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The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs

Antonella Cupillari
Academic Press
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
, on

The teaching of mathematics should concern both content objectives (body of knowledge) and process objectives (ways of doing mathematics). But school mathematics over-emphasises the former and generally ignores the latter. Hence the existence of books like this one.

The ‘body of knowledge’ consists of concepts, facts and skills defined by a particular curriculum stage, which, in turn, may be determined by stereotypical tests and examinations. Therefore, we have the ongoing stultification of mathematical thinking in people young and old.

Process objectives should be (can be) developed through an investigational approach to teaching from the age of five years onwards. Every single aspect of school mathematics can be taught with a strong investigational emphasis, and that includes all arithmetic, geometry and algebra. Central to this is the role of questioning (why? what happens if..?). Also required are skills of communication, which grow with experience of asking and responding to questions.

·         Can you tell me how you worked that out?

·         Why do you think it’s true?

·         What happens if you reverse the digits?

·         Can you draw a diagram to show it?

·         Does that work for all rectangles or only squares? Try it out.

I’ve seen, and worked with, very many kids who can give very good informal explanations for quite general mathematical ideas, such as one 8 year old girl who explained why the sum of two odd numbers will always be even (it was a verbal equivalent of \( (2m + 1) + (2n + 1) = \text{even} + 2\)). Another example concerned a 10-year old boy’s response to the activity of dividing squares into two symmetrical pieces. His method was to find the centre, draw any (squiggly) line from the centre to one side. Turn that line through 180o and the whole squiggly line determines rotational symmetry.

Anyway, this isn’t a lecture on mathematics education, it’s supposed to be the review of a book. But the point is that before coming to a compressed introduction to formal mathematical proof, students should have been psychologically prepared to receive it. And that means an investigational approach to school mathematics and a stage-by-stage introduction to justifying and explaining their viewpoints (which may include early examples of formal proof).

So, this book by Antonella Cupillari seems to have the same purpose of many others — many of which are called something like ‘a transition to advanced mathematics’. It begins with symbolic logic, formal reasoning and goes on to discuss the various sorts of proof involved in mathematics. All of that comes under ‘process objectives’.

The ‘content’ consists of basic set theory, functions, relations, groups, convergence of series and limits of functions. These topics form a context for the consolidation of earlier ideas on proof. As with many other books of this genre, there is nothing about the use of proof in geometry or the calculus done in high school. Why is that?

Unsurprisingly, the book consists mainly of worked examples, exercises and solutions. It is written with great accuracy and a level of enthusiasm necessary for the Herculean task of launching mathematical handle-turners into the world of mathematical thinking.

To conclude: for those required to teach ‘transition courses’, I recommend perusal of this book as a possible course text.

See also our review of the previous edition.

Peter Ruane has taught mathematics to people of greatly varying abilities between the ages 5 and 55. Or rather, he tried to teach them how to learn mathematics.

Chapter 1: Getting Started
Introduction and Basic Terminology
General Suggestions

Chapter 2: Basic Techniques to Prove If/Then Statements
What Does "If/Then" Mean?
Direct Proof
The Negation of a Statement, And/Or
Proof by Contrapositive
Proof by Contradiction

Chapter 3: Special kinds of Theorems
"If and only if" or Equivalence Theorems
Use of Counterexamples
Mathematical Induction
Existence Theorems
Uniqueness Theorems
Composite Statements
Equality of Numbers

Chapter 4: Some mathematical topics on which to practice proof techniques
Basic Set Theory
About Functions
The Basics of Groups
Cardinality and Sizes of Infinity

Chapter 5: Review Exercises
Exercises Without Solutions
Collection of Proofs

Solutions for the Exercises at the End of the Sections and the Review Exercises
Solutions for the Exercises at the End of the Sections
Solutions for the Review Exercises

Other Books on the Subject of Proofs and Mathematical Writing