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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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