Keywords: junior high school mathematics inquiry design method inquiry teaching strategy
Main viewpoints and contents: the inquiry of mathematical problems is different from scientific inquiry; Inquiry questions determine inquiry teaching methods, inquiry
The design of formal questions is an important content of inquiry teaching preparation; Question design includes background design and inquiry point design; explore
The strategies of formal problem teaching are: open teaching strategy, spiral problem presentation strategy and process strategy.
With the implementation of the new curriculum standards, carrying out mathematical inquiry is conducive to cultivating students' innovative consciousness and practical ability. Inquiry teaching is increasingly favored by teachers, which challenges teachers' mathematical concepts and teaching ability. We know that mathematics is the gymnastics of thinking, and the problem is the heart of mathematics. Exploring mathematics undoubtedly pays more attention to thinking activities, and it must be an innovative learning method based on mathematical problems. This puts forward a primary question for our mathematics teachers: what is mathematics inquiry, how to design inquiry questions and how to teach inquiry questions?
First, the meaning of junior high school mathematics inquiry
Scientific inquiry is an exploration and research activity that analyzes knowledge and information, then puts forward scientific propositions, seeks solutions to problems and applies them to practice. Generally, it takes a long process of trial and error, and the task of mathematical inquiry is still to inherit the knowledge of predecessors, which is limited by teaching time and space; On the other hand, because the abstract thinking of junior high school students is developing, the level of thinking can not meet the requirements of scientific inquiry.
Therefore, junior high school mathematics inquiry is not a real scientific inquiry, but mostly a "simulated scientific inquiry". With the support of teachers and students, provide certain background materials, determine the correct direction of evidence collection according to certain clues, make decisions in possible reasonable explanations, and apply the decisions to actual exploration activities. In this process, the background materials, inquiry direction, inquiry content, practical operation tips and other elements are summarized into mathematical inquiry questions, and the activities of explaining decision-making (situational perception, observation and speculation, independent thinking, analogy discovery, summary of viewpoints and conclusions, method exchange and discussion, etc.) are carried out. ) is summarized as the teaching of mathematical inquiry.
Second, the relationship between inquiry questions and inquiry teaching
Considering the teaching factors, inquiry classroom teaching has three basic elements, namely, teachers, students and problems. The relationship between teachers and students is a teaching method and a learning method, and it is a way of human behavior. The problem is the object of exploring behavior. The inquiry between teachers and students revolves around the problem. Inquiry mathematics serves to explore problems, and the presentation and in-depth development of inquiry problems must be supplemented by teaching and learning methods. The teaching process is the inquiry process of problems. Different problems need different teaching and inquiry methods, and the quality of teaching and learning methods affects the inquiry effect. Therefore, inquiry questions determine inquiry teaching methods.
From the teaching goal, inquiry teaching means that students construct knowledge, form basic mathematical thinking methods, understand the general methods and activities of mathematical research, and form skills, methods and abilities on this basis. Their formation must depend on a certain carrier, which is "questioning". Therefore, problems are regarded as the core of learning, and inquiry learning is sometimes called "problem-oriented" learning. The basic goal of inquiry teaching is to enable students to acquire methods and abilities, and the ultimate goal is to pursue the cultivation of innovative consciousness and practical ability. Therefore, the content of inquiry questions is not limited to the inquiry of scientific propositions, but also includes the inquiry of methods to solve and put forward problems, and even includes the summary of experience, the perception of practice and the experience of mathematical life. Just like this, the design of inquiry questions is an important part of inquiry teaching preparation.
We can divide questions into the following categories from different angles. According to the types of mathematical knowledge, it can be divided into formative problems, applied problems, constructive problems, combination of numbers and shapes, and analogical induction problems; From the perspective of certainty of results: open and closed issues; In terms of teaching types: new teaching topics, consolidation of exercises and comprehensive review.
Third, explore the problem design method
Inquiry problem design includes two aspects: one is the background design of the problem, and the background of the problem refers to the process or reason of the problem; On the other hand, it is the design of question inquiry point, which refers to the direction or content of question inquiry and is the core part of question inquiry design.
1, the background design method of problem query.
Problem application background: based on the necessity of inquiry, it is set up to solve a problem and study a mathematical law. Exploring the background of the problem in this way is a problem in itself. Generally, it can be combined with practical application, which can better stimulate students' enthusiasm for inquiry. It is generally used as the background for constructing mathematical laws and exploring problems, for example, solving equations and exploring the properties of functions.
Background of old knowledge and old methods: Introduce old knowledge and old methods, and discover new inquiry problems through extension and analogy. For example, the properties and solutions of one-dimensional linear inequalities are usually explored in the context of equality properties and one-dimensional linear equations, and the basic properties and operations of patterns are usually explored in the context of fractional basic knowledge. Because of this kind of problem, it is easy to activate the original cognitive basis and can better stimulate the individual interests of different students.
Special case background: From the special point of view, it is a small problem to quote many cases as the background to observe and analyze and explore the general law. Background questions have the characteristics of low starting point, easy observation, strong regularity and easy combination of sensibility and rationality, which are easy to arouse the interest of every student. For example, the seventh grade rational number study often uses such a background.
Contradictory background: write a paragraph of material with certain cognitive conflict as the background, which leads to the inquiry question to be discussed. Students' knowledge is constantly assimilated in the constant cognitive conflict. Students' confusion, mistakes and disputes are generally the difficulties for students to learn, and they are also the source of background design for exploring questions. Students are prone to make general mistakes, such as "the distance between any two points on the number axis". According to the limited examples of the combination of numbers and shapes, students think that the absolute value of the difference between the numbers representing points or the sum of two numbers is the distance between two points. Without conscious exploration, it is difficult for students to form general knowledge; For example, it is known that all solutions of X+Y = K2X+5Y = 3 are positive numbers. To find the range of K, some people think that X and Y are greater than 0, then X+Y > 0, that is, K > 0, which makes it easy for students to analyze wrong thinking. It is also easy to cause conflicts between intuitive thinking and abstract thinking, such as solving the equation AX =-2 about X, where A < 0, so the result is X = 2/A, on the grounds that A is negative, and A and -2 are negative and positive respectively, and analyzing the reasons for the errors. In practice, the author designed such a problem and explored it himself, which solved the students' confusion well.
Migration background: provide examples of problem-solving ideas, imitate problem-solving after learning materials, or independently propose and solve problems; Or take the general steps of solving problems as the background, and then elaborate the principles and ideas of inquiry. The exploration of these questions is suitable for students' autonomous learning. For example, solve the one-dimensional linear equation, name each step of the equation and explain the principle of each step; This paper gives the principle that the areas of two triangles with the same base and height between parallel lines are equal, and provides an example to solve the problem, so as to imitate and solve other application problems.
Application background: provide application background, explore questions abstractly or let students make the best decision. Such inquiry questions not only cultivate students' analytical ability.
Ability to process background information and cultivate students' mathematical modeling ability. The prosperity of economic and cultural life has brought a wide range of mathematics problems to mathematics teachers, such as telecommunications, taxis, house mortgage, deposits, stocks, discounted sales, wages, lottery tickets, gambling, transportation costs, taxes, prices, investment returns, engineering costs, tourism prices, shortest paths, the most economical design, cultural relics protection, emergency avoidance, geometric patterns containing aesthetics, etc.
The background of conditional method: one is to provide fixed conditions for the establishment of figures or propositions and the disclosure of conclusions; For example, given a quadrilateral and connecting the midpoints of all sides in turn, ask questions about the graph, give the midpoints of all sides of the diagonal, ask questions about the graph, give the trapezoid connecting the diagonal and its midline, ask questions about the graph and prove the correctness of the conclusion. The second category is conditional transformation and exploration methods. For example, there is a big tree, and the corresponding measurement scheme is designed according to various changing backgrounds. The third category is to throw out all possible conditions and conclusions of the graph, freely combine them, and prove possible propositions through speculation. For example, given the horizontal line of the bottom corner of a trapezoid, the bisector of the other bottom corner of the trapezoid, the sum of the top and bottom is equal to the line connecting the two ends of the waist and the midpoint of the other waist, and these two lines are vertical, can other conclusions be drawn from the combination conditions? This kind of problems can satisfy the development of students' personality and specialty, and play a special role in cultivating students' divergent, unique and creative thinking.
Reading background: provide a background of reading materials containing data or methods, and then put forward reasons or solutions to problems. The common background is the history of mathematics, mathematical stories or problems in the process of mathematical research. For example, the history of Pythagorean theorem, the generation process of irrational numbers, the story of power, the generation process of negative numbers, probability stories and so on. Taking the story process as the inquiry situation, let students experience the process of scientific discovery and feel the method of learning and inquiry. Problems from economic life and daily life. Designing these problems into practical problems will not only help students learn mathematics knowledge and stimulate their interest in learning, but also broaden their horizons in applied mathematics.
2. Explore design issues.
The construction point of design knowledge. Mathematical concepts, properties and laws are the key knowledge contents in our classroom teaching, which are generally formed in the following cognitive order: background materials-forming concepts-abstractly summarizing conceptual features-simple application of features, induction and formation of concepts from concrete to abstract, and the discovery of multiple features is generally the focus and difficulty in learning, so concept formation and features are important problem design points. The steps of exploring design are generally as follows: observe and analyze the materials, list the same examples-find out the common ground of the examples? (Put forward the direction of inquiry)-summarize or express in words or symbolic language-explain its correctness?
Method build point. The construction of a problem-solving method, that is, solving similar problems before exploring the solution of the problem. For example, the construction of the proof method that the sum of two line segments is equal to the third line segment, the exploration of the identification method that two triangles have similar angles on the same side, the construction of the deformation method of finding algebraic values of known equations, and the induction of various methods of graphic equal product. Teachers can design a set of similar variant training questions, and then let students summarize their ideas or methods. The other is the method construction of asking questions. Through creative thinking such as analogy, divergent association and concentrated thinking, we can find new mathematical problems, solve them from limited or special examples, expand association to infinite problems or general conclusions, and transition from simple graphic nature to complex graphic nature. If students associate the judgment of triangle congruence after learning quadrilateral, it will naturally lead to the exploration of quadrilateral congruence judgment method; If you learn a parallelogram, you will compare what a parallelogram is and what its properties are. For example, after students explore various cross-sectional shapes of cubes, they are guided to think about cross-sections of other geometric shapes.
Comprehensive capacity building point. One is the application problem, which is the concentrated expression of comprehensive ability and can fully reflect the process of mathematical modeling.
It is challenging and exploratory, and is suitable for analysis and solution with various models. The other is the problem of constructing knowledge, which can make knowledge systematic and modular, generally embodied in the combination of quadratic function and quadratic equation, the combination of geometric figure and equation, the movement and change of function model and figure.
Fourth, inquiry-based problem teaching method
Through years of teaching practice, the author believes that inquiry-based problem teaching should be carried out from the following three aspects. 1, open teaching strategy
Traditionally, the answer to the question is unique, the solution is modular and the conclusion of the question is "closed". On the contrary, the open question refers to the uncertainty of the elements, thinking methods and strategies that constitute the proposition. Uncertainty may be that there are many ways to solve the problem, the mathematical thinking methods used are multi-channel, the conditions are constantly changing, the background of the problem is diverse, the conclusions are divergent or discussed through multiple channels, and the conditions and conclusions of the problem are freely combined to form an untenable proposition. Such openness determines that the time and space for solving problems must be open, and the inquiry that cannot be completed in class may extend to extracurricular activities and later study. Thinking pays more attention to the participation of various cognitions and the exertion of various comprehensive thinking abilities.
The ways to ask sentences are: what conclusions can be drawn, what are found, what are the similarities and why? What conclusion can be drawn by changing the conditions? What are the possibilities, what kind of thinking, and how do you think? What would you do? According to the material, what problems can you ask and how to solve them?
Because the direction of the question is uncertain or not unique, the method is no longer unique, which can attract students to explore independently and find different answers to the question, and can make students form the psychological motivation of active exploration and creation in solving the problem, and then actively participate in the process of "learning mathematics, doing mathematics and using mathematics", so that students' cognitive structure can be effectively integrated. It can not only better take care of students' individual differences and personality characteristics, but also stimulate students to participate in inquiry easily and effectively. Therefore, teachers should make full use of delayed judgment in teaching and provide students with enough time and opportunities to explore.
2, spiral problem presentation strategy
The internal structure of the problem should conform to the law of students' understanding, from easy to difficult, from simple to complex, from specific problems to abstract, step by step. Only by exploring mathematics can we proceed smoothly. The starting point of most problem design is not to embarrass and identify students, but to make most students accessible and gain necessary experience and achievements, which is in line with students' "nearby development zone" and the reality of students' mathematics learning.
This requires that the problems we design generally have a cognitive basis, which is easy to attract people's attention, simple and clear, and the knowledge involved in solving problems is related to the old knowledge, and the overall control is to a certain extent. It is impossible to have the characteristics that junior high school students must apply a lot of knowledge and have strong emotional motivation, so a lot of intuitive materials must be added to the questions. For example, the application background related to the application of physical and chemical knowledge or related to production, life and social content; Formative problems need to provide a large number of perceptual materials or special cases. General conclusions can be guessed and verified by intuition, proved by reasoning with existing reasoning knowledge, and finally the proved conclusions can be applied to complex problems.
Therefore, problems often appear in the form of a series of questions. The first question should generally be accessible, intuitive, interesting, practical, exemplary and enlightening. The intermediate questions are more and more challenging, and the openness of the questions is gradually increasing, but the thinking methods are basically the same, so as to consolidate the basic knowledge and thinking methods learned by students; Follow-up questions are generally designed as completely open questions.
The topic can be the self-summary of mathematical methods, an independent problem, an unsolved problem in current knowledge, a problem with an unknown conclusion, or the writing of mathematical papers.
Practice teaching proves that inquiry-based problem learning can not only pay attention to students' individual differences, but also satisfy students' achievement motivation in the step-by-step challenge, so that students can get pleasure and sublimation after inquiry in the step-by-step experience, which is conducive to stimulating the internal motivation of learning. When exploring follow-up questions, students can often imitate questions according to the previous question pattern, which is conducive to the cultivation of question awareness and innovation ability.
3. Process strategy
Inquiry teaching is both a learning method and a learning process. Including: (1) creating question situations to guide students to ask questions, or to arouse students to explore through questions; (2) Guide students to guess or assume the problem; (3) Obtaining information about speculation or hypothesis; (4) establishing a mathematical model by using information; (5) Analyze, discuss and think about the model and answer questions; (6) Using knowledge analysis to explain practical problems or expand one's understanding can be summarized as six links: creating problems-guessing hypotheses-obtaining information-establishing models-explaining communication-application expansion.
In teaching practice, teaching forms should be diverse. Some of them include the above six steps and belong to complete exploration. Others include only a part of the above six steps, which is called partial query. Teachers can implement inquiry teaching at all levels according to the teaching tasks and students' learning situation. For example, questions can be raised by students or by teachers or textbooks. Information can be provided by teachers, or obtained through reading, or obtained through surveys and experiments. Therefore, there is no fixed model for the procedure of inquiry teaching. It has different types and modes and should be creatively constructed according to teaching practice.
In the process of mathematical inquiry teaching, establishing conjecture, obtaining information and establishing model are the most important links in the teaching process, which can train students' eyes and hands and achieve the purpose of training their brains. For example, in the application of geometry and function, the emphasis is on movement change and practice. Most of them can simulate the change process with geometry sketchpad software, so that students can increase their experience and feel the spatial change in mathematics practice, which is conducive to students' induction of graphic laws and clear understanding of geometric models. In exploring the nature of geometric figures, we provide a certain amount of comparative data through independent or demonstrative geometric experiments, and focus on observation, comparison, changing trends and summarizing the general laws of data. Finally, teachers provide practical materials to apply these laws, so as to develop students' abilities of numbers, symbols, space, generalization and practical cognition. These explorations can not only stimulate students' hands-on practice, but also trigger students' subsequent thinking and rediscovery.
References:
1, Curriculum Reform Column of Shuguang Education Network (/sxweb/sx-y/lw/sx2.htm) 3. On the Independent Inquiry Teaching of Middle School Mathematics —— Wang Fuying 4. Mathematics Curriculum Standard for Full-time Compulsory Education (Experimental Draft)