What do I have to teach?
How can I best help my students learn this?
This is the first in what I hope will be a series that explains how I think about answering those questions.
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Years of research into teaching and learning have uncovered some basic fundamental principles of how all people learn. This will just be a quick overview of how I've tried to bridge that research with my practice and what I'm calling a pedagogical framework.
But first an excursion into the world of photography. The photographic technique of framing involves finding something that draws the eye, that sits around an object to draw the viewers' attention to that thing. Here's an example.
The lady is the object of this picture the eye is naturally drawn to see her there in the centre as she's framed by the windows of the subway car.
This is a strong example of framing as we see this man walking through the arches and those arches form a frame. The eye is inexplicably drawn towards the centre of the picture where we see the man.
In this case, a more subtle frame. And yet nonetheless it's clear that the horse is the object of the picture because the tree — which we don't really look at, it kind of sits in the foreground, the eye is really drawn towards the horse — provides a frame to see the horse.
What does this have to do with teaching and learning? Just bear with me.
First the three pedagogical principles, drawn mainly from this book: How People Learn. It was written in ... I believe it was 1999 and then it was updated again in 2000. Studying years and years of research they pulled out three fundamental principles of how people learn. As an outgrowth of that book another one was written called How Students Learn, specifically in the areas of History, Mathematics, and Science. These books have been absolutely seminal in forming my thinking and providing the pedagogical framework around which I structure all the teaching that I've done.
Typically when kids come to school they think of the world as flat. They have no reason to think of it as round and they come to school and we tell them the world is round. What studies have shown is that once kids leave school and they're asked to explain what they've learned: "Well the world is round." And their conception of the world is that it's a giant pancake. That's an interesting preconception that kids bring with them when they come to the classroom and teachers need to know that. Because the first principle out of the book is that "students errors and misconceptions based on previous learning" are the first thing that teachers have to try to connect with when they're trying to teach new content.
In this example, one drawn from mathematics, kids are often taught that multiplying is repeated addition. Multiplying is not repeated addition (although that's a good place to start) and they think that the answer to any question, when they multiply, has to be bigger than any of the numbers they started off with. Given a problem like this, fractions, well kids find that really hard, they just mul... they see the two the three and they know they have to have an answer that's bigger than either of them. If we know that that's the preconception that kids bring to the classroom then that can inform our teaching in constructive ways.
The second principle out of the book is that understanding requires not only factual knowledge, knowledge of basic facts, but an understanding of the basic ideas or big ideas of the discipline, whatever the discipline is that happens to be that you're teaching. Because knowledge isn't actually built in hierarchies knowledge is actually built in networks. For example, take something simple like three quarters. It seems like a pretty simple idea. But the number three quarters can be represented in a variety of ways. All of these are equivalent ways to write three quarters. It might have meant money, seventy five percent, it could be the ratio of three to four, point seven five or another way to write the fraction three quarters. Now even this extends beyond that because, "three quarters", well I could have been saying the money, three coins, three twenty five cent coins, which is seventy five cents. You see all these ideas are connected one to the other.
I might have been talking about a piece of cake; that I have three quarters of a cake. And it's implicit in that idea that each of those quarters is the same size. That each piece of cake has to be equivalent in size. But that's not always true with all ratios.
For example the ration of three red Smarties to four blue Smarties. I've got seven Smarties in total. And those Smarties don't all have to be the same size for my ratio of three red to four blue to be true.
These implicit assumptions make learning this material difficult for kids, and we as teachers make assumptions because we understand from the context that these things are clear. It could be seventy five percent off everything. That these ... All of these ideas actually live a network all one related to the other. The underlying assumptions that we make as experienced learners is that we take, from the surrounding context, what the meaning of each of these numbers is. And yet each one is related to the other.
That network has to be made explicit. That network of concepts has to made explicit to students. No matter what fundamental idea you're trying to get across to the students. In this case, the idea, the big idea, is really one of proportion.
The third principle out of the book is that "learning is facilitated through the use of meta-cognitive strategies". The degree to which we can get kids to think about what they're learning as they're learning it will deepen that learning. And they found that this made only moderate increases for high performing students, but for low performing students the use of meta-cognitive strategies made for dramatic increases in their performance.
Error analysis is a great example of this kind of thing; you've probably caught that one really quick. But as you look at this picture and try to find the error look how you're thinking about it. Look how you're paying attention to certain details; finding what's the same, what looks exactly identical? Where is the difference? Where is the thing that stands out one different from the other? And if you pause to reflect even as you're thinking about this now you'll notice that you've deepened your own thought as you look for the error in just something simple, like a picture of three Mounties.
So that's the framework. That's the framing. That's the ... Those are the ideas that sit around any pedagogical approach that you want to take in class. Any time you want to structure or design a learning experience for students these three principles:
Connect with kids preconceptions.
Learning should be networked. Ideas are networked. You need to understand basic facts but also the big ideas around which knowledge is structured.
And engaging kids in meta-cognition as they're learning.
This provides the framework for how we teach.
There's one last thing not to exclude in any of this and that's "community". It's alluded to in that book but not listed as one of the fundamental principles; but "community" is a pretty big deal. Because the degree to which we can get kids working with each other and collaborating and helping each other in their learning, which is what genuine learning looks like, is the degree to which they can deepen and accelerate their own learning.
This provides a wonderful example of exactly what I'm talking about: Why do geese fly in this "V" shape? It seems quite distinctive to see the Canadian geese flying in that pattern. It happens in the Fall, around October, and then again in the month of March as when ... as the geese leave in October and return in March. Why the "V" pattern? Because the flapping of the wings of the lead goose actually provides a little lift for the geese just behind them. And that's true for each goose behind every other goose. And so they're able to fly greater distances. Of course this puts undue pressure on the lead goose. So throughout the flight the geese are constantly shifting positions and rotating their spot in the "V" shape flight pattern. So that different geese take the lead at different times. And by working together the geese are able to fly for much greater distances than any of them would be able to fly on their own. And together they accomplish great things which is the very distant migration patterns of the Canada geese. Of course over short distances, if everybody goes their own way, well, yeah, you could have some success, and when they all land that's what it looks like. And they all kind of come down and each one chooses their own pattern and the "V" is broken up. And occasionally, you know, you need to do that, but by and large, for the most part, it's through that collaboration that great things become possible. And that's part of that framework that I talked about.
It's just like we learned in kindergarten: When you go out into the world, hold hands and stick together.