SHLOMO VINNER THE CONCEPT OF EXPONENTIATION AT THE UNDERGRADUATE LEVEL AND THE DEFINITIONAL APPROACH I. THE ROLE OF DEFINITION IN HIGH SCHOOL MATHEMATICS When we teach math to high school students we try to teach them not only manipulations on algebraic expressions and equation solving but also a little bit of the structure of mathematics and the way it is developed. Many text- books are written from this point of view and many high school teachers emphasize it in their classrooms. When talking about the structure of math- ematics, definition has a very important role in it. In this paper we will deal with one aspect of this role, the role it has in algebra. To be more specific, the role it has in algebra when new arithmetical operations are defined by means of known operations. We will restrict ourselves to exponentiation only. In high school, very often, a lot of emphasis is put on the fact that the operation a x is defined by means of previous operations. This is especially true for x being a non-positive integer or a fraction) Usually, the definitions are given by means of a formula that has an equality sign in it. The student is expected to understand that there is 'some difference' between the formula 'a -x= 1/a x', where x is a whole number, and the formula: aXa y =a x+y, where x,y are (for the sake of simplicity) whole numbers. This idea looks very elementary to math teachers although some of them should have noticed a lot of confusion2 in their classrooms. We are going to examine how some people look at exponentiation a few years after having it in high school. This question has both theoretical and practical importance. Theoretical-because it is interesting to know what remains of one's math studying. Practical - because one is taking additional math courses at the college level and it is important to know what we can assume about his background and the way he looks at mathematics. The students of our sample were all math majors. Therefore, we can assume that most of them were good math students in high school. The question was: Do these students look at exponentiation the same way some of their teachers tried so hard to teach them (either as a result of this teaching or as a result of having some mathematical experience at the college level 3 )? 1 The operation ax when x is a whole number is defined in many cases at the junior high or even at the elementary level when teachers usually do not call the students' attention to the fact that a new operation is defined by means of an old one. 2 We will mention some possible reasons for this confusion in Section 4. 3 All the students had at least three quarters of calculus. Educational Studies in Mathematics 8 (1977) 17-26. All Rights Reserved. Copyright 9 1977 by D. Reidel Publishing Company, Dordrecht-Holland.