Historical Development of Negative Numbers
Author(s):
Mehtap Ozen (presenting / submitting)
Conference:
ECER 2014
Format:
Paper

Session Information

ERG SES C 11, Educational Research

Paper Session

Time:
2014-09-01
11:00-12:30
Room:
FPCEUP - 250
Chair:
Alison Neilson

Contribution

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.

Method

This is a literature review study. In line with the purpose of the study, the databases of Academic Research Complete, Education Research Complete, ERIC, and SocINDEX with Full Text were searched by means of EBSCOhost. In addition, book collection of METU (Middle East Technical University) was searched and the books related to history of mathematics, especially numbers and algebra were investigated. The main keywords for search are “history of mathematics”, “negative numbers”, “history of numbers”, and “history of algebra”. The analysis is based on two main topics; negative numbers in Chinese and Indian Mathematics and negative numbers in the Mathematics of Arabs and Western Cultures. In each topic, historical sketch of the negative numbers were analyzed. The reason behind such a separation into two sections is that there has been a consensus among researchers in which historical development of negative numbers is argued as unsystematic and incoherent (Crowley & Dunn, 1985; Schubring, 2005). Moreover, Chinese and Indian Mathematics was among the pioneer cultures in the acceptance of negative values (Tabak, 2004).

Expected Outcomes

There are three main conclusions arising from the study. The first conclusion is that there is a mutual relationship between negative numbers and algebra (Gallardo, 2002; Heeffer, 2008; Kilhamn, 2011). From the beginning sixteenth century, with the development of symbolic algebra, negative values are interpreted as numbers and as algebraic structure, rather than just quantities (Heeffer, 2008). In other words, algebra had a crucial role in the evolution of negative numbers and the development of symbolic algebra resulted in extending the nature of negative values to algebraic structure (e.g. different meanings of minus sign). The second conclusion is that negative numbers are conceptualized differently in different cultures. Whereas China and Indian mathematics were among the pioneer cultures in the acceptance of negative values, Islamic and Western mathematicians accepted negative values later (Mumford, 2010), which indicates evolution of negative numbers differs from one culture to another. The third conclusion is that there are different ways of conceptualization of negative numbers in the history (Gallardo, 2001, 2002). Gallardo made a historical-critical analysis in which five interpretations of negative numbers emerged, namely, subtrahend, signed number, isolated numbers, relative number, and formal concept of negative number. Being aware of different interpretations of negative numbers could be beneficial to understand students’ conceptualization of negative numbers. The third conclusion is that there has been obstacles in the conceptualization of negative numbers (Schrubing, 2005). Especially, understanding the difference between quantity or magnitudes and negative numbers is among the obstacles emerged in the historical-critical analysis of the texts. In this sense, this study could shed light on learning and teaching of negative numbers and teachers could understand students’ struggles with the concept of negative number.

References

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.

Author Information

Mehtap Ozen (presenting / submitting)
Middle East Technical University
Elementary Education
Ankara

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