Aritmetik ile Cebir Arasındaki Farklılıklar: Cebir Öncesinin Önemi

Öz

Matematiksel kavramlar bir zincirin halkası gibi birbirleriyle bağlantılı olduğundan, bu halkada olabilecek kopmaların ileri matematiksel kavramların öğreniminde zorluklara yol açabileceği bilinmektedir. Özellikle ilköğretimin birinci ve ikinci kademesindeki aritmetik ile cebir bilgisi arasında önemli bir zincir halkası vardır. Aritmetik-cebir arasında kuvvetli bir ilişki olmasına rağmen, aritmetikle cebirin farklı doğalarından dolayı harfleri, sembolleri, matematiksel ifadeleri, eşitlik kavramını, problem çözme yöntemlerini yorumlamada farklılıklar olabilir. İşte bu farklılıklardan kaynaklanan engellerin ve zorlukların giderilmesinde aritmetikten cebire geçiş kuşağı olan “cebir öncesi” kuşağı önemlidir. Bundan dolayı bu farklılıkların ve cebir öncesinin önemine değinilmesi gerekmektedir. Bu çalışmada, aritmetik bilgi ile cebirsel bilgi arasındaki farklılıklar ile “cebir öncesi” kuşağının önemi literatür tabanlı incelenmiş, bu inceleme sonucunda elde edilen sonuçlar araştırmacıların önerileri ile de desteklenerek verilmiştir

Anahtar Kelimeler

• aritmetik bilgi, cebirsel bilgi, cebir öncesi

Differences between Arithmetic and Algebra: Importance of Pre-algebra

Abstract

As it is known; the mathematical concepts are connected like rings of the chain so a rupture in the chain will cause difficulties in teaching mathematics concepts in the future. Especially, this ring is seen between the arithmetic and algebra knowledge in the first and second level of primary schools. Though there is a strong relation between arithmetic and algebra; because of their different natures there are differences in interpreting letters, symbols, mathematical statements, equal concepts and problem solving methods. So prealgebraic period that is the period transition from arithmetic to algebra development is so important in eliminating obstacles and difficulties which are caused from these differences. For this reason, it is necessary to deal with these differences and the importance of pre-algebraic period. In this study we focused on the differences between arithmetical and algebraic knowledge and the importance of pre-algebraic period.

Keywords

• arithmetic knowledge,algebraic knowledge,pre-algebra

Kaynakça

1. Akgün, L. (2006). Cebir ve değişken kavramı üzerine, Journal of Qafqaz University, 17.
2. Akkan, Y. (2009). İlköğretim öğrencilerinin aritmetikten cebire geçiş süreçlerinin incelenmesi. Yayımlanmamış doktora tezi, Karadeniz Teknik Üniversitesi, Trabzon.
3. Alibali, M. W., Knuth, E. J., Hattıkudur, S., Mcneil, N.M. & Stephens,A.C. (2007). A longitudinal look at middle-school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9, 221-247.
4. Armstrong, B. (1995). Teaching patterns, relationships, and multiplication as worthwhile mathematical tasks. Teaching Children Mathematics, 1(7), 446-450.
5. Baki, A. (2008). Kuramdan uygulamaya matematik eğitimi(4.Baskı). Ankara:Harf Eğitim Yayıncılık.
6. Battista, M.T. (1995). Considerations for developing a first course in algebraic thinking. Unpublished doctoral dissertation, Kent State University, Kent, OH.
7. Bell,A. (1996). Problem solving approaches to algebra: Two aspects. In N. Bernardz, C. Kieran & L. Lee (Eds.), Approaches to algebra. perspectives to research and teaching (pp.167-187). Dordretch, The Netherlands: Kluwer Academic Publishers.
8. Booth, L. (1984). Algebra: Children’s Strategies and Errors. Windsor, UK: NFER-Nelson.

9. Booth, L. (1988). Children's difficulties in beginning algebra. In A. F. Coxford (Eds.). The ideas of algebra, K-12 (pp.20–32). Reston, VA: NCTM.
10. Borchert, K. (2003). Disassocation between arithmetic to algebraic knowledge in mathematical modeling. Unpublished doctoral dissertation, University of Washington, USA.
11. Boulton-Lewis, G., Cooper, T., Athew, B., Pilay, H., Wilss, L. & Mutch, S. (1997). The transition from arithmetic to algebra: A cognitive perspective. International Group for the Psychology of Mathematics Education, 21(2),185-192.
12. Carpenter, T.P. & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. [Online]: Retrieved on 11-March-2008, at URL: www.wcer.wisc.edu/ncisla /publications/ index.html,
13. Cooper, T., Boulton-Lewis, G., Athew, B., Wilssi L.,& Mutch, S. (1997). The transition arithmetic to algebra: Initial understandings of equals, operations and variable. International Group for the Psychology of Matematics Education, 21(2), 89-96.
14. Dede, Y. (2003). ARCS motivasyon modeli ve öğe gösterim teorisi'ne (Component Display Theory) dayalı yaklaşımın öğrencilerin değişken kavramını öğrenme düzeylerine ve motivasyonlarına etkisi.Yayımlanmamış doktora tezi, Gazi Ün. Eğitim Bilimleri Enstitüsü, Ankara.
15. Demana, F. &Leitzel, J. (1988). Establishing fundamental concepts through numerical problem solving, In A.F. Coxford(Ed.), The ideas of algebra, K-12,(pp.61-68), Reston, VA: NCTM.
16. English, L.D.& Halford, G.S. (1995). Mathematics education: Models and processes. Mahwah, NJ: Erlbaum.
17. Filloy, E. & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For The Learning of Mathematics, 9(2), 19 - 25.
18. French, D. (2002). Teaching and learning algebra. London: Continuum.
19. Goodson-Espy, T.J. (1998). The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra. Educational Studies in Mathematics, 36, 219-245.
20. Harvey, J. G., Waits, B. K. & Demana, F.D. (1995). The influence of technology on the teaching and learning of algebra. Journal of Mathematical Behavior, 14(1),75-109.
21. Hersovics, N. & Linchevski, L. (1994). A cognative gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59-78.
22. Hersovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. In S.Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra, (pp. 60-92). Reston, VA: NCTM, Hillsdale, NJ: Lawrence Erlbaum.
23. Kieran, C. & Chalouh, L. (1993). Prealgebra: The transition from arithmetic to algebra. In P. S. Wilson (Ed.), Research ideas for the classroom: Middle grades mathematics, (pp. 119-139). New York: Macmillan.
24. Kieran, C. (1990). Cognitive processes involved in learning school algebra. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition, (pp. 96-112). Cambridge: Cambridge University Pres.
25. Kieran, C.(1991). A procedural-structural perspective on algebra research. In Furinghetti, F. (Ed.), Proceedings of the fifteenth international conference for the psychology of mathematics ducation, (2, pp.245–253). Genoa, Italy,
26. Kieran,C. (1992). The learning and teaching of school algebra. In D.A. Grouws (Eds.). Handbook of research on mathematics teaching and learning, (pp.390-419). New York: Macmillan.
27. Kindt, M.M. (2000). Patterns and symbols. In National Center for Research in Mathematical Science Education and Freudenthal Institute (Eds), Mathematics in context, a connected curriculum for grades 5-8. Chicago: Encyclopedia Brittanica Educational Corporation.
28. Linchevski, L. & Hersovics, N. (1994). “Cognitive obstacles in pre-algebra”. Proceedings of the 18th Conference of the International Group for the Psychology of Mathematics Education, 3, 176-183.
29. Linchevski, L. & Hersovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30, 38–65.
30. Linchevski, L. & Livneh, D. (1999). Sctructure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40, 173-196.
31. Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre- algebra. The Journal of Mathematical Behaviour, 14, 113-120.
32. Lodholz, R. D. (1990). The transition from arithmetic to algebra. E.L. Edwards (Ed.), Algebra for everyone,(pp. 24-33). Reston, VA: NCTM.
33. Macgregor, M. & Stacey, K. (1995). The effect of different approaches to algebra on students' perceptions of functional relationships. Mathematics Education Research Journal, 7(1), 69- 85.
34. Malara, N. & Navarra, G. (2003). ArAl project: Arithmetic pathways towards favouring pre- algebraic thinking. Bologna: Pitagora Editrice.
35. Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.). Approaches to algebra, (pp.65-111). London: Kluwer Academic Publishers.
36. Mcneil, N.M. & Alibali, M.W. (2005). Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76, 883-899.
37. MEB, TTKB. (2006). Ortaöğretim matematik dersi öğretim programı ve kılavuzu, Ankara:MEB Basımevi.
38. Mellilo, J., A. (1999). An analysis of students’ transition from arithmetic to algebriac thinking. Unpublished doctoral dissertation, Kent State University, Ohio.
39. Miller, J., O'Neill, M., & Hyde, N. (2010). Pre-algebra. New York: McGraw-Hill Companies.
40. Molina, M. & Ambrose, R. (2008). From an operational to a relational conception of the equal sign: Thirds graders’ developing algebraic thinking. Focus on Learning Problems in Mathematics, 30(1), 61-80.
41. Nathan, M. & Koellner, K. (2007). A framework for understanding and cultivating the transition from arithmetic to algebraic reasoning. Mathematical Thinking and Learning, 9(3), 179-192.
42. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for
43. school mathematics. Reston, Va: NCTM.
44. National Council of Teachers of Mathematics. (1991). Professional standards for teaching
45. mathematics. Reston, Va: NCTM.
46. National Council of Teachers of Mathematics. (2000). Principles and standards for school
47. mathematics. Reston, VA: NCTM.
48. Ohlsson, S.(1993). Abstract schemas. Educational Psychologist, 28(1),51-66.
49. Price, J., Rath, J. N., Leschensky, W., Brame, O.H. & Molina,D. D.(1996). Merrill pre-algebra: A transition to algebra, Westerville, OH:Merrill Publishing Company.
50. Rosnick, P. (1999). Some misconceptions concerning the concept of variable. In Ed: B. Moses, Algebraic thinking: Grades 9-12, (pp.313-315), Reston, Va: NCTM.
51. Sfard, A. & Linchevski, L. (1994). The gain and the pitfalls of reification:The case of algebra. Educational Studies in Mathematics, 26, 191-228.
52. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 21, 1-36.
53. Sfard, A. (1995). The development of algebra: Confront historical and psychological perspectives. Journal of Mathematical Behavior, 14, 15-39.
54. Stacey, K. & Macgregor, M. (1997). Building foundations for algebra. Mathematics in the Middle School, 2, 253 – 260.
55. Stacey, K. & Macgregor, M. (2000). Learning the algebraic method of solving problems. Journal of Mathematical Behaviour, 18(2), 149-167.
56. Stacey, K. (2008). The transition from arithmetic thinking to algebraic thinking [Online]: Retrieved on staff.edfac.unimelb.edu.au/~kayecs/IMECstaceyALGEBRA
57. March-2008, at URL:www.
58. Sutherland, R. & Rojano, T. (1993). Spreadsheet approach to solving algebraic problems. The Journal of Mathematics Behavior, 12(4), 353-383.
59. Swadener, M. & Soedjadi, R. (1988). Values, mathematics education and the task of developing pupils’ personalities: An Indonesian perspective. Educational Studies In Mathematics, 19(2), 193-208.
60. Tabach, M. & Friedlander, A. (2003). “The role of context in learning beginnig algebra”. Proceedings of the Third Conference of the European Society for Research in Mathematics Education, Bellaria, Italia.
61. Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. Edt. D. Grouws, Handbook of research on mathematics teaching and learning, (pp. 495-514). Macmillan Publishing Company, Newyork.
62. URL-1, http://www.uludagsozluk.com. “Aritmetik”. 07 Mart 2009
63. URL-2, http://www.bilgidenizi.net/aritmetik/5458 “Aritmetik”. 11 Aralık 2008.
64. Usiskin,Z. (1997). Doing algebra in grades K-4. In B. Moses (Eds.). Algebraic thinking, grades K- 12, (pp.5-7). Reston, VA: NCTM.
65. Van Amerom, B. (2002). Reinvention of early algebra: Developmental research on the transition from arithmetic to algebra. Unpublished doctoral dissertation, University of Utrecht, The Netherlands.
66. Van Doren, W., Verschaffel, L. & Onghena, P. (2003). Pre-service teachers’ preferred strategies for solving arithmetic and algebra word problems. Journal of Mathematics Teacher Education, 6, 27-52.
67. Vance, J.H.(1998). Number operations from an algebraic perspective. Teaching Children Mathematics, 4, 282-285.
68. Wagner, S. & Kieran, C. (1989). Research issues in the learning and teaching of algebra. Reston, VA: NCTM.
69. Wagner, S. & Parker, S. (1993). Advancing algebra. In P. S. Wilson, Ed., Research ideas for the classroom: High school mathematics, (pp.117-139), New York: Macmillan Publishing Company.
70. Wijers, M. (1995). Using real world contexts to make variables and formulas meaningful. Paper Presented at Area in San Francisco, April 1995: 18.
71. Williams, A. & Cooper, T. (2001). Moving from arithmetic to algebra under the time pressures of real classrooms. In H. Chick, K. Stacey, Jill Vincent, & John Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra, (pp.665-662). Melbourne: University of Melbourne.
72. Witzel, B.S., Mercer, C.D. & Miller, D.M. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research&Practice, 18(2), 121-131.