Matematik Öğretmeni Adaylarının Tamsayılarda Çıkarma İşlemini Sayı Doğrusu ve Sayma Pulları İle Yapabilme Durumlarının İncelenmesi

Yıl 2022, , 205 - 216, 28.02.2022

Ercan Özdemir

https://doi.org/10.24106/kefdergi.833879

Cited By: 2

Öz

Çalışmanın amacı: Bu çalışmada ortaokul matematik öğretmeni adaylarının tam sayılarda çıkarma işlemini sayma pulları ve sayı doğrusu ile yapabilme durumları araştırılmıştır.

Materyal ve Yöntem: Durum çalışması yöntemi kullanılmıştır. Araştırma Türkiye’nin kuzeyinde yer alan bir devlet üniversitesinin ilköğretim matematik öğretmenliği lisans programında eğitim gören 37 öğretmen adayı ile yürütülmüştür. Veri toplama aracı olarak araştırmacı tarafından geliştirilen ve tam sayılarda çıkarma işlemine yönelik 4 soru kullanılmıştır. Çalışmanın verileri iki aşamada analiz edilmiştir. Birinci aşamada öğretmen adaylarının doğru veya yanlış cevaplara ulaşma durumları belirlenmiş, ikinci aşamada ise yanlış cevaplara sebep olan hatalar tespit edilmiştir.

Bulgular: Çalışmada öğretmen adayları tam sayılarda çıkarma işlemlerini her iki modelleme türünde de oldukça başarılı olmuşlardır. Öğretmen adayları sayma pulları ile modellemede en yüksek başarıyı yaklaşık %97’lik oranla pozitif bir tam sayıdan negatif bir tam sayının çıkarılması ((+3)-(-5)), en düşük başarıyı ise yaklaşık %89’luk oranla negatif bir tam sayıdan negatif bir tam sayının çıkarılması ((-4)-(-7)) işleminde göstermişlerdir. Sayı doğrusu ile modellemede en yüksek başarıyı yaklaşık %91’lik oranla pozitif bir tam sayıdan pozitif bir tam sayının çıkarılması ((+2)-(+5)), en düşük başarıyı ise yaklaşık %86’lık oranla negatif bir tam sayıdan negatif bir tam sayının çıkarılması ((-4)-(-7)) işleminde göstermişlerdir.

Önemli Vurgular: Araştırma sonuçlarına göre, öğretmen adayları tam sayılarda çıkarma işlemini sayma pulları ve sayı doğrusu ile modellemede oldukça başarılı olmuşlardır.

Anahtar Kelimeler

Tamsayılar, sayı doğrusu,, matematik öğretmeni adayları, sayma pulu

An Investigation of Prospective Mathematics Teachers’ Ability to Subtract Integers with a Number Line and Counters

Öz

Purpose: In the present study, the ability of middle school prospective mathematics teachers to subtract integers with counters and a number line was investigated.

Design/Methodology/Approach: A case study method was used for the investigation. The research was conducted with 37 prospective teachers who were training in a state university’s undergraduate program on primary mathematics in northern Turkey. Four questions developed by the researcher for subtraction in integers were used as a data collection tool. The data of the study were analyzed in two stages. In the first stage, the answers of the prospective teachers were determined as right or wrong; subsequently, in the second stage, the mistakes that led to the wrong answers were determined.

Findings: The prospective teachers were relatively successful with respect to both modeling types. The prospective teachers had the highest rates of success in modeling with counters, with a rate of approximately 97% in subtracting a negative integer from a positive integer ((+3) − (−5)), and the lowest rates of success in subtracting a negative integer from a negative integer, with a rate of around 89% for the operation ((−4) − (−7)). In modeling with the number line, the highest success rate was approximately 91% for subtracting a positive integer ((+2) − (+5)) from another positive integer, whereas the lowest success rate was around 86% for subtracting a negative integer from a negative integer in the operation of ((−4) − (−7))

Highlights: According to the results of the research, the prospective teachers were relatively successful in modeling the subtraction of integers using counters and a number line.

Anahtar Kelimeler

Integers, Number line, Counters, Prospective mathematics teachers

Kaynakça

• Almeida, R. & Bruno, A. (2014). Strategies of pre-service primary school teachers for solving addition problems with negative numbers. International Journal of Mathematical Education in Science and Technology, 45(5), 719–737.
• Argün, Z., Arıkan, A., Bulut, S. & Halıcıoğlu, S. (2014). Masthead of basic mathematical concepts. Ankara: Gazi Kitabevi.
• Avcu, T. ve Durmaz, B. (2011). Mistakes and difficulties encountered at primary education level in operations related to integers, 2nd ICONTE, Antalya.
• Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. (Doctoral Dissertation). Michigan State University, USA.
• Battista, M. T. (1983). A complete model for operations on integers. The Arithmetic Teacher, 30(9), 26-31.
• Baykul, Y. (2009). Teaching mathematics in middle school (2. Edition). Ankara: Pegem Publications.
• Billstein, R., Libeskind, S., & Lott, J. (2016). A problem solving approach to mathematics for elementary school teachers. (12th ed.). USA: Pearson Education, Inc.
• Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., ve Empson, S. B. (1999). Children’s mathematics: cognitively guided instruction. Portsmouth, NH:Heinemann
• Cemen, P. B. (1993). Adding and subtracting integers on the number line. Arithmetic Teacher, 40(7), 388-389. URL: https://www.jstor.org/stable/pdf/41195814.pdf
• Cunningham A.W. (2009). Using the number line to teach signed numbers for remedial community college mathematics. Math Teaching Res J Online, 3(4), 1–40.
• Durmaz, B. (2017). The success of service and preservice mathematics teachers’ on modeling integer operations, Ahi Evran University Journal of Kırşehir Education Faculty, 18 (3), 171- 192.
• Fischbein, E. (1977). Image and concept in learning mathematics. Educational Studies in Mathematics, 8(2), 153-165. http://dx.doi.org/10.1007/BF00241022
• Fischbein, E. (1987). Intuition in science and mathematics: An educational approach (D.Reidel Publishing Co., Dordrecht), pp. 97-102.
• Gallardo, A., & Romero, M. (1999). Identification of difficulties in addition and subtraction of integers in the number line. In F. Hitt, & M. Santos (Eds.), Proceedings of the twenty-first ınternational conference for the psychology of mathematics education (Vol. I. pp. 275–282). North American Chapter, Mexico.
• Gallardo, A., & Rojano, T. (1994). School algebra. syntactic difficulties in the operativity. In D. Kirshner (Ed.), Proceedings of the sixteenth ınternational conference for the psychology of mathematics education. North american chapter, Baton Rouge, LA (pp. 159–165).
• 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(2), 401–431.
• Hart, K. M. (1981). 'Positive and negative numbers', children's understanding of mathematics. 11-16, pp. 82-87, John Murray, London.
• Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the national council of teachers of mathematics. (pp. 65 - 97). New York, NY: Macmillan.
• Janvier, C. (1985). Comparison of models aimed at teaching signed numbers. Ninth international conference for the psychology of mathematics education; Noordwijkerhout: The Netherlands, p. 135–139.
• Janvier, C. (1983). The understanding of directed numbers. Seventh international conference for the psychology of mathematics education; Jerusalem, Israel; p. 295–301.
• Kilhamn, C. (2011). Making sense of negative numbers. Göteborg, Sweden: Acta Universitatis Gothoburgensis.
• Kubar, A. & Cakiroglu, E. (2017). Prospective teachers’ knowledge on middle school students’ possible descriptions of integers. International Journal of Education in Mathematics, Science and Technology (IJEMST), 5(4), 279-294. DOI:10.18404/ijemst.75211
• Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics. (pp. 33-40). New Jersey: Lawrence Erlbaum As
• Liebeck, P. (1990). Scores and forfeits: An intuitive model for integer arithmetic. Educational Studies in Mathematics, 21(3), 221-239. doi:10.1007/BF00305091sociates.
• Ma, L. (1999). Knowing and teaching elementary mathematics. Mahway, NJ: Lawrence Erlbaum Assoc., Inc. Ministry of National Education (MoNE). (2013). Middle school mathematics lesson (5th, 6th, 7th and 8th grades) curriculum. T.C. Ministry of National Education Board of Education and Discipline, Ankara.
• Miles, M.B & Huberman, A.M. (1994). Qualitative data analysis. Thousand Oaks, CA: Sage.
• National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA.
• Özdemir, E. & İpek, A.S (2020). Use of multiple representations in mathematics teaching. Ed. Melihan Ünlü. New approaches in teaching mathematics with application examples (s. 91-116). Ankara: Pegem Publications.
• Peled, I., Mukhopadahyay, S., & Resnick, LB. (1989). Formal and informal sources of mental models for negative numbers. 13th international conference for the psychology of mathematics education, Paris, France; p. 106–110.
• Peled, I., & Carraher, D. W. (2007). Signed numbers and algebraic thinking. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp.303–327). Mahwah, NJ: Erlbaum.
• Reeder, S. & Bateiha, S. (2016) Prospective elementary teachers' conceptual understanding of integers. Investigations in Mathematics Learning, 8(3), 16-29. DOI: 10.1080/24727466.2016.11790352
• Sowder, J. T., Phillip, R. A., Armstrong, B. E., & Schappelle, B. (1998). Middle-grade teachers’ mathematical knowledge and its relationship to instruction. Albany, NY: State University of New York Press.
• Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for ınteger addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428–464.
• Teppo, A & Heuvel-Panhuizen, M. (2014). Visual representations as objects of analysis: The number line as an example. ZDM Mathematics Education, 46, 45–58. DOI 10.1007/s11858-013-0518-2
• Thompson, P. W. ve Dreyfus, T. (1988). Integers as transformations. Journal for Research in Mathematics Education, 19(2), 115-133. doi:10.2307/749406
• Whitacrea, I., Azuzb, B., Lambb, L.L.C., Bishop, J. P., Schappelleb, B. P., & Philipp, R. A. (2017). Integer comparisons across the grades: Students’ justifications and ways of reasoning. Journal of Mathematical Behavior, 45, 47–62.
• Van de Walle, A. J., Karp, S.K., & Bay-Williams, M.J. (2012)., Elementary and middle school mathematics (Çeviri Ed. Soner Durmuş). Ankara: Nobel Akademik Publications.
• Vlassis, J. (2001). Solving equations with negatives or crossing the formalizing gap. In M. Van den Heuvel-Panhuizen (Ed.), Proceedings of the twenty-fifth ınternational conference for the psychology of mathematics education. Vol.4, 375–382, Utrecht, Netherlands.
• Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in ‘negativity’. Learning and Instruction, 14, 469–484. doi:10.1016/j.learninstruc.2004.06.012
• Vlassis, J. (2008). The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology, 21(4), 555-570. DOI: 10.1080/09515080802285552
• Yıldırım, A., & Şimşek, H. (2008). Qualitative research methods in social sciences (7. Edition). Ankara: Seckin Publications.