*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 our so unaccustomed to fractions that the 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 from the Patricia Kenschaft article, "Racial Equity Requires Teaching Elementary

School Teachers More Mathematics" (Notice 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*getting the division 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.