On technical thought and Advanced Mathematical Thinking by David Tall et al

By Joel K. Pettersson. Added 2022-02-14. Updated 2022-04-24.

Loosely based on some 2012 notes I made after reading this book (which I found in one of LiU's TekNat or HumSam libraries, but later got my own copy of).

A decade ago I read the book Advanced Mathematical Thinking, edited by David Tall, from 1991. It presents cognitive theory and findings about how mathematical thinking and learning works, describing stages of cognitive development and use of new levels of abstraction (each built on the previous), and things which commonly go wrong in learning. While it's focused particularly on mathematical thought and work, I thought then, and still do, that basic insights from it can be applied more widely, especially to other technical thought, e.g. programming.

In the book, the distinction is made between "concept image" and "concept definition", and easily-engaged intuitive vs. more effortful formally strict modes of thought. People tend to rely on the oft-flawed intuitive modeling of how things work that their brains do with experience, instead of consulting formal "concept definitions" even when it matters for technical accuracy, as a lazy inclination part of human nature. This is similar to the broader ideas of "fast thinking" (or "System 1") and "slow thinking" (or "System 2"), as in Thinking, Fast and Slow by Daniel Kahneman, and other research on the "adaptive unconscious" and the limits of rational intellectual power that grew and then became popularized in the 00s and 10s. But here the focus is not decision-making, but instead learning and intellectual work, specifically mathematical.

I think the contents of the book are interesting in relation to other kinds of learning than mathematical learning. A shorter step is to extrapolate to software programming and engineering. More loosely, and more as a matter of personal reflection on life and experience, I've realized a few things about how learning works, though much like the theory would suggest, they are only possible to grok in hindsight, or after enough experience to be able to look back on it and not only think forwards.

On programming, there are several thinking styles involved in it, and mental skills only partially overlap with mathematical work. Functional programming is said to be the most similar in style of thought to doing math. Something like C programming is not only different, but also has its own lower-level rules and operations on top of which higher-level reasoning about programs depends, and some find it easy to master while others never do (similarly to challenges with mathematical learning). Generally, it's intuitively easy to relate the ideas of distorted or wrong "concept images" concerning more basic things with systematic errors leading to bugs or design defects in work done.


Concept images and concept definitions

All experiences, notions, images, etc. we have connected to a concept cause the forming of one or several concept images. In terms of cognitive theory, these are cognitive structures containing all personal knowledge of the concept. They include the mental representations we have of the concept, and all learned connections between things. They are not necessarily accurate, and several conflicting concept images may arise over time, and eventually unify or not do so. Concept images cannot be communicated directly to others in language.

When new intellectual concepts, models, and systems are developed, the mind of their developer first forms personal concept images, as the problem is worked on and understanding grows, as the workings of the mind explore the territory from the bottom-up. By contrast, in a later stage of refinement of the description, a concept definition is a logically valid top-down formulation, as a finished product of intellectual work – communicable to others, and something which future students may perhaps strive to learn to understand.

In learning from the work of others, people instead face concept definitions from the start, and the task of learning from finished top-down products and consulting such as necessary. Such learning requires people not just to memorize a concept definition, but, for understanding to develop, to construct their own concept images through practice, intuition, and experience, and make them soundly match the concept definitions. In this, the concept definitions are a little like blueprints or scaffolds to be matched by concept images which are like buildings erected by the learner.

In the context of mathematics, and as presented in the book, a concept definition is a formal definition or a personal definition playing the same role – and as such, a basis for logical deduction, and also potentially a smaller building block which can be used in further work. People can also work with concept definitions directly, after rote memorizing of them or by looking them up. Done by itself, this amounts to applying rules, or following definitions – pure formal deduction in the context of mathematics – and playing a role similar to that of a computer system. This may just be a mechanical way of slogging away at a formulaic problem, or it may pendulate with intuitive thought, or otherwise be part of the overall experience and complex interplay with concept images which shapes the development of the latter.

Any concept image which logically clashes with a corresponding concept definition as used, when used in place of the concept definition, can be expected to lead to errors in some situations. It easily and often happens that concept images form that work well at some stage of learning, but not past it, due to only capturing part of the picture accurately; good learners need to be able to abandon their old understanding, when experience proves it inadequate, so as to avoid painting themselves into corners.

Furthermore, people are not necessarily aware of all the concept images they have, and different stimuli can evoke different concept images by association. Different situations can prompt the use of different associated concept images and ways of attempting to solve problems. This can hinder further learning when the path of least resistance becomes pendulating between incomplete and distorted understandings instead of working towards a fuller one.

I'm going to extrapolate to e.g. the world of software and say that a concept definition can more broadly be a specification for something, which is clear and detailed enough to be the basis of deductions. But in broadening the concept, and with it the nature of problems and solutions, then the areas of concern in which to deduce things, and in turn how detailed a specification to be used as a basis for deduction needs to be, begins to vary across a large range. At one end is a software manual, which people may need to read in order to nail down a few details on use and configuration. At another end is a specification for the semantics of a standardized version of a programming language, according to which little fragments of software can be dissected and their behavior deduced.

The book notes that mathematical education often over-emphasizes ritualistic use of the finished product – concept definitions – at the cost of focus on the process and concept images and the cultivation of good problem-solving thinking styles in the students. This makes it easier for students to get stuck after developing some confused or otherwise flawed understanding, or to end up with an inflexible and limited ability to recognize and solve problems.

Views and variations on these ideas

The ideas of concept image vs. concept definition originally come from the work of Shlomo Vinner, and have been used in somewhat varied ways in research over years, as David Tall describes on his website.

Shlomo's definition was philosophically based and was a thought experiment to analyse what happens when students focus in different ways on images and definitions. My perception was more humanly based, so that where Shlomo talked about 'the mind' and thought about it as separate from 'the brain' in a cartesian sense, I always thought of the mind as the way the brain works, so that it is an indivisible part of the structure of the brain. Shlomo has always written about 'concept image' and 'concept definition' as being 'two distinct cells' which enables him to make subtle analyses of different ways of employing the two distinct ideas. As the concept definition is a form of words that can be written or spoken, I regard this as part and parcel of the total concept image in the mind/brain. It is up to you to choose which version you want.

Tall basically makes the point that a learned concept definition has been added in some way to the cognitive structure of the learner, and in that sense it overlaps with the concept image which represents the total cognitive structure. In the next section below on modes of thought, which mainly draws on a chapter in the book by Vinner, the focus is however greatly on how concept definitions are used, the use of the 'concept definition' cell seeming to me rather similar to the "slow, effortful thinking" or "System 2" described in other types of research by Daniel Kahneman. Technical thinking essentially seems the combination of the two, the use of strict definitions and of a careful mode of thought which checks each step for errors.

In trying to look more broadly at the ideas from this book, by those two authors and those of other chapters, I'm departing in a different main way from Vinner's idea. I regard the difference in mental processes, and the formalistic application of rules and strict checking for accuracy, the key thing concerning concept definitions and their use. While mathematics students are generally expected to memorize definitions, in programming and other engineering, such thinking often uses external concept definitions instead, retrieved e.g. from technical manuals rather than memorized. The technical thinking is the same, the difference simply being whether or not the definition is located in human long-term memory; like computers, people often play a "game" of formal deduction using rules just handed to them from outside.

Modes of thought in study and work

In describing different modes of thought, the book contains some simple schematic presentations using labeled cognitive "cells", one for the use of the concept definition and one for the use of a corresponding concept image. One or both may be empty, as pure rote memorization means a lack of a concept image, and not knowing or having at hand the precise definition of something means a lack of concept definition. In the below, diagrams from the book are re-created using monospace text and symbols.

Learning and mislearning

When students learn, the ideal for what happens (in the terms described above) is that a concept definition is accurately captured in a developing concept image. This is illustrated as follows...

The cognitive growth of a formal concept
Concept definition -> Concept image

This may or may not happen, and learning is usually not that straightforward. But what happens when a student is faced with the need to improve the accuracy of understanding? Upon being given a concept definition that conflicts with one's present concept image, the book lists three basic things that can happen, in connection with an example. To generalize:

It's easy enough to think of other possibilities too, such as the following which I'll list in addition, and which I think may be recognizable in hindsight to many technologists trying to master new complicated systems and their definitions:

Failures to accurately learn in accordance with a concept definition can be for a variety of reasons. A lack of preparation may make it too difficult, there may be a lack of engagement in the types of mental activity which build a richer cognitive structure, in some cases cognitive dissonance may be an issue, to mention some general ideas. Other books and resources may be better for a broader view of the general psychology. But one very common stumbling block is simply the lack of an effort to be precise and strict in mentally comparing things.

The need for precise thought

[N]o 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.

Even when working on mathematical problems, most of the time most people are centered in a mode of thinking which the book labels "non-technical", or the "everyday life" mode of thought. This seems similar to the idea (described in the book Thinking, Fast and Slow by Daniel Kahneman) of "fast" or "System 1" thinking, and when it's a problem, not making the effort to fully engage in "slow" or "System 2" thinking. The quicker, more relaxed mental path of least resistance is for associations to lead from one thing to the next without slowing down and checking the precise details for correctness.

Honoring technical things as they are properly defined, in study and in intellectual work, requires making efforts to engage in more slow and careful detail-oriented thought, rather than the thought being too purely intuitive. When people operate with the "everyday life" mode of thought, however, then the cognitive processes solely rely on the concept image in order to produce their result, and there's no careful error checking involved. This is illustrated as follows...

Intuitive response
Input -> Concept image -> Output

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.

By contrast, three "ideal" models are sketched out as follows...

Intellectual behavior/answer
Input -> [Concept definition <-> Concept image] -> Output
Purely formal deduction
Input -> Concept definition -> Output
Deduction following intuitive thought
Input -> Concept image -> Concept definition -> Output

Intuition and its development

Intuition is briefly described in terms of the work of several functional "modules" associated with the brain, an idea reminding of more elaborate ideas elsewhere of an "adaptive unconscious" and nonconscious information processing systems forming much (or really most) of the human mind. Intuition is also connected to creativity, especially in the form of the crucial inspiration to often precede, go along with, or follow efforts to work on problems.

Tied to cognitive structures (schemas), intuition develops along with the general understanding and inclinations a person has. But the book focuses on more specific intuitive skills, and the point that mathematical intuition can be trained, as it depends on our concept images. (I think this also goes for other types of intuition.) Our intuitive understanding of concepts develop as we work with them; to the extent knowledge is richly internalized and given substance in the form of concept images, and e.g. the mental representations connected to them, the knowledge will begin to manifest itself in our intuitions. For example, a mind trained in mathematical logic will more often have mathematical intuitions that are logically sound.

Mental representations and problem-solving

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

In the case of mathematical functions, some common representation systems are graphs, algebraic formulas, arrow diagrams, and value tables. In general, such "concrete" representation systems are theorized to be the basis for the creation of mental representations. Visualization, abstract thought, symbolic thought, and exploration of how concepts work and interrelate, 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 lead to rigidity in the problem-solving approach, confining one to the realm of the known, expected and standardized, often unable to solve anything outside of that.

Practicing several ways of concretely representing a concept to oneself gives the option of choosing between several frames of reference during problem-solving. Some problems require transfering information between several different representations in order to be solvable, a basic skill gained with experience.

Experience can increasingly link mental representations used, eventually allowing them to be used in parallel rather than serially, or unifying them further into a whole where focus can "pick out" parts according to need.

The book mentions studies showing that mathematics students generally fail to solve any problems of moderate difficulty if the problems are simply formulated a little bit differently than usual. This seems related to how the structure of education encourages learning by rote and imitation at the expense of other things, for the sake of passing exams, leading often to poorly developed mental representations, and consequently less in the way of more general problem-solving skills.

Generalizing, synthesizing, and abstracting

Generalizing or extending ideas, synthesizing or combining them, and abstracting or newly modeling things and especially their relations, are described as three different, though related, cognitive processes.

To generalize means to extend a knowledge structure for new situations. It often works by analogy from the old context to the new. It can unfold with relative "ease" when it works, because the old understanding is not challenged and doesn't need to change. New and complex concepts may however sometimes have to be formed for the new context, when the old is not complete enough in its role as a template for the new.

To synthesize means to combine pieces of knowledge to form a new and different whole. Previously seemingly unrelated facts may merge into a single picture, and formerly separate islands of ideas and intuitions may begin to merge. The greater the potential for synthesis in a field of knowledge, the more "compressible" the knowledge can be said to be, the "compression" being this merging and combining in the mind of the learner into a single whole of greater meaning. With synthesis comes new insight, and it is noted in the book to be one of the great joys of learning, mathematical knowledge being highly compressible.

Compression of schemas or concept images with synthesis is irreversible; once it happens, the learner has a deeper, more broad-ranging understanding presumably more efficiently "stored" and accessed, but the human brain does not preserve the old version of the schemas involved. This creates a little gap in or barrier to understanding of older and less complete understandings, along with the insight brought by each synthesis. The new cognitive structure creates new possibilities for understanding facts, but it becomes more difficult to understand exactly what other people would need to go through, in terms of a process leading to learning, in order to be able to gain the same understanding.

In other words, learning to the fullest is guaranteed to somewhat reduce your cognitive empathy with those who do not share your new knowledge. As mentioned in the book, this impacts mathematical education, for the mathematician has a lot of synthesized knowledge while the student lacks it. The teaching generally ends up focused more on presenting this finished product than on the long and complicated process that led to it.

To abstract, finally, is linked to both generalization and synthesis; it has 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.

In generalizing, abstraction can model missing parts for the new context, or it may lead to the generalization after a structure begins to be arrived at which is recognizably similar to an old one. As a key to synthesis, abstraction may lead to a structure capable of capturing several other structures, and possibly encompass more than what was already known and explored.

Also playing a part in developing "higher" levels of thought and knowledge is encapsulation of mental processes into mental objects. With enough familiarity and experience with a mental process dealing with some form of mental object (which may also correspond to external computational processes), a concept may be formed for the process and how it relates to its objects which makes a mental object out of the process itself. It allows using the lower-level process as a higher-level object in other mental processes dealing with that type of object. This is the internal counterpart to how various types of mathematical expressions involving objects can, in a different variety of calculation, themselves become objects related in certain ways and used in expressions.

Cognitive development and obstacles

In elaborating on how processes of learning unfold, a metaphor is used in the book for progress between different states of cognitive equilibrium...

[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. Cognitive conflict is unavoidable, and it seems to be so in any cognitively complex field of knowledge. Old concept images, ones that seemed to "work" and "fit", but eventually turned out to be wrong or inadequate, can be stumbling blocks for further learning. On that, 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.

A person working from a foundation of pre-existing flawed understanding may fail to further learn without even understanding why, or at times without even knowing it happens. I think this can happen in even more striking ways outside of mathematics, and outside of technical fields, in the vast and sprawling mess of human ideas and structures. As long as a wrong foundation remains, it may prevent assimilation of new knowledge – but giving it up in order to be able to learn freshly in a new and larger way requires a step to be taken by the learner with some willingness and decisiveness.

Flawed understandings are more easily developed when learning rests heavily on imitation of series of steps to be taken, so that related ideas to a greater extent form by chance in connection with the activity performed. As it was phrased in one passage, in a look at how mathematics students dealt with problems, even the "best" were found to hold many grotesque misconceptions about mathematics.

Mostly as an aside, for making the most of this line of thought in a more generalized way (especially outside of technical contexts), I think it helps to read more about cognitive biases and how they can manifest in life, as for instance in the book What Makes Your Brain Happy and Why You Should Do the Opposite, by David DiSalvo. In a variety of ways, the striving to learn and improve is often at odds with the basic tendencies of human nature to stick with habits, with old frames of reference, and go with quick but less thoughful ways of interpreting situations.