Mathematical reasoning is a process of induction, analogy, judgment and proof of things from the perspective of numbers and shapes. It is an important way of mathematical discovery and an effective tool to help students understand the abstraction of mathematics. "Mathematics Curriculum Standard" points out that students should develop reasonable reasoning ability and preliminary deductive reasoning ability through mathematics learning in compulsory education stage and experiencing mathematical activities such as observation, experiment, guess and proof.

One,

Put forward a conjecture with the help of observation and experiment

Observation can activate students' thinking, and experiment in observation can improve students' hands-on ability, so observation and experiment are important means of mathematical discovery. In teaching, students can be organized to carry out experimental activities such as cutting, testing and doing, so that students can discover its changing law through observation and put forward reasonable guesses. For example, when calculating the circumference of a teaching circle, ask students to draw three different circles with three line segments with different lengths as the diameters, cut them out and roll them for one week at the same time, and get that the lengths of the three line segments are the circumferences of three circles respectively. Ask the students to explore whether the diameter of a circle is related to its circumference. Students found that the smaller the diameter of a circle, the shorter the circumference, the larger the diameter and the longer the circumference. The students concluded that the circumference is related to the diameter. Then organize students to measure the diameter of each circle again, calculate the quotient obtained by dividing the circumference of the circle by the diameter, keep the number to two decimal places, and fill in the corresponding data in the table. By showing the data, the students found the relationship between diameter and circumference, and put forward the conjecture that the circumference of a circle is more than three times the diameter.

Second,

Make a guess by induction.

Mathematics is highly abstract, and abstraction lies in concreteness. In primary school mathematics teaching, many concepts and laws are obtained through inductive reasoning. Many times, incomplete induction is used, and the conclusion drawn by endless induction is not necessarily correct, but it can be guessed and verified by induction. For example, to explore the unchangeable nature of teaching quotient, the teacher first writes a formula 12÷6=2.

Then let the students write some division formulas, and the result is 2. Then, guide students to observe these formulas, and summarize and find the rules. At this time, students may put forward many conjectures: dividend and divisor are divided by the same number at the same time (except 0), and the quotient remains unchanged; Divider and divisor are multiplied by the same number at the same time, and the quotient remains unchanged; Dividend and divisor expand or shrink by the same factor at the same time, and the quotient remains unchanged. On the basis of putting forward conjectures, further guide students to verify and improve.

Third, analogy guessing uses analogy to guess, that is, by comparing the similarities of some aspects of the research object or problem, using analogy to guess or infer. Students have mastered the research method of using analogy to make conjectures, and can draw inferences from others in the class. For example, according to the relationship between division and fraction (both of which have the same division property), the dividend and divisor of division can be expanded or reduced by the same multiple (except 0) at the same time, and the quotient remains unchanged. By analogy, the numerator and denominator of the guess score are multiplied or divided by the same number (except 0), and the size of the score remains unchanged, and the basic properties of the score are obtained. When learning the nature of comparison in the future, students can also use analogy to deepen their memory of comparative knowledge. This lays a good foundation for students to learn the mutual transformation of division, fraction and ratio in the future.

Fourth, example verification Primary school students generally use example verification because of age, knowledge and other restrictions. Example verification is mainly carried out by way of examples. For positive examples, it is more reliable to use incomplete induction to verify the conjecture or to use the original conclusion. You can also cite counterexamples to overturn the original conclusion or conjecture. For example, in the teaching of the sum of triangle internal angles, students can get the sum of triangle internal angles as 180 degrees from textbooks, so that they can operate by themselves and further verify the correctness of the conclusion by two methods. Some students tore off all three corners of a prepared triangle and put them together to form a straight angle. Because a flat angle is 180 degrees, the conjecture that the sum of the internal angles of a triangle is 180 degrees is verified. Some students use a protractor to measure the degree of each angle, and then add up the degrees of the three angles. By measuring several triangles with different sizes and shapes, it is repeatedly verified that the internal angle of the triangle is 180 degrees. In this way, students can verify the accuracy of conjecture in practice and deepen their understanding of knowledge.

There is another very important way in case verification, which is a very important research method. Any conclusion or proposition can be overturned by quoting counterexamples. With the deepening of students' study, students are constantly overthrowing previous conclusions. For example, after students learn the multiplication of integers and decimals, they can know that the product of the two numbers they have learned before must be greater than any one of the factors, as long as a counter example of 4× is given.

0.5=2

, can prove that this conclusion is not established.

Fifth, deductive reasoning.

With the growth of grade, students should combine classroom learning with some effective deductive reasoning methods.

For example, when the fraction becomes a finite decimal,,,,

Ask the students to convert these scores into decimals respectively. Students find that the first three fractions can be converted into finite decimals, and the last two decimals cannot be converted into finite decimals. Guide students to analyze: 25 = 5 ×

5,20=2×2×

5,8=2×2×

2,35=5×

7,63=7×3×3

It was also found that,

25,20,8

The factors of these three denominators all contain only 2 and 5, while 35 and 63 contain prime factors other than 2 and 5. Denominators only contain fractions with prime factors 2 and 5. According to the basic properties of fractions, they can be converted into denominators of 10, 100 and 1000.

Fractions can also be converted into finite decimals; Fractions whose denominator contains prime factors other than 2 and 5 cannot be converted to the denominator of.

10, 100, 1000

The fraction of cannot be reduced to a finite decimal. In this way, on the basis of inductive conjecture, further argumentation and explanation are carried out, and finally a conclusion is drawn.

In a word, it is a long-term task to cultivate students' logical thinking ability in the teaching of mathematics department in primary schools.