Monday, October 18, 2010

Alexander's Fax Machine

I've been meaning to share some activities from this year...both successes and failures not-so-successes.  Here's an activity I did on the second day of school.
Alexander Graham Bell invented the telephone, but if he wanted to “fax” something, he would have to describe in detail the item to be replicated to someone else on the other end of the telephone.  You and your face partner will replicate this.
1.     Everyone should build a wall so your image to be copied and work can’t be seen by your partner.
2.     I will give the window partner a manila folder that has parts cut out.
3.     You and your partner will talk to one another (nothing allowed other than talking…definitely no peeking) and the door partner will attempt to replicate the shape.  The door partner also can’t show her window partner her progress.
4.     Once the door partner has a sketch on paper, she will cut out a manila folder.
5.     Once you have checked and rechecked, show each other your result and see how you did.
6.     Reflect on this process:
How’d you do?
What went well?
What did not go well?
What was hard?
What tools would have made this easier?
What would you do differently next time?

Examples (two columns of originals on the left):
Take away:
language is important
precision is important
articulating confusion is important
checking is important
collaborating and listening is important

Overheard during discussions:
How far from, how big, how long/short, tools for measuring: inches, cm, finger width, nametag width, diagonal, vertical, horizontal, landscape, triangle, isosceles, up/down, positive/negative space, hamburger/hot dog, line, left/right, corner, gap, arrow, width, slope, NW, ___ degree angle, curving, squiggly, acute, obtuse, points
Reflection: * I didn't create the original puzzles with any specific shapes or language in mind...maybe something to think about in the future. * Most students found this super challenging, but I didn't see any of the normal math anxiety that can rear its ugly head. The lack of numbers have anything to do with this? A sense of no black and white right & wrong answer but instead a spectrum of close to far?
*When we switched roles and did this a second time every pair felt better about how they did.  
*I chose not to quantify "how well they did. Felt ok about this decision.

I gained a good bit of insight from their subsequent homework, Alien Encounter.

While I considered this activity a success, I'd still love your thoughts and/or feedback.

Sunday, October 17, 2010

Show Your Work

I can’t speak for other math teachers, but I sometimes fall into a zombie state where I wander the classroom with outstretched arms continuously repeating my math-zombie mantra braaains show your work. Don’t believe me? The best present I’ve ever received from a student (hope I don’t offend any former students reading this who gave me a Starbucks gift certificate) was an art piece by a fourth grader titled “Show Your Work.”

I love my job. But I digress.

I’ve always felt a little bad about telling my students to show my work, partially because I was resistant to doing this when I was a student, especially when I saw showing my work as a pointless exercise. A colleague of mine recently told me a story of her son who, in his early elementary years, would respond to instructions asking him to show his work by drawing his version of The Thinker next to his answer.
I empathize with this stance and sometimes feel that I am in fact going around eating the brains of my students (metaphorically, of course).

Don’t get me wrong or quote this out of context. I think explaining your reasoning is one of the most important parts of mathematics.

This year I have a few particularly studious sixth graders who—off the record—are showing too much work. I wish I had a scanned original, but my version will have to suffice for an example:

So can a student really show too much work?  I think the answer is yes.  First of all, I wouldn’t want students to get turned off by the subject because they felt that they felt forced to do something IF this something isn’t helpful.  Secondly—and this is more subtle—I think that explaining your work can sometimes (I should emphasize sometimes) undermine one of the most powerful aspects of the subject: the ability to use mathematical symbols to tell a story or solve a problem.  The righthand symbolic representation above IS the work.  Here's another example, with the original written by al khwarizmi, an 8th century mathematician everyone should know (although I think it's ok if you don't know how to spell his name):

1200 years ago (translated)
"If some one say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times."
(10 − x)2 = 81x

Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts

100 - 20x + x2 = 81x
x2 – 101x + 100 = 0
(x – 100)(x – 1) = 0
x = 1, 100

Ignoring how the English language has changed (this was translated in the 18th century), I think we’ve come a long ways.

I’ve never had more than a surface level discussion with my students of what I mean by “show your work”.  Students are constantly making decisions about what is necessary (what constitutes reasoning) and what is not (explaining why 1+1=2) and, for the most part, the only feedback I’ve ever given them is “not enough”.  To complicated matters, the same amount of work will sometimes be deficient and sometimes be plenty.  This brings me to audience. I haven’t done this yet, but I am thinking about having a conversation about audience with my kids in the near future.  They’ve been asked implicitly to write for each of these audiences, but I’m hoping to make this even more clear.

Our audiences:
  • The friendly alien (who is intelligent, has a strong command of language, but no experience with mathematics)
  • Classmates who understands the problem, but may have a different solution method (peer review)
  • Classmates who do not understand how to solve the problem (peer assistance)
  • The student
  • The teacher
The audience I want them to write for will depend on my goals of a problem/task/assignment.   Here’s a not-even-close-to-an-exhaustive list of different reasons to show your work:
  • The friendly alien
    - For big picture reflection.   Example: Explain to the alien why THIS is mathematics.
  • Classmates who understands the problem, but may have a different solution methods
    - For problems that lend to multiple solution methods, which I suppose should be every good problem.
    - To share a solution method
    - To compare solution methods
    - Proof
  • Classmates who do not understand how to solve the problem
    - To teach others
    - To reinforce your own understanding (I think Bloom said something about this)
  • The student
    - To ensure your work/reasoning is sound
  • Me- To exhibit mastery

As always, a work in process...