Chapter 3: LANDSCAPE OF MATHEMATICS EDUCATION
1. Domain and Theory: This point suggests that number theory is a field that can have variations depending on its context, such as whether it falls under analytic number theory or complex function theory. Number theory involves discussions on the properties of a sequence of numbers, facts about even or odd numbers, and infinitely many prime numbers.
a. Analytic Number Theory: Analytic number theory is a branch of number theory that utilizes methods from mathematical analysis to study the properties of numbers. It involves the study of the distribution of prime numbers, patterns in number sequences, and other mathematical properties related to numbers.
b. Complex Function Theory: Complex function theory is a field in mathematics that studies functions that take complex numbers as inputs and outputs. Complex functions involve complex numbers, which consist of a real part and an imaginary part. In this theory, you study the basic properties of complex functions, such as singularity analysis, contour integration, and conformal transformations.
2. Theory versus Practice: This point states that mathematical, physical, chemical, or biological theories are not the theories themselves, but rather encompass or discuss the foundations of these sciences. The important question is not whether a theory is true or applicable, but rather how the theory can be understood and applied coherently and consistently.
a. Mathematical Theory: Mathematical theory is a conceptual framework and set of rules that form the foundation of mathematics. It involves abstract thinking, concepts, and mathematical
relationships used to understand and explain mathematical phenomena. Mathematical theory often focuses on logical deduction and provides a basis for solving broader mathematical problems.
b. Scientific Practice: Scientific practice involves the application of mathematical theory and scientific methodology in solving real-world problems. It entails observation, testing, and experimentation to validate and apply mathematical concepts in real-world situations. Scientific practice aims to use mathematical theories and concepts to solve problems, make predictions, and develop practical understanding applicable across various scientific disciplines.
3. Philosophical Background: This point suggests that an understanding of a teacher's philosophical background can influence their teaching style in mathematics. There are differences between horizontal and vertical mathematization in mathematics education, and effective teaching should consider both.
There are two concepts associated with the philosophical background in mathematics education:
horizontal mathematization and vertical mathematization.
Horizontal Mathematization: Horizontal mathematization is an approach to mathematics education that emphasizes the application of mathematics in everyday life and contexts relevant to students.
This approach prioritizes the understanding of mathematical concepts and students' ability to apply mathematics in real-life situations. For example, teaching students how to use mathematics to solve financial, measurement, or statistical problems in daily life.
Vertical Mathematization: Vertical mathematization is an approach to mathematics education that places greater emphasis on the development of a deep understanding of mathematical concepts and the introduction of students to more complex mathematical ideas. This approach highlights
conceptual understanding, problem-solving, and abstract mathematical thinking. The goal is to help students build a strong mathematical foundation and prepare them to understand and master more complex concepts in the future.
Effective mathematics teaching should consider both of these approaches. A teacher who
understands their philosophical background will be able to incorporate aspects of both horizontal and vertical mathematization in their instruction. They will be able to create rich and meaningful learning experiences for students by connecting mathematics to the students' real-world contexts while providing a deep understanding of more abstract mathematical concepts.
4. Philosophy of Mathematics Education: This point explains that there are various approaches in mathematics education based on different philosophies, such as mechanistic, empirical, structuralist, and realistic. Each approach has different characteristics in terms of viewing horizontal and vertical mathematization in mathematics learning.
Here are several approaches in mathematics education based on different philosophies:
Mechanistic Approach: This approach considers mathematics as a set of rules and procedures that students must learn and master. The main focus of this approach is on the mastery of techniques and computational skills in mathematics. In the context of horizontal mathematization, the mechanistic approach may emphasize the routine application of mathematics in everyday life situations. In the context of vertical mathematization, this approach may focus more on understanding mathematical concepts systematically and building a solid mathematical structure.
Empirical Approach: This approach emphasizes the use of direct experience, observation, and experimentation in mathematics learning. Students are encouraged to interact with real objects and real situations to understand mathematical concepts. In horizontal mathematization, the empirical approach emphasizes the application of mathematics in real and relevant contexts. In vertical mathematization, this approach may emphasize understanding concepts through real experiences and mathematical modeling.
Structuralist Approach: This approach emphasizes understanding the structure and relationships between mathematical concepts. Students are encouraged to understand and identify patterns, relationships, and mathematical structures in depth. In horizontal mathematization, the structuralist approach may emphasize the application of mathematics in contexts that reveal structure and relationships. In vertical mathematization, this approach may emphasize understanding mathematical concepts in more complex structural and relational contexts.
Realistic Approach: This approach emphasizes authentic and relevant mathematics learning for students' daily lives. Mathematics is seen as a tool for understanding and solving real-life problems.
In horizontal mathematization, the realistic approach emphasizes the application of mathematics in real-life situations and relevant problems. In vertical mathematization, this approach may emphasize understanding mathematical concepts through solving real problems and authentic contexts.
Each approach in mathematics education has its own strengths and weaknesses, and the selection of an approach depends on the learning goals, students' characteristics, and teaching context. It is important for a teacher to understand these various approaches and apply the approach that suits the needs and characteristics of students in order to create effective and meaningful mathematical learning experiences.
5. Constructivism and "Kant": This point mentions the flexibility of language and the importance of construction in understanding the world. It discusses the development of human worldview and constructive development in the field of mathematics.
Constructivism is the view that knowledge and understanding are actively constructed by individuals through their own mental constructions. In the context of mathematics education, the constructivist approach emphasizes the importance of students actively engaging in constructing their
understanding of mathematical concepts through experience, reflection, and interaction with learning materials.
The term "Kant" refers to Immanuel Kant, a famous philosopher from the 18th century who had significant influence in various fields, including philosophy of education. One of his contributions is in the thinking about human understanding of the world. According to Kant, human understanding of the world is not solely determined by the objectivity of the world itself but is also influenced by the cognitive structure of human beings.
In the context of mathematics education, constructivism and Kantian thinking are related to the concept that students do not merely passively receive information but actively engage in constructing their own mathematical understanding. Language flexibility becomes important in understanding the world and in mathematics learning because language is used as a tool to construct and communicate mathematical understanding.
By adopting a constructivist approach in mathematics learning, teachers play the role of facilitators who encourage students to think critically, analyze problems, formulate hypotheses, and actively explore in constructing their own mathematical understanding. Teachers also encourage students to interact with classmates and use mathematical language flexibly to express and build their
understanding of mathematical concepts.
The constructivist approach and Kantian thinking emphasize the importance of students as active subjects in constructing their mathematical knowledge, appreciating the diversity of thinking and understanding, and strengthening mathematical communication skills. By enabling students to engage in constructing their own mathematical understanding, this approach aims to create more meaningful, relevant, and sustainable learning experiences.
6. Research in Mathematics Education: This point discusses the goals of research in mathematics education and the importance of evolving research methods. It also highlights the significance of comparative research and measurement in education.
Research in mathematics education aims to gain a deeper understanding of the processes of learning and teaching mathematics and to develop more effective practices to support students'
mathematical learning.
Some goals of research in mathematics education include:
a. Understanding the process of learning and teaching mathematics: Research in this field aims to study how students understand mathematical concepts, identify the difficulties they encounter, and discover effective teaching strategies to facilitate mathematical learning.
b. Developing effective teaching practices: Research in this field aims to find the most effective methods and strategies for teaching mathematics, both in terms of facilitating understanding of mathematical concepts and developing students' problem-solving skills in mathematics.
c. Considering individual needs: Research in this field aims to identify individual differences in mathematical understanding and learning, as well as to find teaching strategies that can effectively meet individual needs.
d/ Developing relevant mathematics curriculum: Research in this field aims to contribute insights into the development of mathematics curriculum that is relevant to students' needs, taking into account advancements and trends in the field of mathematics and real-world needs.
Research methods in mathematics education evolve with advancements in the field. Some commonly used research methods include experimental research, qualitative research, action research, and development research.
The importance of comparative research and measurement is also emphasized in the context of mathematics education. Comparative research allows for comparisons between two or more groups of students or teaching methods to determine their effectiveness. Accurate and valid measurement is used to gather data about students' achievement, their understanding of mathematical concepts, and the effectiveness of teaching.
Through quality research and evolving research methods, mathematics education can continuously improve teaching practices, design relevant curricula, and gain a better understanding of how students learn mathematics.
7. Development Research: This point discusses development research that involves changes in educational approaches. It emphasizes that development research is not something new and avoids the negative connotations of the word "traditional."
Development research is a type of research that focuses on development or improvement in the field of education. Its main goal is to create, test, and implement innovations in educational practices that can enhance student learning.
Development research aims to generate new knowledge that can be applied in an educational context. It involves an iterative process that includes identifying problems or needs, designing new solutions or innovations, testing and evaluating those innovations, and implementing the research findings into better educational practices.
It is important to note that development research is not something new in the field of education. As a discipline, education has always involved efforts to improve and develop teaching practices.
However, development research places more emphasis on the scientific approach in the process of change and educational development.
This point also highlights the importance of avoiding negative connotations of the word "traditional"
in the context of development research. Tradition in education does not always have to be seen as something negative. Many traditional practices and methods have proven to be effective in teaching and learning mathematics. However, development research aims to enrich traditional practices with new innovations supported by evidence and scientific research.
In the context of development research in mathematics education, this approach can involve the development of new teaching methods, technology-based learning, the use of games and
manipulatives in mathematics learning, or the development of more relevant and engaging curricula for students.
Development research provides space for educators and researchers to collaborate in developing solutions and innovations that can enhance mathematics education practices. With this approach, mathematics education continues to evolve and adapt to changes in the environment and students' needs.
