6:30 am

Mathematics, for me, is beauty. It is beauty in a stainless steel sense as one follows the rigour of an argument or a proof. It is beauty in an aesthetic sense when one sees the patterns and relationships that follow from a few simple assumptions. And it is beauty in a "sweat on the brow" sense when one sees the results of dedicated effort revealed in a set of skills that give one the power to think about a wide variety of natural and artificial situations in a new way.

What am I trying to do? I am trying to relearn a subject that has been on the sidelines of my life for 40 years: mathematics. My profession used some mathematics when I was analyzing data, but the emphasis was on the proper collection of data and then on how to best present this data in a manner that illustrated the points that one could infer from that particular situation. I cannot recall a single example where I used calculus or modern algebra (i.e groups, rings, fields) in any of my work. I do, vividly, recall using Pythagorus's Theorem, when finishing a basement, and needing to know how to cut an angle of 45 degrees. That was exciting because I saw an application for something that I had learned years earlier. But in general my love of mathematics was unreasonable. I sensed the beauty, but was not in a position, or so I thought, where I could pursue it. Now I have the time to rectify this mistake, and it was a mistake.

In attempting to relearn calculus and modern algebra I have looked at a number of books and web sites. I have a number of highly recommended books on the shelves near me, but I have yet to find one that excites me. The books are excellent in a number of ways, but they all fail to capture the essence of what I find appealing. It is time to see if I can do this.

There are two main issues, as I see it. One is deciding what to read and what to do. The other is to show why the topics and activities that I have selected are worth doing. The first decision is a "big picture" decision: how much time can I reasonably spend on this? My initial answer is about 10 hours a week (i.e. an hour a day plus a little more).

The next decision, also a "big picture" one, is one of readership. Do I simply want to:

a) describe my daily activities
b) have others join me in skipping down the yellow brick road?

Opting for (b) means restructuring and presenting the material so it is a partnership rather than an auto-biography.

I have 5 books that I am reading and sampling from. I do not expect others to have these 5 books in front of them, although a few may try to emulate my activities while selecting and emphasizing different topics.

The first "real" decision it to determine what I should learn first. Even a quick glance at the books around me shows that it is a good idea to spend some time reviewing the basic mathematical pre-requisites before actually engaging in activities that are considered to be "calculus". This can be tricky. You may feel that you are a tin-man, a straw-man, or a lion, and that each of these backgrounds implies both particular strengths as well as weaknesses. All this really means is that the review of pre-requisites may be different for each of us. But at some point, within a few weeks (more or less) we should all be at a similar, although not identical, point. The idea, even before we begin, is to develop a sense of the road map that we are going to follow. Even Dorothy would have benefitted from such a map. The journey may not have been quite as exciting, but there would have been fewer surprises along the way.

Let's have a look at the books I have purchased over the last few years and see how they address the issue of pre-requisites. The first two books are intended for self-study. I am using the Kelley book at the moment, so let's have a look at the topics he covers:

• linear equations and inequalities
• polynomials
• rational expressions
• functions
• logarithmic and exponential functions
• conic sections
• fundamentals of trigonometry
• trigonometric graphs, identities, and equations

Now let's examine Ash & Ash:

• functions
  • graphs
  • trigonometric functions
  • inverse functions & inverse trigonometric functions
  • exponential & logarithmic functions
  • inequalities involving elementary functions
  • graphs of translations, reflections, expansions & sums

The next two books are university text books.

Larson et al:

• graphs and models
• linear models and rates of change
• functions and their graphs
• fitting models to data

Spivak:

• basic properties of numbers
• numbers of various sorts
• functions
• graphs

One of the difficulties with such lists is that until one actually looks at the pages, one still has very little idea of the depth of explanation or the difficulty of the problems that are presented.

Just for the heck of it, here is the first problem in each of these books:

1. Prove the following:

(i) if ax = a for some number a not equal to 0, then x = 1. [Spivak, p. 13]

1. Match the equation with its graph

(a) y = -1/2 x + 2 [Larson, p. 9]

1. Let f(x) = 2 - x^2 and g(x) = (x - 3)^2. Find

(a) f(0) [Ash & Ash, p. 4]

1.1 Solve the equation 3x - (x - 7) = 4x - 5 [Kelley, p. 2]

Let's look at these four books from a different perspective. How many problems are there in the "pre-requisite" chapters. I am assuming a "pre-requisite" chapter is whatever precedes the first mention of limits.

Book	Pages	Exercises
Kelley	121	267
Ash & Ash	39	66
Larson et al	38	260
Spivak	89	104

So far, I have been working through all of the problems in the first three chapters of the Kelley book. This has involved solving a total of 95 problems. If I maintain this approach, I still have 172 problems to work through before I am "ready" for limits and the beginning of calculus.

But all of the above is bean counting.

The real issue is what mathematics is being covered. What ideas, concepts and procedures are being discussed and at what level or depth? There are at least two different ways of describing this. One is to attempt a few sentences that identifies the terms and topics. The second it to reword this in terms of goal and unit objectives that indicate what one should know or be able to do. This is the familiar, "at the end of the section you should be able to ...".

If the terms in the chapter headings are totally unfamiliar to one, then it would seem a good idea to make sure that one can read the material with understanding and that one can do many (not all!) of the problems. The problems are the litmus test. If you have "no clue" as to what the problem is saying, or if you have no idea as to what to do first, then it is a good signal that additional review of the topic is necessary. Mathematics, unlike most topics, is heavily dependent on what goes before. Certain topics are definite prerequisites to other topics.

When I sit down, as I am right now, and push myself to ask "what am I trying to learn?". I come up with three answers:

1. some concepts and ideas, often expressed in a particular notational system

2. some procedures for solving certain types of problems

3. the logic of a mathematical proof.

I also like the order of these three items. For me, they are in the right order in terms of priority.

There is also the question of level of difficulty. Books are deliberately aimed at a particular level of difficulty, which is usually based on a sense of whether the book will be used by majors or non-majors and whether it is a first-year or second- or graduate level course. This makes sense. In the case of a novice, or someone who has not been exposed to the material for 40 years, it then also makes sense to pay attention to the pre-requisites. I am finding the Kelley book to be valuable as I am generally able to handle the problems, and when I am lost, I quickly look at the solution and can usually (so far) follow the approach and move on.

Now to begin drawing a map of the territory. Using Mindmanager software:

Kelley provides the most extensive material for pre-requisites. Therefore I used it as the basic framework for this section of the map. After I have completed all of the pre-requisite material from Kelley, I will have a closer look at the other three books to see if there are some additional topics that I feel I should include.

However Kelley uses problems as the framework for presenting material. I would like to add my own notes which focus on the concepts, ideas and notation of the material. Due to the difficulties of presenting this material (notation, graphs) using technology, I will provide all of this via hand written notes. These notes will be created with a fountain pen and then scanned onto this web site.

8:45 PM

The Humongous Book of Calculus Problems (2006) W. Michael Kelley

Chapter 4 Functions

Here are my scans (original at color - photo, 150 dpi, custom size 8 x 10.5 in, export at width of 600 pixels) of the problems I completed in a one hour session this morning.