Friday, March 23, 2012

I'm angry: Excel has crazy changes!

I was doing a workshop today for a few of my calculus students on using numerical integration techniques. I started to show them how to use Excel, and it had changed!
  1. The columns are no longer lettered, they're numbered, just like the rows. I don't even know how to refer to a particular cell any more!
  2. The formulas no longer refer to cell labels, but to the distance from the formula cell. (This is relative addressing, which was implied before, but not obvious in the formula.) 
  3. The way you click to select has changed too, and I had real trouble typing in my formulas.

Sonia Kovalevsky Day

Her birthday is January 15, but today gets to be her day. Cathy O'Neil (aka Mathbabe) started Sonia Kovalevsky Day at Barnard College in 2006, and it sounds like it's been going strong ever since.

Sonia is one of my heroes. Born in 1850 in Russia, she loved mathematics, fought her parents to study it, and entered into a marriage of convenience so that she could study abroad. She earned a PhD studying under Weierstrass, and still couldn't find work doing mathematics.

Eventually she got a position at the  University of Stockholm, and later won the prestigious Prix Bordin. She also had a daughter, who she raised alone. She died at age 41, when her daughter was only 12.

Mathematician, activist, and mother, Sonia was also a writer, whose novels were published to some acclaim. My kind of woman.

(Read more here and here.)

Sunday, March 18, 2012

Math Movie: Inspirations Shows the Beauty of Math

by Cristóbal Vila

His explanations of the math behind this sweet movie are even more exciting, for me.

(Thanks, Murray.)

Saturday, March 17, 2012

Linear Algebra: Tough Proofs for Determinants

I've been enjoying teaching this course, and I was liking the text, by David C. Lay ... until we got to chapter 3 on determinants. I worked through all the problems last weekend, verifying that everything was easy for me, and read through the proofs. Unfortunately, I thought I got it all, but I was moving too fast, and ended up in front of my class on Monday unable to really do the proofs. Embarrassing!

With my house being broken into later that day (3rd time in 5 months; yes, it's hideous, but nothing was taken this time), I never caught up this week. Yesterday and today, I've spent about 5 hours writing up the proofs for 3 theorems. I still see a few holes, but I'm pretty proud of what I've put together.

My text defines the determinant by expanding on the first row. Looking around online, that doesn't look like a standard definition, but it seems like a fine starting point. From there we want to prove that you can get the determinant by expanding on any row or column. (My text says "We omit the proof to avoid a lengthy digression." Bah! It's not math if you don't know why it's true!) My proof may still have a bit of a hole (regarding which terms are negative), but I think it's more helpful than what I found online.

My proof starts with the definition, expands completely so there are n! terms, each having n factors (which come one from each row, and simultaneously one from each column), observing the symmetry shows that we'd get the same terms no matter what row or column we expand on. The one sticky point is showing that the signs of each term stay the same. I don't think I've quite got that properly proven. Tell me what you think.

det(AB)=det(A).det(B). This proof is done in my text, but I felt it was done badly. I'm following his outline, but writing it up in my own words. I think there's a bit of a hole where I use L*. (The author does this step a bit differently, and I don't like his explanation.) To outline the proof:
  • First, we prove it's true for any elementary matrix times a 2x2 matrix (EA),
  • Then we do induction on the size of the matrix,
  • Last, we show that (almost) any AB can be seen as a series of multiplications by elementary matrices (EB).
Here's my proof. What do you think? Is there a clear way to clean up the induction step?

My 3rd proof was on area of a parallelogram = absolute value of determinant (with column vectors representing adjacent sides of the parallelogram). Not particularly impressive, and I don't have the energy to do the volume proof too. Anyone want to show me a good proof of that? (We have not yet covered dot product or cross product, so it can't reference those notions.) I got this version from a mathematician I spoke with at my math circle a few days ago. I had fun using geogebra to illustrate.

Now, back to my regularly scheduled grading...

Friday, March 16, 2012

The Oakland Math Circle does Spot It

Rules of the game:

Each player gets one card. The deck goes in the middle. When you see a match between your card and the top card on the deck, you call it and collect that card, putting it on top of your pile.  Your card will always have a match with the center card, so it's just a question of who can spot their match first.

(There are 3 or 4 other games, but they're all pretty similar.)

After spending lots of time analyzing Spot It over the winter holidays, I thought it might make a good topic for the Oakland Math Circle. Two weeks ago, when I found out just a few hours ahead that I was scheduled to lead a circle, I jumped on BART, got off in downtown Berkeley, ran across the street to Games of Berkeley, bought 3 more tins of Spot It, ran back to BART, realized I'd left the tins on the counter at the store, ran back and forth another time, and made it to the circle just in time.

I had the students play a few rounds, and then we explored. The kids counted and found that most pictures appeared on 8 cards, but many appeared on only 7 cards. We eventually used Michelle's technique to figure out there were at least 57 pictures. (Pick a particular picture, the heart, for example. Pull out all the cards with a heart - there are 8 - and think about what we now know: 7 other pictures/card*8 cards + the heart = 57 pictures).

After a while, we focused on the question:  
How did the makers of this game make sure that every pair of cards has exactly one match?  

We didn't get much farther the first week. During the second week we had mostly new kids, so we started in the same place. But we got a bit farther, and tried to make our own cards with 4 pictures per card. We also had one group with a kid and an adult who had both worked on the problem before. They found a way to make every card in their deck match every other card. But their deck only had 5 cards. They figured out that they could make a similar deck of 9 cards with 8 pictures each. It wouldn't make a very satisfying Spot It game, though.

Yesterday was my third week of Spot It analysis with the Oakland Math Circle. We got back most of the kids from the first week, and played a few rounds again to get warmed up. (My first week with them, they seemed uninterested in math circle, this time they were really engaged, and much more fun to work with.) Then I let them each pick their color and gave them stacks of half-size index cards (3"x2.5"). They could choose to create cards using numbers or pictures, and were trying to make decks with 4 pictures per card, with one match between each pair of cards.

Lots of folks got the 5 card deck. We started calling that the minimal solution. I realized that was an easier solution to find than the solution that makes a bigger deck. (Although Chris and I never stumbled on it while we were creating our decks over the holidays.) One person pointed out that if they all matched on the same picture, you could have as many cards as you wanted. We called that the infinite solution. Since it would make a super-boring game, we added the condition that you have to use more than one picture for your matches, overall. People were so stuck on the minimal solution, I suggested starting with the infinite solution, and making a bunch of cards, trying to figure out when that would get you in trouble.

It turns out there are 13 cards in what I'll call the maximal solution. I realized that this brought up another question: Are there any symmetrical solutions with more than 5 cards, but less than 13?

One of my questions is whether the cards could use any number of pictures, or if there might be a constraint. (I'm thinking that the number of pictures per card might need to be 1 more than a prime, but I'm not at all sure. Yet.) Another cool aspect of this problem is that it illustrates the mathematical concept of duality. (I can't quite explain that yet, beyond saying there are 8 pictures per card, and we could have 8 of each picture in the deck.)

As I got the students into small groups yesterday, I told them, "I don't do math circles, I do math clusters." I find it's much easier to get lots of participation from the students if they're in small groups. Then my job is cross-fertilization. These last two weeks have been my favorite math circles yet. I think I'm finally getting the hang of it. My eternal thanks to Bob and Ellen Kaplan for helping me get started. I highly recommend their Summer Math Circle Institute, on July 8 to 14, in South Bend, Indiana. [Rodi Steinig, who went last summer, is doing great work in Pennsylvania, and blogs about her circles here.]

Each math circle leader has their own style, so if you're thinking about leading math circles, you need to find cool problems that work for you. My own favorites are: this problem (!), the magic pancake, playing with base 3 and base 8, and Pythagorean triples. You can find lots of math-tested-problems at the National Association of Math Circles site.
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