Continuity of mathematics education between universities and secondary schools based on the theorization of the content of mathematical education

Kenzhebayeva Maira Utenovna, ASUE, Almaty, Kazakhstan.

Abstract

The purpose of this study is to obtain the opinions of teachers in order to form a continuity of mathematics education between universities and secondary schools based on the theorization of the content of mathematical education. A qualitative research method was used in this study. The research team consists of 60 teachers of mathematics who teach in various universities and secondary schools of Kazakhstan. The research data were collected using a semi-structured interview which were developed by the researchers. The research data were evaluated by the analysis method. In the result of the study, most of the teachers of mathematics who participated in the study showed that they weren’t competent insufficiently in additional topics of theorizing in the content of mathematics. The vast majority of teachers stated that they were very inadequate in setting mathematical problems on new topics. The proposals of the majority of teachers of mathematics participating in the study on the creation of continuity of mathematics education between universities and secondary schools are based on the activation of methodological work between teachers of the university, the dissemination of the best practices, generalization of the new topics, as well as the creation of a methodological center, conducting an analysis of the needs for the formation of continuity of mathematics education between universities and secondary schools and conducting additional job training for teachers.

Keywords: theorization of mathematical education, continuity of mathematical education, teachers' opinions.

1. Introduction

In recent years, there have been significant changes both in education and in views of mathematics and mathematical education. Mathematical education is also aimed at educating people who not only know mathematics, but also apply what they know, teach mathematics, solve problems and communicate. Changes in the content of teaching mathematics significantly affect the mathematical education of students.

1.1. Theoretical and conceptual foundations

In today's world, where information spreads quickly and new information is added every day, effective, useful and more constant study of information becomes an element that needs to be explored in the educational process. This is also one of the goals of education, in which students develop themselves in the personal, social and educational spheres and possess the necessary competencies. Due to the increase and development of information to be assimilated, traditional teaching methods are insufficient, and new methods and strategies are needed in the learning process [1].

The skills acquired as a result of effective teaching of mathematics allow you to evaluate the opportunities for practice in everyday life or in various disciplines. People who have mathematical abilities and are able to demonstrate these skills in the tasks they face are necessary in any field.

Given the contribution of mathematics to the development of an individual's thinking, as well as to everyday and real-life situations, individual and social mathematical competence and, consequently, problem solving are becoming increasingly important.

When definitions of the concepts of "continuity" are considered in the literature, it can be seen that difficulties, troubles and problems encountered in everyday life are also expressed by the word interconnection [2]. Mathematics is a formal language that allows us to express our abstract thoughts in the form of systematized knowledge.

The theorization of the content of mathematical education in Kazakhstan consists in the introduction of the concepts of complex numbers, differential calculus in grades 10-11, elements of combinatorics from grades 3-4, elements of mathematical statistics from grade 5, elements of probability theory from grade 9. Application of Newton's binomial formula and its properties in the context of various problems, Pascal's triangle for finding binomial coefficients, finding binomial decomposition with the degree of a natural number.

Mathematics plays a key role in all scientific disciplines, helping to develop theories, model phenomena, conduct statistical analysis and solve complex problems. Mathematical methods are used in various areas of our daily life, including economics, finance, sociology, medicine and technology [3].

1.2. Related research

Changes in the curriculum in 2023 were revised and amended as part of the improvement of the educational program of the AEO "Nazarbayev Intellectual Schools" with a focus on the formation of key competencies. A new version of the curriculum based on the previous version developed by the AEO "Nazarbayev Intellectual Schools" in cooperation with the Cambridge Assessment Council on International Education (CAIE).

Qualitative research is a method that asks questions about the problem that it explores, interprets, and tries to understand the form of the problem in its natural environment [4]. Qualitative research, which uses qualitative data collection methods such as observation, interviews, and document analysis to solve a problem, refers to the subjective interpretative process of perceiving previously known or unrecognized problems and considering problem–related natural phenomena in a realistic manner [5]. In this study, the opinions of teachers obtained in order to form the continuity of teaching mathematics between university and secondary school students based on solving new problems were evaluated in accordance with the method of qualitative research.

2.2. Participants

The research team consists of teachers of mathematics who teach in various secondary schools in Kazakhstan. The math teachers who participated in the study agreed to participate voluntarily in the study. 40 math teachers participated in the study. 26 teachers are women and 14 are men. 10 teachers have work experience from 1 to 5 years, 8 of them - from 6 to 10 years, 7 of them - from 11 to 15 years and 15 of them have work experience of 15 years or more.

2.3. Data collection tools

The research data were collected using a semi-structured interview form developed by the researchers. A literature review was conducted during the preparation of the semi-structured interview form. The questions for the semi-structured interview were prepared based on the results of research in the field and presented for two experts to get their opinions. The questions were rearranged in accordance with the opinions of the experts. Then it was introduced to two math teachers who check if the questions were clear. The math teachers found that the questions were understandable. Two math teachers who participated in this part of the study were excluded from the study's research group. Three interview questions were formed in order to get the opinions of mathematics teachers on the formation of continuity of mathematics education between universities and secondary schools based on the theorization of the content of mathematical education. The interview questions in the form of a semi-structured interview are as follows:

1. How do you assess the competence of schoolchildren in setting new tasks in forming the continuity of mathematics education between universities and secondary schools based on the theorization of the content of mathematical education?

2. How do you assess the formation of continuity of mathematics education between universities and secondary schools based on the theorization of the content of mathematical education?

3. What are the students' suggestions for creating a continuity of mathematics education between universities and secondary schools based on the theorization of the content of mathematical education?

2.4. Data collection process

Personal interviews with math teachers were conducted to collect the research data. The interviews were conducted in secondary schools where math teachers work.

Interviews with teachers were conducted face-to-face and individually.

Semi-structured interview forms were distributed to math teachers and they were asked to answer questions. While the teachers were answering the questions, the researcher was in the interview room and asked the teacher to ask questions if they did not understand something. It took about 25-30 minutes.

The teachers answered the questions in semi-structured interview forms. It took about 5 weeks to conduct interviews with all the math teachers involved in the study.

2.5. Analysis of data collection

The research data were evaluated using the content analysis method. Content analysis requires a more detailed examination of the collected data and identification of concepts, categories and topics that explain this data. Content analysis focuses on the collected data; codes are extracted from events and facts that are often repeated in the dataset or that the participant strongly focuses on, from codes to categories and from categories to topics. In short, data (codes) that turn out to be similar and related to each other are interpreted by bringing them together within the framework of certain concepts (categories) and topics. Content analysis systematically analyzes the content of participants' opinions [4]. The answers given by the mathematics teachers to the questions in the form of a semi-structured interview were divided by the content analysis method and converted into frequency and percentage tables and presented in the Results section in tabular form. Direct quotes from the answers of the mathematics teachers who participated in the study to the questions were made and used directly in the research, provided that their personal information was kept confidential. The opinions of mathematics teachers were presented in the form of X-1, X-2, X-3 encodings....

3. Results

In this section, the answers given by math teachers to questions in the form of a semi-structured interview were evaluated.

Table 1 shows the assessments of the mathematics teachers who participated in the study regarding the students' competencies in solving new problems.

Table 1. Mathematics teachers' assessments of students' competence in solving new problems

Table 1 summarizes the opinions of mathematics teachers who participated in the study on the competence of students in solving new problems while theorizing the content of mathematical education. 2.5% of math teachers gave the answer "very adequate", 5% - "sufficient", 25% - "somewhat sufficient", 55% - "insufficient" and 12.5% - "very insufficient".

The opinions of mathematics teachers who participated in the survey below are the results of a study of students' competencies in solving new problems.

X-3: I find my students very competent. Willing to learn, hardworking and open to self-improvement. I think this is the most important factor. They are open to improving their weaknesses. X-5: Actually, I find it sufficient. I see the efforts of schoolchildren on new tasks. X-33: They are partially enough. They have a negative attitude towards mathematics, which is formed because of their prejudices and fears. Turning this attitude into a positive one will not only change their point of view, but also positively affect their attitude towards learning. X-12: I think they are not enough. They do not have sufficient infrastructure, they cannot easily carry out new training. They don't like the lesson. X-23: I think the students are very inadequate. I find the students' skills in solving new problems sufficient. Similarly, I can state that the theorization of mathematical content is quite high. X-16: I think that students have sufficient skills in setting new tasks. X-18: I find my students sufficient in terms of solving mathematical problems and setting problems. X-32: I think my students are somewhat sufficient in setting new problems based on this problem. X-38: Schoolchildren do not have the ability to draw analogies between tasks. That's why I think they're not enough. X-40: They don't even manage to try different ways to solve the problem. They cannot look at this problem from different angles. For this reason, I find schoolchildren very inadequate in setting new tasks.

On the second question, the answers given by the math teachers to the questions in the form of a semi-structured interview were evaluated and they were similar to the results of the first question. Most teachers do not approve of the theorization of the content of mathematical education, as they conduct them formally.

Table 2. The proposals of teachers of mathematics on the formation of continuity of teaching mathematics between universities and secondary schools, based on the solution of the theorization of the content of mathematical education.

Table 2 shows the proposals of mathematics teachers participating in the study regarding the formation of continuity of mathematics education between universities and secondary schools, based on the theorization of the content of mathematical education, solving new problems of students.

Table 2. Suggestions from math teachers

Table 2 evaluates the proposals of the mathematics teachers who participated in the study regarding the creation of continuity of mathematics education between universities and secondary schools, based on the theorization of the content of mathematical education, solving new problems.

70% of teachers replied that it is necessary to create a lesson environment supported by technology, 80% of them replied that it is necessary to conduct a needs analysis to form the continuity of mathematics education between universities and secondary schools, 60% of teachers should undergo on-the-job training and 20% of them should undergo additional training that will instill students love mathematics. 40% of mathematics teachers replied that they should be taught various ways to solve new problems, 35% of them replied that great attention should be paid to the formation of mathematical concepts, and 55% of them replied that students should be switched from a passive position to an active one in mathematical education. In addition, 20% of mathematics teachers suggested that teacher training policies should be aimed at improving teacher qualifications. The following are suggestions from the mathematics teachers involved in the study to create a continuity of mathematics education between universities and secondary schools based on the theorization of the content of mathematical education and solving new problems.

X-13: I believe that technological support should be used to form the continuity of mathematics education between universities and secondary schools. Teacher deficiencies should also be addressed through on-the-job training. X-27: To form mathematical concepts in accordance with the needs of the era, regardless of technology. With the support of education technologies, the student's transition from a passive student to an active one will be easier. X-35: First of all, it is necessary to instill in students a love of mathematics. This can be achieved through educational technology games.

4. Discussion

Most of the math teachers who participated in the study said that students were insufficiently prepared to solve new problems. The vast majority of mathematics teachers said they were very inadequate at setting new tasks. The proposals of the majority of mathematics teachers involved in the study are to create a technology-supported lesson environment, analyze the needs for the formation of continuity of mathematics education between universities and secondary schools, and conduct on-the-job training for teachers. Some teachers stated that students should receive additional training to instill in them a love of mathematics, various ways of solving new problems should be taught, great attention should be paid to the formation of continuity of mathematics education between universities and secondary schools, and students should be transferred from a passive position to an active position in mathematics education. In addition, some teachers stated that the teacher training policy should be aimed at improving the skills of teachers.

When studying research in this area, it was found that a sequential approach to learning is more effective than traditional approaches to learning in transforming students' attitudes to the math lesson. The succession approach determines teachers' knowledge of the content of higher mathematics. Accordingly, teachers' knowledge of the content of higher mathematics should be taken into account in the process of improving student academic performance.

5. Conclusion

In the modern understanding of education, problem-solving skills are a tool aimed at teaching people to overcome difficulties on their own, adapt to life and contribute to the development of the country. However, while schools place too much emphasis on reading, writing, and math skills, learning, reasoning, and problem solving skills that involve reading and the more complex aspects of basic math are ignored. The creation of continuity of mathematics education between universities and secondary schools based on the theorization of the content of mathematical education, solving new problems is very important from the point of view of educational services. In this regard, this study was aimed at obtaining the opinions of mathematics teachers, they stated that students did not cope well enough with solving new problems. The vast majority of math teachers said they were very inadequate in setting new tasks. The proposals of the majority of mathematics teachers are to create a lesson environment supported by technology, analyze the formation of mathematical concepts and provide on-the-job training for teachers.

6. Recommendations

In the light of the results obtained during the study, the following recommendations were made for teachers of mathematics:

1. Most of the mathematics teachers who participated in the study stated that the students were not well versed in solving new problems, and they were very inadequate in setting new tasks. For this reason, it is necessary to revise the course curricula to meet the needs of students.

2. On-the-job trainings should be organized for teachers of mathematics, additional trainings should be conducted to fill in the shortcomings of teachers, and creative practices should be developed aimed at forming new mathematical concepts by ensuring cooperation between the school and the university.

3. The policy in the field of teacher training at universities should be regulated in order to train teachers who will form the continuity of mathematics education between universities and secondary schools.

4. It is necessary to develop a strategy for the perception of the content of mathematical education, especially the content of a mathematical problem, which require separate research.

References

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