An Example of a "Learning Process" Journal (using the 2 colored box format)

Book: The Nature of Mathematics 10th ed by Karl Smith.

Source: Toronto Thomson Brooks/Cole, 2004.

This is my second day where I am spending a clear hour making 'notes' on Learning more about mathematics. It is 7:15 am (Sunday). I am wide awake and eager to begin. Yesterday did not quite work out as I hoped. After I finished cutting up the firewood and putting it away, I simply didn't have the motivation or energy to get back to the maths.

I now have the book open to the Prologue Problem Set (p. 17-20). There are a total of 60 problems. Clearly I am not going to attempt them all. I am not sure whether I will spend more than an hour on these, but I will set aside the next hour and see how far I get.

The first decision is to decide which problems are most worthwhile for me to try. Thr first 8 problems ask one to write something "In your own words". The remainder appear to be teasers - that get you thinking about problems before beginning the book.

1. IN YOUR OWN WORDS. The title of the prologue begins with "Why study math?" Take a clean sheet of paper and draw a vertical line down the center. In the left column, write down reasons why you should study math, and in the right column, write down reasons why you should not study math. Spend at least 20 minutes filling in reasons. Finally write a summary sentence or two describing what you found. (p. 17).

Why:
1. it is fun
2. it will be good to review the basic ideas since I have not actually tackled a formal math textbook since 1965!
3. this will be a vehicle for testing my own approach to Learning
4. this could lead to a new hobby: recreational mathematics
5. this particular book appears to focus on the basic ideas of various topics. I will actually learn some foundational topics in depth.
6. this may lead to reading other math books that are presently too advanced for me.
7. it will likely remind me of what it feels like to be a student - a serious learning agenda, occasional frustration when I run into difficulties, finding the time given other priorities.
8. I like a good problem, and there are obviously many in this book.
9. I am looking forward to viewing many of the web sites on the topics.
10. Perhaps this meta-learning activity will be of interest to myself as I proceed, as well as to others who happen to see it.
11. The book appears to be very well organized and presented. I am looking forward to seeing it from a pedagogical perspective.

Why not:
1. it will be a commitment to time (time spent on this is time not available for something else).
2. I doubt this will be very useful, since I do not seem to encounter math problems in my daily life that I am unable to handle.
3. All of the topics appear to be fundamental and therefore important from a conceptual perspective but they do not appear to be advanced as in senior undergraduate or graduate level.

Summary sentence: I am clearly excited about the venture: the pluses outweigh the negatives.

The first exercise is much like this overall web site. A time for reflection. I did the activity because I believe there is value in such thought, although at some point one must also focus on the topic and the attendent problems.

Now to pick another problem. The remaining 8 problems are all about the various chronological periods of mathematical history. Interesting but I want to move on. Here is the first math problem:

10. A long straight fence having a pole every 8 ft is 1,440 ft long. How many fence poles are needed for the fence?

The key to this problem is to realize that there is a pole at each end of the fence. This means that there is one more pole than the actual length requires. Now to divide 1440 by 8 and see if is an exact multiple of 8. It is: 8 x 180 = 1440. Therefore I need 180 + 1 = 181 poles.

The first thing I notice is that the problem is phrased in the American system of measurement rather than the metric (SIU - Standard International Units).

There is just a hint of apprehension as I obtain my answer. Now to glance at the end and see if it agrees with the text. Yes. Good. Fascinating! Even though I was confident of my thinking, having to match my answer with the text was anxiety provoking. I did not want to be wrong.

I then began looking at the problems to select another. I immediately focused on the problem with an interesting diagram (15). Unfortunately I also happened to see the answer and that ruined it for me. Having said that, I now realize that the answer for this is not as important as having a clear reason for it. The problem says that it is impossible to trace a route without going over some portion of the route twice.

15. Given the diagram, is it possible to choose a route so that all the permitted streets are traveled exactly once?

The first step (for me) is to redraw the map on a piece of scrap paper. It is easy enough to trace the perimeter, but that still leaves 3 segments. All three of these segments have one point in common. Since this is an odd number, there is no way to travel all three without retracing some path.

I began by drawing a schematic diagram that only captured the geometry of the problem. This was a square, a rectangle and two triangles. That was an important first step since it removed all of the extraneous detail. I tried a few possible paths and then I suddenly realized that I could make a distinction between interior and exterior lines (I have no idea why I noticed this!). However once I noticed this I then realized that there were only three interior lines, and that they all met at a single point. From that observation, I immediately realized that the path was impossible.

I liked this problem! It was not computational, nor algorthmic, but required some form of insight. From here one could begin to see if one could draw other diagrams and eventually come up with some general principles. I believe that this is all part of a branch of mathematics called graph theory, which I have never studied, but which is discussed in the book at a later point.

8:20 am I have spent an hour doing the problems and found it very enjoyable. I think that the purpose of these is to introduce one to many of the topics to be discussed later. I would like to spend more time on these problems, because they are fun, but I am also conscious of the time, and feel that I should press on. I will begin Chapter 1 next.

Mathematics Index