Friday, September 17, 2010

Is there a natural path to formulas?

Can there always be a natural path from the concrete to the formal for all abstract math ideas? This is the question that Dan Meyer was pondering during the Math 2.0 Elluminate session on 8/25/10 so he challenged his audience to help him to find a natural path to solve a perplexing mathematical object (PMO) namely "How do you turn the “rise over run” (counting units) method of finding slope of a line into the more formal/abstract slope formula: Y2-Y1 / X2-X1? (You can hear/watch Dan's 4 minute challenge below.)


In other words, once your students get a handle on finding slope from the graph, how do you get the students to use the formula exclusively? 

Dan said that his students will stick to the “lower level” skill of counting squares on grid paper to determine the rise over run and resist using a more efficient method where they can just plug in numbers into a formula. For example, here is a typical problem which expects you to use the slope formula.

Find the slope of a line that passes through point A:(-4, 2) and B:(8, -3).

Figure 2
The more intuitive and concrete approach is to plot the points and draw the line first, then count the units for rise over run and get the slope that way. But Dan wants his students to use the x and y coordinate values of A and B and substitute them into the slope formula to get  2-(-3)/(-4-8)  equals -5/12 answer. How do you motivate that?

Dan answered his own question in response to this question posed by Colin: "Do any of your students start asking you for the shortcut before they get involved in the problem?"  Dan said he would play "dumb" and see if his students can help him come up with with the formula. But this is only the first step. They still need training wheels until the idea of slope and the formal formula come together in a meaningful way. This can take a long time and students need to be reminded of the connection often before they can just fly with the formula. What is even more important for me is that if my students forget the formula they are able to recreate it from what they know about finding slopes from graphs. That means that the student really understand (or owns) that idea. That's all fine and dandy, but what if progress hasn't been made before the end of the class and the test is the next day? I would postpone the test, but if that's not possible. I would probably give away the formula since unlike the teacher in the comic above I don't have all day to wait for the eureka moment. There will be more teachable moments for me to try again. That's the best I can do. But in the long run I'm not optimistic that I can garner enough such moments to make a significant difference here. The challenge of finding all the natural paths is a tough haul for teachers if the overall trip is still the very scripted, but unnatural road to calculus. There is a better way to get there, but it requires a different way to look at things. We need to make he trip be intrinsically interesting.

Sure, knowing it from memory helps but for students who desperately seek a formula to memorize (like the girl in the comic) I would work on a better way to scaffold it so the student sees the connection between the concrete (the graph) and abstract formula. But this will always be an uphill unnatural effort. It requires something more than just a clever way to bridge the gap between the concrete and abstract. It needs a better context that gives students a reason to want to do it.

The most effective context I found to get kids to “own” this concept and facility with the formula was after playing several rounds of Green Globs. The students learned the necessary skills not because they had to, but because they wanted to. It was very useful for them to learn how to blow up more Globs with one linear function because they would get a higher point total and the reason that is so important more important is because in two weeks they would be competing in the Great Green Globs Contest. You see the more globs you can blow up with one line the more points you get. For example (figure 3) the function y=1.1x + 1 knocked out three globs for a total of 7 points. (1 +2 + 4).

Figure 3
With just a slight change in the slope (figure 4) an additional glob goes down and you get score of 15 (1 + 2 + 4 + 8) points instead. The value of the glob doubles for each additional glob that is hit with one line.

Figure 4

Understanding how Green Globs can inspire students to want to do math for its own sake is what makes finding the slope formula a very natural process and is at the heart of powerful learning.

Here's Guillermo teaching his
classmates about Globs.
Take what happened to Guillermo an unmotivated 8th grade math student who was introduced to Globs by his teacher. Using the slope formula was just the beginning for him. Once he discovered that you can also draw curves a whole new world opened up for him. Here is his report on how he handled an advanced Globs challenge.

At one point I asked Guillermo how many points he would get if hit all the globs with one function. He thought about it. Then I added "I think there is formula you could use." to which he responded "I don't need a formula. I'll figure it out." I thought he would then proceed to add 1 + 2 + 4 + 8 + ... + 4096 using a brute force method. But instead he surprised me with this a couple of days later. (Watch the video below.) Check out his score at the end.


Does Guillermo know the slope formula? I don't know for sure. But then who cares? He's miles ahead of that now. Math will never be the same for him. He will learn what he needs to know when he needs to know it. And that's what Math 2.0 is really all about.

Keith Devlin slide 1
*George Lakoff and Rafael Nunez (Where Mathematics Comes From?) believe that all math can be learned intuitively even calculus if it is taught in way that connects previous learning with current new knowledge. Keith Devlin doesn’t think that’s possible for subjects like calculus. During his opening session presentation** at the NCTM conference in 2004, he said that there is some math that just needs to be learned “top down.”  He poses this challenge to the audience.  

Keith Devlin slide 2

More about Globs
** You will need to download Realplayer to watch Devlin's keynote opening session. His keynote begins at 18:15 in the video. He starts talking about the Lakoff and Nunez book at 46:45.