By Dr. Bijan Riazi-Farzad
Education is not about WHAT you learn,
but what that learning MAKES of you.
but what that learning MAKES of you.
In this part, I will demonstrate how education (by which I don’t just mean schools, colleges and HEIs) is not about teaching facts, but to empower learners to be able to evaluate information (either provided directly through the senses or indirectly through others) by being able to ‘see’ from different perspectives. The facts, or, in most cases, abstract ideas are a means to increasing the learner’s repertoire of thinking styles.
When you learn something new, you haven’t just picked up another fact, like a pebble that you may throw into a bag, you have picked up a cog that will change the way your mental machinery operates. That means that it changes the way you see, evaluate, relate to, use and appreciate your surroundings.
- So, how is learning that the tangent of a circle is perpendicular to its radius going to help me appreciate things more?
- It depends on HOW you learn that.
- What do you mean?
- I mean that memorising this fact to spew out into an exam paper is one way of learning it, which incidentally does its bit in changing your perspective on life, but not in an empowering way.
- How?
- For example, it may lead to the disempowering belief that there are meaningless things that you have to do to get by in this life.
- But you were going to tell me how it helps me appreciate things.
- Oh yes. Let’s imagine that you go into a lesson about circles and the teacher says that the tangent of a circle is perpendicular to its radius.
- Yaaawn!
- Exactly. Now, imagine that you start a discussion by asking, so what?
- That woudn’t go down very well.
- In an ideal world, you wouldn’t have to ask that question because the ‘so what’ would have been explained before the fact, but let’s assume that your teacher has the time and the inclination to talk you through it.
- OK, so what?
- Good! Now, I can explain to you that actually, there is no such thing as a tangent.
- What?
- And there is no such thing as a radius. These are ideas that we have had to invent to allow us to predict and manipulate our environment.
- You mean we just made them up?
- Yes. We made them up. They are abstract concepts that help us use language to talk about things that don’t exist.
- But why?
- For the same reason that we make up other words to help us explain things that don’t really exist.
- Such as?
- How about the notion ‘love.
- Love exists.
- OK. Show it to me.
- I can’t show it to you, but I can feel it.
- How do you know what you feel is what another person feels when they talk about love.
- Erm, well, I assume.
- Exactly. And that assumption is both necessary and very useful, is it not.
- Yes.
- So is the assumption that there is such a thing as a radius.
- But there is such a thing as a radius, it tells us how far the centre of a circle is from its edge.
- But circles don’t actually have radii, but creating that notion allows us to talk about the size of the circle and will help us calculate the amount of dough that we may need to buy and how much it would cost to bake a circular cake that would feed 10 kids at a party, a very practical application of a very abstract idea. We do that with love too.
- What? How?
- We represent the abstract idea of love, with a heart and then we put it on a card and sent it to someone. Here, we are manipulating (in the practical sense of the word) a relationship using symbol for an abstract idea, but look how useful that symbol can be.
- So how is knowing about the radius and the tangent of a circle going to help me then? I mean people don’t need to know these things t bake cake.
- If you do a job that requires it, then you may need to know it, and you never know when that may be. But in fact, if you do go into such a job, you will quickly be able to pick it up from scratch, so it is not about the knowledge itself.
- You haven’t answered my question.
- I’m coming to that. By knowing about tangents and circles and all the interesting ideas that are derived from them, you can now think about shapes from a different perspective?
- I can?
- Let me ask you a question; if you put an egg on a table top, is the surface of the table at a tangent to the egg?
- Yeah.
- But we said that a tangent is perpendicular to the radius of a circle and an egg is not a circle shape.
- That’s an interesting point.
- Well, I agree with you that the table top is at a tangent to the egg, but we are going to need to look at the egg in a different way to get our notion of tangent, as we have defined it, to fit into it.
- An egg is an egg. How can we look at it differently?
- Think of a square.
- Yeah.
- Now imagine that you add a side to it so that you get a regular pentagon.
- Ok.
- Now, keep adding sides, hexagon, heptagon, etc.
- Ok.
- Can you see what’s happening?
- Yes, it’s becoming more rounded.
- Brilliant! What will happen eventually is you keep adding sides?
- It will turn into a circle?
- Exactly. So, a circle is not a one-sided object, it is an object made up of an infinite number of straight lines.
- Is it?
- Well, it would be very useful if, sometimes, we could think of it in this way.
- Yeah, but what has this got to do with the egg?
- It has, trust me! In fact, if you think things through enough, everything has something to do with everything else. But let’s get back to the egg.
- Can’t wait!
- Well, if we can think of a circle as an infinite or, to avoid getting philosophical, let’s just say a very large number of straight lines, then the tangent can be thought of as an extension of one of those lines in both directions. Can you see that?
- OK. Yes, but what has that got to do with the egg?
- Well, if an infinitely small part of a circle can be thought of as a straight line, then, an infinitely small part of a straight line can be thought of as being a minutely curved object that can be part of a circle.
- What has that got to do with the EGG?
- Don’t you see it yet?
- NO!
- The point at which the table top touches the egg is the tangent to A circle, part of which forms part of the shape of the egg.
- What?
- The egg can be thought of as being made up of lots of arcs of circles joined together and the table is tangential to one of these. QED.
- OK, so you can play around with these in your head, so what?
- So, now you se that learning about radii and tangents can give you a completely new perspective on eggs or anything else that you shoes to apply it to.
- So what?
- So, now you can walk into that maths, physics, geography or history lesson and think, “Wow! I wonder how what am learning here is going to change the way I see and appreciate the world, rather than, “I wish I were somewhere else”.
- I’ll need to think about this.
- Don’t do it by yourself. Start a debating society or something and sit down with your friend and ask questions about how your perspective may have changes as a result of what you have been ‘taught’ in the lesson (or by any other of the day’s or the week’s experiences’
In part 3, I will discuss ways in which this kind of perspective could be propagated amongst learners, parents, policy makers and educators.
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