# 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.