# College Math Teaching

## September 24, 2012

Filed under: editorial, how to learn calculus, mathematics education, pedagogy — collegemathteaching @ 9:14 pm

This is going to sound a bit banal, but I think that this is sometimes overlooked.

College students are sometimes under time pressures; therefore it is common for them to view a homework assignment as a task to be “checked off” when completed and a hoop to jump through for a grade.

But that is exactly the wrong approach to take with regards to homework.

Homework is designed to help the student learn the material; that is, the student who does a homework problem should be just a bit smarter and more knowledgeable after completing the assignment (and each problem, for that matter!) than they were prior to starting the assignment.

So, my advice to doing the problems: be sure to do a few problems with your book shut and notes closed; that is how you learn if you really know the stuff or not. Then ask yourself: “why did I do the first step?” “Why did I do the next step?” “Why this approach and not another one?”

After doing the problem (or a set of problems), ask yourself: “what did I learn from this?”

Then, when it comes to review, I suggest writing down some problems on sheets of paper, scrambling the sheets, and then shutting the book and closing the notes. Why?
Many times, 75 percent of the problem is knowing WHAT technique to use.

## February 7, 2012

### Forgotten Basic Algebra: or why we shouldn’t rely on the “conjugate trick”

Filed under: basic algebra, calculus, derivatives, elementary mathematics, how to learn calculus, pedagogy — collegemathteaching @ 7:01 pm

I’ll admit that, after 20 years of teaching at the university level, I sometimes get lazy. But…as I age, I must resist that temptation even though at times I find myself muttering “I don’t have 30 extra f*cking minutes to figure out how to do this…”

But often if I stick with it, it doesn’t take 30 “f*cking” minutes. 🙂

Here is an example: I was trying to remember how to calculate $lim_{z \rightarrow w} \frac{z^{1/3} - w^{1/3}}{z - w}$ and was trying to remember instead of think. I looked at an old calculus book…no avail…then I was shamed into thinking. About 2-3 minutes later it struck me:
“you know how to simplify $\frac{u - v}{u^3 - v^3}$ don’t you?”

Problem solved…shame WIN.

of course things like $lim_{z \rightarrow w} \frac{z^{7/8} - w^{7/8}}{z - w}$ are easily converted to things like $\frac{u^7 - v^7}{u^8 - v^8}$, etc.

This leads to another point. Often when we teach $lim_{h \rightarrow 0} \frac{\sqrt{x + h} - \sqrt{x}}{h}$ we use the “conjugate trick” which only works for square roots. The above method works for the other fractional powers.

## May 10, 2011

### Non-portability of mathematical skill

On my calculus final exam, I gave two questions about a metal plate of uniform density. The plate was easy to describe: it’s boundary was the $x$ axis and the parabola $y = 1-x^2$. In the first question, I asked for $M_x$ (the moment about the $x$ axis and in the second question, I asked for the center of mass (they could use symmetry to deduce $\overline x = 0$). So to find $\overline y$, they needed to find the area (mass) and $M_x$.

What astonished me is that a number of students missed the question “find $M_x$ ” completely but then went on to solve for $\overline y$ correctly!

This says something about the intellectually immature mind, but I am not sure what is says.

## March 10, 2011

### Students: do some problems without your book and notes!

Filed under: how to learn calculus, mathematics education, student learning — collegemathteaching @ 11:22 pm

I’ve found that some students make the mistake by always doing practice problems with their book and notes open. It is oh so easy to convince yourself that you know the material better than you actually do.

You don’t have to look far for instances of people lying to themselves. Whether it’s a drug-addled actor or an almost-toppled dictator, some people seem to have an endless capacity for rationalising what they did, no matter how questionable. We might imagine that these people really know that they’re deceiving themselves, and that their words are mere bravado. But Zoe Chance from Harvard Business School thinks otherwise.

Using experiments where people could cheat on a test, Chance has found that cheaters not only deceive themselves, but are largely oblivious to their own lies. Their ruse is so potent that they’ll continue to overestimate their abilities in the future, even if they suffer for it. Cheaters continue to prosper in their own heads, even if they fail in reality.

Chance asked 76 students to take a maths test, half of whom could see an answer key at the bottom of their sheets. Afterwards, they had to predict their scores on a second longer test. Even though they knew that they wouldn’t be able to see the answers this time round, they imagined higher scores for themselves (81%) if they had the answers on the first test than if they hadn’t (72%). They might have deliberately cheated, or they might have told themselves that they were only looking to “check” the answers they knew all along. Either way, they had fooled themselves into thinking that their strong performance reflected their own intellect, rather than the presence of the answers.

And they were wrong – when Chance asked her recruits to actually take the hypothetical second test, neither group outperformed the other. Those who had used the answers the first-time round were labouring under an inflated view of their abilities.

Chance also found that the students weren’t aware that they were deceiving themselves. She asked 36 fresh recruits to run through the same hypothetical scenario in their heads. Those who imagined having the answers predicted that they’d get a higher score, but not that they would also expect a better score in the second test. They knew that they would cheat the test, but not that they would cheat themselves.

Some people are more prone to this than others. Before the second test, Chance gave the students a questionnaire designed to measure their capacity for deceiving themselves. The “high self-deceivers” not only predicted that they would get better scores in the second test, but they were especially prone to “taking credit for their answers-aided performance”.

Bottom line: frequently quiz yourself by seeing if you can do problems without your notes or seeing if you can write out the proofs without references!

The research, published online Thursday in the journal Science, found that students who read a passage, then took a test asking them to recall what they had read, retained about 50 percent more of the information a week later than students who used two other methods.

One of those methods — repeatedly studying the material — is familiar to legions of students who cram before exams. The other — having students draw detailed diagrams documenting what they are learning — is prized by many teachers because it forces students to make connections among facts.

These other methods not only are popular, the researchers reported; they also seem to give students the illusion that they know material better than they do.