Knowledge Levels

8/2/2019 Knowledge Levels

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KNOWLEDGE LEVELS:3-D MODEL OF THE LEVELS OF EXPERTISE

ABSTRACT

In this paper we present a conceptual model of knowledge levels. There are so far two such

models; the present one does not build on them but, as we have found out later, it is coherent

with both of these and even establishes connection between them. Our model does not aim to

replace the two existing ones; they all serve different purpose and should thus coexist. The

primary purpose of our model is to serve as research framework for forthcoming researches into

the nature of knowledge but it can also be used to estimate the levels of personal knowledge.

Keywords:

knowledge levels, expertise, cognitive schemata, skill acquisition


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KNOWLEDGE LEVELS:3-D MODEL OF THE LEVELS OF EXPERTISE

INTRODUCTION

The inquiry into human knowledge is as old as the human inquiry. Most of what we do can

be traced back, at least, to Plato1. However, this inquiry was only one of the many topics.

However, in the last decade or two it has been announced that the knowledge and the knowledge

worker is the most important for organizations (Davenport & Prusak, 2000; Drucker, 1969, 1993,

2002; Handy, 2002; Nordstrm & Ridderstrle, 2002, 2004; Senge, 1990; Sveiby, 1997;

Tsoukas, 1996), i.e. that by knowledge you make money. Thus in the last two decades the efforts

for learning more about knowledge multiplied; especially in terms of how to use it better for

making money. This paper belongs to this earlier line of inquire; we are interested in knowledge

for the beauty of it and for our passionate curiosity about how we know what we know and what

is this knowledge-thing anyway. Nevertheless, our result presented in this paper may also be

useful for those who are primarily interested in knowledge as means of money-making. We still

think that our primary audience form the researchers that want to inquire about knowledge; i.e.

the model of knowledge levels presented here may serve as a framework for various research

projects about knowledge.

The model we offer in this paper is based on a geometrical metaphor of describing the

various levels of knowledge using the number of points one can see. This metaphor will prove

very useful for deriving conclusions about the nature of knowledge at various levels using the

mathematical nature of the metaphor directly.

1Thus Whiteheads remark that all the philosophy today is a set of footnotes to Plato.


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The paper consists of two parts; in the first one we introduce the two existing knowledge

models and in the second one we introduce our new model. In this second part, after saying a few

sentences about the quite strange origins of the model, in the first section we describe our new

model and in the second one we examine the links to the previous models and some implications.

BACKGROUND KNOWLEDGE

There are two existing models of knowledge levels. One is based on phenomenological

observations of skill acquisition by Hubert and Stuart Dreyfus and the other is based on the

mostly experimental work that Herbert Simon conducted with various collaborators examining

the cognitive schemata in the long-term memory (LTM). Even though others also contributed to

the present models, especially in the case of second one, henceforth we will refer to the first one

as the Dreyfus-model and to the second one as the Simon-model.

Historically the development of the Simon-model of expertise started in the 1950s and it

took its present form in the late 1990s; the Dreyfus-model was almost completely developed in

the second half of the 1980s. The two models had some impact on each other; the more

important one from the perspective of this paper is the Dreyfus-models effect on the Simon-

model. Therefore in this section we first present the Dreyfus-model and subsequently the Simon-

model.

The Dreyfus-Model of Expertise

The Dreyfus-model has two roots; one from each of the Dreyfus brothers. The philosopher

Hubert Dreyfus was interested in the nature of consciousness, offering a phenomenological

alternative to cognitivism (Dreyfus, 2005), as well as in proving that the version of artificial

intelligence, which he calls GOFAI (good old-fashioned AI), is not and will not be realised


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(Dreyfus, 1972, 1992). His brother Stuart, an applied mathematician, was interested in everyday

skills and how they are acquired. The story of the model begins (Kreisler & Dreyfus, 2005: 3)

with the US Air Force asking Hubert Dreyfus to think about skills and he involved his brother

Stuart. They did not engage with the skill we acquire through imitation in childhood, only with

the skill we acquire as adults starting on instructional basis. (Dreyfus, 2004: 2) The Dreyfus

brothers identified five stages at which the acquisition and also the application of skills are

different; they call these novice, advanced beginner, competent, proficient, expert.

(E.g. Dreyfus & Dreyfus, 1986: 51) The following description of the skill levels mainly follows

(Dreyfus & Dreyfus, 1986: 16-51) and uses the examples from it; where other works are used it

is separately indicated.

At novice level the learner needs to be instructed and (s)he can acquire only simple facts

and features that are context independent and then rules for determining action based on these

facts and features. The facts and features at this level need to be so accurately defined that the

novice can recognise them independent of the situation. For instance the novice driver is shown

the stick shift and its positions (context-free facts) and (s)he is told at which speed to shift gear

(context-free features used to define rules for action). The instruction may also include the

distance that is to be kept from the preceding car. The rules for action acquired at this level are

incredibly crude; Dreyfus and Dreyfus compare it to the training wheels of children bicycle: they

are necessary for a while but the sooner you get rid of them the better. Applying the rules

typically requires full concentration from the novices and thus they cannot even respond to

advices if they would receive any during the process; they will do it as they learnt it. Strictly

following the rules, however, leads to poor performance and may even be dangerous.


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Nevertheless, the performance at this level may only be judged by how well the rules are

followed.

The most important characteristic of the advanced beginneris that (s)he recognises the first

situational aspects; i.e. elements that cannot be described in a context-free way. This may happen

in two ways, either (s)he notes such elements herself/himself after seeing sufficient number of

examples or the instructor points them out (Dreyfus & Dreyfus, 1987: 23), i.e. by ostensive

definition. Examples could include learning to recognise the smell of coffee, a particular style of

barking of a dog (meaning ones new dog, not generally dogs), and this is how the advanced

beginner driver learns to identify a particular engine sound. The advanced beginner starts

combining the context-free facts with the situational elements and (s)he now has several rules to

apply in order to determine her/his action. Although the situational aspects cannot be described

in words (i.e. we cannot describe what a coffee smells like and how the engine sounds in

particular situations) but once we have learned to recognise them we can also name them and

talk about them and thus the instructional maxims can now refer to both context-free facts and

situational aspects: Shift up when the motor sounds like its racing and down when it sounds

like its straining. (Dreyfus, 2004: 3) The performance of the advanced beginners is somewhat

improved in comparison with the novice but it is still poor, it is slow, uncoordinated, and

laborious; this level is still a rule follower even though there are now somewhat more rules and

these are based on the blend of facts and situational elements rather than solely on facts. E.g. for

changing the gear the advanced beginner driver uses both the fact of the speedometer and the

situational sound of the engine. Gradually the advanced beginner acquires more and more

context-free facts and situational elements as well as increasing number of rules. But (s)he does

not have any sense of priority; cannot determine which actions are important or which is the


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most important one. The advanced beginner driver, if (s)he is in a situation where two different

things should be done, almost certainly causes accident.

The dominant feature at the level ofcompetence is organisation. Being competent means to

have a goal and see the facts and situational elements, assigning importance to these, and the

importance of particular facts and situational elements may depend on the presence of other facts

and situational elements. The goal provides the perspective for this competent seeing; using

which the competent performer restricts herself/himself to only a few out of the vast number of

facts and situational elements. For instance when the competent drivers goal is to get

somewhere as soon as possible may decide about his route considering the density of the traffic

and the road conditions but disregards the scenic beauty. Stuart Dreyfus gives his own example

from chess, in which field he achieved the level of competence (Class-A) and got stuck there (the

cases of those who got stuck on a particular level are usually remarkably instructive). The

keyword for him was to figure outwhat to do. His teammates, later chess-masters, started to

play fast-chess (5 or 10 minutes a game) and re-play grandmaster games. Stuart (and many

others) did not find fast-chess interesting as it did not allow for figuring out the next move and

they did not enjoy grandmaster games and they could not figure out the grandmasters thinking.

The strictly analytic thinking cut them off from the actual chess experience; and the competence

is hopelessly analytical. There is, however, an interesting consequence of organising, i.e.

determining the action based on the goal in mind. In contrast with the novice and advanced

beginner level, at which both the learning and the action happened in detached frame of mind,

the competent feels responsibility for the outcome of the action, as (s)he has chosen it (and this

choice is an unavoidable for the component performer). So this is the first level where the person

becomes involved. The sense of responsibility makes the successful or disastrous outcome more


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memorable; the joy of success and the sorrow of failure are both unknown to the novice and

advanced beginner. The competent performer also remembers the foreground and background

elements (more and less important aspects) of the situation, risks, opportunities, etc. in a holistic

way. (Dreyfus & Dreyfus, 1987: 26)

The level ofproficiency is the first at which intuition appears. On the previous three levels

everything was done analytically. Similar to the how the advanced beginner started to experience

the situational aspects directly, without being able to describe them verbally, the proficient

demonstrates the ability to directly experience the complete situation. The other characteristic of

this level is the time-effect of situational changes during the course of action. So, the proficient

performer experiences the situation from a specific perspective because of recent events;

intuitively assigning various salience to the context-free facts and situational aspects. The

relative saliences, however, will not remain unchanged; as the events modify the salient facts and

aspects, the relative salience level also changes, and so changes the situation. As the situation

changes, memories of similar situation in the memory of the proficient performer spontaneously

trigger plans that previously worked and anticipations of events that are likely to occur; this is

called holistic similarity recognition. This means that the proficiency is characterised by

intuitive, dynamic understanding of the situation. Nevertheless, not everything is intuitive at this

levelthe intuitive understanding of the situation is followed by analytical choice of the course

of action. For instance, based on her/his previous experience the proficient driver, approaching a

curve in a rainy day, intuitively senses that her/his speed is too high; then (s)he analytically

decides whether to press the break, remove her/his leg from the accelerator or just reduce the

pressure.


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The highest expert level is characterised by complete indwelling; the expert does not

become detached from the situation anymore, (s)he does not devise plans, does not worry about

the future, (s)he apparently does not see problems and solves them that all happens, of course,

but these are not things to do, it is just living. Like when Fred Aster is dancing. The expert is not

more aware of and does not think about the field any more than about her/his body. The exert

drivers does not experience driving a car; (s)he becomes one with the car and (s)he experiences

driving. When things are proceeding normally, the expert does not interfere, i.e. (s)he does not

make decisions or solve problems, just does what works. When action is needed, when a decision

is to be taken or a problem solved, the expert deliberate before acting; only this deliberation is

not analytical, it is critical reflection upon the experts intuition. To demonstrate that experts use

intuition only, Dreyfus and Dreyfus conducted an experiment with the international grandmaster

Julio Kaplan; he was required to add heard numbers at one number per second rate, while at the

same time playing five-second-a-move chess against a master level player. Even though adding

the numbers completely occupied his analytical mind, Kaplan still produced fluid and

coordinated play and held his own against the master in a series of games.

According to Dreyfus and Dreyfus (1986: 35) the most important difference between the

levels of expertise is the gradual shift from analysis to intuition and the grade of involvement:

What should stand out is the progression from the analytic behavior of a detached

subject, consciously decomposing his environment into recognisable elements, and following

abstract rules, to involved skilled behavior based on an accumulation of concrete experiences

and the unconscious recognition of newsituations as similar to whole remembered ones.

The above presented model of Dreyfus and Dreyfus was developed based on

phenomenological observations of car drivers, pilots, second language learners, and chess


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players. Applying the model to skill acquisition of nurses Benner (1984) has provided additional

real-life evidence which is fully consistent with the model. An interesting feature of this model is

that according to it at higher levels of expertise the performance grows from the abstract to more

and more concrete; this is the exact opposite what we observed in childrens knowledge increase

and the opposite of what has been observed since Plato (Mr, 1990: 119) that the learner moves

from concrete facts towards more general and more abstract rules. This is only noted here and

will be revisited later in the paper.

The Simon-Model of Expertise

One way of describing knowledge common in cognitive psychology is as a system of

elements of knowledge called cognitive schemata; as the Simon-model describes knowledge

using cognitive schemata the concept is very briefly revisited here. Simon himself was quite

reluctant using the term and talked about chunks (e.g. Simon & Feigenbaum, 1964) and

templates (Gobet & Simon, 1996b) instead, however the concept of cognitive schemata is more

general and widely accepted and thus it is adopted here. The term schema, in the sense used here,

was originally introduced by Bartlett (1932: 199 ff), although he mentioned that the term

schema was already in use and thus he was reluctant to use it (ibid: 200-201):

I strongly dislike the term schema. It is at once too definite and too sketchy it does

not indicate what is very essential to the whole notion, that the organised mass results of past

changes of position and posture actively doing something all the time; are, so to speak, carried

along with us, complete, through developing, from moment to moment.

The problem with the concept of cognitive schemata is that, we do not really know what

they are, but we know that they are the elementary building blocks of knowledge, because this is


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how they have been defined. Neisser (1967: 8) suggests that the concept of schemata should be

equalled with the concept of information:

in the eyes of many psychologists, a theory which dealt with cognitive transformations,

memory schemata, and the like was not about anything. One could understand theories that dealt

with overt movements, or with physiology; one could even understand (and deplore) theories

which dealt with the content of consciousness; but what kind of a thing is a schema? If memory

consists of transformations, what is transformed? So long as cognitive psychology literally did

not know what it was talking about, there was always a danger that it was talking about nothing

at all. This is no longer a serious risk. Information is what is transformed, and the structured

pattern of its transformations is what we want to understand.

Although, Neissers interpretation might be right, it does not help the present investigation;

the unknown term of schemata was only replaced with the unknown term of information. The

problem with the concept of information is that many (would) like to measure it. This

measurement is typically based on the Shannon-formula (Shannon, 1948; Shannon & Weaver,

1963: 14), which is appropriate for measuring, that is, measuring the capacity of the

communication channel. However, according to Miller (2003: 141) information has nothing to

do with the present conception of information and it has a history of not being applicable in

psychology. A variety of mutually incompatible definitions is available for the information

depending on the discipline and the taste of the author; usually these definitions assume some

kind of new data input which makes sense to someone. Drtos (1997: A-10) expressed the

essence of this problem:


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which common measuring unit could express the amount of information in a satellite

photo, in the flavour of a rose, in Bolero or e.g. in this hypertext? We do not know the answer to

this question yet, maybe there is no answer to it at all.

To avoid the problems and misunderstandings about the concept of information in this

paper the term will be avoided; instead, the schema-description of Mr (1990: 84) is adopted:

Cognitive schemata are units meaningful in themselves with independent meanings. They

direct perception and thinking actively, while also being modified themselves, depending on the

discovered information. Cognitive schemata have very complex inner structures, various pieces

of information are organized in them by different relations. The various schemata are organized

in a complex way in our brains; in the course of their activities they pass on information to each

other and also modify each other continuously.

A cognitive schema can be anything that we know and that forms a single whole,

regardless of the size; e.g. a letter of the alphabet, a word, and even a whole poem is a single

schema. The cognitive schemata (i.e. our knowledge) are in our long-term memory. This leads us

to the next problem about cognitive schemata: they cannot be directly examined. We can neither

put them under a microscope nor measure their weight or size. The only possibility is the indirect

examination, i.e. drawing conclusions about their features on basis of observing their

(inter)actions.2 This can be done only when the schemata are in short-term memory (STM). The

capacity of the STM is, however, limited. This limitation is originally identified by Miller

(1956), this is the magic number 72, meaning that we can have 72 cognitive schemata in our

STM at one time. As we do not have the luxury of examining the LTM directly and the STM is

thus limited, the only possibility is to estimate the number of cognitive schemata in the LTM by

2 This is similar to the measuring in quantum physics. We are unable to directly measure the mass of an electron, butwe can conclude it by knowing how much (kinetic) energy it transmits in collision.


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observing them in action while they are in the limited STM. This is what Simon and his

collaborators did by examining the performance of chess players.

In the 1950s Simon, with Newel (e.g. Newell, Shaw, & Simon, 1958; Newell & Simon,

1972), started using computers to simulate some aspects of human thinking. Based on this Simon

(Gregg & Simon, 1967; Simon & Barenfeld, 1969) developed the conception of chunking; that is

the mechanism of forming hierarchies of cognitive schemata. A simple experiment which we

have carried out with students countless times demonstrates that cognitive schemata naturally

form hierarchies. First we ask them if they know the national anthem, and then ask what is the

10

th

word of it; after several seconds we continue: So you do not know it after all? It is

important in this demonstration not to ask a word before the 9 th (STM limit) and not to wait too

long before the conclusion (so not to allow time to recite it and count the words). The national

anthem is also a single cognitive schema, which is the reason that they cannot respond

immediately; first they have to take it apart this is only possible as schemata are naturally

organized hierarchically. Elementary schemata, that would correspond to letters in the previous

example, are merged to higher level meta-schemata (e.g. Mr, 1990), such as words, and

these into even-higher-level meta-schemata, such as the national anthem and thus we got a

multi-levelled hierarchy. We can also see here that a single schema may belong to various meta-

schemata, e.g. the same word in various poems.

Subsequently, Simon and his collaborators (Chase & Simon, 1973a, 1973b; Gobet &

Simon, 1996a, 1996b, 2000) carried out numerous experiments with chess players of varying

strength from novice to grandmaster, showing that stronger players encode chess positions into

larger chunks, i.e. stronger chess players have more complex and thus presumably higher level

meta-schemata. For a novice the location of a knight or bishop on the board can barely fit into a


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cognitive schema, for a more advance player a schema represents a specific pattern of pieces in a

specific location, for the strongest players a complete position or even a series of positions may

form a single schema. Kasparov claimed that he knows at least 10,000 complete games by heart,

which means that for him a complete game can be a schema. Apart from thus demonstrating the

hierarchical nature of cognitive schemata and showing that the complexity of schemata increases

by the increase of expertise, Simon and his collaborators also showed that the meaningfulness of

a position is of great importance. If the chess positions were taken from real chess games the

stronger chess players consistently performed better than their weaker colleagues but in

meaningless positions the difference became minimal (Gobet & Simon, 1996a) especially with

short presentation times (Gobet & Simon, 2000).

The experiments with chess players also served as basis for Simon (e.g. Chase & Simon,

1973a; Prietula & Simon, 1989: 121; Simon, 1996: 51-110; Simon & Gilmartin, 1973) to

estimate the overall number of cognitive schemata; he concluded that at the highest level of

knowledge one has around 50,000 cognitive schemata. Using a different estimation procedure,

Mr (1990) arrived at a similar number of a few tens of thousands. According to all the

estimations, using various techniques, the highest level of knowledge is in the range 25,000-

100,000 cognitive schemata3. (Simon, 1974: 487) As it was said earlier, cognitive schemata

reside in the long term memory, thus it is reasonable to assume that the estimated number is the

limitation of the LTM.

Apart from the differences in the number of cognitive schemata Simon and his

collaborators (Larkin, McDermott, Simon, & Simon, 1980) also observed qualitative differences;

these are likely to be consequences of the higher complexity of schemata at higher levels of

3This is why knowing at least 10,000 complete games by hart me ans that a complete game has to be a cognitiveschema in Kasparovs knowledge of chess.


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knowledge. The differences they observed is that at higher levels of expertise we are better at

deciding when to use which principle and how to use them, we are flexible adopting our

knowledge to different contexts and we are more likely to develop it further if the context

requires it.

While he gave an estimation of the number of cognitive schemata at the highest knowledge

level and showed some differences between the schemata at lower and higher levels of expertise

as well as identified some differences in terms of what those on higher knowledge levels are

good at, Simon never defined levels of knowledge himself. He talked about Class-A chess

players, experts, masters and grandmasters; the latter three terms he also used for other areas

apart from chess. His interest, however, laid elsewhere he wanted to develop an information-

processing model of cognition and he came a long way towards his goal; he developed his

original model in the 1950s and kept fine-tuning it until his death.

Mr (1990), who was doing estimations of the number of cognitive schemata at the

highest knowledge levels independently of Simon (ibid: 115-118), defined the levels of

knowledge in this approach (ibid: 119-121) taking the idea from the Dreyfus-model. Mr

distinguished four levels, the beginner the advanced student, the candidate-master and the

grandmaster. According to his model, at each knowledge level the number of schemata grows by

an order of magnitude and it takes 2.5-3 years to move one step; which is fully consistent with

the 10-year rule of skill acquisition (Ericsson, 1996), i.e. that it takes 10 years to get from novice

to the highest knowledge level.

Describing knowledge as an image created about the reality, what and how much one sees

of the reality of a discipline depends on the number of her/his cognitive schemata in the

discipline. Having a few ten schemata, the beginner will see only few elements, and (s)he will


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see them indistinctly; (s)he would not know which element to connect to which other one or how

to do it. As (s)he knows so little about the discipline, (s)he will try to apply her/his everyday

schemata. The advanced student, with several hundred schemata of the discipline, will see quite

a lot of elements, though most of them indistinctly; (s)he will manage to connect some of them

assembling smaller-bigger component images, though (s)he will fail to join up the component

images. The expert possesses several thousands of schemata of the discipline; (s)he will see all

the elements and every connection between the elements (or the component images). (S)He will

see every detail that is to see; (s)he will only not see the image itself. The grandmaster of the

discipline has several tens of thousands of professional schemata; (s)he will not see that there are

elements at all; (s)he will see the image itself, losing the details.

As we have seen earlier in this section, apart from having larger number of schemata the

higher level of expertise also means more complex schemata. This is what Mr (ibid: 119-120)

uses to explain how the observations of Dreyfus and Dreyfus that expertise grows towards the

more concrete as well as the more usual observations that it grows towards the more general (as

noted at the end of the previous section) in spite of the apparent contradiction between them can

be both true at the same time. The growing number of schemata is related to the more concrete

nature of the higher expertise while the higher complexity of schemata indicates more general

nature of it; however, this more general nature should not be understood in the abstract sense of

formal logic but in the sense of the specific logic of the discipline.

As both Mr (ibid: 123-124) and Dreyfus and Dreyfus (1986: 21) remark, not everyone

will and can achieve the highest level of expertise; in fact only very few ever do. Those who can

achieve the highest level of knowledge we call talented or, with a more beautiful and more

appropriate term, gifted.


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As we have seen in the present section the two existing models of knowledge levels are

very different and, although they do not particularly support they also do not contradict each

other. The origin of the two models is also different; the Dreyfus-model is primarily based on

phenomenological observations which were to a small extent supplemented by experiments,

while the Simon-model is primarily based on experiments with some additional observations.

What is the same about these two origins is that both are drawing conclusions about the mind

based on behaviour and, as Dreyfus and Dreyfus (1984: 226) note a description of skilled

behavior can never be taken as conclusive evidence as to what is going on in the mind or in the

brain. In the next section we are introducing our new approach which is different from its

origin from both of the previous onesit is developed by pure speculation which was then fine-

tuned on the basis of observations and thought experiments. However, as we will show, it is

consistent with both previous models and thus can claim their supporting evidence as well. Of

course, we acknowledge that speculation is not better way of figuring out how the mind works

than observing behaviour but we believe that it is also not worse.

A NEW CONCEPTION OF KNOWLEDGE LEVELS

Although we were familiar with the previously introduced two models of knowledge

levels, our new model has not been built upon them as foundations. Indeed, for a while we did

not even know that our new model is fully coherent with both of the existing ones. We will show

later that our conception is not only in harmony with the existing models but it can also help

establishing a connection between the two of them. However, the story of the origins of our

model is quite different. The roots are in a casual conversation about the mystical ideas of the


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legendary sage Hermes Trismegistus4 from the ancient Egypt who described the structure of the

world of humans and gods using numbers 1-9. The first five numbers correspond to the world of

humans, then the first four are mirrored on through number five (according to the principle of as

above, so below from the Tabula Smaragdina5) which describe the world of gods this means

that we only need to consider numbers 1-5. Then we combined this with Marcuses (1964)

philosophy on one-dimensional existence; and then we started to examine what would this mean

in terms of knowledge. The result is a model that is fully consistent with the two existing ones,

easy to comprehend intuitively, provides basis for estimating the knowledge level of a person

and it does not contain any of the mystical elements that initiated its conception.

In the first part of this section we are describing the model in its present form and

subsequently discuss the links to the Dreyfus- and the Simon-model as well as some

implications.

The Knowledge Levels in 3-D

Starting from Hermes Trismegistus mystical ideas our model is based on a geometrical

metaphor. The person who just starts engaging with the discipline sees a single point; as a single

point is dimensionless; we call this zero-dimensional (0-D) knowledge. On the second

knowledge level two points can be seen at the same time; two points determine a line, therefore

this is a one-dimensional (1-D) knowledge, and so forth. The strength of this metaphoric

description is that qualitative inferences can be made based on geometrical knowledge; i.e. the

mathematical qualities can inform our intuitive understanding of knowledge levels. By doing so,

our model becomes easy to comprehend, similar to the Dreyfus-model. However, the subtle

4 Often identified with the Egyptian god Thoth.

5 Also known as the Emerald Tablet, a work claimed to contain primordial secret knowledge of HermesTrismegistus.


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differences between the knowledge levels in the Dreyfus-model are somewhat vague; this does

not diminish the explanatory strength of the Dreyfus-model but makes it difficult to use it e.g. for

estimating the knowledge level of a person. The Simon-model (in its Mr-version) is

completely exact; however, it is not very informative in qualitative terms and can be understood

only in a very abstract sense (i.e. apart by those being intimately familiar with the conception of

cognitive schemata). The Simon-model is appropriate for giving accurate estimation of the

knowledge level of a person but such estimations are very laborious and incredibly expensive.

Our model resembles the exactness of the Simon-model in a metaphoric sense by means of

geometry and, at the same time, it has the explanatory power of the Dreyfus-model and beyond.

Furthermore it provides means for intuitive (although not necessarily very exact) way of

estimating the level of personal knowledge. We name the knowledge levels in our model novice,

advanced (beginner), expert, master, and grandmaster; the first two names are taken from the

Dreyfus-model, the last two were used by Simon (see above) and are very near to Mrs names;

the expert level requires a bit of explanation. In the Dreyfus-model the highest level is called

expert, although the master is also used with similar meaning; Simon uses the expert level

sometimes to indicate the highest level, sometimes to describe the level between Class-A

(competent level in the Dreyfus-model) and master chess-player. The reason that we named our

middle-level expert is that we have found that this is the highest knowledge level at which the

knowledge can be articulated and thus expert systems can be built relatively easily. In the

following paragraphs we describe the five levels of knowledge (see Figure 1) using our

geometrical metaphor, demonstrate how inferences can be made based on this metaphor and

provide some examples.


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--------------------------------Insert Figure 1 about here--------------------------------

On the lowest level of knowledge the novice first sees a single point and then more and

more isolated points but one unit remains a single point only; this is the zero dimensional (0-D)

knowledge. (First item on Figure 1.) These isolated points correspond to isolated facts of the

discipline or, if any relationships or rules are learned, these will also be treated as facts therefore

these unquestionable truths we call axioms or doctrines or dogmas (depending on the discipline).

So the acquired knowledge at this level is a list of unrelated concepts and the performance

(applied knowledge) is reiterating these concepts. If there are any formulas (relationships or rules

presented as facts) the only thing the novice can do with them is identifying the actual facts one

by one and substituting them into the formulas; the formulas do not make sense to them. The

complexity level of such knowledge is at the level of elements with no relations between them; it

reminds of a lexicon. To use a business example, the novice can list the leadership roles

(according to a chosen author) without seeing any relationship between the roles or between the

lists. If they can produce something more it is due to their talent or previous knowledge from a

different discipline that they borrow. The high performance at this level can be described as

precise.

On the next level the advanced beginneris able to connect two points using a straight line,

thus we call this one-dimensional (1-D) knowledge. (Second item on Figure 1.) The lines can be

directed from the one point to the other and along such directed lines (arrows, vectors) methods

can be devised, meaning that we can learn how to get from one point to another; these can

correspond to simple causal relations or logical if... then rules. There may be more than two


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points on the same straight line or several unrelated straight lines in 1-D knowledge but no

changes of direction. The acquired knowledge on this level can be presented as isolated causal

relations, using which the advanced beginner can describe processes that are deterministic or

stochastic presented as a set of deterministic ones. A typical area for this knowledge would be

the elementary physics; its deterministic and stochastic systems6 can be described using linear

differential equations. Examples for this kind of knowledge are about accomplishing well-

structured tasks according to the learned recipes, such as applying instant ways of improving

motivation and the advanced beginner will wonder why the instant solutions all too often do

not work. The high performance can be described as efficient.

The expert is able to connect three points into a triangle; as three not collinear points

determine a plane, we call this two-dimensional (2-D) knowledge. (Third item on Figure 1.)

Using three points it is possible to handle circular relationships, such as positive or negative

feedback loops. This enables the expert to modify her/his action along the way and handle

interconnected priorities of the concepts. Gradually the expert may learn how to connect more

and more points but will always remain in a plain. If one sees three dots the simple relations

from the previous level will prove poor for providing satisfactory explanation; this does not

mean, of course, that the expert cannot handle e.g. causal relations (any two from the three dots

may be connected by an arrow); this means that here we can have a richer picture of less rigid

relations through feedback loops or series of interconnected feedback loops that make all sorts of

circular arguments possible. The acquired knowledge at expert level correspond to structures;

complexity level here is that of the chaos, which corresponds to nonlinear differential equations

and the corresponding typical field is biology (especially genetics). This is the level where the

6 Only those stochastic systems belong here which are composed of several deterministic ones; thus this level is notappropriate for e.g. quantum physics, in which the stochastic nature is inherent.


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dissipative structures can be described (e.g. Prigogine, 1997), these dissipative structures enable

balanced states of systems very far from the equilibrium (e.g. metabolism enables us to live far

from the equilibrium which would be death). In business an expert is able to manage individual

processes, such as teamwork processes regarding knowledge and value systems. There will be a

limit to this managing and the expert will be able to recognize these limits but will have no

chance to do something beyond the limits. The high performance can be described as effective.

The master sees four points that form a tetrahedron; this is what we call a three-

dimensional (3-D) knowledge. The master basically sees all the possible relations in the world of

her/his discipline; everything relates to everything and to the whole. One may be surprised by the

ease the master handles those issues the expert struggles with; having in mind that one of the

experts problem is how to handle all those relationships and the master sees many more of

them. However, the master does not only see the relations, (s)he sees the whole as well. The

tetrahedron automatically provides her/him with a perspective and (s)he can also change between

various perspectives easily. Thus the master can easily amend processes that do not work or

adopt processes to radically different contexts. The ways of expressing such high-level

knowledge is strongly shifted towards the softer; and much of it cannot be expressed at all. The

performance of the master can be described as great, but only by the expert; at lower level the

masterpiece cannot be described.

Not many of us can draw in more than three dimensions, or even imagine it. The fifth point

that the grandmastersees does not indicate a new dimension; we imagine this fifth point as being

in the middle of a tetrahedron. It brings the previous four points to harmony; this is the

quintessence. That is why the grandmaster always has a coherent view of the world (not

necessarily the same coherent view at different occasions); this is what enables her/him to see the


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(quint)essence of things; this is what makes the almost instantaneous responses in immensely

complex situations possible. This quintessential point corresponds to parables using which the

grandmaster enlightens her/his disciple about complex pieces of knowledge. The grandmaster

does not see the relations anymore, that all became one. Verbal expression of such knowledge

would be impossible if we would not have the artistic tools at hand, parables, metaphors,

symbols, and even paintings, poems, or music are more adequate representations. The

grandmaster does not know more than the master but (s)he has better sense for the

(quint)essence. This balanced state of the five points seems to be the same as what we typically

understand by wisdom. Interestingly, the grandmaster performance actually makes sense at all

levels of knowledge. Depending whether it contradicts or confirms the lower level knowledge it

will be described asperfector nonsense.

The aim of the description of our knowledge levels using the geometric metaphor was to

offer the taste of how this works. We did not aim for completeness and we think that it would

even be impossible we expect that other people will derive additional explanations from this

metaphor. The Table 1 contains some of the present features and some additional ones that did

not fit into the present description; e.g. about the size and structure of knowledge elements, the

volumes of knowledge that can be transferred at certain levels during a 12 weeks semester, the

sensible mode of transfer and the knowledge media, what percentage is appropriate for being

delivered by e-learning, etc.

Apart from describing the various features of the certain knowledge levels, the geometric

metaphor is also useful to imagine how a person gets from one level to the next one. The novice

sees several isolated points and tries to connect two of them; as (s)he gradually succeeds to this

with more and more point pairs, (s)he climbs to the advanced level. The advance beginner


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identifies more and more collinear points and takes one in the middle and tries pulling it off the

line without breaking the connectionssucceeding with more and more of these (s)he gradually

becomes an expert. Similarly, the expert tries to push the middle of the triangle out of the plane

without breaking what we can imagine as a fabric thus converting her/his triangle into a

tetrahedron; the more (s)he succeeds the more (s)he spends in the master-realm of the discipline.

The master is playing around with the four corners of her/his tetrahedron trying to get them into

a state of balance; when this happens, the quintessence appears, keeping it this fragile balance.

Of course, this fragile state will also break down many times before the quintessence becomes

stable.

--------------------------------Insert Table 1 about here

--------------------------------

Discussion and links to the previous models

Some connections with the Dreyfus-model are directly available from the descriptions. The

isolated context-free facts correspond to the isolated points the novice can see; this is even more

obvious if we consider that apart from the term context-free Dreyfus and Dreyfus (1984: 222)

also use the term interpretation-free in relation with the facts of the novice, which basically

means the same as the axiom or doctrine or dogma. Dreyfus and Dreyfus (1987: 25) mention that

there is a three-dimensional quality of the competence, while in our model competence is 2-D

knowledge. However, the reference system is different, what is indicated is a perspective that

makes the relative salience of the facts and situational elements possible based on their mutual


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relations; in our model the perspective and adjusting the salience levels based on the other points

is possible trough the feedback loopsso there is no contradiction, only a different use of terms.

There are also some easy-to-make connections with the Simon-model. We can, for

instance, easily understand having an order of magnitude higher number of cognitive schemata

on each higher level based on the geometrical analogy: putting the schemata along a line, in a

plane or in a cube can easily do the trick. A less simple but very instructive example is how both

models can lead to the same conclusion refers to the highest knowledge level. Based on his

analysis of cognitive schemata Mr (1990: 193) has concluded that:

Perhaps the feeling of enlightenment can also be conceived as the activization of a

cognitive schema that is simultaneously the meta-level of all our cognitive schemata, but itself

has no meta-level. Unfortunately, I cannot think of any laboratory experiment that could possibly

prove even the faintest aspect of such statement.

This means exactly the same as out fifth quintessence point.

In both the Dreyfus- and the Simon-model it has been mentioned that anyone can only

understand the knowledge which is one level higher than the knowledge level of the receiver.

This is contradicted by the observation (our and others) that everyone seems to understand the

teachings of the greatest masters. Our model can provide a metaphoric explanation to this

phenomenon too: When you see isolated points only, you can only imagine that it should be

possible to get from one to the other the 2-D does not make any sense. Similarly, if you see

point pairs connected by vectors you can imagine that it should be possible to put several of

these one after another and get back where you were but you cannot imagine getting off the

plane into the 3-D. If this is a teaching-learning setting, you can imagine that the 1-D teacher is

pulling the 0-D learner from a point where (s)he is into another one or make the destination point


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blinking when the learner is trying to get there. Similarly, plane shapes can be assembled of

blinking lines and 3-D shapes out of 2-D ones. However, the quintessence point can do a trick

with the tetrahedron; it can highlight various parts or all of it for you, depending on your existing

knowledge: if you can only receive isolated points yet, the corners of the tetrahedron will blink;

if you can receive lines, you will see the edges of the tetrahedron blinking; if you can receive 2-

D shapes the sides will blink and if you are ready for a 3-D piece of knowledge you will see the

whole tetrahedron blinking. This also fits the observation that, although everyone seem to

understand the teaching of the grandmaster, everyone can learn different things from her/his

teachings these newly acquired knowledge pieces may even contradict to each other on lower

levels, so do not fight about which interpretation is right.

The same phenomenon is responsible to what can be observed when two grandmasters

from different discipline talk to each other7: One is telling parables from political economics and

the other from game theory. The eventual listener will almost certainly conclude that they talk

aside each other without paying attention to each other. There is one problem only; they

apparently understand each other very well. And both of the can tell their colleagues later what

the other grandmaster told them and what they have learned from each other. And if some of

these colleagues have listened to the conversation will probably swear that none of this has been

told. It takes (at least) a master to understand that the two grandmasters are actually talking about

their views of artificial intelligence... The parable told by one grandmaster is a quintessence itself

and, instead of working within his own tetrahedron it establishes direct connection to the other

masters tetrahedron and highlights that one. This is the quantum physics of knowledge levels

7 This is a real life story, told as it happened, only without names.


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and it is an amazing experience in itself. It was so inspiring that the listening master conceived

the idea of the blinking tetrahedrons...

CONCLUSIONS

In this paper we have reviewed the two existing models of knowledge levels and offered a

third alternative to them. All the three models have different origins, i.e. the Dreyfus-model is

primarily based on phenomenological observations, the Simon-model is primarily based on

experimentation, and our model is primarily based on speculation. The three models also serve

different purposes or, more precisely, the researchers that use them pursue different and varying

purposes. What is common between the three models is that each of them can serve as

framework for various kinds of research projects. We have also shown that the three models are

in harmony; not only that they do not contradict each other but they are apparently various views

of the same thing and support each other. We have also shown that they may lead to the same

conclusions and that using two of them at the same time may be a chance for additional insight.

The strength of our model is that it is easy to intuitively comprehend; it provides intuitively

obvious and logically correct albeit metaphoric explanations; and it can also serve as basis to

estimate the level of personal knowledge. This may be particularly useful when someone has to

be selected to learn a particular new thing of certain complexity, when people should be grouped

for courses, when teams are being assembled, when ill-structured problems of critical importance

are to be solved.


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FIGURES

FIGURE 1:Knowledge levels for 0-D ot 3-D

0D1D

3Dbut

3D

2D


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TABLES

TABLE 2:Some characteristics of the knowledge levels

dimensions 0-D 1-D 2-D 3-Dpoints 1 2 3 4 5

knowledge

elements

1keyword 2keywordsconnec-tedwith1causal

relation

3keywordsconnectedby3causalrelationsinto1

cycle

4keywordswith4cyclesand1

structure

1meta-concept

volumeof

knowledge12*1=12 12*(2+1)=36 6*3+6*(3+1)=42 4*4+4*4+4*1=36 12*1=12

e-learning all 2/3 1/2 1/3 none

formformulasor

doctrines

linear(differential)

equations

nonlinear(differential)equ-

ationschaos&fractals

topologiesorsets parable

processformulae

substitution

deterministic&

statisticchaotic sensibleheuristic

meta-processof

processcreation

complexity element relation structure process validity

typicaldiscipline lexicon elementaryphysics genetics psychology philosophy

acquiredknowledge

listofunrelatedconcepts

isolatedcausalrelations

(positive&negative)feedbackloops,cycles

setsofconcepts meta-knowledge(essence)

applied

knowledge

iterating

doctrines

accomplishingwell-

structuredtasks

managingindividual

processes

adapting/modifying

processes

cratingnew

processes

knowledge

mediamultipleauthors:manual singleauthors:textbook teamofauthorsledbyamaster:book

knowledgetransfer

e-book classroom+e-learningclassroom

(pub)


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Knowledge Levels