Integer: 1, 2,3 decimal 2. 1, 0.8 score: 1/3,5/6 positive number1,2,3 negative number-1,-2,3.
Please give an example and briefly describe the process of "mathematization".
First, the meaning of "mathematicization"
How do children construct their own understanding of mathematical concepts? How does the understanding difference between children come about? How to promote the development of students' early mathematical thinking and mathematical ability? In order to solve these problems, we need to introduce a new concept-"mathematicization".
Mathematicization is a concept put forward by western scholars in recent years. Specifically, in the process of mathematics teaching, teachers and students work together and interact with each other, so that children can accurately understand the rules and norms needed for mathematical expression or operation, and finally form their own mathematical models about various objects and situations. Mathematicization is very important for the development of students' mathematical thinking and the formation of problem-solving ability. ...
Second, the process of "mathematization"
To study the mathematicization of children, we should go back to the first few years of children's schooling. Since math problems appeared in the classroom, children began the process of mathematization. Next, we analyze children's mathematicization from three aspects: teacher's explanation, student's representation and early formula.
1. Teacher's explanation
There are four birds standing on the wire on the blackboard, and there are three birds beside them. The teacher's task is to make children understand such a picture as a "formula" of 7-3=4. This is the most elementary mathematical problem. In teaching, teachers often decompose these complex relationships into a series of procedures or smaller steps, and spend a lot of time explaining, guiding and correcting them until most students in the class understand these relationships. The following teaching video clips typically reflect this process.
Teacher: What does the number "4" in this formula mean? (Finger number "4")
Health: Is it because there are four birds? (raised his voice and asked)
There are many birds (more than four) here, but what's so special about these four birds?
Student: (Several students discuss at once) They are standing. They come first. They are smaller. They fell asleep.
Teacher: So, what does the number "3" in the formula mean? (pointing to "3")
Student: (Several students) Three birds are flying. They just got here. They went home. No, they flew away!
Teacher: OK, they flew away. So, why do we have to write a negative sign here? (pointing to minus sign)
Health: Because they flew away.
Teacher: So what does the equal sign mean ...?
Student: (Several students say together) How many are left? As a result ... how much is left?
Teacher: Yes. There are always four birds left.
Obviously, through such repeated questions and answers, students can be promoted to connect graphic expression with numerical expression, form the ability to explain rules, and complete the most preliminary mathematization of some students' thinking. To this end, textbook writers have also made a lot of efforts. They usually organize pictures in a concise and easy-to-read way, arrange the objects to be counted, avoid interference items, distinguish them with the simplest and most conspicuous features, and group the contents of the same task.
2. Student representatives
In fact, repeated questions and answers between teachers and students can only help students who are close to the teacher's way of thinking to complete mathematics better. The most fundamental reason is that students have different representations of pictures. The research shows that it is an explanatory reasoning process for students to realize the transformation from pictures to formulas. In the process of learning, students often infer the mathematical relationship in the picture, and then compare it with the judgments of teachers and other students, and finally form their own mathematical model. For example, "1+ 1=2" means that a bird stays and a bird comes. "4-3= 1" means that there are three fewer birds than those that fly away. "7×2= 14" means that seven birds have 14 legs. Birds flying away (leaving a group) can form a reduced pattern, and these birds join another group to form an increased pattern.
Once children have completed the transformation between pictures and mathematical symbols, which is consistent with the "recognized" relationship, they have completed the mathematicization in this respect. But this process is slow and complicated. For different students, there is no one-to-one turning point between realistic pictures and mathematical expression symbols, and the same picture can be understood by different students into many different interpretations. It is worth noting that some teachers regard these early appearances and pictures as self-evident and guide their students with many arbitrary instructions-"just look here" and "you just look here". This vague explanation is beyond the scope of students' acceptance and becomes an obstacle or interference to understanding the original intention. Sometimes, although it is unintentional, it inhibits the formation of students' mathematical concepts and critical reasoning ability, hinders the development of students' flexible meaning attribution ability, and deprives students of the fun of using it. What students finally form is not mathematical ability and thinking, but some mechanical agreements and rules.
3. Early formula
Compared with the concept of representation mathematics, the early "formula" is a more complicated mathematical process. Our investigation shows that the formula "3+2=5" is first introduced in most primary school mathematics textbooks. First, clear operations, including addition, subtraction, multiplication and division; Then there is the equal sign to clarify the meaning of the equal sign; Then there is the result. This order is considered to be a general description of operations and results, which conforms to the order we wrote. When the teacher tried to generalize these early formulas so that students could understand and accept them, related problems appeared. Usually, the teacher first teaches students the commonly used formula "3+2=5", then changes the arrangement order of elements to "5=3+2" and explains: "Now we have five beads, including three red ones and two green ones. Now let's write them down. " Here the teacher did not analyze the order of these elements in the dialogue and the order of writing. So many children wrote "5+3=2" without hesitation, which shows that they have formed a habit pattern of mechanical writing formulas. The teacher explained and guided, repeatedly emphasizing the meaning of "+"and "=", but it was useless. It can be seen that if students are used to one mode, it takes time to use different modes; If some measures are not taken, the result of another training will be as bad and limited as that of the previous training.
Mathematics teaching is a mathematical process. In the process of mathematization, teachers and students * * * cooperate and interact, and * * * complete the process of mathematization.
Third, the psychological and social factors that affect "mathematicization"
Why can some students successfully complete the process of mathematization, while others can't complete their own mathematization? We investigate these factors from the perspective of constructivism psychology and sociology, and summarize them into the following four aspects.
1. Teacher language
Every math activity in the basic education stage should start with daily experience and language, and the language of teachers is very important. First of all, teachers (especially teachers in the basic education stage) should be trained to state mathematical meanings in simple daily language, otherwise they will not be able to skillfully identify the potential mathematical meanings in students' related words in the relevant dialogue with students, and of course they will not be able to promote students to form meanings and improve their corresponding expressive ability. Secondly, teachers should fully negotiate with students through dialogue, so that students' understanding of the "theme" is consistent with teachers' understanding, rather than just letting students get right or wrong judgments. Furthermore, students may ignore teachers' objections and hints, thus lacking the possibility of forming obstacles. Teachers should be good at using challenging language to improve students' thinking and self-control.
2. Students' natural attitude
Due to inheritance and learning experience; Children will unconsciously form some math learning habits, such as habitual representation and thinking strategies. These habitual routines are usually performed subconsciously and form the basis of children's mathematical behavior, which we call "natural attitude". The "natural attitude" is relatively stable, liberating children from the tense and endless decision-making state. If they encounter new problems, students often subconsciously follow the "natural attitude". If they can't understand or solve new math problems, they often need to spend a lot of time to change their "natural attitude", and they also need the help and guidance of teachers.
3. "Interactive mode" between teachers and students
In our math class, mathematization is mainly realized through the interaction between teachers and students. Moreover, because people generally underestimate the significance of interaction between students, this type of interaction has not fully played its role in daily teaching practice. The interaction between teachers and students can be divided into "one-way" and "two-way" "One-way mode" is generally led by teachers, and students are guided to form mathematical habits through repeated drills, and students are in a passive position; There is no chance to reflect on the differences between one's own construction and that of teachers and others, and the result of learning is often mechanical. The "two-way model" is based on the active interaction between teachers and students. For a picture, teachers should think about why they understand it as a "subtraction" model, rather than an "addition", and give students sufficient time to construct, reflect and correct their own model.
4. Cultural factors in the classroom
In fact, mathematics classroom is a mathematical culture, not just an intellectual or psychological activity. These cultural factors are formed by teachers and students in the long-term teaching practice, which exerts a subtle influence on students' mathematicization. These factors include: ① the success and failure of students and all the expected results; ② Students' anxiety about tasks; ③ Perception of teachers' participation and emotion; ④ Students' reaction; (5) Sentences used by teachers and students; ⑥ Behavior patterns recognized in the group, etc. From the perspective of constructivism, teachers should fully consider these factors in order to cultivate and protect a healthy classroom culture. In a healthy classroom full of cultural factors, students' mathematization will be successfully completed. On the contrary, in a classroom with poor culture, even students who actively participate in it have little chance to experience challenges and feel surprises, and students' mathematicization and the development of classroom culture will be damaged.
Fourthly, the enlightenment to mathematics teaching.
1. Teachers should become "builders of cooperation"
Through the analysis of the mathematization process and its influencing factors, we can clearly see the key role of teachers. In the process of students' mathematicization, teachers should play the role of cooperative builders, not just a subjective navigator. In the process of students' conceptualization, teachers should first reflect on the process of forming their own concepts, then analyze the process of students' construction, and guide students to make comparative reflection through dialogue and interactive activities. Students' mathematicization is influenced by individual representation and life experience. Teachers should also use concise and rich life language to make students successfully complete the transformation from life concept to mathematics concept.
2. Let students solve the problem of "self-organization"
In essence, the final result of mathematization is that students construct their own understanding of mathematical concepts and problem situations in their minds. Therefore, teachers should arrange some specific stages in the classroom to let children solve "self-organization" problems, complete "new" tasks in groups, stimulate their creativity, and even teach them some diversified problem-solving principles and strategies. At these stages, teachers and students should also discuss in detail different ways and methods to solve problems and how to find different viewpoints, arguments and arguments, and carefully scrutinize students' oral scores to judge whether various elements in the process of students' mathematicization are appropriate.
3. Correctly treat children's "mistakes"
Children often make some "mistakes" in written tests, homework and classroom answers. Teachers should regard these "mistakes" as inevitable phenomena in the process of teachers and students' active participation and co-construction, as positive signals of students' "getting started", rather than as accidental events that must be deleted immediately. Teachers should seriously study these mistakes, find out the reasons of psychological or social factors behind them, provide students with some comparative experience when necessary, guide students to deepen the construction process, and finally complete mathematics with high quality. For individual "difficult to teach" students, teachers should also analyze their various narratives, exercises and trainings, agree with their positive conclusions even if they are talking nonsense, and gradually improve their "habit patterns" on this basis.
4. Pay attention to the cultural factors in the classroom
In the classroom, teachers tend to be strict with students' mathematical behavior, and relatively despise the cultural factors in the classroom, which often becomes the root of some students' mathematical problems. What teachers and students do in class is isomorphic to the unique culture of this mathematics course, which includes the characteristics of teachers, students and emerging "mathematics" characteristics. The formation of this culture needs two aspects of dynamic support; Teacher-student interaction, student-student interaction. Teachers should start from these two aspects, conduct clear "meaning negotiation" in classroom dialogue, and attach examples of students' daily behavior to create opportunities for students to discuss and solve "self-organization" problems.
What is the Pythagorean number? For example, i=3 j=4 k=5.
i=5 j= 12 k= 13
i=6 j=8 k= 10
i=7 j=24 k=25
i=8 j= 15 k= 17
i=9 j= 12 k= 15
i=9 j=40 k=4 1
i= 10 j=24 k=26
i= 1 1 j=60 k=6 1
i= 12 j= 16 k=20
i= 12 j=35 k=37
i= 13 j=84 k=85
i= 14 j=48 k=50
i= 15 j=20 k=25
i= 15 j=36 k=39
i= 16 j=30 k=34
i= 16 j=63 k=65
i= 18 j=24 k=30
i= 18 j=80 k=82
i=20 j=2 1 k=29
i=20 j=48 k=52
i=2 1 j=28 k=35
i=2 1 j=72 k=75
i=24 j=32 k=40
i=24 j=45 k=5 1
i=24 j=70 k=74
i=25 j=60 k=65
i=27 j=36 k=45
i=28 j=45 k=53
i=30 j=40 k=50
i=30 j=72 k=78
i=32 j=60 k=68
i=33 j=44 k=55
i=33 j=56 k=65
i=35 j=84 k=9 1
i=36 j=48 k=60
i=36 j=77 k=85
i=39 j=52 k=65
i=39 j=80 k=89
i=40 j=42 k=58
i=40 j=75 k=85
i=42 j=56 k=70
i=45 j=60 k=75
i=48 j=55 k=73
i=48 j=64 k=80
i=5 1 j=68 k=85
i=54 j=72 k=90
i=57 j=76 k=95
i=60 j=63 k=87
i=65 j=72 k=97
What do you mean by virtue, ability, diligence, success and honesty? Please give examples to illustrate virtue: thought and conduct;
Ability: working ability;
Diligence: working attitude;
Performance: performance (achievement);
Lian: incorruptible.
Morality, ability, diligence and honesty refer to five aspects involved in the evaluation, inspection and appointment of cadres.
Virtue is obeying laws and regulations. Respect the leadership, take the overall situation into consideration, unite and cooperate, be honest and self-disciplined, be modest and prudent, and pay attention to etiquette. Don't say things you shouldn't say, don't be rude, don't eat invitations you shouldn't eat, don't accept gifts you shouldn't receive, don't do things you shouldn't do, don't build proposals you shouldn't build, and don't exceed the time you shouldn't build. Don't push, don't wrangle, don't be rude, don't be arrogant, don't intercede, put yourself in a correct position and find your role. Be tolerant, kind, easy to get along with, fair, honest and considerate. Can always maintain the collective image, maintain the image of * * *, and maintain the personal image.
It can be to study laws and regulations, principles and policies, professional knowledge, ways of being a man, skills of seeking things, history and skills. Arm yourself with knowledge, explore boldly, be brave in innovation, constantly improve your comprehensive quality and ability to run articles, handle affairs and hold meetings, be able to speak and be good at coordination, establish an image with level and ability, and strive for progress.
Diligence refers to hard work and thrift. Take your career as your responsibility and be ashamed of laziness. Actively be diligent in hand, leg, mouth and brain. Diligently make up for mistakes, cultivate self-cultivation, cultivate self-cultivation, work hard, establish faith and be a man.
Performance refers to results and performance. Show your ability, performance level and shape yourself with your grades. Not only think, say and do, but also dare to think, say and do, even say and do. Only with visible, tangible and tangible achievements can we gain the qualifications of trust, support, understanding and respect, and create development space for our work, study and life with the affirmation of leaders, the recognition of colleagues, the satisfaction of subordinates and the support of the masses.
Morality is the first and the most critical link.
As the old saying goes, "Virtue is the source of water, and it is the wave of water". Virtue, namely morality and morality. Generally speaking, the connotation of virtue refers to the political, ideological and moral character of staff, as well as law-abiding, honesty, professional ethics and social morality. Virtue has different connotations in different historical periods. Different classes have different standards, but no matter which class takes "morality" as the primary criterion for assessment and employment.
At present, the assessment of staff's "morality" mainly depends on whether they adhere to the party's basic line, whether they are loyal to the country, whether they are law-abiding, fair in handling affairs, upright in behavior and noble in moral character.
Deneng diligent and honest Baidu Encyclopedia
If a number is not a real number, what is it? Please illustrate that the imaginary number 3i is an imaginary number, that is, the root sign (-3), that is, 3× root sign (-1
How to find the factor and multiple of a number? (for example) divide this number by the smallest prime number by short division. For example, several factors can be found in 10. 10 is divided by 2 to get 5, and 2 and 5 are factors of 10. 10 is a multiple of 2 and 5.
Please give an example of how to choose phrases related to up; Pick up people by car
Lift, hang, post
Give up. Give up
Take up and start to engage in; engage
Turn up the volume; appear
What is a constant? What are they? Please give an example of 1 2 3 7.5 4/5.
What are the effective ways to reduce students' heavy math homework burden? Please give an example to illustrate that the argument of reducing students' excessive academic burden has been put forward for a long time, and the national education department has repeatedly stressed that the effect of class hours is actually not significant, and even the academic burden of students is increasing. There are many reasons for various situations. If we want to fundamentally reduce students' excessive academic burden, I think we can work hard in the following aspects.
First of all, the reform of the education system, especially the examination and evaluation system, is fundamental.
China has been carrying out quality education for a long time, but the system of selecting talents through examinations has not fundamentally changed, nor has the phenomenon of crossing the wooden bridge with thousands of troops. Moreover, many places still regard the enrollment rate and online rate of key middle schools as the criteria for evaluating a school and a teacher. In this way, schools, teachers and parents discount the exam-oriented education under the banner of quality education, and the so-called reduction of students' burden has become superficial, so it is fundamental to reform the examination system and evaluation system.
Second, it is the key to reform classroom teaching and improve classroom teaching efficiency.
For a long time, teachers' teaching, even some so-called famous teachers' teaching, relies on time, homework and training to improve students' grades. They firmly believe that "what is lost in the embankment will be made up", so those "powerful" teachers in the school will finish their homework first, and other teachers will leave more homework in order to narrow the gap. Over time, students' academic burden is getting heavier and heavier. In order to change this situation, teachers should reform classroom teaching and be efficient for 45 minutes.
When preparing lessons, teachers should carefully study textbooks, grasp curriculum standards, understand students' knowledge reserves, and know where the difficulties and misunderstandings of students' learning are, so as to make classroom teaching targeted and know what to teach and how to teach in each class. In the classroom, teachers should find the starting point of teaching according to students' age and psychological characteristics, stimulate students' interest in learning, guide students to explore, communicate and cooperate, and guide students to learn to learn; In the after-class session, teachers should carefully prepare and choose exercise questions according to the feedback from students in class, so as to make the exercise questions few and precise, aiming at students at all levels. Through practice, every student can be improved and enjoy learning.
The reality that students are overburdened with schoolwork still exists, which requires our education departments, schools, teachers and parents to act together to raise awareness, change their concepts and really consider the healthy growth of students.
Nowadays, it is not a very new problem to reduce students' heavy academic burden and improve classroom teaching efficiency. Reducing the burden is to improve efficiency. How to unify the contradiction between reducing burden and increasing efficiency? The key lies in teachers and classroom teaching reform. As the basic form of school teaching, classroom teaching is the main position for teachers to complete teaching tasks. In order to reduce the burden and increase the efficiency, and improve the teaching quality in an all-round way, we must completely abandon the traditional teaching methods of "cramming" and "cramming" in mathematics teaching, completely abandon the "sea tactics" used to cope with exams, strictly and correctly grasp the teaching materials, and adopt advanced and appropriate teaching methods and means. Speaking in class is accurate, interesting and flexible, giving students enough time to think, do things and speak, discussing and studying by themselves, gradually cultivating students' good study habits, and gradually making students love learning and learn to learn. In order to truly achieve the goal of reducing burdens and increasing efficiency, we must start from the following aspects. One: Clear the purpose and stimulate interest. Curiosity and interest in learning are intrinsic learning motivations. To cultivate students' interest in learning, students should realize the practical value of mathematics, which is of any direct or indirect use to them. When students can realize that learning is a means for them to achieve an important goal, they will have curiosity and interest in understanding. In mathematics teaching, helping students understand the purpose of learning mathematics is an effective way to cultivate and stimulate students' interest in learning mathematics. In mathematics teaching, examples of mathematics widely used in science, production and life are introduced to let students understand the important role of mathematics in the development of society and modern science; This can overcome the one-sided understanding that most students think that learning mathematics is not only to meet the needs of entrance examination, deepen their understanding of the importance of mathematics learning, and thus cultivate and stimulate students' interest in learning mathematics. In the process of mathematics teaching, students should also realize the important role of learning mathematics in improving thinking quality and cultivating logical reasoning ability and imagination. Therefore, it is said that "learning mathematics is a specific medicine to exercise thinking and make people smart". So as to establish the concept of "learning mathematics can make people smart" for students, and thus maintain their interest in learning mathematics in the teaching classroom. Two: diligence and conciseness. Intensive cultivation: Some teachers think that teaching time is directly proportional to teaching effect, and that teachers' classroom teaching and students' extracurricular practice not only make full use of classroom time, but also make full use of students' extracurricular time. So I do not hesitate to squeeze out students' independent homework and feedback time for teaching, thinking that this will improve the teaching effect and achieve good results. They don't know that students can't build new cognitive structure, consolidate knowledge and form skills without their own independent activities. At best, teachers can only let students "understand" but not "know" when giving lectures and activities with teachers and students alone. According to the different requirements of teaching content, teachers should correctly grasp the teaching materials, carefully design the classroom teaching process, fully trust students and take students as the center. For some knowledge that students can understand in textbooks, don't explain it repeatedly, let students explore and practice it themselves, and teachers should design questions around these basic knowledge, so that students can discuss and even argue. According to students' characteristics, basic level and interest differences, different ways and means should be adopted for teaching, so as to enable students to master these basic knowledge. For example, "the difference between a parallelogram and a trapezoid", two groups of parallelograms with parallel opposite sides are parallelograms, and only one group of parallelograms with parallel opposite sides is a trapezoid. In this class, the teacher gives the theorem directly, then explains it, and then illustrates it with examples. On the surface, it seems to meet the teaching requirements. However, this is indeed a very common classroom structure, and it has not got rid of the shackles of the old teacher-centered teaching thought. If we change the design, let students practice observation and give students a concrete and intuitive image; Find the difference by yourself, and get that two groups of quadrangles with parallel opposite sides are parallelograms, and only one group of quadrangles with parallel opposite sides is trapezoid. It will make the whole classroom more vivid, cultivate students' mathematical ability and let students firmly grasp the difference between parallelogram and trapezoid. This will be better than the teacher's direct conclusion. Conciseness: At present, it is still common that homework can't get feedback in class. Many teachers turn classroom homework into extracurricular homework, depriving students of activity time and increasing teachers' burden, which directly affects the improvement of teaching efficiency and gets twice the result with half the effort. We should always pay attention to adjusting our teaching process, the depth of teaching content, the pace of teaching, the choice of teaching methods and the speed of teaching language according to students' information feedback, which are determined by students' feedback on mathematics teaching information in class. It is an important part of classroom teaching. Classroom autonomous homework is an important activity for students to do it in class. On the one hand, it can urge students to apply what they have learned and deepen their understanding of new knowledge in application. On the other hand, it can expose students' shortcomings in applying new knowledge. Class assignments, as the name implies, are done in class and can be fed back. Don't practice across the board in math learning. According to students' characteristics, basic level and interest differences, we should adopt different methods and ways to teach, meet different needs, and let all kinds of students have interest in learning mathematics. When giving lectures to students with poor foundation, we should pay attention to being easy to understand, while for students with good foundation, we should be more logical and profound. The assigned exercises should also be graded according to the students' learning situation. Students with a good foundation can arrange some flexible topics and difficult questions, so that everyone can gain something and make progress. At the same time, we should do a good job in counseling, help them overcome obstacles in their studies through individual counseling, and establish confidence in learning mathematics well. Third, get feedback information and adjust teaching methods. Some teachers think it's better to finish teaching as soon as possible and start reviewing as soon as possible, so they cut off the time for students to do homework and feedback to catch up with the progress. Under the guidance of the concept of examination, the teaching of grasping scores seriously violates the principle of information acceptance and the cognitive law of students, and the result is counterproductive, which is not conducive to the acquisition of knowledge, the formation of skills and the construction of cognitive structure. In the process of teaching, the activities of teachers and students are bound to achieve certain results. If these results are fed back in time, they will become new information for further adjustment of teaching and learning. For students, knowing the results of learning in time can help them get correct information quickly and further adjust their learning methods. Teachers' timely and appropriate evaluation of students' learning is a very important form of feedback and a favorable factor to promote learning and improve learning effect. Teachers know how their teaching effect is from students' knowledge. Get timely and comprehensive feedback, adjust teaching methods, improve teaching effect and promote the formation of students' intellectual skills. In a word, every math class can be carefully designed, and some interesting factors in mathematics can be excavated, so that the math class is full of fun and fun, which fundamentally improves the negative characteristics of the complex and boring mathematics discipline and makes students feel happy in the process of learning mathematics. The cultivation and stimulation of students' interest in learning mathematics, the intensive teaching of teachers, the timely adjustment of students' learning methods and teachers' teaching methods. Only in this way can we really reduce students' schoolwork burden and improve classroom teaching effect.