Tag Archives: education

Analogies not equations, please!

Education, Mathematics, Physics, Science, Science communication, The scientific method, , ,

Have you ever noticed how equations look far more complicated and hard to understand than the concept they represent?

I sometimes get myself stuck having to read other people’s work (it’s the ‘peer review process’) and when I first read it, I am often utterly confused, like a person stumbling around a dark room they’ve never been in before. However, because I am expected to make intelligible commentary, I soldier on until I understand what is being said.

Once you understand something, it is hard to remember what you felt like before you understood it. How did that equation look the first time you saw it? I have been thinking about this…

Let’s consider ‘equations’ – a common part of many technical documents. I have found that I always overestimate how clever or useful the equations really are when I first see them. So what does this mean?

It means that using equations to help teach people we risk turning them off by giving them the impression that the work is harder than it is.

Let me give an example:

Maxwell’s wave equations. These are considered (rightly) to be an cornerstone of physics, as they model the behaviour of waves in the inter-related electric and magnetic fields. When I first read them, they were ‘greek’ to me, literally. Here’s a small one:

maxwell-faraday-equation

Obviously, you need to know more to understand what they are about. You need to know what each symbol represents – and you need to know what the operators (the × in this case) actually do. For anyone who has not specifically studied maths at university would then need to backtrack quite far, because in this case the ‘×’ is not the ‘×’ most folks know and love, its the ‘cross product’ which applies to vectors. That even leaves most science graduates cold, draining the joy of discovery for a few hours or days while you go away to learn (or remember) what the heck that means.

But is it all worth it? Is the complexity of partial differential equations and matrix multiplication really required in order to understand what the equation is describing?

Of course not!

So why are equations always wheeled out to ‘explain’ phenomena? This is a failure of teaching. Of science communication. Surely concepts can be explained much better by the use of anecdotes, metaphors & illustrations?

Scientists working at the bleeding edge of science have to be very precise in their logic, and when communicating with one another, equations are undoubtedly very efficient ways to describe hypotheses. And so, while they are good ways for experts to relate, they make it harder for newbies to “break in”, and are dreadful teaching tools.

The Maxwell equations really just describe how waves propagate in a medium – and really its just the full 3-d version of waves in a slinky, or ripples in a pond. The equations, while drawing on complex (and difficult) maths, are describing something the human brain already has an intuitive grip on, because we’ve seen it!

I’m not suggesting we could do away with equations – they are valuable in the predictions they make for those who already understand what they represent – I am just suggesting that equations should be de-emphasised, and only dragged out when the student starts to feel the need to describe the phenomenon mathematically.

So my message to all university lecturers and text-book writers is: describe a phenomenon with the use of analogy, please!

The scientific method defined (well hypothesised at any rate)

Education, The scientific method, Things Explained, , , , , , , , ,

I recently realised that the jury is out on exactly what science and the scientific method are (or should be, at least).

Some would say that science is the endeavour to understand the world, answer the “how” behind the ocean tides, rainbows or seed germination. So the scientific method is any way we might do this. Sounds reasonable to me.

However, some would say that science is the business of ‘facts’ or ‘truth’ and proofs. We do experiments to ‘prove’ our hypothesis. This is the definition I would like to take issue with.

Theories and facts confused…

I get really agitated when I hear people say that evolution is a ‘fact’. Not because I’m a  nutty young earth creationist (I’m not), because no-one has yet furnished a proof. But, you may argue, there’s loads of evidence, its clearly a fact.

But evidence is not the same as proof.

Even if something is 99.999% sure, it is still not sure.

I think the trouble comes because people are never taught that those ‘theorems’ and ‘proofs’ they learned in maths class are not quite the same as the theories and evidence in the scientific method.

So is maths a science? Well, yes, sort of. But while it can deal with real things, like counting sheep, it actually deals with a sort of imaginary world (the so-called Platonic ‘world of ideas’). The whole of maths is a mental construct with no known (‘proven’) basis is reality. But nonsense, you say, of course there are numbers in the real world! Well so there are, but there are no proofs!

Proofs are only possible is a fully ‘understood’ world, and because the world of maths is underpinned by a set of axioms, it is, more or less, ‘understood’. But the real world in which we live is not like that. We don’t understand how the brain works, we don’t know how many dimensions there are, we don’t even know if there is a god.

So does that mean we don’t know anything? The media (and opponents of science) use this uncertainty to undermine science. “You can’t prove there is no God, because there is!” Hey presto, a proof of God.

No, science and the scientific method doesn’t do proofs and facts. So what does it do?

Let’s consider the old chestnut, evolution. People had a book that explained the marvellous spectrum of life, from the caterpillar to the jellyfish. This was good enough for many years. But some clever folks started to question why God would bother to make different tortoises on different islands, and why He would go to all the trouble of putting dinosaur bones in certain rocks and why he would disguise their uranium-lead isotopes to make them look millions of years old.

So a theory was proposed (Darwin’s natural selection) that explained the incredible story of species and, for good measure, predicted that humans are apes, which went down well in the church.

Since then, loads and loads of observations have been made that confirm the theory (with the odd tweak). Its a theory that would have been easy to disprove. If it was wrong, some animals that couldn’t have logically been explained by the theory would have cropped up. But they haven’t.

But all this evidence is not proof. And the lack of a disproof isn’t a proof.

The same is true for all accepted theories. The sun and the moon are thought to cause the tides. If that a fact?

If you ask a scientist, even a good one, he/she may well say yes, its a fact. Because it is so darn likely to be right. Because there is no good alternative theory. Because non-one is disputing it. Because the maths is just so neat. Because the theory can make predictions. All good reasons to accept a theory. But they do not make it fact.

So we do know ‘stuff’, plenty of stuff, facts to all intents and purposes, but not strictly facts in the sense of logical proof.

So what is the scientific method, then?

Science is the system of theories and hypotheses about the nature of reality that have not yet been disproven and which are ranked by the weight of evidence in their favour.

It is like a model of the world that we are ever refining, chucking out wrong theories, refining the ones that work. The scientific method is that refinement process. Well that is my hypothesis. The truth may be altogether different!

Dumbing down?

Education, In the media, Science, , , , , ,

This is my first ever blog post. Ra-ra and all that, let’s get to the subject matter.

Yes, its going to be one of those repositories for all those thoughts I probably vastly overvalue when I first conceive of them. But as I cannot be objective and they may actually serve some purpose, I might as well pop them on-line.

Topic of today? UK exam scores. Why? I just read some other blog on the subject: http://www.badscience.net/2007/08/calling-all-science-teachers/ and have some feelings on the matter.

I am not particularly qualified to comment on the education system, so I beg of you don’t listen to my ‘opinion’, but rather follow my logic…

Many people have suggested, and most recently in the public eye, Dr Goldacre in his excellent book “bad science”, that exam standards may be dropping in the UK.

I’d like to analyse this statement for the general case (i.e. any population of which a subset write an annual exam in which the questions do not repeat). Let’s try to frame the question of their ease in a less emotive logical statement…

Let’s say we have data that show the pass rate is gradually moving up year-on-year.

This must mean that one or more of the following is true:
i) the population is getting genetically smarter
ii) the population is increasing well prepared by its environment (parents, teachers, peers, the internet, etc.)
iii) the subset of people in doing the test has changed
iv) the questions are becoming better correlated to what people know
v) or last, the test questions are getting gradually ‘easier’ (or the marking more generous)

There may be more, but don’t want the extreme complexity to cloud my (eventual) point.

Now each of these statements is hard to prove without more data – and the only data we seem to have is the test scores (although we do have the tests themselves which may prove useful).

It may well be that people are getting smarter – but we might use some science to tackle that – for example you could argue that evolution cannot work this fast (and I personally doubt that nerdyness is particularly good survival and seduction tool).

But the environment is certainly changing, the subset doing the tests may be drifting, schooling techniques are being constantly refined and the correlation between what’s interesting (celebrities, MMR vaccines) and what’s examined is also hard to pin down.

I would say there is more than enough vagueness to ensure that no-one, no matter how well qualified, could answer the question “are we dumbing down” with any conviction.

However, there is a “but”.

The examiners can set the difficulty of each fresh test to be whatever they want (in theory). They could make it easy and let everyone get A’s, or they could make them so hard that only the brightest “X” percent get an A. Yet what we see, year on year, are slight improvements.

There are (at least) two hypotheses as to what the examiners are doing:
a) they are aiming to make the questions identical in difficulty to the last year, despite the full knowledge that this strategy has, to date, resulted in a gradual trend toward better marks.
b) they are deliberately aiming to get just slightly better results than last year due to some “incentive”

As the examiners for all the subjects are probably a fairly independently minded bunch and as there is no evidence for it, there are good reasons to doubt the latter hypothesis. Occam’s razor would surely favour the former, though we can’t be sure.

So where does that leave us?

We can’t suddenly make the tests harder, thus lowering the number of A grades to what they were years back – that would be unfair, and would mean that future young people will actually have to know more and work harder than their colleagues from the present time to get an A.

Why not simply rank the scores, then place predetermined fractions into each grade? This, incidentally, is what (I believe) was done when I went to school, which was essential as we had several different regional exam boards with different exams, so rankings rather than absolute results were felt more comparable. Isn’t this how IQ tests scores work? This would mean, incidentally, that by definition, exam/IQ scores for a population simply couldn’t increase with time.

Perhaps most attractive is the option to leave things as they are in the UK, and ignore the media circus. After all, what does their opinion matter?