Mathematics teaching material analysis of Grade Seven in Beijing Normal University (Ⅰ) Ⅰ. Analysis of the overall thinking of the textbook.
1. The main contents of this semester are: rational number and its operation, letter representation number, unary linear equation; Rich graphic world, plane graphics and their positional relationship; Data and possibilities in life.
In the field of number theory and algebra, the concept of "rational number" is formed through the expansion of number system. Because of the introduction of negative numbers, the "operation" and "algorithm" of rational numbers are naturally promoted to the object of attention and learning. The number represented by letters is an important feature of algebra, and equation is one of the core concepts of mathematics. Through learning, students realize that the discussion of mathematical problems is carried out within the scope of rational numbers, which lays the foundation for discovering irrational numbers and establishing real number systems.
In junior high school, plane geometry is the main way to learn geometry knowledge. In the rich graphic world, starting from the observation of three-dimensional space objects, making full use of students' rich background experience, in the mutual representation and transformation of physical objects, geometric objects, direct vision and plane graphics, the perception level of geometric graphics is improved and the concept of space is developed. Through observation, operation, thinking and communication, we can accumulate mathematical experience, feel the necessity of learning plane graphics and the foundation of simple graphics, realize that basic graphics are an important tool to describe the real world, and learn to observe the world with mathematical eyes, which can bring endless intuitive sources to real life. In "Plane Figure and Its Position Relationship", the role of understanding the basic concepts of geometry and rational reasoning is highlighted.
Through the discussion of practical problems, Data in Life enables students to understand the important role of data, the process of data processing and the information it expresses, and develop their sense of numbers and statistical concepts. In the chapter of "Possibility", the characteristics of uncertain phenomena are preliminarily understood, and the hidden regularity in random phenomena is understood through experiments, and the random concept is initially formed.
2. The design of teaching materials and the organization of contents have the following considerations.
(1) With the help of examples in life, it is not difficult to realize the necessity of introducing negative numbers and the rationality of forming the concept of rational numbers. The establishment of the number axis gives an intuitive explanation and representation of rational numbers, which can be used as a tool to deepen the understanding of the operational significance of rational numbers in line with the real situation. The concept of absolute value establishes the corresponding relationship between rational number and non-negative number, which is convenient to clearly express the rules of positive and negative number operation. Its geometric meaning is the distance from the point corresponding to the rational number to the origin. The operation of rational numbers, especially the provisions of multiplication and division, does not belong to causal explanation, but hopes that "positive numbers also have negative numbers,&; Hellip& amphellip This is the follow-up number "(spending money to buy grandchildren), which is a rational choice. "The textbook has been treated in detail, which reflects the continuity and inheritance of understanding. The training of operation also adopts the way of game (24 points), focusing on the continuous consolidation and strengthening in the follow-up study.
(2) In the rich graphic world, learning geometric objects does not start from the logical starting point of geometry, but conforms to the process of mathematical history, and has gone through the process from concrete to abstract, and then from abstract to concrete. Starting from the investigation of real-world objects, we abandon secondary factors, decompose simple geometry or basic graphics, and develop geometric intuition and spatial concepts in the process of decomposition and integration. We don't study solid geometry in advance, but learn "mathematicization" through activities. In chapter 4, the concept of geometry is naturally introduced one after another, and the positional relationship and basic properties of simple plane graphics are found through operation and expressed in symbolic language. Textbooks provide a lot of hands-on opportunities, reproduce the process from intuitive action thinking to intuitive image thinking, and make necessary preparations for further development to abstract (logical) thinking stage.
(3) The ultimate goal of statistical learning is to develop students' statistical concepts. The formation of statistical concepts is not spontaneous, nor can it be solved by preaching. Students need to personally participate in such activities, and feel that solving problems requires collecting data, characterizing data, analyzing data, and making appropriate judgments by using the results of data analysis. Therefore, the design of statistics-related content in the whole textbook tries to let students experience the whole process of statistical activities from practical problems, such as the textbook asking "What competitions will you organize to watch in order to attract students to participate as much as possible" and "Are you confident to learn math well?" Driven by these problems, guide students to engage in statistical activities, acquire corresponding knowledge and methods and develop their abilities in the activities.
The ultimate goal of probability learning is to develop students' random concepts, which have multiple levels. Therefore, it is impossible to cultivate students' random concept overnight, and it takes a long process. For this reason, this book aims to make students feel the universality of random phenomena in the real world and the possibility of random phenomena through specific practical activities. As for how to describe them, put them into the second volume of the seventh grade for research. In addition, for randomness, we only pay attention to the feelings in practical activities and don't want to analyze it theoretically. I don't want students to say, "There are three possibilities in this situation and only two possibilities in that situation, so this situation is more likely to happen." This description is actually based on "every possibility is exactly the same", which is already a theoretical calculation. Maybe your analysis in this case is not bad, but if students feel this way when learning probability, it may be easy to put this (equal possibility).
Second, several problems that should be paid attention to in teaching implementation
1. Pay attention to students' understanding of mathematical knowledge.
(1) About the operation of rational numbers, the emphasis is on the understanding of the operation meaning. The understanding of the operation law is obtained in the process of independent exploration. Because we can use computing tools to perform complex digital operations, the cultivation of computing skills mainly focuses on the understanding and flexible application of computing rules. Encourage the diversity of algorithms, because different algorithms may come from different understandings or thinking habits, and they can share resources through communication.
Algebra is a tool to express, communicate and solve problems, and its core is symbols. Through the study of letters representing numbers, students can feel that replacing specific numbers with letters can solve the problem as a whole. Further understanding facilitates formal operations (such as merging similar terms) and the exploration and discovery of laws, which has a direct impact on the understanding of equations.
(2) On the surface, the chapter "Rich Graphic World" seems to have few specific knowledge points. In fact, a space figure can be unfolded and folded through its surface. Using plane cutting and three views to realize the conversion between three-dimensional graphics and two-dimensional graphics. Cultivate students' concept of space by doing, thinking and doing. Through hands-on operation, abstract objects can be simplified and visualized, and rational thinking can be inspired and prompted at the same time. If you cut a cube with a plane, can the section be heptagon? Thinking by doing includes rational analysis and reasoning &; Why mdash& ampmdash can or can't. Developing students' spatial concept and improving their visual thinking ability and level are the main learning objectives of this chapter.
2. Teaching should have an accurate orientation to improve the effectiveness of learning.
(1) In the study of "one-yuan linear equation", students are formally exposed to the concept of equation for the first time. "Equation" is undoubtedly one of the most important concepts in mathematics. Learn to understand the meaning and function of equations, especially to analyze and deal with problems from the viewpoint of equations. Some problems can be solved by "arithmetic", which requires a clear explanation of the meaning of the listed formulas and often requires more intellectual input. The key point of the equation lies not only in the process of solving, but also in the purpose of solving the unknown quantity by establishing the equation. The key step is to treat the unknown quantity (represented by letters) and the known quantity equally, and to seek the structural equivalence relationship between them and express it. The study of equations provides an opportunity to enhance the awareness of mathematical application.
(2) The teaching goal of this chapter is to accumulate experience in mathematics activities and develop the concept of space. The content is close to students' life experience, which is easy to arouse their interest in learning, feel that mathematics is around, and improve their bad impression of mathematics. In teaching, we should fully explore the connotation of mathematics in activities and lead our interest to mathematics. In the activity, students should be guided to think about a series of math problems. For example, in the process of unfolding the surface of a cube into a plane figure, students may encounter many mathematical problems.
Usually, mathematical problems or mathematical thinking can be triggered by vivid and interesting situations, which can provide empirical support for mathematical understanding, but we should cut into the subject in time to avoid "playing peripheral wars" for a long time. First of all, we must grasp the basic orientation of all classes. For example, from different directions, the main purpose is to learn three kinds of views, and to learn the mutual expression of spatial graphics and plane projection. On this basis, students should think about avoiding one-sidedness of problems.
Using information technology to make courseware can produce good results in teaching, but attention should be paid to avoid teaching activities becoming technical demonstration classes.
Teaching material analysis (2) 1 for Grade Seven Mathematics in Beijing Normal University. Analysis of the overall thinking of teaching materials
1. The main contents of this book are: unary linear inequality (group), factorization, fraction; Similar graphs and proofs (1); Data collection and processing.
The unary linear inequality (group) is based on the study of linear equations and linear functions. Therefore, starting from the internal relations of inequality, function and equation, we should think integrally and generally from the aspects of number and form, which provides a broad space for the research and understanding of this chapter.
Factorization is the inverse operation of polynomial multiplication, its main function is to transform the form of algebraic expression, and the change of form also constitutes an identity relationship and an explanation of meaning, which also has an impact on the learning of quadratic equations and quadratic functions.
"Similar graphics" is the deepening and development of graphic congruence content, which provides an opportunity for comprehensive use of various methods to study graphics. Graphic similarity is an intuitive expression abstracted from a large number of similar phenomena in real life. The book only gives the definition of similar polygons, which is the most fundamental. As far as graphics are concerned, triangles can be regarded as the most basic graphics, but the definition of similar triangles is quite special. Since congruent triangles can be regarded as a special case of similar triangles, the nature and judgment of similar triangles can be compared with that of congruent triangles. Through learning, we can feel that the learning of triangles is the basis of understanding and mastering the characteristics of polygons (a general polygon can be regarded as composed of several triangles through "triangulation"), and right-angled triangles are more basic than triangles. As for the position similarity, it is more manifested as "enlargement" and "reduction", from which we can deduce the proportional relationship, or help students understand the meaning of proportion.
Learning "Proof" from "Proof (1)" In the past, the understanding of proof almost became synonymous with "geometry". This set of textbooks moves what is proof and how to prove it to the foreground, which better reflects the duality of mathematics. Mathematics has two sides. Just as mathematics is in the process of creation, it looks like an experimental inductive science. On the other hand, mathematics is a rigorous Euclid science, more like a systematic deductive science. The focus of learning here is on the learning of mathematical proof itself, not only geometric proof, but also improving the learning requirements of mathematical proof. Therefore, the necessity of proof, the meaning of axiom and the significance of proof in this chapter should be the focus of research.
Data collection and processing, on the basis of describing the average level of the last volume of data, further puts forward several measures to describe the fluctuation level of data, so that students can master the characteristics of data more comprehensively, and at the same time puts forward various methods of data collection and feels the idea of sample estimation.
2. This book has the following considerations in textbook design and content organization.
(1) In the "unary inequality (group)", inequality is a mathematical expression of inequality, and there are a lot of inequalities in real life, so that students can learn in a rich practical background. At this time, we should pay attention to the "representation" and "application" of mathematics. In the activity of solving inequalities, we should pay attention to the internal essential relationship between different knowledge, deepen our understanding of the mathematical meaning of equations, functions, inequalities and other knowledge, and deepen our understanding of the structural nature of mathematical knowledge through their mutual interpretation and formal transformation. In this chapter, the section "One-dimensional linear inequality and linear function" is added, and "reading" (the plane area represented by inequality) is set after the sixth section, which increases the depth and flexibility.
(2) Factorization is a further understanding of polynomials. From the operational point of view, it is the reciprocal of polynomial multiplication; From the point of view of the same deformation, it is different forms of the same formula; From the perspective of learning, it is the transformation from operation (process) to object (identity relationship). Textbooks pay more attention to the understanding of the significance and function of factorization, and don't spend too much energy on methods and skills. You don't have to master the "cross multiplication", and the solution of the equation can generally be solved by finding the root of the quadratic trinomial.
(3) Similar graphics are based on the observation, analysis, generalization and abstraction of similar phenomena in the real world, which conforms to students' cognitive laws and embodies the process of mathematicization. The content of this chapter is based on "Similar Graphics &; Mdash Similar Polygons &; Mdash similar triangles mdash Properties of Similar Polygons ",the important knowledge includes: the proportion of line segments, similar figures, similar centers and similar proportions. "similar triangles is the core knowledge of this chapter. This chapter does not require strict geometric proof, but focuses on the exploration, discovery and application of graphic properties. Because geometry is dominated by intuitive thinking, we should pay special attention to the development of geometric intuition and rational reasoning ability.
(4) Proof (1)
Mathematical historian H&; Bull; Eves pointed out that the discovery of geometry experienced three stages in history: unconscious geometry, scientific geometry and demonstrative geometry. Through the observation of natural phenomena and simple technological work, I have inadvertently become familiar with a large number of geometric concepts and facts (such as circles, angles, parallel lines, triangles, distances and the shortest straight line segment between two points); Subsequently, a series of geometric facts are summarized and verified by repeated experiments or practices, which become empirical geometry; When we think rationally about these experiences and question "why", geometry in the form of argument or deduction appears. This development process explains the empirical source of geometric knowledge, and at the same time, we should realize that the results obtained without strict argumentation are easy to make mistakes, and there is no guarantee that there will be no theoretical mistakes. In this chapter, "Are you sure?" Its establishment is to understand the necessity of proof, and its importance lies in the formation of scientific attitude and rational spirit.
According to the requirements of the standard, the textbook constructs a "partial axiom system". Starting from the given axiom (as the starting point and basis of reasoning) and related concepts, the conclusion that parallel lines are related to triangles is proved again through logical deduction. Starting from this chapter, the proof of relevant content should be written in a standardized format. Axiomatic method only needs to understand its basic idea.
(5) Data collection and processing are still carried out in the order of statistical activities: data collection &; Mdash express mdash handles mdash decisions, that is, according to the process of problem solving. Related concepts naturally extend in the actual background, which is easy to understand and apply. In teaching, we should make full use of positive and negative examples to clarify vague understanding or misleading.
Second, several problems that should be paid attention to in teaching
1. Pay attention to students' understanding of mathematical knowledge.
(1) Pay attention to the conceptual differences among linear equation, linear function and linear inequality (group), and pay attention to their internal relations and comprehensive applications (for example, "doing" in the fifth section of Chapter 1 and the second question of exercise 1.6).
(2) The function of factorization is put forward at an appropriate time in fractional deformation and operation. In the fractional equation, we should understand the thinking method of turning the whole into the whole, the reason of increasing the root and the necessity of checking the root. The fractional equation part also provides an opportunity to learn "modeling".
(3) Paying attention to the exploration of graphics can not only discover geometric facts, but also prompt the clues of proof and the methods of producing proof (such as adding auxiliary lines and partial displacement). Intuitive guessing and proof complement each other.
The necessity of geometric proof lies not only in avoiding misjudgment, but also in grasping the logical relationship between knowledge. Logical argument is determined by the nature and characteristics of mathematics. Learning proof is not limited to the specific proposition of learning proof, but embodies a scientific and rational spirit.
2. Pay attention to the infiltration of mathematical ideas in teaching.
Hundreds of years before the birth of (1) Euclidean geometry, people have discovered a lot of geometric facts, including many propositions that use syllogism or proof. Euclid's contribution is not to discover new important geometric facts, but to reorganize these geometric facts logically. At that time, the Greeks formed an idea: logic discipline was a series of propositions obtained by deductive reasoning from a group of recognized original propositions at the beginning of discipline research. When demonstrating by deduction, any proposition must be derived from one or more previous propositions, and the previous proposition must also be derived from one or more previous propositions. Since it is impossible to backtrack indefinitely, and at the same time, it is impossible to cause a logical cycle, then it is necessary to determine a set of original propositions (axioms) that can be recognized, and then all the propositions of the system are completely deduced by deductive reasoning. The original proposition and derived proposition need to use clearly defined technical terms, and the terms also need to be defined by other terms, so it is necessary to determine a group of basic terms (original concepts) and explain their usage. "Geometry is not only a branch of mathematics, but also a way of thinking, which permeates all branches of mathematics &; Hellip& amphellip "(Atia).
(2) Through statistical activities, let students feel that statistics is more about sorting, analyzing and judging data by inductive methods; The data is both true and random; Data processing can be done in different ways, and there is no right or wrong method. What is important is whether we can choose a more scientific and reasonable method according to the actual situation. Sampling is to infer some properties of the population through the information provided by the sample, and sampling is most concerned about whether it can objectively reflect the actual (overall) situation.
Mathematics teaching material analysis of Grade 7 in Beijing Normal University (III): An Overview of Textbooks.
The learning content involves four fields: number and algebra, space and graphics, statistics and probability, and subject learning.
The basic content is to highlight the development stage: all knowledge is just the beginning, and students are not required to achieve &; Laquo standard &; The goal proposed by raquo.
The first chapter is a rich graphic world.
Writing intention &; The Preliminary Development of Students' Concept of Space
Main features: advocating the learning mode from operation to thinking and imagination,
Chapter II Rational Numbers and Their Operations
Writing intention &; Mdash& ampmdash helps students understand the necessity and significance of rational numbers, be able to operate rational numbers, experience the consistency and particularity of "number expansion", and enable students to operate rational numbers.
Main features: highlighting the background and formation process of rational numbers and their operations.
Chapter III Letter Representation of Numbers
Writing intention &; Mdash& ampmdash helps students build a sense of symbols and know algebra.
Main features: the establishment of algebraic expression and its operational significance, the infiltration of function thought, (let students understand and master function thought through data converter)
Chapter IV Plane Figures and Their Positional Relations
Writing intention &; Mdash& ampmdash Understand the basic geometric elements and their relationships.
Main features: Pay attention to the formation process of knowledge and methods. (For example, pay attention to the method of measuring line segments and angles)
Chapter 5 One-variable linear equation
Writing intention &; Mdash& ampmdash helps students understand the meaning of equations, master the methods of solving equations, and understand the basic ideas and processes of applying equations to solve problems.
Main features: pay more attention to the idea of establishing equation model, and reflect the significance of establishing equation model through "finding equivalence relationship"
Chapter VI Data in Life
Writing intention &; Mdash& ampmdash Help students understand the meaning of statistics and cultivate statistical awareness.
Main features: Understand related concepts and statistical processes in the process of solving problems.
Chapter VII Possibility
Writing intention &; Mdash& ampmdash helps students understand the meaning of random phenomena and probability.
Main features: highlighting the method of experimental probability (not from theory to theory, but helping students understand the basic idea of probability through experimental activities)