Parents Night, Abstract Thinking and Evolution

I was at a parents night in my son’s primary school yesterday and the teacher who was talking on the maths curriculum  said something that made me think of Steven Pinker’s paper on the ‘cognitive niche’. Pursuing arguments first made by Alfred Russel Wallace, Pinker asks the question:

“Why do humans have the ability to pursue abstract intellectual feats such as science, mathematics, philosophy, and law, given that opportunities to exercise these talents did not exist in the foraging lifestyle in which humans evolved and would not have parlayed themselves into advantages in survival and reproduction even if they did?”

Pinker proposes these feats are just by-products (‘spandrels’) of evolution, just as many others have suggested that religious belief is also a spandrel.

Pinker suggests that humans’ enhanced ability to think (but not necessarily in an abstract manner), and to cooperate, gave them a survival advantage. On the whole question of abstract thinking he says

“Humans do not readily engage in these [abstract] forms of reasoning. In most times, places, and stages of development, people’s abilities in arithmetic consist of the exact quantities “one,” “two,” and “many,” and an ability to estimate larger amounts approximately. Their intuitive physics corresponds to the medieval theory of impetus rather than to Newtonian mechanics (to say nothing of relativity or quantum theory). Their intuitive biology consists of creationism, not evolution, of essentialism, not population genetics, and of vitalism, not mechanistic physiology. Their intuitive psychology is mind-body dualism, not neurobiological reductionism. Their political philosophy is based on kin, clan, tribe, and vendetta, not on the theory of the social contract. Their economics is based on tit-for-tat back-scratching and barter, not on money, interest, rent, and profit. And their morality is a mixture of intuitions of purity, authority, loyalty, conformity, and reciprocity, not the generalized notions of fairness and justice that we identify with moral reasoning.”

Anyway, to get back to the maths teacher…

While most of what she said was reassuringly sensible and ‘traditional’, there were was one point that she made about early stage maths teaching that got me thinking. She suggested that it is better to avoid making mathematics too abstract too early. And so, the classroom we visited was filled with all sorts of maths ‘toys’.  I would have scoffed at this just a few weeks ago because in general I dislike gimmicks and I have a fairly traditional outlook when it comes to education. But if abstract thinking doesn’t come easily to us as a species, perhaps only some of us will take to it. So maybe we need to be careful about diving too deep too quickly and causing many pupils to drown, only to be left with a minority who, by chance, have been born with a natural aptitude for abstract thinking.

More broadly, maybe we have to accept that not everything to do with maths is cultural or even pedagogical – perhaps mathematics presents problems that are biological in origin.

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Why remembering the rules of maths is important

Just this morning I gave my (third year) students a set of x-y data. During the previous lecture (which was on ultrafiltration) we derived a mathematical model that predicted


where a and b constants.

Their challenge was to use the data to evaluate a and b. Only a small minority of the 30 or so students came up with a solution. Why did such a simple data analysis problem challenge them so much?

My theory is this: the students I teach are learning on a multidisciplinary programme and they do not get immersed in mathematical and computational reasoning to the extent that I would like them to.

Consequently, they do not have the basic rules of mathematics, the grammar of the mathematical language, ‘at the ready’, waiting to be called upon when they have to solve engineering problems. They are not ‘triggered’ to make the leaps that an engineer or a mathematician might make.

So in the above instance they don’t make the first logical step, the step that anyone who is fluent in the language of mathematics would make, i.e., they don’t simply write


Some, but not all, will then be able to extract a and b from the slope and the intercept of a plot of y versus lnx.

This tells me two things: (i) remembering the rules of mathematics is important because having immediate recall is usually the thing that drives mathematical problem-solving and (ii) multidisciplinary programmes are hard to deliver.

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That OECD Maths Graph and Harry Potter

The graph below was recently produced by the OECD. (Full maths teaching report is here.) It positions countries on a coordinate system in which the x-axis represents the extent to which maths teaching is ‘teacher-led’ and the y-axis represents the extent to which students rely  on memory in learning maths. This is all based on feedback from students and it is worth noting that this analysis largely predated the introduction of Project Maths in Ireland.

Ireland was singled out for its reliance on teacher-directed instruction and memorization as a learning tactic. Jo Boaler (@joboaler), a maths education guru from Stanford, tweeted:”Wow, the UK and Ireland top the world in maths memorization. Teachers and students deserve better.”


The first thing to be said is that the y-axis does not in any way represent a pedagogical philosophy, at least not in Ireland. What it actually represents is the extent to which students in Ireland used memorization as a study tactic in advance of the Leaving Cert.

The second, more important, point is that there is no evidence that a country’s coordinate on the above graph correlates with any metric of mathematics achievement. For example, in Pisa 2012, Switzerland was ranked 9 while the Netherlands was ranked 10 – two very different educational philosophies (it would appear) yet very similar outcomes. Likewise, Ireland was ranked at 20 while Slovenia was at 21. France was at 25 while NZ was at 23. Poland and Belgium are at 14 and 15. Germany and Spain look like they should have similar outcomes but, no, Germany was at 16 while Spain was at 32.

Of course, this is all a bit silly because so many factors, cultural and otherwise, are at play when it comes to educational achievement.

In other words, this graph is like that mirror in the Harry Potter movie – it tells you precisely what you want it to tell you. Perfect for the educational ideologue.

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Maths, memorisation and understanding

Much of the debate around the Leaving Certificate in Ireland centres on the issue of ‘rote learning’. Many commentators will express a desire that education should emphasise ‘understanding’, ‘critical thinking’ and ‘creativity’, the implication being that ‘skills’ like these can be developed in a way that doesn’t involve memorisation of some kind.

But let’s look at mathematics and think about how a student becomes ‘good at maths’.  Maths is an interesting discipline because to be good at it requires learning rules ‘off by heart’ in the same way that one learns the rules of grammar when learning a language; but ultimately it also requires one to develop a deeper understanding of what all of those symbols and rules actually mean.

For example, a simple rule that a maths student must know is this:


This is a rule you just need to know.  Of course you can agonise all you want about why this rule is true but to get on the road to being ‘good at maths’, angst is not necessary at this point. As Fields’s medallist, Timothy Gowers has said, “It is quite possible to use mathematical concepts correctly without being able to say exactly what they mean. This might sound like a bad idea, but the use is often easier to teach and a deeper understanding of the meaning often follows of its own accord.

As you progress into the world of mathematics, you will meet all sorts of rules that, once again, you can fret about or you can just accept them as you would a rule of grammar. For example, an important rule that my engineering students often need to able to recall automatically is

If   y=x^n then x = y^(1/n)

Again you can worry about what raising a number to the power of 1/n actually means (especially if n is an irrational number!), but you can also just accept that, once you know the rule, all sorts of algebraic manipulations are possible, as is working out the numerical value of something like 51/6  on a calculator or spreadsheet software. Either way, if you are in the midst of doing an actual calculation, algebraic or numerical, it does simplify the whole process for you if you have remembered the basic rules of powers. If you haven’t remembered the rules (as I find is often the case with my students) you get bogged down, you take wrong turns, and any chance of creative thinking is lost.

As you delve deeper into mathematics and do more advanced topics like calculus, you can still reach a reasonable degree of competence by simply knowing rules and practicing the application of those rules. But eventually you will come up against problems, especially problems where you want to actually apply mathematics, where you will need to go further and go a bit deeper.

It turns out, for example, that calculus, especially differential calculus, is, in many situations, the natural ‘language’ of a host of problems that arise in my subject, chemical engineering. In order to be able to express problems in mathematical form, you do have to have an understanding of what d/dx means. Being adept at the mechanics of differentiation is not enough.  ‘Understanding’ here means remembering that d/dx of something represents the rate at which that something is changing with respect to x. But it’s a deep kind of remembering, one that is formed not by repeating some sort of refrain but by doing lots of practice in which the idea of d/dx representing a rate of change becomes ordinary, not an abstract thing to be learned off.

Ultimately the road to  being ‘good at maths’ is  a long and twisty one and it  really does help to know the rules of the road. Some times those rules are quite deep and  for a while at least you may just have to accept them, or, better still, have them explained to you by a good teacher. But maths is fundamentally  abstract and some rules will only become part of your consciousness by lots of practice.

The fundamental point though is that to become good at maths, you do need to remember quite a bit of stuff!

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Chemical Engineering: science, art and TV

I got into an interesting discussion on Twitter about the difference between engineering and science and thought about this that I wrote a few years ago….


You know what women are like; once they hear you have a PhD in chemical engineering, they’re all over you”. So said a minor character in an episode of the classic eighties television series, Moonlighting, the show that launched the career of Bruce Willis. Obviously the scriptwriters were being ironic and having a little chuckle at a profession that they saw as being the epitome of ‘nerdiness’.

Throughout my life, when I have had to tell people that I am a chemical engineer, I have been met with either a blank stare or some comment or other about that person not liking chemistry, or being useless at maths. Very occasionally, people have seemed rather impressed, believing for some unknown reason that all chemical engineers are geniuses, a very satisfying response indeed. The dread is always that they ask the follow-up question: “What’s chemical engineering?” I usually mumble something like this: “Think of the production of a new drug, an antibiotic perhaps. Scientists have discovered the drug and devised methods for making grams of it. Chemical engineers figure out how to make hundreds of kilos of it – economically.” Moving from the laboratory scale to the industrial scale – process scale-up– is, indeed, an important part of chemical engineering. Chemical engineers routinely apply the sciences of physics, chemistry and, increasingly, biology to bring processes from the laboratory scale to the industrial scale.

But chemical engineers don’t just apply science; they create it as well. Indeed, chemical engineering could be described as the most ‘scientific’ of the engineering disciplines. It is certainly the most molecular. Cornell University’s chemical engineering department even has the title of “School of Chemical and Biomolecular Engineering”.

Chemical engineers have contributed to our understanding of the thermodynamics of both gases and liquids; to our ability to calculate reaction rates of catalytic chemical reactions involving highly complex mixtures of petrochemicals; to the creation of mathematical models of metabolic pathways in bacterial cells; to our understanding of the mechanisms of solute transport through the nanofiltration membranes used in wastewater treatment processes.

Chemical engineers must create the science because there are many areas of science where scientists don’t, or won’t, go. It’s not so much that there are places where scientists fear to tread, but places where scientists have no interest in treading. Chemical engineers have to work with physical, chemical and biological systems that arise, not so much in nature, as in the man-made environment of a production process. Thus, physicists don’t take the same interest in the fluid dynamics of bubbles or drops rising in a chemical reactor as engineers do. Chemists are not so interested in the vapour-liquid equilibrium of mixtures of hundreds of hydrocarbons. Biologists are not so interested in metabolic engineering to improve the yield of a genetically engineered product.

The systems that chemical engineers study are complex and messy. They often defy a purely theoretical approach. Even the simple problem of predicting the pressure drop for flow of water in a pipe network cannot be done theoretically. Therefore, the chemical engineering approach is to combine theory and experiment. Chemical engineering science, for that is what many chemical engineers do, is ultimately driven by the need to find practical answers. The goal is to devise methods, especially mathematical methods, to analyse, design and optimise physical, chemical and biological processes. In that sense, chemical engineering is not a search for ‘truth’ but a search for a workable solution. The art of chemical engineering is in finding the optimum balance between theory and experiment to arrive at that solution.

In the early days of engineering, when the slide rule was at the cutting edge of computation, chemical engineering analysis and design was a wonderful mix of theory and art. Even simplified theoretical analyses required complicated, repetitive calculations, often involving the use of trial-and-error methods. Extraordinarily creative and beautiful – yes, beautiful – graphical techniques were developed that made it possible to do complex calculations in minutes. The famous method for the design of distillation columns, devised by McCabe and Thiele, is a wonderful example of the computational creativity of chemical engineers. It is hard to believe that McCabe and Thiele were both postgraduate students when they unveiled their famous method.

In the age of powerful computers, all scientists and engineers can push the purely theoretical approach a little bit further. But the environments encountered by chemical engineers are just too complicated and the need to combine theoretical and experimental knowledge remains. Chemical engineering is not a dying art.

As a youngster and avid viewer of Carl Sagan’s famous television series, Cosmos, I had dreamed of being an astrophysicist. I settled for chemical engineering because I thought I’d be more likely to get a job! I have no regrets, even if the Moonlighting character was talking nonsense.

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24 Statements of Knowing

Without wishing to go all E.D. Hirsch, here’s my first attempt at presenting an adjunct/alternative to the Junior Cycle’s skills-heavy, knowledge-light 24 Statements of Learning. Some of this might be more appropriate for Leaving Cert but the point is that to be a fully ‘engaged citizen’ (as they say), you really ought to know some stuff.

The student should:

  1. Be numerate, i.e., be able to carry out arithmetical operations employing addition, subtraction, multiplication and division; be able to work with fractions and percentages; be able to work in scientific notation; be able to use a calculator and spreadsheet software as appropriate.
  2. Understand how to present and interpret data that is presented in graphical form
  3. Know the basic rules of algebra and understand the meaning of ‘x’ as representing an unknown quantity
  4. Understand the basic ideas of Euclidean geometry and trigonometry
  5. Appreciate mathematics for its role in world culture as well as it’s applied role in science and business
  6. Know the basic atomic and molecular structure of matter
  7. Understand how the planet Earth is located within the universe and be aware of the main features and magnitude of that universe
  8. Be aware of the basic laws of nature including Conservation of Energy, Newton’s Laws, Gravity, Evolution by Natural Selection (and maybe even Quantum Theory!)
  9. Know about nuclear fission, nuclear fusion and the equivalence of matter and energy
  10. Have a basic knowledge of the geography of the world, including a knowledge of climate in various regions of the world
  11. Have knowledge of the workings of the human body including what a gene is.
  12. Understand how lifestyle affects the workings of the human body
  13. Have knowledge of the various forms of microscopic life including bacteria, yeast, fungi, protozoa and viruses and an appreciation of the role that these organisms play in disease
  14. Have an understanding of the diversity of life on Earth and the impact that man has on that life
  15. Understand the basic tenets of the scientific method
  16. Have a rudimentary knowledge of the major religions of the world and the role of religion and philosophy in world cultures
  17. Have a basic knowledge of world history especially 20th century history
  18. Have read, in whole or in part, some of the great works of literature, including novels, plays and poetry
  19. Understand how the Irish political system works
  20. Have a good knowledge of Irish history especially recent history in Northern Ireland
  21. Have some knowledge of the various political and governmental systems used throughout the world, including democracies, dictatorships (religious and otherwise) and totalitarian regimes of all kinds
  22. Have a good knowledge of English grammar and be able to write meaningful and grammatically correct sentences
  23. Be capable of having a simple conversation in a foreign language
  24. Know about some of the great artists, musicians and composers of history
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Why do school-leavers study science?

The first thing I do with my incoming first year biotechnology class is to give a lecture on the “10 types of scientist”. The lecture is based on this study by the Science Council in the UK, a study that was brought to my attention by Julie Dowsett, programme manager with the pioneering Agri-Food Graduate Development Programme based in UCD and UCC.

I then ask my students to write a 500-word personal statement in which they reveal which of the 10 types of scientists they would like to become. It’s a handy way for me to assess their writing skills and in general I’m pretty impressed, something that makes me think that some, or even many, third level students regress without the supports they have had during their second level studies.

Anyway, the overwhelming message that comes across from the students’ statements is that of impact. Young students don’t claim to be fascinated by the discovery process itself, or to be driven by some sort of innate curiosity, a “need to know”. Instead, they are almost unanimous in their desire to improve the lives of others, especially the sick. Some are honest and admit that they would like to make shed loads of cash in the process!

I have to say that this has surprised me because my view has always been that we should try to attract young people into science by emphasising what science is rather than what science can achieve.

Mind you, it’s early days yet and it will be interesting to see how their views change, or how they don’t, over the coming four years.

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