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Should learning facts by rote be central to education?

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.

Practice counts! So does talent

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).

How to Revise and Practice

References

Campitelli, G., & Gobet F. (2011).  Deliberate Practice. Current Directions in Psychological Science. 20(5), 280 - 285.

Campitelli, G., & Gobet, F. (2008). The role of practice in chess: A longitudinal study. Learning and Individual Differences, 18, 446–458.

Gobet, F., & Campitelli, G. (2007). The role of domain-specific practice, handedness and starting age in chess. Developmental Psychology, 43, 159–172.

Hambrick, D. Z., & Meinz, E. J. (2011). Limits on the Predictive Power of Domain-Specific Experience and Knowledge in Skilled Performance. Current Directions in Psychological Science, 20(5), 275 –279. doi:10.1177/0963721411422061

Hambrick, D.Z., & Engle, R.W. (2002). Effects of domain knowledge, working memory capacity and age on cognitive performance: An investigation of the knowledge-is-power hypothesis. Cognitive Psychology, 44, 339–387.

Hambrick, D.Z., Libarkin, J.C., Petcovic, H.L., Baker, K.M., Elkins, J., Callahan, C., et al. (2011). A test of the circumvention-of-limits hypothesis in geological bedrock mapping. Journal of Experimental Psychology: General, Published online Oct 17, 2011.

Hambrick, D.Z., & Oswald, F.L. (2005). Does domain knowledge moderate involvement of working memory capacity in higher level cognition? A test of three models. Journal of Memory and Language, 52, 377–397.

Meinz, E. J., & Hambrick, D. Z. (2010). Deliberate Practice Is Necessary but Not Sufficient to Explain Individual Differences in Piano Sight-Reading Skill. Psychological Science, 21(7), 914–919. doi:10.1177/0956797610373933

 

Attributes of effective practice

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.

How to Revise and Practice

References

Hallam, S., Rinta, T., Varvarigou, M., Creech, a., Papageorgi, I., Gomes, T., & Lanipekun, J. (2012). The development of practising strategies in young people. Psychology of Music, 40(5), 652–680. doi:10.1177/0305735612443868

Variety is the key to learning

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’.

How to Revise and Practice

References

Kwok, V., Niu Z., Kay P., Zhou K., Mo L., Jin Z., et al. (2011). Learning new color names produces rapid increase in gray matter in the intact adult human cortex. Proceedings of the National Academy of Sciences.

Working memory, expertise & retrieval structures

In a 1987 experiment (1), readers were presented with a text that included one or other of these sentences:

or

Both texts went on to say:

After reading the text, readers were asked if the word sweatshirt had appeared in the story. Now here is the fascinating and highly significant result: those who read that John had put on a sweatshirt responded “yes” more quickly than those who had read that he had taken off his sweatshirt.

Why is this so significant? Because it tells us something important about the reading process, at least in the minds of skilled readers. They construct mental models. If it was just a matter of the mechanical lower-order processing of letters and words, why would there be a difference in responses? Neither text was odd — John could as well have put on a sweatshirt before going out for a jog as taken it off — so there shouldn’t be a surprise effect. So what is it? Why is the word sweatshirt not as tightly / strongly linked in the second case as it is in the first? If they were purely textbase links (links generated by the textbase itself), the links should be equivalent. The difference in responses implies that the readers are making links with something outside the textbase, with a mental model.

Mental models, or as they are sometimes called in this context, situation models, are sometimes represented as lists of propositions, but in most cases it seems likely that they are actually analogue in nature. Thus the real world should be better represented by the situation model than by the text. Moreover, a spatial situation model will be similar in many ways to an image, with all the advantages that that entails.

All of this has relevance to two very important concepts: working memory and expertise.

Now, I’m always talking about working memory. This time I want to discuss not so much the limited attentional capacity that is what we chiefly mean by working memory, but another, more theoretical concept: the idea of long-term working memory.

Think about reading. To make sense of the text you need to remember what’s gone before — this is why working memory is so important for the reading process. But we know how limited working memory is; it can only hold a very small amount — is it really possible to hold all the information we need to make sense of what we’re reading? Shouldn’t there be constant delays as we access needed information from long-term memory? But there aren’t.

It’s suggested that the answer lies in the use of long-term working memory, a retrieval structure that keeps a network of linked propositions readily available.

Think about when you are studying / reading a difficult text in a subject you know well. Compare this to studying a difficult text in a subject you don’t know well. In the latter case, you may have to painfully backtrack, checking earlier statements, trying to remember what was said before, trying to relate what you are reading to things you already know. In the former case, you seem to have a vastly expanded amount of readily accessible relevant information, from the text itself and from your long-term memory.

The connection between long-term working memory and expertise is obvious. And expertise has already been conceptualised in terms of retrieval structures (see for example my article on expertise). In other words, you can increase your working memory in a particular domain by developing expertise, and the shortest route to developing expertise is to concentrate on building effective retrieval structures.

One of the areas where this is particularly crucial is that of reading scientific texts. Now we all know that scientific texts are much harder to process than, for example, stories. And there are several reasons for that. One is the issue of language: any science has its own technical vocabulary and you won't get far without knowing it. But another reason, far less obvious to the untutored, concerns the differences in structure — what may be termed differences of genre.

Now it might seem self-evident that stories are far simpler than science, than any non-fiction texts, and indeed a major distinction is usually made between narrative texts and expository texts, but it’s rather like the issue of faces and other objects. Are we specially good at faces because we're 'designed' to be (i.e., we have special 'expert' modules for processing faces)? Or is it simply that we have an awful lot of practice at it, because we are programmed to focus on human faces almost as soon as we are born?

In the same way, we are programmed for stories: right from infancy, we are told stories, we pay attention to stories, we enjoy stories. Stories have a particular structure (and within the broad structure, a set of sub-structures), and we have a lot of practice in that structure. Expository texts, on the other hand, don't get nearly the same level of practice, to the extent that many college students do not know how to handle them — and more importantly, don't even realize that that is what they're missing: a retrieval structure for the type of text they're studying.

References

Glenberg, A.M., Meyer, M. & Lindem, K. 1987. Mental models contribute to foregrounding during text comprehension. Journal of Memory and Language, 26, 69-83.

Acquiring expertise through deliberate practice

K. Anders Ericsson, the guru of research into expertise, makes a very convincing case for the absolutely critical importance of what he terms “deliberate practice”, and the minimal role of what is commonly termed “talent”. I have written about this question of talent and also about the principles of expertise. Here I would like to talk briefly about Ericsson’s concept of deliberate practice.

Most people, he suggests, spend very little (if any) time engaging in deliberate practice even in those areas in which they wish to achieve some level of expertise. Experts, on the other hand, only achieve their expertise after several years (at least ten, in general) of maintaining high levels of regular deliberate practice.

What distinguishes deliberate practice from less productive practice? Ericsson suggests several factors are of importance:

The acquisition of expert performance needs to be broken down into a sequence of attainable training tasks.

  • Each of these tasks requires a well-defined goal.
  • Feedback for each step must be provided.
  • Repetition is needed — but that repetition is not simple; rather the student should be provided with opportunities that gradually refine his performance.
  • Attention is absolutely necessary — it is not enough to simply mechanically “go through the motions”.
  • The aspiring expert must constantly and attentively monitor her progress, adjusting and correcting her performance as required.

For these last two reasons, deliberate practice is limited in duration. Whatever the particular field of endeavor, there seems a remarkable consistency in the habits of elite performers that suggests 4 to 5 hours of deliberate practice per day is the maximum that can be maintained. This, of course, cannot all be done at one time without resting. When the concentration flags, it is time to rest — this most probably is after about an hour. But the student must train himself up to this level; the length of time he can concentrate will increase with practice.

Higher levels of concentration are often associated with longer sleeping, in particular in the form of day-time naps.

Not all practice is, or should be, deliberate practice. Deliberate practice is effortful and rarely enjoyable. Some practice is however, what Ericsson terms “playful interaction”, and presumably provides a motivational force — it should not be despised!

In general, experts reduce the amount of time they spend on deliberate practice as they age. It seems that, once a certain level of expertise has been achieved, it is not necessary to force yourself to continue the practice at the same level in order to maintain your skill. However, as long as you wish to improve, a high level of deliberate practice is required.

This article first appeared in the Memory Key Newsletter for November 2005

How to Revise and Practice

References

Ericsson, K.A. 1996. The acquisition of expert performance: An introduction to some of the issues. In K. Anders Ericsson (ed.), The Road to Excellence: The acquisition of expert performance in the arts and sciences, sports, and games. Mahwah, NJ: Lawrence Erlbaum.

Successful Transfer

  • Transfer refers to the extent to which learning is applied to new contexts.
  • Transfer is facilitated by:
    • understanding
    • instruction in the abstract principles involved
    • demonstration of contrasting cases
    • explicit instruction of transfer implications
    • sufficient time
  • Learning for transfer requires more time and effort in the short term, but saves time in the long term.

Transfer refers to the ability to extend (transfer) learning from one situation to another. For example, knowing how to play the piano doesn’t (I assume) help you play the tuba, but presumably is a great help if you decide to take up the harpsichord or organ. Similarly, I’ve found my knowledge of Latin and French a great help in learning Spanish, but no help at all in learning Japanese.

Transfer, however, doesn’t have to be positive. Your existing knowledge can hinder, rather than help, new learning. In such a situation we talk about negative transfer. We’ve all experienced it. At the moment I’m experiencing it with my typing -- I've converted my standard QWERTY keyboard to a Dvorak one (you can hear about this experience in my podcast, if you're interested).

Teachers and students do generally hope that learning will transfer to new contexts. If we had to learn how to deal with every single possible situation we might come across, we’d never be able to cope with the world! So in that sense, transfer is at the heart of successful learning (and presumably the ability to transfer new learning is closely tied to that elusive concept, intelligence).

Here’s an example of transfer (or lack of it) in the classroom.

A student can be taught the formula for finding the area of a parallelogram, and will then be capable of finding the area of any parallelogram. However, if given different geometric figures, they won’t be able to apply their knowledge to calculate the area, because the formula they have memorized applies only to one specific figure — the parallelogram.

However, if the student is instead encouraged to work out how to calculate the area of a parallelogram by using the structural relationships in the parallelogram (for example, by rearranging it into a rectangle by moving one triangle from one end to the other), then they are much more likely to be able to use that experience to work out the areas of a different figure.

This example gives a clue to one important way of encouraging transfer: abstraction. If you only experience a very specific example of a problem, you are much less likely to be able to apply that learning to other problems. If, on the other hand, you are also told the abstract principles involved in the problem, you are much more likely to be able to use that learning in a variety of situations. [example taken from How People Learn]

Clearly there is a strong relationship between understanding and transfer. If you understand what you are doing, you are much more likely to be able to transfer that learning to problems and situations you haven’t encountered before — which is why transfer tests are much better tests of understanding than standard recall tests.

That is probably more obvious for knowledge such as scientific knowledge than it is for skill learning, so let me tell you about a classic study [1]. In this study, children were given practice in throwing darts at an underwater object. Some of the children were also instructed in how light is refracted in water, and how this produces misleading information regarding the location of objects under water. While all the children did equally well on the task they practiced on — throwing darts at an object 12 inches under water — the children who had been given the instruction did much better when the target was moved to a place only 4 inches under water.

Understanding is helped by contrasting cases. Which features of a concept or situation are important is often only evident when you can see different but related concepts. For example, you can’t fully understand what an artery is unless you contrast it with a vein; the concept of recognition memory is better understood if contrasted with recall memory.

Transfer is also helped if transfer implications are explicitly pointed out during learning, and if problems are presented in several contexts. One way of doing that is if you use “what-ifs” to expand your experience. That is, having solved a problem, you ask “What if I changed this part of the problem?”

All of this points to another requirement for successful transfer — time. Successful, “deep”, learning requires much more time than shallow rote learning. On the other hand, because it can apply to a much wider range of problems and situations, is much less easily forgotten, and facilitates other learning, it saves a lot of time in the long run!

References
  • National Research Council, 1999. How People Learn: Brain, Mind, Experience, and School. Washington, D.C.: National Academy Press. https://www.nap.edu

1. Scholckow & Judd, described in Judd, C.H. 1908. The relation of special training to general intelligence. Educational Review, 36, 28-42.

Context & the conditionalization of knowledge

Context is absolutely critical to successful communication. Think of the common experience of being a stranger at a family gathering or a meeting of close friends. Even familiar words and phrases may take on a different or additional meaning, among people who have a shared history. Many jokes and comments will be completely unintelligible, though you all speak the same language.

American anthropologist Edward Hall makes a useful distinction between ‘High context’ and ‘Low context’ communications. Your family gathering would be an example of a high context situation. In this setting, much of the meaning is carried in the speakers, their relationships, their knowledge of each other. In a low context situation, on the other hand, most of the meaning is carried in the actual words.

Part of the problem with email, as we all recognize, is that the context is so lacking, and the burden lies so heavily on the words themselves.

The importance of context for comprehension has, of course, profound implications for learning and memory.

I was reminded of this just the other day. I’m a fan of a TV program called NCIS. I only discovered it, however, at the beginning of the third season. After I’d watched it for some weeks, I purchased the DVDs of the earlier seasons. Most recently, I bought the DVD of season 3, which I had, of course, seen on TV. Watching the first episode of that season, which was the first episode of NCIS I ever saw, I was surprised to hear a line which I had no memory of, that was freighted with significance and led me to a much deeper understanding of the relationship between two of the characters — but which had meant absolutely nothing to me when I originally saw it, ignorant as I was of any of the characters and the back story.

The revelation meant nothing to me as a novice to the program, and so I didn’t remember it, but it meant everything to me as (dare I say it?) an expert.

Context is such a slippery word; so hard to define and pin down. But I think it’s fair to say that the difference between the novice and the expert rests on this concept. When an expert is confronted with a piece of information from her area of expertise, she knows what it means and where it belongs — even if the information is new to her. Because of this, she can acquire new information much more easily than a novice. But this advantage applies only in the expert’s area of expertise.

To take another example from the frivolous world of popular culture, a British study of fans of the long-running radio soap opera The Archers were given one of two imaginary scripts to read. One story was representative of the normal events in The Archers (a visit to a livestock market); the other was atypical (a visit to a boat show). These experts were able to remember many more details of the typical, market story than a group of subjects who knew little about the soap opera, but were no better at remembering details for the atypical story. Most importantly, this occurred even though the two stories shared many parallel features and most of the questions (and answers) used to assess their memory were the same. This indicates the specificity of expert knowledge.

Part of the advantage experts have is thought to rest on the ‘conditionalization’ of knowledge. That is, experts’ knowledge includes a specification of the contexts in which it is relevant.

It is surprising to many, this idea that it is not necessarily a lack of knowledge that is the problem — that people often have relevant knowledge and don’t apply it. In reading, for example, readers often don’t make inferences that they are perfectly capable of making, on the knowledge they have, unless the inferences are absolutely demanded to make sense of the text.

Another example comes from the making of analogies. I discuss this in my workbook on taking notes. Here’s a brief extract:

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Rutherford’s comparison of the atom to the solar system gave us a means to understand the atom. The story goes that Newton ‘discovered’ gravity when an apple fell on his head — because of the comparison he made, realizing that the motion of an apple falling from a tree was in some sense like the motion of the planets. These are comparisons called analogies, and analogy has been shown to be a powerful tool for learning.

But the problem with analogies is that we have trouble coming up with them.

Generally, when we make analogies, we use an example we know well to help us understand something we don’t understand very well. This means that we need to retrieve from memory an appropriate example. But this is clearly a difficult task; people frequently fail to make appropriate connections — even, surprisingly, when an appropriate connection has recently come their way. In a study where people were given a problem to solve after reading a story in which an analogous problem was solved, 80% didn’t think of using the story to solve the problem until the analogy was pointed out to them.

It’s thought that retrieving an appropriate analogy is so difficult because of the way we file information in memory. Certainly similarity is an important attribute in our filed memories, but it’s not the same sort of similarity that governs analogies. The similarity that helps us retrieve memories is a surface similarity — a similarity of features and context. But analogies run on a deeper similarity — a similarity of structure, of relations between objects. This will only be encoded if you have multiple examples (at least more than one) and make an explicit effort to note such relations.

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The conditionalization of knowledge is of course related to the problem of transfer. Transfer refers to the ability to extend (transfer) learning from one situation to another (read more about it here) . Transfer is frequently used as a measure of successful learning. It’s all very well to know that 399-(399*0.1) = 359.1, but how far can you be said to understand it — how much use is it — if you can’t work out how much a $3.99 item will cost you if you have a 10% discount? (In fact, the asymmetry generally works the other way: many people are skilled at working out such purchase calculations, but fall apart when the problem is transferred to a purely numerical problem).

Transfer is affected by the context in which the information was originally acquired — obviously transfer is particularly problematic if you learn the material in a single context — and this is partly where the experts achieve their conditionalization: because, spending so much time with their subject they are more likely to come across the same information in a variety of contexts. But the more important source is probably the level of abstraction at which experts can operate (see my article on transfer for examples of how transfer is facilitated if the information is framed at a higher level of abstraction).

In those with existing expertise, an abstract framework is already in place. When an expert is confronted by new information, they automatically try and fit it into their existing framework. Whether it is consistent or inconsistent with what is already known doesn’t really matter — either way it will be more memorable than information that makes no deep or important connections to familiar material.

Let’s return to this idea of high and low context. Hall was talking about communications, in the context of different cultures (interestingly, he found cultures varied in the degree to which they were context-bound), but the basic concept is a useful one in other contexts. It is helpful to consider, when approaching a topic, either as student or teacher, the degree to which understanding requires implicit knowledge. A high context topic might be thought of as one that assumes a lot of prior knowledge, that assumes a knowledge of deeper structure, that is difficult to explain in words alone. A low context topic might be thought of as one that can be clearly and simply expressed, that can largely stand alone. Learning the basics of a language — how to conjugate a verb; some simple words and phrases — might be thought of as a low context topic, although clearly mastery of a language requires the complex and diverse building up of experiences that signifies a high context topic (and also clearly, some languages will be more ‘high context’ than others).

There is nothing particularly profound about this distinction, but an awareness of the ‘contextual degree’ of a topic or situation, is helpful for students, teachers, and anyone involved in trying to communicate with another human being (or indeed, computer!). It’s also helpful to be aware that high context situations require much more expertise than low context ones.

This article first appeared as "Context, communication & learning" in the Memory Key Newsletter for April 2007

References

Reeve, D.K. & Aggleton, J.P. 1998. On the specificity of expert knowledge about a soap opera: an everyday story of farming folk. Applied Cognitive Psychology, 12 (1), 35-42.

About expert knowledge

Principles of expert knowledge

  • Principle 1: Experts are sensitive to patterns of meaningful information
  • Principle 2: Expert knowledge is highly organized in deeply integrated schemas.
  • Principle 3: Expert knowledge is readily accessible when needed because it contains information about when it will be useful.

Do experts simply know "more" than others, or is there something qualitatively different about an expert's knowledge compared to the knowledge of a non-expert?

While most of us are not aiming for an expert's knowledge in many of the subjects we study or learn about, it is worthwhile considering the ways in which expert knowledge is different, because it shows us how to learn, and teach, more effectively.

Experts are sensitive to patterns of meaningful information

A basic principle of perception is that it depends on the observer. What is green to you may be teal to me; a floppy disk to me may be a curious square of hard plastic to you. The observer always sees the world through her own existing knowledge.

An essential part of the difference between an expert and a novice can be seen in terms of this principle. A configuration of chess pieces on a board, seen briefly, will be bewildering and hard to remember for someone with no knowledge of chess, and even for someone with some experience of the game. But to a chess master, the configuration will be easily grasped, and easily remembered.

When chess pieces are placed randomly on a board, the chess master is no better than the novice at remembering briefly seen configurations. This is because the configuration is not meaningful. After tens of thousands of hours of playing chess, of studying the games of other masters, of memorizing patterns of moves, the master has hundreds of stored patterns in his memory. When he sees a configuration of pieces, he breaks it into meaningful elements that are related by an underlying strategy. Thus, while the novice would have to try and remember every single piece and its absolute or relative position on the board, the master only has to remember a few “chunks”.

The master can do this because he has a highly organized structure of knowledge relating to this domain. (It’s worth noting that expertise is highly specific to a domain of knowledge; a chess master will be no better than anyone at remembering, say, a shopping list.)

Expert knowledge is highly organized in deeply integrated schemas.

This sensitivity is thought to grow out of the deep conceptual schemas that experts develop in their area of expertise.

A schema is an organized body of knowledge that enables the user to understand a situation of set of facts. Schema theories include the idea of “scripts”, which help us deal with events. Thus, we are supposed to have a “restaurant script”, which we have developed from our various experiences with restaurants, and which tells us what to expect in a restaurant. Such a script would include the various activities that typically take place in a restaurant (being seated; ordering; eating; paying the bill, etc), and the various people we are likely to interact with (e.g., waiter/waitress; cashier).

Similarly, when we read or hear stories (and many aspects of our conversations with each other may be understood in terms of narratives, not simply those we read in books), we are assisted in our interpretation by “story schemas” or “story grammars”.

A number of studies have shown that memory is better for stories than other types of text; that we are inclined to remember events that didn’t happen if their happening is part of our mental script; that we find it hard to remember stories that we don’t understand, because they don’t fit into our scripts.

Schemas provide a basis for:

  • Assimilating information
  • Making inferences
  • Deciding which elements to attend to
  • Help search in an orderly sequence
  • Summarizing
  • Helping you to reconstruct a memory in which many details have been lost

(following Anderson 1984)

A schema then is a body of knowledge that provides a framework for understanding, for encoding new knowledge, for retrieving information. By having this framework, the expert can quickly understand and acquire new knowledge in her area of expertise, and can quickly find the relevant bits of knowledge when called on.

How is an expert schema different from a beginner’s one?

Building schemas is something we do naturally. How is an expert schema different from a beginner’s one?

An expert’s schema is based on deep principles; a beginner tends to organize her growing information around surface principles.

For example, in physics, when solving a problem, an expert usually looks first for the principle or law that is applicable to the problem (e.g., the first law of thermodynamics), then works out how one could apply this law to the problem. An experienced novice, on the other hand, tends to search for appropriate equations, then works out how to manipulate these equations (1). Similarly, when asked to sort problems according to the approach that could be used to solve them, experts group the problems in terms of the principles that can be used, while the novices sort them according to surface characteristics (such as “problems that contain inclined planes”) (2).

The different structure of expert knowledge is also revealed through the pattern of search times. Novices retrieve information at a rate that suggests a sequential search of information, as if they are methodically going down a list. Expert knowledge appears to be organized in a more conceptual manner, with information categorized in different chunks (mini-networks) which are organized around a central “deep” idea, and which have many connections to other chunks in the larger network.

These mini-networks, and the rich interconnections between them, help the expert look in the right place. One of the characteristics that differentiates experts from novices is the speed and ease with which experts retrieve the particular knowledge that is relevant to the problem in hand. Experts’ knowledge is said to be “conditionalized”, that is, knowledge about something includes knowledge as to the contexts in which that knowledge will be useful.

Expert knowledge contains information about when it will be useful.

Conditionalized knowledge is contrasted with “inert” knowledge. This concept is best illustrated by an example.

Gick and Holyoake (1980) presented college students with the following passage, which they were instructed to memorize:

After students had demonstrated their recall of this passage, they were asked to solve the following problem:

Although the students had recently memorized the military example, only 20% of them saw its relevance to the medical problem and successfully applied its lesson. Most of the students were unable to solve the problem until given the explicit hint that the passage they had learned contained information they could use. For them, the knowledge they had acquired was inert. However, when the analogy was pointed out to them, 90% of them were able to apply the principle successfully.

Much of the information “learned” in school is inert. A compelling demonstration of this comes from studies conducted by Perfetto, Bransford and Franks (1983), in which college students were given a number of “insight” problems, such as:

Some students were given clues to help them solve these problems:

These clues were given before the students were shown the problems. Some of the students given clues were also explicitly advised that the clues would help them solve the problems. They performed very well. Other students however, were not prompted to use the clues they had been given, and they performed as poorly as those students who weren’t given clues.

The poor performance of those students who were given clues but not prompted to use them surprised the authors of the study, because the clues were so obviously relevant to the problems, but it provides a compelling demonstration of inert knowledge.

The ability of students to apply relevant knowledge in new contexts tends to be grossly over-estimated by instructors. Most assume that it will happen “naturally”, but what this research tells us is that the conditionalization of knowledge is something that happens quite a long way down the track, and if students are to be able to use the information they have learned, they need help in understanding where, when and how to use new knowledge.

Differences between experts and novices:

  • experts have more categories
  • experts have richer categories
  • experts’ categories are based on deeper principles
  • novices’ categories emphasize surface similarities3

 

References
  • Anderson, R.C. 1984. Role of reader's schema in comprehension, learning and memory. In R. Anderson, J. Osborn, & R. Tierney (eds), Learning to read in American schools: Basal readers and content texts. Hillsdale, NJ: Lawrence Erlbaum Associates.
  • Bransford, J.D., Brown, A.L. & Cocking, R.R. (eds.) 1999. How people learn: Brain, Mind, Experience, and School. Washington, DC: National Academy Press.
  • Bransford, J.D., Stein, B.S., Shelton, T.S., & Owings, R.A. 1981. Cognition and adaptation: The importance of learning to learn. In J. Harvey (ed.), Cognition, social behavior and the environment. Hillsdale, NJ: Erlbaum.
  • Bransford, J.D., Stein, B.S., Vye, N.J., Franks, J.J., Auble, P.M., Mezynski, K.J. & Perfetto, G.A. 1982. Differences in approaches to learning: an overview. Journal of Experimental Psychology: General, 111, 390-398.
  • Gick, M.L. & Holyoake, K.J. 1980. Analogical problem solving. Cognitive Psychology, 12, 306-355.
  • Perfetto, G.A., Bransford, J.D. & Franks, J.J. 1983. Constraints on access in a problem solving context. Memory & Cognition, 11, 24-31.

1. Chi, MTH, Feltovich, PJ, & Glaser, R. 1981. Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121-152.

Larkin, JH, 1981. Enriching formal knowledge: A model for learning to solve problems in physics. In JR Anderson (ed), Cognitive skills and their acquisition. Hillsdale, NJ: Erlbaum.

1983. The role of problem representation in physics. In D. Gentner & A.L. Stevens (eds), Mental models. Hillsdale, NJ: Erlbaum.

2. Chi et al 1981

3. Taken from The Memory Key.