# Fractions

A number that names part of a whole or part of a group.

## First understandings: the meaning of "half"

### A part

Most children use the word "half" even before they enter school, though not with its precise mathematical meaning. In casual use, the word roughly means "part" of something that is being shared. Young children might well use the word "half" as they share unequally, or even among several people: "half for me, half for you, and half for mommy."

Aside from the imprecision, this first use of "half" treats it as "half *of* something." The first school representations children typically see tend to support the same idea: fractions are still presented as a part of something.

But *half* and *half of* are not the same.

### A number

**A fraction is a number:** *One half* is a number, just as *two* and *seventeen* and *ninety-eight point six* are numbers. *One half* is more than *zero* and less than *one* (and, in fact, half way between those other numbers). *One and a half* is another number, half way between *one* and *two*.

Likewise, *seventeen* is a number that always comes between *ten* and *twenty* (in fact, precisely *seven tenths* of the way from *ten* to *twenty*) and midway between *sixteen* and *eighteen*.

And just as we can refer to *one* seven, *two* sevens, *three* sevens, and so on (with the values 7, 14, and 21 respectively) and *one* hundred, *two* hundreds, *three* hundreds, and so on (with values 100, 200, 300...), we can also refer to *one* half, *two* halves, *three* halves, and so on, and understand their values by looking at their position on the number line (or by some computation).

Each number represents a fixed amount regardless of how it is being used. For example, *seven* doesn't change its value depending on whether it refers to grapefruits or grapes or millions or tenths: in each case it tells how many of the objects (grapes or tenths) we are talking about. The same is true of *one half*.

*Half of something*, though, is not always a number. And noticing that half

*is bigger than half*

**of**a grapefruit*, tells us only about grapefruits and grapes, not about*

**of**a grape*half*as a number. It is, after all, equally true that seven

*is bigger than seven*

**of**the grapefruits*. (And half*

**of**the grapes*of a grapefruit*is not a fraction! It is a

*thing*, not a number. It is breakfast. It is a fraction

*of*the food we might have during the day. But it is not a fraction, itself.)

Is it a distraction to teach "not all halves are created equal"?We think so. Some school curricula go through special effort to teach children that not all "halves" are the same size (i.e., that half a grapefruit is "more" than half a grape). We know of no solid research supporting or rejecting this teaching practice, but we hypothesize that putting effort into teaching this delays children's understanding of fractions.^{[1]}For one thing, children already know that if they're told they can have half a brownie, they should look for the biggest one to split. They also know that if the pieces come out unevenly, they'll want the "bigger half." So, at the minimum, it wastes time to teach this because children know it already. More problematic is that it hides the new idea:one halfis a number; the wordhalvescontinues to focus on the already familiar idea about parts of objects.

**In all ways, fractions behave like other numbers** because they *are* just numbers! The special way in which they are *written* requires us to manipulate the symbols differently in order to perform the ordinary operations of addition, subtraction, multiplication, and division, but the operations, themselves have exactly the same *meaning* for fractions as for any other numbers. (See more details on arithmetic with fractions below.)

to be continued

## Mathematical ideas

*unwritten from here on...*

- unit fraction: developmentally early because invokes the notion of sharing (parts of) objects rather than number; historically early, too (brief ref to Egyptian fractions)
- If we know what 1/5 is, 2/5 is counting fifths (like 2 apples or 2 elephants):
- counting units (in this case unit fractions) is very basic
- closer to "number" but still retains sense of parts
- helps comparison
- reflects the notation: 3/4 and 0.75 are the same number, but 3/4, as a notation, directly counts units and indicates the size of the unit. We have to think about 3/4 to recognize and compare its magnitude with, say 2/3. By contrast, 0.75 directly indicates magnitude, and we'd must think harder to recognize the kinds of units it might be counting.

- fraction of a number (including fraction of a fraction); fraction of a set; fraction of a shape
- Visualizations including number line, bars, circles, manipulative materials
- why
*not*to treat mixed number and fraction differently at early stages (We don't classify 0.5 and 2.5 as different "kinds" of numbers or give them different names; why treat 1/2 and 2 1/2 differently? This is a focus on notation and algorithms, not on meaning, and may impede understanding.) (Some evidence, but needs solid research.)

### Equivalent fractions

- numbers have many names
- recognizing and finding equivalent fractions

### Comparing fractions

- comparing fractions to 1
- comparing fractions to 1/2
- fractions that add to 1
- comparing fractions by looking at numerator
- comparing fractions by looking at denominator
- converting to equivalent fractions with common denominator

### Arithmetic with fractions

- multiplication
- addition, subtraction
- division
- reciprocals

## Terminology

- proper and improper: again a distracting idea when all we mean is <1 or >1
- mixed numbers
- unit fractions

## Special topics

- complex fractions: actually quite
*natural*for children to talk about spontaneously "I've shaded two-and-a-half fifths." Can write it, and the notation is "normal" enough. Mathematically valid. - continued fractions
- Egyptian fractions
- conversion to and from decimals
- ref to decimals article with headline suggesting that the teaching of decimals
*not*be rooted in fractions, but that decimals should be taught afresh, focusing on magnitude first, and only later connecting back to fractions. (Some evidence, but needs solid research.)

## Language and mathematics

- Definition including etymology and taxonomy of types (see topics)
- Developing mathematical language including associated vocabulary

## Psychology and pedagogy

- Cognitive development of underlying and related concepts
- Earlier grasp of half; earlier grasp of unit fractions
- Common difficulties and misconceptions with reference to research

## History and connections

- History of fractions
- Connections to other topics in mathematics, science, social studies, literature (story books), etc.

## References

- ↑ If you know of research, pro or con, please add it here or contact us. If you are a researcher, validating or invalidating this common practice would contribute greatly to teaching.