Some concepts of cognitive structures and learning

Psalehesost

The Living Force
I've been reading a book, Advanced Mathematical Thinking (edited by David Tall) presenting cognitive concepts and research about mathematical thinking and learning. It's from 1991, and so a bit older than the recent cognitive science writings discussed on this board. But there's shades of common understanding to be found.

And, which is the point of writing about it here, there's concepts that I think are interesting in relation to learning in general - specifically, more difficult kinds of learning where understanding must be built in layers. So I will both summarize and extrapolate - the latter being somewhat speculative - and also use terms from this board.

(Edit: 4th revision of summary)


Concept images, concept definitions:

In the context of math, a concept definition is the formal definition - more generally, it is the description that one is supposed to understand. It is an exterior form, like an outline or a scaffold which one's knowledge structure should match in order to be accurate.

All the experiences, notions, images, etc. we have regarding a concept cause the forming of one or several concept images. These include the mental representations we have of the concepts. And be they accurate or not, coherent or not, they are our understanding, and cognitively they represent the essence or substance of our knowledge.

While one mind can "communicate" its knowledge in the form of an outline or scaffold - a concept definition - to another, the essence or substance of the knowledge - one or more concept images - must be "built" by the receiver, such that it accurately "fills" the scaffold or outline.

We may however memorize (not the same as understanding) a concept definition in itself, and (at least in the context of math) work with it directly, using (as the book calls it) "technical" thinking, which seems to correspond to the use of System 2. Such "technical" thinking (in the context of math, formal deduction, checking for errors according to formal rules, and so on) can in itself be mechanical - just formally following a set of rules in a definition and producing a result. It can also interact with mental activity based on our concept images, the new experience brought by the interplay between concept definition and our present concept images changing the latter, and so leading to learning and new insights.

When new intellectual concepts, models and systems are developed, the process is somewhat the reverse: Their developer first builds personal concept images as the problem is worked on, as development proceeds; then, in the final stage of refining the description, the concept definition is created (in mathematics, a rigorous, formal definition). As such, the concept definition can be seen as the finished product of the original intellectual work.

Genuine learning - as opposed to rote memorization - requires not just this finished product, but also something of the process of building an understanding underlying it. An aspect often neglected in education, which presents the finished product to the student while neglecting the cultivation of the process of thinking of the field.

Different stimuli can evoke, by association, different concept images. And so, we can have several different - even contradictory - concept images without knowing it; different situations can prompt the use of different associated concept images, and then we simply think accordingly.


Mental representations and problem-solving:

A mental representation of a concept is a "concrete instance" of the concept; an illustration, example, etc., in terms of which one can think of the concept and work on problems involving it. Mental representations, serving as frames of reference, allow grasping and working with one or several aspects of a concept. They are key parts of our concept images.

In general, "concrete" representation systems (such as, in the case of mathematical functions: graphs, algebraic formulas, arrow diagrams, and value tables) are theorized to be the basis for the creation of mental representations. Visualizing, as well as abstract and symbolic thinking, exploring how concepts "work" and the way they relate to other concepts - these can be involved in producing and enrichening a representation.

A rich representation contains many linked conceptual aspects. Poor representations have too few to allow for flexibility in problem-solving, which tends to confine one to the realm of the known, expected and standardized, leaving one helpless outside it.

One can learn several ways of "concretely" representing a concept to oneself, giving one several frames of reference. (see for instance the examples of representations of mathematical functions above) These can then be switched between according to the specific situation during problem-solving. Learning to do this deliberately is an important step.

With experience, one will often, in working on a problem, transfer information between several representations. This is necessary to solve certain problems. With further experience, the representations may be linked together in the mind and - instead of serially - be used in parallel. Eventually, they may unify into a whole, where one can focus on - "pick out" - specific parts according to need.


Intuition:

Intuition is briefly addressed in terms of the work of several "modules" of the brain - this multi-"module" concept reminds of the more elaborate idea of the adaptive unconscious / System 1. Intuition is also connected to creativity - intuition being crucial as the inspiration to often precede, and/or go along with the perspiration.

And intuition can be trained - it depends on our concept images, so the more we work with and develop them, the more our knowledge will manifest itself in our intuitions. For example, a mind trained in mathematical logic will more often have mathematical intuitions that are logically sound.

(Note: As for the general concept of intuition, there's more to it, in at least two ways: 1. There are some biases ingrained in our mental processing that often lead to faulty intuitions; 2. The capacity for inuition may deepen and change in quality along with our general state of development.)


Generalizing, synthesizing, abstracting - three different cognitive processes:

Generalizing means extending a knowledge structure to new situations. It often works by analogy from the old context to the new. It is often a comparatively "easy" way to grow in knowledge, because the existing information is not threatened and doesn't need to change - sometimes, but not always, new and complex concepts may however have to be formed for the new context. In short, the old serves as a template for the new.

Synthesizing means combining things to form a (new and different) whole - in the context of learning, previously seemingly unrelated facts may merge into a single picture. The more the potential for synthesis in a field of knowledge, the more "compressible" the knowledge is - "compression" being this merging and combining of things in the mind of the learner into a single, greater whole. With it comes new insight, and it is noted in the book to be one of the great joys of learning.

Compression/synthesis is irreversible; once it happens, one has a deeper, more broad-ranging understanding, but it becomes very hard to understand the frame of mind of someone who hasn't yet had that insight. (As mentioned in the book, this impacts math education; the mathematician having a lot of synthesized knowledge and the student lacking it, and the teaching generally being focused more on presenting this finished product than on the long and complicated process that led to it.)

I'm here reminded of Gurdjieff's idea that the greater the difference in level between the teacher and the student, the harder it is for the student. Though probably far from the whole picture, here we have something of a cognitive basis for it.

Abstraction, finally, is linked to both generalization and synthesis; it has the potential for both, and mainly gets its purpose from this. It is however a different mental process, building new mental structures to match and model the structures studied, with an emphasis on the relations between the objects of thought rather than a focus on the objects themselves. For example, "the student is required to focus on the relationships that exist between numbers in order to be able to grasp what a field is, rather than on the numbers themselves".

Something which also plays a part in developing "higher", more abstract levels of thought is encapsulation of mental processes into mental objects. Having become sufficiently familiar with a mental process one has learned which deals with some form of mental object, one may then form for this process a concept which makes it a mental object itself. It can then be used as such in other mental processes on a higher level. And these, in turn, could eventually be encapsulated into objects, and so on...


Different modes of thinking / working on problems:

Even when working on mathematical problems, most people, according to the book, are stuck in a mode of thinking which it labels "non-technical", or the "everyday life" mode of thought. I think this basically amounts to System 1 running the show.

The book presents an idea of two different cognitive "cells" - one for the concept definition (and working with/using it), and one for its concept image(s) (and working with/using it). One or both may be empty; for example, in the case of meaningless memorization of a definition, one lacks a corresponding concept image.

There are some diagrams in the book for processes involving these "cells"; I'll reproduce their content flatly with text and arrows.

"The cognitive growth of a formal concept" - the ideal often expected but which rarely occurs - is illustrated as follows:

Concept definition -> Concept image

If one is given a concept definition that conflicts with one's present concept image, the book lists three basic things that can happen - generalizing from the specific example:
  • The concept image may be changed (expanded, revised, or reconstructed) to match the definition. This is the satisfactory outcome.
  • The concept image does not change. Over time, the concept definition is soon either forgotten or distorted to match the unchanged concept image.
  • The new concept definition is retained alongside an unchanged concept image. The old concept image is used in practice, while the words of the concept definition may be parroted if asked for.

In addition, one can easily think of the following two cases, no doubt familiar in hindsight in the experience of many:
  • The concept definition is misunderstood, and the concept image changes to match this distorted understanding. The memory of the concept definition is distorted over time to match the new and faulty concept image.
  • The concept definition is misinterpreted while its words are retained. The concept image changes and a distorted understanding results; it is used in practice while the words of the accurate definition may be parroted if asked for.

The failure to learn, in the latter cases, could have several reasons. It could be a matter of lack of preparation, or of not engaging in a genuine learning process (settling for imitation and learning by rote), or issues with cognitive dissonance (such as a simple clash of concepts, identification with the old understanding, or other kinds of emotional blocks). These are examples; there could of course be other reasons.

Good, proper intellectual work - and good, proper learning - involves mental activity that is not purely intuitive. When people operate with the "everyday life" mode of thought, however, then the cognitive processes dealing with an input solely reference the concept image in order to produce an output: an "intuitive response".

Input -> Concept image -> Output (intuitive response)

By contrast, three "ideal" models are:
  • Input -> (Concept definition <-> Concept image) -> Output (intellectual behavior/answer)
  • Input -> Concept definition -> Output (purely formal deduction)
  • Input -> Concept image -> Concept definition -> Output (deduction following intuitive thought)

The book says: "no matter how your association system reacts when a problem is posed to you in a technical context, you are not supposed to formulate your solution before consulting the concept definition. This is, of course, the desirable process. Unfortunately, the practice is different. It is hard to train a cognitive system to act against its nature and to force it to consult definitions either when forming a concept image or when working on a cognitive task."

Hence the model of the intuitive response mentioned earlier. "[...] The everyday thought habits take over and the respondent is unaware of the need to consult the formal definition. Needless to say, that in most of the cases, the reference to the concept image cell will be quite successful. This fact does not encourage people to refer to the concept definition cell. Only non-routine problems, in which incomplete concept images might be misleading, can encourage people to refer to the concept definition. [...] Thus, there is no apparent force which can change the common thought habits which are, in principle, inappropriate for technical contexts."


Cognitive development and the process of learning:

A metaphor is used in the book for the progressively shifting states of cognitive equilibrium in learning:

"[...] a dynamic state of equilibrium has a more obvious mathematical metaphor in dynamical systems and catastrophe theory. Here a system controlled by continuously varying parameters can suddenly leap from one position of equilibrium to another when the first becomes untenable. Depending on the history of the varying parameters, this transition may be smooth, or it may be discontinuous. This analogy suggests that [Piagetian] stage theory may just be a linear trivialization of a far more complex system of change, at least this may be so when the possible routes through a network of ideas become more numerous, as happens in advanced mathematical thinking."

There are many mentions of cognitive conflict in the book. Such is necessary - as we know, and it seems to be so in any cognitively complex field of knowledge. Old concept images, ones that seemed to "work", to "fit", but eventually turned out to be wrong or inadequate, can be stumbling blocks. The book quotes the following from Cornu, 1983: "An obstacle is a piece of knowledge; it is part of the knowledge of the student. This knowledge was at one time generally satisfactory in solving certain problems. It is precisely this satisfactory aspect which has anchored the concept in the mind and made it an obstacle. The knowledge later proves to be inadequate when faced with new problems and this inadequacy may not be obvious."

One may fail to further learn without even understanding why - indeed, without even knowing it - so long as a wrong foundation remains and prevents assimilation of new knowledge. Whenever one has a wrong concept image that must be reconstructed, being ready and able to go through cognitive dissonance is essential to progress. Many, lacking this ability, basically end up unwittingly creating insurmountable obstacles to learning for themselves.

For the rest of us, I think - in my experience since reading what I've summarized here - that it helps to know this and to be on the watch, scrutinizing the differences between "concept definitions" and one's "concept images" in every context, and being aware of how System 1 (aka. the adaptive unconscious, aka. the "happy brain") is ready and willing to cling to its habits, its old interpretations - the most likely mechanically formed concept images according to which it, and therefore we, interpret things - until we reconstruct our concept images.



This will have to do for now. If of interest, then as I read on and when I have the time, perhaps there's more of a more general significance to summarize and post.
 

Buddy

The Living Force
I agree that, like being able to debate, an ability of thinking technically, mathematically and 'properly', requires understanding the terms involved. Also, I wonder if the author is using intuitive thought to refer to disassociated heuristic thought done by people too numbed to see the full picture? Like an administrator (Gurdjieff had a bit to say about 'administration' as well) who spends beau-coup time properly filling in the company's time accounting software, yet never realizes he's not tracking the time spent tracking everyone's time. So, technically, his 'totals' are forever incorrect.


[quote author=Psalehesost]
Whenever one has a wrong concept image that must be reconstructed, being ready and able to go through cognitive dissonance is essential to progress. Many, lacking this ability, basically end up unwittingly creating insurmountable obstacles to learning for themselves.[/quote]

True that. Thanks for an interesting post. :)
 

obyvatel

The Living Force
Psalehesost said:
Intuition:

Intuition is briefly addressed in terms of the work of several "modules" of the brain - this multi-"module" concept reminds of the more elaborate idea of the adaptive unconscious / System 1. Intuition is also connected to creativity - intuition being crucial as the inspiration to often precede, and/or go along with the perspiration.

And intuition can be trained - it depends on our concept images, so the more we work with and develop them, the more our knowledge will manifest itself in our intuitions. For example, a mind trained in mathematical logic will more often have mathematical intuitions that are logically sound. (Note: As for the general concept, there's more to this as we have learned, including some biases that won't go away)
...............................
Good, proper intellectual work - and good, proper learning - involves mental activity that is not purely intuitive. When people operate with the "everyday life" mode of thought, however, then the cognitive processes dealing with an input solely reference the concept image in order to produce an output: an "intuitive response".

Here is Dabrowski's view of intuition. IMO it builds on and clarifies the concept

[quote author=Dabrowski]
By intuition we mean a synthesizing mental function or group of functions of a large scope, which grasps various forms of multidimensional and multilevel reality on the basis of stimuli and data which are not sufficient for a global diagnosis and discursive derivation of conclusions. The intuitive ability to grasp complex aspects of reality without sufficient "rational" foundations indicates that intuition operates by means of subconscious or hyperconscious shortcuts.

It seems that this kind of intuition appears only at a relatively advanced stage of mental development and is prepared by the process of multilevel disintegration. The process of multilevel transformations allow insight into various dimensions of reality and wide, synthetic and many-sided apprehension. In a fully rounded human development, intuition closely cooperates with intellectual and discursive functions . Intuition synthesizes the results of empirical and discursive data and creates a new coordinated unit which may become the subject matter of further discursive examination on a higher level.
......
We may conclude that intuition is a complex dynamism of preliminary and final syntheses coordinating emotional, intellectual and instinctive functions. Intuition deserves to be pointed out and emphasized as a fundamental act in every creative work.
[/quote]

Intuition needs to be understood as a multilevel function. At the starting level of primary integration, there is no intuition proper but pattern recognition through undifferentiated sensory perceptions. At the next level, intuition is primitive in nature - it is subject to suggestion and chance and can give rise to magical thinking. Dabrowski considers that real intuition comes into play at the third level of development as mentioned in the previous quote. Fluctuating associative thoughts brought about through some external or self-suggestion would be classified as "apparent intuition" in this Dabrowskian frame of reference.
 

kenlee

SuperModerator
Moderator
FOTCM Member
Dabrowski really nailed the definition of intuition imo. I always thought that intuition is being able to recognize and see significant patterns in events and ideas and then seeing connections between them on multiple levels. Kinda like seeing the points on a triangle, and then seeing how they mathematically relate to each other, and then seeing the entire geometric figure in a flash of insight.

Maybe intuition is something like that but it's more like being able to see significant connections between moments of time (or events) on many levels of reality simultaneously and then seeing them integrated in a flash of meaningful insight. It's like we can see only a fragment of reality at a certain level of consciousness, then we become conscious of that, then our consciousness expands further, then we become conscious of being conscious of that, then from that our awareness continues to expands even further, then even greater fragments of reality are seen with even more significant inter-connections seen, then we become conscious of being conscious of being conscious of that and on and on and our awareness expands outward towards infinity. I always thought intuition was like that.

If I remember correctly Ouspensky in his book Tertium Organum talked about 'relationships' and how it is that, possibly, it's only the relationships between things that are truly real or something like that. Perhaps it's significant relationships, that is, it's significant connections on many levels simultaneously that are truly real and this perhaps is what we feel as the true 'realness' in each moment of time? The feeling of the actual moment still has a feeling of realness to it, something is real there, behind it, even though the moments go by. Perhaps it's this qualitative significance that we feel in each moment and it's this which gives the moments of time the true feeling of realness? All else, perhaps, is like dust in the wind.
 

apacheman

The Force is Strong With This One
Lol, taught about embedded concepts.

I find it amusing that most of European/Middle Eastern cultural descent unconsciously insist on sets of threes. As a Native American, my conceptual framework numbers four, always. Sets of threes always appear incomplete to me.

You missed an obvious fourth outcome:

The world is changed to fit the old concept, the new being killed as too disturbing, the end result being a concept image that changes, but not in favor of the new definition that provoked the change.

Think Galileo.

Anyway, interesting stuff.

Intuition, I think, is dependent upon how broad and how deep your knowledge base is. That is to say how many different individual things do you know and how much do you know about each thing. Then how many systems of things do you know and how much do you know of them. Then how much do you know of the static relationships between things, and systems of things. Finally, (truly there is no finally, but this completes my penchant for sets of four), how much do you know of the dynamics of sets of systems?

That's a lot of knowings, too much to process on a conscious level, so when a problem or event we don't understand consciously confronts us or occurs, somewhere in the basement of our knowledge patterns are sifted through sets and combinations of sets. When an identifiable pattern emerges we begin to follow it to see where it leads: to which set or set of sets. This is intuition: a pointer to where to look for the answer.

When intuition leads us to a blank wall, it represents a hole or gap in our knowledge base. When it leads us to see a solution we have synthesized existing bases into a new thing or set.
 

Psalehesost

The Living Force
apacheman said:
You missed an obvious fourth outcome:

The world is changed to fit the old concept, the new being killed as too disturbing, the end result being a concept image that changes, but not in favor of the new definition that provoked the change.

Think Galileo.

That's outside the context of what happens within the mind of a specific learner when faced with a new concept definition. In its own expanded context, though, it would be another of very many possibilities for what can happen.

There are however also more options within the original context. I just added two. (I just edited the initial post, pasting in an updated version. for the record, I also rephrased some things for clarity and made two minor additions.)


EDIT: Since writing the initial post - having read and summarized that material - I have also changed my view of what the concept definition is.

Stating, as I did originally, that it is basically the thing to be learned in itself is in a way wrong. Because it is just an exterior form. Moreover, there are often several ways to describe the same thing in an exact way, all of which assign it the exact same characteristics.

The concept definition, in relation to knowledge, is more like an outline or a scaffold - it is needed to ensure that the "shape" of the knowledge is correct, but the knowledge itself goes far beyond it. The mind can "communicate" the knowledge in the form of an outline, or scaffold - but its essence, or substance, must be "built" by the receiver, and then it goes beyond the scaffold - "fills" it.

That essence or "substance" is cognitively represented by the concept images. More philosophically, and in connection with intuition, I think concept images may be a representation in our minds of a link to the Platonic realm of ideas - a link grown with the gaining of knowledge. Or so I speculate - here I can go no further at present without wiseacring.
 

apacheman

The Force is Strong With This One
I think you misunderstand the Galileo reference.

I was referring to what occurred in the minds of his judges. His concept and proof was beyond their capacity to accept, so they forced him to recant, thus "killing" the idea. However, the reality he exposed was too obvious, once pointed out, not to entertain. So the judge's perception of the world must change within himself to accomodate two opposed ideas, the old vs new, apparently mutually exclusive. Frequently this dilemma results in the outcome I suggested: mentally restructuring the world to allow them to co-exist temporarily without giving the new its proper validity, altering both image and definition incorrectly. Something like that must have occurred in the minds of the southern slaveholders after the Civil War, vis-a-vis "God's plan" and "His" approval of slavery. Your additions and clarifications address these.

It leads to the second, broader societal effect you noted.

I agree with the scaffolding metaphor. I would extend it a bit to include the tools a concept provides. In some ways I think intuition is backtracking from a conceptual goal one believes exists using a mixture of tool sets from different concepts. Once we know something exists, or we imagine something can exist, our subconscious begins sifting through tool sets to find those necessary to build the scaffold that supports the reality we see or can imagine. Sometimes we "discover" merely the same way that everyone else has done. Occasionally, because we possess a different assortment of concepts, and thus different permutations of tools sets, we discover a truly new way. Intuition is the result of finding which subsets of tools to work in what sequence to get past whatever the difficulty is in erecting the scaffold.

An interesting special case is that in which someone confronts something for which they lack both an image and a definition, something like showing a computer tablet to a Hittite, Mongol, Elizabethan. Or the Great Lakes to an islander or desert dweller. Or a Chiricahua Gan mask, netmending tools, hydrualic mules, and pain to those unacquainted with them. People go through some very interesting mental gyrations trying to process and categorize the unknown.

Truly bad pain is something most who experience it would agree that they lacked both an image and definition for. Whatever their pre-pain concepts were, they were so far from the reality as to constitute feeling good.

Interesting stuff. Having had a unique upbringing and education (attended 13 schools in four states and two countries, skipped a grade), constantly comparing concept images and definitions and their rules of inter-relation was crucial to social survival. Nice to see the process laid out fairly clearly, thanks.
 

Buddy

The Living Force
apacheman said:
An interesting special case is that in which someone confronts something for which they lack both an image and a definition, something like showing a computer tablet to a Hittite, Mongol, Elizabethan. Or the Great Lakes to an islander or desert dweller. Or a Chiricahua Gan mask, netmending tools, hydrualic mules, and pain to those unacquainted with them. People go through some very interesting mental gyrations trying to process and categorize the unknown.

Truly bad pain is something most who experience it would agree that they lacked both an image and definition for. Whatever their pre-pain concepts were, they were so far from the reality as to constitute feeling good.

Apologies, apacheman, but I'm having difficulties grounding what you mean in the quote above. Are you referring to situations where an experience comes prior to a narrative that is then created to explain it?
 

Psalehesost

The Living Force
Just edited in a 4th revision - there is more I want to add, (which would include what the book describes - which is quite damning - as part of the reason for the failure of math education) but this will have to do for the moment. I have to digest things further to summarize it well.

The main addition is a new section on mental representations and problem solving - now the second section - which ties into concept images as well as what follows. Also some expansion of the concept image section. And an added paragraph inspired by apacheman's post, regarding the reasons for failing to learn from new concept definitions.

For now, two points among what I wish to add in the next version:

Math students are, in short, encouraged by the structure of education to "take the deal" of passing courses by imitation and rote, learning to perform rituals (do this, then do this, then do this) instead of actually understanding. Examples of inefficiencies in education include: Lectures mainly convey language, not concept images, which students have to form themselves. And giving many, many examples serves to reinforce existing concept images (including any errors and misunderstandings) rather than to correct students' understanding - unless the misunderstanding is addressed in the specific example, which is generally not the case.

Many students lack in general cognitive development - in 1968, approx. 22% of American college students (according to Ausubel et al, Educational Psychology, a Cognitive View) had reached the "formal stage" (as defined in Piaget's stage theory), meaning they were capable of thinking on a hypothetical "if-then" level - the rest (78%) weren't.
 
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