Some things have been going through my head. There's Kassia's post, Is There Room For Math That Isn't Hard? Also, a conversation on Twitter, one strand in a bigger conversation about intuition in maths learning. Things can get a bit abstract when you're down to 140 characters including names, but there were a lot of great points. Here's one definition of intuition that Kristin posted that I liked:
|"your insights and intuitions as a native speaker..."|
Somehow, it links for me too with a moment in our maths classes in Year 4 this term. I'd read a really interesting post, Making Sense, on Tracy's blog. I had all the Year 4s and I showed them this question:
I asked them to write their thoughts on their whiteboards. All of them, all of them, gave me a numerical answer! That really surprised me. I thought lots would, but all? I showed the classes the video on Tracy's blog afterwards, and very briefly talked about how some questions don't have answers.
Somehow, these things link, in my mind at least, because we need a solid base of intuitions about maths - partly what we call "number sense" - that helps us to deal with both meaningful and meaningless questions, and to tell the difference!
I also reached down Guy Claxton's brilliant book Hare Brain, Tortoise Mind from the bookshelf.
Claxton says there are three processing speeds in the brain. The fastest, faster than thinking, is the kind of response we have when we skid on ice and just do the right thing. It's the sort of processing a concert pianist or an Olympic fencer has to do. Then there's thinking itself, deliberation, which he calls d-mode. But "below this, there is another mental register that proceeds more slowly still. It is often less purposeful and clear-cut, more playful, leisurely or dreamy."
It maybe helps to look at deliberation, the familiar kind of thinking, first. Claxton lists some of its features:
1. is much more interested in finding answers and solutions than in examining the questions.
2. treats perception as unproblematic.
3. sees conscious articulate understanding as the essential basis for action, and thought as the essential problem-solving tool.
4. values explanation over observation
5. likes explanations and plans that are 'reasonable' and justifiable, rather than intuitive.
6. seeks and prefers clarity, and neither likes nor values confusion.
7. operates with a sense of urgency and impatience.
8. is purposeful and effortful rather than playful.
9. is precise.
10. relies on language that appears to be literal and explicit
11. works with concepts and generalisations
12. must operate at the rates at which language can be received, produced, and processed.
13. works well when tackling problems which can be treated as an assemblage of nameable parts.
So, d-mode is how we operate in maths lessons. You could even see mathematics as the place in which it shines most brilliantly.
But what of the slower thinking?
There is evidently a place for it. Here's Henri Poincaré:
"Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable, and traces of it would be found in other cases where it is less evident. Often when one works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, longer or shorter, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind. It might be said that the conscious work has been more fruitful because it has been interrupted and the rest has given back to the mind its force and freshness."(Claxton also gives lots of experimental results from cognitive psychology that demonstrate the effect of slow thinking. I'm glad he does this because words like intuition can sound unscientific, which they're evidently not.)
How to descend from this abstractness then? Is there a place for encouraging slow thinking and intuition in the primary classroom?
A few tentative answers. One: when you ask children what they notice, the pace slows down. There's time for a bit of pondering. Developing this as a regular part of lessons, and the respectful listening and responding that goes with it, allows half-formed and ill-expressed ideas space to breathe and develop.
Daniel Finkel's Prime Climb three times this term. There was no "teaching", apart from, briefly, how to play the game. But I feel that time when students aren't thinking, "I must learn this," is precious. Their hare brain's can be off duty. The games weren't physically slow. Lots of the kids were standing up! But... I hadn't "taught" anything. Slow in that way.
Maybe there's not time for slow thinking in your class. I understand. There's more pressure than ever to pack the learning in, to get the results. And we know ultimately, results will lead to jobs...
So, is there time to slow down?
Is it worth it?
If there is, and it is, what are good ways to do it?
Does it link with number sense?
Does it link with intuition?
Does this help with my meaningless number question?
Do you have any answers? Or more questions?
UPDATE - July 2015
I was really pleased when Gracia, towards the end of the year, came up with this question in class:
She knew it linked back to that "how old is the shepherd?" question we'd looked at before. Still, some people were not getting it. But some were now. As Gracia put it, all that information distracts you; it's like a magic show.
I recently watched Jordan Ellenberg talking about this kind of thing in another guise. I liked what he said:
Also, in The Joy of X by Steven Strogatz:
Other classic word problems are expressly designed to trick their victims by misdirection, like a magician’s sleight of hand. The phrasing of the question sets a trap. If you answer by instinct, you’ll probably fall for it.
Try this one. Suppose three men can paint three fences in three hours. How long would it take one man to paint one fence?
It’s tempting to blurt out “one hour.” The words themselves nudge you that way. The drumbeat in the first sentence — three men, three fences, three hours — catches your attention by establishing a rhythm, so when the next sentence repeats the pattern with one man, one fence, hours, it’s hard to resist filling in the blank with “one.” The parallel construction suggests an answer that’s linguistically right but mathematically wrong.
The correct answer is three hours.
If you visualize the problem — mentally picture three men painting three fences and all finishing after three hours, just as the problem states — the right answer becomes clear. For all three fences to be done after three hours, each man must have spent three hours on his.
The undistracted reasoning that this problem requires is one of the most valuable things about word problems. They force us to pause and think, often in unfamiliar ways. They give us practice in being mindful.