Objects to Think With

Seymour Papert begins Mindstorms with a confession that sounds, at first, like it belongs in a different book. Before he could do arithmetic, he was in love with gears. He played with the differential gear in a car's transmission until he could run it in his head. And then something strange happened: when he later met multiplication, and still later met equations with two variables, the gears were there, waiting, ready to give the new abstractions a shape he already understood.

I believe that working with differentials did more for my mathematical development than anything I was taught in elementary school. Gears, serving as models, carried many otherwise abstract ideas into my head.

He had a name for what the gears were: an object to think with.

What makes an object to think with#

The idea is more than "a helpful example." An object to think with is a thing — physical or computational or notational — that meets three conditions at once.

It is concrete: you can manipulate it, turn it over, watch it behave. It is charged with an idea: built into the way it works is some abstract structure — ratio, feedback, recursion, conservation. And it is appropriable: it's the kind of thing a person can take up, make their own, and form a relationship with.

When those line up, the object becomes a bridge. You reason about the abstraction by reasoning about the object, and because the object is concrete and yours, that reasoning is fluent and confident in a way that staring at the bare abstraction rarely is. You think with the thing.

The turtle as a designed object#

Papert's gears were a happy accident of one childhood. The interesting question is whether you can manufacture objects to think with — build them on purpose, for ideas you want learners to reach. The Logo turtle is his answer, and it's a beautiful piece of design.

The turtle carries geometry. But the genius is the point of view. Because the turtle has a position and a heading, and because you instruct it from its own frame (FORWARD, RIGHT), you can think about a shape by imagining yourself walking it. Papert called this body-syntonic — the geometry is connected to your sense of your own body in motion. A child who can't yet handle the formula for a circle can absolutely walk one, and the turtle lets them turn that walking into a program, and the program into a circle. The abstraction arrives through the body, not despite it.

They're everywhere once you look#

The phrase has escaped education entirely, because the pattern it names is everywhere. So much of human intellectual progress is really the invention of better objects to think with.

• Positional notation. As Kenneth Iverson argued in Notation as a Tool of Thought, writing numbers in place-value decimal turns multiplication into a procedure a child can carry out. The same numbers in Roman numerals are nearly unthinkable to multiply. The notation is an object to think with, and a wildly powerful one.
• The diagram. A graph, a circuit schematic, a Feynman diagram, a subway map — each lets you reason about something by manipulating a picture of it.
• The spreadsheet. A grid of live, interdependent numbers you can poke and watch recompute is one of the great objects to think with of the computer age — which is exactly why it became the medium that turned millions of non-programmers into end-user programmers.
• The model in a programming language. When you represent a domain as objects, or as data structures, or as types, you are building an object to think with — and the quality of your thinking is bounded by the quality of that representation.

The connection to a whole tradition#

This is where Papert's idea opens onto the rest of this site. If the right concrete representation can make a hard idea easy to think, then building better representations is among the most powerful things a tool can do for a mind. That is precisely the dream of the tools-for-thought tradition — Bush, Engelbart, Kay — and it's the deep reason Alan Kay calls the computer a new medium: it is a machine for manufacturing objects to think with, on demand, for any idea you can model.

It also sets a high standard for the software we make. A tool that merely automates a task leaves your thinking where it found it. A tool that gives you a new object to think with can change what you're able to think at all. The next time you reach for a piece of software, it's worth asking which kind it is.

Papert found his gears by luck. The work the rest of this site is about — from the turtle to Smalltalk to live programming — is the attempt to stop relying on luck, and to build objects to think with deliberately, for everyone.