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.

Big Money, Big Oil, and the Mess Corporations Make

Doug Noon at Borderlands has written another post that strikes just the right note for me.   This one is partly about a math skill we all have trouble with, comprehending big numbers. Do you really understand the difference between a million, a billion, and a trillion dollars?

What's a billion? Bill Gates' wealth increased by $50 billion dollars one year. If he wanted to share that income with every one of the 800,000 people in San Francisco, they'd each get $62,500. This video, from David Chandler, online at the L Curve, gives the best visualization I've seen yet of how skewed income distribution is in the U.S.



That kind of money buys whatever it wants. There are two legs or spikes on this graph, a horizontal one representing most of the population, and a vertical one representing most of the income. This graph represents income; wealth is even more skewed*.

David Chandler:

The horizontal spike has the votes. The vertical spike has the money. Who wins, when it comes to electoral politics? Who has influence? Whose interests are being represented in Washington? Can democracy meaningfully exist where the distribution of wealth, and thus the distribution of power, is this concentrated?

The Supreme Court recently affirmed the personhood of corporations (where much of this wealth sits). If BP were really a person, what would that mean? Pretty hard to put the corporation itself in prison for its criminal actions. And pretty hard to control a 'person' who may be immortal and can buy legislators, along with avoiding prison. Kinda like godzilla, maybe?




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*Wealth is what you have; income is what you get, dw/dt (change in wealth with respect to time). The richest one percent are now estimated to own between forty and fifty percent of the nation's wealth, more than the combined wealth of the bottom 95%.

What must be memorized?

I say I have a terrible memory, and that's why I'm in math instead of science (with its myriad names for bones, muscles, organs, chemicals, etc.). I say there's very little to memorize.

But other people disagree. I used to think it was only my students who did badly at math who had the 'wrong' idea that math was about memorizing. Math people will tell you it's not about memorizing, but many of them do think memory has a bigger place than I give it.

What do you think?

Here's what needs to be in your memory, eventually:

• addition 'facts' (if add 1, double, and add 2 to an even are easy, then there are 25 harder facts)
• multiplication 'facts' (2 is doubling, 4 is doubling twice, 5 is easy, 9 can be easy; only 9 facts left)
• circumference and area of a circle (I know which is which by using dimension - circumference is 1D and has r, area is 2D and has r². If I'm unsure of the formula, I can also check reasonableness of my answer by estimation.)
• Pythagorean theorem (a²+b²=c² for right triangles)
• quadratic formula 
• [edited to add ...] sine and cosine, tangent and reciprocal identities (the rest of trig pretty much follows, 6 facts)

That's a pretty short list. What would you add to it?

When I started teaching, I had a BA in math and still didn't have 7x8 memorized. I also didn't have the quadratic formula memorized. Whenever my topic of the day would include quadratic formula, I'd put it at the top of my notes. 7x8 I could figure out in 2 or 3 seconds.

To me, math is all about connections. When I think of slope, I think of a rise and a run (and a little right triangle under the slanty line, showing them). I know steeper lines want a bigger slope, so the rise has to be on top in the ratio that compares rise and run. To me, that's not memorizing.

So tell me what I'm forgetting.  ;^) What more do we have to memorize in math?

Logarithms and Ropes (as found in Mathematician's Delight)

I recently got a copy of Mathematician's Delight, by W. W. Sawyer. I had loved his book Vision in Elementary Mathematics, so I knew I'd like this one. I found out about it through a blog I stumbled upon in my wanderings, where the blogger included this:*

Nearly every subject has a shadow, or imitation. It would, I suppose, be quite possible to teach a deaf ... child to play the piano. ... [The child] would have learnt an imitation of music, and would fear the piano exactly as most students fear what is supposed to be mathematics.

What is true of music is also true of other subjects. One can learn imitation history - kings and dates, but not the slightest idea of the motives behind it all; imitation literature - stacks of notes on Shakespeare's phrases, and a complete destruction of the power to enjoy Shakespeare.

I think that idea, of a shadow subject, will stick with me, and become more powerful for me over time.


Logarithms
I've told my students logarithms were invented in a time when calculators didn't exist, and scientists were looking at lots of data about the planets, trying to discover patterns. Napier invented a way to do multiplication by adding and division by subtracting, a second application of which allows powers and roots to also become questions of addition and subtraction. I don't think this is enough of an introduction to this strange concept.

How did Napier dream this up? Sawyer gives us a glimmering of the sort of inspiration Napier might have had, with this marvelously concrete model for logarithms:

We are all familiar with machines which [we] use to multiply [our] own strength - pulleys, levers, gears, etc. Suppose you are fire-watching on the roof of a house, and have to lower an injured comrade by means of a rope. It would be natural to pass the rope round some object, such as a post, so that the friction of the rope on the post would assist you in checking the speed of your friend's descent. In breaking-in horses the same idea is used: a rope passes round a post, one end being held by a person, the other fastened to the horse. To get away, the horse would have to pull many times harder than the person.

The effect of such an arrangement depends on the roughness of the rope. Let us suppose that we have a rope and a post which multiply one's strength by ten, when the rope makes one complete turn. What will be the effect if we have a series of such posts? A pull of 1 pound at A is sufficient to hold 10 pounds at B, and this will hold 100 pounds at C, or 1000 pounds at D.

Thus, 10⁸ will represent the effect of 8 posts. ... The number of turns required to get any number is called the logarithm of that number. ... So far we have spoken of whole turns. But the same idea would apply to incomplete turns. ... Accordingly, 10^1/2 will mean the magnifying effect of half a turn. ... The logarithm of 2 will be that fraction of a turn which is necessary to magnify your pull 2 times. (page 70)

I had to put the book down here, to ask myself why half a turn wouldn't magnify the pull 5 times - half of ten. As I thought about that, I wanted to know if there would be an easy way, either a thought experiment or a very simple physical experiment (i.e., no special equipment), to prove that this relationship must be multiplicative. That is, how do we know the friction of the rope doesn't just add to our pulling force, so that a certain amount is added at each turn? (Can anyone help me with this?)

If we've decided that the relationship must be multiplicative, then we know that two half turns must multiply to have the effect of one whole turn, and that would mean we need the number that multiplied by itself gives ten. To get to this thought, I had to imagine two posts near one another, with the rope halfway around one, and then halfway around the next.

Why haven't I seen this before?!

I haven't read any more of the book yet, because I keep needing to think more about this cool idea. I look forward to more pedagogical delights as I keep reading this book, and maybe others he wrote. (One list is at the bottom of this page.)



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*W. W. Sawyer wrote this book in 1943, long before feminists began to analyze the effect of using the male for the generic. Although Sawyer uses 'man' and 'he' in a generic sense in other sections (which I've taken the liberty of changing in the second quote I've used), perhaps he was trying to avoid that in this story by calling the deaf child of his music example 'it'. I had real trouble with that, and didn't know how to fix it without messing with his meaning, so I left the meat of the example out. You can go here to see it.