## 2016-12-05

### Observed Belief That 1/2 = 1.2

Last week in both of my two college algebra sections, there came a moment when we had to graph an intercept of x = 1/2. I asked, "One-half is between what two whole numbers?" Response: "Between 1 and 2." I asked the class in general for confirmation: "Is that right? One-half is between 1 and 2, yes?" And the entirety of the class -- in both sections, separated by one hour -- nodded and agreed that it was. (Exception: One student who was previously educated in Russia.)

Now, this may seem wildly inexplicable, and it took me a number of years to decipher this. But here's the situation: Our students are so unaccustomed to fractions that they can only interpret the notation as decimals, that is: they believe that 1/2 = 1.2 (which is, of course, really between 1 and 2). Here's more evidence from the Patricia Kenschaft article, "Racial Equity Requires Teaching Elementary
School Teachers More Mathematics"  (Notices of the AMS, February 2005):
My first time in a fifth grade in one of New Jersey’s most affluent districts (white, of course), I asked where one-third was on the number line. After a moment of quiet, the teacher called out, “Near three, isn’t it?” The children, however, soon figured out the correct answer; they came from homes where such things were discussed.

Likewise, the only way this makes sense is if the teacher interprets 1/3 = 3.1 -- both visually turning the fraction into a decimal, and reading it upside-down. We might at first think the error is the common one that 1/3 = 3, but that wouldn't explain why the teacher thought it was only "near" three.

The next time an apparently inexplicable interpretation of a fraction comes up, consider asking a few more questions to make the perceived value more precise ("Is 1/2 between 1 and 2? Which is it closer to: 1 or 2 or equally distant?" Etc.). See if the problem isn't that it was visually interpreted as decimal point notation.

1. One of the "life hacks" I've developed is thinking of the word "for" as a reverse-order division operator, in other words if, at the dupermarket, I see "3 for $5" I think 5/3 or$1.67 as the price if I buy one. The strategy of the dupermarket is information overload...it is necessary to have some mental macros that translate the language of trade into the rational language that is mathematics...