Sunday, October 31, 2010

Try again! Fail again! Fail better!

Students: Were you feeling discouraged by the difficulty of the problems you tried at our math circle on Friday evening?

If you found yourself struggling and flailing about and feeling lost, that means you are in the right place!

The problems we work on at math circle are supposed to be a LOT harder than the problems you get in regular school classes. Struggling, flailing about, feeling lost a lot--that is what mathematics is all about.

Mathematician Lillian Pierce has great advice on learning mathematics in an interview in the latest issue of Girls Angle Bulletin:

Enjoy math!

Especially, enjoy challenges!

Math is just like any other skill: practice counts.

Also, have courage and confidence in your abilities. Don't shy away from failing to solve something immediately: try out your ideas, make mistakes, learn from your mistakes.

The struggle itself is one of the most important parts of your practice.

Try again!

Fail again!

Fail better!

When you have been beating your head against a problem for a while, and getting nowhere, leave it and move on to other problems, struggle with those for a while, then come back with a new perspective. Sometimes just finding a partner so you can share what you've been trying can help a lot. Even if your partner is just as lost as you are, the simple act of explaining aloud what you've been trying may help to clarify the problem in your own mind. And perhaps you have a great approach but finding a small blind spot your partner can help you spot will make all the difference.

Remember: practice counts! And practice means making mistakes! It is OKAY to make mistakes, lots and lots of mistakes at our math circle meetings. If you are NOT making mistakes at our meetings, you are doing the wrong problems!

Don't be embarrassed about making mistakes at our meetings! All of us make mistakes. Celebrate your mistakes! Learning from your mistakes, your half-baked ideas, your false starts--that is the essence of problem solving.

Our math circle is a place where it's really, really okay to make mistakes! Don't be afraid. Enjoy that space and freedom to make mistakes! It's a treasure! Share your mistakes and what you've learned from them.

Friday, October 29, 2010

Signup for Harvard-MIT Math Tournament LOCAL

Thanks to Emma Willard School for their gracious hospitality in accommodating our math circle's local administration of this November contest to our high school student members.

Please use the form below to sign up.

Monday, October 18, 2010

"maverick mathematicians" and the power of grade school geometry

Earlier this week, the New York Times published this obituary for Benoît Mandelbrot, excerpted below:

Benoît B. Mandelbrot, a maverick mathematician who developed the field of fractal geometry and applied it to physics, biology, finance and many other fields, died on Thursday in Cambridge, Mass. He was 85. ....

Dr. Mandelbrot coined the term “fractal” to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature.

“Applied mathematics had been concentrating for a century on phenomena which were smooth, but many things were not like that: the more you blew them up with a microscope the more complexity you found,” said David Mumford, a professor of mathematics at Brown University. “He was one of the primary people who realized these were legitimate objects of study.”

In a seminal book, “The Fractal Geometry of Nature,” published in 1982, Dr. Mandelbrot defended mathematical objects that he said others had dismissed as “monstrous” and “pathological.” Using fractal geometry, he argued, the complex outlines of clouds and coastlines, once considered unmeasurable, could now “be approached in rigorous and vigorous quantitative fashion.”

For most of his career, Dr. Mandelbrot had a reputation as an outsider to the mathematical establishment. From his perch as a researcher for I.B.M. in New York, where he worked for decades before accepting a position at Yale University, he noticed patterns that other researchers may have overlooked in their own data, then often swooped in to collaborate.


Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain? The answer, he was surprised to discover, depends on how closely one looks. On a map an island may appear smooth, but zooming in will reveal jagged edges that add up to a longer coast. Zooming in further will reveal even more coastline.

“Here is a question, a staple of grade-school geometry that, if you think about it, is impossible,” Dr. Mandelbrot told The New York Times earlier this year in an interview. “The length of the coastline, in a sense, is infinite.”

At last week's middle school math meet, we talked about a special kind of fractals, Pythagoras trees. The particular type of tree we investigated was made entirely out of squares and isosceles right triangles.

We started with the assumption that the biggest square in each fractal was 64, and tried to figure out how the area of the fractal grows as we add layers. One way to solve this problem is to use the Pythagorean theorem over and over and over again. That is the rather laborious approach that teachers came up with this summer at the Bard summer workshop for middle school math teachers.

Max Thomas of Hackett Middle School, however, came up with a very clever alternative approach. He noted that you can figure out that the isosceles right triangle in B1 must have area 16, because you can mentally "fold" it down into the big square below it, and easily confirm that it has area equal to one fourth of the big square. By similar reasoning, you can fold the same triangle in B1 over into the smaller squares in B1 and see that each of those two smaller squares must have an area of 32.

So, the total area of the fractal in B1 must be 64 + 16 + 32 +32 = 144. Can you extend Max's approach to figure out the areas of the succeeding fractals, and the general pattern as you add each layer of the fractals?

You can also investigate other kinds of Pythagorean trees by downloading free interactive applications from Wolfram MathWorld here.

Wolfram alpha also offers many other nice interactive fractal demonstrations here.

As Professor Mandelbrot and Max Thomas have both demonstrated, you don't always need to use high-powered mathematics to get very deep insights into fractal mathematics. Sometimes you don't even need the Pythagorean theorem, though playing with more complicated Pythagorean trees will certainly motivate more use and understanding of that idea. And if you persevere, the study of fractals can take you into more and more complex mathematics.

But first things first. Start with simple grade-school geometry and the Pythagorean theorem. See how far they can take you.

See what you can discover by investigating fractals playfully, starting with simple cases and working you way into the general patterns.

Thursday, October 14, 2010

Math Marketing Done Wrong!

Dear parents of students who attended our first middle school math meeet,

Thank you again for bringing your children to our first middle school math meet! They were a thoroughly delightful group to work with, and they helped our year of math meets get off to a great start!

It has come to our attention that Aileen Leventon, who is apparently a marketing person for a private tutoring business called "Math Done Right," was soliciting parents waiting to pick up their children after the meet, with information promoting that company.

None of the advisors of Albany Area Math Circle know anything about this "Math Done Right" business.

From our perspective, it was "Math Marketing Done Wrong!"

Ms. Leventon did not have our permission or endorsement to market her business to you, nor did she have permission from Hebrew Academy to market her business to you.

Please know that Albany Area Math Circle adult advisors are VOLUNTEERS! We are volunteering our time to create a vibrant mathematical community of kindred spirits who love problem solving together.

We are NOT volunteering our time to provide a marketing platform for a private business about which we know absolutely nothing.

If you encounter anyone attempting to solicit you for a private business at an Albany Area Math Circle event, please bring it to the attention of an Albany Area Math Circle advisor immediately. You can recognize us by our "Tough Traveler" ID badges (see below.)

Mary O'Keeffe
Albany Area Math Circle advisor

Tuesday, October 12, 2010

Thanks to Hebrew Academy of the Capital District ...

... for hosting students from schools all over the Capital District at our October 2010 Middle School Math Meet!

Names and scores of high-scoring students and the value-added whole-is-more-than-the-sum-of-the-parts team award will be posted in this space later this week after the administration window ends.

Thanks to George Reuter and Mike Curry of for writing a great set of problems for our students! Thanks to student coaches Matthew Babbitt and Zubin Mukerjee as well as Albany Area Math Circle advisors Bill Babbitt and Rita Biswas for helping make the meet great. A special thanks to Albany Are Math Circle Advisor and Hebrew Academy math teacher Alexandra Schmidt for all the arrangements she made for us.

Would YOUR school like to host our next monthly math meet, scheduled for Sunday November 7? If so, please get in touch with us by sending an email to

In addition to working on those Math Meet problems, we also talked about fractals and other fun mathematical topics today.

In honor of the recent 10/10/10 day, we talked about powers of ten, powers of two, and 42.

For example, we talked about how 101010 in binary is 32+8+2=42 in decimal, and of course how 42 is a delightfully special number.

(In addition to being the Answer to the Ultimate Question of Life, the Universe and Everything, we noted that 10!seconds = 42 days (exactly!) because

[10x9x8x7x6x5x4x3x2x1 seconds]÷[(24 hours/day)x(60 minutes/hour)x(60 seconds/minute)]=42 days.

We also introduced the concept of "bimal" notation, which is the binary analog of decimal notation.

In bimal, 1/2 = 0.1, 1/4 = 0.01, 1/8 = 0.001, 1/16 = 0.00001, and so on.

This allows us to nicely express the following question:

What is 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + .... (and so on forever)?

In bimal, the answer is 0.111111111111111..... (and so on forever).

But, just as in decimal, 0.999999999...... = 1, in bimal 0.111111111.... = 1.

And we talked about a special kind of fractals, Pythagoras trees. In honor of Dr. Benoît Mandelbrot, who died a few days after our middle school math meet, I have moved our discussion of that topic to a separate post, Maverick Mathematicians and the Power of Grade School Geometry.