Stuff I Always Say
Here are a few things I say over and over again to my students, with good reason, I think:
Hunter-gatherer brain: We have more or less the same brains as our hunter-gatherer forbears did tens of thousands of years ago. This is why math was invented; our instincts were inadequate for some of the harder problems we discovered. We need to continually balance our instincts with mathematical rigor.
- One thing that differentiates us from our hunter-gatherer forbears is paper. Use it! It makes us smarter. Paper is more useful if we can
trust the paper!As math gets more complicated, we humans can't keep it all in our heads. By writing what is true, rather than "scratch work," you can keep track of where you've been and where you're going. You will earn your own trust and learn to trust what you wrote, helping to transcend the limits of your hunter-gatherer brain.
If you want it right, write!Mental math is a wonderful thing, but if you are the least bit confused or unsure of what to do next, start writing. Even if you don't know what to write. Don't skip steps unless you are sure you know what you are doing.
- Use your curiosity to
make it make sense!When you learn something, don't just learn it by rote; think about it deeply until it makes sense to you. That way you'll remember it better and will be able to know when to use it and when not to use it. It's also way more interesting that way.
Be mindfuldoing math. Try to avoid the "What am I supposed to do?" mentality. Instead, take stock of where you are and where you need to be and use all your wits, your
whole brain,to get there. Often it's not so much about remembering as about figuring it out, but that only works if you relentlessly demand that the math you learn makes sense.
Plug in what you know; solve for what you don't!Often a solution is hiding in plain sight and you can find it by this method. A classic case is the problem of finding a slope-intercept equation given two points. What you know are the two points, so plug them in to the formula for slope. Then you know the slope, so plug in one of the points and the slope into y = mx + b, and solve for what you don't know, b. You could figure that out even if you didn't remember the technique. (Of course you don't need to find b at all if you can use point-slope form!)
Check your work!Checking your homework (1) ensures that you are practicing correctly; (2) gives you practice for checking during tests; and (3) gives you a chance to step back and see what you've done.
Say what you mean!I learned this from one of my most influential mentors/professors, John Dyer-Bennet of Carleton College, along with this next one:
Draw a picture!Sometimes a picture can really help guide intuition.
- Coaches say
you play how you practice,and that's true for math too. If you skip steps, rush through calculations, make mistakes, and don't check your work when you do homework (practice) you'll probably do that on the test too, despite your intention to do it better when