First of all, we should establish a new concept of knowledge and education.
The traditional curriculum system believes in the objectivism view of knowledge and regards knowledge as universal, external and accessible truth. The new curriculum establishes a new concept of knowledge, which regards knowledge as a process of exploration and creation, and enables people to get rid of the shackles of the traditional concept of knowledge and move towards understanding and constructing knowledge.
The New Curriculum Standard points out: "Students' mathematics learning activities should not be limited to memorizing, imitating and accepting concepts, conclusions and skills. Independent thinking, independent exploration, hands-on practice, cooperation and communication, and self-study are all important ways to learn mathematics. "Therefore, teachers should fully reflect students' dominant position in teaching, establish a new order of teacher-student interaction, benign interaction and common development, and achieve the goal of cultivating students' innovative ability.
Second, create problem situations and change classroom teaching methods.
The new curriculum standard emphasizes that successful mathematics activities must be based on students' existing knowledge level and experience, and teachers should provide students with a good opportunity to give full play to their mathematics knowledge level, so that students can truly understand and master knowledge in the process of autonomy and cooperation, and truly acquire mathematics knowledge and learning experience independently. Therefore, in mathematics classroom teaching, teachers can mobilize students' initiative to the maximum extent by creating problem situations, protect the seeds of students' innovative thinking, improve the degree of communication and cooperation between teachers and students, and make classroom teaching an activity of cooperation, creation and development between teachers and students.
Teachers must create problem situations according to students' original knowledge level, with moderate difficulty and close to students' knowledge structure level. Modern teaching theory holds that teachers should fully understand students' "zone of recent development" and ask relevant knowledge questions in this zone, so that students can maximize their personal level, promote their sense of participation, and find the "growth point" of new knowledge, so that students can transition from "existing knowledge level" to "future level".
Third, the attempt to create problem situations in classroom teaching
1. Concept teaching
In the past, a common problem in concept teaching was that teachers were satisfied with "explaining the church clearly" and students were satisfied with "understanding and memorizing", ignoring its development and formation process. However, according to the cognitive theory of constructivism, we should change the routine in teaching and let students gradually understand the connotation of concepts and definitions by setting some problem situations.
For example, the instructional design of ellipse concept:
(1) Demonstrate the drawing of stars' orbits and ellipses with multimedia. (2) Ask students to draw an ellipse by themselves with cardboard, string and two thumbtacks prepared before class according to the method of drawing an ellipse in the textbook. (3) What does the picture on the cardboard show? Students answer: The curve is the trajectory of point movement, and the sum of the distances from one point to two fixed points on the ellipse remains the same ... (4) Change the distance between two pushpins under the premise that the length of the rope remains the same, and then draw a picture: What will happen to the drawn picture when the distance between the two pushpins becomes smaller? When two pushpins overlap, what is the figure drawn? When the distance between two pushpins is equal to the length of the rope, what is the figure drawn? When two pins are fixed, the length of the rope is less than the distance between the two pins. Can you draw a picture? After practice and thinking, students can naturally answer the above questions quickly: when = becomes larger, the ellipse becomes flat; When 2c=0, it is a circle; When 2a=2c, it is a line segment; When 2a < 2c, the trajectory does not exist. (5) Give a complete definition of ellipse. (Summarized by students)
By answering the above intuition, practice and questions, students can have a clear and accurate understanding of the concept of ellipse and lay a solid foundation for future knowledge learning.
2. Theorem teaching
In the past, theorem teaching was often just busy with the proof and application of theorems, while ignoring students' own exploration and discovery. Theorem teaching can generally be divided into: (1) creating problem situations, stimulating students' interest in learning mathematics, and clearly finding goals. (2) Inspire and induce students to think positively, analyze and compare with the relevant knowledge and methods they have learned, and explore ways and methods to solve mathematical problems. (3) Verify the conclusion.
For example, the teaching design of "the judgment theorem of straight line perpendicular to plane";
(1) Let the students observe the rectangular model (classroom, etc. ) and guide them to find that the side is perpendicular to the bottom. (Observing intuition) (2) Why is the side of a cuboid perpendicular to the bottom? (Ask questions) Guide students to find two adjacent sides whose sides are perpendicular to the bottom rectangle. (3) A straight line is perpendicular to two straight lines in a plane, so is this straight line perpendicular to this plane? After the students make a judgment, make a model demonstration. (4) If a straight line is perpendicular to countless straight lines in a plane, is it perpendicular to this plane? After the students make a judgment, make a model demonstration. (5) proof.
In this way, it is of great significance for students to experience the process of observation, perception, guess, verification and then draw the correct conclusion, which is of great significance to cultivate their exploration spirit and master and use it accurately.
3. Formula teaching
In the teaching of formulas, we often ignore the process of discovery, formation and demonstration of formulas. Therefore, it is necessary to create problem situations, from special to general, to guide students to explore and discover the laws and summarize them, which will have a "win-win" effect of reviewing old knowledge and cultivating innovative spirit.
For example, the teaching design of the formula of distance from point to straight line is as follows: (1) The distance from point P(x0, y0) to straight line x is d=____. (2) The distance from point P(x0, y0) to straight line y is d=____. ⑶ What are the ways and methods to find the distance d from point P (-3,2) to line 2x-y+ 1=0? And ask for it. (4) Derive the distance formula d= from point P(x0, y0) to straight line ax+by+c = 0 (both a and b are not zero). Guide the use of |? Meaning of |: Take a point on a straight line → Find → Determine the unit vector of the normal vector of the straight line →d=|? |。 (5) Verify that the formula is also applicable when one of A and B is equal to 0.
This change from special to general is in line with students' cognitive development, consolidating old knowledge through solving problems and cultivating students' rigorous reasoning and argumentation ability. Teachers are both designers and collaborators.
4. Problem-solving teaching
In problem-solving teaching, we can't be limited to the topic, and we can't just be satisfied with explaining it clearly to students. By creating a step-by-step problem situation, a complex problem can be decomposed into several related simple problems, which is convenient for students to extract relevant knowledge from the original knowledge structure to solve the problem. At the same time, we can also set open questions, encourage students to participate actively, and cultivate flexibility and broadness of thinking.
For example, in "Prove that for all n∈N*, there is 2 ≤ (1+) n < 3", the following problem chain can be designed:
(1) The proof of this inequality group focuses on the proof of geometric inequalities. (2) What should I associate with seeing (1+)n? (3) After the binomial theorem is expanded, how to replace the variant with the scaling method? (4) How to further deal with the sum formula++? (further enlighten students from the direction and goal, so that students can explore and try independently) (5) reflect on the proof process of this question, what is your experience?
The solution of these problems not only achieves the effect of review, but also mobilizes the enthusiasm and initiative of students.
5. Job evaluation
Homework is the true embodiment of students' thinking in the process of re-creation. Homework evaluation can't just give students correct answers. Teachers should make full use of wrong questions to create problem situations, stimulate students to question and reflect, find them in mistakes and construct them in exploration.
If the function y=x2+(m- 1)x2+m is always positive, the range of m is found.
Students' wrong solution: let t=x2, then y = t2+(m-1) t+m. To make y > 0, you need △ = (m-1) 2-4m = m2-6m+1< 0, and the solution is 3-2.
Classroom design: (1) If m= 10, please ask students to verify whether Y > 0 holds. (2) Multimedia demonstration of students' wrong solutions, and discussion of the reasons for the errors: the condition that Y = AX2+BX+C (A > 0) is always positive is △ < 0. (3) Set variant problem sets to help students internalize and learn from their mistakes: ① Let the function y =1g2x+(m-1)1g+m be positive, then m ∈ _ _; ② Let the function y=4x+(m- 1)2x+m be a positive value, then m ∈ _.
In this way, students will question in the inspection, criticize in the questioning and deeply understand in the criticism.
Fourth, the problems that should be paid attention to
1. Creating problem situations should conform to students' original cognitive level and be dynamic, basic and sustainable.
This requires teachers to make macro and micro analysis of the teaching content, and make secondary processing and creation. Teachers are both developers and researchers.
2. Make clear the leading role of teachers and the dominant position of students, and don't turn "method guidance" into "method teaching".
The genius of guidance lies in that students can discover, determine methods and experience the process independently through guidance, so as to cultivate their ability of independent thinking and lifelong independent learning.
In short, in classroom teaching, it is an effective way to establish a new type of classroom teaching with teacher-student communication, positive interaction and common development. At the same time, it will transform teachers from a single knowledge giver into designers, guides and collaborators of classroom teaching, and will also make students become the main body, activists and creators of the classroom.
(Zhao Tao Middle School in Anxi County)