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.

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?

As a future teacher (currently working on my bachelor’s degree), and a homeschooler of 15 years, this problem helped me engaged first of all, and an example like this would help me to put into action something that I could easily take my kids aside to show them, and they could, in turn, actually use in life. The problem with a lot of the examples in the textbooks I have taught from over the years, is that there were problems that my kids didn’t care about, and I could not give them a good solid example, nor make it applicable to real life. This is something I would use to begin to get their minds thinking about volume before I even introduced that vocabulary, instead of doing it the other way around, or just starting from a textbook problem. I have been a student of Christopher, and he does things like this all the time, meaning he makes math something we use in everyday life. He begins with using an example (which at first seems to make no sense), which is easy to understand, before proceeding to the actual lesson. When he does introduce the lesson, examples like this make the lesson so much easier to understand and apply. With that said, I have learned more in his classes than I ever did in high school, plus I remember what I have learned, and now can apply it. I wished I would have been able to have this knowledge while I homeschooled, because I think my kids would have been able to think that math is something you use every day.

Christopher, I’d like to be more generous; I think your problem is a nice one that (a) engages students in mathematical thinking, and (b) has a real purpose. People fill containers from shop-bought packages every day, and knowing if they will “fill or spill” is pretty important.

Of course, in real life one hardly ever actually calculates the volume of a container, at least in the kitchen. But scale the problem up to, say, a tank to contain some industrial or agricultural product – fertilizer, for example – and knowing in advance if it will spill over could be worth thousands, not to mention a messy problem.

Good work, I say.

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