The Exploring of Visual Imagery: In Their Relation to the Students’ Mathematical Identity
The learning principles usually not visually. The visual understood as an analytical and facilitate an idea. That is the visual is parallel to other representation. The visual ability is not only a tool, or a strategy, or a type of thinking, but also the chain of reasoning to achieve the formal analytic abilities. In this study, the visual ability examined, as a strategy or way of thinking to solve a problem. The question to be addressed is: How do students come to their mathematics identities based on visual imagery? The research conducted at classes of the preservice mathematics education students namely provides evidence for what is their identity relative to their experiences. The results conducted based on explorative studies for the visual abilities. The performance obtained during the teaching and learning, i.e., visualizing and answering analytically or visually, manipulating and answering analytically or visually or by formulas, and visualizing and the answering visually or analytically or based on the conditions. The performance linked to the visual representation and related to intuition underlies formal abilities. The visual perceptions disturbed by prior knowledge, and the level based on optical illusions, so the teaching and learning make a difference between potential and abilities. The analytical affects visual perception and the belief system, so difficult to construct knowledge. However, the level of thinking is different, i.e., not yet formal or new optical illusion. The visual model related to high-level thinking, which distinguished from the analytical thinking model. In the visual model, thinking activities based on the transformations and understood as the other operations in mathematics. The visual model also shows the analytic thinking and hierarchical. The visual and the analytical thinking integrated to develop a richer understanding of mathematical concepts. Through visual thinking, the mental processing was constructed and interpreted as mental objects and processed analytically. The next, exhibits the analytical, consist of the construction process from the visual, namely the reflective abstractions. The visual abilities not related to the duration of courses in the mathematics education department. The longer increases the analytic but in contrary to the visual. Through the learning of the visual abilities, the difficulties in solving the problems decrease, but resistant to the analytic show a specific performance. Visual learning reveals a hierarchical level of the thinking, so the best performance is the highest learning ability. When learning the visual abilities given, the performance increases.
The Exploring of Visual Imagery: In Their Relation to the Students’ Mathematical Identity, Higher Education Research.
Vol. 3, No. 5,
2018, pp. 75-91.
M. W. Matlin, Cognition (third edition). USA: Holt, Reinhart and Winston, 1994.
E. T. Ruseffendi, “Penilaian pendidikan dan hasil belajar siswa khususnya dalam pengajaran matematika,” unpublished.
N. Tadao, “The constructive approach in mathematics education,” paper presented in The 24 th conference of the international group for the psychology of mathematics education, Hiroshima, 23-27 July 2000.
A. H. Schoenfeld, (Ed.), Mathematical thinking and problem solving. Hillsdale, New Jersey: LEA, 1994.
E. Barbin, The epistemological roots of a constructivist. In John A. Malone and Peter C. S. Taylor, Constructivist interpretation of teaching and learning mathematics. Perth: National Key Centre for School Science and Mathematics, 1993.
D. H. Clements, and M. T. Battisda, Geometry and spatial reasoning. In Douglas A. Grouws (Ed.), Handbook of research on mathemtics teaching and learning. USA: Macmillan, 1992.
M. Rif’at, Analisis Tingkat Deduksi dan Rogoritas Susunan Bukti Mahasiswa Jurusan Pendidikan Matematika IKIP Malang, Unpublished master’s thesis, PPS IKIP Malang, Malang, 1997.
R. Zazkis; Dubinsky, Ed; and J. Dautermann, “Coordinating visual and analytic strategies: a study of students understanding of the group D4,” Journal for research in mathematics education, 27 (4), pp. 435-457, 1996.
O’Daffer, G. Phares and B. A. Thornquist, Critical thinking, mathematical reasoning, and proof. In P. S. Wilson (Ed.), Research ideas for the classroom high school mathematics. New York: Macmillan Publishing Company, 1993.
D. H. Clements, S. Swaminathan, M. A. Z. Hannibal, and J. Sarama, “Young children’s concepts of shape,” Journal for research in mathematics education, 30 (2), pp.. 192-212, 1999.
S. Vinner, New mathematics. In D. Tall, The transition to advanced mathematical thinking: function, limits, infinity, and proof. USA: Macmillan, 1992.
E. G. Begle, Critical variables in mathematics education: Findings from a survey of empiric literatur, Washington, D. C.: Mathematics Association of America, 1979.
E. Barbin, The epistemological roots of a constructivist. In John A. Malone dan Peter C. S. Taylor, constructivist interpretation of teaching and learning mathematics, Perth: National Key Centre for School Science and Mathematics, 1993.
E. Galindo, (Eds.), “Visualization and students’ performance in technology-based calculus,” In Douglas T. Owens, Michelle K. Reed, and Gayle M. Millsaps (Eds.), Proceedings of seventeenth annual meeting for psychology of mathematics education, 2, Columbus: ERIC, 1995, pp.321-326.
H. W. Syer, (1953), “ Sensory learning applied to mathematics,” In Suwarsono, Peranan strategi visual dalam pembelajaran matematika, Seminar paper, unpublished.
M. Rif’at, Pengaruh Pembelajaran Visual Dalam Rangka Meningkatkan Kemampuan Deduksi Visualistik pada Soal Berciri Visual, Unpublished doctor’s dissertation, Bandung, PPS UPI Bandung, 2001.
J. Boaler, & C. Humphreys, Connecting mathematical ideas: Middle school video cases to support teaching and learning. Portsmouth, NH: Heinemann, 2005.
J. O. Masingila, (2002). Examining students’ perceptions of their everyday mathematics practice. In M. E. Brenner & J. N. Moschkovich (Eds.), Everyday and academic mathematics in the classroom (Journal for Research in Mathematics Education Monograph No. 11, pp. 30–39). Reston, VA: National Council of Teachers of Mathematics.
S. P. Springer & G. Deutsch, Left Brain, Right Brain. W. H. Freedman & Company New York, 1989.
T. West, In The Mind’s Eye. Prometheus Press. New York, 1991.
J. Freed, (1996) Teaching Right: Techniques for Visual Spatial Gifted Children. Understanding Our Gifted. V8/3, January/February, p3, pp 16-19, p21.
M. Piechowski, Overexcitabilities. In Encyclopedia of Creativity Volume 2. M. A. Runco & S. R. Pritzker Eds. Academic press, 1999, pp 325-334.
L. B. Resnick and W. W. Ford, (1972). The psychology of mathematics for instructions. Hillsdale, N. J.: LEA, 1972.
M. Wertheimer, Productive thinking. New York: Harper & Row, 1959.
St. Suwarsono, Peranan strategi visual dalam pembelajaran matematika. Paper presented in Seminar nasional pendidikan matematika dalam era globalisasi, Program Pasca Sarjana IKIP Malang, Malang, 4 April 1998.
G. Polya, Mathematical discovery. New York: John Wiley & Sons, 1981.
R. M. Gagne, (1984), Kondisi belajar dan teori pembelajaran. Translated by Munandir, 1989. Jakarta: DIKTI.
Z. P. Dienes, Building up mathematics. London: Hutchinson Educational, 1963.
J. S. Bruner, Toward a theory of instruction. New York: Norton, 1966.
W. A. Brownell, “Pshychological considerations in the learning and teaching of arithmetic,” In W. D. Reeve (Ed). The teaching of arithmetic. New York: Columbia University Teacher College Bureau of Publications, 1953.
S. I. Brown, “Mathematics and humanistics themes: sum considerations,” In S. I. Brown and M. I. Walter (Eds.), Problem posing: reflections and applications. New Jersey: LEA, 1993.
W. F. Burger and B. Culpepper, (1993). Restucturing geometry. Research ideas for the classroom high school mathematics, h. 140-142. New York: Macmillan Publishing Company, 1993.
D. Solow, How to read and do proofs. New York: John Wiley & Sons, 1982.
E. Wenger, Communities of practice: Learning, meaning, and identity. Cambridge, UK: Cambridge University Press, 1998.
R. P. Morash, (1991). Bridge to abstract mathematics: Mathematical proof and structure (second edition). USA: McGraw Hill, Inc., 1991.
A. H. Schoenfeld, Mathematical problem solving. New York: Academic Press, 1985.
E. Fischbein, “Image and concepts in learning mathematics,” Educational studies in mathematics, 8, 1977, pp. 153-165.
F. H. Bell, Teaching and learning mathematics (In Secondary Schools). Dubuque, Iowa: Wm C Brown, 1978.
K. Yamaguti, “High school mathematics from a viewpoint of geometric calculation,” unpublished.
Z. P. Dienes, An eksperimental study of mathematics learning. New York: Hutchinson & Co., 1960.
R. Gelman, “Conservation aqcuisition: A problem of learning to attend to relevant attribute,” in Journal of experimental child psychology, 1969, 7, pp. 167-187.
Spitzer, “Without references,” in St. Suwarsono, Peranan strategi visual dalam pembelajaran matematika, unpublished.
C. Janvier, (Ed), Problems of reperesentation in the teaching and learning of mathematics. Hillsdale NJ: LEA, 1987.
G. A. Goldin, “Cognitive representational system for mathematical problem solving,” in C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 125-145), Hillsdale, New Jersey, 1987.
S. Vinner and R. Hershkowitz, “Concept image and common cognitive paths in the development of some simple geometrical concepts,” in R. Kaplus (Ed.), Proceedings of the fourth international conference for the psychology of mathematics education. Berkeley, California, 16-17 Agustus 1980.
G. William, Development and evolution of brain size: Behavioural implication. New York: Academic Press, 1988.
L. Sowder, “Criteria for concrete models’” The aritmetic teacher, 23, pp. 468-470, 1976.
D. Tall, “Intuition and rigor: The role of visualization in the calculus,” in W. Zimmermann and S. Cunningham (Eds.), Visualization in teaching and learning mathematics, 1991, pp. 105-120, MAA.
J. Piaget and B. Inhelder, Mental imagery in the child. London: RKP, 1971.
A. Roe, “A study of imagery in research scientists,” Journal of personality, 19, 1951, pp. 459-470.
R. R. Skemp, The psychology of learning mathematics. Harmoodsworth: Penguin Books, 1971.
H. A. Simon, “Information processing models of cognition,” Annual review of psychology, 30, 1979, pp. 363-396.
A. D. de Groot, Thought and choice in chess. The Hague: Mouton, 1965.
A. Hinsley, J. R. Hayes, and H. A. Simon, “From words to equation: Meaning and representation in algebra word problems,” in P. A. Carpenter and M. A. Just (Eds.), Cognitive process in comprehention. Hillsdale, New Jersey: LEA, 1977.
J. F. Mundy, “Analysis of error in first year calculus students,” in theory, research and practice in mathematical education: Working group reports and collected papers. Nottingham: University of Nottingham, 1985.
S. Susanna, “The role of proof in problem solving,” in A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving. New Jersey: LEA, 1994.
P. M. Van Hiele, “Begrip en inzicht,” in K. P. E. Gravemeijer (Ed.), Realistic mathematics education in primary, 1973, h. 57-76. Culemborg: Technipress.
R. Hersh, (1986). Some proposals for revising the philosophy of mathematics. In T. Tymoczko (Ed.), New directions in the philosophy of mathematics: An anthology (pp. 9-28). Boston: Birkhauser.
J. Hoffman & C. Kugle, (1992). A study of theoretical orientation to reading in relation to teacher feedback during reading instruction. Journal of Classroom Interaction, 18, 2-7.
K. P. E. Gravimeijer, “An instruction-theoretical reflection on the use manipulatives,” in Leen Streefland (Ed.), Realistic mathematics education in primary school, 1991, pp. 57-73. Culemborg: Technipress.
T. Akihiko, “Using manipulatives in problem solving lesson,” paper presented in The 24 th conference of the international group for the psychology of mathematics education, Hiroshima, 23-27 July 2000.
L. C. Seng, “Strategi-strategi untuk pengajaran konsep-konsep matematik,” Jurnal pendidik dan pendidikan, 1 (1), 1979, pp. 42-53.
L. C. Hart, “Some factors that impede or enhance performance in mathematical problem solving,” Journal for research in mathematics education, 24 (2), 1993, pp. 167-171.
W. E. Lamon, Learning and the nature of mathematics. Chicago: SRA, 1973.
E. Warren, “Visualization and the development of early understanding in algebra,” paper presented in The 24 th conference of the international group for the psychology of mathematics education, Hiroshima, 23-27 Juli 2000.
L. K. Silverman, (1994), Teaching Gifted Children With Classroom Adjustment Difficulties. Invited Address to the International Council for Exceptional Children.
M. Rif’at, “Lecturer notes,” unpublished.
G. Masson, “Visual Spatial Learners: A New Perspective,” Understanding Our Gifted. V8/3, January/February, 1996, pp. 11-16.
V. A. Krutetskii, “The psychology of mathematical abilities in school children,” in Jeremy Kilpatrick and Izaak Wirszup (Eds.). Chicago: University of Chicago Press, 1976.
N. C. Presmeg, “Visualisation in high school mathematics,” For the learning of mathematics, 6 (3), 1986, pp. 42-46.