I know on the last post I said that I only had a few good ideas last year, and I’ve just stumbled upon another one. Last year I realized I was resorting to showing *why* a certain operation was wrong by using just integers. For example, sometimes I would see something like:

So, after cringing, I would ask something like “Is this true?”

Edit: Just fixed the equation above to reflect what I meant. I got lost in the latex of it all (and for some reason, the latex parser in wordpress doesn’t like when you copy and paste latex from another site–I’ve had the *exact same* latex work and not work, one right above the other!). Thanks to Steve Grossman for the spot!

They (usually) would recognize their mistake and add the fraction the right way. I found myself doing this so often, that I decided to create an activity where they corrected mistakes (these were real mistakes I found on tests and quizzes–though I didn’t tell that to last year’s group because they were the first class I taught Precalculus to, so they’d realize it was their own mistakes!!) and showed *why* they were wrong using small integers. Here are some examples

Is wrong because:

Since

but which is not equal to .

Or a slightly trickier one (for students):

Is wrong because:

Since

But

As good as the exercise sounds to me, I believe I failed in it last year, mostly because I did not provide enough structure or examples. This year I have plenty of examples, and I am going to have to figure out how to provide more structure for the students. I’ve heard that “error correction” is great for students, and I really think this extra step of *understanding* the error correction is essential, so I really hope that it goes over well!

EDIT: So I’m posting this after I did the exercise, and it went *awesome!* Students were presenting the mistakes and explaining thoroughly why they were incorrect, even going so far as to *explaining what they thought the student was thinking when they made the mistake!* I now see why this kind of error correction is invaluable. The highlight of my day, though, was watching as one group of students (we’ll call them Jack and Jill) was presenting, Jack was explained the problem quickly, and Jill was *watching the other students in the class and looking for comprehension.* When she realized that they didn’t follow Jack’s thought because it went too quickly, Jill stepped in and asked “you didn’t get that, did you?” and proceeded to explain the problem more thoroughly. My students really are becoming teachers. And I’m just sitting back and watching them. Awesome.