Learning, Models and Other Tricks

courtesy Wordle

Have you ever thought why models are so often used in teaching and learning? They formed part of what used to be called teaching aids, more recently termed learning resources.

In the high school that I attended, we had a resource room. Senior students were given access to this room for study purposes and when helping a teacher prepare lessons. The day I was permitted into the resource room I could not stop thinking about the amazing things I saw stored there. It was only years later when I became a teacher that I realised why my teachers had guarded the room and its content so vigilantly and with such reverence.

Effecting learning

I recently ran my blog through Wordle. The result at the head of this post shows what has obviously been the most important thing on my mind when I wrote recent posts. It also made me think that despite my familiarity with the word, learning and how it is brought about is not always looked upon as common ground when discussed in depth with others in the field. Tony Karrer’s recent post has hosted a debate on ways of learning and associated ideas, a follow up to his post on Learning Goals.

Learning by association

The renowned scientist, Dr Jacob Bronowski, who was also an intellectual, expert code-cracker, mathematician and author, gave celebrated lectures on BBC TV in the 1960s. During one of his lectures, he demonstrated how a feat of memory through association could be performed, and he displayed this both by his own memory acumen and with the help of a trained member of his audience.

The 'trick' involves some preparation. A list of (say) 20 commonly known items, personal to the memoriser, is committed thoroughly to memory so that not only the list order can be recalled but also the position in that list of any of its component items.

Once this is accomplished, the memoriser is then shown a series of up to 20 new items in sequence, such as cards drawn randomly from a full pack of playing cards. For each new item shown, the memoriser simply ticks it in the mind by association with each of the previously memorised personal items in sequence.

When complete, the memoriser is asked to name an item by its number in the list and knows what the item is. As splendidly amazing as this act appears when first seen, it is based on learning by association. While it is true that this method has limited application to some learning, it shows how rapid and facile the mind and memory can be in the simple act of learning content.

Models and their place in learning

Learning by association is not a new idea. It is the working part of how learning is assisted when a model is used. It is called upon when a map is used in learning geography or an elaborate digital model is used in learning the function of interior parts of a plant cell. Models work by drawing on previous experiences and learnt ideas, and relating to these when learning something new.

Often the simplest models are the best, even when they may relate to a complicated theory, concept or phenomenon. Though elaborate models may look fascinating, they rarely convey useful learning to the beginner. Learners need to be already familiar with parts of the model itself. As intricate and captivating as the Watson, Crick and Wilkins model of DNA may be, it conveys nothing about its chemistry to those who have never learnt elementary Chemistry.

courtesy NASA

Learning and memory

It is well known that before any skill can be acquired learning a second language, knowledge of vocabulary is fundamental. It is also becoming recognised by educators that the language of a subject, and knowledge of vocabulary in particular, is required for the learner to be able to think in terms of the subject and also to converse about it with others.

Without the vocabulary of a hitherto unknown subject it is impossible for a beginner to acquire any useful subject skill. Motor skills have similar fundamental elements when it comes to the first time learner picking up the ropes of a new skill.

There is no evidence to suggest that the memory required while learning and remembering a vocabulary (content learnt by association) is fundamentally different from that needed to learn and remember the higher skills. When concepts or skills to be learnt become more complicated as the learner progresses, a stage is invariably reached where the learner has to work at them to make the leaps. Once made, these too can be learnt and remembered. Higher thinking skills are required to be learnt to continue to progress. This is often forgotten by the expert who is addressing learning in the subject, the so-called cognitive apprenticeship theory.

Learning is recursively elaborate

A concept, idea or formula learnt in one discipline can find a use in another that’s seemingly unrelated. A child who recognises the relationship between similar patterns of learning in two distinct disciplines makes a cognitive leap. Intelligence is intimately linked with the ability to connect patterns in this way.

The animated equations depicted here are of elementary algebra. They show how the same basic tactic can apply to two distinct areas of learning in Science. A child who masters the simple algebra relevant to this also learns the skill to work within an unbelievably large number of its applications. The scope for it is huge, and it finds use in many common everyday tasks, from a simple calculation in an expenses return to estimating how long a car journey is likely to take.

This recursive application of algebra is by no means a unique feature in learning, for there are many millions of patterns that cross seemingly unrelated disciplines. The ability to understand and recognise these patterns is one that is familiar to the compilers of intelligence tests. It was believed that such tests, according to various scales, could be used to classify and measure cognitive ability. Though there may well be some merit in the idea of pattern recognition being linked to cognitive ability, the means created to measure this fell short of something useful, never mind fairness.

Cutting corners

The recursive nature of learning often compels us to take short cuts that sometimes lead to a misunderstanding that a concept has been learnt. It may even suggest that it doesn’t need to be learnt; the word content springs to mind. I’ll use an example from elementary Chemistry to demonstrate this.

Finding the chemical formula for a simple compound, such as aluminium oxide, can be done a number of ways, all of which have the potential to yield the same answer:

1. use Google - provided the student can apply the search routine and recognises a reputable site when one is brought up, the correct formula might be found,
   
   
2. recall the chemical symbols for the elements oxygen and aluminium, and that oxygen has a valency of 2 and aluminium has a valency of 3, then apply the recalled rule for writing correct chemical formulae,
   
   
3. referring to the periodic table of the elements (or just simply knowing it) the student uses the atomic number of oxygen to write the electron configuration of its atom - having used a similar process to write the electron configuration of aluminium, the student may determine the common valencies of both elements and then apply method 2.
   

Understanding how the formula is found is part and parcel of understanding so-called ‘valency theory’ in Chemistry. Learners who can Google the formula for any simple chemical compound don’t really need to know much chemistry. While method 2 barely touches on some of the principles involved in valency theory, knowledge of how to use method 3 takes the learner closer towards how to apply that theory and to understanding why chemical compounds form between elements in the first place.

Most students who go on to study Chemistry in senior school will learn both methods 2 and 3. They may become so proficient at writing chemical formula using method 2 that they can write several correct formula in the time it takes another student to type in the Google search criteria in following method 1, let alone what’s needed to choose a suitable trustworthy site to browse.

All of the above methods for finding a formula can be learnt and each method has its merit depending on the need. But to say that all a learner needs to know is how to use an Internet search engine to find the formula of a simple chemical compound is not actually learning any Chemistry. Yet this is often used as an argument for not teaching content. There comes a time when the learner just has to face learning some content, and this applies to many distinct disciplines.

Models can be conflicting

One of the many curious phenomena studied in secondary school Science is that of the behaviour of light. This well studied topic requires a series of models to explain how light can behave in different circumstances. A feature of two celebrated models for light, that of the particle or photon and that of the wave, is that neither model explains all the observable properties of light.

While both these models can be used to explain and predict the behaviour of other phenomena not directly related to light, it takes an enlightened learner to understand that they are just models. This peculiarly useful awareness is a higher learning skill. It allows the learner who is very familiar with models used within a discipline to understand their limitations and permits recognition of when a particular model is applicable and when it is not. Recognising that a model and the phenomenon it mirrors are not the same things is extremely important in Science.

The lesser analogy

Unlike the model, an analogy is not trying to depict in any way how the thing or concept exists. It is a direct mapping between unrelated elements of one idea and another. There is no need for there to be any true resemblance between the thing or concept and its parallel used in analogy.

Unfortunately, analogies are often used erroneously as models. For the learner, the analogy is far more involved, the behaviour of one thing being considered while thinking of the associated behaviour of another.

It calls for the most use of imagination, being a parallelism that’s left mainly up to the ingenuity of the thinker. As they are necessarily specific, analogies are severely limited in their broader application.

Facile in association

The mind seems to be facile in the way it can link seemingly unrelated things and learn by association. Perhaps this is why the model enjoys its time honoured place in learning at all levels, for it is so successful.

Models enable a direct mapping of what is seen onto what is being learnt. Good models permit this to be assimilated easily by the learner so that they can apply what’s learnt. Through pattern recognition learners can find further application of what they have learnt.

The enlightened learner, who also understands the difference between the model and the idea, concept or phenomenon that it is mirroring, can flip between models used to reflect these. Introducing the learner to this important difference between the model and what it reflects is the province of good teaching.

Approximate Aims and Nebulous Targets

Tony Karrer, in his recent post Adoption Ideas, brought our attention to the post, Why Doing Things Half Right Gives You the Best Results.

In it, Peter Bregman posits the idea that organisations should aim for imperfection. “I'm not suggesting you settle for imperfect. I'm telling you to shoot for it”, was how he put his advice. Bergman heads a change management firm.

Lyotard's Postmodernism:

In reading Bregman’s post, I was reminded of a conversation I had with colleagues about postmodernism. Few were really aware of what it was. I’d only been made aware fairly recently of the existence of the term when, in 2001, Derek Wenmoth advised me and my teaching colleagues to become more familiar with postmodernism, its existence in society, in homes, in schools and what it meant to our relationship with our students. At that time, I looked on it as a way of thinking that was quite foreign to me. In many ways I still do.

Bregman’s position has strong elements of postmodernism as claimed by Jean-François Lyotard in that the sequential detail and reasons for such detail within the structure of an established process is eschewed. The so-called ‘Grand Narrative’ is cast aside. By its function as a story, it tends to cloud anomalies and unevenness that are naturally present in any community or practice, and so stands in the way of progress.

In doing this, postmodernism instead favours the situational event, dealing with each temporarily as it occurs. There is no need or call for reasoning, or what could possibly be universally acceptable or believed and neither is stability a required criterion.

Analogies:

But the recent conversation brought to mind analogies that helped me when I had to get my head round ways of thinking, strategies and developments that transcended the logic I was more familiar with from the twentieth century. I’ve often used these analogies, almost by way of self protection, in order to avoid the anxt of constantly trying to understand why things were happening the way they were.

The contexts for these analogies are many and varied, and it may well be inappropriate for me to tie them to one specific example; sufficient to refer to Peter Bergman’s contexts.

The analogies are to do with achieving a working success, whether it is of a small project or a larger one, such as a restructuring within an organisation, or any part of these that develops sufficient for there to be a potentially measurable outcome. Having been involved in many different projects that fit this description since the beginning of this century, I feel that, if nothing else, I have some expertise in observing the initiation, development and eventual outcomes of these.

Here is a description of the analogies, comparing the traditional approach (modernism) with postmodernism.

The Project:

To launch a projectile in order to reach a goal called the target.

Traditional:

The target is defined - its position and range established. The launching device is chosen and a suitable projectile with means for propelling is selected according to the target range and conditions.

Past experience with the same or similar equipment is called upon. Some allowance for wind conditions is made. Adjustments to sites are made for the range if necessary and the projectile is launched at the target.

Following successfully meeting the target, or otherwise, there may be some decision made as to how the trajectory may be improved in order to hit the target more accurately in future. What evolves from this is what may be termed a ‘sure fire’ process or strategy.

Postmodern:

The target is defined, though its position and range may not be too definite. The launching device is selected and a projectile launched without a great deal of time spent considering such parameters as direction of aim, range or conditions.

All these minor matters are decided upon and adjusted during the trajectory of the projectile, in much the same way as the Apollo 11 Command Module was navigated in 1969.

The target is then brought more into focus. Provided there is sufficient time for trajectory adjustment before the projectile travels out of range, the target is decided upon. With any luck, the target is met.

There are no repercussions. If the target is met the project is successful. If the target is not met, a new project and strategy to hit a new target is discussed at some later date.

The Simplest Symbolic Language

Finger Painting by Hannah Dear - age 5

Mathew Needleman's Writing Tips #3: Pictures Aren't Just For Babies, brings forward a splendid way to 'unlock details from the brain' by drawing pictures. He made me think:

Drawing pictures comes naturally to us. We’ve being doing it for thousands of years before Darwin. We have the historical evidence to prove it.

What finer metaphor than a drawing for the thing that springs to mind? The word is a metaphor, but is at least twice removed from the image in the memory that it’s used to describe.

Drawing is a primal action - an ability that comes naturally to most. Consequently we see that three-year-olds need no drawing or painting tuition. They don’t have to learn the alphabet of pictures to show us what’s in their minds.

Drawing is a direct mapping, albeit interpretation, of the image that’s in the mind. Once drawn, the picture immediately calls to mind what was seen and done.

The simplest symbolic language; it needs no translating.

Cry Me A River

A lot has been discussed recently in the blogosphere on so-called metaphors for learning. There’s been everything from the idea of neural connections - thinking in terms of models - to what I’d call analogies, where learning is described as building on established structures, the growth of a tree-like organism, or the flow of a stream or river over a terrain.

These wonderfully graphic products of imagination indicate how vibrant our thinking is on something as abstract as learning.

My classical education does not permit me to see easily the bridges that may lie between things I recognise as models and metaphors or analogies, that are often used to explain how learning appears to happen. When it comes to using the terms, however, I often get them muddled.

Neural connections:

The neural connection description for learning is what I’d call a model. At the microscopic level, it is a particle model that explains how connections are made between cells in brain tissue, and is not unlike microscopic circuit connections within a computer chip. Though it explains how the complexity is established within the intricacy of the brain, it does not explain the feature or characteristic of learning.

As a teacher/educator/assessor of student learning, I’m more used to outcomes that are as a result of meeting learning objectives. These, of course, are what we call the assessable outcomes.

On the other hand, I see the river and tree-like ‘metaphors’ more as analogies, where the analogous, known features in a river or tree are used to explain how learning can be thought of as developing or growing through time.

When I read the writing of others on these topics, I am constantly aware that some are talking about models, while some are talking in metaphors, and others are drawing analogies. I get confused. I feel that there is a need to find a distinction between what’s a model, what’s a metaphor and what’s really just an analogy. They are not all the same.

What are models?

Models are the basis for assisting a raft of scientific thinking that has a history going back several hundreds of years. A model can be a physical thing and is often thought of as such. It can be held in the hand, such as a model for a molecule of matter, defined as the smallest particle of a substance, retaining all the known microscopic characteristics of the substance it represents.

It can also be a written thing, such as a mathematical formula or expression that, according to known and understandable parameters, explains how things are seen to behave. A model for how the volume of a cube relates to the length of one of its sides can be described by the equation, v = d x d x d, or v = d³. In general, the model leaves little to the imagination.

Metaphors:

Metaphors are different from models. They don’t necessarily need to be tangible artifacts, nor written expressions or relationships, such as equations. The word comes from its literal use, where something becomes something else.

A metaphor is thought of as having two parts, the target and source. The target is the subject to which attributes are ascribed. The source is the subject from which the attributes are borrowed. “He was a lion in the fight”, is a metaphor for a warrior (target) who was not just like a lion, but became the envisioned replacement - the lion (source).

Metaphors are slick. They permit the mind to think swiftly in terms of the source rather than more cumbersomely in terms of the target.

Metaphors tend to be borrowed from other disciplines. The particle/wave model for light, for instance, has been adopted as a way of describing knowledge, thought of as a thing and a flow. In this way, the model has become a metaphor.

An archaic metaphor for learning was filling jugs. Presumably the jugs were the minds of the students that were to be filled with liquid, which was knowledge. It is a two-in-one metaphor – jugs for minds, liquid for knowledge.

Analogies:

Analogies are quite different from the other two. Unlike the model, an analogy is not trying to depict, in any way, how the thing or concept exists. Nor is it like the metaphor, that tries to make the thing or concept be something else. An analogy is a direct mapping between one idea and another. There is no need for there to be any physical or ethereal resemblance between the thing or concept and its parallel.

So we may well think in terms of a model when joining popper beads to represent neural connections while learning occurs in the brain. We may also think of the metaphor for knowledge as a thing and a flow when considering knowledge management. But the analogy is the most involved of the three.

The metaphor is considered the core of cognition. It calls for the most use of our imagination for it is a parallelism that’s left mainly up to the ingenuity of the thinker, who considers the behaviour of one thing when thinking of the parallel behaviour of another.

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Splitting the Knol

We have the knowledge.
Now we have the knol - a Web 2.0 effort to quantise knowledge.

This is not the first time an attempt has been made to invent a unit of something to do with learning. The invention of the knowledge object, similar to the learning object and now more recently called the learning resource, were endeavours to bring together essential elements of knowledge so that they could be packaged and transported to the learner to achieve a single learning objective.

The matter with Knol:

This latest pursuit, of which Knol is one of its offshoots, is a bit like the quest for an understanding of matter itself. Like the way Science analysed the substance, found the molecule, split the atom, detected the sub-atomic particles, and by all accounts is still pursuing the analysis of these. New Zealand born Nobel prize-winner, splitter of the atom, Ernest Rutherford, would be fascinated today. I wonder if the person who manages to split the knol will win a Knol Prize.

It’s all to do with knowledge, the common fibre that led to the so-called ‘string theory’ of matter and all its sinuous threads. Isn’t it funny how the same patterns run through seemingly different disciplines? We talk about the ‘thread’ of an argument, and how we 'string' together ideas – the 'fabric' of education. How the stuff of science has 'woven' its way into everyday life. How different constructs can 'knit' together. How knowledge is being constantly 'tailored' to fit the learner – never mind the quality, feel the width. And how, for some of us, it is all 'sewn up' when it’s understood.

Well, my understanding of it is that a fair bit of darning is needed to patch the holes and pull the material to meet at the seams. My tartan bag of knowledge is anything but stitched together yet.

But now we have the unit of knowledge - the knol.

Ice-cubes go with the flow:

One model of knowledge, at least from its management point of view, is the idea that it can be a thing or a flow. A bit of knowledge can be looked on as an ice-cube, for instance. How it gets from one place to another can be thought of as the flow of water that’s needed to make the ice-cube. So it can be considered as a thing and a flow. That’s to say it is both at the same time.

This idea is not unlike the way light has been thought of in physics or the way matter has been considered, as a wave and as a particle. It all depends on the situations and how the occurrences are observed.

Frozen knowledge, as ice-cubes, sits in the books in my library at home. When an enquiry is made from the books, bits of the knowledge melt and flow, at least for a short time, into the mind of the reader. There it may solidify again, remaining till it flows to some other recipient or simply evaporates. Of course, here the model tends to founder on the rocks, as nothing really happens to the ice-cubes in the books even if some of them flow elsewhere. It’s nevertheless a cool model.

So what is the knol? Is it a thing and a flow? Let’s see:

Thing, as the ice-cube, sitting somewhere on a server at Google.
Flow, as the digital information when I download it to my PC.
Thing, as the ice-cube, sitting on my PC.
Flow, as I read, observe, listen and understand its content.
Thing, as the ice-cube in my mind, when I think I’ve understood and learnt what there is to learn.
My head hurts.
Flow, as I try to tell someone about it.

What if I’ve got the wrong idea altogether? What if I misunderstood the content of the ice-cube? Is a new ice-cube developed in doing this? Is the new ice-cube useful and worth passing on? Is what’s passed on still knowledge? Could a knol be made out of it? Hmmm.

I think I’ve got some stitching to do on my model for knowledge.
What do you think?

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The Word Is A Metaphor

Tēnā koutou katoa

Welcome to you all

This was originally to be submitted to Michele Martin's post as a comment but due to its length I decided to post it here. In no way is there any intent to upstage Michele's post on How Do You Use Metaphors For Learning?

Key terms: sound, symbol, word, metaphor, target, source, ground, tension

The word as a written metaphor:

The written word, being composed of symbols, is itself a metaphor for the spoken word that’s made up of sounds. Already once removed from the sense of the uttered sound, the written word has to be interpreted before it’s understood.

Despite the thousands, perhaps millions of written words at our disposal, people still invent words – all the time – some of which are never written. Yet for some people the reading of a word new to them can cause them real problems. They may decide, for instance, that a new word has been miss-spelt, relating it to another familiar word, and may lose entirely the sense connoted by the context that it is in.

The best metaphor is spoken:

Exponents of our language such as Shakespeare and Hopkins invented words to convey meanings that could not be conveyed by existing words. Shakespeare is given credit for inventing thousands of words that are in use today. The distinction between Shakespeare’s and Hopkins’ is that Shakespeare’s were heard by his audience, not read by them. Perhaps it’s for this reason that he was so successful in making in-roads to the language with his own words, though the quantity of his writing would also have contributed to achieving this.

So in terms of success, the suggestion is that a good metaphor should be spoken first – a difficult task when using written words to describe a metaphor.

What is the composition of a metaphor?

Wikipedia describes the metaphor in detail and outlines how the metaphor relates to what it's supposed to convey. A metaphor can be thought of as having two parts, the target and source (other terms are used for these - tenor and vehicle). The target is the subject to which attributes are ascribed. The source is the subject from which the attributes are borrowed.

It also speaks of the ground and the tension of a metaphor. The ground relates to what is similar between the target and source. The tension relates to the dissimilarities. The dead metaphor is explained as one that doesn’t work – “the sense of a transferred image is not present . . . dead metaphors, by definition, normally go unnoticed.”

When a metaphor is used for learning, it follows that it must have life. Dead metaphors, whatever application others may say they have, are unlikely to have much use to the teacher. To have life, the metaphor must have ground, sufficient so that the imagination is invoked and as little tension as permits the ground to be envisaged.

Other disciplines:

Mathematics is steeped in metaphor and its symbols are often chosen with metaphor in mind. It’s no accident that the Greek alphabet is so widely used in this discipline, for its letters are also steeped in metaphor – they are the alpha and omega of metaphorical symbolism.

Choosing metaphors that have their source within the scientific discipline can be useful but it can also cause problems. It is often important that the mapping of the source components of a metaphor is clear and accurate. Unless the source components are clearly understood there is danger of poor mapping onto the target that could result in a confusing metaphor. There are also many concepts in Science that are difficult to understand. This is why scientists and science teachers so often use visual models and tangible ones. Not only are they helpful for learners of Science, but they are also useful tools for communicating scientific ideas with others.

Metaphors can mean less learning:

Models like the particle, the wave, the balance and the wheel are often used in Science since these sources are already familiar and understood. With demanding concepts come difficulties associated with understanding and the teaching of them. In attempting to find a model, sometimes the metaphor that’s chosen is as difficult to understand as the concept itself. So it becomes redundant and little is achieved through its use.

Examples of this are attempts to use metaphors for the processes of mitosis and meiosis, biological terms applied to different processes of cell division. Because of the similarity in the look and sound of the words describing these processes there is often confusion as to their application to the related processes.

In introducing these processes to students the teacher has to resort to a related, but nonetheless simple, metaphor in order to convey some understanding of the difference between the terms.

For someone who has been introduced to the concepts of mitosis and meiosis and understands them, for instance, the metaphor ‘sex’ can be of some use. The words ‘sex’ and ‘meiosis’ both contain the letter ‘e’. Meiosis is the special division of cells in the gonads that gives rise to the sex-cells (or half-cells) sperm and ovum.

Mitosis is the term used to describe cell division giving rise to two whole cells. Though it’s a process used in reproduction, it is entirely different from the reproductive processes that involve meiosis. The word mitosis does not contain the letter ‘e’.

In this case, the association of meiosis with the word ‘sex’ related to the context of its use through a common letter ‘e’ can be used to draw the distinction between two similar words mitosis and meiosis. But the metaphor is contrived and is really not an elucidating one, for it conveys only a way of telling which term applies to which process. It does not convey a useful meaning that helps a learner understand the individual processes.

A simple model for metaphor:

Occam’s Razor is a good metaphorical tool that’s helpful when choosing or inventing a metaphor for use in a learning environment.
It says that the number of things needed to explain anything should not be increased more than necessary.

Even the introduction of a metaphor, by its occurrence, is increasing the things that must be assimilated by the learner. If the learner must first discover and understand the source, especially if it encompasses difficult concepts to begin with, the target simply disappears from sight. The whole reason for using the metaphor is lost. So it makes sense that if a metaphor is to be used, it should have a source that is well known and understood. This makes for one less learning task that has to be done to reach the target.

What makes a good learning metaphor?

To find examples of the use of splendid metaphor and how it is employed we need look no further than the successful poets. It is no coincidence that in the traditional teaching of the meaning of metaphor there is often reference to the works of the poets. Their words ring clear, simply from the common nature of their craft. They are well acquainted with the need to find idea from familiar source to target what they wish to convey.

I think that this should apply in a similar way to metaphors that may be chosen as possible learning tools.

The fine delight that fathers thought; the strong
Spur, live and lancing like the blowpipe flame,
Breathes once and, quenched faster than it came,
Leaves yet the mind the mother of immortal song.

Gerard Manley Hopkins

Ka kite anō
Catch ya later