Sunday, 28 December 2014

Intuition and slow thinking

This post is going to be more questions than answers...

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.
Two: games. I got the classes playing 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.

Saturday, 13 December 2014


I touched on knowledge in the last post. I was intimating some kind of process a bit like this, starting at the bottom with exploration:
and ending up with more exploration at the top.

Knowledge was something the Greek philosophers were keen to find, to develop, to pass on. Real knowledge for them was not simply believing the truth, but also having reason to believe it.

So many of our beliefs are not well-founded; so much of what we are told at school we believe because we are told, not because of experience or evidence or conclusiveness. Real knowledge is a precious thing.
So, it seems to me, respecting the genesis of this founded belief is really important. We have all sorts of utilitarian ideas about why schooling, or a particular kind of schooling is good. It's good because we can participate in the economy, we can be internationally competitive. As the new English National Curriculum for maths has it, "it is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment." True as that may be, maths, like all subjects, can be something else, an opening of the eyes, a body of undertandings, of knowledge. Finance doesn't know. Employment doesn't know. How can they guide us? How can they weigh opinions, find truth? Maths, like no other subject, can give a sense of what certain knowledge really feels like. We need that. 

Have you ever read Plato's account of Socrates defence ("The Apology")? I recommend it. It's a really essential read, short, fascinating and pivotal. In it, Socrates describes his enemy-making quest to find what people really know:
Accordingly I went to one who had the reputation of wisdom, and observed to him - his name I need not mention; he was a politician whom I selected for examination - and the result was as follows: When I began to talk with him, I could not help thinking that he was not really wise, although he was thought wise by many, and wiser still by himself; and I went and tried to explain to him that he thought himself wise, but was not really wise; and the consequence was that he hated me, and his enmity was shared by several who were present and heard me. So I left him, saying to myself, as I went away: Well, although I do not suppose that either of us knows anything really beautiful and good, I am better off than he is - for he knows nothing, and thinks that he knows. I neither know nor think that I know. In this latter particular, then, I seem to have slightly the advantage of him. Then I went to another, who had still higher philosophical pretensions, and my conclusion was exactly the same. I made another enemy of him, and of many others besides him.
Read this too, from Jonathan Haidt's great book The Happiness Hypothesis:
In philosophy classes, I often came across the idea that the world is an illusion. I never really knew what that meant, although it sounded deep. But after two decades studying moral psychology, I think I finally get it. The anthropologist Clifford Geertz wrote that “man is an animal suspended in webs of significance that he himself has spun.” That is, the world we live in is not really one made of rocks, trees, and physical objects; it is a world of insults, opportunities, status symbols, betrayals, saints, and sinners. All of these are human creations which, though real in their own way, are not real in the way that rocks and trees are real. These human creations are like fairies in J. M. Barrie’s Peter Pan: They exist only if you believe in them. They are the Matrix (from the movie of that name); they are a consensual hallucination.
This is why it's great that we can find knowledge that's as firm as our direct knowledge of rocks and trees. And young children have this as they explore the world. Papert, points out how Piaget's "genetic epistemology" made us much more conscious of how much young children are learning:
While we can “see” that children learn words, it is not quite as easy to see that they are learning mathematics at a similar or greater rate. But this is precisely what has been shown by Piaget’s life-long study of the genesis of knowledge in children. One of the more subtle consequences of his discoveries is the revelation that adults fail to appreciate the extent and the nature of what children are learning, because knowledge structures we take for granted have rendered much of that learning invisible. We see this most clearly in what have come to be known as Piagetian “conservations”.
(It's interesting re-reading Seymour Papert's Mindstorms, because I find things in there that I say. Now I wonder, did I get that from Papert? Or was there just a similar take on things?)

School can and should be about "understanding rather than turning the handle" (as someone has written on their Twitter profile). There will have to be some handle-turning for sure, but let's keep pride of place for the understanding, the knowledge.

Monday, 8 December 2014

A short conversation

I found this conversation thought-provoking.

I'm not experienced in university maths (I did science), but from what I do know I want to say that there is a continuity, and not just out of respect for the efforts of young children.

Of course, Colin is correct: there are an awful lot of sums in schools, hopefully less than there were when we were at school, but still, many people do think that sums are the thing. And it must get boring getting thought of as a difficult sum doer!

But, but... there is another side to school maths, and one that I would like to see take over as people's picture of school maths.

Take this Kindergarten Interlude that Joe Schwartz posted on the other day.
Here are children in a comfortable situation, getting to know number, its constancy, its patternability, its many possibilities. I like how Joe says, he is there to "poke, push and experiment".

Now this is well-trodden ground, but children are building up their knowledge here, step by step, tentatively at times, in leaps at other times. Sometimes the knowledge will be certain: however you arrange three bears, they will always be three. Other times it will be a hunch: you need four bears before you can have a pattern.

Or take Kristin's post on Articulating Claims in Math, The elementary kids are doing "sums" here, but they're doing a lot more, they're making generalisations about the way sums work:
Now you might argue that proof involves more than this. That it involves formal generalised description. Or that it must be a shared and socially verified knowledge.

But I would say the exploration comes first, then the hunches, then more exploration, then the knowledge, then the sharing.

Consider Polya's treatment of Nicomachus's theorem. How would you first discover this? Polya writes:
Mathematics presented with rigor is a systematic deductive science, but mathematics in the making is an experimental inductive science.
And that is what is happening in many excellent maths classrooms, and could be happening in even more. I like Polya's example, not least because the nine year olds in my class had a go at exploring this last year.
A different kind of activity comes before proof. You get a sense of it in this video:

And proof is not about formal language either. There's the famous Bhaskara proof:
Bhaskara's only word: "See!"
As Keith Devlin says about symbols, "They no more are algebra than a page of musical notation is music."

So, I want to make the case, here as I do in my teaching, that there is a continuity, there is in the best classrooms a relationship with what kids are doing and what mathematicians do.

What do you think?