This weekend I had a chance to do math with some grown-ups. I had a blast, and I think they had a blast — kind of. Maybe. A little bit. Okay, I guess you’ll have to ask them if you *really* want the truth.
Anyway, I introduced the task with this drawing:
I then asked this sequence of questions:
1. How many squares do you see at stage 1? … stage 4?
2. Predict the number of squares you would see at stage 5.
3. Now count the *total* number of squares at stages 1…4, including the squares from all of the preceding stages.
4. Predict the total number of squares you would see at stage 5.
Here’s what they came up with:
Having finished the warm-up, I thrust them into the serious work:
5. Predict the total number of squares you would see at stage 50.
Here was their initial effort. Note the attempt to try to understand the pattern using factoring:
I then showed them this video, which I was inspired to make through Dan Meyer’s Pyramid of Pennies task:
They grappled with that for a while, and eventually made the observation that there are 4x5x9 cubes in the box on the video, which means there are 1/6 as many cubes in the original configuration. This led to the conjecture that there would be 50x51x55 cubes in the stage-50 box (note the part below that is scribbled out):
This was an interesting phase of the problem-solving process, because they had essentially arrived at their answer. But then the question becomes — how do I know if I’m actually right? We had a little conversation about that, during which I suggested to them how they might test their conjecture: use the data you already have! i.e. you have on paper the results for stages #1-4, so test your formula against that.
Since the box on the video was a “stage 4 box,” I suggested that someone draw the “stage 3 box,” and this is the result:
This was a critical move, because now we can actually compare *two different stages* and see if our conjecture works for *both.*
Eventually the team revised their original formula to get 50x51x101/6, as you see on the earlier image. But we *still* don’t know for sure if it’s right! I then suggested that they try to figure out the “stage 5” result, without necessarily drawing a picture. They used their conjecture, then computed the result directly, which was further evidence that they were on the right track. At that point, we talked about how we could have an increased sense of confidence in their conjecture, but was there any way to make *certain?*
We discussed the principle of Mathematical Induction, which requires that we formulate an expression for “stage x,” then formulate it for “stage x+1,” then prove that if the formula holds for stage x, then it must also hold for stage x+1. It was interesting to me to observe that the act of *formulating an expression* for stage x+1 was tough work for the team — I have never taught Math Induction to students before, so I did not anticipate this particular challenge.
We then banged out the algebra. I didn’t include photos for this stage of the work, but I’ll summarize by saying that our efforts were at first messy. For instance, we “proved” that 6x = 6x, and it wasn’t at all clear what that meant. I explained that what we really did was show that our first statement was *equivalent* to 6x = 6x, and since the latter is true, the former is true: Q.E.D. Not very satisfying, if you want the truth. So on further reflection, I saw that what *should* have happened was to transform one *expression* into another, rather than starting with an *equation* and manipulating that.
My takeaway from all of this is: the process of solving problems in math is messy. At every stage there were false leads; places where the team thought “we’re done” but really wasn’t; places where we wondered, “Are we really sure we’re right?” And even that last part was messy: we tried it one way, only to realize that there was a better way later in the day.
Implications? Let’s give our students plenty of time to experiment with math, to solve problems collaboratively, and to enjoy the push-and-pull of problem solving.