Mathematics Curriculum Standard for Ordinary Senior High Schools advocates active and inquiry learning methods. It is clearly pointed out in the standard: "Students' mathematics learning activities should not be limited to acceptance, memory, imitation and practice ... Senior high school mathematics curriculum should strive to let students experience the process of mathematics discovery and creation through various forms of independent learning and inquiry activities, and develop their innovative consciousness. "

How to realize the active inquiry learning mode advocated by curriculum standards in classroom teaching practice? The author's teaching practice shows that: in the teaching process, by carefully designing the interesting and challenging question strings of students, students' eyes can be attracted, their enthusiasm can be aroused, their thinking can be inspired, and students can be guided to actively construct mathematical knowledge. So as to change students' traditional mathematics learning methods and advocate active inquiry learning methods.

First, the question string and its use value

The so-called question string refers to a series of questions designed according to a certain logical structure around a specific knowledge goal and for a specific teaching situation or theme in teaching. Question chain, also known as question chain, refers to a series of questions that meet the following three conditions: (1) points to a goal or revolves around a theme to form a series; (2) Conforming to the inherent logical connection between knowledge; (3) Meet the conditions for students to construct knowledge independently. Effective question strings not only help to stimulate students' strong thirst for knowledge, but also guide students to analyze and solve problems independently, construct knowledge and develop their abilities. So as to realize the active and inquiry learning mode advocated by the curriculum standard.

The following is the author's teaching case of "The Root of Equation and the Zero of Function". Through two well-designed question strings, the author not only successfully completed the teaching task, but also effectively guided students to actively participate in inquiry learning activities. For design purposes, two question strings are given below.

Second, a brief description of the case

1. Question string 1: Give the concept of function zero.

Question 1: From what different angles can we understand the formula?

Student 1: is a binary linear equation.

Student 2: This is a linear function.

Teacher: Formula can be understood in two ways, namely, equation and function.

Question 2: What is your understanding of "being", "being" and "being"?

Student 3: It can be regarded as the root of the equation.

Student 4: It can be regarded as the abscissa of the intersection of the image of the function and the axis.

Teacher: The meaning here is both quantitative and tangible. In fact, there is a new name, called the zero point of function, which is what we will study in this class. (blackboard writing topic)

The teacher asked: We call it the zero point of the function. Why do we choose the name "zero point"?

Student (mass): Because it is passed.

Question 3: How to define the zero point of a general function?

Student (mass): The root of the equation is called the zero point of the function.

Teacher: Good! For the zero point of a function, we can understand it from the angle of number and shape, from the angle of number as the root of the equation, and from the angle of shape as the abscissa of the intersection of the image and axis of the function.

The design intention of the problem string 1: the design of three small problems, the core concept of approaching the zero point of the function layer by layer, and the decomposition of difficult knowledge. Question 1 guides students to understand the formula from two aspects: equation and function, question 2 induces the definition of zero from special to general, and question 3 further guides students to understand the definition of zero from the perspective of number and shape. This reduces the difficulty of teaching, fully mobilizes students' enthusiasm and initiative, helps to guide students to complete the independent construction of knowledge, and makes the teaching of this concept natural, in place and profound.

2. Question String 2: Generation of Zero Existence Theorem.

Problem 1: The quadratic function is known. Does it have zero on the interval (-1, 1)? (The seemingly bland questions ignited the students' thinking sparks and aroused their desire to explore)

Student 1: By finding the root of the equation, it can be judged that the interval (-1, 1) has a zero point.

Student 2: Draw a sketch of the function and then combine it with the symbol, that is, combine it with the symbol. It is found that the function image intersects the axis in the interval (-1, 1), so there is a zero point in the interval (-1, 1).

Teacher: OK! The two students give judgments from the angles of numbers and shapes respectively, while the combination of shape and function images only needs to judge the different symbols of f(- 1) and f( 1), which is very simple.

The teacher asked: Does the quadratic function have zero in the interval (2,3)?

(Most students gave the following answers: Because, it is concluded that the interval (2,3) has zero?

Question 2: Looking back at the problem just solved, can you sum up how to judge whether the quadratic function has zero in the interval (a, b)?

(Students come to the following conclusion after discussion: If it is a quadratic function and the function has zero points in the interval (a, b))

Question 3: Can you summarize the above conclusions? In other words, does the above conclusion still hold true for any function?

(Students think, talk and then answer)

Student 3: Not necessarily. For example, this conclusion is not true for piecewise functions.

Teacher: Let's check it first. It exists in the interval (-1, 1), but the function has no zero in the interval (-1, 1). Well, we found that this conclusion does not hold true for any function. Think about it, class. Can you find other examples?

Student 4: The conclusion just now doesn't hold.

Question 4: Ask students to think about why the above proposition holds true for quadratic functions, but not for the functions just listed by students.

Student 5: The image of quadratic function is continuous, but the image of the function just quoted is broken. Teacher: Great, how thoughtful!

Question 5: Can you fill in appropriate conditions to make the above proposition still hold true for generalized functions?

Student (mass): The image of the function must be continuous.

Teacher: Good. Please summarize the conclusion in a relatively complete language. (Students narrate, teachers summarize and draw a conclusion)

The design intention of Question Series 2 is to guide students to think independently, explore independently, and cooperate and communicate through the gradual progress of questions, so that students can experience the process of observation, induction, promotion, reflection and abstract generalization, deeply understand the existence theorem of zero point, and learn mathematical thinking and learning methods.

Third, the case analysis

The problem is the core of mathematics. It should be said that every math class is inseparable from problems. Problem series is different from ordinary math problems. It is a series of questions designed around a theme or a big problem, which should embody a "chain". This lesson needs to solve two major problems. In order to solve these two problems, the author designed a corresponding question string for each big problem. In classroom teaching, the author gives the question string in an explicit form, that is, labeling "question 1, question 2, ...", which is more eye-catching, convenient for students to concentrate on thinking about problems, and also convenient for students to realize the progressive relationship between problems.

In the teaching of this course, the author thinks that students not only master the knowledge of mathematics, but more importantly, participate in the process of the occurrence, development and formation of knowledge. In this process, students actively think, communicate and cooperate to realize the "re-creation" of knowledge and the meaning construction of mathematical knowledge. At the same time, in this process, I also tasted the sense of accomplishment brought by learning mathematics. Through the communication between teachers and students, I have also enhanced the feelings between teachers and students and better achieved the goals of emotion, attitude and values.

"Teaching is to not teach", the author carefully designed the question string along this line of thought, allowing students to explore independently and actively participate in the whole process of acquiring knowledge, and set up a link from "learning" to "learning" for students.

Bridge, truly implement the "active and bold exploration of learning methods" advocated by the new curriculum standards.

Fourth, the case enlightenment

Based on the case given in this paper and its related analysis, this paper discusses some enlightenment of designing question string.

1. Design diversified question strings as required.

In actual teaching, we can design diversified question strings according to different teaching links or different teaching needs. For example, in the introduction of topics, we can design life-oriented question strings and link them with students' known life experiences, which can not only create a relaxed teaching atmosphere, but also stimulate students' strong thirst for knowledge. In knowledge construction, we can design a refined question string, which can be made much smaller, abstracted into concrete and refined into a question string with a certain gradient and logical structure, so that the learning goal can be concrete, the knowledge construction can be hierarchical, and the thinking activities can be meticulous, thus obtaining clearer new knowledge. In concept discrimination, we can design a comparison question string to guide students to analyze and compare, grasp the * * * and personality of knowledge, and help students identify the subtle differences between knowledge. In problem solving, we can design exploratory question strings, reorganize or deeply process the information provided by questions, guide students to explore the essential characteristics of problems, and constantly explore methods and strategies to solve problems, and so on.

2. The designed question series should be suitable for students' learning situation.

Designing and using question strings is a teaching strategy, which aims to build a platform to push students to the front desk to solve problems. Since students are the main body, the design of question string should of course be aimed at students' reality. Only on the basis of students' existing knowledge, experience and ability can we effectively promote the assimilation of new knowledge and improve the efficiency of teaching and learning. Too difficult questions will make students feel frustrated and lose their enthusiasm and initiative in exploration, while too simple questions will make students feel dull and lose their interest in exploration. Therefore, in teaching, we must design question strings suitable for students' learning situation, guide students' thinking and improve classroom efficiency.

3. The design question string should be hierarchical.

Whether it is a "parallel" structure or a "series" structure, the use of question strings in teaching is essentially a process of guiding students to take the initiative to learn with questions and build their own knowledge system from the outside to the inside. Therefore, the design of question string should be based on teaching objectives, and the teaching content should be organized into groups and interrelated questions, so that the former question can be the basis and premise of the latter question, and the latter question is the development, continuation and supplement of the former question, so that each question becomes the ladder of students' thinking, and many questions form question strings with a certain level and logical structure, so that students can acquire knowledge and improve their thinking ability on the basis of clarifying the internal relationship of knowledge.

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