It was once customary for a dissertation to include a section entitled “Literature Review” in which the author attempted to describe all the research which had been done on the topic of the dissertation. The bibliographic search required to assemble this section was often a large part of the effort involved in preparing a dissertation. This effort was worthwhile, both as basic preparation for beginning research in an area, and as a service to the research community. Such reviews provided a starting point for other researchers interested in quickly becoming acquainted with the literature of a particular area.
The introduction of services like ERIC and Dissertation Abstracts International, especially in their electronic formats, have made the preparation of a literature review much easier, at the same time they have made it largely superfluous. Assembling a list of most of the works in educational research related to proof and proving is now a matter of a few keystrokes, and the information available includes extensive abstracts and detailed information on availability of unpublished manuscripts, research reports, and government documents, dissertations, as well as journal publications. Given this level of service it would be surprising if a researcher interested in proof and proving went to the trouble of requesting a copy of my dissertation, either as an interlibrary loan or from University Microfilms International, just to read my literature review, when the same information, in more detail and more up to date, is available in any university library and over the Internet.
For this reason this appendix is an appendix, and slightly different in form than the traditional literature review. The research which is directly related to mine, and which played an important role in the development of my ideas, is described in the appropriate sections of the main text. Other work on proof and proving which is interesting, but not directly related to my interest in the need to prove, is gathered together here. The one exception to this is the first section, in which I have listed works by researchers who have been very productive, and published their research in a wide range of publications. Rather than list a reference for every occurrence of a researcher’s ideas when I mention them in the main text, I have chosen to list only the most accessible or complete presentations of the ideas. For completeness, and in the event that some sources are not as accessible as I thought, I have listed other publications of these researchers here.
Arsac,
G. (1990). Les recherches actuelles sur l’apprentissage de la demonstration
et les phenomes de validation en France.
Recherches en Didactique des
Mathématiques, 9(2), 247-280.
Arsac,
G., Chapiron, G., Colonna, A., Germain, G. Guichard, Y., & Mante, M.
(1992). Initiation au raisonnement
déductif au collège. Presses
Universitaires de Lyon.
Balacheff, N. (1986). Cognitive versus situational analysis of problem-solving behaviors. For the Learning of Mathematics, 6(3), 10-12.
Balacheff, N. (1987). Processus de preuve et situations de validation. Educational Studies in Mathematics, 18, 147-176.
Balacheff, N. (1990a). A study of students’ proving processes at the junior high school level. In I. Wirszup & R. Streit (Eds.), Developments in School Mathematics Education Around the World, Vol. 2. Reston VA: NCTM. (Originally presented in 1988 at the joint conference 66th NCTM and UCSMP project, Chicago.)
Balacheff, N. (1990b). Beyond a psychological approach: The psychology of mathematics education. For the Learning of Mathematics,10(3), 2-8.
Balacheff, N. (1990c). Towards a problematique for research on mathematics teaching. Journal for Research in Mathematics Education, 21(4), 258-272.
Borasi, R. (1990). The invisible hand operating in mathematics instruction: Student’s conceptions and expectations. In T. Cooney (Ed.), Teaching and learning mathematics in the 1990s: NCTM yearbook 1990. Reston VA: National Council of Teachers of Mathematics.
Bruner, J. (1990). Acts of meaning. Cambridge MA: Harvard University Press.
Davis, P. (1993). Visual theorems. Educational Studies in Mathematics, 24, 333-344.
Hanna, G. (1989a). More than formal proofs. For the Learning of Mathematics. 9(1), 20-23
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6-13
Hanna, G. (1991). Mathematical
proof. In D. Tall (Ed.), Advanced
Mathematical Thinking. Dordrecht:
Kluwer Academic.
Hanna, G. & Jahnke, N. (1993). Proof and application. Educational Studies in Mathematics, 24, 421-438.
Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24, 389-399.
Schoenfeld, A. (1982). Psychological factors affecting students’ performance on geometry problems. In S. Wagner (Ed.), Proceedings of the Fourth PME-NA Conference, (pp. 168-174). Athens, GA.
Schoenfeld, A. (1986). On having and using geometric knowledge. In: Conceptual and Procedural Knowledge: the Case of Mathematics pp. 225-264
Schoenfeld, A. (1987a). Confessions of an accidental theorist. For the Learning of Mathematics, 7(1), 30-38
Schoenfeld, A. (1987b). Understanding and teaching the nature of mathematical thinking. In I. Wirszup & R. Streit (Eds.), Developments in School Mathematics Education Around the World. Reston VA: NCTM.
Schoenfeld, A. (1989). Explorations of students’ mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20(4), 338-355
Tymoczko, T. (1986) Making room for mathematicians in the philosophy of mathematics. Mathematical Intelligencer 8(3), 44-50.
Barbeau, E. (1990). Three faces of proof. Interchange, 21(1), 24-27.
Barbeau mentions the two everyday uses of “proof”: verification, and testing or trying. He also points out that proofs can satisfy a need for explanation in mathematics. He makes a distinction between verification and convincing, which he uses in the sense of explanation. He defines verification as making a mechanically checkable argument, and convincing “a revelation of underlying structure, appropriate level of generality, comprehensiveness, and a degree of satisfaction and appreciation aroused in the listener” (p.24).
Neubrand, M. (1989). Remarks on the acceptance of proofs: The case of some recently tackled major theorems. For the Learning of Mathematics, 9(3), 2-6.
Neubrand quotes Hanna’s (1983) assertion that verification is supposed to be the business of mathematics, but this is practically impossible, and convincing is what mathematicians actually engage in. He states that “a ‘convincing argument’ is not simply a sequence of correct inferences. One always expects some ‘qualitative’ reason, or an intuitive capable basic idea, behind the—nevertheless necessary—single steps of the proof” (p.4). He adds, however, that in mathematics “to be convinced depends on the high standards of argumentation which mathematicians have reached during a long historical development” (p.3).
Wheeler, D. (1990). Aspects of mathematical proof. Interchange, 21(1) 1-5.
Wheeler seems to assert that proving does not create new mathematical knowledge, which would imply that it is not useful for exploration. “We can no longer assume that proofs establish knowledge, because in fact most proofs come after the knowledge of the things they prove. They could be said perhaps to substantiate knowledge or to validate it, or confirm it, but proofs, on the whole, do not establish knowledge” (p.2). He does not believe there is much of a role for proving in mathematics classrooms.
Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359-387.
Chazan reports interviews with high school students describing their views of proofs as evidence, versus their acceptance of examples as verification.
Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the international group for the psychology of mathematics education (pp. 70-95). Cambridge: Cambridge University Press.
Hershkowitz discusses proving in the context of van Heile’s level of understanding of geometry. She suggests that students’ difficulties in proving may stem from a lack of understanding of the necessity to prove in mathematics. She advocates a strategy of presenting proofs to trigger the students’ “intellectual curiosity”, in which empirical discovery acts as a source of a need to verify by proving. “It is a common belief now that inductive, empirical discoveries in geometry are necessary because ... by regarding the generalization as a conjecture in itself, the learner feels the necessity to prove what he or she has conjectured to be true; and ... inductive experiences are the intuitive base upon which the understanding and the generation of a deductive proof can be built” (p.89).
Martin, G. & G. Harel (1989) Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41-51
Martin & Harel (1989) conducted a quantitative study in which they claim: “Many students who correctly accepted a general-proof verification did not reject a false proof verification; they were influenced by the appearance of the argument—the ritualistic aspects of the proof—rather than the correctness of the argument” (p.49). Unfortunately their statistics seem not to back this up. They presented pre-service teachers with two statements, one of which the teachers had seen in class three weeks previously. The teachers were given empirical evidence for each statement, as well as a correct and an incorrect deductive proof. They were asked to rate each on a scale of 1 to 4, with 4 indicating that they felt the evidence constituted a proof. For the statement they had seen in class 52% rated the incorrect proof either 3 or 4, indicating they accepted it as a proof. 75% gave the correct proof either 3 or 4. This would argue that the proof-like form of the incorrect proof was influencing them. If this were the case one would expect a similar result in the case of the unfamiliar statement. In that case however, only 38% gave the incorrect proof either 3 or 4 while 63% gave the correct proof 3 or 4. It may have been that the teachers remembered the familiar statement as correct, and so were predisposed to assume that a proof of it was correct. In that case of the unfamiliar statement the proofs would have been checked more closely, resulting in a drop in the acceptance of the false proof.
Movshovits-Hadar, N. (1988). Stimulating presentation of theorems followed by responsive proofs. For the Learning of Mathematics, 8(2), 12-19, 30.
Movshovits-Hadar proposes presenting conjectures in surprising ways in order to inspire a need to verify and explain in students, in a manner similar to that proposed by Hershkowitz.
O’Daffer, P. G. & Thornquist, B. A. (1993). Critical thinking, mathematical reasoning, and proof. In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp. 39-56). New York: Macmillan.
A general discussion of research and the meanings of the various terms used in the NCTM Standards (1989) to describe reasoning.
Sierpinska, A. (1995). Mathematics: “In context”, “pure”, or “with applications”? For the Learning of Mathematics, 15(1), 2-15.
A discussion of the role of real life contexts in mathematics teaching. Sierpinska critiques the use of social contexts in the teaching of proof, as proposed by Arsac, Balacheff, and Lampert, and suggests instead an apprenticeship model. Her focus seems to be more on the education of future mathematicians than the general population.
Rather than duplicate the efforts of other researchers who have assembled reviews of the literature related to the aspects of proof and proving they find most interesting, I have gathered together here a few works which contain excellent reviews. The two dissertations include the traditional exhaustive literature review, and in addition to the interesting research they report, they also fulfill the traditional purpose of making other researchers’ lives easier. These two are particularly strong in listing older research, which is not always covered in the electronic databases.
Dreyfus, T. (1990) Advanced mathematical thinking. In P. Nesher & J. Kilpatrick (Eds.) Mathematics and Cognition. (pp. 113-134).
An overview of current research.
Smith, E. P. (1959). A developmental approach to teaching the concept of proof in elementary and secondary school mathematics. Unpublished doctoral dissertation, Ohio State University.
Smith’s dissertation was written in the early days of the development of the “New Math” and he sets out clearly the agenda of that movement.
Williams, E. R. (1979). An investigation of senior high school students’ understanding of the nature of mathematical proof. Unpublished doctoral dissertation, University of Alberta.
Williams did his research at the close of the “New Math” era, and he covers most of the research done within that approach to mathematics and mathematics education. His research is a series of statistical studies of students’ performance.