My daughter, a third grader, recently told me a story about a fractions task that was challenging for the kids in her class: How can you split a square into thirds?

To me, the task of dividing a square into thirds seems quite simple. This is how I would do it:

So when my daughter told me that the task was hard for her 3rd grade class, I had to step back from the immediate, easy mental image that came to my own mind and try to imagine what this task might look and feel like to a third grader. 

That’s challenging. I have few, if any, memories of my own experience with math as a third grader. But I am aware of a similar task that seems to be equally challenging for adults: How can you split an equilateral triangle into fourths?

When I give this problem to my college students, it stumps many of them completely, and there are few students who see it immediately. The ah-ha moments come slowly, but they do come, and eventually most students will produce something like this on their paper:

What I find most insightful about this task, though, is what students try that doesn’t work, and a correct solution that students almost never try:

What students try that doesn’t work:	What students almost never try:

Here’s my interpretation of this. First, students have a relatively stable understanding of fourths as “half, and then half again”. The most common version of this is to cut a shape in half vertically and horizontally (the “plus sign” version of a fourth). Second, students have an image of fractions as being composed of congruent pieces, or pieces that are the same size and shape. When students produce a “half, and then half again” image like the two above, it looks wrong to them. They don’t say, “Look, I did it!” They say, “I don’t think this works,” or, “Is this right?” When they come up with the congruent triangle version, they immediately know it works because it fits with their mental image of equal parts.

But the image on the right, the one they don’t come up with, also looks wrong to them, even though it’s not. The four pieces have exactly the same area (which you can easily prove if you understand that the all four triangles have the same height and base length), but they don’t all look the same. They are not congruent—they are the same size, but not the same shape.

In other words, their mental image of fourths and fractions gets in the way of completing the task.

I think something similar is going on with the third grade square task. First, just like adults have lots of mental images of fourths, but not fourths of triangles, I think kids have plenty of experience with thirds, but not thirds of squares. They see thirds of circles, of course, and even thirds of oblong rectangles, and while you’d think the leap from rectangle to square would be easy (a square is, after all, a type of rectangle), it’s not, and that may be partly because of the second point: Kids have lots of experience cutting squares into halves and fourths, and there’s just no way to halve your way to thirds.

Thirds look like this:	Fractions of squares look like this:

Imagining a square cut into thirds requires as much imagination for a third grader as is required for an adult to imagine a triangle cut into fourths.

Here was my daughter’s solution:

She thought it was quite a creative solution, and she was proud of it. But when you just see it drawn on paper, it’s probably hard to tell if it’s a correct solution. In fact, if you’re expecting this:

my daughter’s solution almost certainly looks wrong. So teachers (and parents) need creativity and imagination, too, as well as good questioning and listening skills.

Here’s how my daughter described her solution: “First I drew a line to cut the square in half, but I stopped it part of the way down—more than halfway, but not all the way. Then I drew a line straight across, where I stopped the first line, so there was a rectangle at the bottom of the square and two rectangles at the top. And if you took the rectangle at the bottom of the square and cut it in half and put one half on top of the other, you would get one of the rectangles at the top of the square.”

What you see in her picture is that thirds require three pieces. But what you hear in her description is that she also understands that the pieces must all be the same size, and that she can justify how her intended image does indeed meet that requirement by deconstructing and rearranging the pieces to show that all three are all the same size, even if they’re not congruent.

One of the things I love most about mathematics, and about teaching mathematics, is how both give me the opportunity to expand my imagination, but imagination is so foreign to many of my students’ experiences with school mathematics. Do you have your own examples of mathematical problems or concepts where imagination plays a role? I’d love to hear your ideas in the comments.