I’m taking calculus for the first time right now, and it’s got me thinking about math learning a lot, especially how to build intuition for math concepts and see them as tools for problem-solving in the real world.

As a grade school student, I always liked word problems in math class, for two reasons:

- There’s a heuristic pleasure in reading a passage and figuring out the math problem hidden in it. This is much more interesting to me than just getting the math problem by itself.
- There’s considerably less math per character in a word problem than in a block of equations. Since I suffered from fear of math in school, this was a big plus for me.

But word problems have their downsides, as well, notably that students often hate them. And it’s hard to defend word problems that set up scenarios that are either boring (what do I care about the heights of a group of people I don’t know, or the speeds of hypothetical trains?) or utterly improbable (a menu tells you the price of an item relative to another item’s price—really?).

Why do we even have word problems? They’re an attempt to contextualize math concepts, which on their own can seem abstract. The word problem attempts to answer the question, “How would I use this math?”

What if we contextualized math not with (only) word problems, but with projects that require learners to use math?

A lot could be done with just cardboard building materials. Here’s one example:

We give students the specifications of a structure, and let them figure out how to determine the lengths and widths of a box’s walls when given the volume required. To help them develop intuition for the way the surface area of a box can change while the volume remains constant, we ask them to build a box to hold a certain volume of something small and uniform, like beans, and another to hold a certain volume of something with a less-friendly shape, like pencils. We could even give them a budget for the project (in tokens, pieces of candy, Legos, or whatever currency is appropriate) and charge them per square inch of cardboard they “buy” for the project.

Materials needed (for each student or student team)

- 12-inch ruler
- Pencil
- Scissors or craft knife, per student ability
- Cardboard, enough to make two 14 cubic inch boxes plus plenty of extra for trial and error (if students are using scissors, cereal box-type cardboard is best; if they’re using knives, corrugated cardboard is sturdier and has the benefit of introducing the thickness of the walls into the measurement challenge)
- Adhesive (masking tape, hot glue, etc. per student ability – white glue is not ideal)
- One cup of beans, M&Ms, or similar small, uniform items, in a baggie*
- 12 pens or pencils (since size can vary, try to get utensils that are all the same, and see how many it takes to fill 14 cubic inches first)*
- Currency, if desired, of your choice
- Plastic trays (to catch spills of small items – suggested)

*If it isn’t practical to give every student or team one cup of beans and 12 pencils, have two or three cups of beans and sets of 12 pencils, and let them test their boxes as they have them ready. They’ll need to know the length of the pencils or pens before they start.

Begin by explaining the specifications for the finished products and instructing students to draft a proposal with a budget. How much cardboard will they need to build their boxes, and how much will it cost? When their proposals are complete, let them purchase their materials and get started. After the boxes have been built, test them. Do they hold the items? Are they too small or big? By a little? A lot? Students will, hopefully, have ideas about any shortcomings and be ready to try again.

This project can then lead to others that build on it: for example, the team has now been asked to design a shipping container that will hold as many boxes of M&Ms and pencils as possible within certain limits (footprint must fit on the boat, can’t be more than ½ as tall as wide, etc.).

What were your experiences as a math student? If you teach math now, do you use projects to help students intuit abstract concepts? What about other subjects?