Abstract
Examination of data from several mathematics education research projects has led the author to postulate a form of mathematical reasoning that learners engage in spontaneously and that is not inherently inductive or deductive. Transformational reasoning is generated through the learner's inquiry into how a mathematical system works. This sense of ‘how it works’ may lead to a sense of understanding that may not be provided by inductive and deductive reasoning.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
BalacheffN.: 1987, ‘Processus de preuve et situations de validation’, Educational Studies in Mathematics 18, 147–176.
ChazanD.: 1993, ‘High school geometry students, justification for their views of empirical evidence and mathematical proof’, Educational Studies in Mathematics 24, 359–387.
CobbP., YackelE., and WoodT.: 1993, ‘Learning mathematics: Multiple perspectives-theoretical orientation’, in T.Wood, P.Cobb, E.Yackel, and D.Dillon (eds.), Rethinking Elementary School Mathematics: Insights and Issues, Journal for Research in Mathematics Education Monograph Series, Number 6. National Council of Teachers of Mathematics, Reston, VA, pp. 21–32.
EnnisR.: 1969, Logic in Teaching, Prentice Hall, Englewood Cliffs, NJ.
FischbeinE.: 1987, Intuition in Science and Mathematics: An Educational Approach, D. Reidel, Dordrecht, Holland.
FischbeinE., DeriM., NelloM., and MarinoM.: 1985, ‘The role of implicit models in solving problems in multiplication and division’, Journal for Research in Mathematics Education 16, 3–17.
GruberH. and VonècheJ.: 1977, The Essential Piaget, Basic Books, New York.
HadamardJ.: 1945, The Psychology of Inention in the Mathematical Field, Princeton University, Princeton, NJ.
Harel, G.: 1994, ‘Observations, conjectures, doubts, and facts in the epistemology of proof scheme’, paper presented at the American Educational Research Association annual meeting, New Orleans.
LampertM.: 1990, ‘When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching’, American Educational Research Journal 27, 29–63.
MartinW. and HarelG.: 1989, ‘Proof frames of preservice elementary teachers’, Journal for Research in Mathematics Education 20, 41–51.
MishF. (Ed.): 1991, Webster's Ninth New Collegiate Dictionary, Merriam-Webster Inc., Springfield, MA.
National Council of Teachers of Mathematics: 1989, Curriculum and Evaluation Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA.
SchoenfeldA.: 1989, ‘Explorations of students' mathematical beliefs and behavior’, Journal for Research in Mathematics Education 20, 338–355.
SchwartzJ. and YerushalmyM.: 1985, The Geometric Supposser [Computer program], Sunburst, Pleasantville, NY.
SimonM.: 1989, ‘Intuitive understanding in geometry: The third leg’, School Science and Mathematics 89, 373–379.
SimonM. and BlumeG.: 1994, ‘Building and understanding multiplicative relationships: A study of prospective elementary teachers’, Journal for Research in Mathematics Education 25, 472–494.
Simon, M. and Blume, G.: in press, ‘Justification in the mathematics classroom: A study of prospective elementary teachers’, Journal of Mathematical Behavior.
SimonM. and SchifterD.: 1991, ‘Toward a constructivist perspective: An intervention study of mathematics teacher development’, Educational Studies in Mathematics 22, 309–331.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Simon, M.A. Beyond inductive and deductive reasoning: The search for a sense of knowing. Educ Stud Math 30, 197–210 (1996). https://doi.org/10.1007/BF00302630
Issue Date:
DOI: https://doi.org/10.1007/BF00302630