In fact, mathematics is developed through human activities-problem solving, reasoning, representation, communication and evaluation. Through daily life experience from childhood, children have gradually formed quite complicated informal ideas about numbers, formulas, shapes, data and sizes. It can be said that there is a lot of informal knowledge in the minds of students of all ages. Informal mathematics has many carriers. In practice, I found that "Mathematics Composition" with its unique form, eclectic material selection requirements and flexible expression can provide students with a broad mathematics stage and promote the realization of multiple goals of mathematics.
Trial practice (1)
Therefore, I used the opportunity of assigning winter vacation homework (Grade Four) in 2003 to assign students the task of writing a math story or a math composition. In the guidance, I explained the math composition like this: "The content of math composition is to combine the learning content of this semester and the problems encountered in learning. It refers to a unique solution to a problem (inside and outside the textbook) (the requirement is simpler than the original solution, which is applicable to similar problems), or the perception of mathematics learning, or something interesting and stupid in mathematics. " .
It seems that the children are very enthusiastic about this homework. During the winter vacation, some students sent me emails one after another, asking me to help revise them. After the winter vacation, I summarized the students' math stories and math compositions, and some children's works were quite childlike. I tried to put two students' works on the "Inter-school Exchange" of "Uptown Education Information Port", but they were quickly posted on the Internet. Considering that the children spent a lot of effort, I selected some classmates' works, compiled a math composition report (winter vacation) and sent it to each classmate. The children looked at their hands and thought about the next time. Some asked: Can I write next time? Some asked: Why can't I find my article in the newspaper? ……
When I analyzed the articles handed in by students, I found the following characteristics:
1, there are many mathematical fairy tales. When I assigned my homework, I didn't define what a math story is, but I gave them a math story, Naughty Parenthesis and 7 and 1, which was written by a student under my guidance in advance and in anthropomorphic form, which should be regarded as a math fairy tale. As a result, almost all mathematical fairy tales and jokes that give life to numbers and symbols have been handed in. Of course, this is a simple way to give life to the carriers of mathematical knowledge, such as numbers and symbols. And let them explain their math knowledge by themselves, so that students are willing to adopt it.
2. A considerable number of students have deviated from their understanding and understanding of mathematical stories and compositions. Some students understand a math story as: a story with numbers is a math story. A student changed the name of the hero of a folk story to decimal point, and then organized it in his own language and handed it in, thinking it was a mathematical story. Others think that the composition related to math class is math composition. A classmate described the process of a math class and thought it was a math composition.
According to the students' homework, I realized that for students, math fairy tales and math stories are the same thing, while math composition is a bit confusing.
What are mathematical fairy tales, mathematical stories and mathematical compositions? I looked up some materials and found that since the late 1970s, some mathematical researchers, represented by the famous popular mathematical writer Li Yupei, have published many popular mathematical works, such as Wonderful Curve. Among them, the battle between rational numbers and irrational numbers, sleepwalking in 0 countries and other fairy tales that illustrate and express mathematical knowledge or characteristics in vivid fairy tales, which Miss Li herself calls: mathematical fairy tales and mathematical stories. At the same time, there are similar works abroad (should be earlier than us). For example, there is a particularly famous math popular science book in America called Aha! I had a brainwave. There was a hero named Professor Quibell. He always had some interesting problems and solved them. In 1980s, China often published similar works for students in newspapers and periodicals, often featuring images of Pig Bajie and the Monkey King, wise old man and little Li Ling, and engaged in activities such as crossing the border and traveling. In the activity, you will encounter some interesting math problems and then solve them in interesting ways. At present, this form of work can still be found frequently. Strictly speaking, these works are popular works of mathematics, but they can also be regarded as mathematical stories or fairy tales, because their purpose is to let readers learn or understand some mathematical knowledge.
Mathematical composition can often be seen in some children's newspapers and periodicals, but the author is often an adult, who explains a certain knowledge to students in the form of analysis, analogy and classification, with the aim of promoting readers' understanding of the knowledge.
Based on the above, I think that mathematical composition is first understood from the part of speech, which is a kind of composition, so all means of composition can be used from the genre and rhetoric, but the content must be the process of understanding, feeling, questioning and innovation in the process of mathematical learning, as well as the process of mathematical association or mathematical solution to things and phenomena in life. Mathematical fairy tales and mathematical stories can be used as one of the forms of mathematical composition, just as fairy tales and stories are one of the primary school compositions.
Trial Practice (2)
Based on the above understanding, after the start of school, I began to systematically guide and ask students to do math compositions. According to the actual situation of students, I first put the dividing point on the question of "what to write in mathematical composition".
First of all, what should I write in my math composition?
As mentioned above, a math composition is a composition first, and its genre and rhetoric should be the embodiment of the instrumental characteristics of Chinese class, that is, how to write a math composition is not the task of math class, but what to write is the core issue of math composition. The first thing to do is to let the students separate the math composition from the ordinary composition after accepting the new semester, that is, the math composition that the students give to the math teacher must contain the math that is not in the ordinary composition. Typical non-mathematical compositions and distinctive mathematical fairy tales can be distributed to students as model essays, so that students can make it clear in comparison, analysis and discussion that the mathematicity of mathematical compositions comes from their knowledge, discovery, understanding, perception, questioning and creation of mathematical knowledge in the articles, instead of just replacing the names of the protagonists in ordinary compositions with mathematical nouns. This is a mathematical composition. Similarly, if the word "math" is used in the composition, then use something else. Therefore, if you want to write a good math composition, you should be a serious and responsible person, always pay attention to the math problems you are interested in inside and outside the class, pay attention to the process of observing, operating, discussing, communicating, guessing, summarizing, analyzing and sorting out in your math study and practice, and pay attention to how you experience, discover, explore and innovate, which is reflected in your math composition.
After such comparison and exchange, I arranged for interested students to hand in an article every month. See the following table for details:
March (articles) April (articles) May (articles) June (articles)
1 class (33 students) 28 26 25 1 1
Category II (34 persons) 18 23 20 3
Subtotal 46 49 45 14
Total 154
Judging from the content, there are still many applications introducing some mathematical knowledge or skills in March, such as naughty 0, great decimal point, simple calculation (in fact, only an additive exchange rate is introduced) and so on. In April, the content focused on mathematical papers, such as counting soybeans, how many soybeans are there in one kilogram, and nut shaking experiment. It can be seen that through a period of trial, students can not only write mathematical compositions, but also gradually increase the scope of content and broaden their thinking.
In fact, since May, most students have a clear definition of mathematical composition. So how can we attract students to return to their desks to write math compositions after school?
Second, let students feel the happiness of autonomy.
After all, math composition is a sideline for students after school. If students don't want to write, just hand it in directly, and the quality is definitely not guaranteed. Therefore, in order to make children write happily, it is necessary to take the following measures:
1. guide and let go.
The premise of writing a mathematical composition is to get rid of the restrictions and constraints on students (perhaps this is the biggest advantage compared with ordinary composition, which should have an outline after all, not a mathematical composition), so that students can draw up their own topics, choose their own materials, write their own ways, and write whatever they want, regardless of length and form, so that students can express themselves freely and give full play. In May and June, some students did get a lot of good articles at their fingertips, covering a variety of topics, such as the experimental report-calculating the difference between the estimated number of words and the actual number of words in the Romance of the Gods, and six? My Spending on Children's Day on June 1st, Mathematical Problems in Buying Clothes, Additional Conditions and Today I'm in Charge are all reasonable, and mathematics must not be regretted in the article. There are fairy tales, experimental reports, jingles and narratives in the form, and the enthusiasm and ability of students to create are amazing. This fully shows that mathematics can really help children "learn to use mathematical thinking mode to observe and analyze the real society, solve problems in daily life and other disciplines and sciences, and enhance their awareness of applying mathematics", and mathematical composition can make us clearly see that this goal of mathematics has been realized by these children.
However, some students feel dull and careless when observing mathematical problems. Usually, we should pay attention to guiding and guiding these students to discover the theme of mathematical composition, and it is best to discuss the writing form together. Usually, students' speeches or performances in a certain part of the class are particularly wonderful, and the controversy over a certain problem is particularly great, and a certain problem is creatively solved ... and so on, it can be mentioned that this is a good subject for mathematical composition. For example, reports on counting soybeans, interesting calculations, cries of numbers and calculators, routine or calculators, etc. It's all students' thoughts after class. When you get along with your classmates outside class, once you encounter something that can or must be solved with mathematical knowledge, you should spare no effort to remind and help: what mathematical knowledge can be used to solve this matter, how to solve it, and how to express it with mathematical composition. For example, calculating the total distance skillfully, going to my sister's house and calculating the departure time, whether this is economical or not, are all excavated in chatting and joking with my classmates. In addition, before the third weekend of each month, remind students that it is almost time to collect math compositions once a month. Although there are no propositions or unclear requirements, students can be consciously reminded that some "tidbits" and fragments of this month's math class can be recorded in math compositions, and which activities in schools or classes contain math problems, such as "I explain equations", "interesting natural numbers" and "experimental reports".
The student wrote a math composition under the guidance of the teacher. After being affirmed, it will greatly stimulate his interest in continuing to write, and this interest will make him carry out the next article with a goal and have the desire to complete it independently, thus forming a benign development process from scratch. But in this process, we must pay great attention to the discretion of giving and helping.
2. Combination of guidance and self-transformation.
Students use their leisure time to write math compositions. As the demand side, we must take it seriously. This is not only a respect for students' labor, but also an affirmation and encouragement for the quality of students' mathematical composition. Therefore, after receiving the students' mathematical compositions, the first thing to do is to read them carefully and look for "highlights": the theme, writing techniques and length are all ok. In short, we should find a good place to fully affirm and praise, especially for students who have written hard, and we should publicize it with great fanfare to be worthy of the hard work of the students. Of course, proper guidance is also essential. The main purpose is to further stimulate students' critical thinking consciousness, broaden their horizons and improve their thinking level. Then according to the comments after the article, they will communicate with the students individually, and on the basis of the students' understanding of the teacher's point of view, they will negotiate to modify the article, and then let the students modify it independently. If there are any questions, they will negotiate and revise them again to make every math composition clear. Of course, this process will be very arduous, so it is generally necessary to guide 6~4 students every month, taking into account three levels at a time, so that almost every student must have at least one decent math composition after a semester. The rest is for students to revise carefully according to the teacher's comments, and then communicate with their classmates.
3. The combination of presentation and communication.
Students' mathematical composition can be said to be a combination of wisdom and ability. Teachers' appreciation is definitely not enough. If more people can appreciate their works, they may be inspired to write more works. To this end, students' works can be displayed and exchanged in the following ways.
(1) Classroom presentation and communication
Students' mathematical compositions can be displayed and communicated on a platform.
All mathematical compositions can be displayed on the wall newspaper and blackboard newspaper, and students are required to evaluate them. It can be written on the displayed manuscript or directly communicated with the author. This process should improve what they see and what they are watching.
Some representative, typical and conditional electronic texts can be carefully commented by teachers, compiled into tabloids and printed and distributed to each student, so that students can take them back to enjoy and comment with their parents. In addition to their parents' affirmation, they hope to get their parents' support and help ―― outside school, they can often remind students to pay attention to the discovery and application of mathematics.
Because parents should also participate, the teacher's comments must be accurate after each math composition, pointing out the characteristics and strengths of the article for parents to understand. (The article is attached with the 1~5 issues of mathematical composition newspaper compiled by the author. )
(2) On-campus display and communication
If mathematics composition can be implemented in middle and high grades, it may be better to show and communicate between grades. Senior grades can learn from the childlike innocence and interest of junior grades, and junior grades can learn from the thoughtful and unconventional thinking of senior grades. The form can still be through wall newspapers, blackboard newspapers, self-made tabloids and other forms. Because I only tried it in the fourth grade in practice, this is my idea for the next step. However, I sent the fourth-grade math composition to the students in the third-grade thinking training class I taught)
(3) Online display and communication
When the network has gradually become a part of modern people's life, since there are electronic texts, communication on the internet has become a natural thing. Now many schools in Hangzhou have intranets, which is of course a resource that can be used; Shangcheng District's "Shangcheng District Education Information Port" provides an inter-school communication platform and a column of "Students' Works"; Many educational websites have opened up "educational forums". After registering (generally free of charge) as a member on these websites, you can show students' works, or students can register as members themselves and pay attention to the popularity of their works, because some readers who have browsed will put forward their views and opinions by leaving messages, which is very challenging and exciting for students.
Through the above methods, most students can successfully complete a math composition every month in a semester, so does everyone have to write one every month?
Third, math composition is not a burden of schoolwork.
As teachers, we must never deny that students are different. No matter how unlimited the content, subject matter and length of mathematical composition are, it is still an "impossible task" for individual students. What do we do varies from person to person.
Mathematics composition is not in the mathematics curriculum, so it should not occupy the classroom time generally, and should be arranged as an extracurricular assignment. Students who have spare capacity or are interested in learning can be encouraged to write as much as they want on the basis of finishing one article every month, and can also help students who can't write; For those who don't want to write, communicate with him more, prompt him what he can write and tell him how to write. Over time, maybe he will want to try. If he tries, even if there is only one sentence, praise and encourage him ―― after all, he has finished the extra homework.
In any case, students should not be bored ―― writing well is the eternal theme of mathematics composition.
Understanding after practice
In a semester, in the process of exploring this form of homework, the author gradually got the following understanding:
1. On the Function of Mathematical Composition.
(1) helps students to sort out and show their thinking process of acquiring knowledge, and also helps students to sum up their learning methods and rules. For example, "How many soybeans are there in one kilogram", the estimation process is reviewed and sorted out in the writing process, while "I said to solve the equation" is the summary after learning the "Preliminary Equation".
(2) Provide opportunities for students to explore independently, provide positions for students to express their independent opinions, and create new conditions for cultivating students' innovative consciousness and practical ability. Such as ""six? Children's Day consumption, the math problem of buying clothes, extra conditions, I have the final say today, the economy is not economical … and so on. Are the results of students' independent research and exploration.
(3) Mathematics composition has also changed the situation that the form of mathematics homework is single and the content is monotonous, which is conducive to stimulating students' interest in learning mathematics.
(4) Mathematical composition also lays a foundation for students to write scientific research papers in their future study and work.
I once saw an article written by a student in the New Concept Composition edited by Yang Mingming, to the effect that the math teacher was surprised that he would ask a group of four people to write about 1000 words (on? ), because it is very painful to write a 400-word composition. In fact, it is only in recent years that primary school students, even high school students and college students write investigation reports and case studies. Whether the new curriculum standard requires students to investigate and analyze their life practice in middle and senior grades, if there is no certain preparation, these contents can not be implemented, let alone deepened. Therefore, there are usually forms of homework such as mathematical composition. When students need to write some small topics with academic characteristics, they won't feel at a loss, and teachers don't have to worry about whether to keep such teaching content.
Practice has proved that mathematical composition can better reflect the diversified teaching objectives of the new curriculum standard ―― perhaps it can be considered as an evaluation standard of mathematical ability.
2. Several main types of mathematical composition.
(1) explanation and introduction. The main content is to explain and introduce the characteristics of a certain knowledge point, with the aim of giving people who have not learned this knowledge a preliminary understanding, which is often the first choice for beginners to write mathematical compositions. Sometimes, in order to avoid boring math preaching, fairy tales are often used to explain with cute animals, numbers, symbols and other images. Of course it's hard to write well. Students are advised to read Professor Li Yupei's series of mathematical stories, including Adventures on a Desert Island, Wonderland of Numbers and Detective Axe, which are written in three forms: adventure novels, fairy tales and detective stories, all of which describe the characteristics of mathematical knowledge.
(2) retrospective display type. That is to recall and sort out a knowledge learning process or a mathematical practice activity process, so that readers can use successful experiences and avoid detours when they encounter similar problems after reading.
(3) Experience introduction type. That is, mastering and understanding specific knowledge or summing up experience in solving a problem. It is suggested that students watch Academician Zhang Jingzhong's mathematics lecture album, including the eyes of mathematicians, to help you learn mathematics and new concept geometry.
(4) Exploration and discovery. It is a reflection on the mathematical problems found in the process of learning and applying mathematics. Including two kinds: ① found in the process of learning. 2 found in life.
(5) Innovation and variation. The highest and ultimate realm of mathematical composition. The content is self-evident.
There are still some problems in math composition.
(1) From the content point of view, students like to explain and introduce, like to review and show, and like to introduce experiences, but it seems that they can't explore and discover, innovate and change. It should be said that there is still a long way to go to cultivate students' innovative consciousness.
(2) From the formal point of view, students prefer fairy-tale math compositions, but they have a strong sense of patterns and often use anthropomorphic numbers, symbols and figures. To express their own mathematical knowledge points, but the overall control over the creation of scenes, the refinement of language and the introduction of knowledge points is weak.
(3) From the creative enthusiasm, students are most passionate when writing the first 5~ 10. In that short time, they are very attentive to the study and application of mathematics, always looking for themes and thinking about formats. However, when the blackboard newspaper also went up and the internet died, the enthusiasm declined, and it seemed that it was impossible to maintain that keen observation and passionate imagination. How to make students have the consciousness of constantly surpassing others and themselves has become the key to the endless mathematics composition.
(4) From the teacher's guidance, because mathematics composition is not the content in the classroom, the communication between students and teachers, students and students can only be carried out in the form of "point-to-point", and the possibility of "point-to-face" is extremely small, so the efficiency is relatively low. How to choose a relatively fixed time for class-wide communication and evaluation has become a problem.
(5) From the perspective of subject integration, mathematical composition can be said to be the natural intersection of the two courses, but due to the current situation of the courses, mathematical composition is still a unilateral enthusiasm of mathematics teachers. In the process of guiding students, there are too many instructions and requirements for writing techniques and rhetoric, which seems to be "crossing the line" and impossible. At the same time, for students, composition is an inevitable requirement of Chinese, while math composition is purely interest. When the two conflict, it is very painful for students. Therefore, how to implement "subject integration", improve the guidance efficiency of mathematical composition in English and mathematics courses, and reduce students' academic burden is of interest to teachers and schools.
The implementation of the new curriculum standards has made teachers' teaching content, teaching form and teaching process develop and innovate unprecedentedly. As a new form of homework, mathematical composition has attracted more and more teachers' attention because of its unique value. This paper is willing to provide some thoughts, gains and puzzles of the author in practice to teachers and experts, hoping to play a role in attracting jade.