How to improve student evaluation through mathematical modeling and mathematical inquiry, highlighting the process and motivational role of evaluation.

Students’ mathematics learning activities should not be limited to acceptance, memorization, imitation and practice. High school mathematics courses should also advocate independent exploration, hands-on practice, cooperative communication, reading and self-study, etc. to learn mathematics. These methods include It helps to give full play to students' learning initiative and makes the students' learning process become a "re-creation" process under the guidance of teachers. To truly implement the basic concept of this curriculum into high school mathematics teaching, teachers should base on students' cognitive level and existing knowledge. Some knowledge and experience set up courses that reflect some important applications of mathematics, and carry out learning activities of "mathematical inquiry" and "mathematical modeling", striving to enable students to experience the role of mathematics in solving practical problems and the connection between mathematics and daily life and other subjects. , to promote students to gradually form and develop awareness of mathematical applications, improve practical abilities, and experience the true meaning of mathematics.

Since the second half of the 20th century, the tremendous development of mathematical applications is one of the distinctive features of the development of mathematics. Today’s Knowledge In the economic era, mathematics is moving from behind the scenes to the stage. The combination of mathematics and computer technology enables mathematics to directly create value for society in many aspects. At the same time, it also opens up broad prospects for the development of mathematics. Mathematics education in our country has been developing for a long time. The connection between mathematics and practice, mathematics and other subjects has not been given sufficient attention. Therefore, high school mathematics needs to be vigorously strengthened in terms of mathematical application and connection with practice. In recent years, the practice of mathematical modeling in universities and middle schools in my country has shown that the development of mathematical modeling The teaching activities of mathematics application meet the needs of society, are conducive to stimulating students' interest in learning mathematics, are conducive to enhancing students' application awareness, and are conducive to expanding students' horizons. Under such a curriculum concept, the People's Education Press Curriculum Standards Version B teaching materials provide We have a spring breeze blowing. It is not just a simple change of words, but a prominent embodiment of teaching ideas. The entire set of textbooks has set up a large number of learning activities such as "mathematical inquiry" and "mathematical modeling" to provide basic content. The actual background reflects the application value of mathematics. These courses that reflect the application of mathematics further create favorable conditions for students to form proactive and diverse learning methods. At the same time, they also stimulate students' interest in mathematics learning and encourage students to cultivate mathematics in the learning process. Develop the habit of independent thinking and active exploration.

The author will conduct the following teaching design and attempts on the content of "Function (First Lesson)".

Textbook Analysis

1. The status and role of this lesson

Function is one of the important basic concepts in mathematics. The basic courses of higher mathematics that students further study, including limit theory, differential calculus, integral calculus, differential equations and functional analysis, all use functions as basic concepts and research objects. Other disciplines, such as physics, also use the basic knowledge of functions as a tool for researching and solving problems. It is a re-understanding of the concept of function based on the preliminary exploration of the concept of function, the representation of functional relationships, the position of images, etc. in junior high school, that is, using the idea of ????sets to understand the general definition of functions. The in-depth and improved study of functions and applications is also the basic knowledge needed to further participate in industrial and agricultural production and construction in the future. The study of this chapter plays a decisive role in the mathematics learning of middle school students. And not only in terms of knowledge, but more importantly in mathematical construction In terms of models, it will also be a chapter that will benefit you throughout your life.

2. Teaching focuses and difficulties

Key points: Understand that functions are important mathematical models that describe the dependence between two variables. , Understand the concept of function on the basis of mapping.

Difficulty: Understanding the function symbol y=f(x).

Teaching objectives

1. Knowledge and skill goals:

(1) Help students establish the background of mathematical concepts through different life examples, so as to correctly understand the concept of functions.

(2) Be able to use sets and correspondences Use the language to describe the function, understand the elements that constitute the function, that is, the domain of definition and the corresponding rule; further understand the meaning of the corresponding rule.

2. Process and Method Objectives:

Understand that functions are important mathematical models that describe dependencies between variables. On this basis, learn to use the language of sets and correspondences to describe functions and reproduce the process of generating function knowledge. Experience the process of using mathematical ideas, methods and knowledge to solve practical problems in mathematical modeling.

3. Emotional attitude and value goals:

By creating real life situations, students can get close to real life and pay attention to social reality; feel the role of correspondence in describing the concept of functions, stimulate students' interest in learning mathematics, and cultivate students sentiment, and cultivate students’ scientific spirit of courage to explore.

Teaching process

1. Creating problem situations

Teacher: In junior high school, we have already learned the concept of functions , and we know that functions can be used to describe the dependence between two variables. Today we will further learn about functions and their constituent elements. Let’s look at a few examples below:

Question 1: After a cannonball is fired , fell to the ground and hit the target after 26 seconds. The shooting height of the artillery shell is 845m, and the law of the change of the height h (m) of the artillery shell from the ground with time t (s) is h = 130t-5t2. The following questions are asked:

(1) How high is the cannonball above the ground when it flies for 1s, 10s, and 20s?

(2) When is the cannonball at its highest distance from the ground?

(3) Can you point out the value range of variables t and h? Represented by set A and set B respectively.

(4) For any time t in set A, according to the corresponding relationship h=130t-5t2, does it have a unique height h in set B? Does it correspond to it?

Student: Because I have a foundation in junior high school, I can quickly tell the answers to the first three questions. For question (4), the teacher inspired the students to use sets and corresponding languages ??to describe the dependence between variables: in t Within the changing range, for any given t, according to the given analytical formula, there will be a unique height h corresponding to it.

[Starting from the life problems of multimedia display, reproducing the description function of junior high school variables The concept lays the foundation for later defining functions using sets and corresponding viewpoints. ]

Question 2. The temperature changes measured by a certain city’s meteorological observation station within 24 hours a day are shown in the figure.

(1) What is the approximate temperature at 8 a.m.?

(2) Can you point out the value range of variables t and θ? Represented by set A and set B respectively.

(3) For each time t in set A, as shown in the image, is there a uniquely determined temperature θ corresponding to it in set B? ?

Student 1 answered: The temperature at 8 o'clock in the morning was about 0. C; The value range of t is [0, 24];

The value range of θ is [-2, 9].

Student 2 Answer: For each time t in set A, as shown in the image, there is a unique temperature θ corresponding to it in set B.

Then the teacher asked the students to review the changes in their family life in the past ten years. Which aspects of consumption have changed greatly? Which aspects of consumption have changed slightly?

[Students responded enthusiastically and further mobilized It stimulates students' enthusiasm and personally experiences the process of abstracting practical problems into mathematical models. This is actually a process of advocating doing and using mathematics and paying attention to the formation and development of students' knowledge.]

The teacher raised questions again 3. What data do you think should be used to measure the quality of family life? The slideshow showing the change of Engel coefficient over time (year) shows that the quality of life of urban residents in my country has changed significantly since the "Eighth Five-Year Plan".

t

91

92

93

94

95

96

97

98

99

00

01

r

53.8

52.9

50.1

49.9

49.9

48.6

46.5

44.5

41.9

39.2

37.9

After reading the chart Following question 1 and question 2, describe the relationship between the Engel coefficient r and time t (year) in the table.

Student induction: For any time t (year) in the table, according to the table, there is a unique An Engel coefficient r corresponds to it.

2. Exploring new knowledge

Students discussed the same characteristics of the above examples in groups, and concluded that: they all involve two non-empty There is a certain correspondence between the number sets A and B, so that for each number x in A, according to this correspondence, there is a unique y corresponding to x in B.

[Actual Questions elicit concepts, stimulate students' interest, give students space to think and explore, allow students to experience the process of mathematical discovery and creation, and improve their ability to analyze and solve problems. ]

1. Definition of function

Suppose A and B are non-empty number sets. According to a certain corresponding relationship f, any number x in set A has a unique and certain number in set B. If the value y corresponds to it, then this correspondence is called a function on the set A. Denoted as, where. Domain: The value range of x (number set A) is called the domain of the function; if the independent variable takes the value a, then the value y determined by the rule f is called the function value of the function at a.

Value range: The set of function values ??{y/y=,} is called the value range of the function.

Teachers and students *** recalled the concept of functions introduced in junior high school, which was expressed as follows:

Suppose there are two variables and in a change process. If for each value, there is a unique value corresponding to it, then it is said to be an independent variable and a function.

[We see that the function is defined here from the perspective of motion change, which reflects people’s understanding of it in history. Moreover, this definition is relatively intuitive and easy to accept. Therefore, we follow the steps from the simplest to the deeper and strive to meet the students’ understanding. It is appropriate to introduce the concept of function to this level in junior middle school.]

Teacher: The corresponding rules of functions are usually represented by symbols. Function symbols indicate that for any object in the domain of definition, in Obtained under the action of "correspondence law". In relatively simple cases, the correspondence law can be expressed by an analytical expression. However, in many problems, the correspondence law needs to be expressed by several analytical expressions, and sometimes it is even impossible to use an analytical expression. Express, what expression should be used?

Students: Use other methods (such as lists, images) to express.

Students discuss in groups several aspects that need to be paid attention to when defining functions: (teacher writes on the blackboard)

(1), directionality;

(2) Keywords "any x" "the unique number f(x)".

(3) A , B is a non-empty number set;

(4) Any element in A has a unique element corresponding to it in B; and the corresponding element of the element in B does not need to be unique in A , or not, obviously the value range.

[When teachers explain concepts, they consciously use fonts of different colors on the multimedia screen to highlight key points and mobilize students' non-intellectual factors to understand the concepts. ]

2. Question 4:

(1) Is the following corresponding rule a function on a given set?

①R,g: the reciprocal of the independent variable;

②R,h: the square root of the independent variable;

③R,s: the square of the independent variable t minus 2 .

(2) Are the following set of functions the same function?

①f(x)=x2,x∈R;

②s(t)=t2,t∈R;

③g(x-2)= (x-2) 2. means that this symbol did not appear when learning functions in junior high schools. Several points should be explained:

① It means a function that is, not the product of and;

② It does not necessarily mean An analytical expression;

③ is different from .

3. Example teaching:

The teacher gives an example 1. A watermelon stall sells watermelons, weighing less than 6 pounds. 4 cents per catty, 6 cents per catty for more than 6 catties. Please express the functional relationship between the watermelon weight x and the selling price y.

Explanation: Using the analytical method, the analytical expression of this function should be divided into two types Situation:

At that time,; at that time,.

Teacher: This kind of function is called a piecewise function. We can also use the graphical method to represent it. Ask a student to draw this function Image.

Teacher: Can this functional relationship be expressed in a list? It is inconvenient. Because there are too many weight levels of watermelon, the list is not easy to list.

3. Consolidation Exercise 1: Which of the following graphs can be used as a function graph ( )

Exercise 2: Which of the following functions is the same function as the function?

4. Class Summary

This concludes the research and study of this class. Please review the exploration and gains of this class.

Student 1 , we know the function definition: Assume A and B are both non-empty number sets, then the mapping from A to B is called the function from A to B, denoted as, where, .

< p>Student 2. We know that there are three ways to represent functions: analytical method, list method, and graphical method.

Student 3. We know the three elements of a function: definition domain; value range;

Student 3. p>

The one in is the correspondence rule. The domain of definition is the basis of the function, and the correspondence rule is the core of the function.

Student 4. In this class, we will discuss, cooperate, communicate and other group activities to gain personal experience Through the process of abstracting practical problems into mathematical models and explaining and applying them, I feel that mathematics is everywhere around us.

Teacher: Well said! These are the focus of our class. I hope to see the results of your independent thinking and exploration in the future and show your research style.

5. Modeling Assignment

① A certain nail costs 1 dime and 5 cents each. The amount of money to buy a nail is yuan. Please list the functional relationship between and and draw the graph of the function.

② Mailing package, per kilogram The postage fee for heavy packages is 2 yuan. After the mailing distance exceeds 100km, an additional 2 cents will be charged for each additional 1km. Find the functional relationship between the postage and the number of kilometers traveled by the package.

③ Ask students to record the week's Weather forecast lists the functional relationship between daily maximum temperature and date.

Teaching Analysis

1. Pay attention to the process of forming the concept of functions and understand the true meaning of mathematics

We all We know that mathematical concepts are abstracted directly or indirectly from the objective world, and most of their definitions are given using the method of "problem situation-extracting essential attributes-extending to the general". The concept of function in this lesson is that under the guidance of the teacher, Students appear as explorers and participate in the process of revealing the rules of concept formation, allowing their thinking to experience a cognitive process from concrete to abstract and summarizing the essence of things, and understand the hidden thinking methods in the process of knowledge formation. What you gain is not only the concept of functions, but more importantly, you broaden your thinking space and understand the true meaning of mathematics. While mastering the concepts, your generalization ability is trained.

2. The problem design is open and novel, and penetrates mathematical thinking methods

We all know that students’ original knowledge and experience are the basis of learning, and students’ learning is a self-generated process based on original knowledge and experience. Before learning the concept of functions, students have already been exposed to it in junior high school Functions, teachers are good at using analogy thinking in teaching, grasping the advantages and disadvantages of the two function concepts in junior high school and high school, so that students can experience the organic connection between knowledge and feel the integrity of mathematics. On the basis of students' cooperation and exchanges, students summarized several important aspects of function definition, permeating transformation ideas and induction methods.

3. Exploring teaching material resources and expanding students' exploration space

We all know that mathematics textbooks are the embodiment of mathematics curriculum standards and a selection of the knowledge system of mathematics subjects. They are very convenient for teachers and students to use. In this lesson, teachers did not just stay on the surface of the textbooks, but carefully studied and became familiar with the textbooks. , aiming at the knowledge points in the textbooks, make full use of various teaching resources and organize students to explore to cultivate students' inquiry abilities. This kind of carefully designed inquiry activities can stimulate students' enthusiasm for learning mathematics and improve students' ability to explore and research problems. Ability.

4. Improve teaching and learning methods to enable students to actively learn

Enriching students' learning methods and improving students' learning methods are the basic concepts pursued by high school mathematics courses.

Students' mathematics learning activities should not be limited to memorizing, imitating and accepting concepts, conclusions and skills. Independent thinking, independent exploration, hands-on practice, cooperative communication, reading and self-study are all important ways to learn mathematics. In this teaching section, there are both teacher's teaching and guidance, as well as students' independent exploration and cooperative exchanges. The teacher pays attention to the students' main participation throughout the class, leaving appropriate space and time for students to expand and extend, and inspire students Be interested in mathematics learning and develop good study habits.

5. Focus on mathematical modeling activities and develop students’ application awareness

The famous mathematics educator Freidenthal is talking about When it comes to the application of mathematics, he once pointed out that “mathematical applications should be understood from two aspects: we must not only pay attention to extracting mathematical concepts and principles from practical problems, but also pay attention to using mathematical concepts and principles to deal with practical problems in turn”; “And we must integrate schools into Mathematics is more widely applied to different contexts and backgrounds, and mathematization should be the main method of mathematics teaching." In this lesson, teachers use mathematical modeling activities to guide students to discover problems from actual situations and reduce them to mathematical models to form mathematical problems (that is, mathematicalization of practical problems).

At the same time, it broadens students' horizons and realizes the scientific value, application value and humanistic value of mathematics.