When Simplification Obscures

Wednesday, July 22, 2015

John Cook makes an interesting point regarding a rule you may have had drummed into your head when you first learned fractions:

[It] serves some purpose in the early years, but somewhere along the way students need to learn reducing fractions is not only unnecessary, but can be bad for communication. For example, if the fraction 45/365 comes up in the discussion of something that happened 45 days in a year, the fraction 45/365 is clearer than 9/73. The fraction 45/365 is not simpler in a number theoretic sense, but it is psychologically simpler since it's obvious where the denominator came from. In this context, writing 9/73 is not a simplification but an obfuscation.

Simplifying fractions sometimes makes things clearer, but not always. It depends on context, and context is something students don't understand at first. So it makes sense to be pedantic at some stage, but then students need to learn that clear communication trumps pedantic conventions. [emphasis in original]
This touches on something I have noticed as a parent of an increasingly inquisitive toddler: The need to focus on one lesson frequently requires setting aside a wider context so the matter at hand can be held in mind. It is perhaps harsh to call reducing fractions pedantic, but there is a serious issue here. Teaching such rules as if they must always be followed or come from a vacuum discourages subsequent questioning and integration with other knowledge. A full explanation is likely impractical at the time, but perhaps teachers should more often say something like, "We will be doing things this way because it makes these lessons easier to learn."

-- CAV


Today: Changed "of the time" to "at the time" in last sentence. 


Anonymous said...

Decimal format (0.123) is even better so that fractions can be compared to each other. Unless the denominator is small (2/3) or standardized (90°, 45/365 days) you probably shouldn't use fractions. Who can do mental math with numbers like 29/93?

Gus Van Horn said...

Maybe, maybe not, since decimals combines reduction with conversion to a denominator of 10, 100, 1000, etc. Even non-standard numbers like 93 can be easier in, say, a shared context of things that regularly occur 93 times. (Maybe a company finds it convenient to package 93 of something for some reason or another...)

It all depends on context (as above and in terms of what normal humans can process) and the mathematical background/ability of the audience. Time and again, I see examples of people who don't really understand what "percent" means, and that's arguably better than a decimal since those are whole numbers (or easily-enough rounded to them), and easier to read than decimals. I think Cook's main point is that we have to keep context in mind when using numbers, and that doing things the same way every time can get in the way.