The Math Circle Summer Institute...

... was fabulous! We had morning circles led by Bob and Ellen Kaplan, Amanda Serenevy, and Leo Goldmakher, and afternoon sessions where we led math circles attended by local kids. We also worked on math problems in the evenings, sprawled around our dorm lounge.

I finally led a pretty good math circle there, with high school aged kids, using the Magic Pancake problem. (All the pieces magically change to your favorite kind of pancake, and grow to give you as many yummy bites as you want, perfectly filling your tummy. One cut makes two pieces, two cuts can make 4 pieces, how many pieces can you get with N cuts?) The first year I went, I told a story that used fractions, with kids who were too young for it. The second year, I did base 3 or 8, with junior high aged kids, who I couldn't get to talk. This year my circle finally went well, though I still have lots of room for improvement. My big goal is to learn how to ask more open questions, and let the students guide us along, keeping my mouth shut more, and giving less hints about where I think we should go.

Our morning math circles continued to be my favorite part of the day. I came in late on Monday, and got to hear Bob leading the group as they worked on finding the circumcenter of a triangle. I've never taught geometry, so I don't know this stuff - it was a blast to think about. Next came the incenter, and then the orthocenter. We worked on the topic a bit more on Tuesday, and again on Friday. I'm looking forward to playing with it on my own.

Also on Tuesday morning, Amanda told us a story with Sona Drawings, and then asked: For what size rectangles can we draw one continuous lines around the dots? I knew where that would go, so I got to quietly watch other people thinking it through. I like telling stories, and would love to learn how to do those story drawings. That will be a long-term project.

On Wednesday, Leo led us in thinking about divisibility by 3 and 7. I was very familiar with the first part, and got to watch people thinking again. That was lovely. (I'm not a very good observer, and it's a skill I really want to improve.) The second part was tantalizing. He showed us how you can find out whether 7 is a factor of a large number by removing the last digit, doubling it, and subtracting it from the number formed by the remaining digits.

Example: Is 36,666 divisible by 7? Double the last 6, getting 12. Subtract 12 from 3,666, getting 3654, still too big. Do it again. Double the 4, getting 8, subtract 8 from 365, getting 357, still too big. One more time. 2*7 is 14, 35-14=21. Yes, 7 goes into 21, so that means 7 goes into 36,666.

Buy  why? That's the question! I solved it on Wednesday evening. When I'd done that, Amanda showed me her proof which I liked much better, because it involved modulo 7. I've been playing with modular arithmetic lately, and I like the power it has. She challenged me to make up another way to break a big number up to see if it's divisible by 7. I did it, using mod 7. And then I got excited, thinking about how you could mix and match different techniques. I know that 7*11*13 = 1001, so any number over 1000 can have multiples of 1001 subtracted before using the other techniques. (30030 is a multiple of 7, subtract that from 36,666, getting 6636. Now subtract 6006, getting 630. Yep, multiple of 7. Oops! I didn't even need the other techniques for this one.)

On Thursday, Ellen led us in thinking about Euler's formula for polyhedra (V-E+F=2), as an example of something whose proof is very subtle. Once again, I got to watch people thinking about the beginning steps. If you're interested in this, Dave Richeson's book, Euler's Gem, is a great treatment of it. (And Michael Paul Goldenberg wrote about this session in his post about the Institute. Thanks, Michael, for the reminder that I wanted to write a post.)

What else did we do? My swimming buddy didn't make it back this year, so I only managed to get up early once to swim in the lovely pool. (Missed you, Ellen. Thanks for the cap, Linda.) I ate less than in previous years, but the food was still amazingly good for a big food services operation. And I brought games to share once or twice during our evening gatherings at the dorm lounge. My friend Linda had just given Dizio to my son, and I brought it along. We had a great time with it. We also enjoyed Kataminos, which I mentioned in a previous post.

Come join us next summer!

Vacation Post: Squarus

Abbie, my niece (once removed, or something like that), says her family made their own game up using Blokus pieces. They each have a color and a corner, and the first to make a 9x9 square (uses all but 8 of the little squares possible) wins.


Do you play your own family variation of a good board game?

Vacation Post: Pentominoes

There are two other posts I'd like to write, but they'd take more effort, and I'm on vacation. My son is playing down at the beach with his second and third cousins, with another mom watching. I'm on the porch, with a cool breeze  at my back.

I've just been playing with Katamino, a lovely game I bought a few days ago at Mackinac Kite & Toy, in Grand Haven. It's a way to play with pentominoes in a two-person game. The X and I pieces are left out, each player gets three single squares and a two-square rectangle, and then players take turns picking the ten pentominoes. Then the players race to put their pieces into their side of the board. The whole game takes about two minutes, which is part of its charm.

When I had no one to play with, I started working the solitaire puzzles, which involve putting 3 of the pieces in a 3x5 area of the board, and then 4 pieces in a 4x5 area, and so on. (The board can change size from 3 to 12 rows.)

I've owned a set of pentominoes in a nice 8x8 case for quite a while, and I know there are many ways to put them in so the 4 extra spaces will be symmetrical. But I've never managed to come close to doing it. I just now realized that I should have long ago listened to George Polya's problem-solving advice: Try a simpler problem that has a similar structure. Katamino had me do that. The game includes a booklet with pictures of 7 different workable combinations of 3, and 19 different workable combinations of 4. (12C3 = 220 total combinations of 3, so it's good to know the ones that work.) I've done all the 3-piece puzzles, and I'm almost through the 4-piece puzzles. What fun!

I'm also enjoying some light summer reading. Blue Balliett has written three young adult novels in which Calder Pillay and his two friends solve mysteries surrounding works of art, and marvel at the coincidences in their lives. Calder plays with pentominoes, using them like an I Ching, for inspiration. The third book, The Calder Game, came out in paperback recently. And here's what made me write:

He'd decided on the Start Small, Move to Large approach that his Grandma Ranjana had taught him. It seemed to work with almost everything in life. After all, you couldn't leap to making twelve-piece pentomino rectangles when you'd never made a five-piece one.

Coincidence rules!

Bay Area Circle for Teachers

This is a week-long workshop being held on UC Berkeley's campus, sponsored by MSRI. On Monday afternoon I was able to attend a session led by the wonderful folks from the Algebra Project and the Young People's Project. Imagine my surprise when I saw a post about it on Without Geometry, Life is Pointless. Avery and I haven't met in person, but we were in that same room together, playing some exciting math games. In case anyone else I know online is attending that workshop, I thought I'd write my own post about it.

I'll be going to a dinner with the group this evening, and I'll be leading a session on Thursday afternoon, where we'll play with Pythagorean Triples, my current mathcrush.  ;^)

Anyone else planning to be there?

Challenge: Do a Mathemagical Performance

One of the members of the Natural Math group pointed us to a BBC article, "Taking Maths to the Street." The author of the article (who isn't big on math) and a bunch of math teachers and students get trained to do math tricks and then go out on the street, trying to entice people to watch them. She called it 'busking.' I think that includes putting a hat out for donations, but she didn't mention that part. I was disappointed that the author didn't manage to do her trick properly. Is her article perpetuating the myth that math(s) is too hard for 'regular people' to get?

My math poem challenge back in January was a hit, so let's try it again. This time the challenge is to do some street performing using math. I think it's a great way to show the world that math can be fun. Should we have a way to determine a winner? Should it be the one who gets the most money in their hat?¹

You might want to do this with a few friends. Having a group of you, like they did in the article, might make it easier to attract attention. And maybe it would be a good idea to wear some cool math shirts. I'd like mine to say mathemagician. I wonder if I could find something like that... Yes...

I'd love some discussion here about other tricks that would work well for this. The article mentions two: The first, determining the day of the month of someone's birthday by which of five cards they say it shows up on, is based on binary. The other trick is a variant of Nim - each player takes from one to three post-its off the assistant on their turns, and the post-it with the $20 bill attached must be the last one taken off.

The article's author used 8 post-its. I think this trick would be more mysterious if you started with more. I'd recommend asking the volunteer whether they want to go first or second while you're putting the post-its on your assistant. If they want to go first, you could put on 16 (or any multiple of 4). If they want you to go first, you could put on 17 to 19 (or anything not a multiple of 4).

I could probably go over to SF and try this by the Powell Street Cable Car stop; there's usually a long line there. If I can find some else willing to be a fool for math² with me, I promise to do it at least once. (Is anyone reading this from the Bay Area?)

Who else is game? Step right up! Getcher number here!



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¹Contest ends at the end of summer, September 20.
²I wrote this full of enthusiasm yesterday morning. Now I'm a little nervous. But hey, it's a small commitment, and if it bombs, I don't have to do it again, right?

Fundamental Theorem, continued

I spent an hour yesterday morning, preparing for my tutoring session with Artemis. That's the first time I've done that. I figured there were 3 questions that followed from what we'd done last week:

1. Why does the sum of the first n squares turn out to be n(n+1)(2n+1)/6? (I tried to think about it with drawings, but got nowhere. I looked up "sum of first n squares visual" and found this discussion at a blog called Understanding. This pdf was linked to, and it's the best thing I found. But Jason Dyer pointed me to this much better visual. Now I get it.)
2. What if we wanted that same shape, but could afford a little extra weight, and wanted to find the area out to a variable right edge, labeled R? That was a way to rehash what we'd done the week before, with a little bit more generality.
3. A car part that's under some crazy function.

 When I explained my three questions to Artemis, he said he didn't want to do any of them. I said I was so excited by this stuff, I could stand to do the first question some other time, but I really wanted to look at the second question. I think he was concerned it would be hard. But once we started, he was very excited about it. We figured out that it would be R^3/3. As we took the limit as n (number of slices) approached infinity, he got a kick out of pointing out the parts that would go to 0.

We haven't done many derivatives yet, so he didn't notice that R^3/3 was the anti-derivative of x^2 (at x=R). I asked him to find the area under a triangle with angled side y=x and right side at x=r. That was easy. I wondered what the area under y=x^3 would be, and he saw the pattern, and guessed it. We haven't proved it yet.

We tried to find the area under y=2^x, but we'd never done derivatives of exponential functions yet, so we weren't able to finish that one.

We have one more session before I go on vacation. Maybe we'll tie up some loose ends, or maybe we'll play with something less strenuous. I'll let him decide.

Sneaking Up On the Fundamental Theorem of Calculus

Calculus starts out with two main branches. Finding slopes of curvy lines (aka finding the derivative or differentiating) and finding areas and volumes. It turns out that these two different sorts of questions are inverses to one another (like subtraction undoes addition, and logarithms undo exponents), but that's not at all obvious at first.

Tutoring Artemis has continued to be a joy. We've been exploring all sorts of things, not going in any set direction. Today we started out not knowing what we wanted to do, and played with Guess My Function for awhile. It was fun, but I think we both wanted something that would move us along more. We've done derivatives, so I figured it might be time for the inverse.

I didn't say any of that, I just said I had a problem he might enjoy*. I drew one of our favorite functions, y = x ², and said, "I want to design a part for a toy car for my son. The area under this curve, from x=0 to x=3, is a good shape for the part, but I need to know how heavy it's going to be." We decided to estimate it with rectangles and triangles.

So 1 + 4 (for rectangles) + 1/2 + 3/2 + 5/2 (for triangles) = 9 1/2.

I said the squares were square centimeters, and the metal was 1 gram per square centimeter (is that a reasonable density?), so we knew the part would wiegh (mass) less than 9.5 grams. The curve is always below the stright line segments that the triangles make. So the estimate must be higher than the true value.  I told him we wanted to make sure it was under 9.3 grams, so we might want a better estimate. He was loving it. We did an estimate with rectangles and triangles 1/2 centimeter wide, and got 9.125. That took a while. Lots of fraction practice.

I'm thinking I should have claimed we needed it lighter, like 9.1 grams. But Artemis immediately says, "I want to know how much it will be exactly, let's take the limit." Sure, let's do that...  ;^)

So we did, his way. When I teach this, I always use just rectangles. But they don't brush up against the curve nearly so nicely as rectangles with triangles on top, and he insisted on including the triangles. I was worried it would get too messy, but it wasn't bad. It was interesting to see the area contributed by the triangles drop out when we took the limit at the end. Maybe I'll teach it differently from now on.

When we got to the sum of a bunch of squared terms, he immediately wrote down what the sum would be! It's in the lower right corner in the diagram to the right, that fraction with a mess over 6. Good thing, because I would not have known how to lead him to that - I don't even have that memorized. We would have taken a long, leisurely detour here, and ended with the problem halfway done.

We found that the car part would weigh 9 grams. I'm so glad it worked out to a nice number; he really appreciated the beauty of that.

As he walked to the car, he was telling his mom about finding the weight of a toy car part. I had almost made the mistake of thinking the concrete context I had given it didn't matter to him. It did. He can zoom into abstraction like a rocket, but he does like a good launching pad.

Next week, we can do another problem like this, and look for patterns. I think I can lead us though the Fundamental Theorem without ever 'teaching' it.

[Edit: Next post in the series here.]


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*If my description of our work goes above your head, and you'd like to understand, please ask. I'd love to help you make sense out of it.

Self-Publishing

I'm still hoping to find a publisher for my book. If I don't find one, I've been planning to go with Lulu. Until just now, I hadn't been clear on the cost. I really want some color photos in the book, but here's what Lulu says:

The cost of color books is significantly more than black and white books. When you select the color printing ("Full color pages") option in the publishing wizard, the cost is $0.20 for every page in the book, not just the pages with color.

So a 200 page book would be $40. Nope, I don't think so. That means either Lulu is out, or the color photos are out. (Black and white seems to be 2 cents a page, plus $4.50 for binding. Less than $10 for the book's direct printing costs. But it would be so bland...)

Any recommendations?


[Edited:]
Check my previous post about the content of Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet. Over twenty authors have contributed, and it's looking good.

The recommendations so far  are really helpful, and remind me that I have one more question: Do any of these places use recycled paper?

Mathematical Habits of Mind

Avery (Without Geometry, Life is Pointless) pointed out a very interesting article, Habits of Mind: An Organizing Principle for Mathematics Curriculum* (pdf here). The authors ask "how do we decide what mathematics to teach?" and then suggest that's the wrong question to ask.

For generations, high school students have studied something in school that has been called mathematics, but which has very little to do with the way mathematics is created or applied outside of school.  One reason for this has been a view of curriculum in which mathematics courses are seen as mechanisms for communicating established results and methods - for preparing students for life after school by giving them a bag of facts. ... Given this view of mathematics, curriculum reform  simply means replacing one set of established results by another one. ...

There is another way to think about it, and it involves turning the priorities around. Much more important than specific mathematical results are the habits of mind used by the people who create those results. ... The goal is to allow high school students to become comfortable with ill-posed and fuzzy problems, to see the benefits of systematizing and abstraction, and to look for and develop new ways of describing situations.

 They identify lots of mathematical habits of mind, and give good examples to explain them. Some seem pretty similar to me, so I combined them here.


Students can learn to be...

• pattern sniffers
• experimenters/tinkerers
• describers
• visualizers
• inventors
• conjecturers/guessers

Mathematicians ...

• talk big and think small,   (Trying to understand a new idea? Start with a simple example.)
• talk small and think big, (Start with a simple example, and build a big web of mathematical structure.)
• use functions,
• use multiple points of view,
• mix deduction and experiment,
• push the language,
• use intellectual chants.

They go on to list habits of mind that geometers and algebraists use. I was comfortable with most of their examples in the first half of the article, but often got lost in the second half.

In Avery's post on the new Common Core Math Standards, he noticed two things missing from the standards:  pattern sniffing and problem posing. (A good book on that last is The Art of Problem Posing, by Stephen Brown.)

If the standards were about habits of mind, instead of particular content, maybe I could get over my anti-standards attitude. As I said at Avery's blog,  I'd love to turn the ideas from this article into suggestions for standards; I think that would help us focus our lessons in good directions.


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*I believe the proper word there is curricula, the plural.

Apology

I'll have to rethink having moderation on. All comments are supposed to come to my email. Once in a while they don't, but are stored in my 'dashboard' for the blog. I'd seen this once before, but had forgotten to check recently. I've just accepted a bunch of comments. My sincere apologies for the delay.

I think I'll turn moderation off until I start getting spammed again.