Tag Archives: algebra

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, pointed out a preprint describing Rhie’s work, the two pieces fell into place. Rhie’s lens completes the mathematicians’ proof, and their work confirms her conjecture. So 5n – 5 is the true upper limit for lensed images.

“This kind of exchange of ideas between math and physics is important to both fields,” Rabin told New Scientist.

Indeed, and very cool. The paper that Khavinson and Neumann wrote, with an update that addresses the relevance of Rhie’s result on gravitational lensing, is here.

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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. 

This review was one of the things I was trying to get done last week. It’s gratifying to see a publication process go this fast — I sat down on Tuesday and wrote the review; emailed it in on Wednesday; and it was put up at the MAA yesterday. 

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Filed under Abstract algebra, Math, Scholarship

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, say another 5. Again, put the dividing prime across from the dividend, and the quotient below the dividend. With our example, the array at this stage looks like:

\begin{array}{r|r} \sqrt{50} & 5 \\ 10 & 5 \\ 2 &  \end{array}

In general, continue this process of dividing prime numbers into the lower-left entry in the array, writing the prime across from that entry, and writing the quotient beneath that entry, until you end up with a 1 in the lower-left entry. So the final state of our example would be:

\begin{array}{r|r} \sqrt{50} & 5 \\ 10 & 5 \\ 2 & 2 \\ 1 &  \end{array}

Now, look at the left-hand column of the array. Group off any pairs of numbers you see. Multiply together all numbers which are representative of a pair. In our case, there is only one such pair, a pair of 5’s. Any numbers that occur singly are placed under a radical and multiplied. In our case, that’s the single 2. Then multiply the product of numbers which are in pairs times the radical which contains the singleton numbers. So we end up in our example with 5 \sqrt{2}.

Here’s another example with a larger number, \sqrt{2112}:

\begin{array}{r|r} \sqrt{2112} & 2 \\ 1056 & 2  \\ 528 & 2 \\ 264 & 2 \\ 132 & 2 \\ 66 & 2 \\ 33 & 3 \\ 11 & 11 \\ 1 & \end{array}

There are three groups of 2’s, so outside the final radical we’ll put 2 \cdot 2 \cdot 2 = 8. And the 3 and 11 are by themselves, so under the radical we put 33. Hence \sqrt{2112} = 8 \sqrt{33}.

Pretty clearly, all this method is doing is presenting a different way to do the bookkeeping for doing the prime factorization of the number under the radical. The final step of grouping off the prime pairs and leaving the un-paired primes under the radical is analogous to finding all the squared primes in the prime factorization.

This method is nice and systematic, and I can see why students (and student-teachers) might like it. But it seems to be obscuring some important concepts that students ought to know. With the method of factoring, looking for squared primes, and then removing them from the square root, at least you are dealing directly with the inverse relationship between squares and square roots. The Illini method, on the other hand, uses an approach of “put this here and then put that over there” with minimal contact with actual math. It does work, and it does keep things in order. But do students really understand why it works?

Your thoughts?  What does this method make clearer, and what does it obscure? Should high school algebra teachers be teaching it?

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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 themselves well to this kind of class design. The ratio of text to problems in a typical textbook is probably something like 5:1 — a lot higher than that in some books. When you have a book in the course, it almost forces itself into the center of the class universe and everything tends to revolve around it, and take on its flavor. When the book spends most, almost all, of its pages on stuff for students to read rather than on problems for students to solve, then I guess it’s possible to have a problem-solving oriented class, but you’re going to be swimming upstream the whole way.

It works better, I think, to have no central book — and instead, provide problems via the course notes with just enough information to solve the problems. And if the students need more information, make it an assignment for library research or web queries.

Were there any negative outcomes? Anything you didn’t like as a result of choosing to structure the course in this manner?

There are some important algebra topics, in rings and particularly in fields, that are not going to get the time they really deserve. And I had to cut short or cut out some topics in group theory that are normally standard fare. At least, I see this as a negative; whether it really makes a difference in the long run is yet to be determined.

The way I select students to do course tasks in class basically involves randomly ordering the students and having them attempt the problems one after the other. It seemed like several times, students who had not presented much ended up first on the list on the days they didn’t have something and last on the list on the days they did. Call it bad luck or Murphy’s Law or what-have-you; but I didn’t like how there was no mechanism for making sure the lower-scoring students got more chances to work.

Some students in the class still struggle with basic problem-solving skills and writing proofs. I think they have enough education to carry out successful problem-solving on proofs most of the time. But not having me lecture has meant that they don’t get to see professionally put-together proofs very often unless they go do some reading.

And I think that this course structure caused stress and even ill will among the students who were not used to having so much personal responsibility in their college work. I think that’s an unintended consequence of implementing a course design that is basically sound; I regret that it happened, and I’d like students to have a more uniformly positive experience in the class, but I’m not going to change the basic course design.

Would you do this again?

You bet, although I believe this way of running the class works in some situations and wouldn’t work in others. I thought about running my differential equations class next semester like this, but that course is so focused on methods that a blind application of this course structure onto that course doesn’t seem appropriate. Maybe I’ll come up with some variant that works.

What would you keep the same? What would you change?

I would definitely keep my method for assigning problems to students, my rubric for grading course tasks, and just the overall procedure for running the class sessions that I used. And I’d keep the feature where students get to choose the weights on the various assessments.

I’d do a little more with the course wiki. Right now students are expected to write up their solutions to course note tasks on the wiki, but there is no point value in doing so nor a penalty for not doing so. The exams are open-wiki, though, so there is some incentive for writing results up well. But I think I would make the posting of solutions mandatory and enforce the rule.

I’d also try to have a complete set of notes before the course began. I have been writing things as I go, and it’s led to some snafus I could have avoided.

I might try writing the course notes so that rings and fields come first.

I’d seriously consider having proof techniques be offered as the subject of weekly help sessions or additional course work. Some students are still struggling with basic problem-solving techniques, and they really need more help than what they are asking for.

That’s that for the questions. Any more?

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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 system for choosing who presents what at the board — I won’t get into the details here, but I can do so if somebody asks for it in the comments.

So the class meetings consist almost entirely of students presenting work at the board, where their responsibility is to make their work clear, correct, complete, and coherent — and ruggedized against the questions that I inevitably throw at them.

I was thinking yesterday that this method of doing class has really done a lot of good for the students in the class, in several key ways.

  • Students ultimately rely upon the soundness of their own work. The students can work with others or with print or electronic resources — although with no textbook, they have to learn how to find those resources and tell the good ones from the bad ones, which is a great skill by itself. But it boils down to presenting that work, on your own and with nobody there to bail you out, in front of your professor and peers. I think this is a good antidote to the occasional over-reliance on cooperative learning that we (in education as a whole, and in my department) have. Group work is all well and good, but to be a complete learner you have to be able to rely on your wits and your skills and not just prop yourself up on the strength of peers.
  • Students prepare for class in advance, several days in advance, every night. To do reasonably well on course note tasks, students need to plan on successfully completing 15-20 course note tasks throughout the semester, which comes out to about 1-2 per week. Combine that with the fact there are 8 students in the class all trying to do this, and it’s easy to see that working ahead is really essential. You want to get so far out in front of the class that you have no competition for a particular range of problems. Very often in college, there is no sense that you have to get ready for class the next day — unless there’s an assignment due — and we profs reinforce this by running classes that do not penalize the lack of preparation. (It’s not enough to reward the presence of preparation.) The course design here, though, rewards the students who have read and practiced ahead and learned on their own.
  • Students become skeptical and tough-minded about their own work. It’s quite common in traditional math courses for students completing an assignment to simply barf up something on a piece of paper, hand it in, and see how many points it gets. When you are presenting work before a class, that route leads only to embarrassment. When most of the class time is spent doing these presentations, students learn something I didn’t learn until graduate school — that if you are going to hand something in or present something with your name attached to it, you had better make very sure that it works. I’ve noticed the students anticipating not only the fact that I will be asking them penetrating questions about what they are presenting, but also what those questions are. At that point they are learning to think like mathematicians.
  • Students pay (more) attention to detail, especially terminology and the sensibility of a proof. It’s easy to write a proof or a solution to a problem that has no coherence or sense to it at all — but that incoherence and senselessness vanishes the moment you do something as simple as reading the solution aloud. Which is what these folks are doing every day. Example: A colleague told me a story of a student who was asked whether or not two groups G and G’ were isomorphic. The student answered, “G is isomorphic, but G’ isn’t.”
  • Students base their confidence on the math itself, not on an external authority. Students aren’t allowed to ask me “Is this right?” or “Am I on the right track?” To clarify, they can ask me those questions, but I will only greet them with more questions — mainly, “What justifies this step?” or “How do you know this?” It’s not about me or what I like or what makes me happy with regards to their work — it’s about whether each step of the proof follows logically from the one before it, and whether that logical connection is clearly validated. Students know pretty well now when they have got something right and when they don’t, and if they don’t have it right they have a better sense of what’s missing or incorrect and what they need to do to fix it.

A lot of these effects I’m describing are just embodiments of what it takes to be successful in math after calculus in the first place.

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Filed under Abstract algebra, Education, Higher ed, Math, Problem Solving, Teaching, Textbook-free

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.) Continue reading

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Filed under Abstract algebra, Education, Math, Teaching, Textbook-free, Textbooks