Tag Archives: mistakes

Mistakes

I was just reading a fascinating article about the fear of making mistakes and came across this quote in the article:

“’We translated some textbook pages from a Japanese math textbook,’ Stigler told me, sitting in his office in the rabbit warren that is the UCLA psychology department. ‘There was a really interesting note in the teacher’s edition, and it said: ‘The most common mistake students will make in adding fractions is that they will add the denominators.’ Then it said: ‘Do not correct this mistake. If you correct it, they will immediately stop doing it. But what you really want is for them to take several weeks to understand the consequences of adding the denominators and why that doesn’t work.’” (p. 193)

Wow.  The rest of the article is really great, too.  I suppose in my better moments, I try to lead students to show them why something doesn’t work, but so few of my students have an intuitive understanding of why adding the denominators like that doesn’t work.  And unfortunately at this point I could give them all the time in the world and they will never understand on their own why \frac{\cos(x)}{\sin(x)}+\frac{\cos(x)}{\sin(x)} is not equal to \frac{\cos(x)+\cos(x)}{\sin(x)+\sin(x)} unless they first understand why \frac{1}{1} + \frac{1}{1} is not equal to \frac{2}{2} .  Many of my students, when I show them \frac{1}{1} + \frac{1}{1} will go “oh, I know about that”, but I really get the feeling that they don’t understand why that’s the “rule”.  Perhaps one of these days I should investigate that more closely.  If they had a teacher who taught like the Japanese math textbook suggested above, they would be far less likely to make the mistake pictured above.

Of course, I don’t blame the elementary and middle school teachers.  I was these once, I know.  I would never have the patience to teach using that method.  Heck, I don’t even have the patience to teach those age groups with any method, so please, if you don’t do that, I’m not blaming you.  I just wish I had the patience and foresight to understand when I should employ a tactic like that.  But I feel like it would require an overhauling of the US system of teaching.  Hmm, it’s late and I’ve rambled too much.

Side note: I am in the middle of grading some tests and one student, in all seriousness, has \frac{1}{1} + \frac{1}{1} = \frac{2}{2} .  Seniors in Precalculus.  Sigh.

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Mistakes Game: Gallery Mode

On a recent blog where I talked about error correction, Steve Grossberg reminded me about the Math Mistakes game.  I had used this once before in Physics, but it was less of a success than I had hoped for (I was really excited about it, but the “game” just flopped in class).  However, I hadn’t tried it with my advanced Precalculus class, and I must say that it went over pretty well!

It’s great for the setting of proofs in trig identities, however I had my students create the math mistakes on your standard 2′ x 3′ whiteboard.  Then, the next day, as they finished the quiz, they went one by one over to the “Math Mistakes Gallery” and looked for mistakes.  When they discovered a hidden mistake, they would write it down on their paper.  Once everyone got a chance to look through the gallery, we came back together and discussed what people found.  I even tallied up how many people found each mistake, and the “winner” was the person whose mistake was hidden the best.  Here’s the winner:

A Tricky Mistake!

A Tricky Mistake!  Also hard to read…

We even had a discussion about how good the mistakes were and what made them good!  We decided that the winner won because he put the mistake during multiplying the binomials, and students don’t like to multiply binomials (at least they admit it).

This variation–“Gallery Mode”–is good because it allows students to work at their own pace, AND it allows students to work and think the entire time.  Nobody discovered all of the mistakes, and almost everyone got to at least read all of the proofs.

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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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