For the love of maths

As a young child, I started off good at maths. Not brilliant, but good. To be honest, I was not really interested in it, but I liked being good at things and didn’t like being bad at things, and that was enough to make me stick with learning maths.

When I was 11 I was surprised to find that I was taught maths in a very different way than I had been before, and I found also that I could not see how this new maths worked. It was discovery learning based education. I engaged with the discovery process in good faith, but made very messy notes of what I tried to discover (I was very messy at that age) that my teacher found impossible to assess, and so did not manage to correct and guide my wanderings, so I just got thoroughly lost.

As I wasn’t very interested in maths anyway, it made sense to me at the time to think that the maths I had done up until age 11 had been the maths that my brain could cope with but that it could not go on any further in the subject, so I stopped trying. Many years later I started trying again and eventually I got to be able to do maths far, far beyond what I had studied at 11.

Some people love maths. It makes them feel great. They can’t get enough of it. I am not like that. Even though I have got a lot better at maths I still don’t love it and I really don’t think that I ever will.

For me, if maths was a person, it wouldn’t be someone that I would say that I loved. It would be someone who wasn’t really much fun, someone who I often found it difficult to talk to, and not someone who easily understood me.

But I would also say that if maths was a person then they would be a very, very honest person. Very honest, patient and generous; always ready and willing to help me to do anything that they could help me with- and the more I asked maths for help the more I found out that it could help me; more than I could ever have believed back when I was 11 and giving up on it.

It is only sensible to appreciate a person like that, to value them, and to be grateful towards them. If you actually knew a person who was like that, you would eventually get to thinking that maybe you owed it to them to take some interest in their strange ways of communicating, and to forgive them for not easily understanding you, and even to ask yourself how you might come to have more of the many good things about them.

You don’t have to love maths to do that, you just need to forgive the accidents that were made by your maths teachers.

P.S. I recommend these great insights into elementary maths teaching by The Recovering Traditionalist.


Artificial education

To generalise, it seems that much e-learning related work taking place within educational institutions focuses on two key areas

  • Learning Management System administration
  • Authoring tool content generation

How do these areas relate to learning?

An LMS basically does the following:

  • Presents content to learners in accordance with syllabi
  • Tracks learners’ (instrumental) engagement with content
  • Provides channels for learners to respond to content
  • Tracks assessment of learners’ responses to content
  • Provides (trackable) channels for learners to communicate with teaching staff and with each other

Authoring tools generate content for an LMS, and the content basically does the following:

  • States declarative knowledge about subject matter to be learned
  • Provides opportunities for automated practising of procedural tasks of a basically combinatorial character which are related to subject matter to be learned


Combinatorial learning is acquisition of procedural knowledge, where the procedures involved are easily and effectively reducible to combinatorial processes (such as selecting, matching, ordering, and recalling knowledge items from a collection of such items). Effective reducibility depends on either one of two conditions

  • The collection of items is small
  • The set of combinations that need to be known is small

To illustrate how combinatorial learning differs from generalised procedural learning, if arithmetic was learned through combinations, then answering a question like ‘3+8 = ?’ would be achieved by giving the answer ‘1’, then ‘2’, then ‘3’ etc… until reaching 11.

E-learning using authoring tool generated content with an LMS does not of course need to be as absurdly reductive as this. Combinatorial learning tasks can be automatically monitored such that excessively reductive attempts to complete them can prompt redirection to declarative knowledge content that is concerned with less reductive techniques for task completion and/or to more appropriately learner ability matched combinatorial tasks that would hopefully elicit less reductive responses.

This strategy to avoid reliance on reductive techniques is a significant aspect of what teaching involves. An effective teacher can apply a set of heuristics for detecting overly reductive task solving techniques and matching them with appropriate task redirections and declarative knowledge interventions. Teachers’ heuristics may well be in a large part known only implicitly, and so not readily translatable into automated procedures. This translation process is of many orders of magnitude greater in complexity than the processes of developing authoring tool content and configuring and maintaining LMS operations.

There is a concerning extent to which e-learning practice seems to be ducking the difficult challenge of formally attempting to explicate teachers’ heuristics into theoretical models and is instead focused on assisting in redirection of teachers’ professional efforts into generating content using authoring tools and engaging with LMS management. This approach may be based on an expectation that teachers’ heuristic expertise will, via a sort of aggregated cognitive osmosis, become ingrained in patterns of LMS recorded data from which said expertise could in principle later be statistically extracted. This has much in common with the way that social media platforms create value by accumulating and organising data about their users that can be used to predict their behaviour.

Recent AI research strongly tends to favour data driven approaches over model driven approaches, and recent research has led to practical improvements that appear to justify these approaches. In the case of education though there may be reasons to doubt whether this approach is the best one to take.

Data driven AI has a mimetic character. Recent notable AI achievements have tended concern things like natural language processing and object recognition in images; things that humans learn to do largely implicitly (there have been other, less obviously implicit, achievements of course, such as beating human masters at Go) but which have been difficult to model explicitly/formally.

If data driven AI teaching systems are generated via mimesis of human teaching, then it is presumably worth asking what sort of human teaching is being mimicked and whether this is the sort of teaching that it ultimately makes good sense to want to mimic.

Three principal modes of teaching can be identified- procedural, declarative and discursive.

Procedural teaching has a very long tradition- it goes back to apprentice craftsmen learned their crafts from their masters by observation, imitation, adoption of heuristics and through instrumental experimentation.

Declarative teaching can be nicely summed up by this (edited) extract from Hard Times.

“Girl number twenty,” said Mr. Gradgrind, squarely pointing with his square forefinger, “I don’t know that girl. Who is that girl?”

[her name is established to be Cecilia Jupe]

… “Cecilia Jupe. Let me see. What is your father?”

“He belongs to the horse-riding, if you please, sir.”

Mr. Gradgrind frowned, and waved off the objectionable calling with his hand.

“We don’t want to know anything about that, here. You mustn’t tell us about that, here. Your father breaks horses, don’t he?”

“If you please, sir, when they can get any to break, they do break horses in the ring, sir.”

“You mustn’t tell us about the ring, here. Very well, then. Describe your father as a horsebreaker. He doctors sick horses, I dare say?” “Oh yes, sir.”

“Very well, then. He is a veterinary surgeon, a farrier, and horse-breaker. Give me your definition of a horse.”

[Cecilia Jupe thrown into the greatest alarm by this demand]

“Girl number twenty unable to define a horse!” said Mr. Gradgrind, for the general behoof of all the little pitchers. “Girl number twenty possessed of no facts, in reference to one of the commonest of animals! Some boy’s definition of a horse. Bitzer, yours.”

“Bitzer,” said Thomas Gradgrind. “Your definition of a horse.”

“Quadruped. Graminivorous. Forty teeth, namely twenty-four grinders, four eye-teeth, and twelve incisive. Sheds coat in the spring; in marshy countries, sheds hoofs, too. Hoofs hard, but requiring to be shod with iron. Age known by marks in mouth.” Thus (and much more) Bitzer.

“Now girl number twenty,” said Mr. Gradgrind. “You know what a horse is.”

Blatant declarative teaching has become unfashionable in western education, but may be on its way back in since it has started to be seen by some policy makers as a major factor in the apparent successfulness of maths learning in many Asian countries (not necessarily a valid interpretation). Declarative learning has substantially given way to discursive learning; discursive in the sense of relating to discourse or modes of discourse, the emphasis being on higher order and critical thinking skills.

Some educators (notably David Didau in What if everything you knew about education was wrong?) argue that contemporary western educational methods are largely not genuinely discursive but rather remain declarative, except that the declarative learning occurs indirectly (and hence inefficiently)- learners in practice being coached and supported into superficially reproducing apparently discursive output which on closer examination consists of a heuristically aggregated mass of declarative knowledge that is concerned with discursive learning (learners do not actually learn to think critically but how to convincingly declare that they have thought critically). Didau argues that genuine discursive learning can only develop on a foundation of pre-existing declarative and procedural learning; I happen to agree with him.

Presuming that Didau is correct, then in my opinion the issue of what teaching should be mimicked boils down to whether procedural learning should emerge from declarative learning or vice versa.

The declarative to procedural pathway means first identifying the concrete items and objects that are to be learned about and then trying to understand how these fit together into structures. This is an analytical approach based on primarily acknowledging what needs to be learned and finding ways to accommodate how learners are able to learn to those needs. The primary tool for this pathway is instruction by experts. The experts could in principle be data driven AI teaching systems mimicking human teachers. These AI teaching systems should be able to teach much of what human teachers can, but it is not clear how they would necessarily improve on human teachers (other than in tireless availability). If the data that drove AI teachers was collected from one-to-one tuition (data on this being much less easily available than group teaching data however) then AI teachers could mimic such tutoring (and provide it at much lower cost than human tutors).

The procedural to declarative pathway means first experimenting with how processes of manipulating symbolic items and objects- which could well be facilitated through symbolic projection onto concrete objects that learners can actually (or virtually) manipulate rather than have presented to them declaratively- to generate structures, and then look for analogies between the structures generated and structures that exist in the world that need to be learned about. This is a synthetic approach based primarily on how learners are able to learn and finding ways to accommodate what needs to be learned to match these ways of learning. The primary tool for this pathway is the simulation/game and the community of its players- games providing the opportunities to manipulate, experiment and generate hypothetical structures (to play and to imagine) and communities of players providing the connections to real world structures through their embodied experiences of how these structures (the declarative knowledge about the world) do and do not relate to game generated structures.

The procedural to declarative pathway of course incorporates AI in the design of learning games, but a diverse set of specific model based AI systems, matched with various different real world systems. Each specific model would be designed through explicit processes of consultation with experts in a particular subjects or applications of subjects as well as using data about how learners’ played such games.

What I am suggesting is that the procedural to declarative pathway provides learning potentials that go beyond what could be achieved by automating existing teaching methods, primarily in terms of helping to develop learners’ creativity and problem solving skills. Learners would not just be learning to reproduce existing knowledge but learning how to generate new knowledge, and even how to generate tools for the generation of knew knowledge (by writing their own games).

To some extent, the (LMS + authoring tool generated content = data = learning) approach recognises the potential for new knowledge to be generated through LMS enabling of user generated content, where learners are the users generating the content. Learner to learner sharing has scope for having the same sorts of uses as communities of gamers sharing games and gameplay experiences with each other, and generic user-generated-content models certainly have far greater overall flexibility, but such flexibility can simply end up as a channel for the distribution of pseudo-discursive declarative knowledge- especially if prioritising such knowledge continues to comprise a significant part of educational practice.


I have almost finished reading a really remarkably excellent book What if everything you knew about education was wrong? by David Didau. In praise of the book I can say nothing more flattering than that it is to education what Nassim Taleb’s Antifragile is to economics.

Attempting to summarise WIEYKAEWW in an easily digested overview is to do the book a great disservice; the core premise of the book is that whatever knowledge is learned in a way that tries to avoid the diverse difficulties inherent in learning, is understood only superficially and quickly completely forgotten as if it had never been learned- which it indeed never really had been, only the semblance of the knowledge had been learned.

Learning worthy of the name is that which once it has happened cannot un-happen, changing forever how we understand something about the world (it is difficult, if not impossible, for a functionally literate person in a clear mind to see the letters that form a word in a language that they have learned without internally forming the sound of the word).

Of course, not all knowledge lasts a lifetime. There is a zone between remembered forever and forgotten overnight. Very interestingly though, knowledge learned that was well remembered while it was being at least intermittently re-used, if left unused for a sufficiently long that it finally becomes irretrievable to our memory, is re-learnable significantly faster than it was learned for the first time; faster and better- errors that were never noticed and never understood in the process of learning are recognised and corrected in subsequent learning processes.

Ironically, in saying why WIEYKAEWW isn’t readily summarisable, I have summarised half of its message. The other half is that modern western education systems are largely based around measuring evidence of the semblances of knowledge that are taken in the stead of evidence of knowledge itself- knowledge being something far, far, FAR (incomparably far!) harder to measure.

Obviously, merely reading a summary like this does basically nothing to account for why avoidance of difficulty in learning leads to only the semblance of learning. Even reading WIEYKAEWW probably wouldn’t make someone understand that well unless they had experienced plenty of both successful and unsuccessful learning, and had done a fair bit of teaching too.

However… as an experimental stab at conveying a sense of what semblance learning is like, I am reproducing here a slightly edited transcript of a scene from Monty Python’s The Meaning of Life that deals with becoming acquainted with philosophical questions.


[Mr & Mrs Hendy- a middle-aged American couple on holiday in a restaurant]

Waiter: Good evening… would you care for something to talk about?

          [He hands Mr & Mrs Hendy each a menu card with a list of subjects on.]

Mr Hendy: Oh that would be wonderful.

Waiter: Our special tonight is minorities…

Mr Hendy: Oh that sounds interesting…

Mrs Hendy: What’s this conversation here…?

Waiter: Oh that’s football… you can talk about the Steelers-Bears game, Saturday… or you could reminisce about really great World Series –

Mrs Hendy: No… no, no.

Mr Hendy: What’s this one here?

Waiter: That’s philosophy.

Mrs Hendy: Is that a sport?

Waiter: No it’s more of an attempt to construct a viable hypothesis to explain the Meaning of Life.

Mr Hendy: Oh that sounds wonderful… Would you like to talk about the Meaning of Life, darling…?

Mrs Hendy: Sure, why not?

Waiter: Philosophy for two?

Mr Hendy: Right…

Waiter: You folks want me to start you off?

Mr Hendy: Oh really we’d appreciate that…

Waiter: OK. Well er… look, have you ever wondered just why you’re here?

Mr Hendy: Well… we went to Miami last year and California the year before that, and we’ve…

Waiter: No, no… I mean why *we’re* here. On this planet?

Mr Hendy: [guardedly]… N… n… nope.

Waiter: Right! Have you ever *wanted* to know what it’s all about?

Mr Hendy: [emphatically] No!

Waiter: Right ho! Well, see, throughout history there have been certain men and women who have tried to find the solution to the mysteries of existence.

Mrs Hendy: Great.

Waiter: And we call these guys ‘philosophers’.

Mrs Hendy: And that’s what we’re talking about!

Waiter: Right!

Mrs Hendy: That’s neat!

Waiter: Well you look like you’re getting the idea, so why don’t I give you these conversation cards – they’ll tell you a little about philosophical method, names of famous philosophers… there y’are. Have a nice conversation!

Mr Hendy: Thank you! Thank you very much.

          [He leaves.]

Mrs Hendy: He’s cute.

Mr Hendy: Yeah, real understanding.

          [They sit and look at the cards, then rather formally and uncertainly Mrs Hendy opens the conversation.]

Mrs Hendy: Oh! I never knew that *Schopenhauer* was a


Mr Hendy: Oh yeah… He’s the one that begins with an S.

Mrs Hendy: Oh…

Mr Hendy: … Um [pause]… like Nietzsche…

Mrs Hendy: Does Nietzsche begin with an S?

Mr Hendy: There’s an S in Nietzsche…

Mrs Hendy: Oh wow! Yes there is. Do all philosophers have an S in them?

Mr Hendy: Yeah I think most of them do.

Mrs Hendy: Oh!… Does that mean Selina Jones is a philosopher?

Mr Hendy: Yeah… Right, she could be… she sings about the Meaning of Life.

Mrs Hendy: Yeah, that’s right, but I don’t think she writes her own material.

Mr Hendy: No. Maybe Schopenhauer writes her material?

Mrs Hendy: No… Burt Bacharach writes it.

Mr Hendy: There’s no ‘S’ in Burt Bacharach…

Mrs Hendy: … Or in Hal David…

Mr Hendy: Who’s Hal David?

Mrs Hendy: He writes the lyrics, Burt just writes the tunes… only now he’s married to Carole Bayer Sager…

Mr Hendy: Oh… Waiter… this conversation isn’t very good.

Mr Hendy has a point, doesn’t he?

Of course, this was just the Hendy’s first attempt to talk about philosophy. If though they were involved in a mandatory programme of such opportunities, lasting more than a decade, for which they were regularly assessed and given corrections- then assuming that they didn’t just refuse to cooperate in the process they would eventually learn to have somewhat polished sounding conversations about philosophy (the ongoing opaque impenetrability of the experience might at least inculcate in them a world-weariness suitable for discussing philosophy that holidays with our partners ideally should not do).

My sincere apologies to David Didau for this post.

What is energy anyway?

In Richard Feynman’s book, Surely You’re Joking, Mr. Feynman!,  he critiqued how the concept of energy was starting to be taught in the USA (as part of a more general analysis of science education reforms).

… For example, there was a book that started out with four pictures: first there was a wind-up toy; then there was an automobile; then there was a boy riding a bicycle; then there was something else. And underneath each picture, it said “What makes it go?”

I thought, I know what it is: They’re going to talk about mechanics, how the springs work inside the toy; about chemistry, how the engine of an automobile works; and biology, about how the muscles work.

It was the kind of thing my father would have talked about: “What makes it go? Everything goes because the sun is shining.” And then we would have fun discussing it: 
“No, the toy goes because the spring is wound up, I would say. 
“How did the spring get would up” he would ask. 
“I wound it up” 
“And how did you get moving?” 
“From eating” 
“And food grows only because the sun is shining. So it’s because the sun is shining that all these things are moving” That would get the concept across that motion is simply the transformation of the sun’s power.

I turned the page. The answer was, for the wind-up toy, “Energy makes it go.” And for the boy on the bicycle, “Energy makes it go.” For everything “Energy makes it go.”

Now that doesn’t mean anything. Suppose it’s “Wakalixes.” That’s the general principle: “Wakalixes makes it go.” There is no knowledge coming in. The child doesn’t learn anything; it’s just a word.

What they should have done is to look at the wind-up toy, see that there are springs inside, learn about springs, learn about wheels, and never mind “energy”. Later on, when the children know something about how the toy actually works, they can discuss the more general principles of energy.

It is also not even true that “energy makes it go”, because if it stops, you could say, “energy makes it stop” just as well. What they’re talking about is concentrated energy being transformed into more dilute forms, which is a very subtle aspect of energy. Energy is neither increased nor decreased in these examples; it’s just changed from one form to another. And when the things stop, the energy is changed into heat, into general chaos.

I recently watched the 2016 Royal Institution Christmas Lectures, the subject of which was energy. These lectures broadly reflect the general thrust of the UK physics curriculum’s treatment of the topic of energy. When energy is introduced, the mutual convertibility of different energy types does get stressed, so addressing Feynman’s complaint, but the blanket ‘Energy makes it go’ idea still gets included, just as something that comes in different forms. What all these forms have in common that make it sensible to lump them together under the category of energy is quite a complex question. The given answer is that all types of energy can generate some kind of movement or change, which is rather a vague notion and one that is not very intuitively graspable for some types of energy unless a learner is already familiar with some of the non-obvious kinds of entities that exist in the world that could be moving (individual molecules, compression and rarefaction of matter, free electrons, field intensities). Additionally, a hurdle is set up for when potential energy (which is not anything moving or changing) later needs to be understood.

It ought to be recognised that learners come to physics with a set of preconceptions about energy, based around intuitive concepts, where intuition favours what can be directly sensed-  objects that move, lights that shine, and sounds, particularly monotone sounds. These preconceptions tend to understand moving objects as having energy but light (and by extension electricity) as being energy. Sound is something of a grey area as intuitive links between the motions of certain objects and the generation of sounds can fairly easily be made.

Learners’ preconceptions about energy are merely one aspect of a more general misconception that learners start with, which is a preference for teleological explanations. Teleological explanations are highly intuitive as they are generalisations of personal experience. A person subjectively experiences themself as being a basically autonomous entity that is placed in a world and capable of acting independently of what that world is doing. The actions of autonomous entities are understood as reflecting those entity’s intentions- what it wants to do, what outcomes it desires to see achieved.

Things encountered in the world which do not exhibit complex behaviours are intuitively seen as not purposeful, but an object which is not easily predictable is intuitively ascribed a purposefulness. This ascription is not strongly challenged in science education. Biology is obviously particularly susceptible to this, but less obvious occurrences of it pervade into the rest of science education, as learners are introduced to the realisation that inanimate objects only seem simple because they have not been studied closely or carefully. Matter turns out to have a rich and complex structure that affords detailed explanation.

Teaching the structure of matter, and hence about chemical bonds and reactions, often mentions explicitly (let alone implicitly) that atoms have some sort of teleological drive towards completing their outer electron shells (they ‘want to balance’), which is a specific case of a more generalised notion that systems tend towards their lowest energy state. That notion is ultimately teleological if it does not explain what mechanisms act to minimise systems’ energies.

Physics teaching falls into this thinking trap when it focuses primarily on consideration of individual objects (as much of it does). For example, Newton’s second law of motion gets a lot of attention; a lot of calculations of resultant forces acting on a body are made. Newton’s third law, which brings into the picture that for one body to have been pushed or pulled another body must at the same time have been pushed or pulled in the opposite direction, is added later and usually with considerably less effective understanding. The easy impression for a leaner to take away is that individual objects’ movements are mostly understandable as in some way inherent to those individual objects; obedient to rules, but still treating each object as having some sort of purpose that guides it (but a constrained purpose compared to a human’s). The fact of the movement of an individual object’s inseparability from the combined movement of a group of objects that the individual object is part of- that very important understanding is all too easily missed. When considering the relation of individual objects’ behaviours to the group of objects of which it is a part- this is where the concept of energy can be validly introduced such as to be recognised as incredibly important and useful.

Trying to derive systemic global rules and principles from studying local examples is difficult. The rules and principles can be taught but they may never be deeply understood. Some educators will maintain that learning must use the concrete to lead to the abstract, that phenomena must precede rules. I would argue that the learner imagination is a factor that can modify that progression, and make it more of a general blending process. Tools for exercising and drawing out learners’ imaginations may be able to make their understanding emerge as much from their imagination as from their intuition. I can see particular types of learning games being instrumental in this.

Learning learning games

Many good and well known justifications for game-based-learning already exist. The book that first introduced me to them was The Gamification of Learning and Instruction, by Karl Kapp. It may be a bit superfluous for me to even perfunctorily list the appropriate justifications here, but I will mention at least some basic principles- namely, learners controlling activity flow, learners receiving continual feedback, and task difficulty being adaptive to learners’ actions.

I attempted to incorporate these principles in the first proper learning game that I wrote, which was a substantial part of my e-learning MSc. The game was a visual sequence imitation game (based on the 1970/80s handheld game SIMON), but with the novel feature of including a requirement for learners to extrapolate extensions of as well as to imitate visual sequences; learners needed to have some sort of mental model of the rules by which sequences were generated to extrapolate the sequences with better than chance accuracy. The game can be played here.

Learner control, continual feedback and adaptive difficulty are key elements involved in (many types of) gaming, and there are learning games that pretty much seem to be based around just these principles. I’m thinking here of what are often called brain training games. I played some Lumosity games to get some idea of these games, and while doing that I was reminded of quite a few of the sub-games that I had enjoyed playing in Wii Fit and other Wii exercise games (the only console I have ever had is a Wii, which I originally bought for fitness games so as to get indoor exercise during cold weather).

Brain training really took off around the turn of the century. In 2007, Americans spent $80 million on it. A number of neuroscientists responded to the trend by questioning the learning value of brain training, but by 2012 the spend on it was closer to $1 billion. In 2015, the Federal Trade Commission started suing companies that claimed their brain training products could increase users’ intelligence, on the basis that no evidence of cognitive benefits from playing brain training games could be demonstrated, because brain training users only improved their skills at playing brain training games; the game-play skills were not transferable to anything else (or at least not to anything that was known to be able to measure cognitive abilities).

For a game to help users to learn something other than how to play the game, either of two things are needed-

  1. The game-play must have a significant degree of similarity to some real world activity that is to be learned (so that the game is effectively a kind of simulation).
  2. The game-play must involve the player developing an explicit mental model of something that has a significant degree of similarity to their explicit mental model of some real world activity that is to be learned.

In case (a), the game can teach largely via implicit learning. Learners just have to keep playing the game and in doing so they gradually build up habituated sets of responses and associations that produce desired outcomes in the game. Learners don’t need to know how they brought about the outcomes, and indeed they may not be able to say how even if they wanted to. Surprisingly complex procedures can be learned this way. Children’s early learning of language largely occurs in this way. Very obviously highly artificial procedures (like running a simulated sugar factory to maximise production) have been shown to be implicitly learnable by adults.

Implicit learning is difficult to transfer to new contexts unless the new contexts are superficially similar to the context in which the implicit learning took place. If the new context appears different or is presented differently, even if it is functionally very similar, implicit learning will not transfer to it effectively. There is therefore a trade-off between how effectively an implicit learning game can teach something that transfers well to a real world context and how much that game can incorporate the game principles of user control, continual feedback and adaptive difficulty. Put starkly, the better that the game actually teaches about some real world context effectively, the less fun it is to play (unless the learner finds the real world context intrinsically interesting).

The necessity for this trade-off assumes that real world contexts do not tend to resemble the contexts that are compatible with fun games, which presumably they could be designed to be more like. Imagine perhaps workplaces that look like theme parks; made up of eye-catchingly iconic, highly memorable structures, connected by pulsing motion paths that guide workers to where they are needed and objects that flash selective colour coded prompts showing how to interact with them. Augmented reality could transform mundane environments into something along those lines easily enough.

Case (b) doesn’t depend on changing environments, only how learners represent environments to themselves. This approach seems very natural to me, as it underlies how physics gets learned (when it is learned successfully). A very basic and obvious example- the Sun appears to cross the daytime sky, but is learned to be interpreted as a stationary object viewed from a perspective that changes with the rotation of the surface from which the Sun is being observed.

This learning to see things in non-obvious ways happens by constructing and investigating models, models that have varying degrees of similarity to real world contexts, but seen from very different perspectives.

Traditionally in physics teaching, models are presented to learners to be studied, and when the models have been learned then a process of applying the models to real world contexts is undertaken. Sometimes this approach is reversed and some observations are made and a model is constructed based on these observations.

Neither of these approaches works easily for most learners, because the real world contexts that a learner recognises, or at any rate views as interesting enough to be motivated to pay much attention to, are nonlinear and messy to the extent that a faithful model of them has to be so abstract that it seems superficially only to translate the real world context into a virtually incomprehensible coded puzzle (algebra and calculus usually, if you’re lucky they might have an attractive graphical representation, like fractals). Real world contexts that can be passably faithfully represented by models that are simple enough to be easily comprehensible are unlikely to seem to change much about learners’ perception of real world contexts, and so don’t seem to add anything important and so, to a learner, don’t seem to really matter very much.

To frustrate matters further, a feature of how the real world often works is that a seemingly minor alteration of a simple context or model can sometimes introduce great complexity, so that trying to take a model that is too simple to offer much obvious interest and see if it can explain some interestingly nonlinear real world context by making a few adaptations all too often seems to give a negative result- did not match.

Simplifying the issue greatly- a model simple enough to be easily understood usually seems like a poor match for experienced real life, and what seems worth knowing about real life requires models too abstract for simple understanding, and trying to bring models and life together in a compromise breaks the models.

The gulf between models and real life is hard to bridge unless a learner becomes skilled at manipulating models, so much so that they can understand how to alter models in various ways to try and make them match to different real world contexts while not actually breaking the models. This is where I think game-based-learning can make a powerful contribution to learning in ways that have not yet been very deeply explored.

Games have been created which are about building models. At a superficial level, Minecraft has shown the huge appeal of such games. Construction games that exist tend to provide a set of rules and allow freedom to build structures using those rules. What I am proposing would be construction games in which the rules could be changed, and the players can see how changing the rules changes what the structures that they can build are like. The principle at work in this is that the abstractions inherent in models should be able to be made into concrete objects that can be played with. I think that this principle can apply to the learning of systematic abstractions in general, to mathematics as much as to physics or other sciences.

This is what I am currently inspired by then, the developing of games that make concretion of abstractions possible through play, in which learners develop proficiency at manipulation of models, and which engage learners through the old gamer tricks of learner control, continual feedback and adaptive difficulty.


A surprisingly large number of science students at the college I taught at were very bad at estimating (even after a year or more of science courses) how long one metre is. Asking them to place their hands one metre apart would get an interesting range of responses but almost all significantly underestimated the length of a metre (and when I say significantly I mean that estimates of less than a half a metre were not unusual).

When I had a handy supply of one metre rulers nearby (quite often), I would show one of them at the front of the class and say “This is a metre” and use the ruler to demonstrate that I was getting on for two metres in height. I imagined that this was a simple and memorable enough demonstration to ensure that a brute fact like the length of a metre would be recalled thereafter at least fairly reliably.

It took me a while to finally understand why this wasn’t the case.

The understanding came to me all at once from out of the blue one day while I was running in a park. I realised that I had been insufficiently precise in my language usage, thinking that doing so was unproblematic and highlighted the triviality of recognising the length of a metre.

When I’d said “This is a metre” while holding up a metre ruler, I should have said “This is a ruler that is one metre long”. What I had done was to inadvertently convey the misleading impression that physics teachers, no doubt for their own mysterious and pedantic reasons, called rulers metres. Metre was understood to be an obscure word for a ruler. I presented my physics teacher’s ruler to show how long it was, maybe it might have seemed to the students that I did that to show that my ruler was longer than ordinary rulers because physicists were such geeks about measuring things and needed special long rulers (called metres). When students estimated the length of a metre, they did not think of my physics teacher’s ruler (as they did not see themselves as physics teachers), they thought of the 15 or 30 centimetre ruler that they had in their stationary collection (if they had one of those).

This seems an incomprehensible misunderstanding, but it is comprehensible when taking into account how the world today is different to the one that I grew up in. The world today is not prominently and openly based around objective standards, rather it is based around personalised selections. The world has become shaped by consumer preferences and personalised adaptations. Today’s world is a world of bubbles and echo chambers. It is not automatically sensible to this generation that everyone would share a common unit of measurement that none of them had any part in defining. How could there be only one metre? Surely everyone can choose their own metre? People have their own rulers, in the colour and length that they chose, decorated how they chose. Today’s learners are not necessarily even clear that length itself exists independently of their chosen interactions with it. Quirkily enough, relativity means that they are actually correct about that, but not in any straightforward or convenient way.

Ian Bogost, the influential game designer, has written a thoughtful book called Play Anything, which is centrally concerned with the idea that the activity of playing is ultimately about people appreciating and respecting the limitations of objects that exist independently of themselves. Bogost frequently refers back to an apparently trivial event when he was taking his young daughter shopping in a crowded mall and finding the experience stressful, when he was suddenly made aware that his daughter was responding to the situation of being dragged around the mall by skipping so as not to step on the edges of flooring tiles while avoiding other shoppers’ feet, and while having her movements largely controlled by her father’s decisions- she had made the high stimulus experience of mall navigation into a game which she was enjoying playing. Probably we all remember enjoying this sort of play when we were young children (I know I do). What Bogost makes of this is that the satisfaction of play is not (as has become popularised) the freedom to do anything, but rather the deliberate engagement with a specific set of constraints as a focus of attention- accepting (some little part of) the world as it is rather than introspecting on our dissatisfaction with the mismatch between how the world is and how we would have it be. Games designers have investigated the possibilities of games where anything is possible for the player, and have tended to find that gamers want to act within constraints (but may well not want to recognise the constraints, constraints perhaps working best when they seem to rule out courses of action that were felt to be rejected because they were intrinsically unwanted, not because they were unavailable).

Play is a complex concept. Since formally studying education technology I have become extremely interested in the potential of game-based-learning and have started to think carefully about play, learning and how they relate to each other. There is more to play than Bogost writes about, but he has emphasised an aspect of it that is increasingly overlooked; how this aspect of play relates to learning strikes me as important.

Constraint has been an obvious given in education since the industrial model of mass education existed (and before that, but in different ways). Educational technology has to some extent helped to promote the idea of greater learner autonomy through concepts such as self-directed-learning, learning-on-demand, personalised learning, and more besides. This trend loosely mirrors the general trend of increasing individual choice and rejection of objectively imposed standards that makes the recognition of the length of a metre harder than it used to be. I find myself wondering whether there is a possibility of learners subsuming such individual oriented learning into their ongoing development of their sense of their identity, and hence making this learning more to do with how they, as individuals, would have the world be than about how the world consists of objects that are of interest in and of themselves. Would the playfulness and hence the intrinsic enjoyableness of learning be diminished if this happened?