Language and Mathematics: Bridging between Natural Language and Mathematical Language in Solving Problems in Mathematics

ABSTRACT

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.

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.

Cite this paper

nullIlany, B. & Margolin, B. (2010). Language and Mathematics: Bridging between Natural Language and Mathematical Language in Solving Problems in Mathematics.*Creative Education, 1,* 138-148. doi: 10.4236/ce.2010.13022.

nullIlany, B. & Margolin, B. (2010). Language and Mathematics: Bridging between Natural Language and Mathematical Language in Solving Problems in Mathematics.

References

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[2] 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.

[3] Bloedy-Vinner, H. (1998). The understanding of algebraic language in university preacademic students. Ph. D. dissertation, Jerusalem: Hebrew University.

[4] Brown, G., & Yule, G. (1983). Discourse analysis. Cambridge: Cambridge University Press.

[5] 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

[6] De Lange, J. 1987 (1987). Mathematics insight and meaning. Utrect, Holland: Rijksuniversiteit.

[7] Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht, Holland: Reidel Pub.

[8] 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.

[9] Freudenthal, H. (1991). Revising mathematics education. Dordrecht, South Holland: Kluw-er.

[10] Gee, J. P. (1996). Social Linguistics and Literacy, Ideology in Discourse. Bristol, PA: Taylor & Francis.

[11] 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

[12] 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

[13] Halliday, M. A. K., & Hassan, R. (1976). Cohesion in English. London: Long-man.

[14] Hershkovitz, S., & Nesher, P. (1996). The role of schemes in designing computerized environments. Educational Studies in Mathematics, 30, 339-366. doi:10.1007/BF00570829

[15] Hershkovitz, S., & Nesher, P. (2003). The role of schemes in solving word problems. The Mathematics Educator, 7, 1-24.

[16] 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.

[17] Kane, R. B. (1970). The readability of mathematics textbooks revisited. The Mathematics Teacher, 63, 579-581.

[18] 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.

[19] Kaput, J. J., & Clement, J. (1979). Letter to the editor of JCMB. Journal of Children’s Mathematical Behavior, 2, pp. 208.

[20] Kintsch, W. (1998). Comprehension: A Paradigm for Cognition. Cambridge, England: Cambridge University Press.

[21] 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.

[22] 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

[23] 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.

[24] Nastasi, B. K., & Clements, D. H. (1990). Metacomponential functioning in young children. Intelligence, 14, 109-125.

[25] 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

[26] 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.

[27] 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

[28] 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

[29] Nir, R. (1989). Semantika hivrit mashmaut vetikshoret (Hebrew semantics meaning and communication. Tel-Aviv: Open University.

[30] Ormell, C. (1991). How ordinary meaning underpins the meaning of mathe- matics. Learning of Mathematics, 11, 25-30.

[31] Piaget, J. (1980). Experiments in contradiction. Chicago and London: University of Chicago Press.

[32] Polya, G. (1945). How to Solve it?. Princeton, NJ: Princeton University Press.

[33] 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

[34] Rosnick, P. (1981). Some misconceptions concerning the concept of variable. Are you careful about defining your variables?. Mathematics Teacher, 74, 418-420, 450.

[35] Sarel, Z. (1991). Mavo Lenituah Hsiah (Introduction to discourse analysis). Tel-Aviv: Or-Am.

[36] Schoennfeld, A. H. (1980). Teaching problem-solving skills. American mathematical monthly, 87, 794-805. doi:10.2307/2320787

[37] 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

[38] Van Dijk, T. A. (1980). Macrostructures: An interdisciplinary study of global structures in discourse. Mahwah, N.J.: L. Erlbaum Associ- ates.

[39] Widdowson, H. G. (1979). Explorations in Applied Linguistics. Oxford, England: Oxford University.

[40] 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.

[41] 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

[1] 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

[2] 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.

[3] Bloedy-Vinner, H. (1998). The understanding of algebraic language in university preacademic students. Ph. D. dissertation, Jerusalem: Hebrew University.

[4] Brown, G., & Yule, G. (1983). Discourse analysis. Cambridge: Cambridge University Press.

[5] 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

[6] De Lange, J. 1987 (1987). Mathematics insight and meaning. Utrect, Holland: Rijksuniversiteit.

[7] Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht, Holland: Reidel Pub.

[8] 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.

[9] Freudenthal, H. (1991). Revising mathematics education. Dordrecht, South Holland: Kluw-er.

[10] Gee, J. P. (1996). Social Linguistics and Literacy, Ideology in Discourse. Bristol, PA: Taylor & Francis.

[11] 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

[12] 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

[13] Halliday, M. A. K., & Hassan, R. (1976). Cohesion in English. London: Long-man.

[14] Hershkovitz, S., & Nesher, P. (1996). The role of schemes in designing computerized environments. Educational Studies in Mathematics, 30, 339-366. doi:10.1007/BF00570829

[15] Hershkovitz, S., & Nesher, P. (2003). The role of schemes in solving word problems. The Mathematics Educator, 7, 1-24.

[16] 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.

[17] Kane, R. B. (1970). The readability of mathematics textbooks revisited. The Mathematics Teacher, 63, 579-581.

[18] 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.

[19] Kaput, J. J., & Clement, J. (1979). Letter to the editor of JCMB. Journal of Children’s Mathematical Behavior, 2, pp. 208.

[20] Kintsch, W. (1998). Comprehension: A Paradigm for Cognition. Cambridge, England: Cambridge University Press.

[21] 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.

[22] 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

[23] 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.

[24] Nastasi, B. K., & Clements, D. H. (1990). Metacomponential functioning in young children. Intelligence, 14, 109-125.

[25] 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

[26] 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.

[27] 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

[28] 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

[29] Nir, R. (1989). Semantika hivrit mashmaut vetikshoret (Hebrew semantics meaning and communication. Tel-Aviv: Open University.

[30] Ormell, C. (1991). How ordinary meaning underpins the meaning of mathe- matics. Learning of Mathematics, 11, 25-30.

[31] Piaget, J. (1980). Experiments in contradiction. Chicago and London: University of Chicago Press.

[32] Polya, G. (1945). How to Solve it?. Princeton, NJ: Princeton University Press.

[33] 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

[34] Rosnick, P. (1981). Some misconceptions concerning the concept of variable. Are you careful about defining your variables?. Mathematics Teacher, 74, 418-420, 450.

[35] Sarel, Z. (1991). Mavo Lenituah Hsiah (Introduction to discourse analysis). Tel-Aviv: Or-Am.

[36] Schoennfeld, A. H. (1980). Teaching problem-solving skills. American mathematical monthly, 87, 794-805. doi:10.2307/2320787

[37] 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

[38] Van Dijk, T. A. (1980). Macrostructures: An interdisciplinary study of global structures in discourse. Mahwah, N.J.: L. Erlbaum Associ- ates.

[39] Widdowson, H. G. (1979). Explorations in Applied Linguistics. Oxford, England: Oxford University.

[40] 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.

[41] 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