Some psychologists in B.C. asked a group of students a series of questions. Apparently, the study even caught the attention of Scientific American. The psychologists doing the study reported that there was some relationship between the intuitive mind and the religious mind. One question in particular was a short math question.
A bat and a ball cost $110. The bat costs $100 more than the ball. How much does the bat cost and how much does the ball cost? (note – I’ve changed the wording slightly)
Answer if you wish. The answer is not my point. It’s the process.
There were two common answers given by the students.
The more intuitive minded tend to say the bat is $100 and the ball is $10.
The more analytically minded tend to say the bat is $105.
Of the students involved, the ones that reported being religious also tended to answer the questions with an intuitive mind, mostly. The analytical students tended to report being not all that religious.
This may be important to the psychologists but it’s not my point, for now. Again, it’s the process.
When we read things, or when we are in situations that need our comprehension, or even when we don’t know what to do, we tend to first latch onto things we can already understand. We scan for what is coherent or makes sense to us. Otherwise, we don’t follow the ideas.
When I first read the math problem, my mind went immediately to thinking the bat was $100. But I knew there was a problem with this thinking. I went back through the question and figured it out, but it bothered me that I thought I knew the answer immediately. I didn’t trust the answer I came up with and so went back to the question. What’s the problem with the intuitive answer and what’s the difficulty with the analytical answer?
If we read the math question again, it says right in it:
…The bat costs $100…
When seeking information to latch onto, this looks like a trustworthy anchor. How can we know the bat is $100? Well, it says so right in the question. We might even stop reading right there.
But that’s not what the sentence is actually telling us.
The bat costs $100 more than the ball.
We know even less about the ball than we know about the bat. And what we know of the bat is only in relation to what we don’t know about the ball.
I saw this difference stumble up many of my students when I was teaching math years ago. In math it’s good to list the things you know and the things you don’t know. What you don’t know actually becomes more important than what you do know.
We don’t always want to think this way. It takes more effort. We have built up very convincing and efficient mental blockers that tell us not to bother. We can get along fine not facing what we don’t know. That is, until it doesn’t work.
Math is an aesthetic. It’s a language onto itself. Not everyone shares an appreciation for the beauty of math, and initiation into the group can be tough. The power of math comes in part from being able to work with an unknown. In the question above, we know nothing about the ball, really, but we can still give it a symbolic name and a place in our work – ‘X’.
With some of my students, I had to teach reading comprehension as much as math. We would talk a lot about what was meant by phrases like “more than” or “less than”, and how they could be written in math.
In some cases we rewrote the English sentences into math, step by step. We even talked about grammar – the subject of a sentence and the object of a sentence. I tried everything I could think of.
Maybe I should have taught them the Hero’s Journey.
I didn’t realize until this B.C study that I was teaching them one of the most important lessons of the Hero’s Journey. I was using a more modern, analytical framework, but still:
Face up to what you don’t know. What you don’t know is more important than what you do know. It may mean more effort, and it may mean you have to change your thinking about things, but it’s worth it.
The one thing that seemed to work best with my students? Repetition. I would put them through a model, then lead them through examples, and then test them with some similar practice questions. If they got stuck and needed my help, we would go through a problem together, but then I’d tell them to do the very same question again by themselves.
“Why? It’s done. I have the answer,” the students would say.
“You don’t have the process. It’s about the process.”
What else worked? A primed and structured environment, where they could be challenged by things they didn’t already know, but at their individual levels. Step by step. It was a real joy to see what my students were capable of when they trusted the process and adopted an attitude of first finding their unknowns.
The Hero’s Journey is usually represented by a circle – initiation, trials, then finally return. The process repeats itself; there is always something you don’t know.
It works. It’s costly. It means changing your attitude towards things. It means knowing how to deal with what you don’t know.
Couldn’t that sort of process, or that environment, be useful in a world that’s discovering new information at an exponential pace?
All right. Enough of my soapbox.
What do you think?