. Learning of Mathematics, 11, 25-30.
Piaget, J. (1980). Experiments in contradiction. Chicago and London: University of Chicago Press.
Polya, G. (1945). How to Solve it?. Princeton, NJ: Princeton University Press.
Reusser, K., & Stebler, R. (1997). Every word problem has a solution - the social rationality of mathematical modeling in school. Learning and Instruction, 7, 309-327. doi:10.1016/S0959-4752(97)00014-5
Rosnick, P. (1981). Some misconceptions concerning the concept of variable. Are you careful about defining your variables?. Mathematics Teacher, 74, 418-420, 450.
Sarel, Z. (1991). Mavo Lenituah Hsiah (Introduction to discourse analysis). Tel-Aviv: Or-Am.
Schoennfeld, A. H. (1980). Teaching problem-solving skills. American mathematical monthly, 87, 794-805. doi:10.2307/2320787
Silver, E. A., Shapiro, L. J., & Deutsch, A. (1993). Sense making and the solution of division problems involving remainders: An examination of middle school student's solution processes and their interpretation of solution. Journal for Research in Mathematics Education, 24, 117-135. doi:10.2307/749216
Van Dijk, T. A. (1980). Macrostructures: An interdisciplinary study of global structures in discourse. Mahwah, N.J.: L. Erlbaum Associ- ates.
Widdowson, H. G. (1979). Explorations in Applied Linguistics. Oxford, England: Oxford University.
Woolf, N. (2005). Lilmod lelamed Mathematica leshem Havana beezrat mentorim (Teaching how to teach Mathematics for understanding with mentors). In: R. Lidor, et al (Eds.), Zematim Bamehkar Hahinuhi (Cross-Roads in Educational Research) (pp.223-248), Tel-Aviv: Mofet Inst. Press.
Yerushalmi, M. (1997). Mathematizing qualitative verbal descriptions of situations: A language to support modeling. Cognition and Instruction, 15, 207-264. doi:10.1207/s1532690xci1502_3
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