Vehicle Chat [Invitation]

It all started with a disagreement between Kassia Wedekind and her daughter Lulu.

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Lulu (6) and I have an ongoing debate about "What makes a vehicle a vehicle." She says, "Moves people and stuff from place to place." She's arguing hard for elevator and escalator as vehicles. She loudly announces "All aboard the vehicle!" as we get on elevators. #tmwyk

— Kassia Wedekind (@kassiaowedekind) July 29, 2019

Pretty soon, it wasn’t just elevators we were arguing about, and it wasn’t just Kassia and Lulu weighing in. #vehiclechat was born.

With Math On-A-Stick on the horizon, I put together a prototype concept book, and now I am here to report to you what I have learned from many dozens of conversations about the nature of vehicles.

The book begins with some easy, quick decisions. Dump truck? Yes. Salad? No. (Usually. More on this below.) Airplane? Of course! Next up are increasingly controversial cases, including elevator, horse, and broken-down bus with no wheels.

image credit: Don O’Brien on flickr.com, licensed CC-BY

There are about a dozen subconcepts pertaining to vehicles that people carry around in their heads, but which most people have never spoken out loud. Some of these subconcepts are in conflict with each other; some are independent of each other. Some people cling tightly to one or two of them; others loosely hold four or five.

Here are some examples, in no particular order:

1. Vehicles carry people (not just things).
2. Vehicles must have wheels.
3. No living things are vehicles.
4. Vehicles take people or things from Point A to Point B. (And then there is 4a. Point A and Point B must be different points from each other.)
5. Vehicles must have an engine, motor, or some other dedicated power source.
6. A vehicle must have—as its designed and primary purpose—transportation across the surface of the Earth.
7. Once a vehicle; always a vehicle.
8. If it could be a vehicle, with a little work or repair, then it is now a vehicle.

One thing we know about math learning is that we nearly always work from examples and mental images of things rather than from logical definitions of things. This is why you can have a second grader tell you in all earnestness that a triangle is a shape with three sides, and also tell you that the image below is not a triangle because it doesn’t look like one.

Knowing a definition (three-sided shape) doesn’t guarantee that you’ll use that definition for naming things. A classic paper by Vinner and Tall details this disconnect as being about concept image and concept definition, and it’s related to Kahneman’s Thinking Fast and Slow. Concept images are the tools of the fast judging brain, while concept definitions are the tools of slow analytical mind.

Somewhere into those controversial cases, there is one that makes everybody think. It’s not the same example for everyone, but each person has at least one. My vehicular conversation partner and I will cruise past airplane and tricycle, and then they will pause and smile and look off into the middle distance. That moment is the one I seek—when you notice that your quick reactions aren’t good enough and now it’s time to think—even if only for a brief moment.

Also I have learned that metaphorical examples are more comfortable for many people than close, but not-quite-right examples. This means that I’ll often hear a quick “Yes” to salad (a vehicle for nutrients into my body), but a slow “Yes” or even a “No” to elevator (because it doesn’t have obvious wheels, or has a limited range of travel, or for some other reason). 

Most of all, I have learned that these conversations are really a lot of fun. People smile. They laugh. They think of additional challenging examples. They ask “What about?” and “What if?” They imagine what other people might say, and why particular examples might be controversial.

This is a social game of negotiating meaning, and of noticing that language which seems so precise is often not that at all. Play along with us! Offer up some examples in the comments, or some additional vehicle subconcepts not listed above. Or join the conversation on Twitter.

Here’s a link to download a copy of the book. You’ll want to print two-sided, FLIPPING ON THE SHORT SIDE, and then trim top and bottom with scissors or a paper cutter. The book then assembles tidily as a 5.5 inch square.

And if you like books in development, check out a fun new book about shapes: What Could It Be?