COLLEGE STUDENTS' ABILITY TO PROVE MATHEMATICAL THEOREMS WITH AND WITHOUT TRAINING IN INFERENCE PATTERNS

WITHIN THE last decade the revolution engen dered by proposals for change in school mathemat ics curricula has forced educators to re-examine the mathematics requirements for prospective ele mentary school teachers. The Committee on the Undergraduate Program in Mathematics (2), appoint ed by the Mathematical Association of America, rec ommended that teachers of mathematics in grades kindergarten through six be required to take at least the following: a two-course sequence devoted to the structure of the real number system and its subsys tems, a course involving the basic concepts of alge bra, and a course in informal geometry. In its de scription of the geometry course, the committee suggested that the concept of deduction b e studied through its application to the geometric properties developed during the intutitve introduction of the sub ject. For, if teachers of mathematics were to be effective in the classroom, they must be f am i liar with the techniques, relative merits, and roles of the inductive and deductive approaches to new ideas. The need for teachers to become familiar with these methods of reasoning was emphasized when Deans (5) revealed that all of the recent elementary school programs had placed added importance on both the inductive and deductive types of reasoning as valu able aids to learning and understanding mathematics. In the Fall of 1964 a committee of mathematics instructors at Wisconsin State University River Falls was selected and charged with the responsibil ity to investigate various proposals for curricular change in the courses designed for prospective ele mentary school teachers. The net result of this committee's deliberations was the recommendation of a two-course sequence devoted primarily to the development of the natural numbers, the integers, and the rational numbers, accompanied by a brief intuitive approach to the basic concepts of geometry. This recommendation was approved by the appropri ate authorities and was made effective at the start of the 1965-66 academic year. The content selected and the level of mathematical rigor intended was re flected in the choice of the text (4). The minimal at tention given to the topics from geometry precluded the possibility of introducing deductive methods with this content. Since the deductive method is an in dispensible tool in the establishment of a mathemat ical system, the committee deemed that some at tention must be given to this method and suggested its introduction in conjunction with the topics of sets. There was agreement that the students should have experience in proving theorems appropriate to the content and to their level of mathematical maturity. However, some instructors were of the opinion that the students should learn deductive methods implic itly while proving theorems; others felt that explic it instruction in deductive methods ought to precede or at least accompany the student's experience in proving theorems. This professorial disagreement initiated the investigation undertaken in this study, viz., whether explicit instruction in deduction would improve a student's achievement in proving mathe matical theorems. A thorough study of deduction was impossible and undesirable; this topic had to be made concise with only those aspects of deduction selected which would most effectively aid the student to prove mathemat ical theorems. To this end the use of inference pat terns was considered most appropriate? Definition: An inference pattern is a set of statements leading to a final statement which is said to be inferred from the preceding ones. The final statement is called the con clusion, while those prior to it are called the hypotheses of the inference pattern. Inference patterns are called valid if true hypotheses