Analysis of the planned instruction process aimed at designing methodology developed on the
concepts of the theory of stage-by-stage development of mental actions should provide answers to
a number of questions as well as recommendations on what problems are top priority ones in
organizing instruction. Here are some considerations on these questions.
a. Specifics of subject content. To specify the object of research, let us explain what we mean by
"advanced math". As a rule, it applies to theoretical math disciplines, the study of which, from he
didactical standpoint, is aimed at developing logical operations with mathematical objects as well
as specific mathematic operations that cannot be algorythmized. "Advanced analysis", "abstract
algebra "Foundations of Algebra and Analysis" etc. should be included in the category of
theoretical math subjects. The last course has been selected in our presentation as a model for
this type of courses for several reasons.
First, teaching this course even in a traditional environment demands refined teaching
techniques. Because of its highly abstract nature, there is an ever recurrent problem of
motivation. Second, this course is a core one for almost all categories of students who will need
math in their future professional activities. Third, this course is part of the curricula in a large
number of American and Russian Universities.
b. Course goals and its place in the curriculum. The discipline is the first abstract math course to
be studied. Major goals in teaching this course may include the following:
- First, to give an idea of a procedure of acquiring new knowledge in mathematics based on
the already existing knowledge, i.e. to teach to formulate and prove theorems using examples and
counterexamples, definitions, theorems proved before, the technique of proving theorems
developed previously;
- second, to introduce students to basic structures and concepts of modern mathematics;
- third, to substantiate some of the actions that were developed previously in the course of
study of other math disciplines.
c. Basic actions to be developed in the course of instruction, operational composition and
objects of these actions. Due to the time limitation determined by the place of abstract math
disciplines in the curriculum, mostly general logic operations can be developed to a high enough
degree. They constitute as a whole an action of mathematical proof.
Main objects of general logic actions in studying advanced math courses are notions and
definitions, theorems, examples of objects falling under different notions, counter examples,
problems that are structurally similar to theorems.
Naturally, the description of operational composition of an action of proof is not an easy task,
which requires a separate discussion. Therefore, we shall limit ourselves to a rough model having
to some extent features inherent to every proof and sufficient for further discussion.
The majority of the proof in the subject field reviewed can be presented in the following way:
if object A has properties S1, S2... then it also has property S.
The proof of a theorem starts with singling out its conditions and conclusions - what is given
and what should be proved. At the second stage notions present in the conditions are singled out
and their definitions are renewed. Further on some rules of conclusion are applied, that can be
presented either in the form of (a) "if an object has properties SI, S2... Sn then it has properties
S'1, S'2... S'n", or (b) "in order for an object to have properties S'1, S'2...S'n it is sufficient for it
to have properties S1,S2,...Sn".
Repeated use of this algorithm leads to the proof of the theorem. In contrast with the
systems of artificial intelligence the rules of conclusions of "natural intelligence" are determined by
creative ability to apply the previously proved theorems and to establish hierarchial links between
notions. As we could see the majority of operations that make up the action of the proof belong
to the orientation part of the action.
d. The level of development of actions, described in (c). The initial form of development in
planned instruction. Both specific and general logic actions characteristic of advanced math are
just insignificantly developed at previous stages of training. The majority of notions and concepts
are either new or more abstract and at a relatively higher level than those previously studied.
Thus it would be psychologically justifiable to start developing in the materialized form
general logic and specific for this
subject field actions with objects of an absolutely unfamiliar nature.
e. Peculiarities of traditional mode instruction of this course. As it was pointed out in d.,
instruction in "Fundamentals of Algebra and Analysis" should begin from the materialized form.
All objects of the action or their operations should be presented in this form. The actions themselves should be modeled in materialized form. However, objects of actions of abstract
mathematics, as a rule, are difficult to present in the materialized form. Thus, development of
actions and notions in the framework of traditional instruction usually starts in the external speech
form. This is one of the reasons why students are facing a lot of difficulties in studying these
disciplines, therefore special attention is given (or should be given) to organizing instruction in the
external speech form.
In traditional instruction the problem of development of actions in the external speech form is
solved by organizing the following activities:
- Joint discussion of problems posed by an instructor occupies a significant place in the
classroom.
- Organization of class activities is oriented towards students thinking aloud.
- Students also have a chance to make presentations (often short ones).
Stimulation problems are also solved through organized training, facilitating students
presentations.
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
The Math Forum is a research and educational enterprise of the Drexel School of Education.