Making Sense of Place Value

In 2012, I released my first iOS app, Chocolate Chip Cookie Factory: Place Value, on the App Store. The game is designed to help kids learn place value concepts as they work to ship and deliver cookies to customers as quickly as possible. For more details about the app itself, including where to download it, scroll to the bottom of the article.

A little background

When I was teaching, kids spent about two months (40 instructional hours) per year learning how to do arithmetic with multi-digit numbers. We started with addition in first grade and ended with long division in fifth. Things have changed a bit since the adoption of the Common Core math standards, but I’d say most schools still spend about 200 hours in instructional time on place value—not including decimals.

By the time those kids reach middle school, some of them are traumatized. They react to place value in the same way I react to doing my taxes—except I only spend one weekend a year on my taxes. Most kids end up memorizing the algorithms they need without really understanding them. Even the kids who use math programs emphasizing “conceptual development” don’t seem to develop much intuition.

It’s odd because place value is pretty fundamental. Every early civilization invented systems for counting and recording large numbers and people deal with items in groups naturally and fluently every day. If we are comfortable manipulating groups of items in the real world, it should be relatively simple to apply those same intuitions in understanding place value.

Experiments in third grade

In 2011, an elementary school principal invited me to work with three third graders who were two grades below grade level and slipping farther behind every class. Over a period of two months, I saw them for 45 minutes once a week. To gauge their intuitions about place value, I casually handed them a long division problem to solve.

First, they divided the boxes, stacks, and cookies as far as possible.

Then, they opened the leftover box, took out the stacks, and divided the leftover stacks as far as possible.

Finally, they opened the two leftover stacks, took out the cookies, and divided the leftover cookies as far as possible.

Naturally, given an opportunity to apply basic common sense in an intuitive context, these kids solved the problem easily. As far as they could tell, they hadn’t used any math at all. When I asked them to come up with alternative strategies, they quickly arrived at the standard long division algorithm. They saw the standard algorithm simply as a streamlined version of their original strategy.

It was now apparent to me that kids have plenty of intuition when it comes to working with groups of items in the real world—but we’re preventing them from using that intuition to understand place value in the math class. Instead of requiring kids to construct new and separate mental models for place value in school, why don’t we enable them to build on the existing and fairly sophisticated mental models they already have?

Counting using place value

I wanted to create an app where kids discovered and developed place value concepts through their own experiences; I didn’t want them practicing skills they had learned earlier in class. And since there seemed to be enough materials in the real world for kids to understand place value on their own, I wanted to use something from the real world. I came up with the following scenario: You own and operate a chocolate chip cookie factory. As orders are placed, you fulfill those orders by tapping on individual cookies, stacks of cookies, and boxes of cookies moving past you on conveyor belts. The goal is to fulfill the orders as quickly as possible.

While not advocating for child labor, I believe that kids would develop an intuitive understanding of place value just by working at this factory—with no explicit instruction required. To test my hypothesis, I like to trace the learning progression I might go through if I were a kid. This is not to imply everyone learns the same way; it’s how I confirm there’s at least one viable pathway for self-discovery.

1. How do I fulfill an order for 36 cookies? I have no idea. Let’s see what happens if I add cookies to the order one at a time. Realization: Hey, that works. Phew! Now I can fulfill an order. But that takes a long time. It might be faster if I use the stacks of cookies.
2. How do I fulfill an order for 52 cookies? That seems like a lot of cookies. Let me start by adding stacks: 10, 20, 30… uh oh, I’m starting to get close to 52. Maybe I should switch over to adding cookies one at a time now. Realization: I can fulfill orders a lot faster if I use the stacks. But I switched over to individual cookies too soon. How can I tell when to switch?
3. How do I fulfill an order for 65 cookies? I can start by adding stacks: 10, 20, 30, 40, 50, 60… uh oh, I went too far. Let me go back and try again. Realization: To fulfill an order for 65 cookies, I can add up to 6 stacks. Hey! Is that a coincidence? I wonder…
4. How do I fulfill an order for 83 cookies? If I add 8 stacks and add 3 more cookies… Realization: 83 cookies = 8 stacks + 3 cookies. Yay!

Addition using place value

There’s no math instruction in Chocolate Chip Cookie Factory: Place Value. Kids are meant to learn by doing, seeing, and thinking. If I did provide some instructions, it would be: use common sense and don’t try to do all the math in your head before fulfilling the order—use the boxes, stacks, and cookies to think about and record your work as you go.

1. 32 + 26 = (3 stacks + 2 cookies) + (2 stacks + 6 cookies) Realization: To fulfill an order for 32 + 36 cookies, I can just fulfill the two orders independently, one after the other.
2. 47 + 15 = (4 stacks + 1 stack) + (7 cookies + 5 cookies) Realization: It’s a little faster if I add all of the stacks to the order at once and then all of the individual cookies to the order second. That way, I don’t have to go back and forth between conveyor belts.
3. 38 + 44 = (3 stacks + 4 stacks) + (8 cookies + 4 cookies) = (7 stacks) + (1 stack + 2 cookies) Realization: If I see that I’m going to add 12 individual cookies to the order, I can save time by adding 1 stack and then 2 cookies instead.
4. 65 + 39 = (5 cookies + 9 cookies) + (6 stacks + 3 stacks) = (4 cookies + 1 stack) + (6 stacks + 3 stacks) Realization: Sometimes I might end up adding 10 stacks to the order. What if I add the individual cookies first? Then, instead of adding 4 cookies + 1 stack right away, I can just add the 4 cookies and add the 1 stack when I’m adding all of the other stacks. That way, I’ll see that I’m adding 10 stacks and add 1 box instead.

Play a demo of Chocolate Chip Cookie Factory: Place Value. Note: The experience is much better using a touchscreen.

Using our language brains

Chocolate Chip Cookie Factory: Place Value is designed to enable kids to build on their own natural intuitions about place value through self-discovery. But an essential part of the learning process is not facilitated within the app. As we use our “body brains” to fulfill the orders and we use our “eye brains” to see what’s happening and notice patterns, we also need to use our “language brains” to describe and generalize our thinking.

Ideally, kids would share and discuss their thinking with a mix of peers and adults. As an adult, I would ask probing questions, such as:

• How did you know to do that?
• What were you thinking there?
• What are you seeing now?
• What do you think would happen if…?

Downloading the app

The reason I’m no longer promoting Chocolate Chip Cookie Factory: Place Value is because it’s a 32-bit app and all signs indicate that 32-bit apps will no longer work in iOS 11, which is due to be released by Apple in the fall. I may work on a 64-bit version of Cookie Factory this summer, but I can’t be sure about it.

For a little history, I developed the app using Corona SDK, a cross-platform framework. A few years later, I did learn Objective-C and began work on a native version of the app, but I had overly ambitious plans and got bogged down in CoreData. If I were to attempt a native version of the app again, I’d have to either re-learn Objective-C (it’s been years since I’ve used it) or dive into Swift. I’m tempted to give Swift a try.

There are free and paid versions of Chocolate Chip Cookie Factory: Place Value, so you can try it for no risk. If you visit my website, you will also find a learning guide to help kids transition from the app to paper-and-pencil.

I hope you and your kids have as much fun playing Chocolate Chip Cookie Factory: Place Value as I did creating it!