Creating Problematic Situations, Changing Teaching Methods

The new high school mathematics curriculum is about to be implemented, and the key to the successful implementation of the New Curriculum Standards is to optimize the classroom teaching structure, establish new teaching concepts, and improve traditional teaching methods. The New Curriculum Standard states: "In the teaching process, teachers should actively explore teaching methods and approaches that are suitable for high school students' learning based on the students' knowledge structure and the characteristics of high school mathematics teaching. "Below, I according to the new curriculum concept on the creation of problematic situations in the use of mathematics classroom teaching to talk about a few views.

I. A new view of knowledge and education should be established

The traditional curriculum system espouses an objectivist view of knowledge, which is regarded as a universal, external, truth for people to grasp. The new curriculum establishes a new view of knowledge as an action of exploration and a process of creation, thus freeing man from the grip of the traditional view of knowledge and moving him towards understanding and constructing knowledge.

The New Curriculum Standard states, "Students' mathematical learning activities should not be limited to the memorization, imitation and acceptance of concepts, conclusions and skills; independent thinking, independent exploration, hands-on practice, cooperative exchanges, and guided self-study are all important ways of learning mathematics." Therefore, teachers should fully reflect the students' main position in teaching, establish a new order of teacher-student interaction, positive interaction, *** with the development of the purpose of cultivating students' innovative ability.

Second, create a problem situation, change the way the classroom teaching and learning

The New Curriculum Standards emphasize that successful mathematical activities must be based on the level of knowledge and experience that students already have, the teacher to provide students with a good, able to give full play to the level of mathematical knowledge of the students the opportunity to make the students to be able to independent, cooperative process of real understanding and mastery of knowledge, real independent acquire mathematical knowledge and learning experiences. Therefore, in mathematics classroom teaching, teachers can maximize the mobilization of students' positive initiative by creating problematic situations, protect the germ of students' innovative thinking, improve the degree of exchange and cooperation between teachers and students, so that classroom teaching becomes an activity of teachers and students **** with cooperation and creative development.

Teachers create problem situations, must be based on the students' original level of knowledge, to be moderately difficult, close to the students' knowledge structure level. Modern teaching theory suggests that teachers should fully understand the student's "recent development zone", within this zone to put forward the relevant knowledge issues, enabling students to maximize their personal level of play, to promote the participation of the main body of the student's consciousness, to find the new knowledge "growth points

Three, the classroom teaching problem situation creation attempt

1. Conceptual teaching

In the past, a common problem of conceptual teaching is that the teacher is satisfied with the "clear teaching", students are satisfied with the "listening to understand and remembering", and ignored the process of its development and formation. The process of its development and formation is neglected. However, according to the cognitive theory of constructivism, teaching should change the routine, by setting up some problematic situations, so that students in the exploration of concepts, definitions of the connotation of gradual understanding.

For example, the teaching design of the concept of ellipse:

(1) Use multimedia to show the orbit of the stars, ellipse drawing. (2) Let the students use the cardboard, thin string and two pegs prepared before class to draw their own ellipses according to the method of drawing ellipses in the textbook. (3) What does the drawing on the cardboard illustrate? Students answer: the curve is the trajectory of the point movement, the points on the ellipse to the sum of the distances of the two fixed points remains the same ...... (4) under the premise of the rope length remains unchanged, change the distance between the two pegs, and then draw the graph: when the distance between the two pegs becomes smaller, how does the graph drawn change? What is the graph drawn when the two pegs coincide? What is the graph drawn when the distance between the two pegs is equal to the length of the cord? Can a graph be drawn when the two pegs are fixed and the length of the string is smaller than the distance between the two pegs? After practice and reflection, students can naturally answer the above questions quickly: when = becomes larger, the flatter the ellipse; 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 an ellipse. (Summarized by the students)

Through the intuition, practice, and answer to the above questions, it can make the students have a clear, accurate knowledge and deep understanding of the concept of ellipse, and lay a solid foundation for future knowledge.

2. Theorem teaching

In the past, theorem teaching is often just busy with the proof and application of the theorem, neglecting to let students explore and discover by themselves. Theorem teaching can generally be divided into: (1) create a problem situation, stimulate students' interest in learning mathematics, and clarify the goal of discovery. (2) Inspire and induce students to think positively, use the relevant knowledge and methods they have learned, analyze, analogize, and explore ways and means to solve mathematical problems. (3) Validate the conclusions.

For example, the teaching design of the "theorem of perpendicularity between a straight line and a plane":

(1) Let the students observe the model of a rectangular body (classroom, etc.), and guide them to discover that the side edges are perpendicular to the bottom surface. (Observe the intuition) (2) Why is the side edges of the rectangular body perpendicular to the bottom surface? (Questioning) Lead students to discover that the side edges are perpendicular to both neighboring edges of the base rectangle. (3) A line is perpendicular to two lines in a plane, so is the line perpendicular to the plane? After students have made their judgments, modeling is demonstrated. (4) If a line is perpendicular to an infinite number of lines in a plane, is the line perpendicular to the plane? Students make a judgment and then demonstrate the model. (5) Prove it.

In this way, students experience the process of observation, perception, conjecture, verification, and then obtain the correct conclusion, to cultivate the spirit of exploration and mastery of students and accurate use of great significance.

3. Teaching of formulas

In the teaching of formulas, we often ignore the discovery, formation and argumentation process. Therefore, we should create a problem situation, from the special to the general, guiding students to explore and discover the law and summarize the law, which has the review of old knowledge, cultivate the spirit of innovation, "win-win" effect.

such as "point to the straight line distance formula" teaching design is as follows: (1) point P (x0, y0) to the straight line x distance d = ____. (2) point P (x0, y0) to the line y distance d = ____. (3) What are the ways and means to find the distance d from the point P (-3, 2) to the line 2x-y+1=0? And find it. (4) Derive the formula for the distance d = from the point P (x0, y0) to the line Ax + By + C = 0 (A and B are all nonzero). Guide the use of |? ||'s meaning: take a point A on the line → find → determine the unit vector of the normal vector of the line → d = |? |. (5) Verify that the formula also applies when one of A and B is equal to 0.

This kind of special to general in line with the development of students' understanding, consolidate old knowledge through problem solving, cultivate students' rigorous reasoning, argumentation ability, the teacher is both the designer and the collaborator.

4. Problem-solving teaching

In problem-solving teaching, we can't limit ourselves to talking about the problem, nor can we be satisfied with just giving students a clear explanation. We can break down a complex problem into a number of related simple problems by creating laddered problem situations, so that it is easy 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 to motivate students to actively participate in the development of flexibility and breadth of thinking.

For example, in the "proof that for all n ∈ N*, there are 2 ≤ (1 +) n < 3", the following chain of questions can be designed:

(1) the proof of this set of inequalities, focusing on the proof of geometric inequalities? (2) What should be associated with (1+)n? (3) After expanding using the binomial theorem, how can you use deflation to make a variational substitution? (4) What can be done further with the sum ++...+? (Further inspiration from both direction and goal, let students explore and try on their own.) (5) Reflecting on the process of proving this problem, what is your experience?

These problems are solved to achieve the effect of review, but also to mobilize students' enthusiasm and initiative.

5. homework evaluation

Homework is the real show of students re-creating the process of thinking, homework assessment can not just give students the right answer, teachers should make full use of the wrong example to create a problem situation, stimulate students to question, reflect on, in the error of the discovery, in the exploration of the construction.

As set function y = x2 + (m-1) x2 + m constant positive value, the range of values of m to seek.

Student error: let t = x2, then y = t2 + (m-1)t + m, to make y > 0, need △ = (m-1)2-4m = m2 - 6m +1 < 0, the solution to 3-2 < m < 3 + 2.

Classroom design: (1) if you take the m = 10, ask the students to verify that the y > 0 is constant? (2) Multimedia show students' wrong solution, discuss the reason for the error: set y = ax2 + bx + c (a > 0) constant positive condition is △ < 0. (3) set up a variant group of questions to help students internalize and learn the lesson of the mistake: (1) set the function y = 1g2x + (m-1)1g + m constant positive value, then m ∈ ____; (2) set the function y = 4x + (m-1)2x + m constant positive value , then m ∈ ____.

In this way, students will question in testing, criticize in questioning, and y comprehend in criticizing.

Four, should pay attention to the problem

1. Creating problem situations should be in line with the original cognitive level of the students, and should be trend-motivated, basic and sustainable development

This requires the teacher to analyze the content of the teaching of the macro- and micro-analysis, and a second processing and creation, and the teacher is the developer, the researcher.

2. It is important to clarify the leading role of the teacher and the subjective position of the students, and not to turn "method guidance" into "method teaching"

The brilliance of the guidance is to let the students independently search for, determine the method, and experience the process after the instruction, so that the students can develop independent thinking and lifelong development. This will enable students to develop independent thinking and the ability to learn independently throughout their lives.

In short, in classroom teaching, the teacher timely and appropriate to create a problem situation is to establish the teacher-student interaction, positive interaction, *** with the development of a new classroom teaching an effective way. At the same time, it will change the teacher from a single knowledge transfer into the classroom teaching designers, guides, collaborators, will also make the students become the main body of the classroom, is active, creators.

(Anxi County Numatao Middle School)