Saturday, September 3, 2011

Starting students on logical thinking

Every year, I make time in the first class of mathematics to introduce my students to the kind of thinking I expect from them throughout the year. That is, the inquisitive and logical thinking that'll prompt them to require and enjoy logical soundness in everything they are asked to learn in math class. I do this by showing them Lewis Carroll type syllogisms, such as this one:
  • All babies are illogical
  • Anyone who can handle a crocodile is not despised.
  • Anyone who is illogical is despised. 
I ask the students to form conclusions based on these premises, and every year I find that students are completely stumped by this task. 
Most of them start by questioning the premises. 
"It's true that babies are illogical, but you can be a pretty horrible person and still be able to handle a crocodile..."
I can work with this. After all, it's important that the premises are sound, or else everything is on shaky ground indeed. However, what the students are clearly telling me that they are not able to either do, or understand, the task I am setting before them. In short, they are failing the standard Piaget formal operational stage test.

To be fair, more recent research (after Piaget's) is implying that a disappointingly small percentage (about 20%) of the adult population can do this kind of formal operational task. Also, to be fair, it's probably not that students or adults are "not able", but rather that they don't understand what's expected of them because of a lack of practice with these kinds of tasks. Still, when I showed this video to my psych class (16-17 year olds) a few years back, a large majority of students agreed with the boy instead of the girl. 

So, back to my first class lesson. Walking around the room, I explain to the students what they are asked to do. "Suspend reality for a moment, let's pretend that these premises are correct. Then what can you conclude?" Once students had understood the task, they could solve it with only a tiny bit more prodding.  

I then show them a simple equation, say x + 5 = 7. Students are happy to conclude that x = 2, but now I hope that they are understanding that they are making a logical conclusion based on a premise which might or might not be true. 

I follow this up with the best example of mathematical problem solving that I know of: a Sudoku. Not only is Sudoku great because students are making logical conclusions every step of the way, and in a way such that the reasoning is their own and the "logicalness" of it is readily visible to them, but Sudokus also illustrate many important principles about mathematical problem solving. I'll write about that later.

I think it's important that students get this initial "feel" for logical reasoning. In fact, I ask them to feel it - for me, when something is logically sound, I have this satisfied calm feeling in my stomach. And when it's not, there is a worry, almost like an unpleasant itch. I know some students feel this more than others. Last year, suddenly, a student in the regular (non-accelerated class) simply refused, for weeks, to learn the basic rule of differentiating polynomials because she had missed the class where it was explained why this rule works.  I've encountered this many times already, often in students who are considered to be "weak" at math. Sometimes, when these students' need for logical soundness has been satisfied, they become much "stronger". I like and respect this resistance to learn things without understanding them. 

Throughout the year, I'll be coming back to this initial lesson on logical thinking. I will frequently ask students to explain why something is true, and not just to show me that they know how it is applied. I hope my students will understand, and appreciate, that this insisting on logical reasoning and understanding is not something "extra", something added to the already considerable pressures of their studies. Instead, at least in my experience, 20 minutes of effort at understanding why something is true pays off in exam scores better than hours of practice with more or less routine exercises. After all, in order to fully understand why the rules of logarithms are the way they are, you must fully understand logarithms. In order to solve routine exercises by looking at solved examples, you need only be able to use a formula. 

Of course, this introduction to logical thinking leaves many finer aspects unexplored. When students were attempting to draw conclusions based on the above premises, many offered that "Anyone who can not handle a crocodile is despised" and other illegal moves. Ideally, I'd like to spend more time on these kinds of issues, and in the Mathematical Studies class there is even a section on logic which I really like. Also, students may not be aware what constitutes a logical conclusion in all cases. As Sue recently pointed out, many students are not able to distinguish between an example and a mathematical proof of Pythagoras rule. Such issues can be overcome with counter-examples, but also necessitate a discussion of the difference between inductive and deductive reasoning (and why deductive is so superior! :)). 

Overall, I'm happy with this start. We'll see how it plays out over the coming year. 


  1. I agree with the boy. If you have an unreasonable if - then, it's probably a good idea to ignore it, and use your common sense. I don't think they were testing formal reasoning abilities, but willingness to say what you think the adult wants.

    I think a better test of formal reasoning might be something like Lewis Carroll did. I can't remember his stuff exactly, but something like...

    * All gribbles are plique.
    * Anyone who can droob is not glam.
    * Anyone who is plique is glam.

    Then your common sense doesn't get in the way of following the formal reasoning task. It's a cute puzzle now, worthy of our mental effort.

  2. Well, I think part of the idea of the formal operation stage is to be able to suspend the "common sense" critic for a while and just treat propositions about the world not as facts actually related to the world, but rather as starting points of a game one plays according to the rules of logic. The "gribbles" test would be somewhat misleading, because ultimately the point is that adults should be able to apply logical thinking to meaningful phrases, not just gobbleridook.

    Luria (a colleague of Vygotsky) has that famous example of speaking to illiterate villagers and literate townfolk in Uzbekistan. He asked: if cotton can only grow where it is warm and dry, and England is cold and wet - can cotton grow in England?
    "I don't know, never been there." was the answer he received from the villagers. Here, even though the premises make sense, the villagers were either unable or unwilling to step outside of ones own physical/concrete reality and look at the question more abstractly. I've posed this question to students, and their response was something like "well they have greenhouses don't they?" I don't know if that's comparable to the Uzbeki villagers or not.

  3. Perhaps if the questions are preceded by "once upon a time ...", the kids could let go of common sense, knowing that they've entered a magical place.

    I just got this idea from Zvonkin's lovely Math from 3 to 7, recently translated to English. Here's a quote:
    I ask Gene, “Once upon a time, a worm, a bike, and a plane decided to see who could run faster. It has turned out that the worm runs faster than the bike and the bike runs faster than the plane. Who do you think runs faster: the worm or the plane?”
    Gene gives the correct answer but not before wiggling and giggling and falling back on the sofa.