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:
In addition, one can easily think of the following two cases, no doubt familiar in hindsight in the experience of many:
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:
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.
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.
