FROM THE INTRODUCTION This Handbook is a report on mathematical discourse. Mathematical discourse as the phrase is used here refers to what mathematicians and mathematics students say and write
* to communicate mathematical reasoning * to describe their own behavior when doing mathematics * to describe their attitudes towards various aspects of mathematics.
The emphasis is on the discourse encountered in postcalculus mathematics courses taken by math majors and first year math graduate students in the USA.
The Handbook describes common usage in mathematical discourse. The usage is determined by citations, that is, quotations from the literature, the method used by all reputable dictionaries. The descriptions of the problems students have are drawn from the mathematics education literature and the author's own observations.
Intended audience: The Handbook is intended for * Teachers of collegelevel mathematics, particularly abstract mathematics at the postcalculus level, to provide some insight into some of the difficulties their students have with mathematical language. * Graduate students and upperlevel undergraduates who may find clarification of some of the difficulties they are having as they learn higherlevel mathematics. * Researchers in mathematics education, who may find observations in this text that point to possibilities for research in their field.
The Handbook assumes the mathematical knowledge of a first year graduate student in mathematics. I would encourage students with less background to read it, but occasionally they will find references to mathematical topics they do not know about. The Handbook website at
http://www.cwru.edu/artsci/math/wells/pub/abouthbk.html
contains some links that may help in finding out about such topics.
SOME SAMPLE ENTRIES
Contrapositive The contrapositive of a conditional assertion P> Q is the statement (not Q) > (not P). In mathematical arguments, the conditional assertion and its contrapositive are equivalent. In particular, to prove P > Q it is enough to prove that (not Q) > (not P), and once you have done that, no further argument is needed. I have attended lectures where further argument was given, leading me to suspect that the lecturer did not fully understand the contrapositive, but I have not discovered an instance in print that would indicate that.
Remark 1 The fact that a conditional assertion and its contrapositive are logically equivalent means that a proof can be organized as follows, and in fact many proofs in texts are organized like this:
a) Theorem: P implies Q. b) Assume that Q is false. c) Argument that not P follows. d) Conclude that P implies Q. e) End of proof.
Often P is a conjunction of several statements P1, . . . Pn and the argument in the third step will be an argument that not Pi for some particular i.
The reader may be given no hint as to the form of the proof; she must simply recognize the pattern. A concrete example of such a proof is given under functional knowledge. See also pattern recognition.
Difficulties In contrast to the situation in mathematical reasoning, the contrapositive of a conditional sentence in ordinary English about everyday topics of conversation does not in general mean the same thing as the direct sentence. This causes semantic contamination.
Example 1 The sentence If it rains, I will carry my umbrella. does not mean the same thing as If I dont carry my umbrella, it wont rain.
There are reasons for the difference, of course, but teachers rarely explain this to students. McCawley [1993], section 3.4 and Chapter 15, discusses the contrapositive and other aspects of conditional sentences in English. More about this in the remarks under only if.
Minus The word minus can refer to both the binary operation on numbers, as in the expression a b, and the unary operation of taking the negative: negating b gives b. In current usage in American high schools, a b would be pronounced a minus b, but b would be pronounced negative b. The older usage for b was minus b. College students are sometimes confused by this usage from older college teachers.
Difficulties In ordinary English, if you subtract from a collection you make it smaller, and if you add to a collection you make it bigger. In mathematics, adding may also refer to applying the operation of addition; a + b is smaller than a if b is negative. Similarly, subtracting b from a makes the result bigger if b is negative. Both these usages occur in mathematical writing.
Students sometime assume that an expression of the form t must be negative. This may be because of the new trend of calling it negative t, or because of the use of the phrase opposite in sign.
Objectprocess duality Mathematicians thinking about a mathematical concept will typically hold it in mind both as a process and an object. As a process, it is a way of performing mathematical actions in stages. But this process can then be conceived as a mathematical object, capable for example of being an element of a set or the input to another process. Thus the sine function, like any function, is a process that associates to each number another number, but it is also an object which you may be able to differentiate and integrate.
The mental operation that consists of conceiving of a process as an object is called encapsulation, or sometimes reification or entification. Encapsulation is not a oneway process: while solving a problem you may think of for example finding the antiderivative of the sine function, but you are always free to then consider both the and its antiderivative as processes which can give values and then you can conceive of them encapsulated in another way as a graph in the xy plane.
The word procept was introduced in [Gray and Tall, 1994] to denote a mathematical object together with one or more processes, each with an expression that encapsulates the process and simultaneously denotes the object. Thus a mathematician may have a procept including the number 6, expressions such as 2 + 3 and 2 3 that denote calculations that result in 6, and perhaps alternative representations such as 110 (binary). This is similar to the idea of schema.
See also APOS and semantics.
Power The integer 53 is a power of 5 with exponent 3. One also describes 53 as 5 to the third power. I have seen students confused by this double usage. A statement such as 8 is a power of 2 may make the student think of 28.
