# But Why Is It Wrong?

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:

$\frac{\sin{x}}{\cos{x}}+\frac{\sin{x}}{\cos{x}} = \frac{\sin{x}+\sin{x}}{\cos {x}+\cos{x}}$

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

$\frac { 1 }{ 2 } +\frac { 1 }{ 2 } = \frac{1+1}{2+2}$

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

$\sec^2(x) + \tan^2(x) \sec^2(x) \Rightarrow \frac{\sec^2(x)}{\sec^2(x)}+\frac{\tan^2(x) \sec^2(x)}{\sec^2(x)}$

Is wrong because:

$1+2 \neq \frac{1}{3} + \frac{2}{3}$

Since

$1 + 2 = 3$ but $\frac{1}{3}+\frac{2}{3}=\frac{3}{3}=1$ which is not equal to $3$.

Or a slightly trickier one (for students):

$\sec^2(x) + \tan^2(x) \sec^2(x) \Rightarrow \sec^2(x) (\sec^2(x)+\tan^2(x))$

Is wrong because:

$2 + 2 \times 3 \neq 2(2+3)$

Since

$2 + 2 3 = 2+6 = 8$

But

$2(2+3) = 2(5) = 10 \neq 8$

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.

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2 Comments

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### 2 responses to “But Why Is It Wrong?”

1. I like it, JN. I employ a similar strategy for showing kids why (x + 5)^2 is not (x^2 + 25). Your post reminds me of the “Mistake Game”, which I heard about from Megan Hayes-Golding http://kalamitykat.com/2012/11/25/stolen-pedagogy/, who got it (in a *much* longer post) from Kelly O’Shea http://kellyoshea.wordpress.com/2012/07/05/whiteboarding-mistake-game-a-guide/ (a Physics teacher, you might be interested to know). I recommend the MHG link above in particular, because she mentions all of the favorite things she’s “stolen” from other bloggers.

Thanks as always!

2. Thanks for the idea, Steve. I actually had read about the “Mistake Game” before, and even blogged about it (somewhere). I reeeeally liked the idea for Physics and was super-pumped about it, but when we did it in class, it was a “blah” experience. I think my students just weren’t very creative and I had trouble motivating them to be more creative (examples just weren’t enough for them).

However, I had a little extra time in my advanced precal class today and I remembered the mistakes game since you mentioned it, and we did it and they totally loved it! We didn’t get around to presenting, which we’ll do Friday after the quiz, but they were totally engrossed by the idea and were super-excited by the prospect of catching each other’s mistakes (and hiding them).

So perhaps my expectations were too high for Physics, so I felt like it was a letdown? The other thing is that the “mistakes game” was a perfect follow-up activity to what we had been doing, because they were analyzing other students’ mistakes, and so were already in the right mindset to play the mistakes game. I’ll definitely be trying this more often with this group of Precalculus students!