Tag Archives: algebra

February 26, 2009, 12:12 pm

Why do we overcomplicate calculus like this?

A labelled tangent to a curve. Created to repl...
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In the Stewart calculus text, which we use here, the first chapter is essentially a precalculus review. The second chapter opens up with a treatment of tangent lines and velocities, with the idea of secant line slopes converging to tangent line slopes and average velocities converging to instantaneous velocities taking center stage.

Calculating average velocity is just a matter of identifying two time values and two position values and then performing two subtractions and a division. It is not complicated. Doing this several times for shorter and shorter time periods is also not complicated, and then using the results to guess the instantaneous velocity is a little complicated but not that bad once you understand the (essentially qualitative, not quantitative) idea behind shrinking the length of the interval to get an instantaneous value out of a sequence of…

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September 15, 2008, 11:14 am

Average velocity

Average velocity is another one of those basic calculus (really pre-calculus) topics that, like difference quotients, leave me at a loss for why students have such a hard time with them. There’s a very simple and common-sense definition, namely that the average velocity of an object with position s(t) from t = a to t = b is

\(\frac{s(b) – s(a)}{b-a}\)

(See? It’s just distance = rate * time solved for “rate”.) There are examples in the book and examples on the internet ad infinitum of how to calculate average velocities, and all of these are simple numerical calculations with absolutely no algebra involved. You have to know how to plug numbers into a function and then do basic arithmetic on your calculator. That’s all.

But students get so turned around. They calculate only the position at time t=b. They add up the positions at t=a and t=b and divide by 2 (“average”). They add in…

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June 17, 2008, 6:00 am

Tuesday morning links

June 8, 2008, 8:26 pm

Algebra meets astrophysics

Abstract algebra and astrophysics don’t have much to do with each other, right? Well, perhaps not, after all. Here’s a story about the results from a researcher in gravitational lensing being used to prove an extension of the Fundamental Theorem of Algebra to rational harmonic functions. Snippet:

In 2004, [mathematicians Dmitry Khavinson and Genevra Neumann] proved that for one simple class of rational harmonic functions, there could never be more than 5n – 5 solutions. But they couldn’t prove that this was the tightest possible limit; the true limit could have been lower.

It turned out that Khavinson and Neumann were working on the same problem as [astrophysicist Sun Hong Rhie]. To calculate the position of images in a gravitational lens, you must solve an equation containing a rational harmonic function.

When mathematician Jeff Rabin of the University of California, San Diego, US,…

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May 27, 2008, 11:49 am

Fun with finite fields

For those of you interested, I have a review of Finite Fields and Applications by Gary Mullen and Carl Mummert now posted at MAA Reviews. You can get to it here, although you have to be an MAA member to view it, or else pay $25/year for a nonmember subscription.

If you aren’t an MAA member and don’t want to pay, the bottom line of the review is: It’s a pretty good book. Very good for mathematicians, grad students, and advanced undergrads. Normal undergrads will need patience and perhaps a lot of help with the initial chapter, which is a lot of serious algebra which unfortunately doesn’t appear to make that much of an appearance in later chapters when the applications show up. And what’s with the three-paragraph treatment of AES? On the other hand, lots of neat stuff about Latin squares, including a cryptosystem based on mutually orthogonal Latin squares which I’d never seen before. …

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February 7, 2008, 9:12 pm

The Illini method for simplifying a radical

One of my linear algebra students is an education major doing student teaching. Today he showed me this method of simplifying radicals which he learned from his supervising teacher. Apparently it’s called the “Illini method”. Googling this term returns nothing math-related, so I think that term was probably invented by his supervisor, who went to college in Illinois.

The procedure goes as follows. Start with a radical to simplify, say \(\sqrt{50}\). Look under the radical and find a prime that divides it, say 5. Then form a two-column array with the original radical in the top-left, the divisor prime in the adjacent row in the right column, and the result you get from dividing the radicand by that prime number in the left column below the radical. In this case, it’s:

\(\begin{array}{r|r} \sqrt{50} & 5 \\ 10 & \end{array}\)

Now look for a prime that divides the lower-left term…

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November 8, 2007, 9:46 pm

Questions about the algebra course

Jackie asked a series of good questions about the textbook-free modern algebra course and some of the student outcomes I was seeing in it. I tried to respond to those in the comments, but things started to get lengthy, so instead I will get to them here.

Do you think the results are only a result of a textbook free course?

To repeat what I said in the comments: I think the positives in the course come not so much from the fact that we didn’t have a textbook, but more from the fact that the course was oriented toward solving problems rather than covering material. There was a small core of material that we had to cover, since the seniors were getting tested on it, but mostly we spent our time in class presenting, dissecting, and discussing problems. We didn’t cover as much as I would have liked, but this is a price I decided to pay at the outset.

Most traditional textbooks don’t lend …

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November 7, 2007, 4:13 pm

Five positive student outcomes from the textbook-free algebra class

We’ve got just 4-5 weeks left in the semester and until the textbook-free Modern Algebra course will draw to a close. It’s been a very interesting semester doing the course this way, with no textbook and a primarily student-driven class structure. In many ways it’s been your basic Moore Method math course, but with some minor alterations and usage of technology that Prof. Moore probably never envisioned.

As I mentioned in this lengthy post on the design of the course, students are doing a lot of the work in our class meetings. We have course notes, and students work to complete “course note tasks” outside of class and then present them in class for dissection and discussion. The tasks are either answering questions posed in the notes (2 points), working out exercises which can be either short proofs or illustrative computations (4 points), or proving theorems (8 points). We have a…

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October 3, 2007, 9:19 am

Textbook-free Modern Algebra update

It’s been a while since I last said anything about the textbook-free Modern Algebra class experiment. This is mainly because the class itself is now underway, five weeks into the semester, and it’s only now that I’ve got enough perspective to give a reasonable first look at how it’s going. So, let me give an update. (Click to get the whole, somewhat lengthy article.) (more…)