At present, there is no unified understanding of what is an open question in mathematics, mainly as follows:

(1) exercises with uncertain answers or incomplete conditions are called open questions; (2) Open questions are questions that need to be selected when the conditions are redundant, supplemented when the conditions are insufficient, or the answers are not fixed; (3) Questions with multiple correct answers are open questions. This kind of question gives students the opportunity to answer questions in their favorite way. In the process of solving problems, students can combine their knowledge and skills in various ways to discover new ways of thinking. (4) The only question with different answers is an open question; (5) A question has many different solutions or many possible answers, which is called an open question; (6) The problem does not need to have a solution, the answer is not unique, and the conditions can be redundant.

Open math problems, in popular terms, are topics that give students more cognitive space.

Whether a question is open or closed often depends on the knowledge level of students when asking questions. For example, the question of how many times N people shake hands is an open question before students learn combinatorial knowledge, and a closed question after learning combinatorial knowledge.

Therefore, the types of open-ended questions include the following:

1, conditionally open

For example, as shown in figure 1, to get AD∨BC, you only need to meet the conditions (only fill in one). (Dongguan 02-03 school year, the first semester ended, the first math exam)

For another example, as shown in Figure 2, AB=DB, ∠ 1=∠2. Please add an appropriate condition to make △ ABC△ DBE, then the condition to be added is-.

2. The conclusion is open.

For example, the teacher gives the condition that two straight lines are parallel, and students A, B and C each point out a feature of this condition:

A: cut with the third straight line at the same angle;

B: Intercepted by the third straight line, with equal internal angles;

C: cut by the third straight line, which is complementary to the inner corner of the side.

3. The strategy is open.

For example, in △ ABC (ab > Take a point D) on the AB side of AC and a point E on the AC side, so that AD=AE, and the extension line of the straight line de and BC intersect at point P. Prove:

For another example, please use the method you think is simpler. Students may have the following methods. Method 1: divide directly, add and subtract.

Method 2: The original formula =.

Method 3: The original formula =.

Method 1 is a conventional method; The second method embodies a reduction idea, but it is not simple; Method 3 is converted into the sum of some opposite numbers to calculate, which is obviously novel and simple.

4. Open design

For example, (Course standard: Exercise 5, 13, Mathematics, Grade 7 (Volume I)) A community is greening, and a flower bed should be built on a rectangular open space. Now the design scheme is collected, and the design pattern is required to be composed of circles and squares (the number of circles and squares is not limited), and the flower bed area accounts for about half of the rectangular area. Please draw your design plan and express your design idea in one or two sentences.

For example, open

For example, name a possible event in your life. (Dongguan 02-03 school year, the first semester ended, the first math exam)

For another example, according to your life experience, please give the actual background explanation of algebraic expression 2a. (Dongguan 03-04 school year, the first semester ended, the first math exam)

6. Open to the outside world

For example, (Question 1 18, Grade 7, Mathematics, East China Normal University Edition) has three ordinary cube dice. Throw these three dice. Please name three definite events and three uncertain events.

7. Information disclosure

For example, in a class in Grade One, three students are arguing about who has the best math score. Their five math scores are shown in Table 1, and their average, median and mode are shown in Table 2.

Now these three students all say that they have the best math scores. (1) Please guess and write your own reasons; All three seem to make sense. what do you think of it ? Please use statistical knowledge to make a correct analysis.

8. The solution is open (the following example is omitted)

9. Full opening up

10, scene opening, etc.

The conditions and problems of this open-ended question type are changeable, some conditions are hidden, some conditions are redundant, some conclusions are diverse, and some solutions are rich. Mathematical open questions have the following remarkable characteristics different from mathematical closed questions:

First of all, the mathematics open-ended problem is novel in content, complex in conditions, uncertain in conclusion, flexible in solution, and there is no ready-made model to apply. Topics are extensive and close to students' real life, which is not as simple as closed questions, but depends on memory and patterns.

Secondly, the forms of mathematics open questions are diverse and vivid, some have various backtracking conditions, some seek various conclusions, some seek various answers, and some change, which can reflect the atmosphere of modern mathematics, unlike the single presentation and rigid narration of closed questions.

Third, the solution of open mathematics problems is divergent. Because the answer to the open mathematics problem is not unique, it is necessary to use a variety of thinking methods when solving mathematics problems, and explore a variety of solutions through multi-angle observation, imagination, analysis, synthesis, analogy, induction and generalization.

Fourthly, the educational function of mathematics open questions is innovative, precisely because of its advanced and efficient educational function, which meets the requirements of talent competition in various countries.

Second, the role of mathematics open questions

The core of quality education is to cultivate innovative spirit and creativity. The opening of mathematics entitled "Students' Creative Learning" provides a relaxed and free environment, and its role is reflected in the following aspects:

1, the educational function of mathematics open questions to students;

(1) is beneficial to the cultivation of students' thinking. Students must break the original mode of thinking, expand association and imagination, and think from multiple angles, directions and levels. Their differences in thinking direction and way are conducive to the formation of creative ability. Open-ended questions can be transformed into the explanations of a single teacher for teachers and students to study together, and individual operations can be transformed into collective exchanges and cooperation. Integrating open questions into the classroom can effectively stimulate students to think about problems and actively participate in the process of knowledge construction, thus cultivating students' good mathematical qualities such as flexibility and creativity.

(2) It is beneficial to stimulate learning interest. Mathematics open-ended problems can realize the opening of teaching forms, so that students' learning can be individual competition, cooperative completion, free speech and practical operation. Students study easily and happily in a relaxed and happy teaching atmosphere, which is conducive to stimulating students' curiosity and competitiveness, enhancing the internal driving force of learning and generating a strong interest in mathematical exploration.

(3) It is beneficial to enhance students' innovative consciousness. The answer to traditional closed-ended questions is unique, and students often find an answer without further thinking. In the process of solving open problems, there is no fixed and ready-made model to follow, and students can't find the answer by rote learning and mechanical imitation. Students must fully mobilize their knowledge reserves, actively carry out intellectual activities, and think and explore with various thinking methods. Therefore, open questions can cultivate students' enterprising spirit, strengthen students' innovative consciousness, and are effective tools to improve students' innovative ability.

2. The change of mathematics open questions to teachers;

(1) The change of teachers' ideas caused by open questions. On the one hand, the emergence of open problems and the affirmation of their educational functions reflect the changes in people's ideas of mathematics education; On the other hand, it adapts to the needs of the rapid development era. In fact, it reflects people's pursuit of a new model of mathematics teaching and is a new exploration of mathematics education reform from a new era and historical height. The content of the concept change is:

First of all, the Department of Basic Education of China's Ministry of Education clearly pointed out: "Curriculum is a historical category, and curriculum objectives, curriculum structure and curriculum content will change with the development of the times." Teaching materials should be scientific, basic and open.

Secondly, the concept of mathematics in open classroom teaching is an understanding of the essence of mathematics, and teachers' concept of mathematics directly affects their teaching concept. If a teacher can understand mathematics from a dynamic and comprehensive point of view, then his teaching method will be heuristic, and his teaching philosophy is student-centered.

(2) The change of teachers' role caused by open questions. After the introduction of open-ended questions into the classroom, the role of teachers is defined, that is, in the teaching process, teachers are not the protagonists of teaching activities, but "screenwriters" and "directors"; It is not the imparting of knowledge, but the designer, promoter, demonstrator, organizer and regulator of teaching content and teaching activities.

At the same time, teachers should pay attention to the strategy of "letting go" when opening questions. They should not only boldly "let go" and leave time for students to explore comprehensive and correct conclusions, but also be good at grasping the overall situation and adjusting the degree of "letting go". Teachers will never replace any questions that students can ask; Teachers never hint at questions that students can think about; Teachers will never intervene in the problems that students can solve, and truly "let go" in time to improve the overall efficiency of "let go".