Reacting to Dan Meyer’s “Developing the Question” post

Yesterday (August 14, 2014) I saw this post from Dan Meyer, which is part of what I’m sure will be an interesting series:
Developing The Question: Good Work!

Something made this post stick in my mind for the rest of the day, though even now I have not been able to put my finger on precisely what it was.  After the kids went to bed last night I decided to use one of the problems in the post as a starting point for a little math project this morning.  My hope was that the combination of thinking through and then going through a little talk would help focus my thoughts on the original post.  It failed in that regard, but talking through some of the ideas that were motivated by Meyer’s post was pretty fun nonetheless.

We started with a slightly modified version of one of the problems from the MathArguments180 blog:

Three Questions about three circles

I made the problem with four circles and a square rather than three circles and a triangle so that my younger son wouldn’t have to worry about finding the area of a triangle.  Stepping through the three questions got the project off to a great start:

One of the things about this problem (and all three problems referenced in Meyer’s post) that I thought was interesting was the wide variety of possible extensions. One of my favorites goes back to multiplication. Chopping up a rectangle into smaller rectangles is an neat way to show a geometric interpretation of multiplication. Perhaps a little more surprising is that you can use the exact same idea to show why a negative number multiplied by a negative number is a positive number:

The idea from the negative times a negative piece of the last video has some other fun geometric extensions. By coincidence, yesterday my younger son and I were talking through a neat problem about a cube with some holes in it. He actually built a model of the object out of snap cubes to help him understand the problem and we used that model today. Yesterday’s question was about the number of 1×1 square pieces on the surface, so to mix it up a little today I asked about the number of cubes in the model. My goal here was to show how the “taking away more than you had” idea from the last video gives us a new way to study this cube:

For the last bit of the talk we went back to the whiteboard. I wanted to show an example from geometry that would look similar to the first problem, but might give the boys a chance to understand the “take away more than you have” example in a slightly different setting. They saw the problem in a little bit different way, though. That’s fine, obviously, so I ended up just showing them the “take away more than you have” approach at the end and showing that both approaches lead you to the same answer.

One last bit that I wanted to talk through that I didn’t have time to get to was something from an old 3D printing project we did. At the end of this post – Learning from 3D Printing – are some pictures showing a solution to the “Prince Rupert Problem.”  Learning how to create the red shape in Mathematica was a really fun project whose main idea involved breaking up a space into pieces you already could understand.    Fortunately Mathematica has built in capability to add and subtract shapes in 3 dimensions.

Anyway, I did really enjoy the 3 questions idea from Matharguments180 blog.  It was also fun to talk through some of the various fun and surprising extensions of the ideas from the first problem.  I may update this post if I am ever able to put my finger on why Dan Meyer’s post stuck in my mind, but for now the result of that stickiness will just have to be a fun little math project with the kids this morning.

ALS and the Ice Bucket Challenge

My high school math teacher, Mr. Waterman, died from ALS last year.  There’s almost no way to explain the influence that he had on me as a kid.  From math, to teaching, to simply understanding how one person can have a tremendous impact on another person’s life, I am a different and better person because of him.   Earlier today  I was tagged by another student of his for the Ice Bucket Challenge.  Here’s my video – I hope the publicity from this challenge raises buckets of money for ALS reserach:

For most of this week the Quanta Magazine piece on Maryam Mirzakhani winning the Fields Medal had me reflecting on my time in high school. Here’s the article, it is absolutely fantastic:

A Tenacious Explorer of Abstract Surfaces

The bit that got my attention was this paragraph about an experience she had in high school:

“Eager to discover what they were capable of in similar competitions, Mirzakhani and Beheshti went to the principal of their school and demanded that she arrange for math problem-solving classes like the ones being taught at the comparable high school for boys. “The principal of the school was a very strong character,” Mirzakhani recalled. “If we really wanted something, she would make it happen.” The principal was undeterred by the fact that Iran’s International Mathematical Olympiad team had never fielded a girl, Mirzakhani said. “Her mindset was very positive and upbeat — that ‘you can do it, even though you’ll be the first one,’ ” Mirzakhani said. “I think that has influenced my life quite a lot.”

So, she asked the school principal to provide a problem solving class and a year or two later won a gold medal at the IMO, and followed that up the next year with a perfect score.   Good gracious.  Even several days later I’m not sure I can wrap my head around that.   That has to be one of the most amazing accomplishments (in any field) that I’ve ever heard of in my life.

That insane accomplishment aside, it also shows the influence that principal and the teachers had.  I was pretty lucky, too, to have a great principal,  Dr. Moller,  along with Mr. Waterman.  Dr. Moller had a effectively infinite trust in Mr. Waterman had gave him enormous room to teach how he wanted.  We didn’t have to beg for a problem solving class, for example, because it already existed!   Linear Algebra and Differential Equations, too.  Really amazing and probably not what you’d expect from a high school in Omaha.  Truthfully, though, it was only many years later that I realized Dr. Moller’s role, but the older I get  the more I realize how important it was.  The article about Mirzakhani probably made me understand the luck I had in this area a little better.

There was a nice article about Mr. Waterman in the Omaha World Herald after he died:

John Carl Waterman inspired as Omaha teacher and coach

And a nice letter to the editor from Dr. Moller, as well:

Waterman was excellent instructor

I was John Waterman’s principal during his career at Omaha Central High School (“Waterman inspired as teacher and coach,” Sept. 29 World-Herald).

He had a rare talent for making weak math students believe in themselves and succeed in learning math, often for the first time, and he inspired strong math students to achieve even more than they ever expected. His math teams’ records still are among the best in Nebraska high school competition. He was among the first in the state to recognize computer capabilities for teaching math, and he implemented computer programs and activities that were cutting edge for the time.

John has former students working all over the world who would testify that he was an exemplary teacher taken from us much too early.

G.E. Moller, Omaha

You never know how someone is going to influence your life, but I certainly was the beneficiary of some good luck in high school.  I sure hope the Ice Bucket promotion raises lots of money for ALS research.