"I realised something: he doesn't know numbers. With the abacus, you don't have to memorize a lot of arithmetic combinations; all you have to do is learn how to push the little beads up and down. You don't have to memorize 9 + 7 = 16; you just know that when you add 9 you push a ten's beadup and pull a one's bead down. So we're slower at computation sometimes, but we know numbers."
Richard P. Feynman, from Surely You're Joking....
Is there such a thing as a stupid question? I always remember in the little military training I have done, they say ‘No, not when it comes to weapons’. But what about in general? In a learning environment, people have questions to make sure they can do what they’re supposed to be doing. It would seem logical that they should ask questions to make sure it’s right. However, sometimes these questions reveal that they know what to do, but they don’t know what they are doing.
With the advent of online courses, such as edX and Coursera, learning is beginning a revolution. There's no longer a need to attend lectures - it can be at your fingertips and viewed as many times as you like. Universities are starting to spread their net wider, inviting mature students to attend who have years of experience at work but want to progress (a good model in my opinion). Universities are not necessarily ready for this however. Some have yet to adjust. For example, I was told by one lecturer that:
"You're a second year student. You might be just a few years younger than me, but you're a second year undergraduate, and you don't act like one."
(Well, no, I'm not going to act like a 19 year old, age 35 with many responsibilities and experience).
Yet the danger with all of the above is that it might just be informational learning - the passing of information from a lecturer's blackboard to the student's notebook without anyone thinking in between. It needs to be transformational learning where understanding, a skill and memorisation are all achieved in one go.
Learning can be made exciting, or it can be made crushingly boring. With some clever use of psychology, and the bullet-points above, learning can be multi-dimensional, instead of one-dimensional.
To paraphrase St Francis of Assisi:
Where there is boredom, may we bring excitement
Where there is despair, may we bring hope
Where there is panic, may we bring passion
Where there is error, may we bring patience
Where there are mistakes, may we view them as lessons!
*
From Surely You're Joking...
Richard P. Feynman, from Surely You're Joking....
Is there such a thing as a stupid question? I always remember in the little military training I have done, they say ‘No, not when it comes to weapons’. But what about in general? In a learning environment, people have questions to make sure they can do what they’re supposed to be doing. It would seem logical that they should ask questions to make sure it’s right. However, sometimes these questions reveal that they know what to do, but they don’t know what they are doing.
Here are some examples I have encountered:
When required to enter an angle into a formula, the question was asked:
“For cos 60, do I use the degrees or radians mode?”
On seeing this, A>>B:
“What does chevron chevron mean?”
“What’s a rivet?”: Engineering class
“What does I stand for?”: Electronics class
When asked ‘What shape do you get when you multiply 2 numbers together?’ my student asked:
‘A real shape or an imaginary shape?’
Overheard from one student copying another's work(!):
Student A: "What does that bit mean?"
Student B: "I don't know. The lecturer just told us to put it."
Student A: "Oh, ok." [continues copying]
Overheard from one student copying another's work(!):
Student A: "What does that bit mean?"
Student B: "I don't know. The lecturer just told us to put it."
Student A: "Oh, ok." [continues copying]
And there are many more…these are just the ones I can remember. These questions show that it is glaringly obvious that the students on these courses are able to pass exams, or get their hand held through tutorials, but they don’t really understand what the material is trying to convey. They do not understand the concepts. Their sole focus is passing an exam, so much so, they’ll actually ask, ‘Is this in the exam?’ and if it isn’t, they’ll entirely switch off.
‘Education’ has produced a generation (or more) of people who are not interested in why things are. Just what mark they can get. They do not have a passionate love of learning, but see it as a chore.
Last week I asked my daughter, 'What do you want to be when you grow up?’ Okay, at first, she said ‘A monkey’. But when I explained I meant, 'What job?', she replied, ‘A teacher’. Why? Because then she wouldn’t have to learn, she could just teach!
At age 5 to have already picked up (by osmosis I guess) this attitude, is a damning indictment on the education system.
And this is an international problem, even where it appears that students are doing well in their respective country. Japan
So, this is a chicken and egg situation. Did the students have the attitude before they went to school, or did school plant the idea into the student? Any primary school teacher will tell you that for the first SATs test, the pupils are taught to the test.
And any Montessori teacher will tell you that children intuitively love discovery.
For GCSE maths, the emphasis is getting a C grade, because that is a de facto pass in the eyes of the government. So if you can easily get an A grade, you are largely ignored, and if you’re always going to get an F, you’re ignored. All the resources are thrown into the C/D students. What message does that send?
In spite of all this tinkering, manipulation and ham-fistedness, the national pass rate is still only 57% for GCSE. And that’s to achieve a low percentage on an exam that is essentially taught for 10 years.
For the first time as a tutor, I had a student just recently get a ‘U’ on his first maths module. Why? Because we had limited time and I trusted him to do the work he said he was doing, so I just covered theory. But what was his response? ‘Well…it was a high U’.
The mark shows that he does not understand maths and hasn’t practiced it sufficiently to achieve fluency. But his attitude is fully concentrating on the mark he got – not the bigger picture.
I’ve recently watched a couple of excellent talks by Sir Ken Robinson, and he asks ‘What do we take for granted?’
Thinking about this, I thought: one thing we take for granted is the awarding of grades. Why are exam results graded? Shouldn’t they be pass or fail? You might argue, “Well, that wouldn’t allow differentiation between students!” But why do we differentiate? The idea of education is to give everyone equal skills, opportunity and knowledge. What is this differentiation actually revealing? Intelligence? Hard work? That the parents can/cannot afford a private tutor? Calmness under pressure?
The only fair system would be a pass/fail mark, where to pass you have to achieve 100%. So that’s 100% of any subject on any paper. And if you fail and get 99%, you have to take it again. And again if necessary. What should be recorded, perhaps, is the number of times you have to take it! Driving tests are like this and no-one questions it. Imagine if we saw the grade inflation in driving tests that we saw in national exams – what would be the consequence on the roads?
Incidentally, when I was learning to drive, the first question I asked was ‘How does the clutch work?’. The instructor said that no-one had ever asked him that before. Why? Because they were only concerned about what they had to do to pass the test.
Thinking about this, I thought: one thing we take for granted is the awarding of grades. Why are exam results graded? Shouldn’t they be pass or fail? You might argue, “Well, that wouldn’t allow differentiation between students!” But why do we differentiate? The idea of education is to give everyone equal skills, opportunity and knowledge. What is this differentiation actually revealing? Intelligence? Hard work? That the parents can/cannot afford a private tutor? Calmness under pressure?
The only fair system would be a pass/fail mark, where to pass you have to achieve 100%. So that’s 100% of any subject on any paper. And if you fail and get 99%, you have to take it again. And again if necessary. What should be recorded, perhaps, is the number of times you have to take it! Driving tests are like this and no-one questions it. Imagine if we saw the grade inflation in driving tests that we saw in national exams – what would be the consequence on the roads?
Incidentally, when I was learning to drive, the first question I asked was ‘How does the clutch work?’. The instructor said that no-one had ever asked him that before. Why? Because they were only concerned about what they had to do to pass the test.
If it didn’t matter about failure, I suspect that students would start to develop a love for learning, because virtually all pressure would be removed. In a way, they would have no choice but to begin to love learning, because they will never pass an exam until they understand it fully.
In engineering they have a concept called ‘Poka Yoke’ (sounds like hokey-cokey). What this means is that things are set up intuitively for humans to do without them realising it. Sometimes this is by accident, and sometimes it is intentional. When I tutor maths, I follow the same pattern. Questions are asked in such a way to bring out the intuitive, creative side of a student’s natural ability to use pattern.
Michel Thomas, the language teacher, realised this also. For example, when you learn future and past tenses of French at school, it is a laborious process of memorisation. Michel noticed a pattern and then teaches it like that. The pattern is this. For future tenses, say ‘I’, then the whole verb, then ‘have’.
So for example, I will eat is
Je manger(ai)
I to eat have
For the other persons, tu etc, it follows the same pattern.
For the past tense, put the ‘have’ bit before the verb – which makes sense, since it’s in the past!
So, for example, I have eaten
J’ai manger
In speaking they sound the same. In writing, French speakers will spot that it’s wrong. But that doesn’t matter. We’re learning to speak. Then we can argue about how it appears… and should be J’ai mangé.
So for the past, we put ‘have’ before and for the future, ‘have’ after – makes sense since it will be in the future!
And 90% of the time, this is the case. So in a few sentences you can speak 90% of all all verbs in the future and the past. Now how long does it take to learn this at school? Years? Have most students dropped languages by then, saying it’s boring?
Almost perversely, school does not explain things simply and isn’t interested in trying to!
Another example is multiplication. At primary school, they use something called the ‘grid method’. The goal here is to multiply two numbers together such as 14 x 21.
In many ways, it seems like a reasonable way to learn to do multiplication. It is split up into easier chunks and an answer can be computed with some work. For me, the problems with it far outweigh the benefits. They are:
It’s got the wrong name, and is thereby misleading
It doesn’t teach intuitively being entirely algorithmic
It is prone to error
It makes decimal calculations seem impossible, such as 0.14 x 2.1
- The wrong name?
“Well, it’s a grid Paul, I mean, what else could we call it?”
The name it should have is the ‘Area method’. Why? Because the four multiplications that are carried out calculate the areas of each box. So therefore, we are calculating an area of the whole grid. Incidentally, it is not drawn to scale either, so this is also misleading and confusing. If it was called the Area method, obviously it would then be incredibly simple to teach how to calculate areas and the unit of area! But what did my student ask when I asked her ‘What shape do we get when we multiply 2 numbers together?’ She had no idea. She even thought I could mean an ‘imaginary shape’!
- It doesn’t teach intuitively, being entirely algorithmic
“What the hell are you on about, now?”
Remember the Poka Yoke. If we teach intuitively, and let the student DISCOVER the truth, they’ll never have to remember. It’s like not having to remember not to put your hand on a hot stove. You know the consequences! You don’t put your hand near and think ‘Hmm, there’s something I should remember here…’.
Because the grid method follows a number of steps, it is essentially a computer program. But our brains are not computers. Computers are nothing as compared to our brains. My method of multiplication (once it is understood via the first rule of maths) is so easy that virtually every student exclaims ‘Wow! Hang on, why don’t they teach it like this at school? It’s so easy.’
- It is prone to error
Once the four calculations are made, you then have to separately add them up. Here students often make an error because they don’t align the units, tens and so on, get a ridiculous answer, think ‘Well, that’s the way it’s meant to be’ and think it’s correct. Or worse, ask if it is.
My method has no risk of error as it contains almost no working.
- It makes decimal calculations seem impossible, such as 0.14 x 2.1
I had a student, when on first meeting, was able to multiply 14 x 21 using the grid method. But she didn’t attempt 0.14 x 2.1. Why? Because she didn’t even know where to begin using the grid method. (Also she didn’t realise it was the same answer, but that’s an example that proves my point exactly…the difference between knowing what to do and knowing what you’re doing).
With my method, students can calculate things such as
0.000014 x 0.0021
in seconds. Or do percentages in their head...and have confidence doing so.
in seconds. Or do percentages in their head...and have confidence doing so.
Richard Feynman used to argue that there was a 'pleasure in finding things out'. He used to comment that knowledge is so fragile when you learn by rote, or to pass an exam. Being mischievous, he once had his entire technical drawing class believe that French curves followed a special rule that the tangent of the lowest point is always flat (this is true for all curves and extremely important in maths - something they should have realised*). He was also aghast when asked to review textbooks for the State of California - he remarked there was no discovery, just definitions to memorise with words that had no meaning. [Richard Feynman's comments have been raised in another blogpost of mine!]
So what are the steps to successful learning and teaching?
- Make it intuitive and allow students to discover rather than be told
- Use pattern and symmetry wherever possible
- Reduce fear of failure by allowing multiple tests and no time limits
- Increase pass mark to 100% to legitimately raise standards
- Reduce the number of formal exam landmarks, to avoid rote memorization or teaching to a test
With the advent of online courses, such as edX and Coursera, learning is beginning a revolution. There's no longer a need to attend lectures - it can be at your fingertips and viewed as many times as you like. Universities are starting to spread their net wider, inviting mature students to attend who have years of experience at work but want to progress (a good model in my opinion). Universities are not necessarily ready for this however. Some have yet to adjust. For example, I was told by one lecturer that:
"You're a second year student. You might be just a few years younger than me, but you're a second year undergraduate, and you don't act like one."
(Well, no, I'm not going to act like a 19 year old, age 35 with many responsibilities and experience).
Yet the danger with all of the above is that it might just be informational learning - the passing of information from a lecturer's blackboard to the student's notebook without anyone thinking in between. It needs to be transformational learning where understanding, a skill and memorisation are all achieved in one go.
Learning can be made exciting, or it can be made crushingly boring. With some clever use of psychology, and the bullet-points above, learning can be multi-dimensional, instead of one-dimensional.
To paraphrase St Francis of Assisi:
Where there is boredom, may we bring excitement
Where there is despair, may we bring hope
Where there is panic, may we bring passion
Where there is error, may we bring patience
Where there are mistakes, may we view them as lessons!
*
From Surely You're Joking...
I often liked to play tricks on people when I was at MIT. One time, in mechanical drawing class, some joker picked up a French curve (a piece of plastic for drawing smooth curves--a curly, funny-looking thing) and said, "I wonder if the curves on this thing have some special formula?"
I thought for a moment and said, "Sure they do. The curves are very special curves. Lemme show ya," and I picked up my French curve and began to turn it slowly. "The French curve is made so that at the lowest point on each curve, no matter how you turn it, the tangent is horizontal."
All the guys in the class were holding their French curve up at different angles, holding their pencil up to it at the lowest point and laying it along, and discovering that, sure enough, the tangent is horizontal. They were all excited by t his "discovery"--even though they had already gone through a certain amount of calculus and had already "learned" that the derivative (tangent) of the minimum (lowest point) of
any curve is zero (horizontal).
They didn't put two and two together. They didn't even know what they "knew."
I don't know what's the matter with people: they don't learn by understanding; they learn by some other way--by rote, or something. Their knowledge is so fragile!
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