expertise

Michael Gove is reported as saying that ‘Learning facts by rote should be a central part of the school experience’, a philosophy which apparently underpins his shakeup of school exams. Arguing that "memorisation is a necessary precondition of understanding", he believes that exams that require students to memorize quantities of material ‘promote motivation, solidify knowledge, and guarantee standards’.

Let’s start with one sturdy argument: "Only when facts and concepts are committed securely to the working memory, so that it is no effort to recall them and no effort is required to work things out from first principles, do we really have a secure hold on knowledge.”

This is a great point, and I think all those in the ‘it’s all about learning how to learn’ camp should take due notice. On the other hand, the idea that memorizing quantities of material by rote is motivating is a very shaky argument indeed. Perhaps Gove himself enjoyed doing this at school, but I’d suggest it’s only motivating for those who can do it easily, and find that it puts them ‘above’ many other students.

But let’s not get into critiquing Gove’s stance on education. My purpose here is to discuss two aspects of it. The first is the idea that rote memorization is central to education. The second is more implicit: the idea that knowledge is central to education.

This is the nub of the issue: to what extent should students be acquiring ‘knowledge’ vs expertise in acquiring, managing, and connecting knowledge?

This is the central issue of today’s shifting world. As Ronald Bailey recently discussed in Reason magazine, Half of the Facts You Know Are Probably Wrong.

So, if knowledge itself is constantly shifting, is there any point in acquiring it?

If there were simple answers to this question, we wouldn’t keep on debating the issue, but I think part of the answer lies in the nature of concepts.

Now, concepts / categories are the building blocks of knowledge. But they are themselves surprisingly difficult to pin down. Once upon a time, we had the simple view that there were ‘rules’ that defined them. A dog has four legs; is a mammal; barks; wags its tail … When we tried to work out the rules that defined categories, we realized that, with the exception of a few mathematical concepts, it couldn’t be done.

There are two approaches to understanding categories that have been more successful than this ‘definitional’ approach, and both of them are probably involved in the development of concepts. These approaches are known as the ‘prototypical’ and the ‘exemplar’ models. The key ideas are that concepts are ‘fuzzy’, hovering around a central (‘most typical’) prototype, and are built up from examples.

A child builds up a concept of ‘dog’ from the different dogs she sees. We build up our concept of ‘far-right politician’ from the various politicians presented in the media.

Some concepts are going to be ‘fuzzier’ (broader, more diverse) than others. ‘Dog’, if you think about St Bernards and Chihuahuas and greyhounds and corgis, has an astonishingly diverse membership; ‘banana’ is, for most of us, based on a very limited sample of banana types.

Would you recognize this bright pink fruit as a banana? Or this wild one? What about this dog? Or this?

I’m guessing the bananas surprised you, and without being told they were bananas, you would have guessed they were some tropical fruit you didn’t know. On the other hand, I’m sure you had no trouble at all recognizing those rather strange animals as dogs (adored the puli, I have to say!).

To the extent that you’ve experienced diversity in your category members, the concept you’ve built will be a strong one, capable of allowing you to categorize members quickly and accurately.

In my article on expertise, I list four important differences between experts and novices:

• experts have categories

• experts have richer categories

• experts’ categories are based on deeper principles

• novices’ categories emphasize surface similarities.

How did experts develop these deeper, richer categories? Saying, “10,000 hours of practice”, may be a practical answer, but it doesn’t tell us why number of hours is important.

One vital reason the practice is important is because it grants the opportunity to acquire a greater diversity of examples.

Diverse examples, diverse contexts, this is what is really important.

What does all this have to do with knowledge and education?

Expertise (a word I use to cover the spectrum of expertise, not necessarily denoting an ‘expert’) is rooted in good categories. Good categories are rooted in their exemplars. Exemplars may change — you may realize you’ve misclassified an exemplar; scientists may decree that an exemplar really belongs in a different category (a ‘fact’ is wrong) — but the categories themselves are more durable than their individual members.

I say it again: expertise is rooted in the breadth and usefulness of your categories. Individual exemplars may turn out to be wrong, but a good category can cope with that — bringing exemplars in and out is how a category develops. So it doesn’t matter if some exemplars need to be discarded; what matters is developing the category.

You can’t build a good category without experiencing lots of exemplars.

Although, admittedly, some of them are more important than others.

Indeed, every category may be thought of as having ‘anchors’ — exemplars that, through their typicality or atypicality, define the category in crucial ways. This is not to say that they are necessarily ‘set’ exemplars, required of the category. No, your anchors may well be different from mine. But the important thing is that your categories have such members, and that these members are well-rooted, making them quickly and reliably accessible.

Let’s take language learning as an example (although language learning is to some extent a special case, and I don’t want to take the analogy too far). There are words you need to know, basic words such as prepositions and conjunctions, high-frequency words such as common nouns and verbs. But despite lists of “Top 1000 words” and the like, these are fewer than you might think. Because language is very much a creature of context. If you want to read scientific texts, you’ll want a different set of words than if your interest lies in reading celebrity magazines, to take an extreme comparison.

What you need to learn is the words you need, and that is specific to your interests. Moreover, the best way of learning them is also an individual matter — and by ‘way’, I’m not (for a change) talking about strategies, which is a different issue. I’m talking about the contexts in which you experience the words you are learning.

For example, say you are studying genetics. There are crucial concepts you will need to learn — concepts such as ‘DNA’, ‘chromosomes’, ‘RNA’, epigenetics, etc — but there is no such requirement concerning the precise examples (exemplars) you use to acquire those concepts. More importantly, it is much better to cover a number of different examples that illuminate a concept, rather than focus on a single one (Mendel’s peas, I’m looking at you!).

Genetics is changing all the time, as we learn more and more. But that’s an argument for learning how to replace outdated information (an area of study skills sadly neglected!), not an argument for not learning anything in case it turns out to be wrong.

To understand a subject, you need to grasp its basic concepts. This is the knowledge part. To deal with the mutability of specific knowledge, you need to understand how to discard outdated knowledge. To deal with the amount of knowledge relevant to your studies and interests, you need skills in seeing what information is important and relevant for your studies and interests and in managing the information so that it is accessible when needed.

Accessibility is key. Whether you store the information in your own head or in an external storage device, you need to be able to lay hands on it when you need it. And here’s the nub of the problem: you need to know when you need it.

This problem is the primary reason why internal storage (in your own memory) is favored by many. It’s only too easy to file something away in external storage (physical files; computer documents; whatever) and forget that it’s there.

But what all this means is that what we really need in our memory is an index. We don’t need to remember what a deoxyribose sugar is if we can instantly look it up whenever we come across it.

Or do we?

This is the point, isn’t it? If you want to study a subject, you can’t be having to look up every second word in the text, you need to understand the concepts, the language. So you do need to have those core concepts well understood, and the technical vocabulary mastered.

So is this an argument for rote memorization?

No, because rote memorization is a poor strategy, suitable only for situations where there can be no understanding, no connection.

We learn by repetition, but rote repetition is the worst kind of repetition there is.

To acquire the base knowledge you need to build expertise, you need repetition through diverse examples. This is the art and craft of good instruction: providing the right examples, in the right order.

The thing to remember about Ericsson’s famous expertise research, showing us the vital importance of deliberate practice in making an expert, is that it was challenging the long-dominant view that natural-born talent is all-important. But Gladwell’s popularizing of Ericsson’s “10,000 hours” overstates the case, and of course people are only too keen to believe that any height is achievable if you just work hard enough.

The much more believable story is that, yes, practice is vital — a great deal of the right sort of practice — but we can’t disavow “natural” abilities entirely.

Last year I reported on an experiment in which 57 pianists with a wide range of deliberate practice (from 260 to more than 31,000 hours) were compared on their ability to sight-read. Number of hours of practice did indeed predict much of the difference in performance (nearly half) — but not all. Working memory capacity also had a statistically significant impact on performance, although this impact was much smaller (accounting for only about 7% of the performance difference). Nevertheless, there’s a clear consequence: given two players who have put in the same amount of effective practice, the one with the higher WMC is likely to do better. Why should WMC affect sight-reading? Perhaps by affecting how many notes a player can look ahead as she plays — this is a factor known to affect sight-reading performance.

Interestingly, the effect of working memory capacity was quite independent of practice, and hours of practice apparently had no effect on WMC. Although it’s possible (the study was too small to tell) that a lot of practice at an early age might affect WMC. After all, music training has been shown to increase IQ in children.

So, while practice is certainly the most important factor in developing expertise, other factors, some of them less amenable to training, have a role to play too.

But do general abilities such as WMC or intelligence matter once you’ve put in the requisite hours of good practice? It may be that ability becomes less important once you achieve expertise in a domain.

The question of whether WMC interacts with domain knowledge in this way has been studied by Hambrick and his colleagues in a number of experiments. One study used a memory task in which participants listened to fictitious radio broadcasts of baseball games and tried to remember major events and information about the players. Baseball knowledge had a very strong effect on performance, and WMC had a much smaller effect, but there was no interaction between the two. Similarly, in two poker tasks, in which players had to assess the likelihood of drawing a winning card, and players had to remember hands during a game of poker, both poker knowledge and WMC affected performance, but again there was no interaction between domain knowledge and WMC.

Another study took a different tack. Participants were asked to remember the movements of spaceships flying from planet to planet in the solar system. What they didn’t know was that the spaceships flew in a pattern that matched the way baseball players run around a baseball diamond. They were then given the same task, this time with baseball players running around a diamond. Baseball knowledge only helped performance in the task in which the baseball scenario was explicit — activating baseball knowledge. But activation of domain knowledge had no effect on the influence of WMC.

Although these various studies fail to show an interaction between domain knowledge and WMC, this doesn’t mean that domain knowledge never interacts with basic abilities. The same researchers recently found such an interaction in a geological bedrock mapping task, in which geological structure of a mountainous area had to be inferred. Visuospatial ability predicted performance only at low levels of geological knowledge; geological experts were not affected by their visuospatial abilities. Unfortunately, that study is not yet published, so I don’t know the details. But I assume they mean visuospatial working memory capacity.

It’s possible that general intelligence or WMC are most important during the first stages of skill acquisition (when attention and working memory capacity are so critical), and become far less important once the skill has been mastered.

Similarly, Ericsson has argued that deliberate practice allows performers to circumvent limits on working memory capacity. This is, indeed, related to the point I often make about how to functionally increase your working memory capacity — if you have a great amount of well-organized and readily accessible knowledge on a particular topic, you can effectively expand how much your working memory can hold by keeping a much larger amount of information ‘on standby’ in what has been termed long-term working memory.

Proponents of deliberate practice don’t deny that ‘natural’ abilities have some role, but they restrict it to motivation and general activity levels (plus physical attributes such as height where that is relevant). But surely these would only affect number of hours. Clearly the ability to keep yourself on task, to motivate and discipline yourself, impinges on your ability to keep your practice up. And the general theory makes sense — that if you show some interest in something, such as music or chess, when you’re young, your parents or teachers usually encourage you in that direction; this encouragement and rewards lead you to spend more time and energy in that domain, and if you have enough persistence, enough dedication, then lo and behold, you’ll get better and better. And your parents will say, well, it was obvious from an early age that she was talented that way.

But is it really the case that attributes such as intelligence make no difference? Is it really as simple as “10,000 hours of deliberate practice = expert”? Is it really the case that each hour has the same effect on any one of us?

A survey of 104 chess masters found that, while all the players that became chess masters had practiced at least 3,000 hours, the amount of practice it took to achieve that mastery varied considerably. Although, consistent with the “10,000 hour rule”, average time to achieve mastery was around 11,000 hours, time ranged from 3,016 hours to 23,608 hours. The difference is even more extreme if you only consider individual practice (previous research has pointed to individual practice being of more importance than group practice): a range from 728 hours to 16,120 hours! And some people practiced more than 20,000 hours and still didn't achieve master level.

Moreover, a comparison of titled masters and untitled international players found that the two groups practiced the same amount of hours in the first three years of their serious dedication to chess, and yet there were significant differences in their ratings. Is this because of some subtle difference in the practice, making it less effective? Or is it that some people benefit more from practice?

A comparison of various degrees of expertise in terms of starting age is instructive. While the average age of starting to play seriously was around 18 for players without an international rating, it was around 14 for players with an international rating, and around 11 for masters. But the amount of variability within each group varies considerably. For players without an international rating, the age range within one standard deviation of the mean is over 11 years, but for those with an international rating, FIDE masters, and international masters, the range is only 2-3 years, and for grand masters, the range is less than a year. [These numbers are all approximate, from my eyeball estimates of a bar graph.]

It has been suggested that the younger starting age of chess masters and expert musicians is simply a reflection of the greater amount of practice achieved with a young start. But a contrary suggestion is that there might be other advantages to learning a skill at an early age, reflecting what might be termed a ‘sensitive period’. This study found that the association between skill and starting age was still significant after amount of practice had been taken account of.

Does this have to do with the greater plasticity of young brains? Expertise “grows” brains — in the brain regions involved in that specific domain. Given that younger brains are much more able to create new neurons and new connections, it would hardly be a surprise that it’s easier for them to start building up the dense structures that underlie expertise.

This is surely easier if the young brain is also a young brain that has particular characteristics that are useful for that domain. For music, that might relate to perceptual and motor abilities. In chess, it might have more to do with processing speed, visuospatial ability, and capacious memory.

Several studies have found higher cognitive ability in chess-playing children, but the evidence among adults has been less consistent. This may reflect the growing importance of deliberate practice. (Or perhaps it simply reflects the fact that chess is a difficult skill, for which children, lacking the advantages that longer education and training have given adults, need greater cognitive skills.)

Related to all this, there’s a popular idea that once you get past an IQ of around 120, ‘extra’ IQ really makes no difference. But in a study involving over 2,000 gifted young people, those who scored in the 99.9 percentile on the math SAT at age 13 were eighteen times more likely to go on to earn a doctorate in a STEM discipline (science, technology, engineering, math) compared to those who were only(!) in the 99.1 percentile.

Overall, it seems that while practice can take you a very long way, at the very top, ‘natural’ ability is going to sort the sheep from the goats. And ‘natural’ ability may be most important in the early stages of learning. But what do we mean by ‘natural ability’? Is it simply a matter of unalterable genetics?

Well, palpably not! Because if there’s one thing we now know, it’s that nature and nurture are inextricably entwined. It’s not about genes; it’s about the expression of genes. So let me remind you that aspects of the prenatal, the infant, and the child’s, environment affect that ‘natural’ ability. We know that these environments can affect IQ; the interesting question is what we can do, at each and any of these stages, to improve affect basic processes such as speed of processing, WMC, and inhibitory control. (Although I should say here that I am not a fan of the whole baby-Einstein movement! Nor is there evidence that many of those practices work.)

Bottom line:

• talent still matters
• effective practice is still the most important factor in developing expertise
• individuals vary in how much practice they need
• individual abilities do put limits on what’s achievable (but those limits are probably higher than most people realize).

One of my perennial themes is the importance of practice, and in the context of developing expertise, I have talked of ‘deliberate practice’ (a concept articulated by the well-known expertise researcher K. Anders Ericsson). A new paper in the journal Psychology of Music reports on an interesting study that shows how the attributes of music practice change as music students develop in expertise. Music is probably the most studied domain in expertise research, but I think we can gain some general insight from this analysis. Here’s a summary of the findings.

[Some details about the U.K. study for those interested: the self-report study involved 3,325 children aged 6-19, ranging from beginner to Grade 8 level, covering a variety of instruments, with violin the most common at 28%, and coming from a variety of musical settings: junior conservatoires, youth orchestras, Saturday music schools, comprehensive schools.]

For a start, and unsurprisingly, amount of practice (both in terms of amount each day, and number of days in the week) steadily increases as expertise develops. Interestingly, there is a point where it plateaus (around grade 5-6 music exams) before increasing more sharply (presumably this reflects a ‘sorting the sheep from the goats’ effect — that is, after grade 6, it’s increasingly only the really serious ones that continue).

It should not be overlooked, however, that there was huge variability between individuals in this regard.

More interesting are the changes in the attributes of their practice.

These attributes became less frequent as the players became more expert:

Practicing strategies:

Practicing pieces from beginning to end without stopping

Going back to the beginning after a mistake

Analytic strategies:

Working things out by looking at the music without actually playing it

Trying to find out what a piece sounds like before trying to play it

Analyzing the structure of a piece before learning it

Organization strategies:

Making a list of what to practice

Setting targets for each session.

These attributes became more frequent as the players became more expert:

Practicing strategies:

Practicing small sections;

Getting recordings of a piece that is being learned;

Practicing things slowly;

Knowing when a mistake has been made;

When making a mistake, practicing a section slowly;

When something was difficult playing it over and over again;

Marking things on the part;

Practicing with a metronome;

Recording practice and listening to the tapes;

Analytic strategies:

Identifying difficult sections;

Thinking about how to interpret the music;

Organization strategies:

Doing warm-up exercises;

Starting practice with studies;

Starting practice with scales.

Somewhat surprisingly, levels of concentration and distractability didn’t vary significantly as a function of level of expertise. The researchers suggest that this may reflect the reliance on self-reported data rather than reality, but, also somewhat surprisingly, enjoyment of practice didn’t change as a function of expertise either.

Interestingly (but perhaps not so surprisingly once you think about it), the adoption of systematic practicing strategies followed a U-shaped curve rather than a linear trend. Those who had passed Grade 1 scored relatively high on this, but those who had most recently passed Grade 2 scored more poorly, and those with Grade 3 were worst of all. After that, it begins to pick up again, achieving the same level at Grade 6 as at Grade 1.

Organization of practice, on the other hand, while it varied with level of expertise, showed no systematic relationship (if anything, it declined with expertise! But erratically).

The clearest result was the very steady and steep decline in the use of ineffective strategies. These include:

• Practicing pieces from beginning to end without stopping;
• Going back to the beginning after a mistake;
• Immediate correction of errors.

It should be acknowledged that these strategies might well be appropriate at the beginning, but they are not effective with longer and more complex pieces. It’s suggested that the dip at Grade 3 probably reflects the need to change strategies, and the reluctance of some students to do so.

But of course grade level in itself is only part of the story. Analysis on the basis of how well the students did on their most recent exam (in terms of fail, pass, commended, and highly commended) reveals that organization of practice, and making use of recordings and a metronome, were the most important factors (in addition to the length of time they had been learning).

The strongest predictor of expertise, however, was not using ineffective strategies.

This is a somewhat discouraging conclusion, since it implies that the most important thing to learn (or teach) is what not to do, rather than what to do. But I think a codicil to this is also implicit. Given the time spent practicing (which is steadily increasing with expertise), the reduction in wasting time on ineffective strategies means that, perforce, time is being spent on effective strategies. The fact that no specific strategies can be inequivocally pointed to, suggests that (as I have repeatedly said), effective strategies are specific to the individual.

This doesn’t mean that identifying effective strategies and their parameters is a pointless activity! Far from it. You need to know what strategies work to know what to choose from. But you cannot assume that because something is the best strategy for your best friend, that it is going to be equally good for you.

Notwithstanding this, the adoption of systematic practice strategies was significantly associated with expertise, accounting for the largest chunk of the variance between individuals — some 11%.

Similarly, organization of practice (accounting for nearly 8% of variance), making use of recordings and a metronome (nearly 8% of variance), analytic strategies (over 7% of variance) were important factors in developing expertise in music, and it seems likely that many if not most individuals would benefit from these.

It’s also worth noting that playing straight through the music was the strongest predictor of expertise — as a negative factor.

So what general conclusions can we draw from these findings?

The wide variability in practice amount is worth noting — practice is hugely important, but it’s a mistake to have hard-and-fast rules about the exact number of hours that is appropriate for a given individual.

Learning which strategies are a waste of time is very important (and one that many students don’t learn — witness the continuing popularity of rote repetition as a method of learning).

Organization — in respect of structuring your learning sessions — is perhaps one of those general principles that doesn’t necessarily apply to every individual, and certainly the nature and extent of organization is likely to vary by individual. Nevertheless, given its association with better performance, it is certainly worth trying to find the level of organization that is best for you (or your student). The most important factors in this category were starting practice with scales (for which appropriate counterparts are easily found for other skills being practiced, including language learning, although perhaps less appropriate for other forms of declarative learning), and making a list of what needs to be practiced.

Having expert models/examples/case studies (as appropriate), and appropriate levels of scaffolding, are very helpful (in the case of music, this is instantiated by the use of recordings, both listening to others and self-feedback, and use of a metronome).

Identifying difficult aspects, and dealing with them by tackling them on their own, using a slow and piecemeal process, is usually the most helpful approach. (Of the practice strategies, the most important were practicing sections slowly when having made a mistake, practicing difficult sections over and over again, slow practice, gradually speeding up when learning fast passages, and recognizing errors.)

Preparing for learning is also a generally helpful strategy. In music this is seen in the most effective analytic strategies: trying to find out what a piece sounds like before trying to play it, and getting an overall idea of a piece before practicing it. In declarative learning (as opposed to skill learning), this can be seen in such strategies as reading the Table of Contents, advance organizers and summaries (in the case of textbooks), or doing any required reading before a lecture, and (in both cases) thinking about what you expect to learn from the book or lecture.

On a number of occasions I have reported on studies showing that people with expertise in a specific area show larger gray matter volume in relevant areas of the brain. Thus London taxi drivers (who are required to master “The Knowledge” — all the ways and byways of London) have been found to have an increased volume of gray matter in the anterior hippocampus (involved in spatial navigation). Musicians have greater gray matter volume in Broca’s area.

Other research has found that gray matter increases in specific areas can develop surprisingly quickly. For example, when 19 adults learned to match made-up names against four similar shades of green and blue in five 20-minute sessions over three days, the areas of the brain involved in color vision and perception increased significantly.

This is unusually fast, mind you. Previous research has pointed to the need for training to extend over several weeks. The speed with which these changes were achieved may be because of the type of learning — that of new categories — or because of the training method used. In the first two sessions, participants heard each new word as they regarded the relevant color; had to give the name on seeing the color; had to respond appropriately when a color and name were presented together. In the next three sessions, they continued with the naming and matching tasks. In both cases, immediate feedback was always given.

But how quickly brain regions may re-organize themselves to optimize learning of a specific skill is not the point I want to make here. Some new research suggests our ideas of cortical plasticity need to be tweaked.

In my book on note-taking, I commented on how emphasis of some details (for example by highlighting) improves memory for those details but reduces memory of other details. In the same way, increase of one small region of the brain is at the expense of others. If we have to grow an area for each new skill, how do we keep up our old skills, whose areas might be shrinking to make up for it?

A rat study suggests the answer. While substantial expertise (such as our London cab-drivers and our professional musicians) is apparently underpinned by permanent regional increase, the mere learning of a new skill does not, it seems, require the increase to endure. When rats were trained on an auditory discrimination task, relevant sub-areas of the auditory cortex grew in response to the new discrimination. However, after 35 days the changes had disappeared — but the rats retained their new perceptual abilities.

What’s particularly interesting about this is what the finding tells us about the process of learning. It appears that the expansion of bits of the cortex is not the point of the process; rather it is a means of generating a large and varied set of neurons that are responsive to newly relevant stimuli, from which the most effective circuit can be selected.

It’s a culling process.

This is the same as what happens with children. When they’re young, neurons grow with dizzying profligacy. As they get older, these are pruned. Gone are the neurons that would allow them to speak French with a perfect accent (assuming French isn’t a language in their environment); gone are the neurons that would allow them to finely discriminate the faces of races other than those around them. They’ve had their chance. The environment has been tested; the needs have been winnowed; the paths have been chosen.

In other words, the answer’s not: “more” (neurons/connections); the answer is “best” (neurons/connections). What’s most relevant; what’s needed; what’s the most efficient use of resources.

This process of throwing out lots of trials and seeing what wins, echoes other findings related to successful learning. We learn a skill best by varying our practice in many small ways. We learn best from our failures, not our successes — after all, a success is a stopper. If you succeed without sufficient failure, how will you properly understand why you succeeded? How will you know there aren’t better ways of succeeding? How will you cope with changes in the situation and task?

Mathematics is an area in which this process is perhaps particularly evident. As a student or teacher, you have almost certainly come across a problem that you or the student couldn’t understand when expressed in one way, and maybe several different ways. Until, at some point, for no clear reason, understanding ‘clicks’. And it’s not necessarily that this last way of expressing / representing it is the ‘right’ one — if it had been presented first, it may not have had that effect. The effect is cumulative — the result of trying several different paths and picking something useful from each of them.

In a recent news item I reported on a finding that people who learned new sequences more quickly in later sessions were those whose brains had displayed more 'flexibility' in the earlier sessions — that is, different areas of the brain linked with different regions at different times. And most recently, I reported on a finding that training on a task that challenged working memory increased fluid intelligence in those who improved at the working memory task. But not everyone did. Those who improved were those who found the task challenging but not overwhelming.

Is it too much of a leap to surmise that this response goes hand in hand with flexible processing, with strategizing? Is this what the ‘sweet spot’ in learning really reflects — a level of challenge and enjoyability that stimulates many slightly different attempts? We say ‘Variety is the spice of life’. Perhaps we should add: ‘Variety is the key to learning’.