The theory of stage-by-stage development of mental actions suggests five stages in
organization of instruction.
At the first stage the instructor presents new material. At the beginning of instruction the new
material implies not only new mathematical theory but also general structure of a proof and how it
is reflected in the database of the given mathematical theory, i.e. general principles of design of an
information model of the given subject matter are described.
A specific objective of instruction is identified: development of students personal databases or
elements of these databases. This objective is to be assessed and evaluated in the course of study.
At the second stage the actions are developed in the materialized form. Possibilities of
computer assisted training for this purpose were described in great detail in the previous
paragraph.
The third stage is development of actions in the external speech form.
Each action to be developed starting from the materialized form should go through all the
stages of interiorization including its development in the form of oral or written speech. Since
actions at this stage are in the speech form, work with databases can perform only functions of
control of student's actions.
In the process of work with the database on the given problem a student can plan doing a
number of orientation operations such as identification of notions related to the solution,
identification of theorems to be used in solving the problem, etc. After the orientation operations
are completed, a team discussion of the results of the orientation operations is recommended to
provide opportunities to articulate the performed actions comparing the obtained results.
This process requires development of arguments, providing environment for action
development in the speech form. As a home assignment a student is given a task of writing down
the proof. In this way development of both orientation and executive operations takes place in
the written speech form.
The fourth and the fifth stages are developing actions in the form of internal speech and mental
form. At these stages a student works with the database as with any information system receiving
necessary information on request, constructing a mental scheme of proof, identifying condition,
conclusion, notions and their definitions.
It should be pointed out that at the initial stages of instruction databases are used as major
tools of developing required actions while at the final stages team work and individual work with
an instructor are predominant with databases used as auxiliary support tools. Control system of
databases mostly provides control of student activities.
Note that at each of the last four stages the action is developed at a certain given level of
generalization, completeness and assimilation. Moreover, this level at the second, the third and
the fourth stages is not always the highest possible.
Go on to the next section of this paper.
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
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