Thursday, May 28, 2009

Trig Rummy

My calc students have really been struggling with trig substitution, mostly, i think, due to lack of fluency with trigonometric functions. I'm a big believer in games for skill practice, as they are an engaging context that encourages more practice and ideally offer a chance for forging new connections.

So today we played Trig Rummy. Many of my card games are based on concentration, rummy, or go fish. And occasionally Euchre. (I still think the partially ordered quadrilateral Euchre has a future.) Links for printables are after the rules. I think it would adapt to high school trig pretty easily. I'd replace the calculus relations with graphs and triangle identifications.

Trig Rummy
Objective: the player will practice and gain facility with trigonometric identities and calculus relations.

Players: 2-4

Goal: get the most cards in play.

Set up: randomize cards and deal 7 cards to each player. (You may want to introduce the game with only 4 cards as 7 is overwhelming.)

Play: On your turn you may do one of:
• take any card from the discard pile (not just the top card).
• draw a new card.
After that you may do either or both of:
• pick up any sets you have.
• play any new sets you have.
At the end of your turn, if you did not play a set, discard a card.

• if at any point the discard pile contains a complete set, the first player to otice can cal “Rummy!” and take the set out to play for themselves.
• you can not play matching operator cards, like a pair of d/dx cards.
• There is a wild card, which you can choose to be -1× (times -1), or d/dx, or ∫ ⋅ dx.

End: After the last card is drawn from the deck, each player gets one more turn from the discard pile. Then the cards are counted up and the player with the most cards played wins. No penalty for unplayed cards.

Variation: Play until one player is out.

Example plays:
• Play sin2(x) matching 1 − cos2(x).
• Play d/dx with sin(x) matching cos(x).
• Play ∫ ⋅ dx with sec(x)tan(x) matching d/dx with ln|sec(x) +tan(x)| (since both are equal to sec(x).)

If a player lays a mistaken combination and no one catches it before the next player’s turn, they stay in play but turned face down. If someone does catch it, that player has to pick up the pair.

Feel free to use a trig cheat sheet.

Rules PDF - includes a trig cheat sheet
Trig Cards PDF - for 54 2-sided cards of trig identities and calc relations. Could use for flash cards or other games also.

Tuesday, May 26, 2009

Another Math Blog

Which, come to think of it, would have been a good title for this blog.

I've been cosistently loving a lot of the links given by Y of X. It records the musings of Hendree Milward, a Connecticut math teacher (at Tunxis Community College?). Recently he's posted interesting links to the cool Institute of Figuring and an offbeat math poetry blog. If you read his blog, I won't have to bother putting up those links here.

Monday, May 18, 2009

Stigler on training teachers

Michael Goldenberg over at Rational Math Ed pointed this out on his blog and it's worth spreading out.

See this thread at the Math Forum for a transcript. Dr. Stigler is the co-author of The Teaching Gap and The Learning Gap, which distill research from The International Math and Science Study (TIMSS) into fresh and relevant information for any teacher, but especially math and science teachers.

The big points to me:
-Teaching is cultural => really hard to change
-Teaching needs to change <= student achievement data
-Teaching can only change gradually, but it can change, by focusing on teaching instead of teachers, and evaluating based on students' thinking and reasoning.

A relevant quote: "A review of research by Hiebert and Grouws identifies two features of classroom instruction that are associated with students' understanding of mathematics:
- CONNECTIONS: Making mathematical relationships-among concepts, procedures, ideas-explicit in the lesson
- STRUGGLE: Students spend at least some time struggling with important mathematics."

Check it out.

Wednesday, May 6, 2009

Ken Robinson

So I showed the Ken Robinson video to my class on the first day and it was quite interesting. The video started with medium engagement, but by the end everyone was tuned in. The story about the choreographer (Gillian Lynne) seemed to really connect.

The first connections were to elements of their own stories, and were interesting. Then one of the students brought up a connection with how Robinson describes us as being educated out of creativity. Her sister is interested in being a soloist, and it took ears of retraining or untraining to get her out of what she had been taught to do as a choral member. I'd never heard of that before, but it connected exactly to the point I wanted to make.

They have been educated - and successfully so, as college calc students - in school mathematics, which has almost nothing to do with mathematics as practiced by mathematicians. (Which really needs a descriptor. Real mathematics isn't going to interest anyone. Creative mathematics? Cool vs. school or cruel mathematics?) Instead of being able to repeat what someone shows you, it should be about solving problems that you don't know how to solve. In which situations mathematicians excel, because they are entirely willing to be wrong, and even glad to be wrong if they learn from it. They're willing to, as my friend Dave says (quoting his favorite Australian, Brian Cambourne) to "Give it a go!"

As long ago as 1982, my calc instructor, John Hocking, worked mightily to convince us to have no fear. To be wrong 100 times if it teaches you something. That math was exploratory, and creative. He shared his topological research with us... which was amazing and fun. He's the reason I added the math major in the first place. Often he put it as "blank pages are just waiting to be filled."

Now can I help my calc students to do the same?