Tibor Szalontai, Bessenyei College, Mathematics Department, Nyiregyhaza, Hungary

Although the human learning and teaching came from the pre-historic past, our knowledge on these phenomena is very limited. The psychological studies or experiments of the recent one hundred years on the learning were rather various in research methods, targets and hypotheses;in results and in their level of generalisability. Since in the most cases the researchers were influenced by different (pre)conceptions or 'psychological schools', their answers and rules (discovered) could catch only different narrow features of the learning. (E.g. associational learning theory, mediational learning theory, 'sign-signed type learning' theory, operational learning theory, etc.) In addition, these results were hardly applicable directly in the education and schooling.

So, besides it, an opposite (and maybe a traditional) approach toward the question has been more useful practically, namely the effectiveness of education and teaching-learning, in complexity of the abilities, skills and the whole personality. Indeed, e.g. the research of motivation and attitudes; the research of potentials, abilities and skills; the research of inner and outer conditions of learning (etc.) have added new useful principles for the education and pedagogy to the elder (traditional) principles. In the recent two-three decades several useful psychological experiments have enriched the education: the Soviet school (Vigotsky, Rubinstein), the school of Piaget (interiorization, operationalization), Skemp, Polya, Freudenthal.

The modern schooling is just a part of the widely meant education, although we like to think schooling is the most important and effective part of it yet. We should hold this evidence in our mind to avoid both an overestimating or underestimating of the real possibilities and the role of our present school. We are (of course) rather far from an idealistic 'total' motivation in learning (e.g. the 'guru-chela' connection in the ancient India), but we should know and remember that we are able to teach children the most effectively when as high level of confidental and serene atmosphere and agreement in our common targets are reached as possible.

In the pedagogical literature and school practice, many different teaching approaches or models have been existed[1]. Each of those represents considarable worth in education, but not at the same time and not for any school subject, topic, age, children (or teacher) of course. In addition, each teacher's practice involves several approaches, consciously or unconsciously. We should talk about dominances at individual teachers' practice, and about dominances or trends at certain school types, subjects, ages of students/pupils, etc. within different culture areas or eras, in different countries.

In such kind of meaning we can talk about e.g.

* the mediaval 'scolastica' with the dominance of catetic method

* the universitas of the Reneissance with their 'seven arts' and humanistic world-image

* the first modern schooling of the reformed churches with their 'evergreen' didactic principles summarized by Comenius (the principle of clearity, visuality, graduality, associations; the direction from the simple toward the complicated, from the sensuality toward abstract, from the concrete or particular toward the general, from the surface toward the depth, from the near toward the far, etc) -- which all work even nowadays, tipically in the teaching-learning of natural sciences

* the school of the industrial and later the technical revolutions (started from the anglo-saxon world) with its new dominant principles: 'trial and error', 'little steps', 'immediate confirmation', feedback; pupils' activity, interest, foresight, application, etc.

* the training for sport-movements and work-processes with dominance of the 'psychomotoric phases method'

* the Prussian 'gimnasium' (secondary school) with its militant order and 'classic' teaching style

* the Waldorf kindergarten and primary school with its complex and 'life-style' methods etc.

Mathematics didactics (or 'pedagogy of mathemathics') is a multidisciplinar but independent branch of science, which has very specific features among the other subject pedagogies. It is really not just a kind of applied pedagogy, although at the fore-part of its development it is just elaborating its own scientific language and research methodology now, as Krygowska had stated it frequently since the end of the 70's.[2]

Although the mathematics didactics uses (and applies) the main results and basic principles of the general pedagogy, but has several special principles and results which hardly could be applicable for the general or for most of the other subject pedagogies. In addition, it has several elder or recent questions or problems which are not too interested for the general or other subject pedagogies, or which are not answerable or solvable in the stage of general pedagogy. For example: teaching of mathematical infinity concepts; mathematical ('whole') induction; teaching of implication (A -- B) when A is false; purportive and not purportive models for pre-teaching of definitions; components of the mathematical giftness; etc. The special features are mostly from the curiosity of the mathematics science among (most of) the other sciences (even if real or natural, either human or social ones).

There are several good mathematics teaching approaches, conceptions, styles and class-setting organizations. These have similar or different (sometimes opposite) advantages or disadvantages and effectivity worths. Depending on

* the age or ability level of pupils

* particular didactic tasks; mathematical concepts or problems; written or verbal skills required; and so on.

Some main conceptual trends, for example: investigative, genetical, elementarizative, structural-formalist, problem-oriented, application-oriented, individualized, antroposophical, computer-oriented (recently), etc. can be mentioned in Europe nowadays. None of them are disjunctive to other ones and (so) neither of them appear merely at a teacher's practice. We should just talk on trend or dominancy of any.

Nowadays, the literature of mathematics didactics and the different conceptual trends focus on the developing of pupils' or students' mathematical thinking and problem solving, uniformly. At any ability group within a year cohort or a class. But for the recent decades, national and international measures have shown a high and rising rate of slipping pupils and a downward trend in average achievements, besides of a certain minor but increasing rate of outstanding achievements.

So, meanwhile more spreading of positive mathematics educational ideas toward the teachers, we can notice this fact does not bring good teaching results automatically. The role of teachers and the teaching-learning materials and manuals are continously determinant in success.

At some mathematical topics and ages, several different kinds of 'good' methods can come up, but these usually have different methodological values. So in particular thema, topics, concepts, or in particular didactical tasks (new topic, practice --conditioning or automatizing-- of skills or abilities, etc.) we can (ought to) choose among the possible alternatives. Similarly, it is decidable in almost every case, when a certain theme, topic, concept can be and should be teached (in the meaning of preparation or 'pre-teaching', or of building systematically and making applicable knowledge, abilities and routines). The (student) teachers sould be familiarized with as many of these trends and concrete methods as possible for their methodological culture and innovative or creative abilities.

We have found, developing of mathematical thought of bright, average and slipped children is on the same discovery and communicative learning-teaching basis. All of the pupils and students should get the best methods from the school. There are two main demands seemed to be oppositive to each other:

* demand of (personally) differentiation among the pupils;

* demand for teaching big size groups of pupils at the same time (class-lesson system in schools).

An other demand, which is maybe also in opposition to the first one, is: demand of education for cooperative, communicative abilities. We suggest a compromism and harmonic, balanced ways of teaching and class-work setting.

The differentiation is solvable in the classroom, by home work and (occasionally or regularly) in 'after-school workshop' for gifted children and in 'updating lessons' for slipped children.

We prefer simple appliances for pupils' manipulative activity at the first stage of concept building, in accordance to the 'theory of interiorization' of Piaget. The outer activity is transformed for inner thinking, gradually --by manipulation, 'quasi-manipulative' thinking, thinking, We can confirm the strength-hierarchy of visualisation: lecture; explanation and examples; demonstration by pictures, transparents, figures or graphs; demonstration by mobile visual aids (applicative transparent, phasys-transparent, movie, trick processes (computer); real world demonstration and activity; manipulative activity; movement with own body --in this ascendental order.

We should arrange as many individual work as possible (sometimes in little, heterogenous groups), but within the same topic and problem for the whole class or level group. The individual work may be arranged in one of the 2 or 3 level groups too --in both the phases of practice and the learning of new knowledge. This individualized work is always followed by a whole-class (or level-group) discussion, after each task or problem. The role and aims of the individual work are: chance for trials; intensive thinking; problem solving (intuitions); creativity; developing of written achievement abilities and skills. The role and aims of the common discussion are: developing of verbal abilities and skills; competency feeling; building of own mathematics conceptually; diagnosty and correction of misconcepts and false thinking ways; feedback and modifying of the teaching process.

Mathematics teaching via discoveries means usage of those 'clever', purportive constructions, models and structured problem series which lead the pupils to new predefinitions or definitions, to new preconcepts or concepts by their individual efforts, solving or trials (at least) --by gradual steps. The structure of topics optimally can be built spirally. Several topics and 'preconcepts' appear in early ages and this 'pre-teaching' will lead to a more and more 'correct', mathematized teaching in the higher stages, gradually. An other important feature of this conception is: optimal inner and outer connections and applications.

The role of teacher is very important in arranging this spiral task-series (if there is no a good textbook), in arranging the individual work and common discussion (with the feedback for the pupils and him/herself; diagnosty, evaluation, etc.) and giving the (short) explanations, definitions, confirmations needed. The practising phase is rather similar, but has a new task: optimal developing of the written and oral abilities, the achievements in simplier problem -and type-like tasks) solving, with differentiated acceleration of the pupils. There is a flexible (not too strict) requirement for higher performance of the children --individually if possible. Of course we want to build it on the pupils' interest, competitive feature and 'thirsty for knowledge'. Our future teacher students should learn and practise the technics of effective mathematics lessons. Each pupil should deal with mathematics during at most of the lesson --in an effective deepness, and should reach the teacher's didactic aims. The didactical and educational aims should be agreed --or demanded if possible-- by the pupils.