I wonder if you could do this?
When you've read this sentence, could you close your eyes and, in your mind's eye, see some birds standing somewhere (and when you've done that, open your eyes and read on).
A question:
How many birds were there? Was it a definite number?
(I'm interested. If you reply in the comments, I'd really like to know.)
I got interested in this question after I read this by Borges:
For one, we don't ask children to visualise very much, and this seems like another way to approach thinking, including number and shape.
Another thing is the idea of the indefinite nature of the interior image and its slightness and malleability, its sometimes fleetingness, vagueness.
One of the great pluses about number talks is that we're asking students not just what's 'out there' but what's in there too; How did they see those dots? How did they do that calculation? How are they sure? This kind of introspection is useful, and not just in mathematics.
I asked Sam and Pam to imagine birds on a wire. Then I asked how many there were. Pam said she saw seven, but wasn't sure if that was how many were in the first mental image she'd seen. Sam said it was a number between ten and thirty, but he didn't know the number.
I found these caveats intriguing. Down into the place where numbers aren't definite.
I asked Sarah and Lana. Sarah went on to write a great blog post about it. She says:
Lana said:
and
When you've read this sentence, could you close your eyes and, in your mind's eye, see some birds standing somewhere (and when you've done that, open your eyes and read on).
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A question:
How many birds were there? Was it a definite number?
(I'm interested. If you reply in the comments, I'd really like to know.)
I got interested in this question after I read this by Borges:
"I close my eyes and see a flock of birds. The vision lasts a second, or perhaps less; I am not sure how many birds I saw. Was the number of birds definite or indefinite? The problem involves the existence of God. If God exists, the number is definite, because God knows how many birds I saw. If God does not exist, the number is indefinite, because no one can have counted. In this case I saw fewer than ten birds (let us say) and more than one, but did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, which was not nine, eight, seven, six, five, etc. That integer–not-nine, not-eight, not-seven, not-six, not-five, etc.–is inconceivable. Ergo, God exists."Now, I'm not really into the metaphysical dimension here I'm guessing that Borges' was playing. What interests me is the visualising and the idea of an indefinite number.
For one, we don't ask children to visualise very much, and this seems like another way to approach thinking, including number and shape.
Another thing is the idea of the indefinite nature of the interior image and its slightness and malleability, its sometimes fleetingness, vagueness.
One of the great pluses about number talks is that we're asking students not just what's 'out there' but what's in there too; How did they see those dots? How did they do that calculation? How are they sure? This kind of introspection is useful, and not just in mathematics.
I asked Sam and Pam to imagine birds on a wire. Then I asked how many there were. Pam said she saw seven, but wasn't sure if that was how many were in the first mental image she'd seen. Sam said it was a number between ten and thirty, but he didn't know the number.
I found these caveats intriguing. Down into the place where numbers aren't definite.
I asked Sarah and Lana. Sarah went on to write a great blog post about it. She says:
If, in my minds eye, I see more birds than I can subitize, can I ever truly count them in their original form? Can I capture them? Or will they always be a “clump” somewhere between 10 and 15? When I try to count them, do I change them? By assigning them a number, do I bring them into existence?She asked her husband, who said:
4 pigeons.
There were birds and their existence was switching around. There wasn't a set amount; they were flicking between a few and a couple. I picked four when you said 'how many?' because I knew if I didn't answer you, you would say, 'I need you to tell me a number'.I love the candour of this reply, admitting to the process by which a thing whose existence is switching around becomes a definite number.
Lana said:
@Simon_Gregg my birds move, they fly up and down, I'm like trying to catch them in the air and drag down to the way, they don't oblige— Telanna (@TAnnalet) February 15, 2017
Lana asked her students, who gave quite definite answers:
I've asked a few pairs of my students too. With each pair I asked them to visualise a square first, and then a circle inside the square. They do that straightforwardly, drawing them on whiteboards. Then:
The next pair didn't draw the circles inside the squares. They said they'd seen:
and
D, on my right went on to talk about a quintillion!
I'm struck by how the pairs give similar answers. Small numbers, big numbers, many really big numbers.
I wonder how much what they say reflects any original image? I know that as a boy I wasn't particularly concerned about being truthful. I remember my teacher asking me about a picture on the wall that I'd painted, of a deer under a tree on a hill, 'Did you see it somewhere, or imagine it?' I thought, 'What does he want me to answer?' and said 'I saw it somewhere,' because I thought that was what was required. Actually I'd imagined it. It was only in my teens that I began to discover the pleasure of talking about things as they really are, the pleasure of sharing real thoughts and experiences with all their ambiguities and questions.
In addition, the confabulation of children is delightful and very fruitful. A four year old girl I don't know comes up to me in the playgound. 'Would you like a sweetie?' 'Okay,' I say. She hands me a stone which I pretend to gobble down gratefully, and walks away.
How many birds did you see?
I saw 7 birds. This morning on the way to work a small group of grackles flew down from a wire in a group. I couldn't decide then 7 or 8, but visualizing 7 made me realize it was eight.
ReplyDeleteLove your students images and wide variety of numbers. I'm wondering what it would be like to have them describe their vision to another student and have them draw it.
Thanks. I had to look up what a grackle is!
DeleteI get the impression that visualising shapes is easier for students than numbers of birds. This might be a good way to go, also building up some talking about shape language.
I pictured a reasonably large flock of birds sitting on a wire, probably somewhere around 20 of them. It was the first image that came to mind, a re-construction of what I saw on a recent walk.
ReplyDeleteWhat is interesting is that I wasn't really interested in counting them, I was using all of my energy creating the image to be as vivid as possible as this is something I find difficult.
In a Gattegno seminar on awareness I have just been reading, John Mason talked of 'entering into' an image, and I was taking the opportunity to practice doing that.
I was reasonably successful - it was fleeting but I brought more energy into the imagining - and with this came some of the other aspects of the experience - the fact that this group of birds was one of a number of flocks on a number of wires, it was a dark day by a beach, they were very noisy, I was walking with my daughter - a really experience to re-count, thanks for providing the exercise.
Gattegno suggests that imagination has the function of 'supplying a future for the present', and as such is vital in meeting the 'descending future'. We learn how to do it when we sleep.
“As babies we soon discover that recognition is possible because we can complete the evocation of a fraction of the perception of anything and can trigger a trail of images made from different senses associated with this perception. We cannot fail to notice that associations are many and diverse and that the content of our consciousness is made of images, perceptions, feelings, impacts from the outside, and also a presence of our self in all. Nor can we fail to observe that this stuff is labile and held together from moment to moment by subtle links that can consist of any of the elements on which the self particularly dwells for its own reasons.”
Excerpt From: Caleb Gattegno. “The Mind Teaches the Brain.”
A few thoughts occur to me sparked by this, and John's comment. Our imagination is really fed by our recent experience. If you've been walking on the beach or seen birds that day, it's going to come easier, more vividly.
DeleteAnd maybe imagining birds is harder than other things becuase they typically move around so much.
I'll have to check out the Gattegno book!
I pictured a flock of seagulls at the beach. I love the beach. There were too many to count and, as you pointed out in the previous reply, birds typically fly around a lot - definitely at the beach.
ReplyDeleteMind is an incredible thing. I noticed that after I'd read the article it was impossible to just imagine "some birds" without trying to manipulate the image to a countable number - they could no long just be birds. Birds on a wire was easier since I see them all the time outside my kitchen window and they are usually no more than 4 at a time. I think this is important to the teaching context where we are encouraging students to "image" quantities, not just subitizing, but picturing material demonstrations that use structured materials.
Tens frames are one of my favourite pieces of equipment for such an activity, but also place value blocks, 100s boards, bead strings, almost anything really. Demonstrate a couple of times, for example, what counting on is and how it's used to solve an addition problem with materials, then set up the next problem and ask, "what will I do next?", or cover the materials with a cloth and ask, "what does it look like?". It doesn't take long for them to be able to visualise the whole process without materials.
My experience has been that very few teachers know how (and why) to teach children to image a process or operation but your article shows how simple it can be. Creating a bias, a physical experience by way of demonstration, students can quickly picture the materials and manipulate them appropriately to solve simple problems. Some repeated practice of the thinking sequence with guidance enables the student to form useful internal pictures that lead to facility with mental calculations.
Michele, you're right. Thank you so much for your comment. I wasn't going in this direction originally, but this has all reminded me about visualisation and its potential in the classroom. I like what you say about it bridging from using concrete materials and working mentally. I'm going to be on the lookout for opportunities to weave mental images into our routines...
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