My teaching experience is… limited. I’ll be the first to admit. Nonetheless, even substituting for a semester, there are some things you pick up on pretty quickly. Perhaps some of them are false-lessons to be unlearned later. But maybe not. In any event, I read “amen, brother!” when I read about a new style for teaching math:
Many students were sent to him because they had severe learning disabilities (a number have gone on to do university-level math). Mighton found that to be effective he often had to break things down into minute steps and assess each student’s understanding at each micro-level before moving on.
Take the example of positive and negative integers, which confuse many kids. Given a seemingly straightforward question like, “What is -7 + 5?”, many will end up guessing. One way to break it down, explains Mighton, would be to say: “Imagine you’re playing a game for money and you lost seven dollars and gained five. Don’t give me a number. Just tell me: Is that a good day or a bad day?”
Separating this step from the calculation makes it easier for kids to understand what the numbers mean. Teachers tell me that when they begin using Jump they are surprised to discover that what they were teaching as one step may contain as many as seven micro steps. Breaking things down this finely allows a teacher to identify the specific point at which a student may need help. “No step is too small to ignore,” Mighton says. “Math is like a ladder. If you miss a step, sometimes you can’t go on. And then you start losing your confidence and then the hierarchies develop. It’s all interconnected.”
This was precisely the problem I ran into when trying to teach a second grade girl to approximate and add. And this was how I finally got it through. You simply break it down into as many steps as humanly possible. I wanted to jump ahead straight to “Take 76, round it to 80, then take the 19, and round it to 20, and you get 100,” which was obviously too much. So I stepped back and said “What does 76 round to?” and she had no idea. So… another step back… the number that 76 rounds off to is going to be one of two numbers. Which ones?” and on to “What’s the first number in 76?” “7″ Okay, so take that number, or the next number up, and those are the two possibilities. So what are the two possible numbers you might round 76 up to?” Her first guess was 78, but we got there until I destroyed her confidence.
I’m always skeptical of claims that “any kid can learn up to college level math,” which the article suggests. But I do believe that there is more variability than Half Sigma and the like think. At least there is where there’s motivation, which can be the bigger nut to crack.
The other thought is that this demonstrates the tremendous need for tracking. Take some second graders and try to start with “What’s the first number in 76″, they’re going to go absolutely crazy. This completely and entirely fails to bother some people, but perhaps due to my experiences it does bother me. And it’s a waste of their talent. The notion that “we shouldn’t worry about the really smart kids” because they’ll have the smarts to take care of themselves completely ignores the fact that it’s the smart kids that will be using their education to make this country better for the less smart ones. And while I may disagree with Sigma on the extent to which the left side of the bell curve can be taught, I am in full agreement that you have to approach different aptitudes differently. And just as you don’t want to throw the answers at second graders, like I tried to do, nor do you want to bore the quicker kids to death by starting at a point that is going to be intuitive for many.