4 + 4 Isn’t Simply 8 Bars Put Together

Anuradha is a young woman in her early 20s.

We asked her to teach "doubles" using the example of 4 + 4.

She drew four bars, confirm that number, and then draws four more times next to what was already drawn then counts the total number of sticks she has drawn, then writes the mathematical expression for the problem: 4 + 4 = 8 as shown in the figure 1.

While Anuradha’s drawing did show the correct answer to 4 + 4.

But we weren't very happy with her teaching "doubles".

This representation wouldn’t help students make a direct connection with the doubles fact to learn.

What student can see is simply a group of 8 bars without being able to see its composition. It’s as if she told them the fact without the benefit of a picture at all.

The next time any student sees 8 bars at once, they might not remember the fact that she just taught them. Rather, they might

need to count all bars to know the total.

We suggested

If she leaves a space between the two groups of marks as in the figure 2 so that the students can make a direct connection between this representation of doubles fact to learn.

This revised representation will make much more sense because it will help students recognize that the total number of bars is composed of two equal quantities of 4, and that this total number is 8.

@ ntuition what’s more meaningful to a child is a representation that illustrates the process as well as the result, not simply the result.

For most primary-grade students is a math fact, isn’t an involved one, still children need to see beneath the surface.

To use an analogy, the result part of this problem, 8, is like the upper body of a duck visible above the water surface, whereas the process of the problem is like the duck’s feet hidden under the water. Our task is to expose the duck’s feet so that children can see it’s these feet under the water that drive the duck forward.