How tools become abstractions, The Value of tinkering, and Mathematical pedagogy

I started programming computers when I was about 10; I had no idea what I was doing of course, I just followed the examples in the Apple and TRS-80 books. Meanwhile, part of tinkering with a computer also meant using the stock tools such as the commands to copy files, list directories, or fetch the time. I don't remember the exact day it happened, but there was some point that I realized that the commands like "copy" and "dir" were just programs like the ones I was writing -- that I was capable of making, in principal if not yet in practice, the tools I was using. That realization is a kind of magic moment in mastering a medium -- the moment when you see a tool beyond its immediate utility into the deeper concept that it embodies. You go from a tool-user (imitating the tool's use with variation) to a tool-maker (exploiting the under laying principal).

Tinkering begets insight. For example, when you tinker with woodworking tools you use the tools, hammers, drills, nails, etc. for their intended purpose without considering how they work. A nail is a device which accomplishes the task of attaching two boards, and a first you don't see past that utility. But as you become more intimate with the process at some point it may occur to you that a nail is just a kind of friction joint or that a screw is just a wedge wrapped around in a circle or that a drill is a kind of special scraper. When you see these kinds of things your mind opens up. If a nail is a friction joint then what other kinds of friction joints are possible? You might invent a dowel joint having never seen one before. If a screw is a curved wedge that buries itself into the material, what other ways can a wedge be integrated into a structure?

There are many people who never pass the tool-using stage of their craft. They stay within the existing rules of their medium and are perfectly comfortable there. I don't mean to be critical of that approach. That said, I can't help but think that the reason that some craftsman don't go past the tool-using stage is that for whatever reason they simply haven't had enough "a-ha" moments to internalize the drive to abstraction. One reason for this, I believe, is that most formal pedagogical practices try to shortcut the process of tinkering. There's a strong temptation when teaching something to cut to the chase; many teachers end up acting as if teaching someone something is about giving them answers instead of guiding them to the answer.

While I'm advocating the advantages of coming to understand tools as abstractions, I don't think that this learning process can be far removed from tinkering. As proof, I submit the typical pedagogy of mathematics as an extreme demonstration of how abstraction without tinkering goes horribly awry. To illustrate, imagine that you took a shop class but had never before seen any woodworking -- nails, screws, drills, all of it was totally new to you. Imagine if the teacher began the first day of class by saying: "OK class, this is a hammer. A hammer is characterized by a relatively large hard mass attached to short lever usually, but not always, made of wood. Note the counter-balanced curved metal head called a claw. Everyone pick up your hammer now and follow along with me banging the hammer on the table. One (BANG), Two (BANG), Three (BANG). OK class, now this is a drill. A drill is a helical scraper whereby a pitched screw... blah, blah, blah." Imagine this kind of boring technical analysis of woodworking tools day after day. On the last day of class a pupil raises her hand and asks: "Teacher, I have paid close attention and have learned about all of these tools, but I'm not clear on what it is you DO with these tools." and the teacher responds: "I'm not really either sure as I've never done it myself, but I think people make cabinets and chairs and stuff like that, more importantly all of this *is* on the college entrance exams." The kids think: "What?! You mean this class was about making chairs?!"

While this parable is exaggerated, I think that mathematical pedagogy (and other subjects) is not far from this. Most teachers and students intuit addition and subtraction and after that it's a free-fall into meaningless technical abstraction. By third grade, multiplication and long division become so bogged down in symbol manipulation that almost no child leaves understanding what multiplication and division *are*. Later they'll use them as a tool to perform yet even more technical abstractions but all too often 12 years can go by and the pupil still has no idea what you *do* with these tools other than make more abstract tools. I am convinced that the primary societal accomplishment of current mathematical teaching is to make people hate math. Those few nerds who make it past the abstractions were going to do so even if they hadn't been for the classwork, those who were going to hate it just hate it more, but the majority who might have liked it / used it / appreciated it instead come to despise it after being bludgeoned by it for 12 years. So the net result is that we may be worse off teaching math classes than had we simply done nothing. (My opinions on how to reform this are best left for another day.)

Even in the best case where an excellent and enthusiastic teacher is well practiced and accomplished in the use of the relevant tools of their art, the teaching techniques tend to lack tinkering because of its "out of control" sense. But, without discovering for yourself that a hammer is a solution to a problem, you can't easily appreciate the tool's function and you finally can't abstract the principal of the tool to other problems. If you tinker enough, and if you are lucky enough to have access to someone who has crossed the abstraction bridge, then you can grow from tool-user to tool-maker and, to me, that's the difference between ordinary works and extraordinary works.