Aritmetik ve Cebir Kavramları ile İlgili Farkındalık

Öz

Matematiğin aritmetik ile cebir öğrenme alanları arasındaki bağlantıların kurulabilmesinde ve bu iki alanın doğasından kaynaklanan farklılıkların neden olduğu zorlukların giderilmesinde, öğretmenlerin aritmetik ve cebir kavramları ile ilgili derinlemesine bilgi sahibi olması gerekmektedir. Bu bağlamda çalışmanın amacı, ortaokul matematik öğretmenlerinin aritmetik ve cebir kavramları ile ilgili farkındalıklarını belirlemektir. Bu araştırma nitel bir çalışma olarak tasarlanmıştır. Araştırmanın çalışma grubunu, Trabzon, Gümüşhane ve Bayburt illerindeki ortaokullarda matematik öğretmenliği yapan 15 matematik öğretmeni oluşturmaktadır. Araştırmanın verileri; çalışma grubunda yer alan ortaokul matematik öğretmenleriyle yürütülen yarı yapılandırılmış görüşme tekniği ile toplanmıştır. Görüşme yoluyla elde edilen verilerin analizinde içerik analizi tekniğinden yararlanılmıştır. Elde edilen bulgular öğretmenlerin aritmetik, cebir ve aritmetikten cebire geçiş kavramlarını etkili bir şekilde tanımlayamadığını, bu iki alan arasındaki bağlantıları, ilişkileri ve farklılıkları ayrıt edemediğini ortaya çıkarmıştır.

Anahtar Kelimeler

• Aritmetik,cebir,farkındalık,ortaokul matematik öğretmenleri

Awareness of Arithmetic and Algebra Concepts

Öz

Teachers need to have in-depth knowledge of arithmetic and algebraic concepts to establish the connection between arithmetic and algebra fields of mathematics and to eliminate the difficulties caused by differences inherent in these fields. Within this context, the aim of this study is to determine the awareness of middle school mathematics teachers about the concepts of arithmetic and algebra. This study is conducted as a qualitative research. Participants of the study consist of 15 mathematics teachers who teach mathematics at middle schools in Trabzon, Gümüşhane and Bayburt. Semi-structured interview form was used as data collection tool in the study. In the analysis of the data obtained through the interview, content analysis technique was used. Findings revealed that teachers could not effectively define the concepts of arithmetic, algebra and transition from arithmetic to algebra and that they could not distinguish the connections, relationships and differences between these two fields.

Anahtar Kelimeler

• Arithmetic,algebra,awareness,middle school mathematics teachers

Kaynakça

1. Akgün, L. (2006). Cebir ve değişken kavramı üzerine. Journal of Qafqaz University, 17(1). 25-29.
2. Akkan, Y. (2009). İlköğretim öğrencilerinin aritmetikten cebire geçiş süreçlerinin incelenmesi, (Yayımlanmamış doktora tezi), Karadeniz Teknik Üniversitesi, Trabzon, Türkiye.
3. Akkan, Y., Baki, A. ve Çakıroğlu, Ü. (2011). Aritmetik ile cebir arasındaki farklılıklar: Cebir öncesinin önemi. İlköğretim Online, 10(3), 812-823.
4. Armstrong, B. (1995). Teaching patterns, relationships, and multiplication as worthwhile mathematical tasks. Teaching Children Mathematics, 1(7), 446-450.
5. Attorps, I. (2005). Secondary school teachers’ conceptions about algebra teaching. Paper presented at the Fourth Congress of the European Society for Research in Mathematics Education. Sant Feliu de Guíxols, Spain. Retrieved October 5, 2005, from http://cerme4.crm.es/Papers%20definitius/12/attorps.pdf.
6. Ball, D. L. (1992). Teaching mathematics for understanding: What do teachers need to know about subject matter? In M. Kennedy (Ed.), Teaching academic subjects to diverse learners (pp. 63-83). New York, NY: Teachers College.
7. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.
8. Bednarz, N. (2001). A problem solving approach to algebra: Accounting for reasonings and notations developed by students. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.), Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (pp. 69-78). The University of Melbourne.

9. Bednarz, N., & Janvier, B. (1996). Emergence and development of algebra as a problem-solving tool: Continuities and discontinuities with arithmetic. In N. Bednarz, C.
10. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 115-136). Boston: Kluwer Academic Publishers.
11. 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 dordretch (pp.167-187). The Netherlands: Kluwer Academic Publishers.
12. Booth, L. (1984). Algebra: Children’s strategies and errors. Windsor, UK: NFER-Nelson.
13. 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.
14. Borko, H., & Shavelson, R. (1990). Teacher decision making. In B. F. Jones & L. Idol (Eds.), Dimensions of thinking and cognitive instruction (pp. 311–346). Elmhurst, IL: North Central Regional Educational Laboratory and Hillsdale, NJ: Erlbaum.
15. Burden, P. R. (1982). Implications of teacher career development: new roles for teachers, administrators and professors. Action in Teacher Education, 4(3-4), 21-26.
16. Büyüköztürk, Ş., Çakmak, E. K., Akgün, Ö. E., Karadeniz, Ş. ve Demirel, F. (2008). Bilimsel araştırma yöntemleri. Pegem Akademi Yayınları, Ankara.
17. Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. Retrieved March 11, 2008 from www.wcer.wisc.edu/ncisla /publications/ index.html.
18. Carraher, D., Schliemann, A. D., Brizuela, B., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87-115.
19. 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.
20. Cotton, T. (1993). Children’s impressions of mathematics. Mathematics Teacher, 143, 14-17.
21. Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics teachers’ beliefs structures and their influence on instructional practices. J Math Teacher Education, 12, 325 – 346.
22. 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.
23. English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, NJ: Erlbaum.
24. Even, R., & Tirosh, D. (2002). Teacher knowledge and understanding of students’ mathematical learning. In L. English (Ed.), Handbook of international research in mathematics education (pp. 219–240). Mahwah, NJ: Laurence Erlbaum.
25. Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19 – 25.
26. Franke, M. L., & Carey, D. A. (1997). Young children’s perceptions of mathematics in problem-solving environments. Journal for Research in Mathematics Education, 28(1), 8-25.
27. French, D. (2002). Teaching and learning algebra. London: Continuum.
28. Gomez-Chacon, I. (2000). Affective influences in the knowledge of mathematics. Educational Studies in Mathematics 43, 149–168.
29. Greeno, J. G., Pearson, P. D., & Schoenfeld, A. H. (1996). Implications for NAEP of research on learning and cognition (p. 84). Menlo Park, CA.
30. Greeno, J. G., Pearson, P. D., & Schoenfeld, A. H. (1997). Implications for NAEP of research on learning and cognition. Assessment in transition: Monitoring the nation’s educational progress. Washington DC: National Academy Press
31. Herscovics, 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.
32. Hersovics, N., & Linchevski, L. A. (1994). Cognative gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59-78.
33. Karasar, N. (1998). Bilimsel araştırma yöntemi: Kavramlar, ilkeler, teknikler. Nobel Yayıncılık, Ankara.
34. 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.
35. 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,
36. Kieran, C. A. (1991). Procedural-structural perspective on algebra research. Proceedings of the Fifteenth International Conference for the Psychology of Mathematics Education Genoa, Italy, 2, 245–253.
37. Kieran, C., & Chalouh, L. (1993). Pre-algebra: 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.
38. Lara Roth, S. M. (2006). Young children's beliefs about arithmetic and algebra. Unpublished doctoral dissertation, Tufts University, Medford, Massachusetts, United States.
39. Lee, L. (2001). Early algebra: But which algebra? In Chick, H., K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of Teaching and Learning Algebra, The University of Melbourne, Australia, December, (pp. 392-398).
40. Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra. The Journal of Mathematical Behaviour, 14, 113-120,
41. Linchevski, L., & Herscovics, 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.
42. Linchevski, L., & Livneh, D. (1990). Sctructure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40, 173-196.
43. Lodholz, R. D. (1990). The transition from arithmetic to algebra. E.L. Edwards (Ed.), Algebra for everyone (pp. 24-33). Reston, VA: NCTM.
44. 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.
45. 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.
46. Nathan, M. J. (2003). Confronting teachers’ beliefs about algebra development: Investigating an approach for professional development. Boulder, CO: University of Colorado, Institute of Cognitive Science.
47. Nathan, M. J., & Koedinger, K. R. (2000). An investigation of teachers’ beliefs of students’ algebra development. Cognition and Instruction, 18(2), 209–237.
48. Nathan, M. J., & Koellner, K. A. (2007). Framework for understanding and cultivating the transition from arithmetic to algebraic reasoning. Mathematical Thinking and Learning, 9(3), 179-192.
49. National Council of Teachers of Mathematics (1991). Professional standards for teaching mathematics. Reston, Va: NCTM.
50. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
51. Ohlsson, S. (1993). Abstract schemas. Educational Psychologist, 28(1), 51-66.
52. Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. L. Cortina, M. Saiz, & A. Mendez (Eds.), Proceedings of the 28th annual meeting of the north American chapter of the international group for the psychology of mathematics education (pp. 2–21). Merida, Mexico: Universidad Pedagogica Nacional.
53. Raymond, A. M. (1997). Inconsistency between a beginning elementary school teacher's mathematics beliefs and teaching practices. Journal for Research in Mathematics Education, 28(6), 552-575.
54. Rosnick, P. (1999). Some misconceptions concerning the concept of variable. In B. Moses (Ed.), Algebraic thinking: Grades 9-12 (pp. 313-315). Reston, Va: National Council of Teachers of Mathematics.
55. 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.
56. Sfard, A. (1995). The development of algebra: Confront historical and psychological perspectives. Journal of Mathematical Behavior, 14, 15-39.
57. Sfard, A., & Linchevski, L. (1994). The gain and the pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26, 191-228.
58. Sherin, M. G. (2004). New perspectives on the role of video in teacher education. In J. Brophy (Ed.), Using video in teacher education (pp. 1–28). Amsterdam: Elsevier.
59. Sherin, M. G. (2007). The development of teachers' professional vision in video clubs. In R. Goldman, R. Pea, B. Barron, & S. Derry (Eds.), Video research in the learning sciences (pp. 383-395). Hillsdale, NJ: Lawrence Erlbaum.