What Counts

I remember carefully knocking small, colorful, plastic bears off of a 2x4 or a cigar box or a case of empties in my 1st grade (or was it 2nd grade?) classroom. We were learning addition and subtraction. I would line them up in a row and knock them down, one by one, to arrive at the answer for some complicated mathematical operation like 7 minus 3. I don’t remember counting on my fingers to accomplish the same calculation, but I suppose I did. It just wasn’t as memorable as defenestrating bears.

Using your fingers to do arithmetic is frequently considered bad form. It’s seen as unrefined—almost rude. I imagine that the practice of counting on one’s fingers probably offended the sensibilities of some self-assured school administrator at some point and then caught on as a new pedagogical war to be fought. (“Surely we can’t have children using their hands! That’s for babies! It’s undisciplined or… cheating! They can’t expect to be able to use their hands to do things in real life!”)

As it turns out, kids using their fingers has been linked with better mathematical abilities. Another bit of evidence for embodied and enacted cognition; there is a benefit to literally getting a “feel” for math.

While I suspect that disparaging finger-counting has to do with a feeling that the body is vulgar, there might simply be a general cultural bias against the literal. One insult often leveled at someone who doesn’t understand a verbal explanation is, “Do I have to draw you a picture?!” That might be good, yes; please do. Luckily, this perspective doesn’t seem to extend to math. Using a pencil and paper for ciphering or sketching out mathematical concepts can be vital, since they can quickly become complex. Past a certain point, there’s no easy way of getting around using a symbolic notation to manage it.

But there is some poetry, too, in that we use our digits (fingers) to type digits (numbers) into computers (bright rectangles that steal time from us). And here’s a fun fact: you can count up to 1,023 on your fingers by using them the way a computer would. Each finger represents one bit: on or off, up or down. With both hands in front of you and a slight change of perspective to a base-2 number system, you can count far, far higher than 10. Demonstrate this the next time you’re at a party and see how long it takes for you to be asked to leave.

We count with our fingers, with an abacus (or preferably plastic bears), with tally marks, with numbers, with computers and sometimes quietly in our heads. 

And we still hold up the fingers of our hand to let the hostess know we’re a party of 2.

Touch Play

We’ve been watching The Queen’s Gambit and it’s reminded me of the “touch move” rule: if you touch a piece you’re obligated to use your turn to move it. It discourages a lot of fiddling around with the pieces, trying out moves before committing to one. The game is perhaps more elegant and disciplined as a result.

What I think is most interesting is the effect that even just touching a piece can have. A player might have visualized—incompletely or inaccurately—a set of consequences and side-effects, but the act of simply reaching and grasping a piece can lead to a sudden realization of how the intended move actually impacts the rest of the board. And how the game could play out differently as a result.

Writing feels like this over and over again. I certainly don’t see the words on the page before I write them. I might hear some of them in my head, but it’s not the same as seeing and feeling them play out onto the page or bubble up onto the screen. Often, even if I have a particular outcome in mind, it simply changes as I write it. Sometimes I’m surprised at what ends up on the page after a “move”.

I’m not sure that chess and writing actually have that much in common. But both require touch in order to discover the outcome and the entire game can change after the first move.

Old School

Apparently, world-class mathematicians prefer to work on blackboards more than any other medium. (And some have quite a taste for high-quality chalk.) One might think that blackboards would be considered too restrictive, too slow, too imprecise for high-powered math—clearly a job for high-speed, hi-res, high-tech computers.

But I can see their point: chalkboards are an enormous expanse of real estate to work out both abstract theories and concrete expressions, and you can look at your work up close or at a distance with a field of view that computer screens have yet to match.

And chalk: a substance that allows for quick erasure, deliberate smudging and modification; variation in line weight and shading. My high-school physics teacher had a favorite technique for drawing dotted lines: holding it at a steep angle to the board while “pushing” it would make it skip along the surface. With a little practice, you could produce a long arc of beautifully spaced dashes. Imagine another writing tool that gives you such immediate access to such a wide variety of techniques! (Charcoal, perhaps?)

Working at a blackboard is inherently kinesthetic: standing (sometimes crouching!), moving side to side, toward and away, hand and arm applying a variety of forces depending on the technique used to write or draw. It’s a level of engagement on par with sculpting or painting—the effort is physical, concentrated and the feedback is immediate.

Whiteboards don’t really allow for the same interaction. The scale is similar, but they are too slick and glossy to provide satisfying resistance—unless you count the resistance provided by a dried-out dry erase marker. The range of techniques is limited: just try to smudge a line on a whiteboard! It’s too fragile; it simply vanishes. And even photographing a whiteboard to capture the final result can be problematic because of glare and contrast.

We need more blackboards and lots of chalk—we can do better thinking with them.

Do the math.

Tired of Thinking?

Maybe it’s because you’re trying to do all of it with your brain, instead of letting your body do some of the work.