IV. Modeling of actions in the materialized form

The analysis described in the previous chapter indicated that in organizing instruction in the course "Fundamentals of Algebra and Analysis" it is necessary to solve the following problems. a. Presentation of objects of the action to be developed in the materialized form. The first problem in organizing instruction based on the theory of stage-by-stage development of mental actions with the action passing through all the stages is connected with presenting objects of the actions in the materialized form. As we have already pointed out the objects of actions in studying abstract math disciplines are theorems, notions and their definitions, examples and counterexamples as well as problems structured as theorems.

Practical experience and experiments in teaching these kind of abstract disciplines reveals that any attempts to present objects of study in the materialized form without using information technologies as a rule lead to something very archaic, obsolete and not exciting for students at all, with lack of motivation as an immediate result. Thus if in the process of computer assisted training we can solve both the problem of presenting objects or their models in the materialized form and that of motivation, it is already a substantial improvement compared to traditional forms ofinstruction.

A natural information model of any subject field including mathematical theories is a database or a base of knowledge. Following G. Martin ( Martin, 1978) we consider databases a named complex of data reflecting a stage of an object or a number of objects, their properties and relationships. In fact a database can be viewed as an information model of a given object, and effectiveness of the control system of the object depends on its preciseness and authenticity. A database as well as a recording of a data base are not only notions of information technology but could be presented in a structured form on a computer screen, which could serve as materialized objects of the developed actions. Moreover, these objects can be manipulated both by people and computers.

The screen recordings can be moved all over with the help of a mouse, a certain part can be pasted and transferred to a certain field, named and provided with required field value. In fact, a screen presentation is an object that can be transformed and one can watch the process of transformation. Moreover, in contrast to a conventional text, the structured screen recording along with its name and field name impose significant constraints on its manipulations provided they are meaningful ones.

Thus material representations of objects of the actions to be developed are recordings with the names "Theorems", "Notions and their Definitions", "Examples and Counterexamples", "Problems". The structure of data, recordings, fields in every recording should reflect the corresponding object structure, which this recording materially represents as well as existing interconnections of this theory and demands imposed by organization of the study process. Bearing in mind that the major goal in the course "Fundamentals of Algebra and Analysis" is to teach students to prove theorems, the structure of the database should reflect typical characteristics of every proof, described in p. III.

Hence, record under the name "Theorem" has the following specific fields: "condition", conclusion", "notion of a condition and its definitions", "notion of a conclusion and its definition". Every theorem can also be applied to proving other theorems and for solving practical problems. Therefore, unless a student understands where and how this theorem can be applied, it cannot be stated that it has been completely understood and assimilated. Hence the necessity to have in the record of a theorem fields with the names "examples of problems solved using this theorem" and "theorems which require in their proof applications of the given theorem".

The record's fields named "notions and their definitions" are determined in the same way. It is expedient to have in this record fields under the following names: "Definitions of a Notion", "Previously Studied Related Notions", "Theorems Containing this Notion in their Conditions", "Problems Requiring to Determine if an Object Falls under the Given Definition". Since every exercise has a structure of a theorem, record's fields under the name "Exercise" are similar to those of "Theorem", the only difference being the field under the name of "Model Exercises Solved with the Use of the Given Theorem", which exists in the record "Theorem". Instead of this field a field under the name "Exercise for which specific theorem or group of theorems" should be included into the record "Problem".

b. Presenting developed actions in materialized form. This is the next problem to be solved. As an example, let us consider the operation of determining the structure of a theorem. Experience shows that in teaching theorem proof we should begin with identifying structure elements of a theorem. After we have presented objects of action in materialized form there is no difficulty in materializing this operation as well. In the text of the theorem present in the electronic textbook certain blocks are pasted (conditions, conclusion, definitions of notions of the theorem, etc.) and imported with the mouse to the respective fields in the recordings of the theorem. Similar actions are performed in teaching to identify the structure of a problem, a notion, etc. An important stage in teaching a proof is understanding the structure of a proof.

The key element here is substantiation of the conclusion. Students often face considerable difficulties explaining why we can come to the conclusion that if an object has property A, then it also has property B.

Development of the action of choosing the right theorems for proving or solving a problem at different stage may take different forms. At the initial stage it may be in the form of pasting during the explanation of the proof the part to be substantiated in the electronic textbook and giving appropriate value to the respective name of the field. At more advanced stages a student searches for theorems to be used in the proof and gives respective values to the necessary fields.

We can conclude that there exists a scenario of materialization of a necessary bloc, importing it to the respective part of the screen, giving appropriate value to the field's name. As a whole, this scenario reflects the structure of the proof and is highly effective in organizing instruction provided there exists proper motivation.

c. Orientation basis of an action and motivation of student activities. As it was pointed out in the second paragraph in describing fundamentals of the theory of stage-by-stage development of mental actions, the process of control of student learning activities (which is for us a process of instruction) is based on the orientation part of an action. In terms of the theory of artificial intelligence it means that determining the strategy of performing an action is effective if it is based on the existing declarative and heuristic knowledge about the performed action and the object, and knowledge and procedure of performing an action.

In organizing instruction aimed at work with databases we can assume that at the initial point of instruction students possess some preliminary knowledge acquired before taking this course; a textbook or instructor's explanation that can be viewed as some unstructured knowledge; procedure of acquiring knowledge presented in the materialized form in the structures of records in database.

The objective of the student's work is to structure the unstructured knowledge and retain it. It means that in the process of learning a student creates a personal database which represents in the materialized form the orientation basis of student's actions at each stage of instruction. Control of the learning process of a student is geared to this personal database, which gives a student possibilities for search using key words or names, creating incentives and motivation for study. It should be noted that unstructured knowledge and structure scheme are presented in materialized form: Unstructured knowledge is given in the form of a text in the electronic textbook with sections and blocks that can be pasted and imported to proper parts of computer screen. Structure scheme or its elements can also be presented as cards on the screen.

d. Study process and its control. Since the theory views the process of instruction as control of the study process the natural question arise how it is implemented. First, as it was already pointed out, control is geared to the orientation basis of student's actions presented in materialized form (a personal database).

Control is carried out through requests from a student to the database and vice versa. At the initial stages of instruction the control system of the database guides the learning process of a student providing assignments and requests. Requests are fully determined by the structure of a database and pedagogical objects at every stage of instruction. Here are some of the requests dealing with the record "Theorem" and student's actions regarding these requests.

• "Determine the conditions of the theorem". The student finds the text of the theorem in the electronic textbook and imports it to a certain field called "student's field", paste the conditions of the theorem and imports it to the slot for entering field values in respective record.
• "Determine the notions in the conditions of the theorem".

The students marks the required notions and imports them in the slot for entering field values in the respective record. In this way the development of all orientation operations of the actions of a theorem proof takes place. A set of requests is determined for every record in the database and the students choose their own regime of work. Every action connected with students entering field values into their personal databases is automatically controlled by comparing it with the instructor's database. The results of the control are presented to the student. There is a time limit on filling in each card. Excessive time is a signal for a teacher to provide help.

Initially student's actions are controlled by requests, and the scheme reflecting the structure of presenting knowledge is available for a student on the screen or upon request. At the stage when certain operations develop into a mental form, a student compiles the personal database without computer requests or the scheme of the record on the screen. A student can receive information from the personal database in an automatic regime or through an individual search in the electronic textbook.

Go on to the next section of this paper.

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Mikhail Bouniaev
Southern Utah University
Math/CS Department
351 W. Center
Cedar City, Utah, USA, 84720

E-mail: BOUNIAEV@SUU.EDU

Moscow Pedagogical State University
Department of Mathematical Analysis
Moscow, Russia