Knowledge about the teaching of mathematics Research Review and Implications

Abstract: Knowledge of Mathematics teaching of mathematical knowledge through Research, knowledge of Mathematics curriculum and teaching of Mathematics used in three stages. Mathematical knowledge for teaching Mathematics through the analysis of the core activities of teaching, classroom teaching in the direct Research teachers to use the mathematical knowledge and its impact. It is the knowledge base of effective teaching, teacher education should become the main content.

Paper Keywords: Mathematics, teaching, knowledge, teacher education
First, the mathematics of
Tradition has it that mathematics teachers at least to grasp what he taught mathematics. Class Teaching matured, people began to agree to such a principle: In addition to the teaching of mathematics knowledge, mathematics teachers need to master as the organization of teaching, control of classroom discipline and some teaching knowledge. With the teaching of Research, it was discovered that teachers only know that he taught mathematical terms, concepts, propositions, rules and other knowledge is not enough. ... In addition, teachers should know the subject structure of mathematics. disciplinary structure concept originated from Schwab. He pointed out that the subjects understood the structure of two ways: One is syntactic way (syntactically), is the entity in another way (substantively). The so-called syntactic manner is shown by from the subject to understand the logical structure of the subject structure. For example, the introduction of irrational numbers that can not be incommensurable line segments, the introduction of negative and complex that some of the equations. The former can be seen, the latter can not see, just to maintain the equation has a solution in this thesis The integrity and versatility made an assumption and interpretation. understanding of the meaning of these three concepts, only by generating backward and forward linkages of these concepts can be revealed. The so-called entity to refer to the conceptual design from the subject point of view understand the subject structure. For example, Euclidean geometry and analytic geometry have different conceptual frameworks. Ball subject structure of mathematics knowledge as the knowledge of mathematics. It is the knowledge come from, but also how to develop, and the truth is to confirm , and in turn for where to go.

There are three main dimensions: first, the distinction between conventions and logic construction. Is the right number in the number of axes or we use the ten decimal value system is arbitrary, agreed. And 0 do not define division by zero or any number raised to the power are equal to 1 is not arbitrary, agreed upon, the second is the question of the links within mathematics and between mathematics and other areas of the links is to understand the mathematics of the three basic activities: look for patterns, put forward conjecture to prove assertion confirmed Solution and seek general.

Study of mathematical knowledge, broaden people's mathematical knowledge for teaching with understanding. It shows knowledge of Mathematics Teaching is very complex, in addition to terminology, concepts, rules, procedures, there is structure or mathematics on the mathematics knowledge. this knowledge to determine why teachers teach, choose to teach what and how to teach will have an impact. for example, agreed with the logic of the concept of construction of the teaching strategies would be different, logical construction of the concept of Bixu know how it came about , then why should the definition of this concept, how to define, what it is used, the concept of it with other What is the relationship, which limits its application. And the concept of no such agreement is necessary. However, the effective Mathematics Teaching , only with the knowledge is not enough. It is the lack of consideration for students, teachers, professors can not give a specific group of students to the understanding of teaching. For example, only by derivation to know (+6) = a +2 ab + b on the effective teaching is not enough, teachers also need to know some students over-promotion and easy to remember as distributive +6) = a + b, with a rectangular area that can effectively eliminate the characterization of misunderstandings. students the knowledge and eliminate misunderstanding misunderstanding teaching strategies are obviously not included in the framework of mathematical knowledge, mathematical knowledge for teaching the complexity of the framework required to describe more refined.


Second, analysis of materials
Effective teaching must take into account existing knowledge and knowledge of the students showed the best sequence. In mathematics, the Ma Liping's knowledge packages (Knowledgepackage) is typical of such international studies. Knowledge package is built around a central concept organized a series of related concepts is in the minds of students in the area of ​​cultivation of such a vertical process. (n Knowledge contains three major components: the central concept, the concept of sequence and concept nodes, including the concept of representation, meaning and the establishment of algorithm on these concepts. The next example is less than 20 the number of knowledge of addition and subtraction package (Figure 1). in the knowledge package, the central concept is the 20 to 100 the number of "Subtract", which is more than learning The key premise of the number of addition and subtraction.


Ma Liping's knowledge of the Mainland of China is actually a traditional package materials analysis. The results of such Research is one of the main teaching reference books. It is a curriculum knowledge of teachers on the course than on the analysis of mathematical knowledge closer to the teaching of mathematics. but it is not used when teaching mathematics. It is up to the consideration of teaching teachers, teacher-student interaction have not considered the mathematical needs. teachers in teaching, can not mobilize the knowledge necessarily meet the needs of teaching situation. such as a student teacher is expected to be reflected in the interaction with students does not appear, the reaction of teachers to students as a springboard for follow-up, there is no useless knowledge. Ma Liping generalize knowledge package, and teachers in the classroom teaching of mathematical knowledge to use when there is still a distance, teachers in teaching may apply, or may not have access. teachers need in teaching mathematics well beyond the textbook analysis can provide content. Links to Research Papers Download http://www.hi138.com Third, knowledge of teaching mathematics
Ball created the study of mathematical knowledge for teaching. She Mathematics Teaching through the analysis of the core activities, direct classroom teachers of mathematics knowledge and its impact. Here's a Lesson in Ball to explain their Research methods and results. The class content is more than three-digit subtraction Grade: Joshua ate 16 peas on Monday, Tuesday, ate 32 peas. asked Joshua how much to eat on Tuesday than on Monday, and the pea? students in the problem solving process provides a six solution. Sean successor from 16 to count the number of l7 start back, the answer has been the number to 32. ba that half of 32 is 16, the answer to that is 16.Betsy aids 16 and 32 (beans) eleven paired about 32 the number of teaching aids that do not match the remaining beans get an answer. Mei approach is that 32 of the beans directly from the take away 16, count the number of remaining on the line. Cassandia provides a standard subtraction algorithm, Scan inspired , offers another solution: 16 +16 = 32, the whole class, students try every means to identify similarities and differences between these solutions. L6JBall that the core activities of teaching this lesson to deal with related mathematical knowledge and control of class discussions. knowledge association and symbols related to the specific model, the subtraction and addition is how related, subtraction of the "more" and "take away" and explain how it related, aids the characterization of how it translates into symbolic representation, Betsy's paired comparison Law How backward into Sean count method, Betsy methods and Mei's method of coordination to control the first in class discussion and interpretation of clues, and promote the development of the right way, followed by the performance of a problem in the hold. such as suspended Riba's argument. Riba's thesis is correct, but to make the other students can understand what he means, but also need to add a few steps reasoning. But this reasoning and use it a few steps to prove Sean's conclusions more than three students understanding.

Ball teachers need to use these classes were summarized mathematical knowledge. In addition to the traditional textbook analysis provided by the sign bit subtraction algorithm and its place value system behind, the teachers also need additional knowledge. First need to know the two Characterization of model species (such as subtraction 32-16:? missing addends and the addition of 16 +? = 32) are equivalent. Second, but also know some of the characterization of this problem: for example, like Sean's number from 17 to 32, or Mei 32 years away from l6 months and so on. Third, teachers also need to have a profound mathematical perspective to examine, analyze and coordinate a variety of student solution. Finally, teachers also need some knowledge about the mathematical arguments. Through the above analysis , Ball pointed out that the analysis can only provide the teaching materials used as part of mathematical knowledge, most of the rest only in the analysis of the core activities of Mathematics Teaching can be.

IV Revelation
1. Mathematical knowledge for teaching is the knowledge base of effective teaching. It's mathematics and mathematicians, materials drawn from the analysis of mathematical knowledge is not the same. It has a useful teaching mathematical understanding, this understanding focuses on students on the concepts and misconceptions. student understanding of specific content is different, and teachers need to reconcile different ways of understanding the students and between them flexible and comfortable way to convert the knowledge to guide students to further organize and promote their existing knowledge on the basis of effective learning.

2. Mathematical knowledge for teaching is a high point of view of mathematical knowledge, it is linked with a more profound concepts and methods. Ball's third grade Lesson only two-digit subtraction step down, but the core of classroom teaching through the analysis of mathematical activity show, hide away in the abdication of subtraction is the equivalent of higher mathematics, the same structure, and characterization of similarity between the concepts of transformation. from the structures, said the first five solution is isomorphic, solution and the first five The last addition is equivalent to missing addends. However, the interpretation of the first four solution model is different, there are three is the "take away" model, one is the "more" model. only from the mathematical structure, sort out these solution of the relations in order to effectively guide students in different ways to distinguish between the conversion and the similarities and differences of these methods to promote students to efficiently organize their own mathematics knowledge. Hong Kong's "Classroom Learning" also confirmed the participation of experts in Mathematics Teaching and research activities, can improve the effectiveness of classroom teaching.

3. Knowledge of teaching mathematics there is a certain structure. The first is to understand the knowledge of students. As demonstrated by Ball's Lesson, the students understanding of the abdication subtraction there are different ways, at different levels and some of the misconceptions is that knowledge of teaching starting point. the students are the bottom-up knowledge as a starting point to expand the knowledge taught to students already constitute new knowledge and concepts and methods of internal network links, this is the secret to improve teaching efficiency, followed by the teaching strategy. Like Ball's Lesson shows, the students understand the variety, is necessary for teachers to use appropriate strategies to control the class discussion, different ways of coordinating and promoting the development of the right way, stay with the problem, which is to improve classroom teaching an important means of efficiency, and the third, control and feedback of knowledge. Teachers need to provide clues and explanations, correcting student misconceptions, and promote the participation of students in self-evaluation to promote students to further streamline the rationalization of knowledge, the fourth, curriculum knowledge. like Ma Liping's revealed by the concept of knowledge packages, the best sequence of specific topics presented, and its context and horizontal linkages with other disciplines, is a teacher for teaching mathematics knowledge base. Gu Ling Yuan's study also revealed that identify the knowledge of a discipline fixation point the distance between their potential and build features for students, with the appropriate sequence of gradient structure, the foundation is to improve teaching efficiency and, finally, the overarching purpose of the concept of teaching. like a step down subtraction, like Ball, as the experience of students streamline or rationalize the direct subtraction of the Law professor to step down, depending on the understanding of mathematics teachers, mathematics epistemological beliefs as well as the most valuable of specific mathematical knowledge of students to judge. Of course, these elements are from different dimensions to illustrate the teaching the properties of mathematical knowledge, their relationships and improve teaching efficiency of the mechanism of topics from classroom teaching experience needed to proceed further conceptualization.

4. Get used to this kind of teaching of mathematical knowledge in classroom teaching of mathematics is a core activity for analysis. These activities include the identification of what students already know, selection and management of teaching mathematical concepts of characterization, evaluation, selection and revision of textbooks in all kinds of teaching strategies to make decisions, find the resources required to support this decision, the discussion of the control students. Mathematics Teaching in the core activities of the framework can be more targeted analysis of the mathematics classroom. can grasp the nature of representation, precipitated Knowledge of Mathematics Teaching for teacher education to provide content to support professional development for teachers. Teachers using Ball's method to analyze their own classroom, to ignore the minor details, misses the main point, and gradually improve their teaching of skills.

5. Teacher education, especially in-service training should be combined with specific Lesson. If Ball does not analyze their own classroom, is not found in the subtraction of the class to step down, will the hidden mathematical concepts such as isomorphism. Isomorphism have different interpretations of the proposition, which students receive a different interpretation models have as easily in the teaching strategies are very different. This tells us that education, especially in-service training of teachers, teachers need guidance with specific Lesson, rather than general preaching to educational philosophy. Mr. Gu Lingyuan the survey also proved that the demand for teachers. It should be noted that the teaching of the teaching of mathematics knowledge is situational, personal aspect, not all propositions of the public knowledge, teachers also can not copy the learning needs of students for their own characteristics and qualities to be transformed. Despite this, the teaching of mathematics reflects the professional mathematics teachers is to math teachers and mathematicians, mathematics education theorists distinguish between the professional knowledge, professional development of teachers of mathematics in real terms. teachers accumulation and learning specific topics, teaching with the teaching of mathematical knowledge, is the main form of professional development. reposted elsewhere in the Research Papers Download http:/ / www.hi138.com