Mathematics as a discipline covers a vast body of knowledge and requires understanding of a wide range of principles, techniques and facts. Trying to memorize these by rote is in many ways a fool’s errand due to the extent of the task, but may also be fundamentally misguided. There’s an idea that highlighting something in a textbook is the easiest way of not remembering it. Kind of like how I said that telling somebody else your goals makes you less likely to meet them, highlighting something is an easy way of selecting something important and symbolically satisfying the need to acknowledge its significance. The brain is great at settling for “good enough,” and so it takes this as a signal to forget about the highlighted text until further notice*.  The same can happen with mathematical learning when you’re trying to memorize by rote, by repeating a phrase, or copying it out. I tried copying out some things from memory, and it’s more effective than trying to copy them out from the opposite page, especially if you keep coming back to them, but it’s not the right idea for much of mathematics.

There are things called trigonometric identities, equations using angles inside triangles and telling us about the relationship between angles and the length of the sides. They’re incredibly useful and are fundamental to much of advanced mathematics, but there’s a lot of them and they can be hard to remember. One way of trying to remember them is to copy them out and then practice writing them from memory, using up a lot of paper, and probably having to stop and go back a bunch of times. At the end of it, maybe you’ve got them locked in, or maybe they’ll slip, but you probably won’t understand the significance of the ideas.

The thing about the trig identities is that they’re a hundred percent logical, and are based fully on much simpler principles. It can be hard to see the connection between the simple ideas and the more complex expressions of them, but working through the logical chain of reasoning can really help solidify not only your knowledge but your understanding of the identities. This is what I’ve been doing for the past few weeks, apart from teaching students, because I mostly teach year eleven maths and the trig identities are a little bit more advanced than what they’re doing, so I figured working through the proofs is a good way of establishing the underpinnings of the techniques they use in their geometric reasoning NCEA standard. The other useful thing is that once you understand the trig identities, you can see one version of an equation containing trigonometric functions and instantly, or at least quickly, see it or rearrange it as another, equivalent expression which might be simpler to operate with. This can really help with higher mathematics like calculus.

Sometimes, rote memorization is the only way, or it’s all you need. The times tables might be a good example, although personally I never learned them by rote. I’ve been using chunking and skip-counting instead, to relearn them and try to increase my speed. But I think, just like highlighting something and then forgetting it, theres an important idea that we could touch on later in another blog post. It’s about how the brain settles for “good enough.” There’s a saying, “practice makes perfect,” but I prefer to think “practice makes permanent.” If you practice something improperly, then you’re only reinforcing your improper technique, and the more you repeat, the worse it gets. This means experience isn’t necessarily the recipe for success. When you’re first learning something, it might be too difficult to perform the task properly, and so you find some approximation which actually makes the task slower, less efficient or ultimately more difficult, but which at the time operates as a short-cut. If you don’t, at some point, consciously unlearn this habit, then it can become strongly ingrained into your technique, and leave you handicapped for the long-term. So; let’s try and work logically, and slowly, through the problems we face, especially if they’re mathematical, and to understand why we’re doing each step. This way, our mathematical knowledge builds on itself, has strong foundations, and there aren’t any inexplicable gaps where we accept things “just because,” or idiosyncratic ways of completing some problems which might actually make them more difficult. Of course, this is simply an ideal, and generally impossible to meet fully in practice, but as a mindset it can be helpful to go back over things we think we’ve already got down, and re-examine our understanding or our assumptions, to make sure there’s not room for improvement somewhere. Periodic re-examination of our approach to difficult problems, and a willingness to unlearn bad mental habits, will stand us all in good stead.


*Theoretically, you’ll come back to the text and see what you highlighted, but that may or may not happen, and I’ve seen people highlight more than fifty percent of a page before, such that “if everything is special, nothing is.”


People who are good at remembering large amounts of stuff, like multiple digits of pi or whatever, tend to have interesting mental techniques they use to improve their skills. A couple of these, chunking and back-chaining, work well with the memorization of sequences. If you take a long series of digits and break it up into chunks, such that each small chunk has what seems like some kind of internal relationship, then you can memorize each of these chunks relatively easily. The working memory of the average person can only contain something like seven items, but if you break something up into chunks then each of those chunks can be one of the seven or so items you hold in your head at a time. If you can memorize the chunks, and then memorize the connections between them, you’ve got it down.

The problem can often be the connection between the chunks, where you’re reciting something and one chunk is complete but you’re unsure what comes next. I find back-chaining useful in this instance, where you look, not just at how things go forward, but how they go backwards, and practice connecting each chunk not just with the next but with the one before it. I’m currently using this and skip counting to relearn my times-tables.

Some of you from my generation or earlier might remember learning times-tables by rote in primary school. These days the curriculum is a little more broad and perhaps in some ways shallower, so if somebody misses their times tables we don’t like to necessarily punish them and so there might not be the same emphasis. This is probably good, in some ways, but it looks like there might not be the same facility with basic arithmetic in the children of today than yesteryear. Anyway; I never learned my times tables properly, back then. If I recall correctly, the learning process involved chanting “seven times seven is forty nine; seven times eight is fifty six.” I could do that fine, but I hadn’t actually memorized most of them, and instead was just figuring it out as I went along, because usually the time it took to state the complete sentence was enough to get the relatively simple calculation done.

But skip counting is a little different. We all probably know what that is, or at least we can recognise it when the skip size is two. It’s as simple as going “two-four-six-eight-…” up to whatever number. But it’s a little more complicated to go “three-six-nine-twelve-…” or “…twenty-one, twenty-eight, thirty-five…” with the various larger numbers. I find that, when I practice skip-counting with bigger skip sizes, I naturally start chunking the sequence, such that “fifteen-eighteen-twenty one” forms one chunk and “twenty-four, twenty seven, thirty” forms another. Sometimes, though, the connection between chunks isn’t as strong as the internal structure of the chunks themselves, which is where back-chaining becomes useful, going down through the sequence, and sometimes forming alternate chunk regimes, so that there’s a different way of breaking up the sequence which creates an overlap between the existing chunks.

The task I’ve set myself is probably relatively unimportant, but I like to keep my brain busy and have a few mental projects to work on just as a form of exercise. The process of chunking, however, is less trivial, and can be used as a great way to categorise, organize and internalize large amounts of information for efficient retrieval. Most of us do it instinctively to some extent, otherwise we’d never be able to remember anything of any significant size, and so that’s part of why books have chapters and songs have verses. But becoming conscious of the technique and finding new ways to apply it can probably improve our abilities in mutliple areas.

Goal Setting

If you wanna improve your life, keep your goals a secret. Or at least, if you don’t, be warned that it may decrease your chances of actually meeting them. This seems somewhat counter-intuitive, and goes against some of the conventional wisdom and motivational advice out there. Especially with goals like quitting smoking, people are encouraged to tell others about their plans, to “make themselves accountable” for sticking to them. This feels good, because telling other people about your goals provides the same dopamine boost, the same sense of satisfaction, that the brain would produce after actually meeting the goal in question. The problem is that this makes your brain less “hungry” for that same boost, and takes away one source of motivation which can help you meet your goals. So try not to short-circuit your plans by telling them to other people, at least not prematurely.

The other thing about goals is that there are two kinds, at least. The normal idea of a goal is an achievement-based goal, where it’s about where we get to, where we’re going. The other way of looking at it is a behavioural goal, which isn’t about where we’re headed but about how we get there; it focuses on the journey, not the destination. So an achievement goal would be “I want to write a book.” A more realistic version of this would be “I want to write a book by the end of the year.” A behavioural version of this goal would be “I want to write a thousand words a day.” These behavioural goals are more useful for many people than achievement goals. If you plan to write a book at a specific time then you plan to spend the intervening time writing a certain number of words with a certain amount of regularity. The achievement goal is impossible without your first meeting the behavioural goal. There’s a quote, that “People don’t really decide their future; they decide their habits, and their habits decide their future,” which I find summarizes the whole idea about these two types of goals succinctly.

Behavioural goals are related to the growth mindset I outlined in a previous post, while achievement goals are kind of like the static mindset I talked about. They’re similarly opposed and fall into the same kinds of thinking. Static ideas and achievement based goals do seem to be much more common than the growth mindset and behavioural goals, but are markedly less effective. Another important point is that you should set yourself small, realistic goals to meet today, or tomorrow, or this week. If you give yourself a specific time and make your goals easy, maybe even fun, without setting the bar too high, then you get your brain into the habit of following through and seeking that completion reward, rather than just enjoying the rewarding feeling of having a plan without the need to follow through with it.

The other thing is that even if your goals are behavioural, they can still be unrealistic. Saying “I want to work out every day and never make a mistake and study for two hours every night” is pretty much impossible, and setting yourself up to fail like this makes it less likely that you’re going to succeed in the future. This means there’s something like a meta-level of behavioural goal setting. I want to behave in positive ways which can, over time, improve my understanding and physical and mental fitness, but I also want to accept my mistakes and see them as part of the learning process, rather than evidence that it’s not going to work, and beyond this I also want to be able to slowly and realistically improve the consistency of my behaviour, rather than planning to start a new, high-powered routine immediately and being incredibly disappointed and demotivated if I fail to stick to it after the first week.

So, to get to where you want to be, it might help to plan to behave in ways which lead to success, to de-emphasise specific goals about what grand accomplishments you’re going to make, to give yourself small goals in the short term which are easy for you to meet, to increase the challenging intensity of your goals over time, and to keep the whole thing a secret, at least until it’s well under-way and you’ve already started to receive some of those mental rewards rather than getting all of your satisfaction from the positive ideation itself, and the related affirmation you get from telling your friends.

High Flat Skill Profile

An interesting piece of popular science with a spurious pedigree is the idea that brain function is determined by hemispheral activity. We say that the left hemisphere is for quantitative, precision thinking such as mathematics and spatial reasoning, whereas the right brain deals with holistic activity, recognising patterns and governing intuition. So we talk about somebody being either left- or right-brained, with right-brained people being creative, disorganized and impulsive while the left-brained are fastidious, numerate and boring. This belief isn’t just imposed on others, but governs self-belief as well.

It turns out that none of this holds much weight. Many of us go through our education emphasizing either the arts or sciences, with students who consider themselves more quantitative shying away from softer tasks like writing essays or analysing literature, and those who find painting a picture or writing poetry more interesting than solving equations tending to ignore the more rigorous subjects. There’s an idea that you’re good at either one or the other of these, and like many prominent ideas it’s gained a degree of accuracy simply by virtue of how seriously it’s been taken. Since we tend to believe these things and allow them to govern our life decisions, they can determine what we end up learning and being good at.

But it’s not true that you can only be good at one or the other. It turns out that, in terms of actual ability, whether innate or developed, those of us who are able to understand complex mathematical subjects also do well in tests of verbal intelligence, and vice versa. There’s a strong correlation between these abilities, but often, by the time we’ve completed our formal education, we’ve learned much more of one of these than the other. Testing students who’ve graduated or are attending university, you find that most of them favour one of these areas more than the other. Those of us who do okay at both have what’s called a “high flat” profile, scoring in the top deciles for both mathematical reasoning and understanding of language.

To me, this isn’t much of a surprise. So far as I’m concerned, both of these categories of tasks are variants of symbol manipulation. They each involve the construction and modification of meaning based on a set of underlying rules. The rules of natural language may seem inconsistent compared to the hard and fast rules of mathematics and formal logic, but the degree of fuzziness is something which can itself be precisely quantified. Linguistics as a science is one of the most rigorous disciplines you can study, and along with mathematics and information theory is one of the “formal sciences,” which collectively provide the framework with which the rest of our knowledge can be described.

So what does all this cold, hard analysis have to do with those of us who enjoy the musicality of poetry and prose, or the transformative imagery of our favourite fiction? To me there’s a richness available in both mathematics and language, a complexity and elegance which combine to create something beautiful. Although to many people mathematics seems dry and too much like hard work, much of what it reveals can be not only useful but surprisingly aesthetically pleasing.

So what educational impact does this have? I do think individual students demonstrate divergent interests and skills, and that some of them respond more readily to English or mathematics instruction. But this doesn’t mean they’re completely resistant to one or the other. For children with a proclivity for the order and neatness of mathematics I find I can appeal to them by focusing on the structural elements of grammar and the logical consistency of the underlying rules from which meaning is constructed. For students who are more interested in the arts or who have a phobia of numbers, mathematical instruction has to be flexible, and I keep in mind the multiple possible approaches to solving the same problem. Each student develops their own understanding of a mathematical topic, and the beauty of mathematics comes from its consistency, such that divergent approaches to the same question converge on a single set of correct answers. So long as a student can understand what they’re doing and how they find the solution to a problem, there is a surprising amount of room for innovation, unique approaches, and the integration of wider knowledge.

Cognitive Load

I heard a story once about Albert Einstein, who, after a hard day’s work revolutionizing physics, left the university and went on his way, but ended up lost and confused, unsure where to go. It was like his brain had been working so hard that he couldn’t find his own house anymore; the supreme difficulty of whatever thought process Einstein was engaged in at work meant he had almost nothing left in his gas tank for the performance of even the most simple mental tasks, like recalling a piece of information he used every day. There’s no evidence that this story is true, and it’s about as extreme an example as you can get, but the phenomenon of cognitive load is clearly real and relevant to those of us who are interested in education.

Cognitive load is like the mental energy we use to perform difficult tasks. This can take the form of  solving  a mathematical problem, forcing ourselves to do routine household chores when we’d really rather watch TV, or biting our tongue when somebody annoying is provoking us. Apparently, all of these tasks take the same kind of mental energy. It’s effectively the same thing as stress; all of these situations act like a work-out for our brains. Just like you need a rest between physical resistance exercises if you want to avoid damaging your muscles through over-training, your brain can only take so much mental stress before its performance drops and the effect on your development becomes negative.

There are a number of strange phenomena which seem to be related to this concept. Apparently, if people are put in a difficult situation and have been making stressful ethical decisions, for instance making sacrifices for the benefit of others, they are less likely in the immediate future to resist their impulses. It appears that putting yourself second for a long time makes it harder for you to do the apparently ethical “right thing” next time. This works the same if you were doing some other kind of mentally difficult thing, like trying not to eat a tempting sugary treat, working on a complex intellectual exercise, or waiting for something. In any of these situations, people were less likely to exhibit restraint if they were tested immediately afterwards.

So it looks like our brain only really has one gas-tank, and it fuels pretty much any difficult thing we do. This relates to how much effort it makes sense to put in each day. It seems like today’s society glorifies the state of being busy, the rushing from one engagement to the next. Our lives are over-scheduled and we’re expected to be productive much of the time. It looks like our genetic heritage might not be entirely suited to this. Our brain can probably only manage about four hours of useful work a day, for most of us, and the rest of what we do is largely on autopilot. It’s important not to over-exert ourselves mentally, and to save our energy for the important stuff so we don’t get drained working on all those other things that take up our time and cognitive capacity. A healthy balance is required, such that we experience sufficient stress to promote growth, and we exert effort in areas which will reinforce our learning and help us continue to develop, but without imposing expectations on ourselves which could end up being counter-productive.

Growth Mindset

The idea of intelligence as an innate quality which determines an individual’s ability to learn new skills and perform well on cognitive tasks has gained some widespread popular acceptance. When we find ourselves impressed by somebody’s performance on an intellectual task we often say things like “Wow, you’re really smart,” with the implication that their performance is evidence of intellectual giftedness. IQ tests purport to measure this idea of general intelligence and are pretty robust, statistically, giving relatively consistent results between tests and reliably ranking one individual compared to another. But the idea of general intelligence is problematic and in some ways potentially counter-productive.

Developmental and academic testing has shown that there are two competing ideas of intelligence, and the one outlined above represents what’s called a static, as opposed to a growth, mind-set. The static mind-set defines performance in terms of ability and sees success as evidence of talent. The growth mind-set is focused not on ability but effort, and sees success as evidence of growth. While the jury’s still out on exactly how much of intelligence is based on innate ability and how much can be modified by effort, most evidence points to effort being significantly more important, and there is another benefit of thinking this way. It turns out that if you believe in the importance of effort, you’re likely to gain a wide range of cognitive benefits. Individuals who favour a growth mind-set enjoy a boost in performance on cognitive tests, and children who are praised in terms of effort seem to develop more readily than those who are praised in terms of ability.

Multiple tests have been performed and replicated on children, and the results are quite striking. One of these tests involved children drawing pictures, and then being praised either for their apparent talent or for their effort. If they were told “Wow! What a great picture; you must have worked really hard!” then they were more likely to continue drawing for fun, trying new techniques and developing their skills than children who were told “You have a natural talent for drawing.” Another experiment involved a series of more formal tests, which began with something relatively easy, designed to encourage the students to do well, and continuing with the option for children to select more difficult tests or continue at the same difficulty level. If, after their initial success, children were praised based on their ability, then they were less likely to choose more difficult tasks later on. They tried to play it safe and stick to things they knew they could do well. Children who were praised based on their efforts tried to take on more advanced tasks and challenge themselves.

The results of these experiments and others seem to indicate a pretty clear overall trend. Children praised based on personal qualities such as intelligence, while feeling good about achieving well, also viewed their failures as evidence of stupidity. They were more afraid of performing poorly and more likely to stick to easy tasks. Children praised according to effort were more interested in doing well, not just looking good, and had an understanding that they could change what they were capable of by applying themselves through increased effort. They were also less likely to compare themselves to the other children.

These developmental trends are pretty clear, and the significance for all of us is obvious. Beyond the implications for children, pretty much anybody who thinks this way is more likely to succeed. It may not always be true that intelligence is malleable and you can grow from your effort, but believing in it and acting like you can change what you’re capable of acts like a self-fulfilling prophecy. For children who are still developing, or for those of us who are interested in doing the best we can in whatever tasks we set ourselves, the growth mindset offers significant benefits.

The Renaissance Man

People these days talk about how you need a trade, anything so long as it’s specialized. We live in an age of super-specialists, where people spend entire careers developing their skills. They do an apprenticeship, an undergraduate degree, a graduate course, on-the-job training. We have high level experts within sub-sub-disciplines, like somebody’s not just a physicist any more, not even just a nuclear or theoretical physicist, but a member of a team specializing in a particular application of string theoretic mathematics to some intractable problem. We all use products which are designed to make everyday life easier, like computers, but at the same time the operating systems and user interfaces of these products can be so complicated that tertiary institutions offer certification programs in their proper usage. Many of us drive a car, but gone are the days when most of us could fix them; this isn’t just because we’re too lazy to learn, but because when you look inside the engine you see what looks like a spaceship’s innards; all covered in shiny plastic and neatly encased, with that telltale label “no user serviceable parts.”

This is all echoing through into our educational paradigm, where from an increasingly early stage we start pushing people to develop the skills they need for a specific career path. Louis C.K., the famous comedian, talked about the American concept of the technical high-school, describing it as the place where dreams go to die. A technical high-school is an institution for teenagers where they learn skills related to a series of trades, as opposed to focusing on more general academic skills. C.k. talks about how we tell kids they can do anything, that any one of them could grow up to be the president of the united states, so long as they were born there, but by the time they go to a technical high school that’s been narrowed down to a much shorter list, like, you can be these eight things. Pick one of these and that’s what you’re going to do for the rest of your life.

So we live in a world where many of us define ourselves by what we do for a living, like saying, instead of “I teach,” “I am a teacher”. We also live in a world built by specialists, where their hyper-specialized skills go into producing ubiquitous products that most of us have only the most limited understanding of or ability to modify, improve or repair, let alone conceive and create.

This hasn’t always been true. While there have always been trades, of a sort, and division of labour, the educational system which our own is based on, the old English style, was one based on what were called the classics. The idea was that at one of these schools, everybody was given what some these days call the burden of a classical education. A classical scholar could read, understand and speak some form of Latin, possibly classical Greek as well, and was conversant in a body of canonical literature that included, I suppose, such lights of the western tradition as the King James Bible, probably the works of shakespeare, and then some of the ancients, like Plato, maybe, or virgil, Homer. Boring as anything, really, but everybody learned the same thing, and an educated man had a definable set of  skills and body of knowledge which identified him and informed the way he spoke, the references he made and the company he kept.

This whole idea of a classical education is defined by the historical context within which it emerged. The need for an educated public became apparent when the development of the printing press meant kings and rulers could disseminate mass propaganda to the entire population, communicating directly to individual, private citizens en masse for the first time. This ability was relatively ineffective, however, if the masses couldn’t read, and at the same time it was necessary to educate a class of writers who would produce the material for the kings. This process of educating the populace meant, eventually, that the autocratic style of rule became much more difficult as the people were no longer kept weak and disorganized through their own ignorance, but like any transformational technology the press had to be embraced, for fear that if one ruler failed to do so, somebody else would be doing it instead and thereby gaining an intolerable advantage.

But this whole process, the press, mass education, and mass industrialization and standardization in general, were all products of the changing times. The middle ages had ended with the renaissance, which means rebirth. This period was named as the rebirth of knowledge, of scholarship, as the civilization of Western Europe looked back to the time of the Roman Empire and the other so-called classical civilizations, when the continent had been ruled by one government, the infrastructure was organized and there was a stable economy and an educated class of scholars and clerks. This had all been supposedly lost, although really it just shifted elsewhere*, and the resurgence of new ideas in the 14th to 17th centuries was an attempt to regain the heights achieved by the ancients and, eventually, to surpass them. The press itself originated in China, but its impact was perhaps most fully felt in the Europe of the early industrial era.

The renaissance, with its emphasis on education and striving after ancient ideals, is responsible for the term renaissance man. Its closest synonym is perhaps polymath, and it means somebody who excels in multiple fields. The ideal of a renaissance man was somebody who could fight a duel, speak multiple languages, recite poetry, play musical instruments and win a debate. He should be a student of multiple disciplines and current in as many fields as possible. This was perhaps easier to do, back then, when many noblemen led lives of leisure, without the pressure to earn a living, and when the total body of knowledge available was probably a fraction of what currently is produced in a single day. But this ideal influenced the development of the education system, especially in England, where we and much of the world get our current system from.

This system developed in a time of mass production and industrialization, when these ideas were new and shiny and appealing. The King needed an educated populace and a standardized, one-size-fits-all style of rote learning and harsh discipline was instituted. This ethic, as well as having a generalist focus, was also reminiscent of an assembly line, with the pupils being stamped into a mold established by generations of their fellows going through before them. While this has succeeded in reliably producing large numbers of young men who can recite classical Latin, it has proven less than entirely adequate for the modern world of specialization and diversity, and enjoys significantly less favour now than it once did.

Despite all this, the idea of a generalist education, a fundamental set of skills with which to face the modern world, is still interesting and may be of some merit. While Latin may no longer be the highest priority of modern educators, the hyper-specialization of academic and technical disciplines may in some ways be counter-productive. Inter-disciplinary research and collaboration has yielded fruitful results on numerous occasions, with surprising insights from outsider knowledge workers being able to shake up disciplines which previously had been experiencing stagnation. The specialized jargon of each area of research is supposedly intended to allow the convenient communication of equally specialized concepts, but may in fact do little more than indicate the learned status of the speaker within their own branch of the academic hierarchy. This kind of jargon is especially rampant in the humanities, and can serve as a barrier to collaboration.

This disjoint set of jargons and technicalities can leave many effectively disenfranchised from our highly technical world, and this means a generalist understanding of science would be highly useful. It would be advisable, however, to avoid creating a class of jacks of all trades, masters of none, and finding a way instead to give the students an advanced understanding and high level of fluency in the most general set of skills practical, rather than merely giving them a low-level introduction to a broad yet disconnected array of possible skills. To do this it behooves us to examine the academic sphere for the most general and abstract set of topics and find the best way of communicating these to the student.

A friend of mine teaches guitar, and says that the solo from Michael Jackson’s beat it, played by Eddy Van Halen, features so many techniques executed with such elegance to produce such a catchy and appealing sound in its brief, minute or so execution, that it provides an exemplary learning tool. I always thought it’d be cool to do something like that but in a more general sense. He figured that once a kid learned the Beat It solo, they could use that as a practice exercise, improving one technique or another until they could play the whole thing and make it sound like the CD. Once they had that down, they would have learned a whole set of skills which they could extend to almost any other song. My friend was trying to develop a better solo, something even more feature-rich than Beat It but which would still sound cool and be fun to listen to rather than just flashy. This makes me wonder how you could teach something like that in the academic world.

The solution I came up with was to focus on two main things, English and mathematics. English, if understood and developed correctly, allows you to communicate and understand almost any idea. Mathematics allows you to express more concepts than English can, with greater precision but with less ease. Mathematics is an enormously broad topic, and research indicates that unlike many other disciplines, developing one skill in mathematics won’t necessarily improve any others. Many concepts within mathematics are logically related and they form a giant tree of dependencies, but there are still many distinct branches which need to be individually practiced.

English, on the other hand, is much less systematic, and different elements of the study of the English language mutually reinforce each other. I have, however, a great affection for the form of the essay, and evidence shows that writing free-form essays presenting the individual’s understanding of a scientific topic has an unparalleled effectiveness in the memorization of the material covered. The essay is also a difficult thing to get right, requiring both style and diligence, and brings together multiple skills. The essayist needs to be able to present an argument, be conversant with spelling and grammar, appeal to the audience and maintain readability and interest.

A combination of mathematical practice and free-form essay writing strikes me as a good basis for any child’s education, and can be combined with less rigorous influences such as student-directed learning to create an overall package. The idea behind student-directed learning is that children learn best when pursuing knowledge which interest them directly. To this end I’d propose allowing students to study pretty much whatever they want, so long as they can write an essay about the topic and are willing to submit their written work for assessment in terms of their compositional skill as well as their knowledge of their chosen subject. This way, they can develop not only their understanding of their chosen subjects, but also their skill as a writer and and their general academic ability.

This idea of combining language and mathematics into a complementary whole makes me wonder about what area covers both of these. To get this kind of coverage you need to look at things from the most general and abstract level possible, which is the level of the so-called formal sciences. These include linguistics, computer science, cognitive neuroscience, analytic philosophy and abstract mathematics. They’re all descriptions of informational structures, and the same basic essence defines all of them, whether the information is being processed or communicated by a machine or a human brain, or whether it exists in the abstract world of mathematical objects, as lines of code in a computer program or in the symbolic features of a natural human language. These are all pretty much symbol manipulation regimes, and it strikes me that this maximally abstract level of study provides an opportunity to be both entirely general and highly specialized, simultaneously cutting edge and open to anything. It’s in this area where I can see the possibility for some kind of grand unified theory of knowledge or whatever, some way to describe everything in one common language instead of this almost Biblical babel of competing theoretical and conceptual lenses through which to interpret reality.

So that’s pretty much the stuff that I find the most interesting. It sounds dry and highly abstruse but it covers the most fascinating topics, from the study of the brain’s function and the results which are finally freeing psychology from its Freudian roots and turning it into something resembling an actual science, to the mathematical physics that describes the universe at the largest scale and points to exciting new possibilities for exactly how the world works. Some kind of unified theory of symbol manipulation and information system would allow students to better  understand people, concepts, technology and science. If it’s possible to take these abstract disciplines and break them down into step-by-step progressions and simplified, bite-size chunks, to make them accessible while maintaining an awareness of the advanced, cutting-edge applications which make them exciting, then this could be a great educational tool.

 *We talk about the fall of Rome as if the entire civilization collapsed in the first half of the first millenium AD, but really there was an entire Eastern half that continued for hundreds of years. We refer to these people as the Byzantines, after Byzantium, their capital city, but at the time the Byzantine Empire self-identified as Roman and Byzantium, or Constantinople (modern day Istanbul) was just another Rome. The Empire had been split in two and divided between two emperors when it grew too large and unwieldy to maintain under one banner, and the continuation of the Eastern Empire isn’t some trivial remnant. The classical age flourished and the knowledge and technological development from the times of the ancients continued in the hands of the Eastern Empire, to begin with, and later the Arabs. The strong tradition of scholarship within Islam is clearly evident to any student of history and religion, and the Muslim imperative to study God’s world and the laws of nature so as to better understand God’s creation compelled them to faithfully preserve and diligently extend many of the ancient Greek scientific texts. This process of development generated many innovations in fields as diverse as engineering, astronomy and mathematics. In fact, though the name “Arabic numerals” is a misnomer, it is the Arabs who taught the European civilization its current base-10 number system, which they had in turn originally learned from the Indians. Not only this, but many modern mathematical concepts have arabic names, which many people may not recognise. Terms such as Algebra or algorithm come respectively from the Arabic Al-Jabr and the Latin Algoritmi, a transliteration of the name al-Khwārizmī. The universities of the Islamic world have long been centres of great scholarship, and the oldest continuously operating tertiary institution is the University of Ez-Zitouna in Tunis, which was founded as the Ez-Zitouna madrassa in 737 C.E.

The Vagaries of the English Language

The English language, like the English nation, has come into contact with an extraordinary range of other cultures and languages. The Telegraph recently reported that Britain has invaded or fought with nine out of ten countries in the world, leaving only twenty two nations historically unmolested. Similarly, England itself has experienced waves of immigration, invasion, settlement and occupation. All of this is reflected in the diversity and complexity of the English language, little pieces of history echoing through the ages to leave their mark in the form of various linguistic peculiarities.

Despite all this, some of the grandest claims as to English’s sheer size and complexity are in fact plausible sounding misconceptions. Steven Fry, the popular actor and public speaker, has been quoted in support of the oft-repeated but erroneous claim that the English language has the largest vocabulary of all languages, ever, throughout history. It’s presumed by the people who adhere to this belief that because of its sheer complexity and the history of extra—linguistic contact the English language has gone through that it has taken on the characteristics of, and many of the words from, these various other languages. Not only is this supposed to have left English with a “larger vocabulary” than any other language but it is sometimes claimed this sheer volume of words is larger than all the other languages combined, or some similar piece of hyperbole.

A relatively simple thought experiment can discount the possibility of such suppositions. Different languages can be classified according not only to their linguistic family in terms of relatedness to other tongues but also to their functional nature. English is what we call an analytical language, as opposed to languages like German which we call synthetic. This means that in English we modify meaning by changing word order and appending adjectives, adverbs and conditional clauses to the statements we make, whereas in synthetic languages a lot of this work can be done by combining words into new neologistic constructions. We’re all familiar with German’s comically over-complex words formed from the Frankesteinian combination of what might account for an entire sentence in English. These new words aren’t necessarily new in the sense that they need to be added to the lexicon, because German speakers have a clear and established set of rules for how these compound neologisms are formed, such that the meaning of each is often apparent so long as the speaker is familiar with its component words.

The comparison of English to German may seem extreme, with the monstrosities of German linguistic synthesis such as Donaudampfschifffahrtsgesellschaftskapitänsmütze seeming almost absurd*. But there are all kinds of similarly confounding complexities to the question of exactly what counts as a word. Think of the relatively simple case of different verb forms in English; to run, running, ran, did run etc. Or think instead of the differing degrees of modifiable adjectives, such as big, bigger, the biggest. Are these multiple forms of the same word, or is each a word in its own right? What about compound words? I used German because it’s probably familiar to most people, but there are also less obvious but more instructive examples of synthetic languages, such as Czech.

In Czech, for one root you get something like twelve words for free. If I’m addressing you in the second person, I add the suffix “-íš,” pronounced “-eesh,” whereas if I’m speaking in the first person I use “-im.” There are different forms of the word if I’m using it in the general sense, different tenses, and depending also on conjugation. The language also sometimes dispenses with vowels, resulting in apparently unwieldy clusters of consonants like the children’s tongue twister “strč prst skrz krk.” As difficult as that kind of thing is for English speakers to get their tongues around, however, the apparently complex rules of the Czech language are in fact almost entirely consistent, and compared to English the spelling is almost entirely logical and phonetic, albeit in a weird, uncannily altered version of the Latin alphabet. It’s rated one of the hardest languages for other people to learn to speak, but English is right up there with it.

One factor often making it difficult for learners of English as a foreign language is that of its countless inconsistencies and irregularities. One historical peculiarity is that in the past, many of these rules didn’t exist at all. It was only with the implementation of the printing press that familiar standardized spellings began to emerge. Printers used mounted blocks to transfer letters and words to the page, and it was more efficient to have a block for an entire word than to arrange it every time with individual letter blocks. This combined with the existence of multiple rival publishing houses to generate a profusion of standardized and yet differing sets of spellings for commonly used words. The need to create these blocks also led to certain rules for the creation of sensible spellings, so that one house would create word blocks according to one principle and another to a quite different rule. It is from this chaotic evolution that we get the written word of today, with the added confounding factor of differing dialects and their canonical texts, such as American English and Webster’s dictionary.

Webster is the reason many American spellings differ from their British counterparts, in that Webster decided it would make more sense to start spelling in phonetics, according to some kind of logical plan. But this didn’t always stick, and a lot of his simplifying influence has been lost or rejected as people have clung to the old ways. He still managed to push a few of his innovations through though, such as the change from “colour” to “color”, “favour” to “favor” or “realise” to “realize.”

Beyond rivalry between printing houses and dictionary publishers for the right to wrest control of the English language once and for all from the chaotic inconsistencies that had determined its evolution to that point, there is one fundamental schism at the heart of English’s history which leaves it and many other European languages greatly affected. This is the fact that English is a Germanic language, but from the time of the Roman occupation a thousand or so years ago it has been written in a Latin script. There are some linguists today who argue that English has effectively become a Latin language, shifting from one historical branch of the Indo-European language family to another by virtue of accretion. The sheer weight of Latinate terminology which has injected itself into English, as well as the fact that its script is based on the Latin alphabet, means that English as we know it today has both historical and incidental connections to Latin. It could be argued that in some sense English is more “purely” Germanic, that its origins come from the Celtic tongues spoken in the British isles before the Roman occupation, but questions of what came first are difficult to settle given that English’s evolution into its present form has involved from multiple branches of linguistic development merging over time. How can we determine that one is the original branch, simply by virtue of its continuous occupation of the same physical location, when the language itself has expanded out to become what is effectively the universal auxiliary language for almost all international affairs? Is the most defining aspect of English its history in the land of England, or its history as a promiscuously mongrelized meta-tongue?


*This example is actually a joke, but in some scientific disciplines such as chemistry, where names are required for enormously complex chemical compounds, the length of these constructions can be virtually unlimited.

Secondary English Tuition

Hi, my name’s Tobias and I’m the new English tutor here at Emerson Willard. My job is mostly to extend students’ writing ability and go beyond what they learned in the primary course. By now they hopefully have a functional level of spelling and grammar and so I focus on a combination of revising these basics to find problem areas, and trying to get the students to write. That second part is actually most of the program.

I learned to be an English teacher in Prague, at the Edua Languages TEFL course. My trainer there, an American, was talking about the basic approach required to help people learn English. Most of his student teachers were already university graduates, and some of them even had master’s degrees in certain disciplines, and so most of us thought of ourselves as able speakers and writers of academic English already. But John asked us, “If you were a baseball coach, would you get up in front of your team and say ‘Look how many home-runs I can hit?’ Or would you try and get the players swinging the bat for themselves?”

Most of the task of being an English teacher was eliciting content from the students, while maintaining an actual language goal in mind for them to develop. Maybe that day you wanted to work with them on the difference between “had had” and “had,” or difficult verb tenses like “will have been.” Whatever it was, the important thing was being able to engage the students and get them thinking about the language goals you had for them.

After gaining my qualification as an English teacher I promptly went on to never work as a TEFL teacher, ever again. I’d given several practice lessons throughout the course of the four week program in order to gain my qualification, but after that I eventually left the country and came back to New Zealand. Apart from that brief stint in the TEFL world, I’ve also worked as a physics demonstrator and a private tutor for university students. As a physics demonstrator, my job was to show university physics students how to perform experiments and then to help them when they had trouble and mark their work when it was done. As a private tutor I spent most of my time with students from Saudi Arabia, going over their written essays and finding the obvious mistakes and opportunities for stylistic improvements, while at the same time helping them with whatever course material they were struggling with. This often involved taking the subject matter from their prescribed texts and processing it into bite sized chunks to help their comprehension during test preparation.

In all of this and the course of my own academic studies, I’ve found that the most powerful revision and memorization tool is the essay. The process of writing an essay requires not only the recall but the organization of a disparate array of facts and concepts to form a coherent and easily readable argument. The essayist must take items scattered throughout their conceptual space and render them into a one dimensional string, putting one word and one paragraph after another. At the same time they have to exercise theory of mind, and understand how the reader will approach the text they’re creating, how to make it engaging and comprehensible. They need not only to present the facts but get a point across, and there are few other techniques which bear these complementary features.

This isn’t just my hunch, either. A 2011 study, published in the journal Science showed that using essays as a form of retrieval practice was one of the strongest techniques for encouraging the memorization of information, out-performing rote memorization or more modern techniques such as mind-maps. This process required that students write an essay based on material they had learned and that in the act of writing they attempted to recall what they had studied, rather than looking it up again. This meant the essay writing acted as a form of retrieval practice, and the process of manipulating the information as well as actively recalling it helped to cement the information in the learners’ brains while also establishing rich connections to multiple areas due to the necessity of fitting the information into a logically structured text. This wasn’t limited to the arts, either, as the original study was carried out with students a week after reading brief passages about scientific concepts.

The surprising applicability of tools from the English department in studying scientific topics is of particular interest to me. My academic history covers a wide range of topics but right now I’m working on a degree in English and mathematics. The double major isn’t just because I’m a glutton for punishment but because the interdisciplinary approach has always been attractive to me. The areas that interest me the most are the so-called formal sciences, the overlapping set of disciplines including linguistics, mathematics and philosophy from which more specialized areas are developed. At the core of all this are the questions of how information works, how it can be communicated and how it is understood, from which such disparate areas of study as psychology, computer science, physics, and language all draw much of their theoretical underpinnings. Some of the deepest and most intractable unexplained problems in science today are the questions of consciousness, cosmology and comprehensibility. In short, what is the mind, why is there anything and why does any of it seem to make sense? These questions, esoteric as they may seem, are all related by concepts drawn from areas such as information theory and abstract mathematics, and the most recent and promising answers to many of them come at times from unexpected quarters. The current consensus in physics, for instance, is moving towards the ideas of the “mathematical universe,” the universe as an abstract mathematical object, and the holographic principle, the idea that our apparently three dimensional world is really the projection of a two dimensional space.

These developments are as exciting as they are intriguing, and I hope to be able to convey something of my passion for knowledge to my students. Another interesting development is that student-directed learning is one of the most powerful educational paradigms, in that when allowed to direct their own course of study students tend to learn more than when a course is imposed upon them from above. While my approach involves a fair amount of simply pushing the student to achieve a greater level of understanding and fluency in their writing, I also hope to be able to engage students’ interests from a wide range of subject areas and find topics which appeal to them, using the process of learning to write as an opportunity to learn about other concepts as well.