Sunday, October 17, 2010

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):

 Date 1200 years ago (translated) Current Problem "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 Solution 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 = 81xx2 – 101x + 100 = 0(x – 100)(x – 1) = 0x = 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
• The student
- To ensure your work/reasoning is sound
• Me- To exhibit mastery

As always, a work in process...

1. Hilarious - I'm going to use the alien in my class.

2. You didn't ask for a suggestion...so feel free to ignore completely...yet I think it might be helpful.

A few years back my students and I had a very frank conversation about "Show your work" and all the gray hair I was getting by saying it so often. Long conversation later, to the students in all came down to "I have to show this so that you know I didn't cheat. You don't really need to see it." I was horrified because that was not the intent. I wanted to see the steps in their thinking. One student said, "Well then why don't you just say that. Jeez, you want us to be precise, so why aren't you?!?"

I identify with the "love my job" sentiments.

Anyway, from then on I have always asked students to show the steps to their thinking. Amazing what a huge difference a few words have made. Students have been willing to show how they thought about the problem/task and then compare with each other.

3. @Norlin: Steal away! Glad you found my alien audience amusing. My students do also...especially once they realize that our alien is a benevolent ruler.

@A: While it's true that I didn't ask for suggestions, they are always welcome. Especially spot on suggestions. Your comment reminded me of another reason that I've asked my students to show work...to increase their score on a test. Blah...And I don't know if this was intentional, but I like your distinction between "show your work" and "show your steps to your thinking" or "show your reasoning."

4. I really like this post, and It got me thinking a lot about the value of showing work in the subject I reach, physics. I wrote up a post with some of my musings: On the value of showing work and corrections.

6. I forget where I first saw this idea, but I have found it extremely useful.

Instead of exhausting myself repeating the zombie math teacher mantra, I've started rephrasing problems (especially for my Algebra 1 students) so that students have no *choice* but to show their work.

What is the point of arguing with the demands of a problem like this one?

Ex. #1: The solution to 3(x+2) +5 = 20 is x=3. Demonstrate how and why this is so.

My sense is that what's most frustrating (and enervating) is getting into a contest of wills with an adolescent. So for the sake of one's own sanity, why not reframe the interaction in a way that will work well for both parties?

This problem has something for everybody. It gives me the demonstration of problem-solving, computation, and mathematical reasoning I need to assess their progress. And they get a clear task statement, along with the training and practice they will need to develop thorough and appropriate problem-solving habits.

Anyway, it's been working so far.

Elizabeth

7. The classic (super-annoying) response is: "But I already know the answer. Why do I have to do anything?"

Kids who have been trained that the only important thing in math class is to find the answer tend to have a difficult time conceptualizing that that is not the be-all and end-all of mathematics. I've found that the "good" students sometimes have the most difficulty with this idea

8. As a teacher in training I have appreciated the original post and all of the comments. They will stick with me as tools of the trade - especially the choices: of audience, of words, and of approach to the problem. Giving them the answer and asking them to demonstrate how and why is a great paradigm shift. Especially if you throw in a wrong answer once in awhile to appease the contrarian inside. Another way of building confidence through play; keeping them on their toes.