Determination of Students’ Beliefs is one of the Aspects of Competence Oriented System of Mathematics Teachers’ Methodical Preparation

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5.1.1. Analysis of the Data

The questionnaire shows that the structure of students' beliefs about the nature of Mathematics is miscellaneous. There are features of “Mathematics as a Cognition Process” as well as “Mathematics as a Set of Formulas and Procedures” in it. We compare the indexes according the answers of Ukrainian (U) and Russian (R) students (Table 2).

Table 2. The Comparison of Indexes According to the Answers of Ukrainian (U) and Russian (R) Students

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The comparison shows that among future Mathematics teachers of there are more individuals (by 23%) who share idea about mathematics as “A Set of Formulas and Procedures”. There are fewer differences in the item of the necessity to solve mathematical problems in order to develop creative thinking and preparedness to the independent discovery of new mathematical facts. The comments of Russian and Ukrainian students concerning the expression “When solving mathematical tasks you need to know the correct procedure or else you would be lost” do not differ significantly; and they show the students' belief to learn according a model, when the presentation of the correct scheme for solving a problem is necessary for successful learning. Most of Ukrainian students (82,2%) focus on the activity of memorization and application of definitions, formulas, mathematical facts and procedures while learning Mathematics. At the same time, 99.3% of students consider learning Mathematics to be a creative and cognitive activity that is the basis for the implementation of new discoveries.

5.2. Beliefs about Learning Mathematics
5.2.1. Obtaining the Data

This area includes questions about the appropriateness of particular instructional activities, questions about students’ cognition processes, and questions about the purposes of mathematics as a school subject.

The following scales of students' beliefs were distinguished concerning the peculiarities of learning mathematics [^[1], p.92]: 1) learning mathematics is under the guidance of a teacher; 2) pupils learn mathematics mostly be the way of independent education and cognitive activity (Table 3).

Table 3. The results of Ukrainian students' questionnaire “From your perspective, to what extent would you agree or disagree with each of the following statements about learning mathematics?”

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5.2.2. Analysis of the Data

The questionnaire shows that the structure of students' beliefs about learning Mathematics is miscellaneous. There are features of beliefs about teacher's priority guidance in the process of acquiring mathematical knowledge by the pupils (beliefs 1), as well as beliefs about pupils' independent work as a dominant of their educational and cognitive activity (beliefs 2). For example, 66.3% of respondents agree with the statement “B. Pupils need to be taught exact procedures for solving mathematical problems” that forms the scale of the first belief. At the same time, even more students agree with the statements that form the second scale: “H. Teachers should allow pupils to figure out their own ways to solve mathematical problems” (80.6% in agreement zone), “N. It is helpful for pupils to discuss different ways to solve particular problems” (83.9% in agreement zone).

The relation of Ukrainian students to the statement “А. The best way to do well in mathematics is to memorize all the formulas” causes an anxiety (43.9% in agreement zone).

To compare with the data of Russian investigators, agreement zone with such a statement is formed by 21.1% of future Russian teachers. Besides, an alarming fact is that almost one-third (27.1%) of Ukrainian students agree with the statement “ Non-standard procedures should be discouraged because they can interfere with learning the correct procedure” (Table 4).

In general, we can say that Ukrainian students demonstrate higher index concerning “Pupils learn mathematics mostly under the guidance of a teacher” if compared with Russian students; and, respectively, lower index concerning “Pupils learn mathematics by the way of active independent education and cognitive activity”. Thus, the traditions of authoritarian pedagogy among Ukrainian students, future Mathematics teachers, are rather strong.

Table 4. Some Indexes Comparison of Ukrainian (U) and Russian (R) Students’ Answers

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5.3. Beliefs about Mathematics Achievement
5.3.1. Obtaining the Data

This area taps into future teachers’ beliefs about various teaching strategies used to facilitate learning of mathematics. Other questions explore beliefs about how mathematics learning may take place, and yet others explore the application of attribution theory to teaching and learning interactions (e.g., innate ability for learning mathematics).

To determine this available beliefs, the students were proposed to answer the question: “To what extent do you agree or disagree with each of the following statements about pupil achievement in secondary mathematics?”. The variants of answers and the results of questionnaire are in Table 5.

Table 5. The results of Ukrainian students' questionnaire “To what extent do you agree or disagree with each of the following statements about pupil achievement in secondary mathematics?”

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5.3.2. Analysis of the Data

The questionnaire shows that while most Russian students (80%) generally consider that mathematical abilities are constant and cannot be developed or improved in education process, the corresponding index for future Ukrainian teachers is 25.5%.

Thus, the vast majority of future Ukrainian teachers (65,5%) believe that pupils' mathematical abilities can and must be developed in education process, and education efficiency is achieved due to not only pupils' abilities but corresponding efforts of all participants of education process (80,9%).

At the same time, students' gender beliefs are noticed. 60,1% of our respondents agree with the statement “In general, boys tend to be naturally better at mathematics than girls”.

5.4. Beliefs about Preparedness for Teaching Mathematics
5.4.1. Obtaining the Data

Table 6. The results of Ukrainian students' questionnaire “Please indicate the extent to which you think your teacher education program has prepared you to do the following when you start your teaching career?”

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The fourth area of belief relevant to TEDS-M concerns the extent to which future teachers perceive their teacher preparation has given them the capacity to carry out the central tasks of teaching and to meet the demands of their first year of practice. The items in these scales are therefore designed to explore different areas of teacher preparation impact. At the end of the questionnaire, a direct question is used to confirm these views. To determine how future teachers are confident in their preparedness to professional activity, they were proposed to answer the question “Please indicate the extent to which you think your teacher education program has prepared you to do the following when you start your teaching career?” The variants of answers and questionnaire results are in Table 6.

5.4.2. Analysis of the Data

The questionnaire shows that the vast majority of future teachers are rather confident at the start of their professional career; however, a set of a teacher’s professional functions within which young specialists are not quite confident, may be distinguished:

• G. Challenge pupils to engage in critical thinking about mathematics (22,9 %);

• D. Use questions to promote higher order thinking in mathematics (25,5 %);

• F. Establish a supportive environment for learning mathematics (21,1 %);

• I. Provide parents with useful information about your pupils’ progress in mathematics (21,4 %);

• K. Incorporate effective classroom management strategies into your teaching of mathematics (28,7 %);

• L. Have a positive influence on difficult or unmotivated pupils (33,6 %);

• M. Work collaboratively with other teachers (26,3 %).

According to the students’ point of view, the study of problem “zones” of their future professional activity will promote the improvement of pedagogical and methodical training system of future specialists.

5.5. Beliefs about Program Effectiveness
5.5.1. Obtaining the Data

The efficiency of pedagogical education program was assessed by the way of a number of questions:

“In general, how efficient was the program of your preparation to learning Mathematics in basic and special school, in your opinion?”;

“In your teacher preparation program, how often did you have the opportunity to learn to do the following?”;

“During the school experience part of your program, how often were you required to do each of the following?”

Table 7. The results of Ukrainian students' questionnaire “In general, how efficient was the program of your preparation to learning Mathematics in basic and special school, in your opinion?”

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The answer variants and questionnaire results are in Table 7.

As we can see, the vast majority of students generally consider the program of their preparation to teaching Mathematics to pupils of the basic and special school to be efficient (93,4 % and 85,6 %, respectively).

However, 14.4% of the students consider the preparation to the work of Mathematics teacher at special school to be not rather efficient and need to be improved; the corresponding index for the basic school if significantly lower – 6.6%.

The obtained data allow making a conclusion about the topicality of the complex investigation of the problem of future Mathematics teacher methodical preparation for special school that would consider the specific nature of professional functions and tasks which a teacher faces at senior stage of general secondary education.

It is a generally known fact that knowledge criterion is practice, and knowledge efficiency is provided with the experience of its application. Taking into account this fact, the students were proposed a question “In your teacher preparation program, how often did you have the opportunity to learn to do the following” for more detailed assessment of this aspect of future specialist training program. The variants of answers are in Table 8.

Table 8. The results of Ukrainian students' questionnaire “In your teacher preparation program, how often did you have the opportunity to learn to do the following”

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The questionnaire results show that the vast majority of students gets the experience of guiding pupils' learning process while learning Mathematics (90,3 %) and the experience of applying various pedagogical technologies that would influence upon pedagogical skills of young specialists and enrich their pedagogical knowledge.

At the same time, much more attention is paid to practical preparation of future teachers to the work with gifted children than with children having emotional, behavioral or cognitive disorders (65,7 % and 50,1 %, respectively); however, medical, psychological and sociological studies show the increasing number of children having related disabilities.

These aspects should be counted in the process of methodical preparation of the future Mathematics teachers giving them an opportunity to form and enlarge the field of subject experience of interaction with such children.

International experts stress that while assessing the efficiency of education program it is important to consider the succession of theoretical instruction in higher schools and pedagogical practice.

The questionnaire for future teachers included questions that allowed the students to express their pinion concerning how often they used knowledge and skills that were acquired theoretically (Table 8, Table 9).

Table 9. The results of Ukrainian students' questionnaire “During the school experience part of your program, how often were you required to do each of the following?”

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