What is a Fraction?

A couple of Calgary Math Ed students have announced on LinkedIn that they are starting a Concept Study of Fractions.

This is a good idea as the topic is often challenging for students and I suspect that one reason lies in its language.
“A fraction is part of a whole” is consistent with the use in chemistry and with the concept of “proper fraction” sometimes introduced in math classes. Some would interpret “fraction” as equivalent to “ratio” (including ratios of irrationals) but others restrict to ratios of integers and some of these would include 3/2 as a fraction but not 6/3 (with the idea that a fraction is a rational number that is not an integer). And yet others would deny that a fraction is a number at all – with the word “fraction” referring to a way of naming a number rather than the number itself.

The difficulty of a not insignificant jump in the understanding of quantity is compounded by seeing all these various usages at the same time.

(and not even just from different sources – many teachers are themselves confused enough to keep switching from one to another in the course of essentially the same discussion)

On the other hand the desire for precision can lead to an apparent arbitrariness which is also off-putting to students. So I suspect that another major problem is the fact that some teachers, in their justified zeal to emphasize the need for precise definitions, often fail to emphasize the equally important issue of *scope* of a definition.

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