My geometry students have actually started asking to find out about what we are learning in Calculus class. It all started when I realized(thank you Randi Engle) that showing my students how geometry would be useful in real world Calculus problems might increases their ability to transfer knowledge to new situations. I’d already been showing kids where geometric formulas appeared in the long Calculus problems left over on the board from the class before but now I had research based impetus to really spend some time trying to explain to kids why Geometry was invented and what it has to do with physics, velocity, motion, science, etc… I taught them that the integral symbol meant area under a curve and had them do some simple integrals of linear functions first. Then I got a bit more ambitious and had them solve a problem which gave a velocity versus time graph and asked a number of questions about position and acceleration. They did surprisingly well, especially one particular 9th grader who makes fantastic logical leaps. He tends to solve challenge problems first and get really excited about pushing his brain. I hope that he’s thinking that this is one of the first classes where he’s truly been able to use all of his reasoning abilities. There are a few students who know that Calculus is not going to be on a test any time soon and sit back and relax while other kids dig into the hard problem. I know that my methods of show and tell with the Calculus problems are not ideal and that I should have more carefully designed a learning environment where I’d be able to check for understanding a bit better. The enthusiasm and willingness to ask questions by the vast majority of the class seems to outweigh the cost of not having a carefully structured worksheet that allowed me to see if kids were getting things. I think kids like lessons that they see developing on the fly. it’s as if they have gotten the teacher off track and are learning something outside of the curriculum and somehow that is more exciting.
Today I tried to teach the students that there are infinitely many prime numbers. I think I did a fairly good job at dispelling some notions about prime factorization and divisibility but I’m pretty sure that only 15% of the class understood the mechanics of the proof by contradiction. I didn’t even try a repeat performance the next period and am actually considering finding an SAT problem involving number theory so the lost and confused kids don’t think I wasted 45 minutes of their time confusing them. I suppose it makes sense that people struggling with basic two column proof would find proof by contradiction subtle and confusing but I suppose it is worth doing something really hard once in awhile just to push the limits of what is possible.
Eighth period wanted to do logic puzzles and I gave them one my good math major friend taught me over Pitas late Friday night after a thrilling econ lecture. (You can tell how cool I am right?)
Suppose you have eight bottles and one of them is poisoned. Your job is to figure out which one using three mice. If you give the mice the poison they will die in exactly 24 hours but not have any symptoms until then. You have 24 hours to find the poison meaning you are not able to give the mouse some poison and then wait and see what happens before giving it a different bottle.
How can you find the poison? It’s fine to mix bottles, put the mice in cages, or do anything else that seems logical in the situation. I promise the answer is quite deductive, involves binary numbers and is not going to make anyone groan in frustration.
Here are the two logic problems my kids brought to class.
A man was found dead in a room with a rock. How did he die?
You meet two men on a fork in the road. One always tells the truth and the other always lies. One road leads to death. What one question can you ask them to figure out where to go?
The answer to the first is that the man is superman and the rock is kryptonite. The answer to the second is something you’d be able to figure out without asking questions or introducing additional stories or circumstances.
The take away question for me is: What are my kids learning from all of this? They do know how to count to 31 in binary on their hand now and how that relates to finding poison and computers. Is knowing this better or worse than knowing a theorem about circles and chords? None of it is on the SAT directly so does it matter that I pick what seems to make them happiest? Joy=investment= learning?
I read an interesting article today that sums up so many of my feelings about mathematics. It was written by a mathematician who quit teaching at the university to teach “real” math to students. Math that involves conjectures, ideas thinking, justification, proof, beauty.
“Why aren’t we giving our students a chance to even hear about these things, let alone giving them an opportunity to actually do some mathematics, and to come up with their own ideas, opinions, and reactions? What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources— beautiful works of art by some of the most creative minds in history— in favor of third-rate textbook bastardizations?
The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Hereis how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.
But a problem, a genuine honest-to-goodness natural human question— that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of mankind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them). A good problem is something you don’t know how to solve. That’s what makes it a good
puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions.”
I’m not sure how I feel about the part where he says schools of education are complete crocks but I do like his belief that mathematics is beautiful and interesting and can arise naturally from the world. I’m glad that I’m going to enter a school of education and spread my profound appreciation of mathematical beauty to the world. His opinions about geometric proofs made me feel better about the time I spend doing logic problems instead of proofs. I do need to come up with some more problems that are actually related to Geometry however. And at least take a peek at an SAT because those problems tend to be decent. (Or at least more likely to assess real understanding than what I find in most textbooks.)
He has a vision for math classes rooms where kids solve problems and learn skills so that they can solve more problems. He worries about the lack of mathematical ability of most teachers. But I suppose I’ll be able to fix all this. Or at least in a few situations. And I have a new problem for tomorrow.
Check out the complete article at this website.