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Procedía

Social and Behavioral Sciences

ELSEVIER Procedía - Social and Behavioral Sciences 163 (2014) 132 - 141

CESC 2013

The model object-product-cognitive operation through mathematical education

Costicä Lupu *

"Vasile Alecsandri" University of Bacäu, 157 Märä§e§ti Street, Bacäu, 600115, Romania

Abstract

The aim was to investigate the efficiency of the model to operationalize D'Hainaut with applications in mathematics, the formation of concepts, theorem proving, problem solving.

Hypothesis can be reduced to the claim that the use of operationalization model object, product, cognitive operation in mathematics education will lead to the formation of effective skills and problem solving approach to mathematics and the development of thinking. From this hypothesis derives research objectives: presenting mathematical problem solving examples using this model, observation and application of these approaches in solving mathematics teaching practice days, achieving a systematic research using questionnaire and method tests the effectiveness of the model operationalization. The research was conducted in the second semester from February to May 2013, in the process of observation and evidence in support of teaching practice lessons with 46 students in the third year of the Department of Mathematics - Informatics, University "Vasile Alecsandri" in Bacau through systematic observation on how it is assisting 46 math lessons and support of 46 lessons followed by testing a sample group of 260 middle school students and application of questionnaires to 65 teachers from different disciplines. This study deals with such aspects as: the object-product-cognitive operation model in mathematical education; the analysis and classification of problematic tasks; the building of certain learning skills characteristic of the mathematical content; the comparative analysis of the essential characteristic of the learning methods. This study is concerned with aspects such as: the model object-product-cognitive operation through mathematical education; achieving a systematic observation regarding the initiation in the particularities of mathematical activity; gaining knowledge of the elements and priorities of certain fundamental notions and configurations; using concepts and theorems in solving mathematical problems. © 2014 The Authors.PublishedbyElsevier Ltd.This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/). Peer review under the responsibility of the West University of Timisoara.

Keywords: object, product, cognitive operation, learning abilities, problem solving, problematic situation

* Corresponding Author name. Tel.: +004-072-358-6101 E-mail address: costica_lupu@yahoo.com

1877-0428 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/3.0/).

Peer review under the responsibility of the West University of Timisoara.

doi:10.1016/j.sbspro.2014.12.298

1. Introduction

The impact of mathematical education is two-sided: on the one hand, the learner gains knowledge, on the other hand he builds those skills which are involved in work, developing the necessary abilities to take part in such an educational activity. Thinking is formed through mathematical education. Other factors are also involved in building thinking, but the role of Mathematics is essential.

Mathematical education has a double effect: on the one hand, the learner acquires knowledge and, on the other hand, builds those skills which are engaged in work, develops the forces required by this type of education. Mathematical education builds thinking. Of course, other actions contribute to forming thought as well, but the role of mathematical education is paramount.

This article presents a study with applications in Mathematics, regarding the building of certain specific learning skills, which may lead to demonstrating theorems, solving problems etc. and progress of the students' thinking in middle-school education. This article presents a study on D'Hainaut's operationalization model, with applications in Mathematics, leading to the formation of notions, demonstration of theorems, problem solving etc. and development of thinking at middle-school students.

D' Hainaut's operationalization model is characteristic of the mathematical domain and allows the adaptation of the model object-product-cognitive operation, to the context of specifying the pedagogic intention associated with content elements. For description on an operational level, D'Hainaut proposes the model of the basic intellectual act, according to which stating an objective should describe a student's activity by referring to three aspects: the object subjected to the student's activity; the product resulted at the end of the activity; the cognitive operation, characterized by the circumstances in which the activity is conducted.

2. D'Hainaut's operationalization model in mathematics

2.1. The model object-product-cognitive operation

D'Hainaut's operationalization model is characteristic of the mathematical domain and allows the adaptation of the model object-product-cognitive operation, to the context of specifying the pedagogic intention associated with content elements. For description on an operational level, D'Hainaut proposes the model of the basic intellectual act, according to which stating an objective should describe a student's activity by referring to three aspects.

• the object subjected to the student's activity;

• the product resulted at the end of the activity;

• the cognitive operation, characterized by the circumstances in which the activity is conducted.

More specifically, a cognitive operation is a mental activity, which in the context of an intellectual act, ensures the correspondence between a given object and a certain product, possibly through the intervention of an operator. The nature of the cognitive operation depends upon the operator's degree of availability in relation to the student's cognitive repertoire.

The following categories of cognitive operations are defined:

a) Reproduction^ The subject, placed before an object which is identical with the object from the learning situation, provides the same product.

b) Conceptualization^ understanding - The subject, placed in front of an object, provides an answer which is valid for the class to which the object belongs, on condition that in the learning situation the product should not be associated with the object.

c) Application - The activity through which the student provides a determined product for a given object, belonging to another class, without this particular object having been associated with this specific answer during training, given the fact that the class of the object has been associated with the class of the product.

d) Exploitation - Consists in extracting a determined piece of information, of content, out of a situation.

e) Mobilization - Consists in extracting from the cognitive repertoire one or several pieces of information (the product), which respond to one or several precise conditions, without there being a previous association between these conditions and this product.

f) Problem solving - Is the cognitive activity consisting in creating a product by starting from a certain object, on condition that the product, object or solving procedure may present a certain degree of novelty.

Generally, one objective of mathematical education refers to describing a class of tasks (questions, exercises, problems, problem situations) which we expect that the student may be able to solve by the end of a training unit (group of lessons, intra-interdisciplinary module). Such a task may be characterized by the following aspects:

a) the task information (conditions, hypotheses) is what the student is being offered;

b) what should be demonstrated or found, is what the student should obtain, the answer which should be given;

c) the methods, procedures, knowledge used to obtain the solution.

The task information and the methodological indications regarding what should be found or demonstrated forms the object upon which the student's activity will be exerted.

The result obtained by solving the task, the solution itself, which may be a number, function, demonstration etc., represents the product of the activities.

The procedures used in providing the answer, which may be more or less obvious, with a higher or lower degree of novelty, in relation to the learning situation, form the operational skills and abilities. The effective application of these in obtaining the product, a certain object having been given, form the cognitive operation.

The object, as well as the product of the activity, belongs entirely to the domain of Mathematics. The object-product perspective alone is not enough for the use of objectives in designing training. In the process of finding the solution, the intellectual activity, which the students perform, is characterized by the relation established between the object-product from the learning situation and the object-product from the evaluation situation. This process involves various cognitive operations, elements which constitute intellectual qualities of the individual engaged in problem solving as a distinct intellectual phenomenon.

Starting from this analysis, we may retain the following intermediary objectives of Mathematics: initiation in the specific of Mathematical activity; knowledge of the elements and characteristic properties of certain fundamental notions and configurations; using the properties to build the demonstrations for certain sentences or to determine certain measures regarding the elements of such a configuration.

2.2. Initiation in the specific of mathematical activity

The organization of training should make the student understand the following aspects:

1) The activity's inherent particularity is the placing of a real situation within a model, which would replace reality, and within which concepts would have a well-defined signification.

2) The activity implies the adaptation and acceptance of certain principles, so that the judgements made may refer not only to these principles, concepts or other demonstrated propositions, by complete ignorance, at the moment of the demonstration, of the modelled real situation.

Thus, to establish the truth of a statement regarding the object of the model, we can resort only to procedures that had been defined within the model, but not to the modelled real situation. The problem is not to completely exclude the real situation from the discussion, but to highlight the distinction between issuing a conjecture by inspecting some aspects of the real situation and establishing the respective statement, by reasoning, based on acknowledged or already demonstrated principles. The students should be made to understand the fact that making a drawing on a sheet of paper and performing a reasoning are two different activities.

Knowing the elements and properties of certain fundamental notions and configurations

a) To understand the statement of the sentences from Algebra, Analysis or Geometry, it is necessary to know the elements and properties of certain fundamental notions and configurations: segment, triangle, rectangle, circle, polyhedron etc. On coming across a certain term in a text, the student should be able to substitute a property of a certain notion or geometrical configuration, although this may not be clearly demanded by the statement.

b) To know the terms characteristic of Geometry (median, height, bisector, middle line, chord, tangent etc.), requires that the student should be able to perform the following actions:

• - being given the notion, indicate the properties, methods or solving stages;

• - being given the name, reproduce the definition;

• - being given the name, point out the respective object within a certain configuration;

• - being given the definition, indicate the name of the respective object;

• - being given an element of a geometrical configuration, indicate its name.

c) To know the geometrical concepts and theorems. By geometrical concept we refer to those notions of Geometry which express essential traits shared by an entire class of geometrical objects (congruent triangles, similar triangles, parallelogram, inscribed quadrilateral etc.). Regarding these categories of content, we propose the following tasks:

• - being given the name, reproduce the definition;

• - being given the name, enumerate the properties of the respective object (even those which are not part of the definition);

• - being given the name, produce or recognize examples or counterexamples of the concept;

• - being given a property, draw a distinction among the concepts, according to whether they have or not this property (discrimination between related concepts);

• - enumerate the minimal requirements that an object should meet in order to constitute an example of the concept (procedures used to demonstrate the fact that two lines are parallel, that a quadrilateral is a parallelogram, that a line is tangent to a circle);

• - being given the name of a theorem or an indication regarding its content, reproduce the statement of the theorem;

• - reproduce the demonstration of a theorem, in the form in which it had been presented in class;

d) To understand a mathematical text, which may represent the statement of a task, or a set of statements which may constitute, together, a demonstration, relies, first of all, on knowledge of the concepts and theorems, as it had been previously methodologically defined. Besides knowledge of the disparate elements, it also implies a series of integrating skills, which may enable the student to:

• - make the drawing associated with a given configuration, introduce notations, transpose the hypothesis and the conclusion in the language of the introduced notations;

• - being given a statement, recognize or produce the properties which may be deduced, as conclusions, if the respective statement may be regarded as a hypothesis (by recalling a certain theorem, definition etc.);

• - justify each step of a demonstration by referring to a hypothesis of the demonstrated sentence, a previously demonstrated theorem or other steps of the same demonstration;

• - identify that part of the hypothesis which has not been involved in the demonstration;

• - seize the formal logical structure of the statements and demonstrations (expressing geometrical sentences like an implication, replacing the expression "p if and only if q" within a conjunction of implications, formulating the converse or converses of a given sentence).

2.3. Using concepts and theorems in solving problems

To understand the content of this objective, we should clarify the notion of problem. In German etymology, proballein means faced with a barrier, an obstacle. According to the Greek etymology, the word problem designates the act of being challenged into searching, discovering the solution, this being the result of elaboration through thought and not of the standard application of an algorithm. According to P. P. Neveanu, the problem occurs like a cognitive obstacle in the cognitive and technical relations established for this purpose, which outline the domain of problem solving. To this effect, we shall resort to a famous definition: the term of problem designates a difficulty which cannot be removed and which demands conceptual or empirical research. In both cases, any socio-human problem includes the following aspects:

i) a problem should be regarded as an object, conceptually different from, but epistemologically similar to a statement;

ii) the act of asking a question by means of a set of interrogative or imperative sentences (the linguistic aspect of the problem).

The study of interrogation is undertaken by psychology (including the psychology of science), whereas the study of questions regarded as linguistic objects (namely, sentences ending with a question mark) belongs to linguistics. In our case, we shall designate by the term problematic task a statement which contains certain information with which the student is provided and in which the student is asked to demonstrate a mathematical fact or find the measure of a

certain element of the configuration, on condition that the solving may involve a certain amount of initiative from the part of the solver.

Example 1: Any point on the bisector of a segment is situated at an equal distance from the ends of the segment. Let there be segment [AB], raising the perpendicular to AB through O. For an arbitrary point M on this bisector, we demonstrate that A MAO = A MBO (c.c), therefore MA= MB. It follows that not any task proposed to the student is a problematic task.

In Geometry, there is a wide variety of problematic tasks. To control the domain and to be able to make the decisions needed in an operation of defining objectives, problematic tasks should be analyzed and described in relation to certain criteria.

According to D'Hainaut, problem solving is an essential thinking process, a phenomenon which is complex in terms of the processes involved as well as the diversity of the situations contained. In his view, problem solving is a cognitive activity which may be described through the following attributes:

1) - the subject is placed in an initial situation, which contains the object of its activity and a problematic situation;

2) - the subject deals with this object, meaning that he applies a process or a set of cognitive operations to it;

3) - the subject reaches the final situation which contains the product of his activity, meaning the very solution to the problem.

Example 2. In any triangle, the bisectors of the sides are concurrent.

The first attribute: - the initial situation: any triangle; - the object of the activity: the bisectors of the sides; - the problematic situation: are they concurrent?

The second attribute:

a) By reduction to the absurd we demonstrate the concurrence of two bisectors: Let there be a A ABC, d1 and d2 the bisectors of segments (AB), respectively (BC). Assuming that d1 and d2 are not concurrent, it results that d1//d2. Because d2 ^ BC, it results that d1 ^ BC. But d1 ^ BA. Therefore, through B, there pass two distinct right perpendiculars d1 (absurd).

b) Let there be d1 ^ d2={O}. From the property of the points of the bisector, it results that (OA)=(OB)=(OC).

The third attribute: the product of the activity, point O belongs to the bisector of the segment (AC), meaning that

all the bisectors of the sides of triangle ABC pass through O, therefore are concurrent.

D'Hainaut believes that an activity falls into the category of problem solving if at least one of the following factors present a certain degree of novelty for the student: (a) - the class of initial situation; (b) - the cognitive processes involved in the solving process; (c) - the class of the final situation.

3. Building learning skills characteristic of mathematics

3.1. The model object-product-cognitive operation through mathematical education

The object is generally a geometrical configuration, which is assumed to have certain properties, consisting in metric or non-metric relations between the elements of the configuration. Generally, it is required to be demonstrated that, in the specified conditions, the configuration has a certain property P. We shall note the

procedures which may be used for this purpose with p1, p2,.....pn. We should mention the fact that not any

procedure is suitable for any configuration, as well as the fact that there may be configurations for which the respective property may be highlighted through several procedures.

The product obtained at the end of the student's activity is a demonstration, meaning a succession of sentences, accompanied by the respective justifications, these being constituted from the information related to the task, the configuration of a previously demonstrated sentence or other sentences from the same demonstration.

The cognitive operation represents a mental activity, which ensures, within an intellectual act, the correspondence between a given object and a certain product, possibly through the intervention of an operator. The nature of the cognitive operation depends upon the operator's degree of availability in relation to the student's cognitive repertoire.

3.2. Analysis and classification of problematic tasks

Adopting the criterion object - product, we obtain a first classification of the problematic tasks which may be regarded as objectives of teaching Geometry. The tasks shall be classified into categories, according to the criterion of what is to be demonstrated/found, each category requiring the indication, in a more or less accurate way, of the range of procedures from among which the student should select the adequate procedure to solve a given task.

The first category, that of basic tasks associated with a configuration, includes the tasks for which the solving procedure is obvious and no decision is expected from the student regarding the selection of this procedure. In these tasks, the student is asked to apply certain algorithms, a procedure learnt in a typical situation, and in obtaining the answer there are involved specific intellectual skills related to a certain mathematical content. For each geometrical configuration, it is necessary to establish a more or less well-defined inventory of the associated basic tasks.

Among these non-specific intellectual skills, with a significant contribution to solving problematic tasks in Geometry, we mention:

• recognizing, distinguishing, within the statement of the task, between what is given (the data, the hypotheses of the task) and what is to be demonstrated or calculated;

• recalling information relevant for the respective task;

• recognizing the part of the problem which has already been solved or which is obvious;

• redefining or reformulating what is being required so that the solving procedure may be determined more easily (replacing what is to be demonstrated by an equivalent condition);

• deducing, from the hypotheses of the task, certain immediate consequences, by replacing the mathematical terms with their definitions, seeing whether the conditions for the applications of certain theorems are met;

• selecting, from among these consequences, those solved for what is being required, because not all the information which may be extracted from the hypotheses would be useful during the solving process (distinguishing between relevant and irrelevant information);

• investigating not only the hypothesis, as we have shown above, but also the conclusion, by recalling certain relevant information, properties whose demonstration is enough to highlight the required property;

• demonstrating certain relations - if in the configuration resulting from the problem's data there are no elements needed for the achievement of an auxiliary construction;

• reviewing the hypotheses of the task and verifying one's activity, at a given moment during the solving process, in case there is no foreseeable solution and in the context of an unused hypothesis which may suggest how to continue the solving process;

• replacing what is required to be demonstrated or calculated by an equivalent condition and selecting the equivalent condition so that it may suit the data of the task;

• comparing, throughout the solving process, the partial results obtained and what is being required to be demonstrated, so that according to the result of this comparison, the subsequent optimal development may be selected out of the alternatives existing at a given moment.

3.3. Considerations regarding the building of certain learning skills characteristic of mathematical content.

One of the pedagogical purposes of the educational process, stipulated in the Law of Education from Romania, No.1/2011, is the student's acquisition of the learning strategies demanded for constant (self) training. One of the specific informative objectives of intellectual education is cultivating higher cognitive attitudes. Among these, there are knowledge integration - skills - strategies - intellectual skills in intradisciplinary and interdisciplinary cognitive structures. To achieve the pedagogic purpose and the specific informative pedagogic objective mentioned above, the teacher should use, in his training activity, teaching-learning methods which may favour learning through discovery by means of heuristic and research methods.

Some abilities which are formed and developed at students during the teaching-learning activity are shared by a series of educational disciplines. These include: skills in measuring, observation, experimentation, calculus, extracting knowledge from an activity, independent from the literature.

The methodology of forming intellectual skills and interdisciplinary practices relies on psychology data related to the types of cognitive orientation and on research on action structure. The guiding basis represents a significant part

of the psychological mechanism of the action. Some psychologists (Galperin, P.) distinguish between 3 types of cognitive orientation of the action (incomplete orientation, complete orientation and wide range orientation) and, accordingly, 3 types of orientation in terms of working tasks. Each type determines, univocally, the result and course of the action.

Incomplete orientation overlooks a series of essential conditions for execution and resorting to trials and errors is necessary, which makes the subject lack self-confidence. This type of orientation relies on models of the results of the action. For example, the teacher presents the definition of the image of a function: f : A ^ B , Im f = {y / y e B, y = f (x),vx e A} and asks the students to determine immediately the values of m for which:

f : R ^ R, f (x) = 4x 2 + (2m + 4)x + m + 2 , m G R, verify im f s [-3,2] .

The students search for the solutiba2stArting-fTom the definition, mark f(x) with y, bring to the same denominator, group the terms according to the powers of x and apply the condition that x is real, A > 0 , completing the task but, finding, nevertheless, that the solving process is cumbersome and time consuming.

Complete orientation presents the models of the action results and provides all the indications referring to the correct execution (design) of the action for limited domains of its applicability.

If the teacher asks the students to solve the inequation: — 3 < 4x + (2m + 4)x + m + 2 ^ 2 , and to solve the system of inequations: J m _ 6m ~ 55 < 0 ,

They will obtai4 ihe "so2lutio"n1m e (_4,0)much faster. In this case, learning occurs fast mancl wi4mutce0rors.

In the wide sphere orientation of the action, the subject builds independently his concrete orientation for the action. In this case, the first option is to learn based on a plan (procedures) which may enable the highlighting of essential elements (main stages) and of the conditions required to perform the task correctly. To do this, students should be taught how to extract, from the proposed material, properties and essential relations which may eventually serve as orientations (essential elements or main steps) in solving any task. Learning by the third type of orientation is less complex as compared to the other types and requires, at the beginning, a wider time span than the time needed for the first two types of orientation. By contrast, subsequent tasks are performed correctly and independently. Finally, the time needed for learning is less that the time needed for learning through the first two orientations. Besides, students make fewer mistakes when learning by means of the third type of orientation.

The method of complete orientation ensures the students' active participation in establishing the structure of the action and the logical succession for performing the operations of which the action consists. The method of complete orientation ensures a scientific argumentation for the structure of the action. It is recommended that, in the teaching-learning activity, this method should be used only for building certain interdisciplinary learning skills at students.

The method of wide range orientation should be used for building concrete skills: the direct measuring of physical sizes, the design and testing of electrical wiring, the interpretation of graphs or drawings etc.

The issue of constantly increasing efficiency requires that students should build learning skills with a high degree of generalization, that is, skills which are formed during the teaching-learning activity of a discipline and which may be applied in the teaching-learning activity of other disciplines and, subsequently, in self-education and practical activity. Among these activities, we should mention, first of all, those focused upon learning:

Instrumental (independent study, mental calculus, oral calculus, handling measuring tools or geometrical tools etc.).

Operational (observation, measuring, figuration, drawing, experimentation etc.).

Creative (solving, demonstration, research, improvement, construction, etc.).

4. Research methods

During study we used the following methods and research techniques:

• the method of analyzing scholar documents created by respecting the obtained info from the documents study;

• the comparative method is the research method classified in the category of research methods by evaluating

questionnaires that use comparison in finding the adequate solutions at the investigated problems;

• the content analysis made possible the comparison of results, by recordings, drawings, etc., using quantitative techniques such as the time-table of using the info lab, the lecture handbook, the list with the AEL disciplines and classes, diagrams or qualitative techniques, semantic matrixes;

• observation in pedagogical practice was a method of research based on the direct support of lectures that use D' Hainaut's operationalization model;

• pedagogical research has used methods of mathematical statistics, in variant stages.

4.1. Participants

The study was conducted in 2013 on a sample of 36 participants, 19 are feminine and masculine 9, third-year students from the Faculty of Mathematics, University "Vasile Alecsandri" of Bacau.

4.2. Research Objectives

Students have the following objectives:

• - Use of the operationalization D'Hainaut in mathematics leads to the formation of students' thinking and develop an appropriate course in mathematical problem solving approach;

• To show that, regardless of the field, using the model to operationalization in solving problems should characterize every man, in many cases, school, family and society;

• - Establish a research study on the effectiveness of model effectiveness in problem solving methods and combining traditional methods with active-participative methods;

• - Promoting the idea that using model operationalization D'Hainaut in mathematical problem solving leads to the development of thought, creativity, feelings and attitude, intellectually competitive spirit.

4.3. Research Methods and Techniques

Research methods and techniques used were: - Observation teaching; - Communication; - Analysis of school documents and student products; - Interview; - Questionnaire; - Statistical techniques for data processing.

4.4. The research hypothesis

The research started from the assumption that: if the students will use to the operationalization model object -capacity - items proposed by D' Hainaut's they will form and grow among middle school students, skills and abilities to solve problems and exercises and lessons will be more effective and better student outcomes.

4.5. Results and their interpretation

Through experiments conducted on a test proposed by students of mathematics faculty at the University "Vasile Alecsandri" of Bacau, activity analysis of 260 students from classes V-VIII National Pedagogical College "Stefan cel Mare" of Bacau, presented in:

Table 1. Test performance results.

Note Ratings Frequency

9 - 10 very good 100

7 - 8 well 90

5 - 6 enough 60

1 - 4 insufficient 10

Test results that reflect student performance on the test determined the overall mean 7.50

Table 2. Histogram reflects the test results of students.

120 100 80 60 40 20 0

verygood good enough insufficient

5. Conclusions

Pedagogical practice has enabled the establishment of the following conclusions:

• Increasing learning efficiency requires that students should build general learning skills, which may be formed during the teaching-learning activity from other disciplines;

• Mastery of such skills is better if the same method is applied in the teaching-learning activity of several disciplines from their training process;

• The most important learning skills, which should be given special attention in school are: independent work with handbooks and corpora;

• Independent performance of calculi, solving, measuring, observations, experiments, applications.

The rapid building of the learning skills, required for performing observations and experiments, is achieved if the teacher engages the students in analysing the structure of the observation, experiment and practical applications as forms of activity and in finding ways to solve particular problems.

The analysis of the 260 students in grades 5-8 from "§tefan cel Mare" Pedagogic National High-school has shown that the students who display learning skills characteristic of Mathematics: tend to learn more in a shorter time; have got 18% higher grades in mathematics; have been able to apply concepts and theorems efficiently in solving problems. The percentage of the students who showed their interest in studying Mathematics following the initiation in the particularities of mathematical activity has grown from 73,1% to 89,5%. In middle school, 58% of the seventh graders and 48% of the eighth graders believe that their knowledge of the elements and properties of a fundamental notion or configuration has helped them in achieving high quality papers and improving their attitude towards their school. The students have obtained better results in the Mathematical tests from the National Evaluation contest.

The students from the specialization of Mathematics, from "Vasile Alecsandri" University of Bacâu, have discovered, during the lessons of pedagogical training, that building learning skills characteristic of Mathematics leads to forming the students' thinking and develops a corresponding behaviour in approaching the solving of Mathematical problems, may motivate students not interested in Mathematics, whereas teachers have changed their way of thinking regarding the lesson design activity.

References

Ausubel D.P. Robinson F.G, (1981). Learning in school, an introduction to the psychology of teaching, Bucharest, Teaching and Pedagogical Publishing House.

Cerghit I., (2002). Alternative and complementary training systems. Structures, styles and strategies, Bucharest , Aramis Publishing. Cristea S. (2003). Foundations of science education. General theory of education. Letter Publishing Group / Point International, Chisinau, Bucharest.

Darfler W. & McLane R.R, (1986). School Subjects Mathematics I, in: B. Cristiansen, Hawson GA, M. Otte, Perspectives on mathematics

education, Edition D. Reidel Publishing Company, Holland. Dewey J., (1992). Foundations for a science of education, Bucharest , Didactic and Pedagogical RA Publishing House. D'Hainaut L., (1977). Des Objectifs aux fins de l'education, Paris , Fernard Nathan.

D'Hainaut L., (1981). Education and lifelong learning programs, Bucharest , Didactic and Pedagogic Publishing House.

Lupu C. (2014). The psycho-pedagogical paradigm of discipline didactics , Saarbrücken, Germany, LAMBERT Academic Publishing.

Lupu C., (2008 ). Pedagogical paradigm of school discipline didactics, Bucharest , Publishing Didactic and Pedagogic, RA.

Piaget J., (1982). Psychology and Pedagogy, Publishing House Didactic and Pedagogic, Bucharest.

R.M., Gagne, L.J., Briggs, (1977). Principles of training design, Bucharest , Didactic and Pedagogic Publishing House.

The Landsheere V. and G., (1979). Defining the objectives oof education, (trans.), Bucharest, Didactic and Pedagogic Publishing House.