Descriptive Model of Learning Capability

Viktor Dörfler Department of Management Science

University of Strathclyde Glasgow, United Kingdom

Email: viktor@mansci.strath.ac.uk Abstract In this paper the impact of the existing knowledge on the knowledge increase is examined. As the result of the investigation the descriptive model of learning capability is introduced. The starting point is Polanyi’s personal knowledge while the organizational knowledge is neglected. The knowledge is described with cognitive schemata and the system approach is applied onto this description. The most important factors are identified and used to present the behaviour of the model. The model may be considered as a component for a model of learning ability and can also provide basis for a simulation. Keywords: knowledge increase, learning, cognitive patterns, cognitive schemata, personal knowledge Introduction The problem domain is the knowledge increase. Its boundaries are fuzzy, as in any problem domain, though some of them may be made sharp. It is also necessary to make some of them sharp otherwise it is impossible to define the component problem which is solved in this paper. Beyond these boundaries, further component problems are to be solved but in this paper they are not investigated. The knowledge increase depends on several factors (e.g. the willingness of the person to increase his knowledge and his attention during the knowledge increase), though in this paper only one of them is examined: the impact of the present knowledge of the person to hi knowledge increase. I call the model of this factor the learning capability. The Increase of Personal Knowledge I start from the Polanyi’s approach that organizational knowledge does not exist (Polanyi 1962a), only the personal. Therefore the investigations on organizational knowledge are not considered. This does not mean that I neglect the organizational memory (Davenport – Prusak 2000). I also know that one of the main roles of the knowledge management is to separate the knowledge from the person, creating knowledge-bases (Davenport – Prusak 2000, Nonaka – Takeuchi 1995). However, both the knowledge-bases and the organizational memory are passive; the active knowledge only exists in the head of the member of the organization – the person. The knowledge bases, the organizational memory and the knowledge of the members compose the knowledge capital of the

organization. All of them are important though this investigation engages with the personal knowledge only. Polanyi’s famous classification divides the knowledge into codified (explicit) and tacit. (Polanyi 1966) These are not disjunctive categories. Once we learned a grammar rule we are able to put this knowledge into words. Later we forget the definition but while writing we use the rule perfectly. The tacit knowledge may increase when some explicit knowledge becomes tacit or by training, as in case of skills. However, the most important part of the increase of the tacit knowledge is what happens in master-apprentice relation. The precondition is that the follower must believe that the master knows: “… the methods of scientific inquiry cannot be explicitly formulated and hence can be transmitted only in the same ways as an art, by the affiliation of apprentices to a master.” (Polanyi 1962/b) This paper is concerned with explicit knowledge only. Minsky (1982) distinguishes the special knowledge from the common sense. Common sense is out of the domain of this investigation. We know much more about the increase of the special knowledge then about the increase of the common sense, partly as it begins before the representation is formed. The most of common sense is tacit and also acquired tacitly in personal relationship of master-apprentice or similar e.g. parent-child. Otherwise we have no reason to presume that the increase of the special knowledge and the common sense differs. This investigation is limited to the special knowledge. Investigating the Capability for Knowledge Increase In this paper deduction is used; it’s my firm belief that it is necessary: “I believe that theories are prior to observations… I do not believe, therefore, in the ‘method of generalization’… I believe, rather, that the function of observation and experiment is the more modest one of helping us to test our theories…” and “I do not believe that we ever make inductive generalisations… I believe that the prejudice that we proceed in this way is a kind of optical illusion…” (Popper 1986) Russell came to the conclusion that induction does not lead to new scientific results, and it also fails as logical construction. “… induction is a bodily habit, and only by courtesy a logical process.” (Russell 1992) Roszak has found the roots of induction in the early empiricism, when scientists were looking for a neutrally innocent approach of pure observation and data collection. But the empiricism itself is an idea — prompted to Descartes by an angel. (Roszak 1994) The examination of the schemata is possible only indirectly. We can neither put them under the microscope nor can we measure them. The only thing we know is that they are the basic units of knowledge; as schemata are defined so. Indirect examination means that we can draw conclusions about the attributes of the schemata only from their interactions. This is similar to the measuring process in quantum physics: we are unable to directly measure the mass of an electron, but we can conclude it by knowing how much (motion) energy it does transmit in a collision. Accordingly to the characteristics of the problem domain experimenting is not only impossible but useless as well. The experiments in natural sciences are based on similarity (equality!?) of conditions and the consequent generation of certain impacts.

However in social/human sciences it is impossible to have similar conditions, as the first realization of the experiment will change them. Furthermore the artificial isolation may eliminate the most important impacts. (Popper 1986) However, phenomenological observations and thought experiments can be and were used. The source of the new knowledge I call the available knowledge, which I define as the knowledge admitted by the paradigm – therefore I do not investigate the effect of the paradigm. (Kuhn 1996) The Cognitive Schema The basic unit of knowledge is the cognitive schema. The component-knowledge appears if the schemata are organized. Relations among the schemata are constantly changing. The term was invented by Bartlett (1932) and it is widely accepted (e.g. Neisser 1982, Mér� 1990 and Baddeley 1999). “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.” (Mér� 1990) If we have to complete a task, to make a decision or to solve a problem, our schemata form an ad hoc structure. (Baracskai 1999) The structure remains until we have completed the task, made the decision or solved the problem. If the forming of the ad hoc structure is accompanied by a deeper understanding, a bigger unit of knowledge, a component-knowledge evolves. Component-knowledge evolves by forming a meta-schema which describes its structure and the schemata (some also dissolved in it). On the basis of the description stored in the structure, the meta-schema is at any time able to re-create the schemata together with their relations. A schema may belong to various component-knowledges and to various meta-schemata accordingly, by this means the meta-schemata may overlap. They may also merge or incorporate new schemata. Afterwards they will cover a larger domain. Meta-schemata form hierarchies, so the new meta-schema can evolve not only from schemata but also from other meta-schemata. Gobet and Simon (1996a, 1996b), Mér� (1990) and Baracskai (1999) investigated the levels of knowledge, which are described with the number of cognitive schemata. In each discipline the beginner level is at few ten schemata, the advanced at few hundred, the expert at few thousand and the (grand)master at few ten thousand. The mentioned authors use different names for the certain levels, though they can be identified by the number of cognitive schemata. A master has a few ten thousands schemata in a certain discipline. The chess master as the Simon verified (Gobet-Simon 1996a) recognizes few ten thousand positions, which are obviously meta-schemata. Kasparov knows at least ten thousand complete games by

heart; these do not consist of moves but of positions. This means that he merges the schemata into meta-schemata and after into a higher level meta-schemata, etc. The difference between the expert and the master level of knowledge means more than increase of the number of schemata by an order of magnitude; there is also a qualitative difference and it is not only the existence of meta-schemata. Meta-schemata surely exist on expert level furthermore they may appear on advanced level as well. Baracskai says that in the head of a master the discipline frames up in a whole. It means that a special schema is formed – I call it super-schema –, which incorporates all the meta-schemata of the discipline. The nature of the super-schema differs from that of the meta-schemata, as when meta-schemata incorporate schemata and/or each other, the original ones disappear. The super-schema’s ingredients do not disappear. The knowledge of the master is tacit. The highest level of explicit knowledge appears at the level of the expert. Therefore the expert knowledge is the highest level which can be modelled (Velencei 1998), and it is so done in this paper. All these statements apply for the long-term memory (LTM). There are other kinds of memories: the sensory memory and the short-term memory (STM). The environmental input arrives first to the sensory memory and then it is transferred to the STM and then to the LTM. (Baddeley 1999, Eysenck 2001) I do not engage with the sensory memory at all, and the STM is only important for this investigation as it indicates a new boundary: the examination of cognitive schemata (meta-schemata) is possible only while they are placed in STM and the capacity of the STM is about seven schemata. (Eysenck 2001) During the knowledge increase new component knowledge (a schema) is absorbed. The process of its absorption to the sensory memory, its passing to STM and then to LTM is not investigated in this thesis, instead of process, states are examined. The knowledge and the knowledge increase of a master fundamentally differ from the others’. This is caused by the existence of the super-schema. The master sees the whole reality through his discipline, which means through the super schema. Therefore during the knowledge increase of the master his whole knowledge changes. In this paper I highlight and partly explain the difference between the knowledge increase of the master and the others. Applying the System Approach To examine the impact of the present knowledge to the knowledge increase I applied systems approach to the personal knowledge described by cognitive schemata. Boulding (1984) has defined eleven levels of systems of which human knowledge belongs to the ninth level, while our knowledge is more or less complete about positive feedback systems on the third level. There is no reason to believe that a higher level system, which we do not know, is similar to the lower level system, which we know. Therefore I adopt the simplest definition of system: “A system can be defined as a set of elements standing in interrelations.” (Bertalanffy 1968) I consider the relations among the elements of the system constantly changing, the stabile relations forming sub-systems, and the system has input/output relations with its environment, however, I examine only the input. I also use mathematical markings, functions, integrals, differentials; however they serve only as a

language to give more elegant and easy-to-handle description of the model. Applying the system approach I start from the following definitions:

1. The examined object (Q) is personal knowledge, before the knowledge increase I call it the present knowledge (Q0), after the knowledge increase I call it the increased knowledge (Q1). The knowledge, that is absorbed during the knowledge increase I call the new knowledge (�Q)

2. The environment is the available knowledge (�) 3. Interactions among the examined object and the environment:

a) X: � → Q the effect of available knowledge on personal knowledge, the source of the new knowledge.

b) Y: Q → � the effect of personal knowledge on available knowledge, if there is something in the increased knowledge that the available knowledge lacks, it can increase the available knowledge – in this study I will avoid this subject;

4. Component-knowledge, elements and relations: a) Qi i = (1, …, m): the ith component-knowledge – vectorially: Q is an m-

dimensional vector; b) ej j = (1, …, n): the elements are the cognitive schemata – vectorially: e is

an n-dimensional vector; c) RQ

k,l k,l = (1, …, m): the relations among the component-knowledges – vectorially: RQ is an m××××m-dimensional tensor;

d) Rek,l k,l = (1, …, n): the relations among the schemata – vectorially: Re is

an n××××n-dimensional tensor; 5. Structures (see Figure 1):

a) Se(i) = Se

(i)(e(i), Re(i)) i = (1, …, m): the microstructure of an ith component-

knowledge (e(i) is the vector of the schemata of the ith component-knowledge, Re

(i) is the tensor of the relations among the schemata of the ith component-knowledge) – vectorially: Se m-dimension vector, the projections are composed similarly to the above;

b) SQ = SQ(Q, RQ): the structure of the whole knowledge, the macrostructure, the projections are composed similarly to the above – the examination of the whole knowledge is impossible (the causes explained later), therefore I have not examined the macrostructure;

c) S = S(SQ, Se, Q, �, X, Y): the global structure, which describes the whole knowledge, the environment and the interactions between them; similarly to the macrostructure, it cannot be examined therefore I will omit it.

Re-thinking the definitions some simplifications can be made without loosing the complexity. It’s been stated that the examination of cognitive schemata may be done only when the schemata (meta-schemata) are placed in the STM. Humans have to have more than seven meta-schemata (the capacity of the STM) as everyone knows more than seven things at least on some elementary level. E.g. we know how to talk, write, ride a bicycle, make coffee, etc; thus the relations among component-knowledges along with macrostructure and global structure of knowledge cannot be examined.

Figure 1: The microstructure, the macrostructure and the global structure.

If we accept that:

1. meta-schemata describe the schema structure of component-knowledges, that 2. the macrostructure and the global structure of personal knowledge cannot be

examined; and that 3. meta-schemata are not organized in single-level structure but hierarchically

multi-levelled; then we can swap the examination of existing microstructures on different levels with the examination of meta-schemata. As in the structures of component-knowledges (in meta-schemata) there may be meta-schemata too along with the schemata, we also need to consider the relations among the component-knowledges (Q). So, even though, we do not know the whole macrostructure (SQ), some parts of it we do. Therefore in the followings instead of structures (Se, SQ) I will use meta-schemata (M). It is of no relevance to distinguish the relations among component-knowledges (RQ) or relations among schemata (Re). From now on I will use a single relation matrix, which includes both the relations among schemata and the relations among component-knowledges. For distinction I use a marking without index (R). I continue the examination in regard to the following conditions:

• one person (i.e. the personal knowledge of a person) is to be examined, • the new knowledge is a particular new knowledge, • the examination happens „here and now”.

It is an interesting thing, that these conditions perfectly mach the egocentric particulars defined by Russel (1992). “The four fundamental words of this sort are ‘I’, ‘this’, ‘here’ and ‘now’.” The Learning Capability We can approach the problem solving, by covering the problem domain with the required knowledge. Some parts of it are covered with the existing knowledge (Q0). The uncovered area can be covered with the new knowledge (�Q). When the new knowledge is absorbed we get the increased knowledge (Q1). The problem domain does not have sharp bounds, neither does the knowledge. If we start from the problem domain, we cannot know if our increased knowledge will cover it. However, for each piece of knowledge an applicable problem domain (or a component-problem) can be found.

Therefore I examine backwards, considering the problem domain (P1), which is covered with the increased knowledge (Q1) at the end. (Figure 2)

Figure 2: Absorption of the new knowledge.

If the knowledge would be simply additive, than the increased knowledge would simply happen:

QQQ ∆+= 01 [1]

But that’s wrong. Our schemata exist only through their relations. We cannot have a schema, which is not connected to others ones. This also means that we can only learn things that can be connected to our existing knowledge. If a schema is connected to the other schemata, then they will affect each other. Let the Q0 be the knowledge that can cover the P0 problem domain; than the �Q, covers the �P problem domain. The increased knowledge (Q1) covers the P1 problem domain. Then:

PPP ∆⊕= 01 , where CBA PPPP ∆⊕∆⊕∆=∆ [2]

Instead of addition I apply distinctive marking, because the additive problem domain is nonsense. The marking means “the A, the B and the C component-problems together”. Using these markings, the new knowledge is presented as:

( )dqqfQP�

∆

=∆ [3]

Here, the dq is an infinitesimal unit of the knowledge. Since schemata are considered to have limited sizes, the integration could be substituted with a sum. However, this mathematical description is easier to handle and knowledge is easier to interpret if it is connected to a problem domain. As I am not going to calculate neither the previously mentioned nor the forthcoming integrals there is no benefit in summative description. The existing knowledge is presented similarly to the previous:

( )dqqfQP�=

0

0 ; if it is not a scientist, then ( ) 1lim0

=→

qfPP

[4]

The formula means that knowledge is not changing if we do not learn something new. This observation serves as a starting point to draw conclusions about the f(q) function. The scientist increases his knowledge not only by absorbing new knowledge but also by rearranging his existing knowledge. Therefore limes of f(q) on P0 will not equal 1. It does not mean that this type of the knowledge increase takes place on P0, because the covered domain will also change. The existing knowledge – not for scientist only – changes during the knowledge increase:

( ) 0*0

0

QdqqfQP

≠= � [5]

To describe the increased knowledge, we only have to consider the integrals at the P1 problem domain:

( ) QQdqqfQP

∆+≠= � 01

1

[6]

I will not determine the function f(q) in the form of an equation. However we can make important conclusions about its character. I stated that the elements of knowledge are the schemata (p), with relations among them (R), and these relations can be organized in a structure, described by the meta-schemata (M). For the working hypothesis I use:

( ) ( )MRpfqf ,,= [7]

The knowledge increase is a process, which happens in time. We may observe that different people learn the same new knowledge at different speed. The same person acquires different pieces of new knowledge at different speed. Therefore the partial differentials of the mentioned variables by time are also likely to be worth considering. For these I will use the following markings:

t

pp

∂∂

=� , t

RR

∂∂

=� and t

MM

∂∂=� , where t is time [8]

Let’s examine the six variables, to determine, which are the needed ones for the function f(Q):

• p : Are there any schemata to which the new knowledge can be connected? – As we can absorb only the schemata, which can be connected to existing ones. So, I shall consider the schemata in the function.

• p� : I consider the schemata to be permanent. We either have a schema for something or not. So the speed of changing the relations among the schemata has no significance. I will not consider it.

• R : The relations among the schemata are changing fast. The relations that are more stabile have a roll in structures, so they are described by meta-schemata. Therefore I will not consider the relations.

• R� : The changing speed of relations is of high importance. If we have schemata to which we can connect the new knowledge, it is crucial how fast the relations evolve. Thus I will consider the differentials of the relations by time.

• M : It is important to know if there is a structure in which the new knowledge fits. Furthermore the more stabile relations are described by meta-schemata. Consequently I will consider the meta-schemata.

• M� : The creation and the modifications of meta-schemata happen with enlightment, that is to say under a zero time. Suddenly the picture is formed. Since I do not engage with enlightment, the changing speed of meta-schemata will not be considered.

What does it mean for the interpretation of function f(q):

( ) ( ) ( )MRpfMMRRppfqf ,,,,,,, ���� ≈= [9]

What is the meaning of the considered variables? • Does the person have schemata to which’ the new knowledge can be connected? –

Is he capable to learn it at all? • At what speed is the person able to build in the new schema among the existing

ones? – How fast will he learn it? • What kind of structure will receive the new schema? – How deeply will he learn

it? The function, answering these questions, I call the learning capability, and I use the following marking for it:

( ) ( )MRpcqc ,, �≈ [10] Behaviour of the Learning Capability In the following three figures it is shown how the learning capability depends on the three considered variables. I emphasize, these are not plots of the functions but only conceptions, in other word pictures. In every figure only one input is considered to be variable, the others are constant.

��������

����� ���������

��������

����� ���������

Figure 3: Learning capability by the number of schemata.

The learning capability by the number of schemata is similar to a power function (Figure 3). It comes from the observation that talented disciple needs about two years to increase

the number of schemata by ten times. Therefore the picture applies only to the person, who is talented to absorb the new knowledge.

�����

����� ���������

�����

����� ���������

Figure 4: Learning capability by the speed.

The faster somebody changes the relations among his schemata, faster he will absorb the new knowledge. Therefore the first part of the relation is linear. However above certain speed it does not bring further increase, so the curve will flatten with a horizontal asymptote. (Figure 4)

�������������

����� ���������

�������������

����� ���������

Figure 5: Learning capability by the meta-schemata.

Meta-schemata, which contradict the discipline where the new knowledge belongs, block the absorption of the new knowledge. The more of these meta-schemata, the stronger the blockade will be. By decreasing the number of contradictory meta-schemata and increasing the number of consistent meta-schemata, the learning capability suddenly increases. The worst is to have many contradictory meta-schemata, though the best is not to have many of them, only few of them. This can be explained by the hierarchical nature of the meta-schemata, so having a few meta-schemata is a higher level of meta-schemata in a certain discipline. Therefore, when finally the function flattens to a horizontal – similarly to the previous picture – there are a few meta-schemata only, on the highest level. (Figure 5)

Conclusion The goal of this paper is to explain the impact of the personal knowledge on the knowledge increase. The increase of the personal knowledge of a single person is examined, while acquiring a particular new knowledge. The increase of explicit knowledge is considered, which happens at expert level. Applying the system approach to the knowledge description using cognitive schemata, the model of learning capability is developed. Examination of the factors of the model showed, that some of the factors can be omitted without loss of initial complexity. The remaining factors are the cognitive schemata, the relations among them and the meta-schemata. The model has been developed using strict limitations. These limitations are to be revisited – perhaps using further observations – it is likely that the model can be extended beyond the initial boundaries. The model has strong explanatory features and can also be considered – together with learning willingness and attention – as a component of a more complex model of learning ability. It can, as well, serve as basis for development of a simulative model (i.e. it would be a component of a simulative model), actually a simulation with DoctuS Knowledge-Based Expert System (http://www.doctus.info) was used to validate the model. Today knowledge engineers use unknown heuristics to determine the learning capability of experts towards a new knowledge. The developed model can serve as a formalized framework for such assessment. References Baddeley, A. D. (1999) Essentials of Human Memory, Sussex: Psychology Press. Baracskai Z. (1999) A profi vezet� nem használ szakácskönyvet (The Master of

Leadership), Nyíregyháza (Hungary): “Szabolcs-Szatmár-Bereg megyei Könyvtárak” Egyesülés.

Bartlett F. C. (1932) Remembering: A Study in Experimental and Social Psychology, Cambridge: Cambridge University Press.

Boulding, K. E. (1984) The World as a Total System, London: Sage. Davenport, T. H. and Prusak L. (2000) Working Knowledge: How Organizations Manage

What They Know, Boston: Harvard Business School Press. Eysenck, M.W. (2001) Principles of Cognitive Psychology, Sussex: Psychology Press. Gobet, F. and Simon, H. A. (1996a) Recall of random and distored chess positions:

Implications for the theory of expertise. Memory & Cognition, 24(4), 493-503. Gobet, F. and Simon, H. A. (1996b) Templates in Chess Memory: Mechanism for Re-

calling Several Boards. Cognitive Psychology, 31(1), 1-40. Kuhn, T. S. (1996) The Structure of Scientific Revolutions (3rd ed.), London: The

University of Chicago Press. Mér�, L. (1990) Ways of Thinking: The Limits of Rational Thought and Artificial

Intelligence, New Jersey: World Scientific. Minsky, M. L. (1982) Why People Think Computers Can’t. AI Magazine, 3(4), URL

http://www.ai.mit.edu/people/minsky/papers/ComputersCantThink.txt Accessed 28 Nov 2003.

Neisser, U. (1982) Memorists, in Neisser, U. (Ed.), Memory Observed: Remembering in Natural Context (pp. 377-381), San Francisco: W. H. Freeman and Co.

Nonaka, I. and Takeuchi, H. (1995) The Knowledge-Creating Company, New York: Oxford University Press.

Polanyi, M. (1962a) Personal Knowledge: Towards a Post-Critical Philosophy, London: Routledge.

Polanyi, M. (1962b) The Republic of Science: Its Political and Economic Theory. Minerva, 1, 54-74.

Polanyi, M. (1966) The Tacit Dimension, Gloucester: Peter Smith. Popper, K. R. (1986) The Poverty of Historicism, London: Routledge. Roszak, T. (1994) The Cult of Information: A Neo-Luddite Treatise on High Tech,

Artificial Intelligence, and the True Art of Thinking, London: University of California Press.

Russel, B. A. (1992) Human Knowledge: Its Scope and Limits, London: Routledge. Velencei J. (1998) A szakért� tudása (The Expert Knowledge). Vezetéstudomány, 29(10),

18-26. von Bertalanffy, L. (1968) General System Theory: Foundations, Development,

Applications, New York: George Braziller.