Mathematical thinking processes for the pythagorean theorem of the secondary school students

Yıl 2022, Cilt: 3 Sayı: 2, 87 - 99, 30.12.2022

Esra Akarsu Yakar Süha Yılmaz

Öz

In this study, mathematical thinking processes on geometry of 6th, 7th and 8th grade students who have the same geometric thinking levels were examined. Two students from each grade level were selected for the study. The geometric thinking levels of the students were determined as the third level. In addition, the algebraic thinking levels of the students were discussed. The three worlds of mathematics was used in the research. The mathematical thinking processes of the students were examined in terms of the embodied world, the proceptual world and the formal world. The Pythagorean Theorem was chosen as the geometry subject. Two-stage semi-structured interviews were conducted with the students. In the first part of the interviews, the verbal expression of the Pythagorean Theorem was directed to the students. In the second part, an activity was presented for them to discover the theorem in a real-life situation. As a result, while the students had difficulty in explaining the theorem in the verbal expression, they were able to express it more easily in a real-life situation. 7th and 8th grade students were more successful than 6th grade students in demonstrating the processes of the three worlds of mathematics.

Anahtar Kelimeler

Geometric thinking, Mathematical thinking, The three worlds of mathematics

Kaynakça

• Açan, H. (2015). Examining 8th graders'knowledge construction processes in transformation geometry [Unpublished master’s thesis]. Dokuz Eylul University.
• Açıl, E. (2015). Investigation of middle school 7rd graders abstraction processes of the concept of equation: APOS theory [Unpublished doctoral thesis]. Ataturk University.
• Akarsu, M. (2022). Understanding of geometric reflection: John’s learning path for geometric reflection. Journal of Theoretical Educational Science, 15 (1) , 64-89. DOI: 10.30831/akukeg.952022
• Alkan, H. & Güzel, E. B. (2005). Development of mathematical thinking in the student teachers. Gazi University Journal of Gazi Education Faculty, 25 (3) , 221-236. Retrieved from https://dergipark.org.tr/tr/pub/gefad/issue/6755/90835
• Alyeşil, D. (2005). The effect of geometry learning with the method of conception charts supported problem solving to the 7th class students' geometry thinking level [Unpublished master's thesis]. Dokuz Eylul University.
• Arslan, S. & Yıldız, C. (2010). Reflections from the Experiences of 11th Graders during the Stages of Mathematical Thinking. Education and Science, 35(156), 17-31.
• Bağdat, O. & Saban, P. (2014). Investigation of the 8th grade students’ algebraic thinking skills with solo taxonomy. International Journal of Social Science, 26 (2), 473-496.
• Bal, A.P. & Dinç Artut,P. (2020). Developing the Mathematical Thinking Scale: Validity and Reliability. Journal of Çukurova Education Faculty, 49 (1), 278-315.
• Breen, J. J. (2000). Achievement of Van Hiele level two in geometry thinking by eight grade students through the use of geometry computer-based guided instruction. Dissertation Abstract Index, 60 (07).
• Bobango, J. C. (1988). Van Hiele levels of geometric thought and student achievement in standard content and proof writing: The effect of phase-based instruction. Dissertation Abstract Index, 48 (10).
• Burton, L. (1984). Mathematical thinking: The struggle for meaning. Journal for Research in Mathematics Education, 15 (1), 35-49.
• Chan, C.C, Tsui, M.S, Chan, M.Y.C. & Hong, J.H. (2002). Applying the structure of the observed learning outcomes (SOLO) taxonomy on student’s learning outcomes: An empirical study. Assessment and Evaluation in Higher Education, 27 ( 6).
• Creswell, J. W. (2013). Research design: Qualitative, quantitative, and mixed methods approaches (4nd ed.). Thousand Oaks, Sage.
• Doğan, M. & Güner, P. (2012, June). Examination of pre-service primary school mathematics teachers' skills of understanding and using mathematics language. Paper presented at the X. National Science and Mathematics Education Congress, Niğde University, Niğde.
• Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. O. Tall (Ed.). Advanced Mathematical Thinking. USA: Kluwer Academic Publishers. 25-41.
• Fidan, Y. (2009). The geometric thinking levels of primary grade 5 students and the effect of geometry teaching with discovery learning on geometric thinking levels of students [Unpublished doctoral thesis]. Dokuz Eylul University.
• Fidan, Y. & Türnüklü, E. (2010). Examination of 5th grade students’ levels of geometric thinking in terms of some variables. Pamukkale University Journal of Education, 27 (27), 185-197.
• Freudenthal, H. (1973). Mathematics as an educational task. The Netherlands:Riedel Publishing Company.
• Fuys, D., (1985). Van Hiele levels of thinking in geometry. Education and Urban Society, 17(4), 447-462.
• Groth, R. E. & Bergner, J.A. (2006). Preservice elementary teachers' conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8 (1), 37-63.
• Gündoğdu Alaylı, F. (2012). The investigation of thinking process of primary 6th ,7th and 8th grade students in the studies of composing and decomposing shapes in geometry and determination of their levels in this process [Unpublished doctoral thesis]. Dokuz Eylul University.
• Hannah, J., Stewart, S. & Thomas, M. O. J. (2016). Developing conceptual understanding and definitional clarity in linear algebra through the three worlds of mathematical thinking. Teaching Mathematics and Its Applications. 35, 216-235.
• Henderson, P. B., Fritz, S. J., Hamer, J., Hitcher, L., Marion, B., Riedesel, C. & Scharf, C. (2002). Materials development in support of mathematical thinking. The 7th Annual Conference on Innovation and Technology in Computer Science Education, Working Group Report, ACM SIGCSE Bulletin, 35 (2), 185–190.
• Jukić, L. & Brückler, F.M. (2014). What do croatian pre-service teachers remember from their calculus course?. Issues in the undergraduate mathematics preparation of school teachers, 1,1-15.
• Karakarçayıldız, R. Ü. (2016). Geometrical thinking levels and polygons classification skills of 7 th grade students and relationship between them [Unpublished master's thesis]. Eskisehir Osmangazi University.
• Kashefi, H., Ismail, Z. & Yusof, Y.M. (2010). Obstacles in the lerning of two-variable functions through mathematical thinking approach. Pocedia Social and Behavioral Sciences, 173-180.
• Keskin, M. , Dag, S. A. & Altun, M. (2013). Comparison of 8th and 11th grade students behaviours at mathematical thinking, Journal of Educational Science , 1 (1) , 33-50. Retrieved from https://dergipark.org.tr/tr/pub/jedus/issue/16124/168704
• Kidron, I. (2008). Abstraction and consolidation of the limit procept by means of instrumented schemes: The complementary role of three different frameworks. Educational studies in mathematics, 69, 197-216.
• Köse, O. (2018). Determination of SOLO taxonomy levels of mathematics teacher candidates with high level saptial ability accordingto thinking structures [Unpublished master’s thesis]. Selçuk University.
• Liu, P. H. (2003). Do teachers need to incorporate the history of mathematics in their teaching?. The Mathematics Teacher, 96 (6), 416-421.
• Liu, P. H. & Niess, M. L. (2006). An exploratory study of college students’ views of mathematical thinking in a historical approach calculus course. Mathematical Thinking and Learning, 8 (4), 373-406.
• Lucas, U. & Mladenovic, R. (2008). The identification of variation in students’ understandings of disciplinary concepts: The application of the SOLO taxonomy within introductory accounting. Higher Education, 58 (2), 257-283. doi:10.1007/s10734-009-9218-9
• Martínez-Planella, R. & Triguerosb, M. (2019). Using cycles of research in APOS: The case of functions of two variables. The Journal of Mathematical Behavior, 53, 1-22.
• Mason, J., Burton, L. & Stacey, K. (2010). Thinking mathematically. Pearson Education Limited.
• Miles, M, B. & Huberman, A. M. (1994). Qualitative data analysis: An expanded Sourcebook. (2nd ed). Thousand Oaks, Sage.
• Mistretta, R. M. (2000). Enhancing geometric reasoning. Adolescence, 35 (138), 365-379.
• Miyazaki, M., Fujita, T. & Jones, K. (2017). Students’ understanding of the structure of deductive proof. Educational Studies in Mathematics, 94, 223-239. doi: 10.1007/s10649-016-9720-9
• MoNE (2018). Mathematics Course Curriculum (1st-8th Grades). Ankara: MEB.
• Mubark, M. (2005). Mathematical thinking and mathematical achievement of students in the year of 11 scientific stream in Jordan [Unpublished doctoral thesis]. New Castle University.
• Mudrikah, A. (2016). Problem-based learning associated by action-process-object-schema (APOS) theory to enhance students’ high order mathematical thinking ability. International Journal of Research in Education and Science (IJRES), 2 (1), 125- 135.
• National Council For School Mathematics (2000). Principles and standarts for school mathematics. Reston, Va: Author.
• Olkun, S., Toluk, Z. & Durmuş, S. (2002, September). Geometric thinking levels of primary school and mathematics teaching students. Paper presented at the 5th National Science and Mathematics Education Congress, ODTÜ, Ankara.
• Özcan, B.N. (2012). The investigation of knowledge construction process of primary school students while developing geometric thinking levels [Unpublished doctoral thesis]. Dokuz Eylul University.
• Polya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton University Press.
• Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition and sensemaking in mathematics. In Grouws, D. (Ed), Handbook for Research on Mathematics Teaching and Learning, (334-370), MacMillan.
• Ramdhani, S. & Suryadi, D. (2018). The analogical reasoning analysis of Pesantren students in geometry. Journal of Physics Conference Series,1132 (1).
• Stacey, K. (2006). What is mathematical thinking and why is it important? Progress report, Collaborative Studies on Innovations for Teaching and Learning Mathematics in Different Cultures (II)-Lesson Study focusing on Mathematical Thinking, University of Tsukuba.
• Tall, D. O. (1995). Cognitive growth in elementary and advanced mathematical thinking, in D. Carraher and L. Miera (eds.), Proceedings of XIX International Conference for the Psychology of Mathematics Education, 1, 61–75, Recife, Brazil.
• Tall, D. (2002). Advanced mathematical thinking. Kluwer Academic Publishers
• Tall, D. (2006). A Theory of mathematical growth through embodiment, symbolism and proof. Annales De Didactique Et De Sciences Cognitives, 11, 195-215.
• Tall, D. (2007). Developing a theory of mathematical growth. Zdm Mathematics Education, 39, 145-154.
• Tsamir, P. & Dreyfus, T. (2002). Comparing infinite sets – A process of abstraction: The case of Ben. Journal of Mathematical Behaviour, 21, 1-23.
• Tall, D., Lima, R. N. & Healy, L. (2014). Evolving a three-world framework for solving algebraic equations in the light of what a student has met before. The Journal of Mathematical Behavior, 34, 1-13.
• Tuluk, G. & Dağdelen, I. (2020). The Investigation of geometric thinking levels of fifth year students who started to primary school at different ages. International Journal of Scientific and Technological Research, 6 (9).
• Türnüklü, E. & Özcan, B. (2014). The relationshıp between students' construction of geometric knowledge process based on rbc theory and van hiele geometric thinking levels: Case study. Mustafa Kemal University Journal of Graduate School of Social Sciences, 11 (27), 295-316.
• Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry, Final Report, Cognitive Development And Achievement in Secondary School Geometry Project. Chicago: University Of Chicago.
• Vandebrouck, F. (2011). Students conceptions of functions at the transition between secondary school and university. Proceedings of the Conference CERME 7, University of Rzeszow, Poland.
• Van De Walle, J. (2004). Elemantary and middle school mathematics: Teaching developmentally (5th edition). Boston: Allyn & Bacon.
• Van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Academic Pr.
• Yeşildere, S. (2006). The Investigation of mathematical thinking and knowledge construction processes of primary 6, 7 and 8th grade students who have different mathematical power [Unpublished doctoral thesis]. Dokuz Eylul University.
• Yeşildere, S & Türnüklü, E. (2007). Examination of students’ mathematical thinking and reasoning processes. Ankara University Journal of Faculty of Educational Sciences (JFES) , 40 (1) , 181-213 . DOI: 10.1501/Egifak_0000000156