H. Cordenthal (1905- 1990)
[America] g. Paulia (1887- 1985)
[Rui] J. Piaget (1896- 1980)
] D.P. Ausubel
Bloom
6. 1 Freudenthal Mathematics Education Theory
The representative work "Mathematics as an Educational Task"
The basic characteristics of mathematics education (reality, mathematicization and re-creation);
-Situational problems are the platform of teaching.
Mathematicization is the goal of mathematics education.
-The conclusions and creations that students get through their own efforts are part of the educational content.
-"Interaction" is the main way of learning.
Intertwining of disciplines is the way to present the content of mathematics education.
What is truth in mathematics education?
Reality in mathematics education-Mathematics comes from reality, exists in reality and is applied to reality. Every student has his own different "mathematical reality".
One of the tasks of mathematics teachers is to help students construct mathematical reality and develop their mathematical reality on this basis.
Examples are life-oriented and questions are situational.
Teaching with "Realistic Mathematics"
First, the concepts, operations, rules and propositions of mathematics are all formed from the actual needs of the real world, which is an abstract reflection of the real world and a summary of human experience.
Second, the object of mathematical research is a quantitative model abstracted from similar things or phenomena in the real world.
Thirdly, mathematics education should provide different levels of mathematics knowledge for different people.
What is mathematicization?
In the process of observing, understanding and transforming the objective world, people use mathematical thinking methods to analyze and study various phenomena in the objective world and organize them-that is, the process of organizing the real world into mathematics is mathematicization.
Mathematics teaching is mathematics teaching.
Abstraction, axiomatization, modeling, formalization and so on can all be regarded as mathematization.
Mathematical form: practical problems are transformed into mathematics; From Symbol to Mathematicization of Concept
Basic process P 168
"Re-creation" of Mathematics Learning
The process of students "recreating" learning mathematics is actually a process of "doing mathematics". Its core is the representation of mathematical process.
Mathematics learning is a process of experience, understanding and reflection. It emphasizes the importance of student-centered learning activities to students' understanding of mathematics, and emphasizes that stimulating students to actively study and do mathematics is an important way for students to understand mathematics.
6.2 Paulia's problem solving theory
Representative works: How to Solve Problems, Discovery of Mathematics, Mathematics and Guess.
"Every college student, every scholar and especially every teacher should read this fascinating book"-Van de Walden.
The fundamental purpose of middle school mathematics education is to "teach young people to think"-purposeful thinking and productive thinking, including formal thinking and informal thinking.
The b way to learn is to explore and discover by yourself.
Table on How to Solve Problems-Example1(p171~173)
6.3 Constructivism Theory of Mathematics Education
What is mathematical knowledge?
The way of mathematics learning: copy and construction.
The main features of mathematics learning from the perspective of constructivism;
The process of students' self-construction of knowledge is irreplaceable by others.
-Learning is actively and meaningfully constructed according to experience.
-Re-encode new knowledge and build your own understanding.
Understanding \ Situation \ Problem \ Reflection \ Construction
Application examples of constructivist teaching principles
Dobinsky et al., an APOS theory developed in the practice of mathematical education research (taking the concept of function as an example)
The steps of traditional mathematics concept teaching: defining the concept (definition, name, symbol); Classification; Consolidate; Applications and connections
Mathematical concept has the duality of process and object, which is not only the object of logical analysis, but also a mathematical process with realistic background and rich implications. Therefore, we must return to nature, reveal the formation process of the concept, and understand a mathematical concept from the aspects of realistic prototype, abstract process, ideological guidance and formal expression, so as to make it conform to the educational principle of students' active construction.
APOS theory (taking the concept of function as an example)
Action stage: understanding a function requires an activity or an operation. Understand the meaning of function through operation activities.
Process stage: integrate the above operation activities into a functional process. x？ x2，x？
Object process: regard the function process as an independent object. Addition, subtraction, multiplication, division and compound operations of functions
Schema stage: the concept of function is stored in the brain as a comprehensive psychological schema, forming a knowledge system (integrity).
APOS theory (taking algebraic concepts as an example)
The essence of algebraic expression is "indefinite element", and numbers can be operated like numbers.
Answer: Understand specific algebraic expressions through operational activities.
P: the process of experiencing algebra
O: Formal expression of algebraic expression
Establish a comprehensive psychological schema. Psychological representation of algebraic expression in students' minds: concrete examples, operation process, mathematical idea that letters represent a kind of numbers, and definition of algebraic expression can all be used.
6.4 "Double Basis" Mathematics Teaching in China
The characteristics of "double-base" teaching theory are attaching importance to logical deduction;
1. Operating speed
Memory knowledge
3. Moderate formalization
4. Variant training
"Double Basis" mathematics teaching strategies: problem introduction, teacher-student interaction, consolidation exercise, heuristic; Strengthen oral English and practice more; Small steps, small turns and small slopes; Large capacity, fast pace and high density
Learn and carry forward the traditional "double foundations of mathematics" in China, but be wary of "excess foundations". Clearly "inherit the tradition". National is also international.
Reform should go from the surface to the essence. Consciously understand the problems in the Standards and textbooks. Establish the relationship between "general education idea" and "law of mathematics education" and "return to nature" to reveal the essence of mathematics.
Accumulate a large number of classic cases of mathematics teaching. Mobilize and respect the enthusiasm of bottom-up reform. Protect copyright and respect creation.
The Framework of "Double Basis" Mathematics Teaching in China
Five target levels: mathematical concept, thinking mode and mathematical method.
4-layer development layer
Mathematical modeling; Inquiry learning; Mathematical cultural thinking; Reflection and questioning; Open teaching; Problem reform ...
Three-layer "double-base" layer
Current situation of double bases (quantitative analysis); The definition of double base; The state in the standard ...
Two-layer teaching experience
"return to nature"; "talk in detail and practice more"; Variant practice; Logical discrimination; Examination-oriented training ...
1 cultural background
"rice culture"; "Confucian culture"; "Examination culture"; "textual research culture"; "Practice makes perfect"
Basic principles of mathematical structure
(1) Subject principle: Students are the subject of mathematics learning activities.
(2) The principle of adaptability: Teachers should proceed from the reality of students.
(3) Construction principle: Students construct from the original experience world.
(4) Guiding principle: Teachers are designers, participants, instructors and evaluators of mathematical construction activities.
(5) the principle of problem solving
The main arguments of constructivist learning view
(1) Knowledge cannot be taught by teachers alone, but can only be actively constructed by students according to their existing knowledge and experience;
(2) Constructive activities are the combination of students' relatively independent creative activities and the process of communication and interaction in the "learning isomorphism" composed of teachers and students;
(3) The learning process of mathematical knowledge is a process of "giving meaning" and also a process of "cultural inheritance" (the process of understanding).
The main arguments of constructivist teaching view
Teachers should not be regarded as "imparting knowledge", but as promoters of students' learning activities;
It poses a severe challenge (complete negation) to the traditional teaching method design theory.
The math teacher asked, "What is math?" And the concept of "how to engage in mathematical research" has a direct and important impact on the teaching concept;
Pay attention not only to the conclusion, but also to the analysis of the process;
Change "problem solving" into "mathematical thinking" and take it as the center.
"Double Basis" Teaching under the Constructivism Teaching Concept
Accurately grasp the constructivist view of mathematics education and promote the scientific and effective teaching of "double basics" in mathematics;
First, students' learning and teachers' teaching are a unified process, and the concepts of learning and teaching should be regarded as a whole;
Secondly, basic mathematics skills are of great significance in the process of mathematics learning;
Third, teachers should establish a correct "student view" and "understand two sides" (teaching materials and students): tolerance, adaptation, cherish and creation.
Fourth, the central task of teachers is to "design carefully around the theme".