Today my Geometry students presented their awesome projects which explained how to measure the field at our school with trigonometry and a cross staff. All of them had invented a different way to do it and some were shockingly close to the correct answer of 330 feet from baseball diamond to baseball diamond. I was so glad that I’d given them the freedom to solve the problem like they wanted to because some of the best solutions were entirely different than the idea I’d copied from reading about the triangulation of India in “Trigonometric Delights” by Maor. If only all trig books could be this awesome it might not be such a hated and dull subject in high school. (Though I’m sure that there is much more to math that the textbook-the solution to our math woes couldn’t be as easy as choosing the correct book.)
During the presentations today one group showed up how they had measured the sides of a quadrilateral that they made with cones on the field. They said that they didn’t measure between the bases because they didn’t have time after they figured out how the cross staff worked and returned from illnesses. They claimed that they had achieved the goals of the project because they’d learned how to use the law of sines. Another group had followed the rubrics suggestion to use 3 or more triangles to solve the problem and came up with an answer that was 100 feet off. They calculated that an error in 3 degrees of measurement in each angle could result in 100 feet of error in final measurement. They maintained that they had completed the goals of the project because they met the rubric requirement of multiple triangles and checked their work and analyzed their errors. A third group joined the rather heated discussion and claimed that their project was the best because they had come within 10 feet of the correct value. They were shot down by another group who said that because they had only made one right triangle they didn’t use the law of sines and therefore did not learn what they were supposed to. There seemed to be a general consensus that the law of sines practice was the primary goal of the project. I asked “couldn’t solving a real-world problem using math be the main goal of the project?” Some thought that the law of sines was quite important. I countered “I hate to admit this as a math teacher but most of you will probably not use the law of sines in your life unless you have a surveying, engineering, or architectural job. You will all be creative problem solvers.” And it got me to wondering just what we were learning and how to assess it. This project counted as their test grade for the law of sines unit. I’m pretty sure that most of them know how to use it though I did uncover plenty of misconceptions during the process. I think that kids learned more because we were able to discuss and correct misconceptions that led to nonsensical answers. I wished I’d been videotaping the loud and rowdy fight about the point of the assignment because it said a lot about what students perceive the point of math education to be. To calculate things. To find answers. To practice formulas. To solve problems. To get accurate answers. To solve an interesting problem. To solve a problem for the sake of fulfilling a rubric.
In other news, someone stopped by to collect my census data today and when i gave him my phone number he asked where in Eastern Washington I was from. It turns out that we both went to the same college, both majored in math, had the same professors and both sell metal art for money. Forty-five minutes after we finished collecting census data we were out in my garage looking at a bunch of wood I’d collected talking about art projects we could make in his garage down the street. He’s also going to graduate school but in mechanical engineering. I proposed we have math parties. Fun Fun. I’m not usually blown away by “small-world” coincidences because I realize how likely it is for social groups to overlap when you do the math. But at one point in the conversation he said “wow, we are like the same person.” There are some great “six degrees of separation” lessons that show how matrices can be used to calculate how likely it is for you to have a common friend with someone you meet on an airplane. There is a fantastic Kevin Bacon number website that calculates how many movies link Kevin Bacon to any other actor. The math involved is fascinating and definitely captures kid’s interest.
Wikipedia, always my favorite lesson planning resource, has a neat article explaining the idea. http://en.wikipedia.org/wiki/Small_world_experiment