Zhang Qihua, male, born in June, 1976, from Haimen, Jiangsu. 65438-0997 teaches at Haimen Experimental Primary School. In 2004, he was transferred to beijing east road Primary School as the director of the teaching department. Has been committed to the exploration and practice of mathematics classroom culture, and participated in the compilation of the Jiangsu Education Edition primary school mathematics textbook. He was awarded the title of Nantong backbone teacher and Nanjing outstanding young teacher.

Mies van der Rohe is one of the greatest architects in the 20th century. When asked to describe the reason for his success in one sentence, he only said five words, "Success lies in the details". The same is true of successful classroom teaching. Grasping the details correctly is the key to a good class.

In the teaching of "Preliminary Understanding of Fractions", Mr. Zhang Qihua hid the bisector in the textbook (sketch), first took out the first one, and told the students to color all the pieces of paper, which could be represented by the number "1". Ask the students to estimate that the colored part is the score now.

Some students guess 1/3, and some students guess 1/2. After the courseware is verified, the color part is 1/3. The teacher continued to show the third note and asked the students to estimate. Many students estimate that it is 1/6 at once. How does the teacher ask students to communicate? Any tips? Originally, the students compared the 1/3 of the third note with the 1/3 of the second note, and found that the color part was only half of it this time, so they decided to use 1/6 to represent it.

The teacher immediately concluded: "You see, estimating by observation and comparison is a good thinking strategy!" " "This little detail is thoughtful. However, the excitement does not stop there. Next, Teacher Zhang uses this small piece of paper to make a big article, so that students can observe the colored parts and corresponding numbers here and talk about their findings. Some students found that the same piece of paper, its 1/3 is larger than 1/6; Three out of 1/3 and six out of 1/6; The more copies the average score is, the smaller the colored ones are ... the students are chattering and their thinking is extremely active. This is a classroom full of spirit. From the preset lesson plan to the dynamic generation, from the cultivation of students' estimation consciousness, to the comprehensive training of mathematical thinking strategies, and then to the organic infiltration of extreme thoughts, simple content carries rich mathematical connotations, and all the highlights come from teachers' attention to details.

From this perspective, the author also found that in the teaching of exchange method, Mr. Zhang is the creator of teaching materials, not the consumer.

Teacher Zhang told a "chop and change" story first, and then asked the students what they wanted to say.

Combined with the students' speeches, the teacher wrote on the blackboard: 3+4=4+3.

Teacher: Look at this equation. What do you find?

Health 1: I found that the sum of the positions of the two addends remains the same. (The teacher is writing on the blackboard)

Teacher: What about the other students? The teacher's findings are similar to his, but slightly different. (The teacher immediately indicated: exchange the positions of 3 and 4 and keep the same) What do you want to say compared with the conclusion we gave?

Student 2: I think the conclusion you (the teacher) gave only represents a special case, but his (student 1) conclusion can represent many situations.

S3: I agree with him (S2), but I don't think it's good to draw the conclusion that "the positions of two addends are interchanged and remain the same" only by this formula on the blackboard. What if the other two numbers add up and the positions are not equal? I still think your point of view is more accurate and scientific.

Teacher: Indeed, it seems a little hasty to draw the conclusion that "the positions and identities of the two addends have been interchanged" based on a special case. But we might as well take this conclusion as a guess (the teacher will immediately give birth to ""in the conclusion given by 1). Change it to "?" )。 Since it's a guess, then we have to-

Health status: verification ...

Cao Yiming, a teacher from the School of Mathematical Sciences of Beijing Normal University, thought when commenting on the class: From the whole class, "addition and associative law" is just a connection. "Is there an exchange law for subtraction?" "What about multiplication and division?" Other new problems are all new growth points born in the original contact. Integrated together, the knowledge of switching law as a concrete operation is weakened, while the switching law itself, the dialectical relationship between change and invariance, the thinking route of "guess-experiment-verification" and the mathematical association from "this knowledge" to "that knowledge" are highlighted one by one, which has become a higher pursuit beyond knowledge in mathematics classroom. When we appreciate the tranquility of children in meditation, the confusion in doubt, the joy in epiphany, the agitation in argument, the surprise in listening, the fluency in argument, and the joy after success in class ... a class that enjoys speculation is full of brilliance because of Mr. Zhang's attention to details.

Based on this kind of thinking, I also found that paying close attention to learning trends and making effective use of students' classroom resources are also the magic weapon for teacher Zhang to lead students into the realm of thinking. In the practice of students writing Thirty-six divisors, he deliberately chose two different works to comment on:

The divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.

The divisor of 36: 1, 36,2,18,3,12,4,9,6.

First, he asked two children to introduce their methods of finding divisor: the first child said "one-to-one correspondence method" and the second child used "pairing method, two to two". Teacher Zhang quietly asked other students to compare which method is the best, and why? Many children naturally feel that the "comparison method" is good, and it is not easy to lose the answer by looking for one by one. Teacher Zhang was not satisfied with this "consistency" and immediately asked, "Isn't the first method worthy of recognition?" This humorous question solved the dilemma of the first child. The children meditated and reflected independently, and finally had an epiphany. Finally, he asked the children who used the "one-by-one method": "If you continue to find factors, which method are you going to use?" In this teaching detail, Mr. Zhang explained the "comparison" method incisively and vividly: in the first level of comparison, students learned to get the idea of "optimization" between different methods; The second level of comparison, learned the idea of "dialectical analysis", can not simplify the problem; The third level of comparison has gained the idea of "appreciating and learning from others". Only by amplifying the advantages of others can we enjoy the fruits of wisdom. Three "comparisons" are not only the teaching of mathematical methods, but also the infiltration of ideological values.

Use a smart heart to perceive, use a pair of smart eyes to capture, and use the educational feelings of "squatting and walking in" to lead students to touch the wonderful mathematics, which lies in the subtleties. Teacher Zhang's deep processing of teaching materials and meticulous treatment of texts can capture students' wonderful moments such as questions, ideas and unique opinions at any time, making the classroom a stage for teacher-student interaction and spiritual dialogue, and a time and space for teachers and students to jointly create miracles and awaken their sleeping potential.

-Published in the 5th edition of China Education News, June 2007 15.

Zhang Qihua Teaching Art Series (II)

Wisdom of evaluation: like fragrant wild flowers blooming all the way.

Chen Huifang

"Listening to Zhang Qihua's class is very comfortable, relaxed, pleasing to the ear, very comfortable ..." This is the teachers' * * knowledge, which may be related to Mr. Zhang's rich humanistic background and solid language skills, especially his fresh and natural, free and easy evaluation language. Counting his math class, we can hear:

When a classmate put forward different opinions, Mr. Zhang did not ignore the psychological feelings of the previous classmate, but smiled and said to him, "Are you happy that someone challenged you?" "Happy!" The students answered confidently.

When the exercises are presented, Teacher Zhang will ask with warm eyes, "Students, are there any difficulties?" So, who will speak first? "When showing students' works, Teacher Zhang will ask with concern," What are you going to give this assignment? " "Is there anything to add? Do you want to talk about his methods? " Then turn around and tell other students that there is no need to be superstitious about others. When there is no other answer, Teacher Zhang will remind everyone: "There are no different ideas to say." His words made people feel warm.

We also enjoyed such a set of shots:

Teacher: Look! The folding and tearing just now really created an axisymmetric figure in mathematics. To tell the truth, mathematics is sometimes as simple as that. If I remember correctly, everyone is familiar with axisymmetric graphics. There should be some axisymmetric figures in the plane figures we know.

Show the practice of axisymmetric figure, and let the students judge whether it is axisymmetric figure. )

Teacher: Before practicing, I want to give you some advice. Sometimes, don't trust your eyes too much. What looks like an axisymmetric figure may not be, but what doesn't look like it may be.

The teacher asked the students to make a bold guess based on their experience and choose the one they are most sure of. They can also combine the learning tools in their hands, in groups of six, and stack them together to test their guesses. Students communicate in groups, and there is a debate about whether a parallelogram is an axisymmetric figure. )

1: I think the parallelogram is an axisymmetric figure. If you cut it along the height, it can be put together to form a rectangle. After being folded in half, the left and right sides can completely overlap.

Health 2: I don't think parallelogram is an axisymmetric figure. After the parallelogram is folded in half, the figures on both sides can't completely overlap, so I don't think so.

Teacher: (I walk over to shake hands with student 2) I shake hands with you not because I agree with you, but because I thank you for creating two different voices for the class. Think about it, how monotonous it would be if there was only one voice in our classroom!

After the students practiced again, the first student changed his mind and learned that the parallelogram is not an axisymmetric figure. )

Teacher: Your concession will bring us closer to the truth!

(Next, the teacher guides the students to find the symmetry axis of symmetrical figures. )

Teacher: It is said that practice makes true knowledge. Mathematics is to dig deep. Don't you want to go into these five figures and say something? This trapezoid is an axisymmetric figure, but ...

There are more silent words than vocal words. Intelligent evaluation suddenly caught the students' heartstrings and activated their thinking. The students stared at the five numbers, kept looking and arguing. The teacher's wonderful narration has undoubtedly become the propeller of students' thinking.

His comments are very philosophical. When discussing whether a two-digit number composed of nine beads can be divisible by nine, students immediately mistakenly think that eight has such a law. "Is this really the case?" Teacher Zhang guides the students to think further. When students find that eight beads can't work and seven beads can't, they have the wrong idea of "nothing else" Teacher Zhang said, "Don't just deny everything." A few words, relax.

In the lesson of "Understanding the Circle", some students exchanged experiences in drawing a circle and said, "Our group tied a rubber at one end of the rope, grabbed the other end of the rope and shook it, and a circle also appeared." For such an unexpected generation, Teacher Zhang commented: "Although this method failed to' draw' a circle on white paper, their creation is still wonderful, isn't it?" There was warm applause in the classroom. This applause comes from the appreciation and encouragement, acceptance and recognition of students' hearts, which is the flow of true feelings.

Teacher Zhang's language is full of magnetism, and it is often said that "we can feel her feelings before she plays", which contains endless interest. For example, "ellipsis came too late", "think while doing homework, and then make a decision" and "don't jump to conclusions", he always calls on students to think positively.

A student left out 2 when writing a factor of 36. Facing the students' mistakes, Teacher Zhang said humorously, "Do you have anything to say after reading it?" "Listen to how he found it." "There are a lot of people didn't miss a.. I'm sure they have a knack. Let's take a look! " ..... A short spiritual dialogue, in line with the students' inner eye movements, is full of love.

"People who touch people's hearts, don't worry about feelings first." Some people say that the extension of language is the fluency of thought, the beauty of language comes from the delicacy of thought, and language is the most beautiful flower of wisdom in the world. In class, I often hear teacher Zhang's praise: "It's very associative!" "Very good!" "Oh, it's amazing!" "Great!" A casual comment, an encouraging word, what he said is humorous, funny, affectionate or wise, which can always make students learn Emotional adjustment to reach the best state and make them have a positive psychological tendency of autonomous learning. With fluent teaching language, teaching charm of poetry, songs and paintings, he created a beautiful artistic conception by rendering, aroused the shock of the soul by true feelings, cleared away the heavy confusion by enlightenment, and induced far-reaching thinking, so that the classroom was always filled with a rich humanistic atmosphere connected with the bud of life. He inspires students' insights with true feelings, inspires students' innovation sparks with self-confidence, and leads students to the wonderful realm of mathematics learning with poetic interpretation. There is always a scene of "listening to the sound of tender bamboo jointing" in class. This unique and charming classroom evaluation interprets the new roles of teachers and students and skillfully interprets the classroom. Sharing his class, we clearly feel that wisdom is like a fragrant wild flower, which blooms all the way in the education and life, and we will see bloom every time we take a solid step. ...

-This article was published in the 5th edition of China Education News on June 29th, 2007.

Zhang Qihua Teaching Art Series (3)

Creating interesting teaching magnetic field with situation

Chen Huifang

Teacher Zhang Qihua is good at creating situations in the math class and embedding the teaching contents into the colorful life background.

In the process of understanding "cuboid", "cuboid's length, width and height" as a knowledge point, teachers usually tell students directly. However, when Zhang Qihua was teaching, he created such a problem scene: If the 12 sides of a cuboid were erased, can you still imagine the size of the cuboid? What if you erase two, three or more? Try it and see that there are at least a few sides, so as to ensure that you can imagine the size of a cuboid. When students choose three sides: length, width and height in unison after trying, exploring, operating and optimizing mathematical activities, the specified mathematical common sense "length, width and height" is "activated" at this moment. Teacher Zhang Qihua believes that "brain creation" like this can restore the inherent vitality of mathematical concepts, and its cultural value is greater than the giving and receiving of concepts. This interesting teaching situation based on problem research gradually leads to new problems from a problem, and students always study around the problem, thus realizing the climbing of thinking. In this teaching process, students are looking for ways, knowing laws and mastering methods instead of just knowing the "length, width and height" of a cuboid, which is undoubtedly valuable for subsequent learning.

Teacher Zhang Qihua believes that the real situation should stimulate students' sense of participation, concern and initiative, and guide students to immerse themselves in the "environment" of exploration, thinking and discovery. It certainly needs a specific scene as the background and carrier. However, whether the presentation of the scene can effectively arouse students' sense of cognitive imbalance, problem awareness and cognitive conflict, and whether the scene itself can attract students to actively participate in the exploration and thinking of problems still needs further exploration.

Based on this mathematical idea, Teacher Zhang showed a photo of himself walking upright when he was 1 years old when he was teaching "Preliminary Understanding of Fractions". He asked the students to guess who the child in the photo was. A student said excitedly, "I think it's Teacher Zhang."

Teacher: Good eye! This is my 1 time. Observe carefully. (Animation demonstration: the height is about 4 times the height of the head)

Teacher: Did you find that the height of a baby's head at the age of/kloc-0 is about a fraction of its height?

Health: 1/4.

Teacher: What will you be like when you grow up?

The teacher showed his upright photo and demonstrated it with animation: the height of his head is about 65438+ 0/7 of his height.

Teacher: Now, how much is the height of the head?

Health: 1/7.

Teacher: Actually, the scores corresponding to different ages are different. The student is about 10 years old this year. So, how high is his head for a child about 10 years old? Do you want to know?

Health: (excitedly) think!

The teacher immediately invited a student to the stage, and the other students estimated together.

The height of students' guessing head is about 1/5 of the height, some think it is 1/6, and some say it is close to 1/7. Teacher Zhang told everyone that it is normal to estimate errors. As for 10, the height of a child's head is about a fraction of his height. After class, students may wish to check the information. The student returned to his seat, and the other children were still very interested and beaming.

I think at this time, the teaching situation of guessing scores is created by a photo, and its "meaning of drunkenness is not wine". Not to mention the novelty and vividness of the theme, the key is that students have deepened their understanding of the score in a series of operational activities such as taking a look, comparing and estimating. This introduction organically broadens students' cognitive horizons, so that they can truly feel the extensive application of scores in daily life and truly experience the value of learning scores.

In the introduction part of the new curriculum "Factor and Multiplicity", Mr. Zhang created operational situations, skillfully used models to construct knowledge and revealed conceptual connotations. The "Exchange Method" class creates story situations at the beginning, and builds a thinking platform for the new class learning. In "simple statistics", create scenarios for students to conduct field investigations and enhance their understanding of statistical methods and values; When teaching "understanding integers", it begins with dialing games. In the process of dialing numbers, arouse students' memory of counters, counting units, numbers and other related experiences.

It is true that how to create an effective teaching situation under the background of the new curriculum reform has always been the focus of attention. In Mr. Zhang's math class, no matter whether it is a pleasing fairy tale or a novel and interesting operation situation, the design of each class is based on students' different cultural backgrounds and life experiences, and efforts are made to tap the fresh activities that may occur in real life, with the situation as the highlight, emotion as the link, thinking as the core and life world as the source.

When creating teaching situation, Mr. Zhang broke through the fortress of subject classroom, created the thermal effect of classroom with the integration of various subjects, expanded the extension of learning activities, made learning activities three-dimensional, and allowed students to accumulate culture and humanistic spirit while acquiring knowledge. He drives and encourages students to think deeply with questions, realizes the collision of wisdom and the enjoyment of experience with classroom dialogue between teachers and students, and creates a dynamic and interesting teaching magnetic field with effective interaction between teachers and students, which may cause recognition, association, speculation or doubt ..... and make students' knowledge tend to be rich, complete, accurate and profound.

-This article was published in the 6th edition of China Education News on July 6th, 2007.

Zhang Qihua Teaching Art Series (IV)

Poetically pursuing mathematical culture all the way

Chen Huifang

When we talk about Zhang Qihua, we can't help talking about mathematical culture.

Zhang Qihua often wondered whether mathematics could fundamentally change a person and make him more powerful and spiritually cultivated. How does mathematics study affect and nourish students' life and spiritual growth? Therefore, he regards teaching as a part of his life. In class, he set up a stage for children to show themselves, and began to stack, cut and spell, talk and discuss in groups, so that children could feel the aesthetics and imagination and the natural beauty of mathematics in the process of experience. This kind of communication between teachers and students means dialogue, participation, open mind and personality. The teaching process has become a process of sharing and understanding, and the lives of teachers and students always flash in the classroom.

In the process of "understanding the circle", he fully demonstrated the beauty and cultural atmosphere of the circle with the help of the wonderful water patterns, sunflowers, auras, electromagnetic waves and beautiful circles in nature, and showed the beauty of the axisymmetric figure from the aspects of symmetry in paper-cutting, symmetry in architecture, symmetry in famous signs and symmetry in Guilin mountains and waters. Perhaps the beauty of mathematical culture, which has just begun to be understood, depends more on something other than mathematics, integrating the wonderful display of nature, science, society, culture and media. However, in many links of Factor and Multiplication, Mr. Zhang's profound thinking on mathematical culture and his high concern for cultural tension are reflected.

We might as well do a lens playback: teacher: the students' ideas are very valuable! Indeed, among the natural numbers within 100, 60 is not big, but it has the most factors. It is this characteristic of 60 that makes it occupy an important position in the history of mathematics and astronomy. (Display information: We all know that 1 hour = 60 minutes, 1 minute = 60 seconds. However, historians have found through textual research that the rate of time progress is set at 60 because "among natural numbers within 100, 60 has the most factors, and * * * has 12". It is said that this can make many calculations about time very simple. )

Teacher: Well, I didn't expect the progress rate between hours, minutes and seconds to be set at 60, which has a lot to do with the number of factors in our mathematics. The wonder of mathematics is sometimes unbelievable! In fact, as a small branch of number theory, similar wonderful mathematical phenomena abound in the field of factors and multiples. Here, the teacher also wants to introduce a special number to everyone, that is 6. Want to know why?

Health: Yes.

Teacher: Then let's do a little experiment together! First, write down all the factors of 6; Second, remove 6 itself and add the remaining factors. What did you find?

Health: (unexpectedly) the result is still equal to 6.

Teacher: Because such numbers are special, mathematicians call them perfect numbers. 6 is the first perfect number. Never underestimate these figures, because they are very rare. Want to know what the second perfect number is?

Health: Yes!

Teacher: Tell me, more than 20 and less than 30. Cooperate in groups to see which group finds the second perfect number first! The students cooperated in groups, and soon, several groups found the second perfect number of 28, and the excitement was beyond words.

Teacher: Actually, people's interest in digital exploration is endless. When they found the second perfect number, people began to look for the third and fourth ... in this way, one perfect number after another was constantly found. At this time, the courseware soundtrack is presented in turn: 496, 8 128, 33550336, 8589869056. ...

It is not difficult to find that there is almost no perfect number in the process of leading children to find the "perfect number", which highlights the difficulty of mathematicians in finding it, which is undoubtedly the guidance of mathematical spirit. Then, in the grandeur of ancient Roman architecture, Mr. Zhang told the children that this building has experienced thousands of years of vicissitudes because it hides the secrets of multiples and factors. With the beautiful and harmonious melody flowing slowly, Teacher Zhang reminded the children that the harmony between notes comes from the relationship between multiples and factors. Isn't this the charm display of mathematics? It is conceivable that children have realized the application value and magical power of mathematics from their hearts with rich mathematical conjectures and the magical beauty of Greek architecture, music and perfect numbers. When they were surprised by the perfect figures and excited by the diligence and wisdom of the ancient people in China, the seeds of loving the motherland, science and mathematics have quietly sprouted. Isn't this the power of mathematics?

At this point, I still remember the interesting advertisement that Mr. Zhang brought to everyone at the end of the class "Preliminary Understanding of Fractions". After the boy divided the cake into four parts in winter and winter, he found that a * * * had eight small partners. He had a brainwave. He cut a knife horizontally from the middle and divided the cake into eight parts equally. At this moment, the ninth boy appeared. What shall we do? Dongdong split his share in two and gave him 1 ... a small advertisement, which contains rich mathematical connotation and profound humanistic care, paid attention to students' emotional experience in time, consolidated their understanding of scores, and awakened their love, innocence, friendship and responsibility. Students not only gain knowledge, but also acquire noble character and beautiful mind. This kind of culture represents students' understanding and experience of the world, and shows students' unique values, ways of thinking and behaviors. Perhaps this is what Teacher Zhang said: "Be a good person and enjoy the spiritual strength endowed by mathematics"!

In Zhang Qihua's lecture "From Simple to Profound", I also learned how to infiltrate statistical thoughts into Simple Statistics. In the process of "discovering laws", how to seek common ground from changes and rise to the mathematical thought of "one-to-one correspondence"; How to carry out the coordinate thought in Positioning Method, especially the calculation of the back area of irregular figure-calculus thought infiltrated in the process of curve to straight line. ...

With a classical and aesthetic feeling, Mr. Zhang Qihua pays attention to the improvement of students' mathematical thinking and the cultivation of mathematical thinking mode, and pays attention to the organic infiltration of mathematical spiritual quality, which not only enriches the connotation of mathematical culture, but also makes theoretical exploration and practical research on the future mathematical culture, opens up new ideas, shows new opportunities and depicts a new future.

Nowadays, in his math class, we can feel the source, history, spirit and strength of mathematics anytime and anywhere. It seems that what appears in front of us is no longer a thin textbook or two, but a picture of mathematics with a long history. On the surface, mathematics is boring, but it has a hidden and profound beauty, a combination of sensibility and rationality. Mathematical beauty is the perceptual and rational presentation of the essential strength of mathematical science, the presentation of human essential strength through human mathematical thinking structure, a real beauty and a scientific beauty that shows humanistic spirit.

"I like to travel, because travel has witnessed a gesture, a gesture of constantly walking and thinking. I am willing to be a walker in the journey of mathematics education. " This is Zhang Qihua's heartfelt words. I am convinced that Zhang Qihua will add many new "spiritual elements" to mathematics culture; For mathematics education, in his carefully interpreted wisdom classroom, it will be more full of vitality, poetic humanity and more agile and elegant.

-This article has been published in the 6th edition of China Education News on September 4, 2007.