As we know, mathematical conjecture refers to a specious judgment of unknown quantity and its relationship according to known conditions and basic knowledge of mathematics, which undoubtedly plays an important role in promoting the development of mathematics, the cultivation of exploratory thinking ability and the formation of personality quality. Newton once said: there is no great discovery without bold conjecture, and many of Einstein's inventions and theories are also produced by some conjectures. Judging from the process of students' mathematics learning, guessing is a good preparation for students' effective learning, including students' knowledge preparation for new learning or practice, positive motivation and good feelings. In mathematics learning, conjecture is a means to verify whether conjecture is correct. It is the core of the new round of curriculum reform to let students actively participate in the learning process, actively acquire knowledge and cultivate students' innovative consciousness and practical ability. Dare and be good at guessing is the premise of innovation. In primary school mathematics classroom teaching, it is an effective way to cultivate students' innovative consciousness by encouraging students to imagine and question boldly and cultivating students' reasonable guess. Guess can play a unique role in primary school mathematics classroom teaching, which can shorten the time for students to solve problems, give students the opportunity to discover mathematics, exercise their mathematical thinking and stimulate their interest in learning mathematics.
Second, the application of conjecture in primary school mathematics classroom teaching
(A) create conditions for students to have the opportunity to guess.
At the beginning of class, teachers can create some contradictory situations or design some games according to the connection between old and new knowledge, so that students can guess the content of new knowledge and feel that new knowledge is not only a cognitive requirement, but also an emotional requirement. For example, when teaching the "Preliminary Understanding of Multiplication", first swing it with a stick, and you will get 3+3+3 = 9, 2+2+2+2+2+2+2. 3+ 3+ 3+ 3= 12, 5+ 5+ 5= 15 and then compare each addend of these four formulas. These four formulas can be divided into several categories. What categories? (Add the same addend) Then affirm the classification of students and say, "These formulas are all questions about finding the sum of several same addends. Now just add several identical addends to a single digit, such as eight 9s and six 7s.
Add them up, and the teacher can quote the figures in one bite. Can you believe it? Who will test the teacher? "As soon as the students heard that they were going to test the teacher, they came up with some difficult problems to beat the teacher, but the teacher could quickly answer them. At this time, the teacher seized the opportunity and mobilized the students' enthusiasm for learning. "Your questions are the sum of several identical addends, and the teacher will make them soon. Guess what you will learn in this class today? " Students guess that they may have learned the correct and fast algorithm for finding the sum of several identical addends, and then new knowledge will naturally appear. The concept of "multiplication" is abstract, but the teacher's design makes the classroom atmosphere very active. Teachers always test students, but today students test teachers. The distance between teachers and students has suddenly become smaller, which not only has a democratic learning atmosphere, but also makes students have a strong psychological demand for learning new knowledge, eager to know the mystery, and makes good knowledge preparation and psychological preparation for the teaching of new knowledge.
Students are by no means a blank sheet of paper. The teacher taught them everything. They can accurately infer new knowledge on the basis of certain knowledge. At this time, if students lose no time to guess and think, there will be unexpected gains. When teaching "area and area unit", when students master the two area units of "square meter and square decimeter", I ask students to guess "what is the area unit larger than square decimeter?" What is the area unit of 1 square meter? "Without the teacher's teaching, the students themselves mastered the square meter area unit through knowledge transfer.
In normal teaching, teachers should design more questions with multiple answers and problem-solving strategies, encourage students to guess boldly from many aspects and angles, stimulate students' innovative consciousness, and create opportunities for students to guess. This is very valuable, because the solution of problems often appears in the form of assumptions first, and innovation is possible only with certain assumptions.
(A reasonable guidance, so that students are good at guessing.
Everyone has the potential to guess. When a person's thinking is activated, he will be excited and eager to know the answer to a question, and he will often guess first to meet his demand for knowledge. As a teacher, we should skillfully conceive, carefully ask questions, create problem situations, mobilize students' full enthusiasm and positive thinking, and reasonably guide them so that they have the desire to guess and acquire knowledge actively and creatively. However, a reasonable guess comes from a certain imagination, which is inspired by all kinds of knowledge. If students want to learn to guess and be good at guessing, they must be guided reasonably to dabble in various fields of knowledge and help them form a good knowledge structure with the help of life experience, because every guess of students is an expansion of their life experience and existing knowledge.
In the teaching of "Possibility", because students have some life experience, the activities of touching the ball in groups are specially designed. First, each student in each group randomly draws a ball from the schoolbag, then puts it back in the schoolbag, stirs it, touches it again, and then guesses according to the result of touching the ball: What color balls may be put in these schoolbags and why? According to their own life experience, the students quickly got the result of guessing. One group of students touched the red ball and the yellow ball in the bag at the same time, so the students guessed that there might be a red ball or a yellow ball in the bag. Another group of students found that all the red balls in the bag were red, so they guessed that the bag might be full of red balls. Then the teacher asked, "Is there a yellow ball in this bag?" Why? "The students have a heated discussion. Students actively participate in the formation of "possibility" knowledge by touching the ball. The knowledge gained in this way is effective and valuable.
Whether the conjecture is reasonable or not indicates the level of one's guessing ability. In teaching, we should not only help students to communicate the relationship between knowledge and build a knowledge network, but also consciously infiltrate some mathematical thinking methods, so that students can understand and use them flexibly and guide them to sum up thinking methods constantly, thus enriching their thinking experience. In addition, we should design certain mathematical situations or activities to guide students to make full use of their own life experience and existing knowledge and experience to make themselves good at guessing.
(2) Verify the conjecture and let students experience the joy of success.
Students think positively and guess boldly in class, and their sense of innovation is stimulated. But if you want to know whether the guess is valuable, reasonable and correct, teachers must also guide students to verify it carefully so that students can experience the joy of success. This is an indispensable process. Because the learning of knowledge can not be limited to the acquisition of conclusions, students should not only know why, but also know why, and practice makes true knowledge. If they pass the verification, if you find this conjecture wrong, you should immediately adjust your thinking and re-analyze. Only by guiding students to combine conjecture and verification organically can conjecture be meaningful. If students are only allowed to guess, their understanding can only be ignorant in the end, or they have a little knowledge of whether the students' guesses are correct, teachers will not answer, guide students to participate in the process of knowledge formation, and let students explore and verify themselves. At this time, it is best to give students enough time to ask them questions. Choose materials to do experiments according to your own ideas, let students do it boldly, and encourage students to write down what they see. Only by asking questions, participating and suggesting, can teachers guide students to the principle of concepts step by step, and consciously observe and record students' performances in experiments, materials, methods, language expressions, conclusions and discoveries, so as to make targeted induction and summary.
For example, when teaching "the characteristics of numbers divisible by 3", the teacher asked: "We already know the characteristics of numbers divisible by 5, so what characteristics might a number divisible by 3 have?" Some students immediately said his guess without thinking: "The numbers of 3, 6 and 9 can be divisible by 3." The teacher did not evaluate his guess, but led everyone to verify it quickly. Some students suggested that "19,29 cannot be divisible by 3." This guess is obviously wrong. After the failure of guessing, the students realized that they could not guess according to the original experience. Ten digits can still be divisible by 3 after unit switching, such as12,21,15,51. The teacher immediately showed a set of numbers: 345, 354, 435, 453, 534, 543. After calculation, the students found that they can all be divisible by 3. Therefore, the students participated in the verification of this conjecture ... In the process of this conjecture-verification-re-guess-re-verification, the students' thinking gradually improved from one-sidedness.
Students can best develop their creativity and exert their potential in the process of discovering problems, guessing, trying and finally seeking methods. It is precisely because of twists and turns that the final conclusion is that the all-round development of precious students is the ultimate goal of our curriculum reform. The process of giving students a chance to guess, experience guessing, verify and succeed in primary school mathematics classroom teaching is full of "learning, learning and living"
Third, the problems that should be paid attention to in the application of conjecture in primary school mathematics classroom teaching
The cultivation of innovative consciousness is not a problem that can be solved in one class or two classes, and it must go through long-term training to reach the other side of victory. In primary school mathematics classroom teaching, students can learn creatively only if teachers' teaching methods directly affect students' learning methods. It is necessary to push students to the main body, attract students with the charm of knowledge, and give students the opportunity to guess, dare to guess and be good at guessing.
There are also some specific requirements for using guessing teaching method in primary school mathematics classroom teaching, that is, teachers should pay attention to creating a democratic, harmonious and equal learning atmosphere for students. Educator Rogers pointed out: "The general conditions conducive to creative activities are psychological security and psychological freedom." Psychological research also shows that good emotions can inspire students' spirits, while bad emotions will inhibit students' intellectual activities, because primary school students' guesses are blind in most cases, and they often have some strange and unusual things. In that case, students' whimsy will be stifled by scruples. When students have guesses, we should not accuse them of "guessing" or "talking nonsense" because they don't know the truth clearly, but should fully praise and encourage them and help them think patiently. Over time, students will have no worries when they encounter new problems, and dare to guess and think. A democratic and relaxed teaching environment is for students. Teachers should dare to let students fully discuss. In fact, innovative ideas often lie in students' views. When students discuss enthusiastically, divergent thinking is often the most active time. Students' thinking sparks will begin to bloom, various conjectures will arise, and then innovative ideas will emerge. Teachers can only create democratic leisure for students! The harmonious learning atmosphere makes them relax physically and mentally, which leads to active thinking and novel guesses. The leading role of teachers is to help students build up the courage to dare to guess, master the methods of being good at guessing, cultivate the scientific spirit of seeking truth from facts, and let students know that any guess is reasonable and correct only after being verified as a teacher. If we can establish this brand-new concept and use this advanced consciousness to control our classroom, we can cultivate students' innovative consciousness and practical ability in primary school mathematics classroom teaching.
Mathematics curriculum standards point out that students' mathematics learning should be realistic, meaningful and challenging. These contents should be conducive to students' active observation, experimental speculation, verification, reasoning and communication. Efforts should be made to gradually cultivate students to dare to guess and be good at guessing in mathematical activities, and to obtain mathematical guesses through observation, experiment, induction and analogy, so as to finally realize the formation process from focusing on results to focusing on results and knowledge, from focusing on knowledge accumulation teaching to developing creative teaching.