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.

It is 5:50 am (Wednesday). I am feeling quite energetic this morning.

Perhaps it is due to the fresh snow (unlikely) or to the fact that I am now organized for three different LEARNING ventures (Math, Afghanistan, Art History) or to the fact that we are beginning to make progress with the online data analysis course I am working on.

Now to get my coffee and continue with the problems for section 1.1 Problem Solving.

I have reviewed most of the Level 3 problems. Many of them I recognize and can solve in my head. A few simply don't seem to be worth the effort. I think I will move on and look at the Problem Solving problems.

Problem Solving

58. Thoth, an ancient Egyptian god of wisdom and learning, has abducted Ahmes, a famous Egyptian scribe, in order to assess his intelectual prowess. Thoth places Ahmes before a large funnel set in the ground. It has a circular opening 1,000 ft in diameter, and the walls are quite slippery. If Ahmes attempts to enter the funnel, he will slip down the wall. At the bottom of the funnel is a sleep-inducing liquid that will instantly put Ahmes to sleep for eight huse if he touches it. Thoth hands Ahmes two object: a rope 1,006.28 ft in length and the skull of a chicken. Thoth says to Ahmes, "If you are able to get to the central tower and touch it, we will live in harmony for the next millennium. If not, I will detain you for further testing. Please note that with each passing hour, I will decrease the rope's length by a foot." How can Ahmes reach the central ankh tower and touch it?

1) understand the problem.

What information is important? The chicken head could be used as a weight and then Ahmes could try to throw the head through the opening of the ankh. But this is a distance of 500 ft. which is much further than the length of a football field. Impractical. Ignore the chicken head for the moment at least.

Why is the length of the rope given so precisely?

2) create a plan.

Suppose Ahmes simply walked around the circumference of the funnel. This would solve the problem since the rope could then be pulled tight and it would form a "bridge" from the ankh in the center to the ankh where he is standing as he begins.

3) carry out the plan.

What is the circumference of the funnel? C=2pr

C = 2 * 3.14 * 500 = 3140 ft.

4) create a new plan

As before. The rope does not have to remain on the circumference.

5) carry out the plan

Thus Ahmes only needs to keep it fairly taut as he walks around the circumference.

6) Check.

The rope needs to be 1000 ft long when he is at the furthest point, so he has an extra 6.28 ft to play with. No problem.

I was surprised at how "easy" it was, once I thought of the idea of walking around the circumference.

6:25 am I still have half an hour of my alloted time for this LEARNING activity. I will begin reading the next section.

1.2 Inductive and Deductive Reasoning (p. 16 - 24)

"Studying numerical patterns is one frequently used technique of problem solving." (p. 16)

"Given a difficult problem, a problem solver will often try to solve a simpler, but similar, problem." (p. 17)

Not a long section. The material is only 6 pages.

I have just noticed the sub-section titles:

• A Pattern of Nines
• Order of Operations
• Inductive Reasoning
• Deductive Reasoning

The order of operations is ingrained in me, so this is very familiar.

"... first observing patterns and then predicting answers ... is called the inductive method" (p. 18)

"... deductive reasoning involves reaching a conclusion by using a formal structure based on a set of undefined terms and a set of accepted unproved axioms or premises. The conclusions are said to be proved and are called theorems." (p. 19)

Once again, this is familiar territory.

I have glanced at the problems at the end of this section and they all appear fairly straight forward, so I think I will give them a pass, at least for the moment.

I will begin section 1.3 Scientific Notation and Estimation next time.

6:50 am

I have picked up my pace as I am finding the material familiar at the moment. There is no point in being too pedantic. It is a fairly thick book and I am sure I will soon have to slow down and think more carefully.

Having said that, it has been useful to review the following topics:

1) Polya's 4 steps for problem solving
2) the difference between induction and deduction
3) the importance of looking for patterns in numbers.

I may have skipped this latter topic too quickly. Thus I do not have some of these patterns "in my mind" as "facts". If I did, then I might recall them at a later time when they could provide me with some alternative insights into potential patterns.

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