**Question: Are young children able to reason algebraically?**

This experiment comes from a recent study published by researchers at Johns Hopkins University.

**Materials**

1. A set of about 20-25 each of four different kinds of small objects (buttons, beads, fish crackers, cereal, pennies, etc.)

2. Two identical, opaque cups (paper cups work well)

3. A stuffed animal with a name

**Setup**

Separate the objects into piles somewhere that will be accessible to you, but out of your child’s view. (You may want to lay out the objects on your table, then hide them behind a book.) For three of the objects, you will want 12 in the first pile (those 12 will go in the “magic cup”), and anywhere from 5 to 9 in the second pile. For the fourth object (I used chocolate chips), you will want 12 in one pile and 4 in the other.

** How to Conduct the Experiment**

Tell your child you are going to play a number game with the stuffed animal (ours is Puppy). Tell your child that Puppy (or whatever the animal is named) has a magic cup, and show them one of the cups. Tell them that any time you put the cup on a pile of something, it adds the same number of that object to the pile, no matter what the object is. Say, “Let’s try and see what happens.”

Out of your child’s sight, place 12 of the first object (I used pennies) into the magic cup Place the remaining pennies in front of Puppy. Point at the pennies and say, “See Puppy’s pennies?” Then bring the magic cup out, without showing the child the pennies inside. Tilt it upside down over the pile of 5 pennies, then lift it to reveal the (now larger) pile of pennies. Say, “It worked! Shall we try again?”

Remove the pennies and do the same thing with the next two objects (I used cherries and Cheerios). The key is to move fairly rapidly, without rushing. You want to discourage counting, and you want to make sure that your child never actually sees the objects in the magic cup until they have been added to the pile in front of the animal.

Now that you’ve laid the groundwork, you’re going to see if your child can tell how many objects the magic cup creates. Discretely place your fourth objects (chocolate chips in this case) into the two cups, 12 in one and 4 in the other. Bring both cups out and tell your child that you found two cups and you’re not sure which is Puppy’s magic cup. Turn both cups over onto the table and lift them to reveal the 12 objects and the 4 objects. Ask your child to tell you which is Puppy’s cup. You should realize that, at this point, your child has never seen the contents of the magic cup—only the starting amount and the final amount. Will she now be able to tell which of the two is the amount that was added to each pile?

**What is the purpose of this experiment?**

The researchers who conducted this experiment already knew that children are born with number sense that does not need to be taught. They wanted to know if that same number sense extends to algebraic reasoning.

The task in this experiment may seem simple, but it’s not. There’s a big difference between being able to think about a situation where the unknown quantity is the result of an operation, such as 5 + 12 = ?, and being able to think about a situation where the unknown quantity is within the operation, such as 5 + ? = 17. Identifying the change amount (5 + ? = 17) rather than the result amount (5 + 12 = ?) is something that older kids often struggle with when they first begin algebra.

In this experiment, the children were too young to solve an algebraic situation symbolically. When given equations like 5 + ? = 17, and asked to choose between the numbers 4 and 12, they just guessed. But when the magic cup story presented children with the exact same situation in a non-symbolic form, the children were suddenly successful.

This experiment reveals that your child’s brain is already wired to do simple algebra! Knowing what our children are capable of can give us confidence as parents who want our children to achieve their potential.

**What if my kid failed the experiment? Does it mean my child will struggle with algebra?**

Of course not! For one, it’s hard to perform an experiment perfectly, and you only had one go at it. And maybe your child was tired or hungry or distracted. Maybe they would have gotten the right cup 4 out of 5 times (better than chance), and you happened to see the one they got wrong. More important than the outcome of your particular experiment is the opportunity you gave your child to think algebraically just by doing the experiment with her. Now give them more opportunities!

*Reference: Kibbe, M. M. and Feigenson, L. (2014). Young children ‘solve for x’ using the Approximate Number System. Developmental Science. doi: 10.1111/desc.12177*

I remember seeing this when you originally posted, but never got around to doing the experiment. Just now reminded of it by @earlymath on twitter. Quickly taking an opportunity while I’m alone with my youngest (33 months old), I gathered:

– coins 12 and 9 (didn’t have enough of one type, so we used a mix of denominations)

– dice 12 and 7 (selected only 20-sided from our pound-o-dice collection)

– trio blocks 12 and 6(I just used 1x1x1 cubes)

– polydrons 12 and 4 (used only square frames)

I told her that Jimbo, her doll had a magic cup and then did the experiment in the order above (coins, dice, trio blocks to set the pattern and polydrons to test). In order to have time to prep the trio blocks (third round), I asked her to put the dice away in their box, so this might have skewed the results. I didn’t see her counting them and know that she can count to 10 but has errors in the low teens anyway.

When we came to the last step (which cup produced 12 polydrons and which 4) she instantly pointed to the one that produced 12. I asked why and she said that one made “5” while the other only made “3.” I asked what that meant and she said “it is more.”

That’s very interesting! I’m glad you tried the experiment, and I think it’s interesting to see the result from a child who is a little younger than the age in the study. I’m looking forward to trying with my son when he is a little older.