Mathematics Chronology

6:40 am I continue to play. My math books are all back on the shelves. I want to begin setting up a concept math that is a blend of my sense of the conventional structure of mathematics, combined with topics that are of special interest to me.

Here is a map that I made yesterday:

One idea is to place a date-stamp in the bottom right corner.

I have reorganized some of the categories to improve the display. More importantly, I have added 3 high priority sections that highlight my current interest and activities. These revolve around three books: Fearless Symmetry which is about prime numbers and galois group theory, The Road to Reality which is about the relationship between mathematics and physics, and Calculus, which suprisingly is also about numbers.

The small + signs in front of some of the topics indicates that they can be expanded to show sub-topics (when one is using the MindManager software).

I now have a clear map of what I am Learning. I can see at a glance where my three primary areas of interest fit into the big picture.

Now to fit in one more activity: Harold Jacobs book "Mathematics: A Human Endeavor". Now that I have 4 activities identified, I need to change the priority icons to a rank ordering.

So far, so good. I am delighted with these maps. I have a much better sense of context now. The remaining task is to actually make some notes for each of these four chapters.

Harold R. Jacobs (1970) Mathematics: A Human Endeavor Chapter 1 The Mathematical Way of Thinking Lesson 1 The Path of a Billiard Ball This lesson is one of my favorite activities! It brings back memories of when I created a simulation of this activity using Logo. I recall spending hours playing with different variations (shape of table, angle and starting position of ball) and trying to predict what would happen next. Looking back on this now, it would have been about 20 years ago, I would have been well advised to have paid closer attention to Jacobs first three lessons. My approach was quite exciting, in large part because the computer did the actual graphing so I could quickly try different experiments. Jacobs' approach uses pencil and graph paper and is much slower and more tedious. But Jacobs acts like a mathematician whereas I acted like a boy enthralled with the technology. The difference is nicely captured by a brief article by Dave Pratt (Micromath Spring 1991, p. 28-9) where he described working with a class of children using Logo to draw fractal trees. "The idea was just as stimulating as I had hoped. The children produced some very impressive trees, and a forest developed on the classroom wall. ... So why did I feel uneasy? The children experimented freely but with little planning or control. They were becoming excited about the product, but I was becoming increasingly concerned about the thinking process. ... some of the best looking work was produced by children who had little understanding of the procedure itself." This time I will try to act like a mathematician. It is not too late. There are two issues to keep in mind while exploring the path of a billiard ball. One has to do with the variety of possible situations, and the other has to do with keeping track of what happens. One might begin, as Jacobs does, by simply trying a few possibilities. He restricts himself to rectangular tables with integers for the lengths of the sides, and to only 45 degree paths and a starting position of the lower left corner. Even so, the variety of rectangular shapes makes a wide ranging different types of paths. Underlying all of this is something critical. Curiosity. One must be intrigued by the situation. One needs to spontaneously generate questions. Jacobs tries to give a few examples. For the 8 different rectangles he provides, he asks: On which table does the ball have the simplest path? Can you explain why? What do you notice about the paths on tables 5 and 6? Can you explain? Do you think the ball will always end up in a corner? If the ball starts from the lower-left corner, do you think it can end up in any of the four corners? Jacobs then goes on to comment on the complexity of the situation. He notes that the shape of a rectangular table depends on the width and the length and invokes a strategy that would be approved by both Piaget (formal operations stage) and Polya (think of a simpler problem). He keeps one variable constant and explores the effect of varying the other. 9:10 am I will continue with this later ...