Recursive Critical Thinking

I was going to entitle this piece "critical thinking squared" as a cute way to imply critical thinking about critical thinking, but the imprecision bothered me. Squared means multiplication by self, and that's not the same as applying a process to itself. What multiplication means in this context isn't precise either, but you can possibly make a sense of it by considering a combinatorial factorization into dimensions like critical thinking = (analysis, creativity, communication). If we abbreviate critical thinking = CT, then CT² might look like a matrix:

	Analysis	Creativity	Communication
Analysis	Analysis	Creativity * Analysis	Communication * Analysis
Creativity	Analysis * Creativity	Creativity	Communication * Creativity
Communication	Analysis * Communication	Creativity * Communication	Communication

This assumes that each dimension is idempotent (meaning S*S = S), and that "*" is some way of combining the two dimensions. You still have to figure out what the Creativity * Analysis combination means, but at least you have a way to produce detail from the squaring operation. But this is all rather silly, and the reason I don't like the "CT squared" idea.

Here's a better way to think about it. If you talk about talking, that isn't (talking)², but rather talking(talking), expressed here as a function that takes itself as an input.

This is even practical. For example, you could write a function get_loc(...) in the C programming language to take a function and return its address in memory. Then you could ask for get_loc(get_loc) to retrieve its own location. This sort of thing is called recursion, and it's a big deal in computer science. In common parlance, we might stick the prefix meta- in front of the concept to show that it's recursive, as in metacognition, which in the right context we justifiably call a noncognitive trait: the reflective practice of thinking about one's own thinking process. More on the relationship between CT and noncogs later. First, let's take a closer look at the role of recursion in problem solving.

In a few stolen moments this morning I was sipping an iced latte, enjoying a cool breeze, and trying to make some progress on a research project regarding survival in a certain abstract sense. You can read the actual paper here, but in a nutshell you can imagine an environment that poses survival challenges to organisms (all abstracted into computer language, which you can learn more about and download a simulator here). There are two different questions regarding the complexity of a given environment:

1. What's the simplest thing that can survive the given conditions?
2. What's the simplest recursive process that can find the solution to #1?

Here, recursion means that some process can be tried over and over again, feeding the output of the last iteration into the input of the next. Like natural selection, for example, acting recursively on the gene pool to blindly hone the fitness of the survivors.

Let me give a more down-to-earth example, as a simple "critical thinking" problem. Suppose Tatiana works all day in retail, and part of her job is to calculate sales reductions for coupons, sale prices, and so on. In addition she has to add sales tax to total amounts. For reasons known only to management, they skimped on her point of sale (cash register) and these functions are not included. So all day long she has to do stuff like:

1. Find actual cost of an item by reducing for sale price, coupon, etc.
2. Sum adjusted prices
3. Calculate tax
4. Add to get total

This is pretty tedious and prone to error, so there's an advantage to having the best possible way of doing this. In the context of my framing questions, we should ask:

1. What's the best way of doing her job?
2. How do we find it?
   

In practice, we have to address the second before the first. The second we might call a critical thinking exercise, requiring analytical and creative thought.

Tatiana need not be reflective. She probably already has a solution, and may not care that it's not optimal. I see this all the time in real stores. A clerk wants to reduce an item by 15%, say. Most often they multiply the price by 15% (.15) on a calculator, write down this number, and then subtract from the original price. Sometimes I tell them a quicker way to do it: just multiply the original price by .85 and you're done.

In a complex environment, you're never really finished with the "how do we find a better solution?" step. The question itself is recursive: "how do we find better ways of finding better solutions?" Mathematics is full of this sort of thing. You can follow the chain of meta-thought all the way up to something called category theory, where logic itself can be generalized (logic about logic).

So without really thinking about the definition, the idea of "critical thinking" can get you rapidly into the deep part of the pool. For me, it's very important to keep straight the difference between knowing a good solution to a problem and finding a good solution to a problem. I don't think this is as appreciated as it should be. The first is analytical/deductive and the second is creative/inductive, and they require very different preparations.

Think of an old-timey telephone switchboard operator, plugging and unplugging wires all day.
(photo courtesy of Wikipedia). There's a vast difference between knowing how to operate the switchboard and knowing how to design one or improve existing designs. The link between these two questions, as with the ones above is the "why" operator. Here's a possible chain of why-iterations for thinking about telephones:

1. Q: Why are you moving those wires and plugs around?
   A: I'm operating a switchboard according to the procedures I've been trained in
   
2. Q: Why is there a switchboard?
   A: To facilitate telephone calls.
   
3. Q: Why are telephone calls useful?
   A: So people can communicate across distances.
   
4. Q: Why do people need to communicate across distances?
   A: So they can lead better lives.
5. Q: Why do people need to live better lives?
   

Each one of these increasingly general domains has its own problems and solutions. Solving the general ones can make the specific ones go away. If we keep asking why (see this related article), we end up with very general questions like "what problem does my existence solve?" and "why does anything exist?" I've tried to portray this recursion graphically below.
I'm assuming that the creation of knowledge is scientific (what art does is something different from what I mean here). I've quoted Bertrand Russell before on this point (here):

All definite knowledge--so I should contend--belongs to science; all dogma as to what surpasses definite knowledge belongs to theology. But between theology and science there is a No Man's Land, exposed to attack from both sides; this No Man's Land is philosophy.

Philosophy, theology, and other avenues of inquiry that remain immune to the scientific method are lumped together at the bottom of my graph. Wouldn't it be nice if we showed our students of critical thinking how this works? The unveiling of the breadth of meta-thought ought to be a stunning moment of realization for an undergraduate. Consider the following question and meta-question:

Why is the sky blue? (proposed answer here)
Why ask why?

From a scientific question we arguably leap directly over all of science to a philosophical one. Not only that, to my eyes it seems like a fixed point under meta-recursion. That is:

Why ask "why ask why?"? is the same as Why ask why?

Which would make the question the most profound one possible, I suppose. This could be a great starting point for a course on critical thinking. Note that I kind of cheated in my one-step leap to "why ask why?" Figuring out how is your meta-cognition homework. :-)

How does all this fit with existing literature on critical thinking? A colleague recently pointed me to the 1988 publication "The Delphi Report" on critical thinking, which arguably kicked off recent interest in the teaching and assessment of said skill. In the executive summary, which is linked to the title, a consensus statement reads:

We understand critical thinking to be purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation, and inference, as well as explanation of the evidential, conceptual, methodological, criteriological, or contextual considerations upon which that judgment is based.

This is quite different from the line I've taken above, isn't it? Maybe it's not even useful to use the term "critical thinking" for both. In the above definition, which is the kind usually used, it's described as an activity with a particular type of result. The activity itself is not described here other than purposeful, self-regulatory judgment. These are noncognitive descriptors, please note. In fact, the definition elaborates on this point with a vivid description of the thinker:

The ideal critical thinker is habitually inquisitive, well-informed, trustful of reason, open-minded, flexible, fairminded in evaluation, honest in facing personal biases, prudent in making judgments, willing to reconsider, clear about issues, orderly in complex matters, diligent in seeking relevant information, reasonable in the selection of criteria, focused in inquiry, and persistent in seeking results which are as precise as the subject and the circumstances of inquiry permit.

As far as I can tell, in practice most programs in critical thinking don't actually pay much attention to the "affective" or noncogitives listed. But that's not too unexpected--academics in general seems to be allergic to modeling and teaching personal attributes. I find it increasingly odd that this is so.

Into the meat of the executive summary we do find particular cognitive skills. You'll see these or similar ones in rubrics and learning taxidermy.

1. interpretation
2. analysis
3. evaluation
4. inference
5. explanation
6. self-regulation

I don't see how self-regulation is cognitive, but maybe it is in a metacognition sort of way (self-reflection). One of the findings is that evaluating one's own thinking is a way to improve it.

Although they don't get around to saying it this way, the authors note the importance of analytical skills:

Although the identification and analysis of CT skills transcend, in significant ways, specific subjects or disciplines, learning and applying these skills in many contexts requires domain-specific knowledge.

Knowing how to solve an urgent problem while sailing is different from solving one while flying a plane.

This debate is important. Lots of institutions put "critical thinking" on their to-do list. Good definitions should lead to good implementations and good assessments.

Although I appreciate the value of the work that's been done in traditional meta-critical thinking, I don't much like the result--those lists of vague terms like interpretation and evaluation. I know they can be used successfully, and they can probably produce a good curriculum and assessment. But to me they're just a disjoint collection of loosely-defined techniques that a committee came up with. There's no underlying structure or theory. No way to make sense of it all by asking the meta-question: why is critical thinking the way it's described in "The Delphi Report?" You can only answer that the experts agreed that this is what it should be. In Russell's description, this makes it dogmatic. And if critical thinking has any value at all, it's to question dogma, no? That makes it ironic, but we can't judge too harshly on this account. Even Karl Popper freely admitted that his system could not be proven to be self-consistent (i.e. prove that nothing is every really proven). You have to start somewhere.

It may just be my bias coming from a computer science/math background, where a more natural schema is the study of algorithms and complexity, but I'd like more than the opinion of a panel of experts. I want to keep asking why until there is a self-consistent answer, if possible. In the meantime, here's my recipe for teaching CT:

1. Analytical thinking in various domains
2. Creative thinking in domains where analytical thinking is well-developed by the student
   
3. Training in recursive thought [ including CT(CT) ]
   
4. Communications skills (otherwise, what's the point?)
   
5. Noncognitives like intellectual honesty and open-mindedness and willing use of the above skills.
   

I know the first two can be effectively assessed; I've never tried to do the third because it just occurred to me today. The last one ought to be on everyone's to-do list.