First, the characteristics of "thinking after class"
Senior high school mathematics "thinking questions after class" should meet the following main characteristics.
1 question
Because mathematics "after-class thinking questions" are based on one or several questions with mathematical thinking value designed by teachers, students can deepen their understanding and mastery of what they have learned in the process of solving problems through independent exploration, so questions are the formal characteristics and the most typical characteristics of "after-class thinking questions".
2 Openness
The teaching goal of "after-class thinking questions" in mathematics is not limited to the completion of teaching content, but starts from the comprehensive quality of mathematics such as students' spirit of mathematical exploration, desire for knowledge, interest in research and willpower cultivation. The openness of teaching objectives determines the diversification of content organization and form of "after-class thinking questions", and also determines the diversification and individualization of evaluation and feedback methods and application of results of "after-class thinking questions". Openness is the content feature of "thinking after class".
3. Motivation
Mathematics "thinking questions after class" has certain mathematical thinking value. It is not a simple examination of knowledge and skills, but guides students to new goals and encourages them to carry out trial and inquiry activities. Sometimes it is a challenging small research topic, which can arouse students' interest and desire to explore, so motivation is the emotional feature of "thinking about questions after class"
Second, design the strategies and methods of "thinking after class"
There are many wonderful "after-class thinking questions" in the new mathematics curriculum experiment in senior high school, which can be extended and deepened; Or far-reaching, guiding exploration; Or set suspense to make people think; Or connect with reality, feel the application and so on. The following is a high school mathematics textbook published by Jiangsu Education Publishing House, and an experiment is being conducted in Nanjing. Taking the teaching content of mathematics as an example, this paper discusses the strategies and methods of setting "thinking after class" in senior high school mathematics classroom teaching.
1 Expand "Thinking after class"
When setting up "after-class thinking questions", teachers can proceed from students' reality, according to their actual knowledge level, cognitive ability and knowledge structure, appropriately extend and expand the content of mathematics teaching in the form of questions or inquiry questions, dig out the connotation and help students deepen their understanding and mastery of knowledge.
Example 1 Equation of Circle (Lesson 2) (Compulsory 2) Thinking questions after class:
(1) The ratio of the distance between a given point M(x, y) and two fixed points O (0 0,0) and a (-2,0) is 2. What relation should the coordinates of point m satisfy? Can you tell me what the trajectory of starting point M is?
(2) According to the example 1( 1), complete the following mathematics questions of the 2008 Jiangsu College Entrance Examination:
Meet the conditions AB=2, AC = x/2ab AABC maximum area is _ _ _ _.
This thinking problem extends the teaching content and actually introduces "Apollonius Circle" and "apollonius Trajectory". Because "apollonius Circle" has appeared frequently in recent years' college entrance examination mathematics papers all over the country, combining college entrance examination mathematics questions can effectively stimulate students' interest in inquiry.
Example 2 "Inductive Formula of Trigonometric Function (Class 1)" (Required 4):
In the inductive formula of (1) trigonometric function, can another set of formulas be deduced from any two sets of formulas 2, 3 and 4?
(2) What is the special positional relationship between the terminal edge of angle A and angle β? Can you explore the relationship between their trigonometric functions?
This thinking question extends the content to be studied, that is, the relationship between implicit formulas in trigonometric function induction formula. Through students' after-class exploration, they can not only master and use formulas, but also experience these methods of studying trigonometric function induction formulas again, which also provides materials and space for students to further explore trigonometric function induction formulas.
Example 3 The first n sums of geometric series (lesson 1) Thinking questions after class (compulsory 5):
Find the sequence: 1? 2+2? 22+3? 23+…+n？ Sum of 2n,
This thinking problem extends and expands the important method of studying the summation formula of equal proportion series, that is, dislocation subtraction. In the derivation of the first n terms and formulas of geometric series, the formula can be directly obtained by dislocation subtraction, but in this question, a new geometric series is constructed by dislocation subtraction. Therefore, this problem, as a "thinking problem after class" in mathematics, has the value of method expansion.
2 Migrate the "after-school thinking questions" of the application
The transfer of applied "thinking questions after class" mainly involves the proper transfer and application of mathematical knowledge and methods, including solving mathematical problems and practical problems with mathematical knowledge. Setting the migratory and applied "thinking after class" can not only improve students' ability and level of solving problems, but also cultivate students' awareness of application and innovation.
Example 4 "Basic inequality AB ≤ A+B/2 (A ≥ 0, b≥0)" (required 5):
The two sides of a rectangle are A and B respectively. The area of this rectangle is 3 larger than the circumference. Find the area range of this rectangle.
In this thinking question, because both A and B are positive numbers, after listing the equation ab=a+b+3, the basic inequality can be transformed into a quadratic inequality about ab. The purpose of setting this thinking question is to improve students' ability to analyze and solve problems by using basic inequalities.
Example 5 "Monotonicity of Functions (Class 1)" (required 1):
A proper amount of sugar is completely dissolved in a bowl of water. If the mass of this bowl of water is 1kg, the mass of sugar is xkg, and the concentration of syrup is y, try to write the functional relationship between y and x, and explain the feature that "the more sugar is added, the sweeter the syrup is" with the monotonicity of the function.
This thinking question is a simple application of monotonicity of functions. Because it is related to practical problems, it can stimulate students' interest in learning "after-class thinking questions" and help students further understand the concept of monotonicity of functions.
3 Echo "Thinking after class"
"Thinking after class" can be started from two aspects: one is to echo the teaching content or method of this class, and the other is to echo the teaching content or method of the next class.
Example 6 "Standard Equation of Ellipse" (optional 2- 1) Thinking questions after class:
(1) If the abscissa of a point on a circle remains unchanged and the ordinate becomes half of the original, is the resulting curve an ellipse?
(2) How to study the geometric properties of an ellipse with the help of its standard equation?
The question (1) in this example is different from the definition of ellipse in the textbook. It is a transformation method, but it can help students understand ellipses from the perspective of transformation, which echoes the teaching content of this lesson. The second question of this example begins with the geometric properties of ellipses, which echoes the teaching content of the next class.
Example 7 "Average Change Rate" (Elective 2-2) Thinking questions after class:
The displacement s and time t of a moving particle satisfy s(t)=t2. How to describe the velocity of a particle at t= 1 (Displacement in meters and time in seconds)
The function of thinking questions in this example is to guide students to think about how to describe the real questions from the average change rate to the instantaneous change rate after class, and also to pave the way for students to learn the instantaneous change rate in the next class.
4 Operation experiment "Thinking after class"
The "thinking after class" of operation experiment is to set up some operation experiment activities, so that students can deepen their understanding and perception of knowledge and methods in the operation experiment, thus deepening their understanding and developing their mathematical thinking.
Example 8 the positional relationship between a straight line and a plane Thinking questions after class (Lesson 2) (Compulsory 2):
(1) As shown in figure 1, please use a triangular piece of paper to do the experiment: fold the paper over the vertex A of AABC to get the crease AD, and put the folded paper vertically on the desktop, so that BD and DC can contact the desktop. ① Is the crease advertisement perpendicular to the desktop?
② How to make crease advertisements perpendicular to the desktop?
(2) Can we design a tetrahedron whose four faces are right triangles?
In this example, the question (1) requires students to operate, and will be constantly analyzed and adjusted during the operation until the correct answer is obtained; Question (2) requires students to constantly construct and try graphics. In the operation experiment, students can deepen their understanding and mastery of some common graphics, and further clarify the positional relationship between lines, lines and surfaces, and surfaces and surfaces in some special graphics.
5 Questioning and correcting "thinking questions after class"
Setting "after-class thinking questions" by using common mistakes in solving problems can arouse students' doubts and reflections. These "common mistakes" are important resources in mathematics teaching.
Example 9 "Slope of straight line (class 1)" (compulsory 2):
Is the following judgment correct? Please provide a justification for the answer.
(1) If the straight line 1 passes through point P (3 3,2) and point Q (m,0) (m is a real number), the slope of the straight line 1 is 2/3-m;
(2) If the straight line 1 passing through point C (2,4) intersects with line segment AB, and the coordinates of point A and point B are A(-3, -2) and B (3 3,3) respectively, the range of the slope of the straight line 1 is [-7,5/6].
This example comes from the most common mistake that students make when learning this part of the content, that is, ignoring the fact that the slope of a straight line does not exist. Through students' thinking after class, let students further understand and know the slope of the straight line.
6. Get information-based "thinking after class"
By arranging such thinking problems, teachers can let students use their spare time to consult various books and periodicals, or surf the Internet to find information to solve "thinking problems after class", which is helpful to enrich students' ways of learning and exploring problems.
Example 10 "number system extension" (optional 1-2):
Is imaginary number illusory? Is imaginary number useful in real life?
To complete this thinking question, students must consult various books and periodicals or online. In the process of solving this thinking problem, students can further understand the expansion process of number system, understand the role of the contradiction between actual demand and mathematics in the expansion of number system, and feel the role of human rational thinking and the connection between number and the real world.
7 "Thinking after class" based on micro-topics
"Thinking after class" based on micro-topics means that teachers design some mathematical inquiry questions and ask students to solve challenging and comprehensive problems related to mathematics or life experience through independent exploration and cooperative communication around these mathematical problems, thus developing students' problem-solving ability.
Example 1 1 Application of Basic Inequalities (Category II) (Compulsory 5):
As we all know, when the cross-sectional area of the channel is constant, the smaller the wet circumference, the greater the flow. There are two designs to choose from:
Fig. 2 cross section is isosceles △ABC, AB=BC, and wet circumference l1= ab+BC;
Fig. 3 shows an isosceles trapezoid ABCD, AB=CD, AD∨BC, ∠ bad = 60, and wet circumference l2=AB+BC+CD.
If the areas of AABC and trapezoidal ABCD are both S.
(1) Find the minimum values of l 1 and l2 respectively;
(2) To maximize the flow, the best design scheme is given. This kind of "thinking after class" is closely related to the mathematics content that students are learning, which can make students go through the process from problem to function, and then through learning and comparing the relationship between the two functions, the solution to the problem is obtained, in which the main method to solve the maximum problem of function is to use the basic inequality and the boundedness of sine function. This kind of "thinking after class" highlights the value of mathematics application and can play a positive role in improving students' learning style.
Third, set the attention points of "after-class thinking questions"
1 should be practical for students.
Students are the main executors to complete the "after-class thinking questions" in mathematics, which determines that the setting of "after-class thinking questions" should conform to the students' knowledge level and ability level, and it will lose its effectiveness if it is too easy and too difficult, so that students should be "right in one jump". At the same time, we should pay attention to the differences between students and have flexible requirements for different students, so that each student can get the development he deserves.
2 overall design and planning
For the design of "after-class thinking questions" in a certain stage of mathematics, teachers should have an overall plan, break through different emphases and difficulties according to the development of students in different periods, and highlight the core concepts and thinking methods of mathematics.
3 should pay attention to feedback evaluation
The evaluation of students' completion of mathematics "after-class thinking questions" should not only pay attention to the right or wrong results, but also pay attention to students' attitude towards mathematics "after-class thinking questions", whether they have thought about it or not, and emphasize the value of the process itself. Paying attention to process evaluation also requires teachers to pay attention to the difficulties encountered by students and guide them how to overcome them. In addition, teachers should give students more opportunities to show, express appreciation for students' achievements in completing the "after-class thinking questions" in mathematics, and encourage students not to be afraid of difficulties and establish self-confidence.
Of course, the methods and forms of setting "after-class thinking questions" in mathematics should be eclectic and diversified. "Thinking after class" in mathematics teaching also needs to further enrich its connotation and expand its extension, so that it can truly become an effective teaching method and truly serve the development of students.
(Editor Liu Yongqing)