|There are differences between definition, postulate, theorem, conjecture, and observation.|
|Choice is involved in making definitions and postulates. These choices determine what are theorems.|
|A proof proves the statement for all of the drawings that satisfy the given.|
|All Euclidean geometry has not been created/discovered/developed.|
Some people may find it strange to imagine that doing a proof can lead to conjectures, yet as one tries to explain a phenomenon, by deductive proof or otherwise, one sometimes comes to understand the key aspects of the phenomenon. In this way, one may understand that the phenomenon is more general than was thought, or one may find conditions that may be relaxed, resulting in new conjectures.
Bell, A. ( 1976). "A study of pupils' proof explanations in mathematical situations". Educational Studies in Mathematics, 7, 23-40.
Brown, S., and Walter, M. ( 1983). The art of problem posing. Philadelphia: The Franklin Institute Press.
Chazan, D., and Houde, R. ( 1989). How to use conjecturing and microcomputers to teach high school geometry. Reston, VA: National Council of Teachers of Mathematics.
Chi, M., and Bassok, M. ( 1989). "Learning from examples via self explanations". In L. Resnick (Ed.), Knowing, learning, and instruction. Hillsdale, NJ: Lawrence Erlbaum Associates.
Clement, J. ( 1983, April). Observed methods for generating analogies in scientific problem solving. Paper presented at the annual meeting of the American Educational Research Association. Montreal.
Gentner, D., and Gentner, D. R. ( 1983). "Flowing water and teeming crowds: mental models of electricity". In D. Gentner and Stevens (Eds.), Mental models. Hillsdale, NJ: Lawrence Erlbaum Associates.
Hawkins, J., and Sheingold, K. ( 1986). "Microcomputers and education". In Curbertson and Cunningham (Eds.), The 1986 Yearbook of the NSSE. Chicago: University of Chicago Press.
Holland, J., Holyoak, K., Nisbett, R., and Thagard, P. ( 1986). Induction: Processes of inference, learning, and discovery. Cambridge, MA: Massachusetts Institute of Technology.
Kidder, R. ( 1985, April 19). "How high-schooler discovered new math theorem". Christian Science Monitor.