# Convergence of sequences and series: Interactions between visual reasoning and the learner's beliefs about their own role

Article

- 504 Downloads
- 41 Citations

## Abstract

This paper examines part of a set of students who were followed during their first-term, first-year studies in formal definition-based real analysis at a British university. It explores the approaches to problems about convergence of sequences and series made by students who have a tendency to include visual imagery in their reasoning. We explore links between the students' mathematical behavior in solving these problems and their perception of their roles as learners. We develop a theory in which the tendency to visualize, coupled with the students' view of their role, can be used to account for their mathematical behavior.

advanced mathematical thinking beliefs convergence definitions grounded theory proof real analysis representations sequences visualization

## Preview

Unable to display preview. Download preview PDF.

## REFERENCES

- Alcock, L.J.: 2001,
*Categories, Definitions and Mathematics: Student Reasoning About Objects in Analysi*s, Unpublished Ph.D. Thesis, Mathematics Education Research Centre, University of Warwick, UK.Google Scholar - Alcock, L.J. and Simpson, A.P.: 2001, 'The Warwick Analysis project: practice and theory', in D. Holton (ed.),
*The Teaching and Learning of Mathematics at University Leve*l, Kluwer, Dordrecht, pp. 99-111.Google Scholar - Alcock, L.J. and Simpson, A.P.: 2002a, 'Two components in learning to reason using definitions',
*Proceedings of the 2nd International Conference on the Teaching of Mathematics (at the Undergraduate Level*), Hersonisoss, Greece, published at www.math.uoc.gr/ ∼ictm2/.Google Scholar - Alcock, L.J. and Simpson, A.P.: 2002b, 'Definitions: dealing with categories mathematically',
*For the Learning of Mathematics*22(2), 28-34.Google Scholar - Arcavi, A.: (2003) 'The role of visual representatios in the learning of mathematics',
*Edu-cational Studies in Mathematics*52, 215-24.CrossRefGoogle Scholar - Bell, A.W.: 1976, 'A study of pupils' proof-explanations in mathematical situations',
*Educational Studies in Mathematics*7, 23-40.CrossRefGoogle Scholar - Burn, R.P.: 1992,
*Numbers and Functions: Steps into Analysi*s, Cambridge University Press, Cambridge.Google Scholar - Copes, L.: 1982, 'The Perry development scheme: A metaphor for learning and teaching mathematics',
*For the Learning of Mathematics*3(1), 38-44.Google Scholar - Cornu, B.: 1992, 'Limits', in D.O. Tall (Ed.),
*Advanced Mathematical Thinkin*g, Kluwer, Dordrecht, pp. 153-166.Google Scholar - Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K. and Vidavovic, D.: 1996, 'Understanding the limit concept: Beginning with a coordinated process scheme',
*Journal of Mathematical Behaviour*15, 167-192.CrossRefGoogle Scholar - Davis, P.J.: 1993, 'Visual theorems',
*Educational Studies in Mathematics*24, 333-344.CrossRefGoogle Scholar - Davis, R.B. and Vinner, S.: 1986, 'The notion of limit: Some seemingly unavoidable mis-conception stages',
*Journal of Mathematical Behaviour*5(3), 281-303.Google Scholar - Dreyfus, T.: 1991, 'On the status of visual reasoning in mathematics and mathematics education', in F. Furinghetti (ed.)
*Proceedings of the 15th International Conference on the Psychology of Mathematics Educatio*n, Assisi, Italy, pp. 33-48.Google Scholar - Dubinsky, E., Elterman, F. and Gong, C.: 1988, 'The student's construction of quantification',
*For the Learning of Mathematics*8(2), 44-51.Google Scholar - Fischbein, E.: 1982, 'Intuition and proof',
*For the Learning of Mathematics*3(2), 9-24.Google Scholar - Fischbein, E.: 1987,
*Intuition in Science and Mathematics: An Educational Approac*h, Kluwer, Dordrecht.Google Scholar - Gibson, D.: 1998, 'Students' use of diagrams to develop proofs in an introductory analysis course', in A.H. Schoenfeld, J. Kaput, and E. Dubinsky (eds.),
*CBMS Issues in Mathematics Education II*I, pp. 284-307.Google Scholar - Ginsburg, H.: 1981, 'The clinical interview in psychological research on mathematical thinking: aims, rationales, techniques',
*For the Learning of Mathematics*1(3), 4-11.Google Scholar - Glaser, B.: 1992,
*Emergence vs. Forcing: Basics of Grounded Theory Analysi*s, Sociology Press, Mill Valley, CA.Google Scholar - Harel, G. and Sowder, L.: 1998, 'Students' proof schemes: results from exploratory studies', in A.H. Schoenfeld, J. Kaput and E. Dubinsky (eds.),
*CBMS Issues in Mathematics Education II*I, pp. 234-283.Google Scholar - Hughes-Hallet, D.: 1991, 'Visualisation and calculus reform', in W. Zimmermann and S. Cunningham (eds.),
*Visualisation in Teaching and Learning Mathematic*s, MAA Notes 19, pp. 121-126.Google Scholar - Johnson-Laird, P.N. and Byrne, R.M.J.: 1991,
*Deductio*n,Lawrence Erlbaum Associates, Hillsdale, MI.Google Scholar - Kosslyn, S.M.: 1995, 'Mental imagery', in S.M. Kosslyn and D.N. Osherson (eds.),
*An Invi-tation to Cognitive Scienc*e, 2nd ed.,*Volume 2: Visual Cognitio*n, MIT Press, Cambridge, MA, pp. 267-296.Google Scholar - Krutetskii, V.A.: 1976, 'Type, age and sex differences in the components of mathemat-ical abilities', in V.A. Krutetskii (eds),
*The Psychology of Mathematical Abilities in Schoolchildre*n, Chicago University Press, Chicago, pp. 313-343.Google Scholar - Lakatos, I.: 1976,
*Proofs and Refutations: The Logic of Mathematical Discover*y, Cambridge University Press, Cambridge.Google Scholar - May, T.: 1997,
*Social Research: Issues, Methods and Proces*s, Open University Press, Buckingham.Google Scholar - Mason, J.J.: 2002, 'Researching your own practice: The discipline of noticing', Routledge Farmer, London.Google Scholar
- Monaghan, J.: 1991, 'Problems with the language of limits',
*For the Learning of Mathe-matics*11(3), 20-24.Google Scholar - Moore, R.C.: 1994, 'Making the transition to formal proof',
*Educational Studies in Mathematics*27, 249-266.CrossRefGoogle Scholar - Nelsen, R.B.: 1993,
*Proofs Without Words: Exercises in Visual Thinkin*g, Mathematical Association of America, Washington DC.Google Scholar - Piaget, J.: 1975,
*The Development of Thought: Equilibration of Cognitive Structure*s, The Viking Press, New York.Google Scholar - Pinto, M. and Tall, D.: 2002, 'Building formal mathematics on visual imagery: A case study and a theory',
*For the Learning of Mathematics*22(1), 2-10.Google Scholar - Presmeg, N.C.: 1986, 'Visualisation and mathematical giftedness',
*Educational Studies in Mathematics*17, 297-311.CrossRefGoogle Scholar - Schoenfeld, A.H.: 1985, 'Making sense of "out loud" problem-solving protocols',
*Journal of Mathematical Behaviour*4, 171-191.Google Scholar - Schoenfeld, A.H.: 1992, 'Learning to think mathematically: Problem solving, metacognition and sense making in mathematics', in D.A. Grouws (ed.),
*Handbook for Research on Mathematics Teaching and Learnin*g, Macmillan, New York, pp. 334-370.Google Scholar - Sfard, A. and Linchevski, L.: 1994, 'The gains and pitfalls of reification-the case of algebra',
*Educational Studies in Mathematics*26, 191-228.CrossRefGoogle Scholar - Sierpinska, A.: 1987, 'Humanities students and epistemological obstacles related to limits',
*Educational Studies in Mathematics*18, 371-397.CrossRefGoogle Scholar - Skemp, R.R.: 1976, 'Relational understanding and instrumental understanding',
*Mathemat-ics Teaching*77, 20-26.Google Scholar - Strauss, A. and Corbin, J.: 1990,
*Basics of Qualitative Research: Grounded Theory Proce-dures and Technique*s, Sage, London.Google Scholar - Tall, D.O.: 1991, 'Intuition and rigor: the role of visualization in the calculus', in W.S. Zimmerman and S. Cunningham (eds.),
*Visualization in the Teaching and Learning of Mathematic*s, MAA Notes No.19, Mathematical Association of America, Washington DC, pp. 105-119.Google Scholar - Tall, D.O.: 1995, 'Cognitive development, representations and proof',
*Proceedings of Jus-tifying and Proving in School Mathematic*s, Institute of Education, London, pp. 27-38.Google Scholar - Tall, D.O.: 1997, 'Functions and calculus', in A.J. Bishop, K. Clements, C. Keitel, J.Kilpatrick and C. Laborde (eds.),
*International Handbook of Mathematics Educatio*n, Kluwer, Dordrecht, pp. 289-325.Google Scholar - Tall, D.O. and Vinner, S.: 1981, 'Concept image and concept definition with particular reference to limits and continuity',
*Educational Studies in Mathematics*12, 151-169.CrossRefGoogle Scholar - Thurston, W.P.: 1995, 'On proof and progress in mathematics',
*For the Learning of Mathematics*15(1), 29-37.Google Scholar - Vinner, S.: 1992, 'The role of definitions in teaching and learning mathematics', in D.O.Tall (ed.),
*Advanced Mathematical Thinkin*g, Dordrecht, Kluwer, pp. 65-81.Google Scholar - Watson, A. and Mason, J.: 2002, 'Student-generated examples in the learning of mathematics',
*Canadian Journal of Science, Mathematics and Technology Education*2(2), 237-249.CrossRefGoogle Scholar - Williams, S.R.: 1991, 'Models of limit held by college calculus students',
*Journal for Research in Mathematics Education*22(3), 219-236.CrossRefGoogle Scholar

## Copyright information

© Kluwer Academic Publishers 2004