kitchen table math, the sequel: we begin

## Wednesday, June 23, 2010

### we begin

Deﬁnition of a Fraction

Recall that a number is a point on the number line (§5 of Chapter 1). This chapter deals with a special collection of numbers called fractions, which are usually denoted by m/n, where m and n are whole numbers and n ≠ 0. We begin by deﬁning what fractions are, i.e., specifying which of the points on the number line are fractions. The deﬁnition will be both clear and simple. If you ﬁnd it strange that we are making a point of giving a deﬁnition of fractions, it is because this is something thousands (if not hundreds of thousands) of teachers have been trying to get at for a long time. Most school textbooks and professional development materials do not bother to give a deﬁnition at all. A few better ones at least try, and typically what you would ﬁnd is the following:

Three distinct meanings of fractions — part-whole, quotient, and ratio — are found in most elementary mathematics programs.

[snip]

Such an explanation is unsatisfactory for several reasons. To say that something you try to get to know is three things simultaneously strains one’s credulity. For instance, if I tell you I have discovered a substance that is as hard as steel, as light as air, and as transparent as glass, would you believe it? Another reason for objection is that a fraction is being explained in terms of a “ratio”, but most people don’t know what a ratio is.2 In addition, while we are used to the idea of a division a ÷ b where a is a multiple of b (see §3.4 of Chapter 1), we are not sure yet of what 2 ÷ 3 means. So to use this to explain the meaning of 2/3 does not seem to make sense. Finally, we anticipate that fractions would be added, subtracted, multiplied and divided, and it is not clear how one goes about adding, subtracting, multiplying and dividing a part-whole, or a quotient, or a ratio.

This is why we opt for a deﬁnition that is both simple and clear.
(pdf file)
H. Wu
Department of Mathematics #3840
University of California, Berkeley
Berkeley, CA 94720-3840

Anonymous said...

Catherine,

This is Chapter 2 ..... chapter 2 of what??? I've got to get my hands on the rest!!!

Richard I

Catherine Johnson said...

Hi Richard - This is a chapter of the textbook Wu is writing for teachers (to be used in education school courses, that is).

I see 2 chapters posted on his website: Hung-Hsi Wu

There is also a 60-page paper called The Mathematics K-12 Teachers Need to Know

Bostonian said...

"We begin by deﬁning what fractions are, i.e., specifying which of the points on the number line are fractions."

I don't see how such a definition is going to help students in the early elementary grades, since the only points on the number line that are not fractions are "irrational" numbers such as pi, e, and square roots such as sqrt(2), and I would not expect students to learn about those kinds of numbers before they learn about fractions.

Allison said...

This was the material at the Middle School Mathematics Institute, for *teachers* grades 4-8.

So first, this isn't for *students directly* per se. It's for teachers who then learn how to adapt their material to their classroom, in age-appropriate ways.

So what early elementary grades are you referring to? If you mean k-3, then yes, you don't have to introduce it as yet on the number line.

But I don't understand your point at all about what is and isn't on the number line, because the number line isn't rationals, it's reals. Beyond that, I don't understand, because k-3 students should be taught numbers on the number line, and learn how the operations work for those numbers.

No one is suggesting that a fraction is defined as "any point on the number line", since that's false.

He defines a fraction in a specific way. That specific way wasn't something catherine posted. Is that the problem?