An Examination of Middle School 7th Grade Students’ Mathematical Abstraction Processes

Year 2019, , 233 - 256, 30.04.2019

Elif Kılıçoğlu , Abdullah Kaplan

https://doi.org/10.18009/jcer.547975

Cited By: 2

Abstract

In this study, abstraction processes of 7th grade students were examined. In addition, it has been tried to explain how the implementation of this process affects the students' academic success. For this purpose, the experiment and control groups were formed. While the current teaching program was applied to the control group, the experimental group was taught with the ACE teaching cycle which is the application dimension of the theory based on the abstraction philosophy. It can be stated that such a study is shaped according to semi-experimental method. The application was carried out in the 7th grade of a state secondary school in the province of Erzurum in the 2014-2015 academic year, with a total of 31 students in the experimental group and 32 students in the control group. Both quantitative and qualitative data were obtained in the study. The achievement tests developed by the researcher for the quantitative data and the interview forms developed by the researcher for the qualitative data were used as data collection tools. In addition, the application in the experimental group was recorded with the help of a camera for later review and these records were used for support of qualitative data. The involvement of the researcher as a participant in the process provided the opportunity to obtain observation notes, and these observation notes were added to the qualitative dimension of the study. Therefore, in this study, it was tried to provide reliability by data diversity. The analysis of the quantitative data was done by statistical tests and the analysis of the qualitative data was done by descriptive analysis method. At the end of the research, it is seen that students’ abstraction level of the equation subject is better in the group that ACE teaching cycle is applied than the other group. Furthermore, it is seen that teaching in application process keeps the students’ interest and motivation alive. According to the results obtained it can be said that classroom activities based on students’ abstraction process may be necessary for a qualified learning.

Keywords

Mathematical abstraction, ACE teaching cycle, mathematical equation, mathematics teaching

An Examination of Middle School 7th Grade Students’ Mathematical Abstraction Processes

Abstract

In this study, abstraction processes of 7th grade
students were examined. In addition, it has been tried to explain how the
implementation of this process affects the students' academic success. For this
purpose, the experiment and control groups were formed. While the current
teaching program was applied to the control group, the experimental group was
taught with the ACE teaching cycle which is the application dimension of the
theory based on the abstraction philosophy. It can be stated that such a study
is shaped according to semi-experimental method. The application was carried
out in the 7th grade of a state secondary school in the province of Erzurum in
the 2014-2015 academic year, with a total of 31 students in the experimental
group and 32 students in the control group. Both quantitative and qualitative
data were obtained in the study. The achievement tests developed by the
researcher for the quantitative data and the interview forms developed by the
researcher for the qualitative data were used as data collection tools. In
addition, the application in the experimental group was recorded with the help
of a camera for later review and these records were used for support of
qualitative data. The involvement of the researcher as a participant in the
process provided the opportunity to obtain observation notes, and these
observation notes were added to the qualitative dimension of the study.
Therefore, in this study, it was tried to provide reliability by data
diversity. The analysis of the quantitative data was done by statistical tests
and the analysis of the qualitative data was done by descriptive analysis
method. At the end of the research, it is seen that students’ abstraction level
of the equation subject is better in the group that ACE teaching cycle is
applied than the other group. Furthermore, it is seen that teaching in
application process keeps the students’ interest and motivation alive.
According to the results obtained it can be said that classroom activities
based on students’ abstraction process may be necessary for a qualified
learning.

Keywords

Mathematical abstraction, ACE teaching cycle, mathematical equation, mathematics teaching

References

• Abels, M., de Jong, J. A., Dekker, T., Meyer, M. R., Shew, J. A., Burrill, G., & Simon, A. N. (2006). Ups and downs. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in context. Chicago: Encyclopeadia Britannica, Inc.
• Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, E., K. (1997). The development of students' graphical understanding of the derivative. The Journal of Mathematical Behavior, 16(4), 399-431.
• Asiala, M., Dubinsky, E., Mathews, D. M., Morics, S., & Oktac, A. (1997). Development of students' understanding of cosets, normality, and quotient groups. The Journal of Mathematical Behavior, 16(3), 241-309.
• Bass, E., J., & Montague, J., E. (1972). Piaget-based sequences of instruction in science. Science Education, 56(4), 503-512.
• Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A. & Zaslavsky, O. (2006). Examplification in mathematics education. In J. Novotna (Ed.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague, Czech Republic: PME.
• Cooley, R. (2002). Writing in calculus and reflective abstraction. Journal of Mathematical Behaviour, 21, 255-282.
• Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme. Journal of Mathematical Behavior, 15, 167- 192.
• Çetin, İ. (2009). Students’ understanding of limit concept: An APOS perspective. Unpublished Doctoral Thesis. Middle East Technical University, Institute of Educational Sciences, Ankara.
• Çetin, İ., & Top, E. (2014). Programlama eğitiminde görselleştirme ile ACE döngüsü. [ACE cycle in programming education by using visualization]. Turkish Journal of Computer and Mathematics Education, 5(3), 274-303.
• Davydov, V., V. (1990). Types of generalisation in instruction: logical and phsycological problems in the structuring of school curricula. In: J. Kilpactrick (Ed.). Soviet studies in mathematics education, (p. 2). Reston, VA: National Council of Teachers of Mathematics.
• Dienes, Z., P. (1967). On abstraction and generalization. Harvard Educational Review, 31, 281-301.
• Dubinksy, E. (2000). Mathematical literacy and abstraction in the 21st century. School Science and Mathematics, 100(6), 289-97.
• Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical Thinking, (pp. 95-123). Dordrecht, The Netherlands: Kluwer. Retrieved July 18, 2018 from: http://www.math.wisc.edu/~wilson/Courses/Math903/ReflectiveAbstraction.pdf.
• Dubinsky, E., Weller, K., McDonald, A., M., & Brown, A. (2005). Some historical ıssues and paradoxes regarding the concept of infinity: An Apos-based analysis: part 1. Educational Studies in Mathematics, 58(3), 335-359.
• Frorer, P., Hazzan, O., & Manes, M. (1997). Revealing the faces of abstraction. The International Journal of Computers for Mathematics Education, American Mathematical Society, 3, 234-283.
• Garcia-Cruz, J.A., & Martinon, A. (1997). Actions and invariant schemata in linear generalizing problems. In: Pehkonen E. (Ed.). Conference of the International Group for the Psychology of Mathematics Education, 2, 289-296.
• Kabael, T., U., & Tanışlı, D. (2010). Cebirsel düşünme sürecinde örüntüden fonksiyona öğretim. İlköğretim Online, 9(1), 213-228.
• Kathleen, M. (1999). Active learning and situational teaching: How to ACE a course. Clinical Laboratory Science, 12(1), 35-41.
• Kindt, M., Dekker, T., & Burrill, G. (2006). Algebra rules. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context. Chicago: Encyclopeadia Britannica, Inc.
• Kindt, M., Roodhardt, A., Wijers, M., Dekker, T., Spence, M. S., Simon, A. N., Pligge, M. A., & Burrill, G. (2006). Patterns and figures. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context. Chicago: Encyclopeadia Britannica, Inc.
• Kindt, M., Wijers, M., Spence, M. S., Brinker, L. J., Pligge, M. A., Burrill, J., & Burrill, G. (2006). Graphing equations. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context. Chicago: Encyclopeadia Britannica, Inc.
• Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students’ understanding of core algebraic concept: Equivalence & variable. National Science Foundation, 37(1), 1-9.
• Liu P., H. (1996). Do teachers need to incorporate the history of mathematics in their teaching. The Mathematics Teacher, 96(6), 416.
• Maharaj, A. (2013). An APOS analysis of natural science students’ understanding of derivatives. South African Journal of Education, 33(1), 458-477.
• Meel, E., D. (2003). Models of theories of mathematical understanding: comparing Pirie and Kieren’s model of the growth mathematical understanding and APOS theory. CBMS Issues in Mathematics Education, 12(2), 132-181.
• Milli Eğitim Bakanlığı [MEB]. (2014). İlköğretim matematik 7 ders kitabı. [Elementary mathematics 7 textbook]. Ankara: Ada Matbaacılık. [Ankara: Ada Typography].
• Murray, M., A. (2002). First-tıme calculus students discovering the product rule: functıon, notatıon and apos theory (Dissertation doctoral thesis). University at Albany, New York. Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers. The Netherlands: Kluwer
• Özmantar, M., F., & Monaghan, J. (2007). A dialectical approach to the formation of mathematical abstractions. Mathematics Education Research Journal, 19(2), 89–112.
• Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan.
• Sezgin-Memnun, D. (2011). İlköğretim altıncı sınıf öğrencilerinin analitik geometri’nin koordinat sistemi ve doğru denklemi kavramlarını oluşturması süreçlerinin araştırılması [The investigation of sixth grade students? construction of coordinate system and linear equation concepts of the analytical geometry using constructivism and realistic mathematics education]. Yayınlanmamış Doktora Tezi. Uludağ Üniversitesi, Eğitim Bilimleri Enstitüsü, Bursa. [Unpublished Doctoral Thesis, Uludag University, Institute of Educational Sciences, Bursa]. Retrieved from https://tez.yok.gov.tr/UlusalTezMerkezi/tezSorguSonucYeni.jsp.
• 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, 22, 1-36.
• Soylu, Y. (2008). Yedinci sınıf öğrencilerinin cebirsel ifadeleri ve harf sembollerini (değişkenleri) yorumlamaları ve bu yorumlamada yapılan hatalar. [7th grade students’ interpretation of algebraic expression and symbol of letters while doing these interpretations]. Selçuk Üniversitesi Ahmet Keleşoğlu Eğitim Fakültesi Dergisi [Selcuk University Journal of Ahmet Kelesoglu Education Faculty], 25, 237-248.
• Skemp, R. (1986). The physcology of learning mathematics (2nd. Ed.). Harmondsworth: Penguin.
• Tabachnick, B. G., & Fidell, L. S. (2007). Using Multivariate Statistics (5th ed.). New York: Allyn and Bacon.
• Tall, D. (1999). Reflections on APOS theory in elementary and advanced mathematical thinking, Proceedings of the 23rd Conference of PME, Haifa, Israel, 1, 111–118.
• Tzirias, W. (2011). APOS theory as a framework to study the conceptual stages of related rates problems (Dissertation masters thesis). Concordia University, Canada.
• Wachira, P., Roland G. P., & Raymond, S. (2013). Mathematics teacher’s role in promoting classroom discourse. International Journal for Mathematics Teaching and Learning, 13(1), 1-38.
• Weller, K., Arnon, I., & Dubinsky, E. (2011). Preservice teachers’ understandings of the relation between a fraction or ınteger and ıts decimal expansion: Strength and Stability of Belief, Canadian Journal of Science, Mathematics and Technology Education, 11(2), 129-159.
• Weller, K., Arnon, I., & Dubinsky, E. (2009). Preservice teachers' understanding of the relation between a fraction or ınteger and ıts decimal expansion. Canadian Journal of Science, Mathematics and Technology Education, 9(1), 5-28.
• Wijers, M., Roodhardt, A., Reeuwijk, M., Dekker, T., Burrill, G., Cole, B.R., & Pligge, M. A. (2006). Building Formulas. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in context. Chicago: Encyclopeadia Britannica, Inc.
• Yeşildere, S., & Akkoç, H. (2011). Matematik öğretmen adaylarının şekil örüntülerini genelleme süreçleri [Pre-service mathematics teachers’ generalization processes of visual patterns]. Pamukkale Eğitim Fakültesi Dergisi [Pamukkale University Journal of Education], 30(2), 141-153.
• Yıldırım, A., & Şimşek, H. (2011). Nitel araştırma yöntemleri. [Qualitative research methods]. Ankara: Seçkin Yayıncılık. [Ankara: Seckin Publishing].
• Yılmaz, R. (2011). Matematiksel soyutlama ve genelleme süreçlerinde görselleştirme ve rolü [Visualization in mathematical abstraction and generalization processes and its role]. Yayınlanmamış Doktora Tezi. Uludağ Üniversitesi, Eğitim Bilimleri Enstitüsü, Bursa. [Unpublished Doctoral Thesis, Uludag University, Institute of Educational Sciences, Bursa]. Retrieved from https://tez.yok.gov.tr/UlusalTezMerkezi/tezSorguSonucYeni.jsp.
• Yin, R., K. (2011). Oualitative research from start to finish. New York: A Division of Guilford Publications.