Abstract

In this essay we will try to provide grounds for six theses:

If regarded as an expert, teacher must be expert in various domains. The theories needed for teacher's expertise can be divided on the basis of four questions:

We can classify the main educational theories along these lines. Philosophy of education is not a theory among other theories. Its position is 'beneath' or 'beyond', as is often mentioned in terms of its 'integrative'('beyond') or 'fundamental'('beneath') role. Being integrative and fundamental, philosophy of education explores inter-theoretical relationships and most general intra-teorethical issues. An important issue to be settled in philosophy of education is the question of determining dependence relations among theories.

The fixed points in complex net of integrated pedagogical theory and practice belong to theory of learning and accepted educational aims¹. It seems that there is a common agreement that theories of learning and development ('Whom ?') influence the answers given for questions 'What to and how to teach?'. We recognize in the educational aims ('Why?') the same kind of influence on curriculum and teaching methods².

We can locate educational aims in cognitive domain in a spectrum defined by two extreme points: adoption of knowledge and development of skill to acquire knowledge. Cognitive ends conceived in absolute isolation from each other represent absurd positions. We can not develop skill of acquiring knowledge without acquiring knowledge about specific topic. On the other hand, we can not acquire knowledge about anything without a skill to acquire it. If we take term 'learning' to mean 'knowledge acquisition' than our spectrum for setting educational aim consists of positions³ between two extremes, namely, 'learning to learn' and 'mere learning'.

Skill develops through exercise. In order to develop the skill for knowledge acquisition one must engage in the learning process. Logical priority belongs to 'learning by exploring', not to 'learning by being told'⁴. Therefore, the teaching method that we should adopt for the purpose of developing primal cognitive skill must resemble real process of acquiring knowledge. The teaching is symbolically controlled process of learning. Guided exercise in learning by exploring must be organized in a way that does not block activity of the student while maintaining at same time the pattern of effective learning acquisition.

There are many arguments in favor of the thesis about our inability to surpass incomplete state of our knowledge. Philosophy of science and meta-theory in general have provided lot of arguments of that kind.. Here is an incomplete list of arguments:

and for formal sciences,

The list should be longer, but two points suffice for the thesis we are discussing. They are: the impossibility of conclusive verification (1) and non-equivalence of notions of 'provability' and 'truth'(6). We must reject classical idea of inverted relation between building and presentation of a theory if we accept (1) and (6.). The theory grows from the bottom, inductively, but, once finished, it becomes deductively organized, from the top downwards. We should present finished theory in its completed form. Pedagogically and epistemologically justified presentation of theory should not take the form of a presentation of a product. Rather, reconstruction of theory's genesis is more congenial to the nature of knowledge.

Epistemology, philosophy and history of science and logic give us strong reasons to regard knowledge acquisition as never-ending process. Knowledge is not a product and we should not present him as such.

Those stages form a kind of spiral, since cognition is a process, not a product⁸.

We partly accept Popperian disregard for induction: there is no induction starting from theoretical neutral description of reality. There are nevertheless specific cognitive processes involved in hypothesis formation (induction) and hypothesis selection (abduction) which are not deductive in nature. The logically first step in cognition is formation of hypothesis. Deductive step follows in which we deduce a singular statement about not yet observed state. We check the derived statement against empirical evidence⁹. We should refute the hypothesis if the observation does not confirm it, else we must repeat deduction step. We should not conceive these repetitions as endless loop. There is vague criterion of rationality that classifies a theory as 'well-confirmed'. The same endless procedure applies to more general hypotheses (logically connected to examined one). It is hard to tell how many confirmations must a hypothesis and its more general ancestors receive for a theory to be 'well confirmed'. The principle of rationality imoposes restrictions on confirmation process. In reality, the case of refutation is more common outcome. In that case we encounter a stage in which we must refute the hypothesis. In more realistic modeling refutation is ultimate cognitive action. Falsification of hypothesis leads to further cognitive action of testing all the statements that imply it. The sentences that imply falsehood can not all be true¹⁰. Removal of hypothesis is not always a rational choice since it might involve a grave reconstruction of knowledge base. It seems that in reality a process of 'belief revision' is more common. The sub-process of hypothesis formulation is the most complex one and requires separate treatment (to be given later ).

We have put forward the thesis: 'By accepting the discovery as cognitive skill the means for that end are to be sought in the way the teaching is organized'. Since the skill develops through exercise it follows that appropriate teaching process must be homomorphic to real discovery process. In our model of discovery we must distinguish the following components: logic, reality and background knowledge. Scientist embodies background knowledge and logic. The process can now be analyzed by looking which component comes to fore in each stage: background knowledge¹¹ gives rise to hypothesis, logic deduces statements about not-yet-observed facts, reality verifies or falsifies deduced statement, logic removes hypothesis.

Teaching which is homomorphic to real learning process gives the teacher only those roles which in real process belong to the reality and to background knowledge. It the only way in which a student becomes involved in process of discovery similar to real one. The notion of 'learning by being told' presupposes that something has been 'learned by exploring'. The skill that enables the latter is common to all the people. By learning natural language each human subject shows the ability to 'learn by exploring'. In the mainstream in philosophy of education many authors have recognized the importance of homomorphism of learning and teaching, either implicitly (for example, Socrates) or explicitly (for example, Dewey). There is no need to regard teaching methods of discovery and discussion as related to particular educational philosophy, say progressivism, or to consider it suitable only for particular age.

The picture shows parallelism (homomorphism) in learning and teaching: left column graphically depicts cognitive changes that are the same for both processes. We describe those changes in middle and right column. We can observe the homomorphism: for each stage of learning there is correspondent stage in teaching.

The process of hypothesis formulation is more complex. The problem does not lie in the selection of possible explanans since there are none of them at our disposal. The problem is how to generate one. We can represent hypothesis formulation in terms of an 'inductive algorithm'. 'Inductive algorithm' presented here as an example¹² embodies the principle of so-called Occam's razor : we should choose the simplest hypothesis (see step 2.).

Let's describe an algorithm which in given description space tries the find the answer which object exemplifies a certain property. Positive examples are objects having the property (goal predicate), while negative examples are objects lacking it.

Let's suppose that description space does not include attribute volume. It would be possible than to find two objects with identical description and different classification. This example does not give such a possibility: we have relevant attribute in our description space - namely volume. Still, algorithm does not give a clue how to widen up a description space. We can construct a new attribute as combination of already given attributes (specific density for example). Unfortunately, that is not always the case.

Two points are important here:

Note that the use of an algorithm in 'hypothesis formulation stage' can not account for emerging of relevant hypothesis. Relevance of hypothesis depends on previous knowledge or on introducing of new theoretical term or terms. Both things are items of knowledge. Therefore, this stage (sub-process) of knowledge acquisition does not reduce itself to 'clerical procedure' (?Turing's machine') unless we are willing to accept regressus in infinitum.

Still, we can employ some heuristic strategies. The most important is use of background knowledge as a source of metaphors: find a familiar structure that is similar to certain degree to the one being explored and look for homomorphisms¹³.

From the standpoint of formal learning theory we can develop similar argument regarding computational properties of hypothesis formulation sub-process. Usual paradigm¹⁴ tries to give a formal reconstruction for following concepts: a theoretically possible reality; an intelligible hypothesis about reality; the data available about any given reality, were it actual; a scientist (or child); a successful behavior by a scientist working in a given, possible reality. In that framework a scientist (or a child) is identified with a function y from set SEQ of sentences in given language L (natural language or language of nature) to a hypothesis N (about grammar or laws). We can prove the following proposition¹⁵: "Let S be any countable collection of functions from SEQ to N (conceived as scientists). Then there is an identifiable collection L of languages such that no member of S identifies L." We can give the interpretation for the proposition along the following lines. There is a class of learning problems that 'computable functions' can not solve. By 'computable function'¹⁶ we mean a function computable on a theoretical machine ('Turing machine'). The proof rests on lemma that there are uncountable many possible hypotheses and only countable many Turing machines (computable hypotheses). This lemma is theorem in the theory of recursive functions. If we interpret the proposition in the light of previous 'inductive algorithm' we can see that there is a way to enumerate hypotheses that we can formulate relative to given description space. On the other hand, there is no way to enumerate the hypotheses that can be formulated within other vocabularies, i.e., formulated relative to different space of description.

Do we encounter again one amongst other 'pedagogical paradoxes'? It does not seem so. It is learnable how to learn to a certain degree: some metacognitive skills can develop provided suitable learning environment. Critical thinking, use of metaphors, strengthening of logic can develop in learner but they do not guarantee that she will become successful independent learner. Still they make independence, firstly, and successfulness of learning more probable.

Post-modern notion of loose connection between theories and reality seems to diminish the value of science. Meta-theoretical considerations renew the value of science. The real value of science does not lie on the side of 'truthfulness' but on the side of creativity. The analysis presented in this essay favors inclusion of wide spectrum of subjects into curriculum and extensive use of teaching methods of discovery and discussion.

¹ Rousseau was the first to discover dependence relations between educational theories in his Emile. Three kinds of 'education' (by people, nature and things) must converge towards common end. We can not have influence on the development ('education by nature'). It follows that educator (education by people) must create educational situation (education by things) in accord with the developmental stage (education by nature).