There’s a fantastic story about Isidor Rabi (I think), who won the Nobel Prize for his work on nuclear magnetic resonance (NMR). NMR is an indispensable technique in chemical analysis, and also the basis of MRI scanners in hospitals. This story shows that Rabi must have had the admirable trait of being secure enough in his intellect to not mind appearing ignorant. It comes from ‘Eurekas and Euphorias’ from OUP, a book that I lost many years ago, so what follows may be largely made up. But this is the internet in the era of fake news, so let’s continue.
Rabi and colleagues were (in my memory at least) working on the cavity magnetron, a device that creates microwaves, specifically for use with radar. When someone asked “How does it work?” there was much scratching of heads, until one member of the group announced “It works just like a whistle.” There then followed an awkward silence, until Rabi asked “How does a whistle work?” Cue more scratching of heads…
The process of thinking of one concept in terms of another is called ‘analogy’. We will use the term ‘target’ for the object/concept that we want to understand; the helpful object/concept to be compared to is the ‘analogue’. In the story above, the scientists were drawing an analogy between the workings of a cavity magnetron and a whistle; the cavity magnetron is the target, and the whistle is the analogue. As well as being funny, this story helps to introduce some points about the nature of analogy in science. An analogy usually involves identifying relations between two things. The relations can be superficial (so they break down if we try to take the analogy beyond surface similarity), or deep (so that the analogy holds across a range of situations or concepts).
In the rest of this post we will argue that:
- Analogy can be useful in education and to make discoveries
- Great care must be taken when using analogy in education
- Scientific revolutions happen when people run out of analogies
Using analogy in science education
Analogies can help us understand difficult concepts. Imagine meeting two Martians called Martian 1 and Martian 2 (obviously that actually is the naming convention on Mars…). They are experts in astronomy and celestial mechanics, but know absolutely nothing about the world on the microscopic scale. For some time you have been persuading them that everything in the universe is made of atoms. And then one day Martian 1 asks “So what are atoms made of?” Knowing about their prowess in mechanics, you decide to compare an atom to the Solar system, and before you know it they have grasped the idea of lighter electrons revolving around a more massive atomic nucleus. You have just educated extra-terrestrials.
To educate them, you used an analogy with two aspects to it:
- An electron orbits a nucleus in the way that Earth orbits the Sun
- A nucleus is more massive than an electron in the way that the Sun is more massive than Earth
These two aspects of the analogy are summarised in the diagram below.
Using analogy to make discoveries
Martians 1 and 2 go home, wishing you well and promising to send a postcard (if you were born in the 21st century you might need to look up what one of those is, but hey, this is a story about Martians so how much can it matter?).
The two Martians go their separate ways and think long and hard about their new knowledge. They wonder what properties atoms might have that you didn’t get round to telling them. Martian 1 uses his knowledge of the solar system and reasons as follows:
“I know Earth orbits the sun, and I know it does this because between them there is a mutual attraction due to gravity.
Gravity is a force that acts along the line separating the two objects, and follows the inverse-square [latex](\frac{1}{r^2})[/latex] law.
In fact, the sun and Earth orbit their common centre of mass, but because the mass of the sun is so much greater than the mass of the Earth, for all practical purposes we can say the planet orbits the sun.
Perhaps electrons orbit the nucleus because there is an analagous inverse-square force between them. It can’t be gravity because atoms are so tiny, but there might be another force that behaves the same way.”
This reasoning is represented in the following diagram.
Martian 1 has just hypothesised the inverse-square electrostatic attraction between positive and negative charges, using nothing but analogy. Once Martian 1 finds a way to test this hypothesis, then that is real science.
The analogy is now deeper than it was before because the connection of the [latex]\frac{1}{r^2}[/latex] force is a more fundamental and causal relationship than the observation of orbiting.
Science is full of examples of discovery being made through analogy. One good example is Carnot using the analogy of falling water to hypothesise that temperature difference in steam engines might determine efficiency. You can find more on this example in Historical Shifts in the use of analogy in science (Genter and Jeziorski). It is a really good place to go if you want a next step after this post, and any rigour in this post is largely due to them. They explain how making analogy can be a logical procedure, not just a loose comparison between superficially similar situations.
Here are two other examples, gathered from the Stanford Encyclopedia of Philosophy , which could be your next stop after Genter and Jeziorski:
- In 1769 Priestley sought to explain the experimental observation that a hollow electrically-charged sphere has no electrical influence (no electric field) inside it. Nowadays this is known ‘Faraday cage’ effect, but Priestley’s work pre-dates Faraday’s. Priestley reasoned that an analagous situation is the lack of gravitational force inside a hollow sphere of uniform density. This was known to be the case because gravity is an inverse-square force. Priestley suggested by analogy that electrical forces are also therefore of the [latex]\frac{1}{r^2}[/latex] form. This is the conclusion that Martian 1 arrived at, also by analogy, but by a different analogy!
- In 1934, the pharmacologist Schaumann was testing drugs; he observed that one of the compounds – meperidine – had an effect on mice that was previously observed only with morphine: it induced an S-shaped tail curvature. By analogy, he conjectured that the drug might also share morphine’s narcotic effects. Meperidine, indeed, turned out to be an effective painkiller, known as Demerol.
Stretching an analogy past its breaking point
While Martian 1 was reasoning about forces, Martian 2 came up with another relation between the sun and planets:
“The sun provides the heat and light that supports life on Earth.”
The diagram below shows Martian 2 following the same logical process as Martian 1.
However, the resulting hypothesis this time is erroneous (as Martian 2 would find if it designed and performed an experiment to test it). To suggest that atomic nuclei sustain life on their electrons is not true; the analogy has been stretched past breaking point.
Notice that Martian 2’s suggestion was as reasonable as Martian 1’s in the sense that it, too, concentrates on relations between the sun and Earth. It was just less true. Neither Martian said “the sun is yellow, so atomic nuclei are yellow”, because they realised that the colour of the sun is an attribute of a single object, not a relation between objects.
Saying that one thing is like another, is just that. It’s not saying they are equivalent. The better/deeper the analogy, the larger the number of ways they are alike, but they are still not ‘the same’.
And yet we can fall in to the trick of thinking they must be. The analogy might have served well up to a point, and we may be emotionally invested in it because it helped us learn something. So we start thinking that the target always works in exactly the same way as the analogue. Our poor Martians might do pre-University Chemistry and find that the ‘electrons orbiting nuclei’ model cannot explain everything that needs explaining. Martian 2’s hypothesis may have been ‘less true’ than Martian 1’s, but it turns out that the [latex]\frac{1}{r^2}[/latex] classical force idea isn’t the whole truth either. It explains some things, but not others, such as atomic spectra.
Despite their chemistry course, the Martians might well be intellectually/emotionally reluctant to give up the [latex]\frac{1}{r^2}[/latex] analogy (especially Martian 1 who was justifiably proud of creating it). It can be hard to conceive that something might not be true, when it has been so persuasive thus far. When the Martians do finally manage to replace their world-view with an improved model of atoms, they are likely to consider that they have previously been lied to, and have now been told the truth. They might not realise that their new model is just that – it is still a model, and it too might need replacing as evidence, and their stage of education, requires.
Analogies as educational tools should be used with caution
We have seen that analogy can be used to understand concepts, and discover new ones. We have seen that the deeper and more systematic the relations between the target and the analogue, the more useful the analogy is likely to be. But when we are using analogy as a tool to educate others, another factor is overwhelmingly important in determining the analogy’s usefulness. It is a factor that seems not to be talked about much. That is, ‘whether the analogue itself is well understood by the learner’.
Even if an analogy is applicable, it might be useless as an educational tool if the analogue is no better understood than the target. A classic example of this is the tendency of physics textbooks to explain electrical circuits using the analogue of pressure in water pipes. Now, electricity is difficult to explain because, first, it is invisible, happening within materials via microscopically small charge carriers, and secondly, because it contains abstract concepts (such as potential). So it would seem to cry out for an easily understandable analogue in the macroscopic world that can make it easier to visualise. But water in pipes isn’t it. Using water in pipes is enticing because, unlike electricity, it is at least visible. However, water flow is not routinely taught in schools and so many students have no more idea about how water behaves in pipes than they have about how electrons behave in wires. The analogy might be valid (up to a point), and seem beautifully elegant to the author, but it isn’t useful educationally if it doesn’t contribute to the student’s mental schema. This leads to another interesting point: analogies may become more or less useful educationally with time, as curricula change. If water flow in pipes becomes a staple of school science in the coming decades, then the analogy may become useful again (although its validity will remain untouched).
Likewise, “it works like a whistle” didn’t help Rabi’s group understand the magnetron (although it did point out that their understanding of whistles needed some work). As Holyoak and Thagard (1995) caution: ‘Without guidance from a teacher, analogy is often a trap for the unwary novice, rather than a stepping stone to expertise.’ We didn’t fall into this trap with the Martians: we were careful to use an analogy in which the learner was expert in the field of the analogue.
Another note of caution for educators: just because we understand where the limits of an analogy are, why should we expect our learners to grasp this point straight away? If we introduce an analogy that revolutionises their understanding of something, why wouldn’t they want to hang on to it, milk it for all it’s worth, and potentially stretch it well beyond breaking point? I suggest teaching not just the analogy, but the benefits and limitations of the act of analogy, to help learners recognise what is going on.
Scientific revolutions
So if much scientific discovery relies on analogy, generalising from well-characterised situations, what happens when we find a situation that is so weird or new that we have nothing to compare it to? How do you explain something, when there is no analagous situation? All you can do is make up something genuinely and fundamentally new – that is a scientific revolution.
In the late 19th century, the Michelson Morley experiment established that light appears to travel at the same speed for all observers, regardless of the observer’s motion. There is no comparable situation for that. Let’s say a plane can fire bullets at 1000 m/s. If it travels at 1500 m/s and opens fire, it doesn’t shoot itself down! Instead the bullets travel at 2500 m/s, because the velocities add. This addition of velocities happens for all examples of motion at normal everyday speeds. But light does not travel at a ‘normal, everyday speed’; and it travels at 300 000 km/s, whatever the speed of the object emitting it. The explanation of these facts was accomplished by Einstein in his special theory of relativity, a genuine scientific revolution.
Quantum mechanics is another example of a scientific revolution born from a situation in which the analogies had run out. Learning quantum mechanics is not easy, but perhaps we make it harder on ourselves because we are all so obsessed with making connections to other things that behave similarly. So when learning quantum mechanics we search for analogues. But there really aren’t any. Maybe much heartache would be saved by taking the approach of ‘here are some rules, they won’t make sense to you because nothing that exists on your length scale seems to follow them, but these are the rules and you just need to put up with it…’ Leonard Susskind’s book ‘The theoretical minimum’ takes this approach, and although it’s not an easy read, it’s an illuminating one for that very reason.