Conjecture verification is an important mathematical thinking method. As Friedenthal, a Dutch mathematical educator, said, "Real mathematicians often make various conjectures with mathematical intuitive thinking and then confirm them." Therefore, in primary school mathematics teaching, teachers should pay attention to the infiltration of guessing and verifying thinking methods, so as to enhance students' ability to actively explore and acquire mathematical knowledge and promote the development of students' innovative ability. So, how to infiltrate the thinking method of conjecture verification in teaching? Here are some examples:
Lesson 65438 +0: Teaching Fragment of "Calculation Formula of Rectangular Area"
1, operational awareness.
Multimedia demonstration: the plane of the rectangle is gradually changed from Figure ① to Figure ② (the width of the rectangle is constant and the length is enlarged); Figure 1 gradually becomes Figure 3 (the length of the rectangle is constant and the width is enlarged); From Figure ① to Figure ④ (the length of the rectangle is enlarged, and the width of the rectangle is also enlarged).
Figure ①
Figure ③ Figure ④
Figure ②
Students observe and think: ① What has happened to the area of the rectangle? ② From the demonstration, what do you think the area of the rectangle is related to?
Preliminary impression: the area of a rectangle is related to its length and width.
Students take out 24 pieces of square paper 1 cm prepared before class, and the teacher provides the following experimental record form (one for each):
chart
Length (cm)
Width (cm)
Area (square centimeter)
long
square
shape
Ask the students to make as many rectangles as possible with these 24 pieces of paper, and fill in the record form one by one according to the data such as length, width and area.
2. Make assumptions.
Guide the students to observe the data in the table and think independently: ① What are the lengths and widths of these figures? ② What is the area of these figures in square centimeters? ③ What do you find between the length, width and area of each figure?
Exchange discussions and form a preliminary guess: the area of a rectangle = length × width.
3. Verify the law.
Teachers timely guide: Can the area of all rectangles be calculated by "length × width"? Can you give an example to verify whether your findings are correct? If you want to know whether your conclusion is correct, what methods can you use to verify it? (Doing Math and Swing)
Displays a rectangle 5 cm long and 3 cm wide. Ask the students to guess and put it in a small square of 1 cm 2 to see what the area is and whether the results are consistent.
Students are divided into groups to give examples to verify again.
4. summary.
Students talk about the formula of rectangular area with each other, and then summarize the formula: rectangular area = length × width.
Thinking: What does "length× width" actually mean in the area formula?
Students draw a patchwork rectangular plan and hide the area unit. Imagine how many area units there are in each row of a rectangle, a * * *, and how many area units there are.
Lesson 2: Teaching fragment of "The Basic Nature of Ratio"
1, creating perception.
(1) The quotient invariance of recall division and the basic properties of fractions.
(2) Talk about the relationship between ratio, division and score.
(3) Calculate the ratios of 3∶4, 6∶8 and 9∶ 12, and get 3∶4=6∶8=9∶ 12.
By observing and analyzing the changes of the first and second items of "3∶4=6∶8=9∶ 12", what did the students find? The first and second terms of the ratio are multiplied by 2 or 3 at the same time, and the ratio remains unchanged; The first and second items of the ratio are divided by 2 or 3 at the same time, and the ratio remains unchanged.
2. Make assumptions.
Guide students to think: according to the findings just now, consider the basic nature of the score and the invariance of the divisor: what are the properties and laws between two ratios with equal ratios?
Students exchange reports and form a conjecture: the first and second items of the ratio are multiplied or divided by the same number at the same time, and the ratio remains unchanged.
3. Verify the law.
Do all the proportions have such a changing law? Can you find a way to verify it? After verification, students exchange ideas.
Student A: According to the relationship between ratio and division and fraction, I think ratio should have similar properties.
Student B: I wrote the ratio in the form of a fraction. According to the basic properties of the fraction, I found that the ratio does have this law.
Health C: I took my guess just now as an example, and then worked out the ratio of the two ratios, and found that the guess was correct.
Student D: I write the ratio in the form of division, and deduce that the ratio does have such a property according to the invariance of the division quotient.
4. summary.
Who can summarize the basic properties of ratio in one sentence? What is "the same number"? Why? Then the basic nature of the teacher-student induction ratio is that the first and last items of the ratio are multiplied or divided by the same number (except 0) at the same time, and the ratio remains unchanged. Then ask the students to illustrate this property with examples.
In the above two classes, students not only obtained mathematical conclusions through the process of perception-hypothesis-verification-induction, but also gradually learned the thinking method of obtaining mathematical conclusions-guess verification, which improved their ability to actively explore and acquire knowledge and enhanced their confidence in learning mathematics well.
First, perception-sowing the seeds of thinking methods
Perception is the beginning of children's cognition. Without correct perception, it is impossible to understand the nature and laws of things. Psychological research shows that the richer the students' perception and the clearer the representation, the more they can discover the laws of things and acquire knowledge. Therefore, in teaching, we should provide students with sufficient perceptual materials that can reveal the law, guide students to do, think, say and move, so that students can acquire rich perceptual knowledge in doing, calculating, thinking, saying and fancy, establish a clear representation, build a bridge between materialization and internalization of knowledge structure, and urge students to form a preliminary guess. For example, the following links can be designed for the teaching of "the sum of the internal angles of a triangle":
1. Students draw three different triangles at will (an acute angle, a right angle and an obtuse angle).
2. Measure the degree of each inner angle of the triangle and fill in the table.
triangle
∠ 1
∠2
∠3
Sum of degrees of three internal angles
acute triangle
right triangle
Obtuse triangle
3. Students report the sum of the angles of the triangles they draw. Write it on the blackboard 180, 179, 18 1 ... and then ask the students to guess what the sum of the three angles of the triangle is.
In this way, by drawing → measuring → filling → calculating → saying, students initially perceive the sum of the internal angles of the triangle. At this point, I guess the sum of the internal angles of the triangle is taken for granted.
Second, the hypothesis-spread the wings of thinking methods
Assumption is an initial and unconfirmed judgment of what students perceive, and it is an important link in the process of students acquiring mathematical knowledge. Paulia once said, "Once a child expresses some conjectures, he associates himself with the problem, and they will be anxious to know whether his conjectures are correct.". So I am actively concerned about this issue and care about the progress in the classroom. " Therefore, teachers should give students enough time and space to observe, think, analyze and reason freely with their own way of thinking according to their own perception, gradually rise from perceptual to rational, and then communicate and discuss with each other to form reasonable assumptions. For example, when teaching "Finite Fractions", first provide a set of fractions:,,,, and let students think while calculating, and then guess: Can a fraction be converted into a finite fraction, and which part of this fraction is related to it? What kind of relationship may there be? In this way, after some right or wrong guesses, students form a * * * knowledge: if the denominator only contains prime factors 2 and 5, a fraction can be reduced to a finite decimal. But this * * * knowledge is only a hypothesis and cannot be used as a final conclusion. Further verification is needed to test whether the hypothesis is universal.
Third, verification-a way to grasp the direction of thinking
Mathematics in primary schools generally does not need strict argumentation. Therefore, regarding whether students' assumptions are universal, we can start with students' life experience and thinking level, and provide enough exploration time and space for students to carry out independent and cooperative exploration activities, experience their own process of trying, exploring and verifying, and gain the ability to verify what they have learned. For example, in the teaching of "the sum of the internal angles of a triangle", after students put forward a preliminary guess, they are guided to explore and verify in operation:
1, folding: according to the experiment in the book, three different triangles are folded respectively, and the sum of the inner angles of the triangles is 180.
2. Spelling: Cut off the three angles of each triangle and put them together to form a right angle, and the sum of the inner angles of the triangle is 180.
3. Calculation: Divide the square paper into two identical triangles diagonally, and calculate the sum of the internal angles of a triangle as 180 from the fact that the four angles of the square are 90× 4 = 360.
It is worth noting that when students' guesses are wrong, teachers should not immediately deny or remind them. But to guide students to give examples to verify. When necessary, teachers can cite counterexamples, so that students can find their own guesses in the verification, adjust their thinking direction, and put forward hypotheses again.
Fourthly, induction-harvesting the fruits of thinking methods.
After verification, teachers should seize the opportunity to guide students to talk, discuss and communicate with each other and reach a * * * understanding. On this basis, let students reason and sum up their knowledge accurately. In induction, students should be guided to understand the universality of the conclusion and every sentence in the conclusion. For example, when summing up "the basic nature of ratio", let students think and discuss: Is "the same number" any number ok? Why? After students accurately summarize the basic nature of ratio, let students illustrate this nature with examples, and then guide students to apply this nature. This not only deepens students' understanding of knowledge, further consolidates and grasps knowledge, but also cultivates students' ability to solve practical problems.
Newton said: "Without bold speculation, there can be no great discovery." Bruner also believes: "The process of experiencing and discovering learning materials is the most valuable thing for learners in some problem situations." Practice has proved that attaching importance to the infiltration of guessing and verifying thinking methods in teaching will help students quickly discover the laws of things and obtain clues and methods to explore knowledge, which will undoubtedly give students great psychological satisfaction and pleasure, enhance their confidence in learning mathematics well, and stimulate their initiative and participation in learning, so as to better develop their creative thinking and improve their autonomous learning ability and their ability to analyze and solve problems.
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