Saturday, May 18, 2013

Talking with Children: Shape Centers

I have been spending considerable effort looking for situations to "mathematize" in my daily interactions with students. Sadly, upper-level students are so mark and answer focused that they spend little time wondering about emerging problems with me.

This is not the case with my 8-year old friend.

While he was inventing his word problem, he stumbled upon the idea of a middle. Specifically, he told me that four train tracks met in the middle. I quickly asked him if three tracks could also meet in a middle. He responded with an annoyed, "of course".

A problem was born.

I drew a circle and asked him to find the center. He traced his way around the circumference and made some chords across the shape. When he was satisfied, he placed a dark dot in the interior of the center and smiled. "There".

I asked him how he knew he was right. He told me that he found the point that was the same distance to each edge. (At this point I thought about asking him how many edges a circle had, but I decided to stay the course.)

I drew a square.

We repeated the process. He used the vertices and tried to find a point that was equidistant. When I asked him if he could use the sides, he discovered that it yielded the same point. He was a little weirded out, but we pressed on. 

I drew a rectangle.

This time his strategy changed. He couldn't seem to find a way to be the same distance from each side. He placed a point and tried to measure by "pinching" the space with his thumb and index finger. Finally, he erased all his points and traced the perimeter of the shape. He then continued to trace another rectangle inside that one. And another, and another. As he worked, it was obvious that his brain was miles ahead of his hand because the lines got sloppier and sloppier. 

When he was done, his rectangle looked like this: (I asked to keep his work)
He proudly explained that tracing his way into the middle of the shape provided a perfect center. I was fascinated by this process, and asked him to apply it to his square.
Interesting. I drew a triangle, and he hesitated. I could tell that he thought triangles had no center. Nonetheless, he pressed on:
I was about to add a shape when he stopped.


He went on to explain that a center had to be a dot, and a dot had to be a circle. Using his method, he was creating smaller and smaller rectangles, squares, and triangles. He went on to show me that the dot in the middle of his triangle is just another triangle if "you put it under a microscope". In fact, he stated that there could never be a center to any shape because the dot would be round and the shape would not be. It could therefore not be the same distance from each side. 

I was impressed with his ability to iterate into the infinite. Not only was he constructing geometric knowledge, he was now struggling with the basic structures and how they related to infinity. (Infinity has been his favourite topic for years). 

I asked him if a circle has a center.

He waited and responded... "No".

He explained. Each circle could have a "dot" as a center, but that dot is just a smaller circle, and must have a "smaller dot" as it's center. Each dot can always get smaller "until it is only a molecule thick".

And if that is the case, "then there can't be a middle for anything". He slumped his shoulders with the weight of this reality. 

Here we have a young boy struggling with his personal construction of logico-mathemaitics as juxtaposed to the notation that is meant to describe it. He has been introduced to the formality of "points", "edges", and "centers" without ever having the opportunity to construct a deep understanding of what a center is. He is left at a crossroads between his own construction of reality and what the school worksheet tells him.

It is a sad predicament--one created by the burdens of time, curriculum, and the dreaded "test".


Tuesday, May 7, 2013

Talking with Children: Word Problems

My wife and I spend a lot of time with friends who have three young children. I spend most of that time engaged in a combination of trampoline dodge ball and mathematical discourse. I have begun to document the snippets of conversation on the "Talking with Children" page, but decided that larger ideas warrant their own post.

The middle child is most willing to think mathematically. During one of our conversations, he decided to turn the tables. What resulted is a wonderful look into a child's perception of what "mathematics" does. 

Him:  Maybe you can answer my question?
Me:   Sure. What is it?
Him:  Ummm... (literally scratches head)

I could tell that he was reaching for straws, but just as I was going to suggest some possible pathways, he began detailing his problem. What followed was a hodge-podge of textbook drivel. Roads met at points, trains travelled at speeds, the vehicles took neatly-timed breaks, and the whole thing finsished with a simple question:

Will the trains crash?

After the initial digestion of the problem, I asked him to draw it out so I could get a visual. Despite major changes in the question, he managed to draw this diagram and detail the problem:

His "textbook ready" diagram
Ok... um. There are 4 trains. One of them is 3km long. Another is 2km long. Another is 5km long. Another is 1km long. The 3km train goes 2mph. The 2km train goes 3mph. The 5km train goes 3mph and the 5km train goes 3mph.
All the tracks are five miles long. The 5km train stops at a stoplight for one minute. The 3km train stops for 30 seconds. Will the trains crash? 
There are so many awesome things going on here. First, we must keep in mind that the child's natural curiosity has been polluted by school mathematics. He constantly shows desire to make sense of his world through mathematics, but when required to produce a question, he can only think of manufactured, one-size-fits-all textbook problems. He hasn't even covered the mathematics of rates but knows that, when doing "math", it is important to list every detail neatly and orderly. He knew that each track needed a length and each train needed a speed. He even managed to mix up his units--for good measure. (Pun intended).

Second, the child makes the problem more difficult by adding more details. When I asked him if he could make the problem easier, he told me I could take away one of the trains. To him, more details always equals more difficulty. I asked him to make the problem more difficult, he paused and then told me that an asteroid hits Train 1.

Third, he assumes that there is no real basis for the questions his teachers ask him. He knows that they include real life objects (such as trains, stoplights, and asteroids), but has no inkling that mathematics is used to model and describe real life. He also has a sense of mathematical neatness. The entire situation is tied up with one, nice question: Will the trains crash?

We have to be constantly aware of what are students are perceiving as important details. We have built an educational culture where all the details are readily available to those who are willing (or are literate enough) to search for them within drawn out paragraphs of textbook banter. Students need to be presented mathematical situations where they are required to make decisions, gather resources, and apply strategies. I'm worried that school is not offering opportunities for students to reason and model mathematically. What would your students create if asked to write a math question?