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.
- 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 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
- 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...