1. The meaning of open-ended questions
The following are some scholars' comments on open questions: (1) An exercise with uncertain answers or incomplete conditions is called an open question; (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 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) Problems with many different solutions or possible solutions are called open problems; (6) The so-called open question "The question doesn't have to have a solution, the answer doesn't have to be unique, and the conditions can be redundant".
Looking at the above discussion, the description of the conditions of open questions is: incomplete; Can be redundant; Extra needs to be selected, insufficient needs to be supplemented, and so on. The description of the answers (conclusions and solutions) to open questions is: not fixed; There are many kinds; Not unique; Not necessarily unique; Not sure; No need for a solution, and so on.
As can be seen from the above, although there are various descriptions of the conditions of the question, there is a consistent view on the answer: the answer is not unique. The author thinks that the "conclusion" of the (1) problem is relative to the "condition" of the problem within the problem system, and cannot be confused with the concept of the "answer" of the problem, and the "answer (solution)" of the problem is relative to the whole problem; (2) There are not too many restrictions on the conditions of questions, and there is a relaxed environment for answering questions, but the requirements are diversified and colorful. This is the meaning of openness. Therefore, the author thinks that open-ended questions can be simply described as: questions with different answers are called open-ended questions. A notable feature of open-ended questions is the diversity of answers (multi-level).
Whether a question is open or closed often depends on the knowledge level of students when asking questions. Step 2 explore the problem
In the discussion, open questions and exploratory questions are often confused, which will have an impact on the study of open questions, so it is necessary to distinguish them. Generally speaking, inquiry questions refer to questions that students need to explore, guess and prove because the conditions are complete and no conclusion is given. Of course, the intersection of open problem set and exploration problem set should be non-empty. Second, the educational value.
Some teachers have introduced open-ended questions into middle school mathematics classroom teaching and studied the educational value of open-ended questions. At present, the educational value of open questions has not attracted wide attention. We must study it carefully and publicize it actively, so as to reach a consensus.
Main value: it is beneficial to cultivate students' creative thinking ability and teach students general scientific methods.
Creative thinking is people's thinking in creative activities. Creative activities have two most remarkable characteristics, one is initiative, novelty and uniqueness, and the other is sociality, that is, the products of creative activities should have social value. Creative thinking ability refers to the ability to use known information to produce some novel, unique and socially valuable thinking products. Closely related to creative thinking is divergent thinking. Divergent thinking is a form of thinking that produces information from given information and various outputs from the same source, that is, expands from various possible directions of the problem and explores various solutions to the problem. The opposite of divergent thinking is centralized thinking, that is, thinking that combines all kinds of information to produce only one answer, which is usually applied to problems with only one correct answer. Creative thinking is not the same as divergent thinking, it is the unity of divergent thinking and concentrated thinking, but creative thinking is usually more or first manifested as divergent thinking. The process of creative thinking is often to find solutions to problems through divergent thinking and then concentrated thinking. The improvement of national quality, the cultivation of creative talents, and the exploration of knowledge innovation and technological innovation ability mainly depend on education. With the rapid development of science and technology and the arrival of the era of knowledge economy, education plays a more important strategic role in the economic competition and social development in the 2 1 century, which provides new opportunities and severe challenges for the reform and development of education. The traditional educational concept in China is to instill knowledge, not to inspire and cultivate people's learning ability, creative thinking ability and independent development ability. Students often get good grades in books and study hard, but their practical ability is poor and their innovative spirit is insufficient. We must deepen the educational reform, from the traditional education centered on imparting and inheriting existing knowledge to the modern education centered on cultivating students' innovative spirit. We should not only teach students knowledge, but also teach students to use knowledge to solve problems, explore knowledge and learn to learn on the basis of knowledge education. We must establish an innovative education system based on all-round quality education and innovative talents with innovative consciousness.
Training mechanism of core talents.
Compared with mathematics teaching in other countries, mathematics teaching in our country has the remarkable characteristics of attaching importance to the teaching of basic knowledge, the training of basic skills and the cultivation of mathematical ability, so our students have solid basic mathematical skills and a higher overall mathematical level. However, China's mathematics teaching also has its shortcomings. Two outstanding problems are that students' consciousness of applying mathematics is not strong and their creativity is weak. Students often don't ask mathematical questions, abstract practical problems into mathematical problems, apply what they have learned to practical problems, and don't know much about the actual background of what they have learned. Students have a strong mechanical imitation ability to solve some common mathematical problems, but when facing a new problem, there are not many methods, and they don't know enough about scientific thinking methods such as observation, analysis, induction, analogy, abstraction, generalization and conjecture, so it is difficult to find and solve problems. We should strive to inherit the fine tradition of mathematics teaching in China and make up for the shortcomings at the same time. First of all, we should introduce more practical problems in the teaching process, attach importance to cultivating the ability to abstract practical problems into mathematical problems, teach students the practical background of mathematical content, teach students to observe and understand problems from a mathematical point of view, and cultivate students' mathematical consciousness; Secondly, to truly establish students' dominant position in the learning process, it is necessary to provide opportunities, create conditions, let every student actively participate in the learning process, allow and encourage students to express various opinions, and cultivate students' creative spirit. We advocate the introduction of open questions in mathematics teaching to meet the requirements of the times.
Because the answer to open questions is not unique, it provides students with more possibilities to put forward novel and unique methods, which is conducive to cultivating students' extensiveness, flexibility and originality in obtaining various answers, thus cultivating students' divergent thinking; Cultivate students' profundity and rigor in the process of finding the optimal solution of multiple answers, so as to cultivate students' concentrated thinking. Cultivate students' creative thinking ability in the process of divergence and concentration.
In addition, in the process of solving open problems, we can often learn some general scientific problem-solving methods: drawing, introducing symbols, and analyzing data in tables; Classification, special case analysis and generalization; Transformation; Analogy and association; Modeling; Discuss and work separately; Prove, cite counterexamples; Simplified method (conclusion and method); Estimate and guess; Look for different solutions; Inspection; Popularization, these are common methods to solve problems, but also general scientific methods.