Callum writes:

Filling a container with salt, sorry but…is this kind of thing inspiring to students learning maths? Isn’t this all rather pointless – you filled a container with salt – well done!

Touché.

But can we agree that my salt problem is no worse than any of the following?

These were gleaned from a quick sample of middle-school, remedial college and mathematics for elementary teachers textbooks on my shelf. I grabbed four books off my shelf that I thought might have volume problems in them and found a problem for the gallery in each of them. No cherry-picking here. And I swear I didn’t leave out any compelling applications of volume of a cylinder.

But even if we agree it’s no worse, that’s not a very strong argument in favor of the problem.

So is there any aspect in which it might be better? I think there are several.

### intuition

The initial question-*will it fill or spill?* admits student guesses. Students will have a hunch about the answer, and an intuitive sense of why it will or will not fill or spill.

This contrasts with these other problems. Students are asked to *find a volume* for the sole purpose of finding a volume. Not in order to answer anything some more meaningful question. And not even I have an intuitive sense of the volume of 678 flapjacks.

Plus, the question can come from the students. *They* can ask whether it will fill or spill; I don’t think I’ll have to. And then we’ll need to find some volumes in order to make a good prediction.

### reality

These are real containers. Perhaps not very compelling containers (although I’m a big fan of vintage Tupperware). But real containers nonetheless. Unlike anything in the problems above, these are objects in their daily lives.

Perhaps this is a sign of my hopeless math geekdom, but I am pleasantly surprised every time I refill my salt container that *it fits perfectly*. No leftover salt; no space left in the container. A perfect fit. I imagine some of that enthusiasm will be contagious in the classroom. And perhaps inspire some students to look at the containers in their own homes a little bit differently and wonder which ones are “bigger” than others.

### the answer

Did I mention that the salt fills the container perfectly? And that we can see it happen before our eyes?

I’m not looking to draw eyes away from the Super Bowl with this problem, nor to cause students to switch their major. But I hope they’ll be a bit more invested in the outcome than they are in the textbook problems above.

### intuition again

Here’s an interesting task from the math for elementary teachers book.

from Beckmann, S. (2010). Mathematics for Elementary Teachers. Boston: Pearson.

My instinct is that, at middle school, where the salt task would be appropriate, this will still be part of some students’ intuition. It is much more abstract to run the calculations and see that they are very, very close than to run them and then see that closeness play out in the physical world.

I’m not hoping to draw students to mathematics with this problem; I’m hoping to get them engaged for a lesson on volume.

But we’ll see. I’ll be using the problem with my future elementary teachers in a few weeks. This is not a population that is already sold on math (although by this late in the semester, I’ve reeled them in pretty well). I’ll report back.

And I welcome further critiques.

How *do* we make volume compelling?