ERG SES C 11, Educational Research
Negative numbers is one of the most difficult concepts for students in mathematics education. Several research has indicated students’ difficulties in the conceptualization of negative numbers (Ball, 1993; Fischbein, 1987; Gallardo, 2002; Hativa & Cohen, 1995; Stephan & Akyuz, 2012; Vlassis, 2004). In particular, students mostly have difficulty in understanding the minus sign, understanding operations with negative numbers, and understanding the difference between negative numbers and early number concept. For example, Vlassis (2004), in the study with 8th grade students, indicated that most of the students conceptualize the minus sign as a sign of operation, in particular as subtraction and most of the difficulties of students in solving equations mostly depends on the difficulties with negative numbers rather than algebraic structure of the equations. In another study, Fischbein (1987) describes students’ obstacles in learning of negative numbers. Students’ understanding of negative number contradict with their early number concept due the fact that nature of negative numbers does not fit the notion of existence. Similarly, Hativa and Cohen (1995) identified several misconceptions of students in solving addition and subtraction problems. The reason behind these misconceptions is that students mostly confuse the meaning of minus sign as a sign of operation and as a sign of number. Furthermore, researchers argued the reasons why students have difficulties in learning negative numbers. In this regard, one of the powerful approaches in order to examine the reasons behind problems or difficulties regarding a mathematic conpcet is historical-critical analysis in which the development of mathematical concepts is investigated in relation to its historical development (Gallardo, 2001). This allows researchers to discover paralellism between obstacles in the conceptualization of the mathematical concept in the history and students’ struggles with that concept (Sfard, 1991). Sfard described the historical development of numbers as “a long chain of transitions from operational to structural conceptions: again and again, processes performed on already accepted abstract objects have been converted into compact wholes, or reified” (p.14). In this sense, consistent with the reification theory of Sfard (1991), the historical evolution of negative numbers could give clues for such a transtion in concept formation from operational to structural through which number system is extended to negative numbers and negative values are conceptualized from quantities to numbers. Moreover, this study would be an attempt to explore epistemological obstacles for learning negative numbers, which could shed light on understanding of students’ difficulties in learning negative numbers and reorganize learning environments. Furthermore, being aware of difficulties in the conceptualization of negative numbers in the history, teachers could understand students’ difficulties with the negative numbers.
Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 93(4), 373-397. Crowley, M. M. & Dunn, K. A. (1985). On multiplying negative numbers. Mathematics Teacher, 78(4), 252-256. Gallardo, A. (2001). Historical-epistemological analysis in mathematics education: Two works in didactics of algebra. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspective on School Algebra (pp. 121-139). Dordrecht, The Netherlands: Kluwer Academic Publishers. Gallardo, A. (2002). The extension of the natural-number domain to the integers in the transition from arithmetic to algebra. Educational Studies in Mathematics, 49, 171–192. Fischbein, E. (1987). Intutition in Science and Mathematics: An Educational Approach. Dordrecht, Holland: Reidel. Hativa, N., & Cohen, D. (1995). Self-learning of negative number concepts by lower division elementary students through solving computer-provided numerical problems. Educational Studies in Mathematics, 28(4), 401-431. Heeffer, A. (2008). Negative numbers as an epistemic difficult concept: Some lessons from history. In C. Tzanakis (Ed.), History and pedagogy of mathematics conference. Satellite meeting of international congress on mathematical education, Mexico City, México. Kilhamn, C. (2011). Making sense of negative numbers. Unpublished doctoral thesis, University of Gothenburg, Gothenburg. Mumford, D. (2010). What’s so baffling about negative numbers? A cross-cultural comparison. In C. S. Seshadri (Ed.), Studies in the history of Indian Mathematics. New Delhi, India: Hindustan Book Agency. Schubring, G. (2005). Conflicts between generalization, rigor, and intuition: Number concepts underlying the development of analysis in 17-19th century France and Germany. Heidelberg: Springer. Sfard, A. (1991). On the dual nature of mathematical conceptions. Educational Studies in Mathematics Education, 22(1), 1-36. Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428-464. Tabak, J. (2004). History of mathematics: Numbers: Computers, philosophers, and the search for meaning. New York, Facts on File, Inc. Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in `negativity´. Learning and Instruction, 14(5), 469-484.
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