Danielson, doing his best Howard Beale:
THE STEPS WIN, PEOPLE! The steps trump thinking. The steps trump number sense. The steps triumph over all.
Let me add to the conversation the category of “steps that are correct but useless.” These are great. They come from a conversation I had, like, fifteen minutes ago with a teacher named Leah Temes here at NWMC 2014.
Leah teaches Algebra II. We were talking about solving systems of equations. It’s really easy to teach the solution to a system like this as a series of correct, useful steps:
2x + 3y = 10
5x – 3y = 4
- Add the second equation to the first one.
- Solve for x.
- Substitute x in either equation to solve for y.
- Check that pair in the other equation for full credit.
Leah said she was tired of seeing her students mimic those correct steps without understanding why they worked. So instead of showing her students steps that were useful and correct, she asked them if she was allowed to add the following two equations:
2x + 3y = 10
5 = 5
2x + 3y + 5 = 10 + 5
Is everything still correct? Yes.
Was that useful? No.
This experience awakened her students to a category of steps in addition to the correct and useful ones they’re supposed to memorize and the incorrect and useless ones they’re supposed to avoid – correct and useless steps.
Alerting your students to that category of steps may make math seem less intimidating and more interesting. Math isn’t any longer a matter of staying on the right side of a line between the incorrect and correct steps. There’s another region out there, one that’s a bit less tame, a place for explorers, a place where the worst thing that can happen is you did something right but it just wasn’t useful. That category of steps also requires justification – “how do you know this is correct?” – which can help bend the student away from memorization and back towards understanding.
BTW. All of this implies a fourth category of steps – incorrect but useful. Can anybody give an example?
I do a similar thing when solving equations in one variable by asking students if I can add 1,000,000, let’s say, to each side of an equation… or if I can subtract 27 from both sides… or divide both sides by 200… etc. etc. We talk about what is “legal” (have we followed the rules of algebra and the concept of “balance” and equivalence?) and what is “helpful” (have we done something “legal” that helps us isolate the variable so we can solve this thing?”) Exaggerated examples like adding 1,000,000 to both sides seem to make an impression on kids.
I have long been a fan of deliberately sabotaging a solution to something that I might be doing on the board so that somewhere down the road things become obviously wrong. This is so students can start to develop strategies for what to do when this happens.
Many will tell you that it’s important for students to make mistakes (in fact, that they learn the most when they do). But that sometimes runs counter to what they see in class. That is, a teacher demonstrating flawless execution of mathematics. Even some of our best students often won’t even attempt a problem unless they are sure they will get it correct. If they are ever going to become comfortable with making mistakes as part of the normal process then we have to include managing those mistakes as part of our day to day in class.
[It's] incorrect but useful to estimate things like area problems, in order to find out a ballpark figure and check if you’ve done the math right.