Date: Oct 28, 2012 9:21 AM Author: Robert Hansen Subject: The Teaching of Fractions Here is a link to what H.Wu wrote last year regarding the teaching of fractions and the Common Core Standards...

http://math.berkeley.edu/~wu/CCSS-Fractions.pdf

I disagree with much of it.

When I think of teaching a subject I start with the final goal in mine, stated in the simplest terms possible. For example, if I were planning a section in probability (for algebra students) my goal is to arrive at...

1. Probability is the ratio between the number of ways an expected event can occur and the number of ways any event can occur.

2. The difficulty in determining probability lies in determining the number of ways an expected event can occur, the number of ways any event can occur, or both.

So when I think about teaching fractions, I have a goal...

1. A fraction is a number, represented by the quotient of two numbers, the top number being the numerator and the bottom the denominator.

2. Fractions can be ordered (placed on a number line) and two fractions are equal if their quotients are equal (the numerators and denominators can be different).

3. Fractions can be added, subtracted, multiplied and divided, just like numbers, but the mechanics are more involved because fractions involve the quotient of two numbers.

In Wu's defense, he is using the common core standards as a guide so he was bound to screw this up. The common core standards are what I call 3rd generation standards. The first generation started in the late 80's through the 90's (before the NCLB made it a national past time), the second generation started in the 2000's during the heyday of NCLB and 3rd generation (the Common Core Standards) evolved from all of that.

As a collection of topics the standards are (just) ok. If you or I sat down and listed all of the micro points to arithmetic we would come up with a similar list. The problem with the standards is sequencing. I and others have said this a number of times before. The sequencing is not systematic in its progression. Somewhere during the evolution of "standards making", someone made a very bad judgement call and it became popular. They decided that these topics should be arranged in threads, simultaneously occurring at the same time. So today, the students (and Wu) are tasked with up to 6 threads all occurring at the same time. Number, operation, algebra, geometry, fractions, reasoning, etc. It's like eating a 6 course dinner, not one course after another, but all at the same time. It is very difficult for a student to reverse engineer all of this back into a meaningful and fulfilling meal of mathematics. Either they are very bright, they have parents or tutors to reverse engineer it for them, or they fail. Sadly, the only option available to most is the last one.

Back to fractions...

I would not start teaching fractions until multiplication and division (with whole numbers) are firmly established.

After that is established the steps are rather straightforward...

1. Introduce the concrete idea of a fraction using pizzas, rectangles, etc.

2. How much is 1/5th of 20? How much is 2/5th of 20? Is 2/5th of 20 bigger than 1/5th of 20? Is 2/5th bigger than 1/5th?

3. How much is 1/5th plus 2/5th of 20?

4. How much is 1/10th of 20? How much is 1/2 of 1/5th of 20? Is 1/2 of 1/5th the same as 1/10th?

5. How much is 2/10th of 20? How much was 1/5th of 20?

6. How much is 1/4th of 12? How much is 1/3rd of 12? Which is bigger?

7. How much is 1/4th plus 1/3rd of 12?

The only thing I didn't cover is division by a fraction. That we withhold till the students are comfortable and fluent in the above operations. Also, we have decimals coming up, which I would delay until the later stages of the above.

If you don't believe that these standards are a mess for a student to navigate, look what they did to Wu (who I adore by the way).

Bob Hansen