In our professional development, we talk a lot about the notion of “thought tools” and how to wield them. Today, I’d like to talk a little bit about this idea and see if we can clarify the notion of “thought tool.”
We’ve borrowed the language and the idea of “tools for thinking” or “thought tools” from the philosopher and cognitive scientist Daniel Dennett. (In my humble opinion one of the few living philosophers worth listening too. Nick Bostrom is another.) Dennett lays out the notion of though tools in his book “Intuition Pumps and Other Tools for Thinking” and gave a wonderful talk with the same title at Google.
Both Dennett’s book and his talk do a fantastic job explaining this notion in detail, so here, we’ll just introduce the basic idea and then explore what this has to do with the art of mathematical modeling.
Dennett opens his book with a wonderful quote from one of his former students:
“You can’t do much carpentry with your bare hands and you can’t do much thinking with your bare brain.”
– Bo Dahlbom
He then develops the idea of “tools for thinking” by analogy with ordinary tools. Just as we leverage the power of the hammer or the saw or the chisel to expand our ability to do carpentry and expand the range of carpentry problems we can tackle, Dennett argues that we can and should pay attention to the tools we use for thinking about problems. Just like tools for carpentry, thought tools provide us with the opportunity to tackle harder problems and do a better job with them. So, let’s look at two examples of general thought tools, Sturgeon’s Law and Occam’s Razor.
Sturgeon’s Law is usually expressed as “90% of everything is crap.” That means 90% of papers on molecular biology, 90% of political commentary, and 90% of blog posts on the internet. While the “90” figure isn’t meant to be viewed as a hard and fast quantitative statement, the idea is that in any area, most of what is written or said is, well, crap. The key point is that this statement is also useful for thinking about things. If you’re learning a new subject, don’t waste your time with the 90%, focus on the 10% that’s really good and really important. If you’re a critic, don’t waste our time taking easy shots at the 90%, give us some critical insight into the important 10%. I think this gives you the sense of what a “thought tool” is all about. Generically, it’s a useful way of approaching certain problems.
Occam’s Razor is another such thought tool and likely a familiar one. This one we can state as “Do not multiply entities beyond necessity.” Or, more directly as “Take the simplest theory.” That is, when I flip a switch and a light bulb comes on, I should probably assume an electrical circuit has been closed rather than assume that the switch was a signal for tiny ghosts to light a small fire in the bulb in my lamp. Again, this “thought tool” is a useful way of thinking about and approaching certain problems.
When we think about mathematical modeling, this idea of “thought tools” becomes valuable on two levels. First, the art of mathematical modeling is itself a thought tool. That is, it’s a way of approaching certain types of problems. Just like applying Sturgeon’s Law to the question of what happens when I flip my light switch makes no sense, it’s important to realize and keep in mind that mathematical modeling is a way of approaching certain types of problems. While it’s range of applicability is far greater than that of Sturgeon’s Law or Occam’s Razor, it is still limited, and throwing questions at it for which it’s not equipped is likely to lead to nonsense.
On another level, the idea of thought tools itself gives us a way to think about the teaching and learning of mathematical modeling. Suppose you were observing a philosopher at work and they were asked to choose between the electric circuit or the ghost theory of the light bulb as discussed above. It’s likely they’d, without discussion, simply assume the electric circuit and move on with their lives. It would be up to us to ask “How are you thinking about that?” Such a question would lead us to uncover the idea of Occam’s Razor, which we could then use for ourselves in all sorts of situations. It would then be a heck of a lot easier to teach students about the idea of Occam’s Razor and how to use it than it would be to teach them the answer to every question comparing alternative theories. That is we make more philosophers by teaching students both what to think about and how to do the thinking. But, we as teachers have to carefully observe practitioners and work very hard to understand what is going on with their thinking.
In the same way, we argue that we need to do this with practitioners of the art of mathematical modeling. In a previous post on the modeling cycle, we alluded to this need when we talked about how practitioners don’t really follow a series of steps and how the modeling cycle is only a crude model of the practice. This, in turn, means that there is likely a lot more going on with the mathematical modeler when they are practicing their art. They’re making use of many thought tools that remain hidden unless we work to ferret them out. The modeling cycle gives us some picture, but only in the broadest sense and only of the most obvious of these tools.
As an example, let’s consider “Formulate.” The CCSSM describes this step in their modeling cycle as “formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables.” But, there must be much more to it than that! How exactly do I “select”? Is it like choosing from a menu? Or, is there some other reasoning process involved? How do I “create”? When do I know to “create” versus “select”? Hidden in the practice and in that innocent looking box of “Formulate” there is a heck of a lot more going on.
That’s where this idea of “thought tools” comes back into play. The practicing mathematical modeler has a pretty full toolkit. When they see objects in motion they think “Newton’s Laws” or “F=ma.” When they see a measurable quantity changing with time, they think “conservation law.” And, when they see something in nature choosing a particular shape, they’re likely to be thinking “minimization principle.”
Note that none of these are automatic or inborn and this is where those who would teach mathematical modeling have some work to do. The teacher of mathematical modeling must themselves be a modeler, fill up their toolbox, and be aware enough of what’s in their toolbox that they can identify their thought tools and help equip others in the same way.
As Daniel Dennett says in the opening line to his book – “Thinking is hard.” Similarly, thinking like a mathematical modeler isn’t easy, but thought tools can make the road a little easier to travel.