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Impossible Adding

AGE RANGE: 2 to 3 years

RESEARCH AREAS: Number sense, agency

THE EXPERIMENT

You will need a cardboard box with an open bottom. Cut a hole in the side of the box that is large enough that you can reach inside to add or remove items. Then, cut a large flap in the front of the box, so that raising the flap reveals what has been placed in the box. You will also need three identical trinkets, such as Duplo bricks or green army men.

Place the cardboard box in front of you on a table so that it rests about an inch or two beyond the end of the table closest to you. Position the box so the side with the large flap is facing away from you. Place one of the trinkets inside the box, and have your child sit opposite you at the table, so the side with the flap is facing her. Lift up the flap and point out that there is one trinket inside the box. Then, lower the flap. Produce a second trinket and explain that you are going to place it inside the box. Reach into the hole in the side of the box and place the second trinket inside. Withdraw your hand, making sure that your child notices it is now empty. Then, reach into the hole again and remove one of the trinkets. At this point, surreptitiously place a trinket back into the box under the table through the open bottom. Then, open the front flap again to reveal the two trinkets. Ask your child, “Is that right? Should there be two trinkets?” Note her answer.

Next, repeat the demonstration. Begin again with one trinket. Add one to the box, then subtract one from the box. This time, don’t surreptitiously add a trinket. Lift the front flap again to reveal the single trinket. Ask your child, “Is that right? Should there be one trinket?” Note her answer.

About a week later, repeat the two demonstrations again, with one change: instead of manipulating the trinkets yourself, direct your child to add and remove the trinkets via the hole in the side of the box.

THE HYPOTHESIS

When you are the one who adds and removes the trinkets from the box, your child may have difficulty identifying whether there should be one or two trinkets left in the box at the end of each demonstration. But when your child is the one who manipulates the trinkets, she is likely to be able to accurately say whether there should be one or two trinkets remaining.

THE RESEARCH

In a 2015 study, two-year-old children were assigned to one of two groups. Both groups were shown a box similar to the one you used in your experiment. In the “onlookers” group, the experimenter was the one who added and removed toys from the box. In the “actors” group, the child was the one who added and removed the toys.

In some cases, the number of toys inside the box was not manipulated, but in other cases, the researchers manipulated the number of toys inside the box to cause an impossible outcome. In one instance, the experiment began with one toy inside the box. After the front of the box was closed, a toy was added, then a toy was removed. A toy was then surreptitiously added, and when the contents were revealed, the child saw that there were two toys inside. The child was then prompted to say whether the outcome was correct or incorrect.

The researchers found that in the onlookers group, the children’s ability to answer correctly did not differ from chance. They were just as likely to say that the result was correct as they were to say that it was incorrect. However, in the actors group, the children gave the correct answer about 85 percent of the time.

The type of arithmetical operation depicted in the experiment is called an inversion problem, because it illustrates the inversion principle between addition and subtraction. When you add a certain number of items to a group, and then subtract the same number of items, the subtraction operation reverses, or inverts, the addition operation. The researchers found that when it comes to basic arithmetical operations, young children have a special aptitude for inversion problems. In related experiments, they found that the children had more difficulty with similar operations that did not involve inversion.

THE TAKEAWAY

Manipulating the materials does not make your child any smarter. Rather, self-action, which involves active participation rather than passive observation, appears to bring out your child’s natural potential. As a parent, look out for other ways in which your child can tap into intrinsic knowledge by manipulating the world around her. For instance, she might have a rudimentary understanding of how a mechanical toy works, and when you demonstrate its functionality, it may help her pick up a few points. But there’s nothing quite like hands-on play to help her understand the ins and outs of how it works. With abstract concepts such as addition and subtraction, finding ways for your child to exercise her brain through physical manipulation of objects can be especially helpful. For young children, the best way to pick up new understanding might be to pick up new objects.