Long Division and Everyday Learning

Recently, my friend Alec and I have been exchanging emails on the role and effectiveness of everyday learning. Alec believes that all learning should look like everyday learning. While I don’t necessarily disagree with that, I’m not sure that everyday learning is as effective as Alec makes it out to be. It may be a good place to start, but I feel that we need to do much better than that.

So what is “everyday” learning? I’ve been waiting for Alec to define that for me since he’s the one that brought it up, but let me take a stab at it. For me, everyday learning typically has the following characteristics:

1. It is learning that occurs to meet an immediate need
2. The feedback that you receive while learning is nonjudgmental
3. You have plenty of opportunities to apply what you have learned

When you take on a home improvement project and start going to Home Depot every day, you quickly learn how the store is laid out. Your need is immediate: it’d be a big waste of time if you had to wander the entire store to find one item. The feedback is fairly nonjudgmental: either you can find what you need quickly and efficiently or not. (Okay, I do feel like everyone in Home Depot can tell I’m a noob when I’m wandering around like a lost sheep, but my need almost always outweighs the small discomfort I experience.) And you have plenty of opportunities to apply what you have learned: you make multiple trips and only learn the layout over time.

In terms of its everydayness, how you learn something shouldn’t matter. If you download a map of the local Home Depot and study it for hours the night before, ask a friend who knows the store like the back of her hand to go with you and show you the ropes on your first few trips, or go in blind… it is still everyday learning. In terms of effectiveness, my gut tells me that everyday learning is most effective when there is lots of immediate feedback and when you can take self-concept out of the picture.

As a teacher and curriculum developer, there are things that I do to make school learning more like everyday learning. In general, students are introduced to long division in 4th-grade and master it in 5th-grade. But what if you delivered a truck full of cookies to eight 2nd-graders on a playground and told them that they could keep the cookies only if they could divide the cookies evenly amongst themselves?

When they open up the truck, the kids find ten pallets:

I’m willing to bet that some enterprising kid would suggest that they start by each taking one pallet. When the two remaining pallets are unpacked, the kids find twenty crates:

This time, four of the kids suggest that they each take one, and then two, of the crates. When the four remaining crates are unpacked, the kids find forty cartons:

They quickly divide up the forty cartons, each taking five, leaving them with one pallet, two crates, and five cartons each, with nothing left over:

The kids would immediately recognize this as 125,000 cookies if they had ever played Chocolate Chip Cookie Factory: Place Value, but that’s not the point. They just applied the long division algorithm to divide up a million cookies on their own in under half an hour.

Now, if this was really an example of everyday learning, these 2nd-graders would work at some kind of chocolate chip cookie distribution center and they’d be doing these division problems all day. And they’d get very good at them. At first, their solutions would be ad hoc (each problem would be solved as a unique case), but over time, they would start to generalize and find an efficient solution for all problems. How long would that take? Some kids would start doing it right away; others would only do it subconsciously over months.

As a teacher, I can speed things up by designing and sequencing activities to stretch their thinking and by helping them raise things to a conscious level. For example, if a 2nd-grader had to divide three stacks of cookies two ways, he might immediately start by dividing two stacks and opening the third. This would be so obvious to him that he would do it without thinking about it.

But as his teacher, I could ask: “Why don’t you open up all three stacks before dividing them?” This would cause him to pause, think about it, and reply: “Well, that would be silly because you can already tell that each person is going to get one stack. It is faster and easier to start by dividing the big things first.”

I could then follow up by asking him to divide 4 boxes, 6 stacks, and 2 cookies three ways. He might start by dividing three of the boxes and the six stacks before unpacking the leftover box. We could discuss different strategies and explore “what-if” problems together. We could even talk about the benefits of grouping (place value) and how things would be different if everything at the cookie distribution wasn’t in groups of ten. Finally, we could invent a notation system for doing these division problems out on paper. Why would you ever want to do that? Well, imagine that, instead of being on the floor and moving boxes, crates, and pallets around manually, the distribution center became automated and you had to move things around symbolically in a separate control room. How many 2nd-graders would make that transition on their own instead of being laid off and having to find low-wage jobs in the service sector?

These things could happen in everyday learning if you had a good mentor or the distribution center had a surprisingly good training program. But I would say that these opportunities for generalization, reflection, and extension aren’t typical in everyday learning. Remember how I said that some 2nd-graders would start generalizing on their own right away? How did they get that way? Someone modeled it for them, they liked it and internalized it, and started doing it for themselves all the time. My goal as an educator is to guarantee that everyone, whether learning in school or in everyday life, has this same opportunity. I think this only happens if we collectively raise our expectations for what learning looks like so that these models, mentors, and coaches are everywhere for everybody.

When I think about teaching long division using chocolate chip cookies, I’m not thinking about bringing everyday learning into the classroom. I don’t see this curriculum as an example of everyday learning. To me, it is applying basic learning theory. These 2nd-graders have rich and powerful mental models for working with groups. They’ve been working with groups of things in their everyday lives for years. I’m simply leveraging those mental models because it would be incredibly foolish not to. It would take years of arduous work to build parallel mental models for the same concepts in the classroom, and then you’d have two disconnected ways of thinking: one way of thinking for everyday life and a separate way of thinking for the classroom. Who would want that? Crazy.