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 5:40 am (Monday). I am feeling quite energetic this morning. It is still dark outside.

Now to get my coffee and continue with the problems for section 1.3 Scientific Notation and Estimation.

The BC cartoon is a nice piece of logic.

• "The sun is 93,000,000 miles away.
• "How do you know that?"
• "If it were further away, there would be more than 365 days in a year." (p. 25)

Once again, I am familiar with scientific notation, but the section on calculators caught me by surprise.

• There are 3 types of calculator ( arithmetic, algebraic and RPN). These are classified by how they handle the order of operations.
• There are also 3 types of notation for writing large numbers: fixed point, scientific and floating point.
• 745 (fixed point form) is written as 7.45 02 (floating point form). The exponent is written as an integer following the base.

It took a few minutes, but I finally realized that RPN stands for Reverse Polish Notation. This is normally used in advanced courses on symbolic logic. Why do we not see it in "regular math"?

I have just added a note to myself to add a comment on RPN in the NCTM paper I am writing on notation.

In addition to the three forms of notation mentioned in the book, there are the many different conventions for different programming languages. Floating point form (which is new to me) is necessary to accommodate the small screen of a calculator. Similarly, the conventions of programming languages are to accommodate the restrictions imposed by computers which require unambiguous expressions. As humans we are more flexible (e.g. whether multiplication is specified by an x, a dot or is implicit). But this may be part of the difficulty for young children - we are inconsistent. But learning to deal with inconsistency is also important. However we should at least make this inconsistency explicit!

The section on estimation was familiar but the example on the grains of wheat on a checkerboard caught me by surprise. It was not the math, but the extra effort the author took to see how large the answer was: how large is a grain of wheat, how large is a bushel and what is the annual wheat production for the US!

1.3 Scientific Notation and Estimation (p. 36 - 39)

Level 1

1. In your own words. What do we mean by exponent?

An exponent is a value that indicates the number of times that the base occurs when multiplied by itself. (i.e. it is the number of factors of the base). Alternatively it is the (number of times - 1) that the base is multiplied by itself. This can get tricky if the exponent is not an integer.

I really like question 1. It forces the issue of whether one really understands the idea.

1.3 Scientific Notation and Estimation (p. 36 - 39)

Level 1

24. How many hours are there in 365 days? (estimate)

24 x 365 is about 25 x350 is about 1/4 of 35000 which is 9000.

The last step was surprising. I suddenly realized that 25 is the same as 1/4 of 100 and that this fact would help me simplify the expression.

6:40 am

I want to spend another hour on these problems.

Mathematics Index