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.

– John

Thanks for bringing Daniel Dennett into the conversation. He is one of the few philosophers today who is able to make a meaningful contribution to a field most often associated with the humanities that embraces scientific thinking. Introducing philosophy into a discussion about mathematics does, however, invite an expanded view of the enterprise in which we are engaged.

Almost half of who we are (let’s say 40%) is inherited – we are born with a DNA and certain innate capacities we share with our parents and the rest of humanity.

The other “half” (again let’s say about 40%) is assimilation – we assimilate ideas and behaviors from our parents, peers and culture in ways which we are often unaware. We speak the language we hear, dress a certain way, belong to certain groups, etc.

That leaves a small percentage to a few other factors that make us who we are. Some percentage (let’s say around 10% – just by computing the time we spend in formal schooling and training compared to the length of our lives) is the result of transmission – we teach each other things we have learned. We pass our learning on to the next generation. That’s the enterprise in which we, as teachers, are involved. It also explains why we can’t do everything considering the inheritance and assimilation the students bring to the classroom.

Another percentage (let’s say around 10%) is acquisition – we take actions on our own – we put ourselves out there – we do things that change who we are. The physical models John puts together in his basement are a way for him to learn something on his own – through his own experience. Some people over estimate this and too often think of themselves as “self-made”.

Then there is that illusive 1% that is sometimes referred to as emergence. Each of us, through accident, luck or hard work is unique – some precious part of 1% of who we are is just found in us – apart from our genetic inheritance, cultural assimilation, pedagogical transmission, or even conscious effort. An emergent property of a system (of us in this case) is one that is not a property of any component of that system, but is still a feature of the system as a whole. Just as life, consciousness, and civilization were once emergent properties that sprang from a system in which they hadn’t existed before, we have the potential to add a small piece to the larger system that wouldn’t exist without us.

Marty – if you haven’t read the book “Curious” that I mentioned in another post, you should, I think you’d really like it. Some of the ideas you mention about are explored there as well, but with an eye toward the importance of curiosity.

One of the things that really stuck with me was the discussion of the innate curiosity with which we’re all born and the degree to which our very early experiences and environment impact that in a measurable way. For example, if an infant points at objects and the parent shows them the object and names the object, the child rapidly builds a vocabulary and learns that interacting with others is a valuable way to gain information. If they’re ignored they quickly learn this is not a useful way to gain information and stop pointing. The same is true for the questioning stage later on. Reminds me of the 40% you talk about as assimilation. We assimilate both good AND bad things from others, the habit of curiosity being a really key one!

Thanks for the reminder about the book “Curious”. The book reinforces Carol Dweck’s concepts of fixed and growth mindsets. We are born curious but we must “use it or lose it.” Curiosity can be squelched by rigid pedagogy and it must be nurtured and practiced to grow. I’m thinking that the idea of “Change Just One Thing” – nurturing questioning and curiosity – may be the key to transforming education. Change just one thing – help students develop their ability to ask questions and have life-long curiosity.

The paperback version comes out in December – http://www.amazon.com/Curious-Desire-Know-Future-Depends/dp/0465079962/ref=sr_1_1?ie=UTF8&qid=1438805222&sr=8-1&keywords=curious