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First understandings: the meaning of "half"
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 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.
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 of a grapefruit is bigger than half of a grape, tells us only about grapefruits and grapes, not about half as a number. It is, after all, equally true that seven of the grapefruits is bigger than seven of the grapes. (And half 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. 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 half is a number; the word halves continues 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
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.)
- numbers have many names
- recognizing and finding equivalent 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
- addition, subtraction
- proper and improper: again a distracting idea when all we mean is <1 or >1
- mixed numbers: we don't claim 0.5 and 2.5 are different kinds of numbers, so treating 1/2 and 2 1/2 as different is also a distraction.
- unit fractions: unit fractions are an essential building block: the meaning of three eighths or twenty-three eighths depends on knowing what one eighth is
- 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.