THE USE OF EXAMPLES IN MATHEMATICS TEACHING: BASIC DEFINITIONS, CONCEPTS AND APPROACHES

Year 2019, , 569 - 586, 01.07.2019

Duygu Ören Vural , Fulya Kula

https://doi.org/10.17240/aibuefd.2019.19.46660-427934

Cited By: 3

Abstract

The use of examples in
mathematics and its contributions to the learning of mathematics has been
receiving both international and national attention recently. Mathematical
examples are widely studied in the international literature with their
classifications or the development and use of theoretical frameworks. However
examples were limitedly studied in the national level. This study aims to
review the literature of mathematical examples from a historical and
pedagogical approach. The various classifications of examples and the
theoretical framework are introduced. In addition the national studies
conducted about the topic are reviewed to draw an overview about the trends.
  According to the
results of this study, although there has been increased national interest in
the subject in the last two decades, it has been determined that the studies
carried out are limited and generally examined at secondary and undergraduate
levels. The results of the national studies examined point out the finding of
the limited and poor use of examples in the teaching contexts.  The
current study also gives insights for future research. The most basic
suggestions are the extensive use of examples in teaching settings and books
and the effects on student learning and widespread studies in the primary
level.

Keywords

Mathematical Examples, Personal Example Space, Literature Review

MATEMATİK ÖĞRETİMİNDE ÖRNEKLER: TEMEL TANIM, KAVRAM VE YAKLAŞIMLAR

Abstract

Matematikte örnek kullanımı ve örneklerin matematik
öğretimine katkısı ulusal ve uluslararası alanyazında son yıllarda ilgi gören
çalışma başlıklarıdır. Uluslararası alanyazında örneklerle ilgili
sınıflandırmalar ve kuramsal çerçeve oluşturma çalışmaları yaygın olarak
mevcuttur. Ulusal alanyazında ise örnek kullanımı sınırlı şekilde
incelenmektedir. Bu çalışmada örnek kullanımının matematikte ve matematik
öğretimindeki tarihsel gelişimi ve pedagojik sınıflandırma çalışmaları
incelenmiştir. Ayrıca matematikte örnekler ile ilgili geliştirilen kuramsal
çerçeveler derlenmiş ve konu ile ilgili ulusal çalışmalar incelenmiştir. Bu
çalışmanın sonuçlarına göre ulusal alan yazında son yirmi yılda konuya olan
ilgi artmış olsa da yapılan çalışmaların sınırlı sayıda olması ve genellikle
ortaöğretim ve lisans seviyelerinde incelendiği belirlenmiştir. İncelenen
ulusal çalışmalar doğrultusunda matematik öğretiminde sınırlı ve zengin olmayan
örnek kullanımı dikkat çekicidir. Ayrıca bu çalışmada elde edilen bulgular
doğrultusunda ileride yapılacak çalışmalar için kimi önerilerde bulunulmuştur.
En temel öneriler ise çalışmaların daha yaygın olarak kitaplarda ve derste
zengin örnek kullanımının ve bunun matematik öğrenmeye katkısının incelenmesi
ve ilköğretim seviyesinde yapılacak çalışmaların daha yaygınlaştırılmasıdır.

Keywords

Matematiksel Örnekler, Kişisel Örnek Alanı, Alanyazın Derlemesi

References

• Akkoç, H. (2006). Fonksiyon kavramının çoklu temsillerinin çağrıştırdığı kavram görüntüleri. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 30, 1-9.
• Alcock, L. and Inglis, M. (2008). Doctoral students' use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69:111-129.
• Alkan, S. (2016). Matematik öğretmenlerin kullandıkları örneklerin sınıflandırılması ve öğretimsel açıklama boyutlarıyla ilişkisinin incelenmesi. Yayınlanmamış doktora tezi, Karadeniz Teknik Üniversitesi, Trabzon.
• Alkan, S., & Güven, B. (2018) Ders kitaplarında kullanılan örnek türlerinin analizi: Limit konusu. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 1-1.
• Alkan, S., Güven, B., & Yılmaz, Ş. (2017). The types of examples teachers use in teaching function concept. Bayburt Eğitim Fakültesi Dergisi, 12(23), 367-384.
• Anthony, G. (1994). The role of the worked example in learning mathematics. A. Jones ve diğerleri. (Ed.). SAME papers (s. 129-143). Hamilton, New Zealand: University of Waikato.
• Antonini, S., Presmeg, N., Mariotti, A. ve Zaslavsky, O., (2011). Examples in mathematics thinking and learning from an educational perspective. ZDM Zentralblatt fur Didaktik der Mathematik, 43.(3).
• Asiala, M. Brown, A. DeVries, D. Dubinsky, E. Mathews D. ve Thomas K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6, 1-32.
• Avcu, R. (2014). Exploring middle school mathematics teachers’ treatment of rational number examples in their classrooms: A multiple case study.Yayınlanmamış doktora tezi, Orta Doğu Teknik Üniversitesi, Ankara.
• Aylar, E. (2014). A Study on the forms of perception of 7th grade students towards the concept of proof. Journal of Education and Future, 5, 39-56.
• Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. D. Pimm (Ed.).Mathematics, Teachers and Children, (s. 216-235), London: Hodder and Stoughton.
• Bell, E.T. (1967). Men of Mathematics. New York: Simon and Schuster.
• Bentley, P. (2008) The Book of Numbers: The Secret of Numbers and how They Changed the World.
• Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A. ve Zaslavsky, O. (2006). Exemplification in mathematics education. Proc. 30th Conf. of the Int. Group for the Psychology of Mathematics Education içinde (s. 126-154). Prague, Czech Republic: PME.
• Bills, L. ve Watson, A. (2008). Editorial introduction (Special issue on the role and use of examples in mathematics education). Educational Studies in Mathematics, 69(2), 77–79.
• Bogomolny, M. (2006). The role of example-generation tasks in students’ understanding of linear algebra. Yayınlanmamış doktora tezi, Simon Fraser University, Burnaby, Canada.
• Boyer, C. B. (1949). The History of the Calculus and Its Conceptual Development. New York:Dover Publications.
• Boyer, C. B. (1968). A History of Mathematics. New York: Wiley International Edition.
• Breen, S., O’Shea, A. ve Pfeiffer, K. (2016). Students’ views of example generation tasks. Teaching Mathematics and its Applications. doi: 10.1093/teamat/hrv017
• Bruner, J., Goodnow, J. ve Austin, A. (1956). A study of thinking. New York: Wiley.
• Burn, R. (2002). The genesis of mathematical structures. P. Kahn ve J. Kyle (Ed.). Effective learning and teaching in mathematics and its applications. (s. 20-33). Kogan Page.
• Charles, R. (1980). Exemplification and characterization moves in the classroom teaching of geometry Concepts. Journal for Research in Mathematics Education.11(1), 10-21.
• Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education (6th ed.). New York, NY: Routledge.
• Colburn, W. (1826). Intellectual arithmetic: Upon the inductive method of instruction. Boston, USA: Reynolds ve Co.
• Confrey, J. (1991). Steering a course between Vygotsky and Piaget. Educational Researcher, 20(2), 29-32.
• Cook, J. P. ve Fukawa-Connelly, T. (2015) The pedagogical examples of groups and rings that algebraists think are most important in an introductory course. Canadian Journal of Science, Mathematics and Technology Education, 15(2), 171-185.
• Courant, R. (1981). Reminiscences from Hilbert’s Gottingen. Mathematical Intelligencer, 3(4), 154–164.
• Dahlberg, R. ve Housman, D. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics, 33, 283-299.
• Davis, R. (1984). Learning mathematics: the cognitive science approach to mathematics education. Norwood, NJ, USA: Ablex.
• Davis, P. ve Hersh, R. (1981). The Mathematical Experience. Brighton UK: Harvester.
• De Morgan, A. (1831). On mathematical instruction. Quarterly Journal of Education,1, 264-279.
• Dienes, Z. (1960). Building up Mathematics. London, UK: Hutchinson Educational.
• Dienes, Z. P. (1963). An Experimental Study of Mathematics Learning. London: Hutchinson.
• Dreyfus, T. (1991). Advanced mathematical thinking processes. D. Tall (Ed.). Advanced mathematical thinking (s. 25-41). Dordrecht, The Netherlands: Kluwer.
• Duran, M , Kaplan, A . (2016). Lise matematik öğretmenlerinin türevin tanımına ve türev-süreklilik ilişkisine yönelik pedagojik alan bilgileri. Erzincan Üniversitesi Eğitim Fakültesi Dergisi, 18 (2), 795-831.
• Edwards, A. W. (2011). Using example generation to explore undergraduates' conceptions of real sequences: A phenomenographic study. Yayınlanmamış doktora tezi, Loughborough University, Leicestershire, UK.
• Evans, R. (1973). Jean Piaget: The man and his ideas. New York, USA: Dutton.
• Feynman, R. (1985). “Surely you’re joking, Mr Feynman!”: Adventures of a curious character. New York, USA: Norton.
• Feynman, R. P. (1986). Personal observations on the reliability of the shuttle. Report of the Presidential Commission on the Space Shuttle Challenger Accident,2, 1-5.
• Fukawa-Connelly, T. P. ve Newton, C. (2014). Analyzing the teaching of advanced mathematics courses via the enacted example space. Educational Studies in Mathematics,87, 323–349.
• Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.
• Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht, The Netherlands: Reidel.
• Gagné, R. (1985). The conditions of learning (4th basım). New York: Holt, Rinehart and Winston.
• Gillings, R. (1972). Mathematics in the time of the Pharaohs. Reprinted 1982. New York, USA: Dover.
• Goldenberg, P. ve Mason, J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69, 183-194.
• Gökbulut, Y. ve Ubuz, B. (2013). Sınıf Öğretmeni Adaylarının Prizma Bilgileri: Tanım ve Örnekler Oluşturma. İlköğretim Online, 12(2), 401-412.
• Gökkurt, B. ve Soylu, Y. (2012). Üniversite öğrencilerinin matematiksel ispat yapmaya yönelik görüşleri. Eğitim ve Öğretim Araştırmaları Dergisi,1(4), 56-64.
• Güler, G., Kar, T., Öçal, M. F. ve Çiltaş, A. (2011). Prospective mathematics teachers’ difficulties in proof. Procedia Social and Behavioral Sciences, 15, 336-340.
• Güler, G., Özdemir, E. ve Dikici. R. (2012). Öğretmen adaylarının matematiksel tümevarım yoluyla ispat becerileri ve matematiksel ispat hakkındaki görüşleri. Kastamonu Eğitim Dergisi, 20(1), 219-236.
• Gürsel, G. (2013). Matematik öğretmeni adaylarının cebir öğrenme alanındaki ispat süreçlerinin incelenmesi. Yayımlanmamış doktora tezi, Atatürk Üniversitesi Eğitim Bilimleri Enstitüsü, Erzurum.
• Gray, E. ve Tall, D. (1994). Duality, ambiguity, and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116–140.
• Hershkowitz, R. (1989). Visualization in geometry-two sides of the coin. Focus on Learning Problemsin Mathematics, 11(1), 61-76.
• Hershkowitz, R. (1990). Psychological aspects of learning geometry. P. Nesher ve J. Kilpatrick (Ed.). Mathematics and cognition. Cambridge, UK: Cambridge University Press.
• İncikabı, L. ve Tjoe, H. (2013). A comparative analysis of ratio and proportion problems in Turkish and the U.S. middle school mathematics textbooks. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi (KEFAD),14(1), 1-15.
• Kant, I. (1998). Critique of Pure Reason. Guyer, P. ve Wood, A. W. (Ed.). Cambridge University Press.
• Kinach, B. M. (2002). A cognitive strategy for developing pedagogical content knowledge in the secondary mathematics methods course: Toward a model of effective practice. Teaching and Teacher Education, 18(1), 51-71.
• Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. V. Richardson (Ed.). Handbook of Research on Teaching (4. basım, s. 333-357). Washington DC, USA: American Educational Research Association.
• MacVicar, D. (1879). A complete arithmetic, oral and written: designed for the use of common and high schools and collegiate institutes. Montreal, Canada: Dawson Bros.
• Marton, F. ve Booth, S. (1997). Learning and awareness. Hillsdale, USA: Lawrence Erlbaum.
• Marton, F. ve Trigwell, K. (2000). Variatio est mater studiorum. Higher Education Research and Development, 19 (3), 381-395.
• Marton, F. ve Tsui, A. (2004). Classroom discourse and space for learning. Marwah, NJ, USA: Lawrence Erlbaum Associates.
• Marton, F., & Tsui, A.B.M. (Eds.). (2004). Classroom discourse and the space of learning. Mahweh, NJ: Lawrence Erlbaum Associates.
• Mason, J. (2006). What makes an example exemplary: Pedagogical and didactical issues in appreciating multiplicative structures. Number theory in mathematics education: Perspectives and prospects, 41-68.
• Mason, J. ve Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277–290.
• Mason, J. & Watson, A. (2001). Getting Students to Create Boundary Examples, MSOR Connections, 1 (1), 9-11.
• Mason, J. ve Watson, A. (2008). Mathematics as a constructive activity: Exploiting dimensions of possible variation. M. Carlson ve C. Rasmussen (Ed.). Making the connection: Research and practice in undergraduate mathematics(s. 189–202). Washington, DC: MAA.
• Michener, E. (1978). Understanding understanding mathematics’, Cognitive Science, 2, 361–383.
• Minsky, M. (1975). A framework for representing knowledge. P. Winston (Ed.). The psychology of computer vision (s. 211-280). New York, USA: McGraw Hill.
• Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in mathematics,27(3), 249-266.
• Oktaç, A. ve Çetin, I. (2016). APOS teorisi ve matematiksel kavramların öğrenimi. Bingölbalı, E., Arslan, S. ve Zembat, İ. Ö (Ed.). Matematik Eğitiminde Teoriler (s.163-181). Ankara: Pegem Yayıncılık.
• Özer, Ö. ve Arıkan, A. (2002). Lise matematik derslerinde öğrencilerin ispat yapabilme düzeyleri. V. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi, 2, 1083-10989.
• Özkaya, M., Işık, A., & Konyalıoğlu, A. (2014). İlköğretim matematik öğretmenliği öğrencilerinin sürekli fonksiyonlarla ilgili ispatlama ve ters örnek oluşturmaperformansları. Middle Eastern & African Journal of Educational Research, 11, 26-42.
• Peled, I. ve Zaslavsky, O. (1997). Counter-examples that (only) prove and counterexamples that (also) explain. FOCUS on Learning Problems in mathematics, 19(3), 49-61.
• Pestalozzi, J. H., Holland, L. E., Turner, F. C., & In Cooke, E. (1898). How Gertrude teaches her children: An attempt to help mothers to teach their own children and an account of the method. Syracuse, N.Y: C.W. Bardeen.
• Petty, O. S. ve Jansson, L. C. (1987). Sequencing examples and non-examples to facilitate concept attainment. Journal for Research in Mathematics Education, 18(2), 112-125.
• Piaget, J. (1970). Genetic epistemology. New York, USA: Norton.
• Pirie, S. ve Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26,165-190.
• Polya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton, USA: Princeton University Press.
• Polya, G. (1954). Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics. Princeton University Press.
• Polya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving. New York, USA: Wiley.
• Record, R. (1632). The Ground of Arts: teaching the perfect worke and practise of arithmeticke, both in whole numbers and fractions. London: Harper, Thomas.
• Rosch, E. (1975). Cognitive representations of semantic categories. Journal of Experimental Psychology: General, 104, 192-322.
• Sağlam Kaya, Y. (2017). Öğretmen adaylarının matematiksel örnekleri algılayışları üzerine bir metafor analizi. Bartın Üniversitesi Eğitim Fakültesi Dergisi, 6(1), 48.
• Schwarz, B. ve Hershkowitz, R. (1999). Prototypes: Brakes or levers in learning the function concept? Journal for Research in Mathematics Education, 30(4), 362-389.
• 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.
• Sierpinski, W. (1998). Elementary theory of numbers (2. basım). The Netherlands: Elsevier Science Publishers.
• Sinclair, N. Watson, A. Zazkis, R. ve Mason, J. (2011). The Structuring of Personal Example Spaces. Journal of Mathematical Behavior,30 (4), 291-303.
• Skemp, R. (1969). The psychology of learning mathematics. Harmondsworth, UK: Penguin.
• Skemp, R. R. (1979). Intelligence, learning and action. Chichester: Wiley.
• Skinner, B. F. (1938). The behavior of organisms: an experimental analysis. New York.
• Sowder, L. (1980). Concept and principle learning. R. Shumway (Ed.). Research in Mathematics Education (s. 244-285). Reston, VA, USA: NCTM.
• Spencer, H. (1878). Education: intellectual, moral and physical. London: Williams ve Norgate.
• Stylianides, G. J., ve Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 314-352.
• Sun X. (2011). “Variation problems” and their roles in the topic of fraction division in Chinese Mathematics textbook examples, Educational Studies in Mathematics, 76 (1) 65-85.
• Swetz, F. (1987). Capitalism and arithmetic: The New Math of the 15th century. (Trans. Smith). LaSalle, USA: Open Court.
• Tall, D. ve Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169.
• Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mateamatics, 26 (2), 229-274.
• Thorndike, E. L. (1922). The psychology of arithmetic. New York, USA: Macmillan,
• Thorndike, E., Cobb, M., Orleans, J., Symonds, P., Wald, E. ve Woodyard, E. (1924). The psychology of algebra. New York, USA: Macmillan.
• Ubuz, B. ve Kırkpınar,P. (2000). The role of example in the formation of mathematical concepts. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 19, 134-138.
• Ubuz, B. ve Gökbulut, Y. (2015). Sınıf Öğretmeni Adaylarının Piramit Bilgileri: Tanım ve Örnekler Oluşturma. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi (KEFAD), 16 (2), 335-351.
• Vinner, S. (1983). Concept image, concept definition and the notion of function. International Journal of Mathematics Education in Science and Technology, 14(3), 293-305.
• Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. D. O. Tall (Yay. Haz.). Advanced mathematical thinking. Dordrecht, The Netherlands: Kluwer Academic Publishers.
• Watson, A. ve Mason, J. (2004). The exercise as mathematical object: dimensions of variation in practice, BSRLM, 107-112.
• Watson, A. ve Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ, USA: Erlbaum.
• Watson, A. ve Shipman, S. (2008). Using learner generated examples to introduce new concepts. Educational Studies in Mathematics,69 (2), 97-109.
• Whitehead, A. (1911). An introduction to mathematics (reprinted 1948). Oxford, UK: Oxford University Press.
• Wilson, S. (1986). Feature frequency and the use of negative instances in a geometric task. Journal for Research in Mathematics Education, 17(2), 130-139.
• Wilson, S. (1990). Inconsistent ideas related to definitions and examples. Focus on Learning Problems in Mathematics, 12, 31-47.
• Winston, P. H. (1975). Learning structural descriptions from examples. P. Winston (Ed.). The psychology of computer vision. New York, USA: McGraw-Hill.
• Witmer, T. (Trans.) (1968). Ars Magna or the Rules of Algebra, Girolamo Cardano. New York, USA: Dover.
• Yüce, M. (2017). Lise öğrencilerinin matematik dersi kapsamında örnek üretme becerileri. Yayımlanmamış doktora tezi, Hacettepe Üniversitesi, Ankara.
• Zaslavsky, O. ve Peled, I. (1996). Inhibiting factors in generating examples by mathematics teachers and student-teachers: The case of binary operation. Journal for Research in Mathematics Education, 27(1), 67-78.
• Zaslavsky, O. ve Zodik, I. (2007). Mathematics teachers’ choices of examples that potentially support or impede learning. Research in Mathematics Education, 9, 143–155.
• Zazkis, R. (2001). From arithmetic to algebra via big numbers. Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (s. 676-681). Melbourne, Australia: University of Melbourne.
• Zazkis, R. ve Chernoff, E. (2008). What makes a counterexample exemplary? Educational Studies in Mathematics, 68(3), 195-208.
• Zazkis, R. ve Leikin, R. (2007). Generating examples: From pedagogical tool to a research tool. For the learning of mathematics,27(2), 15-21.
• Zazkis, R. ve Leikin, R. (2008). Exemplifying definitions: a case of a square. Educational Studies in Mathematics, 69 (2), 131-148.
• Zeeman, E. C. (1965). Topology of the brain, in Mathematics and computer science in biology and medicine. London: H.M. Stationary Office.
• Zeeman, C. (2005). Three-Dimensional Theorems for Schools. The Mathematical Association, Leicester.
• Zodik, I. ve Zaslavsky, O. (2008). Characteristics of teachers’ choice of examples in and for the mathematics classroom. Educational Studies in Mathematics, 69(2), 165-182.