Typical Case Analysis of Junior High School Mathematics Teaching
I will only report to the teachers my personal experience of mathematics teaching from four aspects, with the help of teaching case analysis. These four aspects It is:
1. Achieving the integration of three-dimensional goals in diversified learning activities; 2. Dynamic adjustment of presets and generation in the classroom teaching process; 3. Thinking about mathematics learning topics; 4. Thoughts on classroom questions.
First of all, take the teaching of "Pythagorean Theorem" as an example to talk about how to achieve the integration of three-dimensional goals in diversified learning activities
Case 1: "Pythagorean Theorem" 》Classroom teaching of one lesson
The first link: exploring the teaching of the Pythagorean theorem
Teacher (show 4 graphs and tables): observe and calculate square A, For the areas of B and C, complete the table. What did you find?
Area of ??A
Area of ??B
Area of ??C
Figure 1
Figure 2
p>Figure 3
Figure 4
Student: It can be seen from the table that the sum of the areas of the two squares A and B is equal to the area of ??the square C. Moreover, it can be seen from the figure that the sides of squares A and B are the two right-angled sides of the right triangle, and the side of square C is the hypotenuse of the right-angled triangle. Based on the above results, it can be concluded that the two right angles of the right-angled triangle The sum of the squares of the sides is equal to the square of the hypotenuse.
Here, the teacher designs problem situations to allow students to explore and discover the close relationship between "number" and "form", form conjectures, actively explore conclusions, and train students' inductive reasoning ability and the idea of ??combining numbers and shapes. It has been naturally applied and penetrated, and the "area method" also paved the way for the proof of the following theorem. Double-based teaching is embedded in the learning situation.
Second link: Teaching to prove the Pythagorean theorem
The teacher gave each group hard work to make right triangles and square pieces of paper, first divided into groups to explore the puzzle, and then communicated and displayed. Allow students to develop new abilities in practical inquiry activities (trying to discover the rules of puzzles and proofs: the area of ??the same figure is represented in different ways).
Student presentation summary
Through group exploration and demonstration of proof methods, students are allowed to connect their existing area calculation knowledge with the algebraic expressions to be proved, and try to understand the construction through geometric meaning Graphics allow students to deeply understand mathematical thinking methods and improve their innovative thinking abilities in the process of exploring proof methods.
The third link: teaching using the Pythagorean theorem
Teacher (show the picture on the right): The picture on the right is a figure composed of two squares
. Can it be cut and assembled into a new square with the same area? If so, who can cut it the least times?
Student (show the picture on the right): It can be cut and assembled into a new square with the same area
. Assume the side lengths of the two original squares
are a and b respectively, then the sum of their areas is
a2+ b2. Since the area remains unchanged, the area of ??the new square
should be a2+ b2, so as long as it can be cut out Take two right triangles with a and b
as right-angled sides. Just put them back together into a square with side length a2+ b2.
Problems are the heart of mathematics, and the core of learning mathematics is to improve the ability to solve problems. The teacher's setting of questions here is not only a flexible application of testing the Pythagorean Theorem, but also a comprehensive application of the Pythagorean Theorem exploration methods and proof ideas (the idea of ??combining numbers and shapes, the method of area cut and complement, the idea of ??transformation and reduction), so as to allow Students develop innovative abilities in problem solving.
The fourth link: Excavation of the cultural value of the Pythagorean Theorem
Teacher: The Pythagorean theorem reveals the quantitative relationship between the three sides of a right triangle, and sees that number and shape are closely linked. It has a unique role in cultivating students' mathematical calculations, mathematical conjectures, mathematical inferences, mathematical arguments, and the use of mathematical thinking methods to solve practical problems. The Pythagorean Theorem was first recorded in the "Zhou Bi Suan Jing" in ancient my country in the 11th century BC. In the ancient Chinese book "Nine Chapters of Arithmetic", the principle of "incoming and outgoing complement each other" was proposed to prove the Pythagorean theorem. In the West, the Pythagorean theorem is also known as the "Pythagoras theorem". It is one of the core theorems of Euclidean geometry and an important basis for plane geometry. The proof of the Pythagorean theorem has attracted many mathematicians and physicists at home and abroad at all times and in the past. Celebrities, artists, and even the President of the United States have also devoted themselves to proving the Pythagorean Theorem. Its discovery, proof and application all contain rich mathematical humanistic connotations. I hope students can check relevant information after class, understand the history of mathematics development and the stories of mathematicians, feel the value and spirit of mathematics, and appreciate the beauty of mathematics.
The three-dimensional goals of the new curriculum (knowledge and skills, processes and methods, emotional attitudes and values) build a goal system with rich connotations from three dimensions. Each goal in the course operation can be related to the three If the dimensions are connected, educational value should be obtained in these three dimensions.
2. Dynamic adjustment of default and generated in the classroom teaching process
Case 2: Years ago, on page 70 of the "Supporting Workbook" of the first volume of Lujiao's seventh-grade mathematics , encountered a fill-in-the-blank question:
Example: Suppose a, b, and c represent three objects with different masses. As shown in the figure, the two balances in Figure 1 and Figure 2 are in equilibrium. In order to make the third balance (Figure ③) also in a balanced state, an object b?
a
a
b should be placed at the "?"
c
Picture ① Picture ②
a
c
Picture ③
Through the survey, only a very small number of students have filled in the answers to this question. I don’t know if they can really solve it. I need to explain it.
The design ideas I explain are as follows:
1. Guide the equilibrium states in Figure 1 and Figure 2 to be expressed with mathematical formulas (symbolic language - mathematical language) (Mathematicalization of real-life problems - mathematical modeling):
Figure ①: 2a=c+b. Figure ②: a+b=c.
Therefore, 2a=(a+b)+b.< /p>
Available: a=2b, c=3b.
So, a+c = 5b.
The answer should be 5.
I I think my thinking is rigorous and well-founded. However, when I asked students to present their ideas, I was surprised.
Student 1 thinks like this:
Suppose b=1, a=2, c=3. Therefore, a+c = 5, the answer should be 5.
The students used the special value method to solve the problem. Although the special value method is also a mathematical method, there is a lot of uncertainty and students cannot be allowed to just stay on the surface of this superficial thinking. Facing this "new starting point" in the teaching promotion process, I must deepen students' thinking, but I cannot undermine their self-confidence and must protect students' thinking results. Therefore, I immediately gave up the prepared explanation plan and made adjustments based on the results of students' thinking.
I first commented positively on the method of student 1 and affirmed the positive role of this way of thinking in exploring problems. When the self-confidence of the students who did the same thing was beyond words, I then put forward this A question:
"How do you think of assuming b=1, a=2, c=3? Can a, b, c be assumed to be any three numbers?"
Some students answered immediately without thinking: "It can be any three numbers." Some students also held negative opinions, and most of them were doubtful. All the students were intrigued by this question, and I took the opportunity to point out:
"Verify it."
The whole class immediately began to think and verify. For about 3 minutes, the students began to answer this question:
"b=2, a =3, c=4 is not possible, and the quantitative relationship in Figure 1 and Figure 2 cannot be satisfied."
"B=2, a=4, c=6 is also acceptable. ”
“It’s ok when b=3, a=6, c=9, and the result is the same. ”
“It’s ok when b=4, a=8, c=12. The result is the same."
"I found that as long as a is 2 times of b and c is 3 times of b, the quantitative relationship in Figure 1 and Figure 2 can be satisfied, and the result must be 5."
At this time, the students' thinking has risen from the specific to the general. In other words, in this process, the students' inductive reasoning has been trained, and they have a deeper understanding of the special value method, represented by letters. Discover the rules, and then get a=2b, c=3b. Therefore, a+c = 5b. The answer should be 5.
My purpose has not been achieved, so I will continue to ask questions:
"We have listed a lot of data and found this conclusion. Can you also find a more concise method from the quantitative relationship itself in Figure 1 and Figure 2?" The students fell into deep thinking again. When I inspected each group When "Figure ①: 2a=c+b. Figure ②: a+b=c." appears in the picture, I know that the students' thinking is about to be in line with rigorous logical reasoning.
Do we all have this feeling? Classroom teaching design has the characteristics of both "reality" and "possibility", which means that there is no "inconsistency" between the classroom teaching design plan and the development of the teaching implementation process. The relationship between "architectural drawings" and "construction process" means that the classroom teaching process is not simply a process of executing the teaching design plan.
At the beginning of classroom teaching, we may first choose a starting point to enter the teaching process. However, as the teaching unfolds and the multi-directional interaction between teachers and students, and between students and students, multiple problems will continue to form. A "new starting point" for teaching based on different students' development status and teaching promotion process. Therefore, the starting point of classroom teaching design is not unique, but multiple; it is not determined and unchanging, but generated from the preset; it is not rigid and unchanging according to the preset, but dynamically adjusted.
3. Thoughts on a mathematics exercise lesson
Case 3: A teacher’s exercise lesson, the content of which is “Special Quadrilaterals”.
The teacher designed the following exercises:
A
O
F
E
B
H
G
C
Question 1 (Example) Connect the midpoints of each side of the quadrilateral in sequence, What kind of quadrilateral is the resulting quadrilateral? and justify your conclusion.
Question 2 As shown in the picture on the right, in △ABC, the midlines BE and CF
intersect at O, and G and H are the midpoints of BO and CO respectively.
(1) Verify: FG∥EH;
(2) Verify: OF=CH.
O
F
F
p>
A
E
C
B
D
Question 3 (Extended Exercise ) When the original quadrilateral has what conditions, the midpoint quadrilateral is a rectangle, rhombus, or square?
Question 4 (Extracurricular homework) As shown in the picture on the right,
DE is the median line of △ABC, AF is the median line on side BC, DE and AF intersect at point O.
(1) Verify: AF and DE bisect each other;
(2) When △ABC has what conditions, AF = DE.
(3) When △ABC has any conditions, AF⊥DE.
F
G
E
H
D
C
p>
B
A
The teacher first asks students to think about the first question (example question). The teacher guides students to draw pictures and make observations before entering into proof teaching.
Teacher: As shown in the figure, the conditions E, F, G, H
are the midpoints of each side, which can be thought of as the triangle median
line theorem , so connecting BD, we can get EH and FG are parallel and equal to BD, so EH is parallel and equal to FG, so the quadrilateral EFGH is a parallelogram. Next, ask students to write out the proof process.
After only five or six minutes, the teaching of the proof process was completed "smoothly", and the students did not find it difficult. But when students are asked to do question 2, only a few students can do it. Question 3 is more difficult for students. Some imitate the examples and draw pictures to observe, but they cannot get special quadrilaterals such as rectangles; some draw rectangles first, but the vertices of the rectangle are not the midpoints of the sides of the original quadrilateral.
Lesson evaluation: The selection and design of the exercises in this course are relatively good, covering mathematical knowledge such as the median line theorem of triangles and the properties and determination of special quadrilaterals. The main methods used are: (1) Study mathematics through activities such as drawing (experiments), observation, conjecture, and proof; (2) Communicate the connection between conditions and conclusions, realize transformation, and add auxiliary lines; (3) Because the exercises have There is a certain degree of openness and diversity of solutions, so thinking must also have a certain depth and breadth.
Why do students still not know how to solve problems? The poor foundation of students is one reason. Is there any reason in terms of teaching? I personally feel that there are three main problems:
(1) Students’ thinking has not been formed. The teacher only tells how to do it, not why. The teacher stated all the proof ideas, but did not guide students on how to analyze, depriving students of thinking space;
(2) Lack of induction of mathematical ideas and methods, and failure to reveal the essence of mathematics. There is a situation where you can solve this question after being told, but cannot solve another question;
(3) Question 3 is a dynamic conditional open question. Compared with Question 1, it is reverse thinking and requires high thinking. Students Difficult to grasp, teachers lack necessary guidance and guidance.
Correction: Based on the above analysis, the teaching design of question 1 can be improved as follows:
First of all, for the beginning of the teaching of example proof, a "serialization" thinking question is proposed:
(1) What are the methods for determining parallelograms?
(2) Can this question directly prove that EF∥FG , EH=FG? If it cannot be proved directly, indirect proof is usually considered, that is, the positional relationship of EH and FG is determined with the help of the third line segment. (Parallel) is related to the quantitative relationship. Analyze, which line segment has such a function?
(3) Since E, F, G, and H are the midpoints of each side, what mathematical knowledge can you associate with it?
(4) Are there any ready-made triangles and their median lines in the picture? How to construct?
Design intention: The above questions (1) activate knowledge; question (2) implies the necessity of adding auxiliary lines and penetrates the thinking method of indirect problem solving; questions (3) and (4) guide students to discover auxiliary lines The specific method of the line.
Secondly, after the proof is completed, the teacher can guide the induction:
We call the quadrilateral ABCD the original quadrilateral, the quadrilateral EFGH the midpoint quadrilateral, and draw the conclusion: the midpoint of any quadrilateral A quadrilateral is a parallelogram; auxiliary lines communicate the connection between conditions and conclusions, achieving transformation. A diagonal line of the original quadrilateral communicates the position and quantity relationship of a set of opposite sides of the midpoint quadrilateral.
This kind of communication comes from the diagonal of the original quadrilateral, which is also the common side of the two triangles with the midpoint side of the quadrilateral as the median line. From this, we can feel that the person who plays this communication role is often Common elements in the graph, therefore, we must pay attention to this common element in the proof.
Then, add a "transition question": What conditions are met for the original quadrilateral to make the midpoint quadrilateral a rectangle? Teachers can prompt you to think:
What kind of parallelogram is a rectangle? Considering the characteristics of this question, which method do you choose? Consider a right angle, the positional relationship of a set of adjacent sides of a quadrilateral with its midpoint. When the position and quantity relationship of a set of adjacent sides changes, the position and quantity relationship of the two diagonals of the original quadrilateral also change.
According to the revised teaching design, I changed classes and retaught this lesson. The effect was obvious. Most students successfully solved the problems, and different solutions appeared for several questions.
Enlightenment: Teaching exercises and examples is the key. The relationship between examples and exercises is the relationship between the outline and the outline. In teaching examples, teachers should guide students to learn to think, reveal mathematical ideas, and summarize problem-solving methods and strategies. You can try the following methods:
(1) Activate and retrieve mathematical knowledge related to the question. The activation and retrieval of knowledge are due to topic information, such as associating knowledge with conditions and linking knowledge with conclusions. The activation and retrieval of knowledge marks the beginning of thinking;
(2) Enlighten thinking where there are obstacles in thinking. Thinking comes from problems, and mathematical thinking is an implicit psychological activity. Teachers should try to adopt certain forms to highlight the thinking process, such as designing relevant thinking questions, breaking down obstacles in questions, and inspiring students to think effectively.
(3) Summarize thinking methods and problem-solving strategies in a timely manner. From a methodological perspective, the significance of teaching mathematics exercises is not in the exercises themselves. Mathematical thinking methods and strategies are the essence of mathematics. Exercises are only the carrier of learning methods and strategies. Therefore, a summary of methods and strategies is necessary. The summary of question 1 makes it easy to solve question 2. Question 2 is to change the convex quadrilateral ABCD of question 1 into a concave quadrilateral ABOC. The essence of the two questions is the same. When students solve problem 3, they try to imitate problem 1. This is a problem of problem-solving strategy. The conditions for question 1 are determined and can be discovered through drawing and observation. For question 3, the graphics must be discovered through reasoning before the graphics can be drawn.
4. Pay attention to the art of asking questions in class
Case 1: An open class - "Properties of Similar Triangles". In order to understand the students' mastery of the determination of similar triangles, two Questions:
(1) What are similar triangles?
(2) What are the methods for determining similar triangles?
After listening to the students’ fluent and satisfactory answers, the teacher started teaching the new lesson with satisfaction. What do teachers say about this?
C
B
A
In fact, the students answered only some superficial memory knowledge and did not indicate whether they Really understand. The question can be designed like this:
As shown in the figure, in △ABC and △A?B?C?,
(1) It is known that ∠A=∠A?, add one Suitable
C?
A?
B?
Conditions such that △ABC∽△A?B?C?;
(2) It is known that AB/A?B?=BC/B?C?; add a suitable
condition so that △ABC∽△A?B?C? .
To answer such a question, you cannot just rely on rote memorization. Only by truly mastering the determination of similar triangles can you answer it correctly. Such questions can play a role in reflection, activate students' thinking, and improve the effectiveness of teaching.
Case 2: In a class about the determination theorem of a rhombus (it means that a quadrilateral whose diagonals bisect each other perpendicularly is a rhombus), after the teacher drew the graph, there was a conversation:
Teacher: In quadrilateral ABCD, do AC and BD bisect each other perpendicularly?
B
C
A
D
Student: Yes!
Teacher: How do you know?
Sheng: This is a known condition!
Teacher: So is the quadrilateral ABCD a rhombus?
Sheng: Yes!
Teacher: Can the conclusion be proved by proving that triangles are congruent?
Student: Yes!
How do teachers feel? In fact, the teacher had already specified that congruent triangles should be used to prove that the sides of a quadrilateral are equal, and the students began to prove it without much thinking. The so-called "guidance" essentially became "indoctrination" in disguise. Although it looks lively and lively on the surface, in fact it is just a formality and is not conducive to students' active thinking.
You can revise the question design like this:
(1) What methods have you learned to determine rhombuses? (1. A set of parallelograms with equal adjacent sides is a rhombus; 2. A quadrilateral with four equal sides is a rhombus)
(2) Are both methods possible? Is there any way to prove that sides are equal? (1. Properties of congruent triangles; 2. Properties of perpendicular bisectors of line segments)
(3) Which method is simpler to choose?
Case 3: Teaching fragment of "Linear Equation of One Variable":
Teacher: How to solve the equation 3x-3=-6(x-1)?
Student 1: Teacher, I can see it before I start calculating, x =1.
Teacher: Just seeing it is not enough, you have to calculate it according to the requirements to be correct.
Student 2: First divide both sides by 3, and then... (interrupted by the teacher)
Teacher: Your idea is right, but you should pay attention to it in the future. When learning new knowledge, remember to solve it according to the format and requirements of the textbook, so as to lay a solid foundation.
How do teachers feel? When this teacher asked questions, he interrupted the students' novel answers midway, only satisfied a single standard answer, and blindly emphasized the mechanical application of a few steps and "general methods" to solve problems. As everyone knows, the answers of these two students were indeed creative. Unfortunately, this spark of creative thinking that occasionally flashed was not only not cared for, but was easily denied and suffocated by the teacher's "standard format". In fact, even if the student's answer is wrong, the teacher must listen patiently and give motivating comments. This can not only help students correct their misunderstandings, but also encourage students to think actively, stimulate students' different thinking, and thus cultivate students' thinking. ability.
Some teachers leave too little time for students to think after asking questions, and students do not have time to think deeply. As a result, they do not answer questions or answer questions that are not what they are asked; some teachers ask questions that are too narrow, and most students become foils and are left out. On the other hand, in the long run, students who have been left out gradually lose interest in asking questions, no longer listen to the teacher in class, and lose motivation for learning.
As for asking questions in class, I feel that we should pay attention to the following issues:
(1) When asking questions, we should pay attention to all students. The content of the questions should be designed from easy to difficult, from shallow to deep, and should be multi-layered. Different questions should be asked to students of different levels;
(2) Questions must have the value of thinking, and classroom questions must choose a Asking questions at the "optimal intelligence level" will allow most students to "jump and reach";
(3) The form and method of asking questions should be flexible and diverse. Pay attention to changing the angle of questioning, guide students through the process of trial and generalization, fully disclose their spirituality and show their personality, so that students can get the results of their own inquiry and experience the joy of success, so that "cold, wordless" mathematical knowledge Through "process" becomes "fiery thinking".