Mathochism: Becoming independent

One woman’s attempt to revisit the math that plagued her in school. But can determination make up for 25 years of math neglect?

I went to office hours today before class, because I had a few questions. I’ve been going over limits again, helped by a take-home review supplied by the professor.

The problems on this review are far more challenging, and far more rigorous, than what I encountered in the book. And boy, I wish I had had it before the test! Mastering these problems would have prepared me so much better, really reinforcing my understanding of limits.

Oh well. I feel somewhat vindicated by the fact that the Calc Professor told me she wished she had given us that review during my office visit. I also felt vindicated when she agreed that the homework problems in the book are at times too simple.

She also told me that this “simple homework, harder tests” thing is part of a larger departmental policy. By Calculus 1, students are expected to be more “independent,” and expected to seek out information on their own. Bottom line: There will be no more spoon-feeding or hand-holding.

Side note: The Calc Dementor appears to take this policy to the extreme. The Calc Professor, while abiding by this policy, seems to sympathize and try to smooth over the transition, so students like me won’t be completely demoralized. Can you guess which approach I appreciate more?

Considering calculus is a college-level course (in spite of many high schools offering it), I can understand the policy. In college, you’re responsible for your own learning and are expected to have good study habits. Help is available, but you have to identify what you need, and to have made an effort first.

However, I don’t think it is effective to teach concepts using the simplest examples, give homework featuring examples that are just a bit harder, then testing on the hardest examples. Why not teach the simpler and harder examples, give homework featuring the harder and hardest examples, followed by helping students with the hardest examples as needed, i.e when they’re well and truly stuck, and then test on the hardest examples?

I mean honestly, why is it so wrong to want a better idea of what you’re facing? If I may resort to metaphor, while I suppose it is possible (though certainly painful) to run a marathon when you’ve only ever run a 5K, it seems more productive to run a half-marathon before attempting a marathon.

In preparing for this calculus marathon, I’ve been looking for harder examples involving limits. I’ve been looking for ones involving fifth roots and absolute values and other often nasty permutations that make the clear murky. I’ve also been looking for more examples of piecewise function limits, because I really, really need to practice those.

I can’t say I have been wildly successful at finding either. The book, as mentioned before, is useless, at least in the first chapters. Excellent resources like Khan Academy, Wolfram Alpha, etc. have been disappointing, though I appreciate the videos and practice problems. The review, like I mentioned above, helped a lot, but I wish I could dip deeper into that well. A UC Davis website had some interesting stuff that really helped my understanding of infinite limits and sandwich theorems and delta epsilon intervals. If only they had one on piecewise functions!

I know many of my kind readers are math professors. Can any of you recommend a book, or site, or series that gives me some semi-nasty to truly nasty examples of limits to labor over? I’m not asking for worked-out solutions to go with them, but answers would be nice, so I know if I’m on the right track. If I get truly stuck, I will go to a tutor or the professor.

Sigh. I never thought I would see the day where I would be begging to solve harder math problems. Are the seas boiling? The four horsemen galloping? The locusts swarming?

All text copyrighted by A.K. Whitney, and cannot be used without permission.

• I have to say that the policy of giving simpler homework problems – and then testing on the most difficult problems – seems counterintuitive to the learning process to me. I get that you are supposed to be able to seek out the knowledge yourself, but how can you know what you’re not understanding unless you are faced with it?

• akwhitney

Precisely. I have spent my evening (apart from watching ghost shows on SyFy) looking for practice worksheets online. I’ve found some promising ones, but not nearly enough!

• I use Stewart’s 6th edition calculus book (ISBN 978-0495553793 if you want to look it up), but any somewhat-recent edition of that book should be very similar. It covers all the material I would consider essential to calculus, and the problems range from basic to problems even my best students might struggle with. There’s a lot of problems to choose from, and most of the odd answers should be in the back. You can also purchase a complete solutions guide if you find yourself struggling. Stewart is one of the standard texts for teaching calculus, and for good reason.

I would also look into whether or not you can use WebAssign for the text even if you’re not enrolled in a class that uses it. It’s online homework that automatically checks your answers, and can be set to display both answers and solutions after you’ve finished working on a section.

• Here’s a really mean question about limits and continuity. Suppose you have a function defined as follows:

( 1/2^n if x is a number whose denominator in least terms is 2^n
f(x) = |
( 0 otherwise.

For example f(1/2)=1/2, f(25/40)=1/8 (because 25/40=5/8=5/[2^3]), but f(1/3)=0 and f(1/6)=0 (because the denominator is, though a multiple of two, not a power of two).

Where is the function continuous? Where is it discontinuous? (You should use try answering using the definition of continuity that relies on limits; your answer should strike you as really strange.)

• Err. WordPress ate my formatting. That was supposed to basically show a function defined piecewise as “f(x) is 1/2^n if x is a number whose denominator in least terms is 2^n, and is zero everywhere else.”

• akwhitney

Thanks, Antonia! And Trold — urrgh. My eyes are crossing just looking at that thing. I will torture myself with it later.

• Would you like a (slightly obscure) hint about Trold’s problem?

• akwhitney

Sure! Sorry this has been in limbo for so long — I was out of town, and off the Internet.

• a comment about Trold’s example:
(a) is f(x) continuous at x = 1/2? you know f(1/2) = 1/2; if c is any other number close to 1/2, is f(c) close to 1/2?
(b) is f(x) continuous at x = 1/3? you know f(1/3) = 0; if c is any other number close to 1/3, is f(c) close to 0?
(b) is harder than (a), and gets at some wierd things (at least to me) about real numbers.

• My hint is: look at a ruler (or a tape measure). One that uses English units rather than metric.

If that’s not enough, think about where the function takes on the values 1, 1/2, 1/4, and 1/8. Think about how you would plot those points. Hopefully the connection with the ruler will become apparent. With that in mind, you can tackle the question of continuity (perhaps along the lines that bill suggests).