Keywords: open questions of divergent thinking in middle school mathematics
Open-ended problem is a new type of problem in mathematics learning, which is relative to the traditional closed-ended problem. The core of open question is to cultivate students' ability of logical thinking, reasoning and continuous exploration, and to stimulate students' consciousness of independent thinking and innovation.
First, the concept and characteristics of mathematical open questions
"Open math problem" refers to a kind of math problem with uncertain answers or incomplete conditions, which can stimulate students' divergent thinking.
Mathematics open questions generally have the following characteristics:
1. It is often associated with practical problems. When answering, students are required to use mathematical language to make it mathematical, that is, to establish a mathematical model.
2. There is no ready-made problem-solving model, and some answers may be easy to find by intuition, but in the process of solving, it often needs multi-angle thinking and exploration.
3. In the process of solving, new problems can often be introduced, or problems can be summarized to find more general and general conclusions.
4. The answers to some questions are uncertain, but what matters is not the diversity of the answers themselves, but the reconstruction of the cognitive structure of the subject in the process of answering questions.
Second, the classification of mathematics open questions
(A) according to the classification of unknown elements in mathematical propositions
Mathematical proposition can be generally divided into three parts according to the form of thinking: hypothesis-reasoning-judgment.
1. If the unknown element is a hypothesis, it is a conditional open problem.
Example 1. Proposition A: A triangular pyramid whose bottom is a regular triangle and the projection of the vertex on the bottom is the center of the bottom is a regular triangular pyramid. The equivalent proposition of proposition A is that the base is a regular triangle and the triangular pyramid is a regular triangular pyramid.
Analysis: This is a conditional open question that needs to be completed. According to the conclusions and requirements given in the question, think from different angles, and finally complete the conditions and get the answer.
2. If the unknown factor is reasoning, it is an open question of strategy.
Example 2. What other methods can be used to compare the sizes of 47 and 5 1 1?
Analysis: the topic gives the conditions, but the strategy of how to infer the conclusion is unknown.
3. If the unknown factor is judgment, it is a question to be concluded.
Example 3. The square of a number can be expressed as the product of four consecutive odd numbers. Find all numbers with this property.
Analysis: this question gives certain conditions, and the conclusions that meet the conditions can be varied. Only by careful analysis, comprehensive thinking and flexible use of the relationship between quantitative operations can we get the answer.
4. Some questions only give a certain situation, and their conditions, problem-solving strategies and conclusions need to be set and found in the situation. This kind of problem can be called comprehensive open-ended problem.
Example 4. On a rectangular wasteland 50 meters long and 30 meters wide, if you want to open up a part as a flower bed, if the area of the flower bed is half of the rectangular area, please give your design.
Analysis: rectangular flower beds are required in the question, and you should seek your own assumptions according to the conditions. Therefore, you can use your imagination and creativity to design and give yourself creativity.
(2) According to the structural type of the answer to the question.
1. Finite exhaustible type, that is, the answers to the questions can be enumerated one by one.
Example 1. Please design three different division methods to divide the right triangle into four small triangles, so that each small triangle is similar to the original right triangle.
Analysis: This question has a low starting point and many solving strategies, with 10 answers.
2. Finite chaotic type. The answer to the question is definitely limited in theory, but either it is difficult to exhaust the answers one by one because of the existing level of understanding, or people think that the exhausted work is meaningless, and the answer structure is temporarily chaotic (for example).
3. Infinitely discrete type. The answer to this kind of question is usually to classify the answers properly and list the typical solutions of each type of answer.
Example 2. Two students, A and B, play the game of "throwing the ball into the basket". It is agreed that each student will play five games, and each game will throw a ball into the basket outside the designated line. If he fails to score once, he can throw it again and so on. But you can only vote six times at most. When you throw, the game is over and the number of pitches is recorded. When six shots are missed, the game is over and marked with "*". The situation of two people pitching in five games is as follows:
(1) In order to calculate the score, both parties agree that the game marked with "*" will score 0. The calculation method of other game scores must meet two conditions: ① The more pitches, the lower the score. ② The score is positive. Please choose a specific scheme, and convert the pitching times n of other innings into scores m through formulas, tables and language descriptions. According to the agreed requirements. (2) According to the above agreement and the plan you wrote, calculate the scores of each game between Party A and Party B, fill in the form, and judge who voted well from the perspective of average score.
Analysis: The answer to this question is infinite in theory, but there are not many meaningful answers. This problem is to let students experience the relativity of statistical data: the victory or defeat of Party A and Party B depends not only on their actual performance, but also on the standard of scoring. Different data processing methods will lead to different evaluation results.
4. Infinitely continuous. The answers to the questions are distributed in some real numbers, or some geometric figures that can change continuously. The mathematical method to describe this change is usually to introduce parameter representation.
Example 3. Please simplify x first? 3-x？ 2x？ 2-x- 1-x？ 2x+ 1, and then select a number that makes the original formula meaningful and substitute it for evaluation.
Analysis: This is an open question to examine the basic knowledge. The knowledge points to be examined are: simplification of algebraic expressions and conditions for meaningful algebraic expressions. After simplification, as long as the substitution number is not 0,-1 and 1.
(3) Classification according to the operation mode of the target.
1. Legal exploration type. This is a question of finding the law. Explore various conclusions under established conditions or relationships.
Example 1. Calculate (1+13) (1+18) (1+15) ... (1+65438)
Analysis: Observing the topic, we can see that the formula is the product of some 1 plus a unit fraction, and the denominator of the unit fraction is 3,8, 15...99, that is,1× 3 = 3,2× 4 = 8,3× 5 = 65438+.
2. Quantitative design. It is a kind of question type that turns general problems into common problems in mathematics application.
Example 2. Same as in Example 3.
Analysis: This question is not only a comprehensive open question, but also a quantitative design question. It is a quantitative design problem involving graphic design and quantitative calculation.
3. Classified discussion types.
Example 3. A headmaster led three outstanding students from the school to spend their summer holidays in Beijing. A travel agency said: If the principal buys a full ticket, the rest of the students can enjoy a half-price discount. Travel agency b said: all air tickets, including the principal, are 60% off. If all the tickets are in 240 yuan. ① Suppose the number of students is X, and a travel agency charges Y? B travel agency charges Y B? , respectively, calculate the cost of travel agencies. (2) When the number of students is what, the fees charged by the two schools are the same? ③ Discuss which travel agency is more favorable in terms of the number of students X. ..
Analysis: Classification is a basic mathematical method. This question is separated by two travel agencies, A and B. According to the meaning of the question, discuss the relationship between them so as to get what you want.
4. Mathematical modeling. Mathematical modeling is an important part of mathematical learning to cultivate students' awareness of mathematical application.
Example 4. A factory and a raw material 360? , b raw material 290? . It is planned to produce 50 products A and B from these two raw materials. As we all know, it takes 9? , b raw material 3? , you can profit from 700 yuan; To produce type b products, type a raw materials are needed. 10, the second raw material? , you can make a profit 1200 yuan. What is the plan to arrange the number of production pieces of products A and B as needed? Please design it.
Analysis: This problem integrates operation, scheme design and mathematical modeling. There is more than one plan, but you should choose the best one.
Three. Concluding remarks
In a word, the discussion on the classification of open questions in mathematics is helpful for us to deeply understand the concept of open questions and grasp the openness of questions. At the same time, it also helps students to grasp whether the open questions of mathematics are suitable for classroom teaching, help students change the way of setting open questions and help classroom learning. The embodiment number of mathematical open questions
The thinking method of scientific research embodies the formation process of mathematical problems. It provides students with time and space to explore and know themselves accurately.
References:
[1] Zhang, et al. Research on solving mathematics problems in middle schools. Changchun: Northeast Normal University Press.
[2] Yu Qiushi. Open problems in middle school mathematics textbooks. Middle school mathematics teaching reference.
[3] Cao Mubin. Research on mathematics open problems and their solutions. Mathematics teaching in middle schools.
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