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Students' mental prototypes for functions and graphs
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Tall, David and Bakar, Md. Nor. (1992) Students' mental prototypes for functions and graphs. International Journal of Mathematical Education in Science and Technology, Vol.23 (No.1). pp. 3950. ISSN 0020739X

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Official URL: http://dx.doi.org/10.1080/0020739920230105
Abstract
This research study investigates the concept of function developed by students studying English Alevel mathematics. It shows that, while students may be able to use functions in their practical mathematics, their grasp of the theoretical nature of the function concept may be tenuous and inconsistent. The hypothesis is that students develop prototypes for the function concept in much the same way as they develop prototypes for concepts in everyday life. The definition of the function concept, though given in the curriculum, is not stressed and proves to be inoperative, with their understanding of the concept reliant on properties of familiar prototype examples: those having regular shaped graphs, such as x2 or sin*, those often encountered (possibly erroneously), such as a circle, those in which y is defined as an explicit formula in x, and so on. Investigations reveal significant misconceptions. For example, threequarters of a sample of students starting a university mathematics course considered that a constant function was not a function in either its graphical or algebraic forms, and threequarters thought that a circle is a function. This reveals a wide gulf between the concepts as perceived to be taught and as actually learned by the students.
Item Type:  Journal Article 

Subjects:  L Education > LB Theory and practice of education Q Science > QA Mathematics 
Divisions:  Faculty of Social Sciences > Institute of Education 
Library of Congress Subject Headings (LCSH):  Mathematics  Study and teaching, Functions 
Journal or Publication Title:  International Journal of Mathematical Education in Science and Technology 
Publisher:  Taylor & Francis Ltd. 
ISSN:  0020739X 
Date:  1992 
Volume:  Vol.23 
Number:  No.1 
Page Range:  pp. 3950 
Identification Number:  10.1080/0020739920230105 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  Barnes M 1988: “Understanding the Function Concept: Some Results of Interviews with Secondary and Tertiary Students”, Research on Mathematics Education in Australia, 2433. Breidenbach D., Dubinsky E., Hawks J, and Nichols D. (in preparation): “Development of the Process Concept of Function”. Department of Education and Science 1989: Mathematics in the National Curriculum, HMSO, London. Dreyfus T. and Vinner S. 1982: “Some aspects of the function concept in college students and junior high school teachers, Proceedings of the Sixth International Conference for the Psychology of Mathematics Education, Antwerp, 1217. Even R. 1988: Prospective secondary mathematics teachers’ knowledge and understanding about mathematical function, unpublished Ph.D. thesis, Michigan State University. Malik M.A. 1980: “Historical and pedagogical aspects of the definition of function”, Int. J. Math. Ed. Sci. Technol. 11,4, 48992. Markovits Z., Eylon B. & Bruckheimer M. 1988: “Difficulties Students have with the Function Concept”, The Ideas of Algebra, K12, N.C.T.M. 1988 Yearbook, 4360. Smith E.E. 1988: “Concepts and thought”, in The psychology of human thought, editors Sternberg R.J and Smith E.E., Cambridge University Press, 1949. Tall D.O. 1985: Supergraph, (software for BBC compatible computers), Glentop, London. Tall D.O. 1990: “The transition to advanced mathematical thinking: functions, limits, infinity and proof”, Handbook on Research in Mathematics Education, N.C.T.M., (to appear). Vinner S. 1983: “Concept definition, concept image and the notion of function”, The International Journal of Mathematical Education in Science and Technology, 14, 293305. 
URI:  http://wrap.warwick.ac.uk/id/eprint/511 
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