Shen Jiangang from Xiaoshan Middle School in Zhejiang Province

I. Introduction

Academician Wu Wenjun, a famous mathematician, pointed out in the article "Mathematics education can't start from the requirements of training mathematicians" that "any mathematics should talk about logical reasoning, but this is only one aspect of the problem, and it is more important to use mathematics to solve problems and solve mathematical problems in daily life and other disciplines. All the questions given by the school have answers. What is known and verified is clear and must be done. However, when you come to society, you don't know the answers to most of the problems you face in advance, or even whether there will be answers. This requires cultivating students' creativity and learning mathematical methods to deal with various practical problems. "

In the newly compiled mathematics syllabus for senior high schools, research topics are described as: in-depth discussion of some mathematical problems, or research on some problems in daily life and other disciplines from the perspective of mathematics. Mathematical research projects emphasize students' practical activities, pay attention to students' active exploration of problems, advocate the real experience of learning process, comprehensively cultivate students' mathematical literacy and humanistic literacy in an open and free environment, and realize people's internal promotion and subject liberation.

How to implement inquiry learning in mathematics. How to effectively change teachers' teaching methods and students' learning methods? This is a difficult problem faced by every math teacher at present, and it is also the key to the current math teaching reform. Incorporating research topics into the syllabus effectively promotes the development of research-based learning, makes teachers' understanding of research-based learning more specific and clear, and makes its implementation more directional and safe. By studying the research project theory in the new syllabus and combining with the reality of our school, all the teachers in the mathematics group of our school have made some attempts in developing research projects by using modern information technology.

Second, the understanding of the subject objectives

The purpose of setting up research learning in China is to change the way students passively accept knowledge, build an open learning environment, provide opportunities to acquire and apply knowledge through multiple channels, form a positive learning attitude and good learning strategies, and cultivate innovative spirit and practical ability. Under this general goal, infiltrating into the mathematics discipline, the outline puts forward four specific teaching objectives for the research topic:

Learn to ask questions and clarify the direction of inquiry. Research projects focus on students' subjective status, and students, as practical subjects, choose and determine their research direction from their lives according to their own interests and needs. In this way, it is easy to change passive acceptance of knowledge into active acquisition of knowledge, and change simple rote memorization into flexible application and expansion.

Experiencing the process of mathematical activities. Constructivist learning theory holds that the world exists objectively, but the understanding and significance of the world is determined by the individual himself. Learning is not a simple process from ignorance to knowledge, but mainly a process in which learners actively construct meaning on the basis of the original cognitive structure through a comprehensive and multifaceted experience of the perceived object. Therefore, it is particularly important to grasp and experience abstract mathematics.

Cultivate innovative spirit and application ability. How to embody the requirements of innovative education in subject teaching has always been the focus and difficulty of reform. The research project seeks a breakthrough from the design and organization of teaching, focusing on students' "how to learn" and "how to learn", helping students to obtain healthy and perfect learning strategies, so that cognitive ability and non-cognitive ability can be fully developed. Enable students to cultivate scientific attitude, innovative consciousness and innovative thinking methods in the process of personal practice.

Reflect research results in the form of research reports or small papers, and learn to communicate. Compared with other disciplines, mathematics pays more attention to expression, and abstract problems are vividly expressed, which is also an important part of mathematics learning. Organizing students to reflect research results in the form of writing research reports or small papers not only enables them to learn the format of research reports and papers, but more importantly, enables them to learn mathematical expressions. An accurate and scientific expression is equivalent to a mathematical re-creation.

Third, several entry points of the implementation subject.

To carry out research projects, the first task is to choose the learning content, which should be based on the learned mathematical knowledge and closely combined with the actual life and production. The theme should not only stimulate students' interest in learning, but also consider students' learning characteristics and learning level, so that students can gain something and think. In practice, we mainly cut in from the following points.

1, from the research topics, reading materials, exercises and other places given in the textbook.

In the new textbook, reading materials, exercises or research topics are arranged after almost every chapter. Reading materials are often a brief introduction to the generation and development of knowledge in this chapter, and students can be required to collect information and make detailed reports through the internet, libraries, expert interviews and other means. Internship assignments often give students an idea, asking them to ask a question, design a solution, investigate and collect data, and analyze and solve the problem. The research topic gives the research content and requires students to conduct research and draw conclusions.

For example, in the reading material "A Brief History of Logarithm and Exponential Development" on page 85 of the first volume of the new textbook, we guide students to do further research on things they are interested in on the basis of reading materials. After the homework, some students used the website given in the textbook: www.0-100.com.cn/2/21/02/0741.htm to further explore the development of logarithm, and some students became interested in the mathematicians mentioned in the material. They went to the library to look up information.

If our teacher can be a conscientious person, then we find that there are many materials in the textbook worthy of students' understanding and learning. Through the excavation of these materials, students can not only understand the history of mathematics development, but also learn from some mathematics masters what they can't learn in class.

2. Cut in from the extension of knowledge.

Due to the limitation of syllabus and textbook content, many contents in the textbook cannot be expanded and extended excessively, so we can choose some contents that can arouse students' interest without increasing their burden as a subject.

For example, in the process of learning conic curve, the searchlight experiment is used to make students interested in the optical properties of parabola and stimulate their research enthusiasm.

Another example is to guide students to understand Fibonacci series by observing the distribution law of sunflower seeds in the chapter of "Series".

In particular, it should be pointed out that the extension of knowledge should be based on the principle of not increasing students' burden and interest, and at the same time, students' mastery of extended knowledge should not be used as the evaluation standard, but only on their experience in the process of inquiry.

3. Cut into the open questions.

Because of the openness of conditions and conclusions, open-ended questions are easy to give students broad thinking space and freedom of research, so that they can put forward their own ideas and methods at different levels of experience and ability. For example, in our "colorful midpoint trajectory" project, a scene map is set on the geometry drawing board: circle O and moving point P and fixed point A on the circle, and students are required to explore the trajectory of midpoint M in P A. We encourage students to boldly change their own conditions and conclusions and conduct in-depth research. In this way, students' divergent thinking can be fully demonstrated and students can have an innovative experience.

4, from the application of knowledge in life.

The new mathematics syllabus clearly stipulates students' ability to solve practical problems: they will put forward, analyze and solve mathematical problems with practical significance or in related disciplines, production and life; Can use mathematical language to express problems and communicate, and form the consciousness of using mathematics. It can be seen that applying knowledge to solve problems in life has been understood as an effective way to improve students' quality. Moreover, because of the wide application of mathematics and the colorful life, students can find many math problems that they are interested in. By studying these problems, they not only deepen their understanding of mathematics knowledge, but also enhance their sense of mission and responsibility to society from their participation in social problems.

For example, the household electricity consumption is 0.53 yuan per kilowatt hour, and each bottle of gas is 53 yuan (3 1.5 0.5 kg). Who is more cost-effective using electricity or gas?

Another example: according to the latest bank interest rate, how to save money for maximum profit?

Another example: it is the turn of a family on the floor to collect water charges. Because the time when the toll collector copied the user's sub-meter was later than the time when the water company copied the floor master meter, the total number of sub-meters was more than the meter reading. How to deal with it more reasonably?

We constantly remind students to be conscientious, learn to look for math problems from around them, and encourage students to boldly enter the society, learn to investigate and learn to discover.

Four, the use of modern information technology to carry out projects.

The integration of modern information technology and disciplines has achieved changes in the presentation of learning content, students' learning methods, teachers' teaching methods and the interaction between teachers and students, providing a rich teaching environment and powerful learning tools for students' learning and development. The rational use of office tools, campus network, internet and mathematics teaching software provides a strong guarantee for the development of mathematics research courses.

1, Application of Office Tools and Software

Microsoft's Office tool software is loved by users all over the world because of its good interface, simple operation and powerful functions. The rational use of information technology in the process of developing mathematical research projects will greatly facilitate students to collect and process mathematical data, enrich the ways of exchanging research results and stimulate students' research enthusiasm.

In practice, we know that in the information technology course of senior one, students have basically learned some basic operations of Word and Excel. At the same time, we use lectures to let students know mathtype in detail in the slide making software of Word and Powerpoint. Although teachers and students spent some time learning to use these tools, later, we found that the use of tools aroused students' interest, promoted the development of courses and improved the quality of research. For example, Wu Zhijian, a student in Class 0307, used Excel to make the collected traffic meters of various vehicles into vivid and intuitive graphics in his paper "Let Numbers Speak with Images-Talking about the Application of Statistical Graphs", thus showing the hidden laws. At the seminar of the research topic "Functions in Life" in Class 7 and Class 8, Grade 2004, a cooperative team used Excel to process the collected data, and then used the learned function images to fit according to the images, so as to determine the fitting function, and then used the fitting function to predict future events. There are also some cooperative groups that make good use of Powerpoint to show their research process vividly, intuitively and clearly with presentations, and have achieved good results.

Wu Zhijian drew a motorcycle traffic map with a line chart.

The pie chart in Wu Zhijian shows the percentage of traffic flow of various vehicles.

2, the use of the network

Because of the richness of information resources, the infinite openness of time and space and the sharing of information resources, the network itself provides an excellent platform for the development of mathematics research courses. On this big platform, students are exposed to a lot of information, and the chances of resonance are correspondingly increased, so that the content involved is closer to themselves, and the exploration of knowledge is more conscious, deeper and internalized.

During the development of the topic "Interest Rate in Life", we made some attempts in the application of network.

Our teacher first made a theme website on this topic. The website opened several small columns, and set up the problem scene of the topic in the column of "Question Raising" to induce students' research motivation. Show the cases made by teachers in the "Examples" column to stimulate students' inspiration; In the "link" column, provide some materials and exploration ways to help students determine the exploration direction.

We made a research-based learning theme website.

In the experiment, Lin of Class 02 10 became interested in the personal housing loan calculator () on the website of China Industrial and Commercial Bank, and he wanted to explore its calculation formula. After many experiments, calculations and verification, he finally came up with the formula:

Where: X is the monthly repayment amount (equal amount), A is the monthly loan interest rate, N is the loan term (year), and M is the total loan amount.

Although this formula is very common, it comes from his own discovery and exploration, which has gone beyond the meaning of mastering knowledge itself for him. The network gives students the opportunity to discover problems and explore knowledge.

Gao became interested in various deposit methods of banks. Assuming that there is 10,000 yuan to deposit10 years, he found that depositors can choose one-year, three-year and five-year deposit methods. Then he calculated the profits of three deposit methods respectively, compared the list, and found that the method of transferring once every five years was the most profitable. Finally, he explained the rationality of the bank's interest rate setting. From the understanding of various deposit methods, to the calculation of interest rates and tax rates, and then to the consideration of bank interest rate setting, all actions are mastered by students themselves. As the main body of the research, students can really feel the ups and downs in the implementation of the project.

Comparison chart of principal and interest earned by three deposit methods (1* 10 means one-year deposit 10, with 9 transfers in the middle).

Curriculum networking is the necessity of technological development. In this regard, our exploration is still very limited, such as the remote interactive function of the network and the sharing of resources. We are full of expectations for the future of the network. We imagine setting up our own mathematics website and opening up a research topic column so that research topics can be shared by everyone.

3. The application of mathematics teaching software "Geometry Sketchpad".

The geometric sketchpad takes points, lines and circles as basic elements. Through the transformation, construction, calculation, animation and tracking of these basic elements, it displays or constructs more complex graphics, animates more abstract mathematical objects, and makes people know the unchangeable relationship of mathematical objects in dynamics. In the course of the project, guiding students to observe, guess, verify, reason and communicate is helpful to stimulate students' enthusiasm for exploring mathematical problems, ensure the effectiveness of exploration, enable students to accumulate rich mathematical experience, and is conducive to the formation of mathematical innovation consciousness.

In the process of developing the theme of "function in life", teachers guide students to study the following problems with the geometric sketchpad:

As shown in the figure, a real estate company plans to build a residential park, rectangular GHCK, on the rectangular plot ABCD. In order to protect the cultural relics area △AEF, the vertex G of the rectangular park cannot enter the protected area. Known AB=200m, AD= 160m, AE=60m, AF=40m. How to plan the park to ensure the largest area?

In practice, students use the sketchpad to make a simulation diagram of practical problems at the ratio of 1∶4000, and take g as any point of the line segment EF. Using the calculation function of sketchpad, the length of line PG and the area of rectangular GHCK are calculated. Drag the G point, and the size of the area will change accordingly. When the area is the largest, use the displayed area value to find the position of the G point. One of the students skillfully made a point in the coordinate system with the length of PG as the abscissa and the area of right-angle GHCK as the ordinate, and then made the trajectory of the point around the point G, and the law of the area changing with the point G was clearly revealed.

In regular teaching, the general teacher always talks about this kind of application problems, guides students to establish functional relationships, and uses the maximum value of functions to solve problems. On the surface, it seems understandable. Then, if we think from the students' point of view, students actually lack perceptual knowledge of the given problem, neglect the exploration of various problem-solving methods and miss the opportunity to have more experience.

The simulation map made by middle school students in the study of the maximum area of the park

When students use sketchpad to solve problems, we find that they have not established a functional relationship. Instead, the area of G-point in each position is calculated by experiment with the help of the quick operation function of the drawing board, and finally the answer is found and the problem is solved by the simplest method. Obviously, the students who make images have creatively solved the problem. During the experiment, we also found that many students were not satisfied with the answers. On the contrary, finding the law in the experiment urges them to explore the law and make them return to the method of using functions. But at this time, their understanding of function method and the application of mathematics in life are quite different, which is probably the significance of setting up research topics.

Verb (abbreviation of verb) project implementation procedures and cases

Different research topics have different research methods and implementation procedures due to different contents involved. The following are the implementation steps when we do the internship assignment behind the new textbook "Practical Application of Functions". We think it is representative.

1, publicity and mobilization; By showing the "practical application of function" in the internship done by the last classmate, students' desire to participate can be stimulated and their general operation ideas can be understood.

2. Learning transcendental knowledge; Software Powerpoint, Excel, word, geometry sketchpad, fitting function, data statistics and other related knowledge learning.

3. Composition and task layout of the cooperation team; Set up a project team of 3 to 7 people in the form of free combination, elect the team leader, discuss the project content and implementation plan, and announce the discipline and requirements for external transfer.

4, investigation, research, grasp the first-hand materials; Go outside to investigate as planned, or go online or go to the library to check the information.

5. Collate and analyze data, and make images and tables; Make a preliminary arrangement of the collected data, and then make images or tables by using the geometric sketchpad or Excel.

6. Fitting the obtained image with the learned function, and obtaining the analytical expression of the fitting function through calculation; If you can't find a suitable simple function, students are advised to use piecewise function to fit it.

7. Make some preliminary expectations for some practical problems by using the obtained functional relationship; The main purpose of obtaining the function relation is to make some expectations, which can encourage students to make all kinds of bold expectations for practical problems and suggest students to make reasonable explanations for the expected results.

8. Write an internship report; You can write an internship report according to the format of the textbook or you can write it yourself.

9. Use Powerpoint to make the results report and hold the results exchange meeting. According to the practice report and combined with the research experience, each project team is required to make a presentation, show and communicate at the exchange meeting. The presentation teacher should let the students play the leading role, mobilize one person from each project team to speak and encourage other students to ask questions. Wonderful speeches and humorous questions can make every student feel the knowledge and happiness brought by the project.

Case: Investigation Report on Per Capita Income of Urban Households in Xiaoshan District (050 1 Class Chen Kai Group)

Survey data (per capita income of urban households in Xiaoshan District 1986- 1998)

Drawing analysis:

Function fitting:

Phenomenon analysis: According to the picture, the per capita income growth can be divided into three stages:

1 growth rate. 86-92 was slow, because the reform was not strong enough and the mind was not completely open.

2. From 1992 to 1994, due to1992 talks in southern Deng Xiaoping and the convening of the 14th National Congress of the Communist Party of China, the reform and opening up reached its climax and the growth rate was obviously accelerated.

3. From 1994 to 1998, with the increase of income base, the growth rate slowed down, but it still increased at a relatively fast speed.

It is predicted that the per capita income of urban families in Xiaoshan District will continue to grow at a faster rate.

VI. Significance of implementing this project

1, scientific research projects can stimulate students' vitality. From the establishment of the topic, to the organization and implementation of the topic, and the exchange of results, students have devoted great enthusiasm, which is a learning atmosphere that we rarely see in our usual teaching activities, because they have to find their own topics and go out for research, and challenging work stimulates them and inspires their vitality. Many times, we are really surprised by the creativity of students. In the process of completing the practical application of the function, if a student studies the votes of a singer's songs in the first few weeks, and then obtains the law of the function, and then uses the function to predict the time when the song falls off the list. For another example, some students collected the GDP growth rates of China, Japan and the United States in recent years, and then predicted the time when China's GDP would overtake those of the United States and Japan. Different students do what they are interested in according to their different hobbies, which arouses their enthusiasm.

2. Research projects give every student the opportunity to exercise various abilities. The implementation of research projects is a process of testing students' various abilities. The implementation of the project needs the close cooperation of team members, and students need to learn to get along, communicate and cooperate. Going out to investigate requires social skills, writing investigation reports and exchanging results. What is needed is the ability to express.

3. The research project has deepened students' understanding of the essence of mathematics. It is a good inspiration for students to use mathematical methods and ideas to solve problems in real life, which is conducive to understanding mathematics and prompting them to learn mathematical knowledge, methods and ideas better.

Seven. Thoughts triggered by topics

In practice, we find that the development of a project often needs many processes, such as mobilization in advance, preparatory knowledge training, project organization and start-up, guiding implementation, achievement exchange and so on. In the middle, we need not only the participation of subject teachers, but also the cooperation of various departments of the school. It is a systematic work, which needs the overall planning and reasonable arrangement of the school.

In order to give students more experience opportunities, the development of the project must have sufficient time to ensure, but in practice, many times, teachers are unable to do this. Because under the heavy academic burden, students really can't take care of their subjects, let alone make in-depth exploration and extensive contact. The research topic calls for free students and a free sky.

In the process of guiding students, our deepest experience is that teachers should learn, and the knowledge of modern information technology and related disciplines needs us to reserve and update. "To give students a glass of water, the teacher must have a bucket of tap water."

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