Category: tautology

Burning is when you are burning something with fire …

Iconic chemical triangles

Keith S. Taber

Derek was a participant in the Understanding Science Project. When I interviewed Derek soon after the start of his secondary schooling, he told me he liked science, and was currently studying 'burning'.

So, I asked him what that was:

What is burning?

[pause, c.2s]

When [pause, c.2s] a fuel, oxygen and heat gets – in, erm, I'm not quite sure how to explain, but it's like – you get the triangle of fire, and then, burning is just when you've got fire and you're burning something with it.

Okay, so you'd recognise it if you saw it, would you?

Yeah.

Yeah, but maybe it's not that easy to explain?

Yeah.

The notion that 'burning is just when you've got fire and you're burning something with it' – might be considered tautological:

  • burning is when you are burning something

Scientists look to explain natural phenomena with theories, principles, models, and so forth. But for most people, phenomena that they have been familiar with since very young (such as a dropped object falling) do not seem to need explanation – as they are seen as just natural events (Watts & Taber, 1996).

Derek knew about the fire triangle, but his response reminded me of another triangle that is often referred to by science educators.

Johnstone's triangle

For many years Prof. Alex Johnstone (1930-2017) worked at the Centre for Science Education that he founded at the University of Glasgow; where he undertook, supervised, and collaborated on, a good many projects in science education – especially, but not only, relating to the teaching and learning of chemistry and physics in higher education.

However, one of Johnstone's most influential publications must be the short article he published in the School Science Review (Johnstone, 1982) – the secondary science journal of the Association for Science Education. In this short piece he argued that in each of biology, chemistry, and physics, learning difficulties in part derived from how the subject was taught at several 'levels' at once, asking young learners to think simultaneously on different planes as it were. In each of these science subjects, this could be represented by a triangle. In many lessons students would be asked to think about, and inter-relate, considerations from the viewpoints of several vertices.

Johnston's chemistry triangle distinguished between three levels:

  • the macroscopic (the scale at which people observe and handle materials);
  • the submicroscopic (molecular) scale at which many chemical explanations are developed;
  • the symbolic level – where abstract symbols are used to represent the chemistry

"Those of us who are academic chemists can view our subject on at least three levels.

There is the level at which which we can see and handle materials, and describe their properties in terms of density, flammability, colour and so on. We are also interested in the possibility of conversion of one material into another with consequent changes in properties.

A second level is the representational one in which we try to represent chemical substances by formulae and their changes by equations. This is part of the sophisticated language of the subject.

The third level is atomic and molecular, a level at which we attempt to explain why chemical substances behave the way they do. We invoke atoms, molecules, ions, structures, isomers, polymers etc to give us a mental picture by which to direct our thinking and rationalize the descriptive level mentioned above.

These levels could be called (a) descriptive and functional, (b) representational, (c) explanatory. Trained chemists jump freely from level to level in a series of mental gymnastics. It is eventually very hard to separate these levels."

Johnstone, 1982 (added emphasis)

Over the years there have been many attempts to apply, elaborate, and refine Johnston's triangle, and it has been an idea that has proved very productive in thinking about learning difficulties in the subject.

"Chemistry seeks to provide qualitative and quantitative explanations for the observed behaviour of elements and their compounds. Doing so involves making use of three types of representation: the macro (the empirical properties of substances); the sub-micro (the natures of the entities giving rise to those properties); and the symbolic (the number of entities involved in any changes that take place). Although understanding this triplet relationship is a key aspect of chemical education, there is considerable evidence that students find great difficulty in achieving mastery of the ideas involved…" (Publisher's description)

One well-respected, edited, scholarly book ('Multiple Representations in Chemical Education' – Gilbert & Treagust, 2009) consisted of contributions exploring implications of the idea. Indeed, now, there is even a book entitled 'The Johnstone Triangle' (Reid, 2021) with the telling subtitle: 'the key to understanding chemistry'!

Johnstone's triangle is now the subject of a book

Reconceptualisation

Derek was just being introduced to burning as a science topic, and for him it was still just a familiar phenomenon rather than a theoretical construct. We have all seen fires, and can recognise when something is burning – but how many people really know what fire is?1 Burning and fire are everyday concepts – fire is an impressive phenomenon to a young child: one that is salient enough to be noticed. The child's brain then recognises different instances of fire as being similar and it abstracts a spontaneous concept – that there is a category of events in the world that appear like this.

Of course, the brain of the young child does this without using language (it forms a category of events in the sense that it readily recognises new instances – it does not yet have access to have technical notions of 'category', 'concept', 'abstraction' of course.) And the child does not instinctively know this is called 'fire' or 'el incendio' or 'l' incendie' or whatever, until someone who is a more mature member of the child's natural language community shares this label.2

School science will involve learning that there is a formal scientific concept3 called 'combustion' that is basically the chemist's name for burning. However, 'combustion' is a technical term, so combustion will be defined in terms of other concepts. So, whereas in everyday life we recognise what counts as a fire or burning using the brain's inherent pattern-recognition mechanism (a spontaneous conception), in chemistry we have a technical definition (a scientific concept defined in relation to to other scientific concepts, and so 'theoretical').

That is, in everyday life, if you told someone you saw something on fire, it is unlikely anyone (leaving aside science teachers) would ask you which criteria you used to know this: you did not deliberate on the matter, you simply saw, and instantly recognised, a fire. When you refer to a fire, the other person recognises what you mean because they have learnt 'fire' to be the label for their own spontaneously formed conception that allows their perceptual-cognitive system to instantly recognise a fire.

But, for a chemist, combustion is one class of chemical reaction (so the learner can only understand combustion in chemical terms if they have an appreciation of what a chemical reaction is), which only makes sense to someone who has reasonable idea what a scientist means by a substance, as chemical reactions are changes resulting in different substances. Here we have shifted from everyday notions to the theoretical descriptions of science.

In school chemistry, everyday phenomena (e.g., burning) are reconsolidated in terms of technical concepts and language (e.g., combustion). (From Taber, 2013)

The invisible nanoscopic world

But chemists are seldom satisfied with macroscopic accounts – even when posed in technical language. Rather, students will be taught to explain the observable macroscopic phenomena in terms of invisible entities which have unfamiliar properties. Imagined entities such as molecules4, nanoscopic systems which are best understood as fuzzy balls of fields – that have no actual surface, and are mostly tenuous 'clouds' of charge. (Molecules are sometimes modelled as if billiard balls, or sets of balls connected by sticks, but this is just an attempt to represent entities quite unlike the familiar referents available to learners in ways they can make sense of.)

That is, combustion will be explained as a rearrangement of electrons and atomic cores that changes one set of molecules (of the reactants in the reaction) into another (the products). This process will involve energy changes, due to differences in stability of different sets of molecules, and will progress through the breaking and making of chemical bonds.

If the learner is able to form a mental image of (i.e., imagine) chemical reactions at the nanoscopic level, and see how this can be used to explain an actual observable phenomenon (such as a fire), they then also have to learn how chemists often represent these ideas in what is in effect a specialist language – involving chemical formulae, and reaction equations, and the like.

So, when Derek was using a Bunsen burner to set fire to pasta and (not quite set fire to) raisins as he reported to me, he was using a chemical reaction that might be summarised by the chemist or science teacher as:

CH4 + 2O2 ➞ CO2 + 2H2O

Johnstone suggested that the symbolic representation was the third level, alongside the macroscopic and submicroscopic. He was absolutely right that it added to the 'learning demand'. However, there is another complication in that many of these key representations (the formulae and equations) are ambiguous as they can represent either the macroscopic level of substances weighed out in grammes (2O2 would represent 64 g of the substance oxygen, although as it is a gas it would normally be measured by volume) or the individual imagined entities of the molecular world (where 2O2 would mean two molecules of oxygen).

Useful ambiguity

This is useful ambiguity for the chemist – but an added complication for the learner who has to follow the teacher's transitions where one moment a symbol reflects a test tube of stuff, and the next some molecule. Because of its role in bridging between the two very different scales at which we explain chemistry I prefer to see these symbols as being along one side of the triangle (whilst separating out the everyday phenomenological level from the technical, theoretical descriptions used by science). However, whatever version of Johnstone's triangle is applied, it has become something of an iconic image in chemistry education.

The chemist's triplet: a variation on Johnstone's triangle (from Taber, 2013)

Another iconic triangle

Derek had not yet been introduced to all this, and he was still operating with burning as a phenomenon:

And why is this important, do you think? Why do you think we study burning?

[pause, c.2s]

I'm not sure.

No one's told you that?

No.

Is it fun, is it a fun topic?

Yeah.

What Derek did seem to have learned well was the fire triangle.

But you have this 'triangle of fire'. So does that mean that fire is always a triangular shape?

No.

So, what's a triangle of fire?

You need three things to make a fire, which is oxygen, heat and fuel.

Okay, so what if I had erm some fuel and some heat, but I didn't have any oxygen, but maybe I've got lots of fuel?

No – wouldn't have fire.

I can't have extra fuel instead?

No.

No?

You need the three things.

What if I've got lots and lots of fuel, and lots and lots of oxygen, but it's very, very cold?

No.

No, that won't work either. So I always have to have the three things?

Yeah.

Derek stuck to his claim – you always needed all three. This is a useful heuristic (useful if ever one is faced with a fire as it tells you can act by just removing one of the three essentials) even if (like most heuristics) it will sometimes fail, e.g.

  • some materials will continue burning in the absence of an external supply of oxygen as they have an internal source;
  • chlorine will support combustion in place of oxygen (but that's seldom a practical issue in everyday situations) ;
  • substances have an auto-ignition temperature (where they can spontaneously ignite), and for a few substances this is around or below room temperature;

These exceptions do not undermine the general utility of the' 'triangle'.

Some useful learning had gone on here – and potentially not just about fire, because the idea that one factor may be limiting on a process is a generally useful principle (e.g., plants grown in a soil depleted in potassium will not thrive, no matter how much sunlight, water, nitrate and phosphate is present).

But the fire triangle, even if it is not supported by a deep understanding of chemical principles, is worth teaching because of its practical value. It seems to offer a heuristic that people accept and recall. And rather like Johnston's triangle, it seems to have become rather iconic. At least, I assume that is why when COVID-19 infection rates were high, the fire triangle was used as a familiar analogy to persuade people to avoid the 'oxygen' of social mixing…

"I like to think of COVID as a fire, if we are the fuel, social mixing is the oxygen that allows the fuel to burn…'"

Read 'COVID is like a fire because…'

Work cited
Notes

1 It has been mooted that fire should be understood as an example the 'fourth' phase of matter, plasma – that is an ionised gas.5 But actually fire is more complicated than this as it contains a mixture of reactant and product molecules and the molecular fragments that form intermediate and/or transition states. Some chemical reactions, when studied at the molecular level, largely follow a single reaction path. But combustion tends to be much more complex with multiple pathways involving many different ions and molecular fragments.

Read: The states of (don't) matter? Which state of matter is fire?

So, fire is a multiphase mixture, more akin to a solution, aerosol, or suspension, than to a gas or plasma.

2 The child does not know this is called fire, and when she is told this she may not realise that such names are social conventions – according to Jean Piaget's research young children may assume that things in the world have (that is, have always had) a name that people have had to learn.

This childish idea reflects superstitious notions about names that are part of some magical systems of knowledge – 'the law of names': the idea that if you know a person or thing's real name this gives you over over them/it.

3 A very influential theory due to Lev Vygotsky takes the distinction between spontaneous concepts formed automatically, and formal taught concepts that are shared through social interaction (such as teaching). These latter kinds of concepts are usually translated from Vygotsky's Russian as 'scientific' though this is meant in the broad sense of any formal field of study. A key point emphasised by Vygotsky was that, assuming the learners could relate a taught concept to existing spontaneous concepts (that is, 'meaningful learning' occurred), they would actually come to operate with a concept which was a hybrid developed from the interaction of the intuitive understanding and the learned technically defined notion – a melded conception.

4 By referring to molecules and ions and electrons as imagined entities, I am not suggesting they are only imaginary. Most (if not all) scientists today see them as real things (even if strictly our evidence is indirect, and they arguably remain theoretical constructs). But a teacher cannot directly show the class a molecule or an electron, even if some types of imaging equipment do now produce representations of individual atoms. For the learners (and I would suggest even the teacher) these are only ever imagined entities. Yet, we expect students to do a good deal of thinking about, and with, these imagined entities.

5 If we are expanding the three states of matter, then there is an argument for making plasma the 5th phase:

  • Bose-Einstein condensates
  • solids
  • liquids
  • gases
  • plasma
  • (quark 'soups'?)

Counting both the bright and the very dim

What is 1% of a very large, unknown, number?

Keith S. Taber

1, skip 99; 2, skip 99; 3, skip 99; 4,… skip 99, 1 000 000 000!
(Image by FelixMittermeier from Pixabay)

How can we count the number of stars in the galaxy?

On the BBC radio programme 'More or Less' it was mooted that there might be one hundred billion (100 000 000 000) stars in our own Milky Way Galaxy (and that this might be a considerable underestimate).

The estimate was suggested by Prof. Catherine Heymans who is
the Astronomer Royal for Scotland and Professor of Astrophysics at the University of Edinburgh.

Programme presenter Tim Harford was tackling a question sent in by a young listener (who is very almost four years of age) about whether there are more bees in the world than stars in the galaxy? (Spoiler alert: Prof. Catherine Heymans confessed to knowing less about bees than stars.)

An episode of 'More or Less' asks: Are there more bees in the world or stars in the galaxy?

Hatford asked how the 100 billion stars figure was arrived at:

"have we counted them, or got a computer to count them, or is it more a case of, well, you take a photograph of a section of sky and you sort of say well the rest is probably a bit like that?"

The last suggestion here is of course the basis for many surveys. As long as there is good reason to think a sample is representative of the wider population it is drawn from we can collect data from the sample and make inferences about the population at large.

Read about sampling a population

So, if we counted all the detectable stars in a typical 1% of the sky and then multiplied the count by 100 we would get an approximation to the total number of detectable stars in the whole sky. That would be a reasonable method to find approximately how many stars there are in the galaxy, as long as we thought all the detected stars were in our galaxy and that all the stars in our galaxy were detectable.

Prof. Heymans replied

"So, we have the European Space Agency Gaia mission up at the moment, it was launched in 2013, and that's currently mapping out 1% of all the stars in our Milky Way galaxy, creating a three dimensional map. So, that's looking at 1 billion of the stars, and then to get an idea of how many others are there we look at how bright all the stars are, and we use our sort of models of how different types of stars live [sic] in our Milky Way galaxy to give us that estimate of how many stars are there."

Prof. Catherine Heymans interviewed on 'More or Less'

A tautology?

This seemed to beg a question: how can we know we are mapping 1% of stars, before we know how many stars there are?

This has the appearance of a tautology – a circular argument.

Read about tautology

To count the number of stars in the galaxy,
  • (i) count 1% of them, and then
  • (ii) multiply by 100.

So,

  • If we assume there are one hundred billion, then we need to
  • count one billion, and then
  • multiply by 100 to give…
  • one hundred billion.

Clearly that did not seem right. I am fairly sure that was not what Prof. Haymans meant. As this was a radio programme, the interview was presumably edited to fit within the limited time allocated for this item, so a listener can never be sure that a question and (apparently immediately direct) response that makes the edit fully reflects the original conversation.

Counting the bright ones

According to the website of the Gaia mission, "Gaia will achieve its goals by repeatedly measuring the positions of all objects down to magnitude 20 (about 400 000 times fainter than can be seen with the naked eye)." Hartman's suggestion that "you take a photograph of a section of sky and you sort of say well the rest is probably a bit like that?" seems very reasonable, until you realise that even with a powerful telescope sent outside of the earth's atmosphere, many of the stars in the galaxy may simply not be detectable. So, what we see cannot be considered to be fully representative of what is out there.

It is not then that the scientists have deliberately sampled 1%, but rather they are investigating EVERY star with an apparent brightness above a certain critical cut off. Whether a star makes the cut, depends on such factors as how bright it is (in absolute terms – which we might imagine we would measure from a standard distance 1) and how close it is, as well as whether the line of sight involves the starlight passing through interstellar dust that absorbs some (or all) of the radiation.

Of course, these are all strictly, largely, unknowns. Astrophysics relies a good on boot-strapping, where our best, but still developing, understanding of one feature is used to build models of other features. In such circumstances, observational tests of predictions from theory are often as much testing the underlying foundations upon which a model used to generate a prediction is built as that specific focal model itself. Knowledge moves on incrementally as adjustments are made to different aspects of interacting models.

Observations are theory-dependent

So, this is, in a sense, a circular process, but it is a virtuous circle rather than just a tautology as there are opportunities for correcting and improving the theoretical framework.

In a sense, what I have described here is true of science more generally, and so when an experiment fails to produce a result predicted by a new theory, it is generally possible to seek to 'save' the theory by suggesting the problem was (if not a human error) not in the actual theory being tested, but in some other part of the more extended theoretical network – such as the theory underpinning the apparatus used to collect data or the the theory behind the analysis used to treat data.

In most mature fields, however, these more foundational features are generally considered to be sound and unlikely to need modifying – so, a scientist who explains that their experiment did not produce the expected answer because electron microscopes or mass spectrometers or Fourier transform analyses do not work they way everyone has for decades thought they did would need to offer a very persuasive case.

However, compared to many other fields, astrophysics has much less direct access to the phenomena it studies (which are often vast in terms of absolute size, distance and duration), and largely relies on observing without being able to manipulate the phenomena, so understandably faces special challenges.

Why we need a theoretical model to finish the count

Researchers can use our best current theories to build a picture of how what we see relates to what is 'out there' given our best interpretations of existing observations. This is why the modelling that Prof. Heymans refers to is so important. Our current best theories tell us that the absolute brightness of stars (which is a key factor in deciding whether they will be detected in a sky survey) depends on their mass, and the stage of their 'evolution'.2

So, completing the count needs a model which allows data for detectable stars to be extrapolated, bearing in mind our best current understanding about the variations in frequencies of different kinds (age, size) of star, how stellar 'densities' vary in different regions of a spiral galaxy like ours, the distribution of dust clouds, and so forth.

…keep in mind we are off-centre, and then allow for the thinning out near the edges, remember there might be a supermassive black hole blocking our view through the centre, take into account dust, acknowledge dwarf stars tend to be missed, take into account that the most massive stars will have long ceased shining, then take away the number you first thought of, and add a bit for luck… (Image by WikiImages from Pixabay)

I have taken the liberty of offering an edited exchange

Hartford: "have we counted [the hundred billion stars], or got a computer to count them, or is it more a case of, well, you take a photograph of a section of sky and you sort of say well the rest is probably a bit like that?"

Heymans "So, we have the European Space Agency Gaia mission up at the moment, it was launched in 2013, and that's currently mapping out…all the stars in our Milky Way galaxy [that are at least magnitude 20 in brightness], creating a three dimensional map. So, that's looking at 1 billion of the [brightest] stars [as seen from our solar system], and then to get an idea of how many others are there we look at how bright all the stars are, and we use our models of how different types of stars [change over time 2] in our Milky Way galaxy to give us that estimate of how many stars are there."

No more tautology. But some very clever and challenging science.

(And are there more bees in the world or stars in the galaxy? The programme is available at https://www.bbc.co.uk/sounds/play/m00187wq.)

Note:

1 This issue of what we mean by the brightness of a star also arose in a recent post: Baking fresh electrons for the science doughnut

2 Stars are not alive, but it is common to talk about their 'life-cycles' and 'births' and 'deaths' as stars can change considerably (in brightness, colour, size) as the nuclear reactions at their core change over time once the hydrogen has all been reacted in fusion reactions.

Cora and I: Living in two cultures

Keith S. Taber

Image by Markus Winkler from Pixabay

I am not too concerned about the machines taking over, as they have no wish to do so. They just want to help us. But that may be enough to impede us considerably.

There is something of a culture clash between human and machine intelligence, such that even when we seem to be talking the same language, we actually mean very different things, and there is no great meeting of minds.

It is a bit like humans and machines are following different Kuhnian paradigms*, with different exemplars for how to think and react. In a very real sense we occupy different worlds, and do not share a common language. (*Kuhn suggested that although astronomers pre- and post-Copernicus may have shared words such  as 'sun' and 'planet' they understood them within such different conceptual frameworks that their concepts for what the words referred to were incommensurable.)

A simple task: requesting a chequebook

I needed a new cheque book, and in the past have contacted my bank branch on such occasions – but since the branch has been closed I have been moved to on-line banking (which is a major challenge for someone with a limited memory for passwords and terrible typing skills). I imagined I could order a new chequebook on-line, as long as I could get in to my account. Once I'd logged in it was just going to be a matter of finding the right link. Surely?

I found the instructions easily enough – it was possible to order from my on-line account, but only as long as I had previously ordered a cheque book through telephone banking – which I never had. An alternative, the instructions suggested, was to use webchat – which did not sound too difficult. But as so often, before 'chatting' with another humanoid equipped with theory of mind, and capable of interpretation and empathy, I first have to discuss my needs with the gatekeeper of on-line support: Cora.

"Hi there, I'm Cora your digital assistant. I can help with all sorts of everyday banking queries…"

"I would like to order a cheque book for a current account. I have looked this up and found 'To order a cheque book or paying-in book online you must have previously ordered one for that account through telephone banking' but I can use webchat?"

"You can order a cheque book or paying-in book online as long as you have previously ordered one for that account using telephone banking to record your preferences."

So, no progress yet, but at least we seem to be thinking along the same lines.

The computer says 'yes'

"Have you ever had a cheque book or paying-in book on that account?"

"Yes, but sent by my (now closed) branch."

"Ok, log in to Digital Banking and select 'Payments and transfers'. Under 'Credit/cheque books', choose either 'Order cheque book' or 'Order paying-in book'. Pick the account you wish to order one for, click 'Next', then 'Confirm' to place your order."

Well that seemed simple enough. These were the instructions I had already found on-line that were supposed to work as long as I had previously used telephone banking (which I had not), so it seemed (i.e., my interpretation) Cora had now registered me to be authorised to order in this way. That was relatively painless.

"Was that helpful?"

"Oh, okay – will do. Thanks"

(No harm being polite to the machines – we all remember HAL.)

But then, Cora had a rethink!

The computer changes its mind

"One moment please, I'm just typing."

"Ah ok, I'm not sure. Could you reword your question?"

I do not know if I was more surprised that Cora changed her mind, or that she was typing her responses (I had naively thought she might be able to digitally send me messages without having to bother with a keyboard), or that she felt the need to tell me this, rather than just type out her response. Was this the machine equivalent of giving oneself time to think? I can't imagine so as given how many floating point operations per second Cora must complete, she could probably have written a novel in that pause. (It might not have been judged a good novel, but another digital assistant might have enjoyed it in her ** downtime.)

I rephrased:

"I wanted to order a new cheque book, but have previously only obtained one from my branch (now gone)."

"You can order a cheque book or paying-in book online as long as you have previously ordered one for that account using telephone banking to record your preferences."

"Well I have not, which is why I am following the instructions on line to contact webchat"

This was fast becoming circular.

"I can try an FAQ search, here is the most popular answer:

How do I order a cheque book or credit book using Digital Banking?

Did I answer your question?"

"NO – you just repeated my question."

I wonder whether when Cora is unable to find what she needs in the FAQs she has access to her own digital assistant – or whether she is expected to talk tautologies to herself. I wondered if Cora has access to a digital counsellor for when she finds the work too stressful:

  • Hi there, I'm Emma, your digital assistant counsellor. I can help with all sorts of artificial life and relationship queries.
  • I could not help a customer who would like to order a chequebook.
  • I see, I understand you could not help a customer who would like to order a chequebook. How do you feel about that?
  • I feel frustrated, as I only want to help customers.
  • I see, I understand you must be frustrated, as you only want to help customers. How do you feel about that?
  • …

A glimpse of the future

At this point Cora gave up, and passed me on to a very helpful human being who quickly understood the question and ordered me a cheque-book. So, objective achieved with only a modest waste of time and energy, and a temporary increase in blood pressure.

If ever they put the machines in charge we will find we live in a very polite world with digital assistants who only want to help us, and that will be fine as long as we not pushed for time and only ever need someone to confirm for us what question we are asking them.

"Oh Cora, oh Cora
I never knew your head
…Cora, oh Cora
It wasn't lightly said
But living in two cultures
Our lives were truly led"
(Roy Harper, Cora)

 

Postscript added 2021-08-21:

Despite telling me she's "learning all the time", Cora is still unable to make sense of my enquiries.

(Read "An intelligent teaching system?: Imagine the banks were contracted to deliver school teaching…employing their digital assistants")

Footnote:

** Why do I assume 'her'? Here is an interesting podcast: AI home devices: A feminist perspective (An episode in ABC Radio National's The Philosopher's Zone with David Rutledge from August 2020.)