Inquiry learning is a brand-new research topic in the field of educational theory and practice in recent one or two years. Research-based learning strengthens the connection between students and social development, which will fundamentally change students' learning methods and provide time and space guarantee for students' all-round development and training creative talents. With the promotion of theoretical research and educational policy, research-based learning has been paid more and more attention in educational practice. Because inquiry learning has just been put forward, its theoretical research and practical exploration are far from enough, so there are often many puzzles in the specific operation: what is inquiry learning? What is the goal to be achieved? How to implement it? This paper reflects and summarizes some problems in junior high school mathematics inquiry learning.

[Keywords:] junior high school mathematics, inquiry learning, reflection and summary

Text: Inquiry learning is a brand-new research topic in the field of educational theory and practice in recent years. Research-based learning strengthens the connection between students and social development, which will fundamentally change students' learning methods and provide time and space guarantee for students' all-round development and training creative talents. Inquiry learning refers to the learning activities in which students choose and determine inquiry topics from their own study life and social life under the guidance of teachers, and actively acquire knowledge, apply knowledge and solve problems. As a compulsory course, research-based learning has been included in the national nine-year compulsory education curriculum plan, aiming at cultivating students' scientific literacy, innovative spirit and practical ability. This paper discusses and thinks about some problems in junior high school mathematics inquiry learning.

1. Curriculum View of Inquiry Learning

The inquiry learning of junior middle school mathematics mainly refers to the learning way that students acquire knowledge, apply knowledge and solve problems in a way similar to scientific research under the guidance of teachers. Here, students' inquiry learning is carried out under the guidance of teachers and in the environment of class collective teaching, which is different from individual spontaneous and individual inquiry activities in the process of self-study. Teachers' main roles are organizers, guides and collaborators in mathematics learning. "In a way similar to scientific research", that is, let students reveal the law of knowledge and solve problems through the process of "observing and comparing, discovering and asking questions, guessing and solving, trying to answer and verifying". Its essence is to let students learn the thinking mode and research method of scientific research, and use the existing mathematical knowledge and mathematical thinking method to solve problems, so as to cultivate students' ability to explore, acquire knowledge and solve problems actively, cultivate students' innovative thinking and practical ability, and cultivate students' broad thinking.

1. 1 Attach importance to students' independent activities and realize the change of learning style.

Inquiry learning adapts to junior high school students' desire for independence, hoping to experience success and gain recognition in independent activities. It provides and creates a "soil" and a good atmosphere for students' active exploration, independent operation and free expression, in which students innovate and practice, and establish an active learning mode of active discovery, independent thinking and focusing on solving practical problems. Students learn through experience and creation, and realize the change and development of cognition, emotion, attitude and concept in a subtle way.

1.2 Starting from students' interests, enter the process of exploring questions.

Interest is the starting point of students' inquiry activities, and the satisfaction of interest needs to be realized in the process of inquiry. Paying attention to students' interests and turning them into questions that can be explored can effectively guide students to actively participate in inquiry learning activities and meet their development needs. Such learning activities ignite students' thinking sparks, and are also conducive to cultivating students' questioning attitude and critical spirit, gaining the feeling and experience of learning freedom and happiness, and recognizing and acknowledging self-worth. For example, teachers and students prepare for a day trip and ask students to design a more reasonable car rental plan based on statistical data; Questions like where is the best shot for a football player (regardless of other factors) can stimulate students' interest.

1.3 Provide an open learning space to explore and develop students' various intellectual potentials.

Every student has various intellectual potentials, but there are also obvious individual differences. The openness of inquiry learning enables students to understand social life, know the world and discover themselves through various inquiry methods according to their hobbies. At the same time, it also provides students with opportunities to show their personality and develop their talents in an open situation. For example, when teaching 1 hectare and 1 square kilometer, let students measure and experience their own sizes. Lead the students to the playground, visually measure a square with a side length of 100 meters, and feel the size of 1 hectare; Take to the street, measure the length of 1000 meters, try to estimate where the other two vertices with this side as a square are, and experience the size of 1 square kilometer. Then estimate the size of the urban area, and let the students calculate the population density of the urban area in combination with the knowledge learned in the "Society" class, and put forward the planning suggestions for residents' entertainment and fitness places. Through independent practice, students can experience the size of 1 hectare and 1 square kilometer in the largest space, and feel that mathematics is close to them.

1.4 Teachers and students * * * explore new knowledge together, and the course becomes a process of interactive promotion between teaching and learning.

The implementation of inquiry learning is an interactive process between teaching and learning. In order to establish a new relationship between teachers and students, teachers should realize the transformation from simple knowledge imparting to organizers, instructors, promoters and participants of students' inquiry learning. In this new relationship, teachers and students explore new knowledge and develop their thinking, ability, emotion, values and behavior.

2. The goal of inquiry learning

Research-based learning focuses on changing students' learning methods and cultivating their innovative spirit and practical ability. The main goal is to satisfy students' interest in learning and improve their ability and character. It emphasizes students' practical application of knowledge and skills, the formation of ability and the acquisition of experience. It also emphasizes that students can deepen their understanding of learning value through personal experience and be sublimated in ideological consciousness, emotional will and spiritual realm. Specifically, according to the age characteristics of junior high school students, the goal of inquiry learning in junior high school emphasizes the following points:

2. 1 Get the experience of personally participating in the inquiry activities.

Inquiry learning emphasizes that students can experience the practical process of problem inquiry through independent participation in inquiry learning activities, gain the initial experience of scientific inquiry, deepen their thinking and perception of natural, social and life problems, stimulate their interest and desire in exploration and innovation, and gradually form the psychological quality of loving questions, being diligent in thinking and being willing to acquire new knowledge in inquiry.

2.2 Improve the ability to find and solve problems.

Research-based learning pays special attention to improving students' ability to find problems from observing and thinking about life and actively explore through practice. Including: finding and determining the subject (project) to be explored; Put forward exploration ideas and carry out exploration activities independently; Get the conclusion of the inquiry, make a preliminary prediction on the development of things (problems) or put forward appropriate countermeasures; Show or exchange inquiry process, achievements, experience, etc.

2.3 Cultivate the ability to collect, analyze and use information.

Inquiry learning is an open learning process. Through inquiry learning, students should learn how to organize and summarize, judge and identify the value of information and make rational use of information; Learn to use the information obtained to describe or explain things (problems) and make appropriate explanations.

2.4 Learn to cooperate and share

Inquiry learning provides a good space for interpersonal communication and cooperation. Students should learn to cooperate and cultivate team spirit in the learning process. Including: independent thinking and initiative in the cooperation group, and willingness to help and cooperate with partners; Consciously abide by the norms of cooperation and correctly treat the relationship between individuals and groups; Be able to coordinate interpersonal relationships in cooperation, actively communicate with peers and share information, ideas and achievements. For example, "the relationship between the monthly living expenses of resident students and their family income", "the number of weeks in school and the number of times to go home" and so on. The presentation, analysis, modeling, answer and summary of each question should be the crystallization of students' active knowledge, unity and cooperation.

3. The content of inquiry learning

According to the law of students' cognitive formation and development, junior high school mathematics inquiry activities can be divided into:

3. 1 Formative inquiry refers to some typical materials in the formation of design knowledge as inquiry questions: these materials can be the process of putting forward mathematical concepts, formulas, theorems and laws; The process of derivation, analysis and demonstration of the conclusion; The process of knowledge occurrence, development and formation; The exploration process of solving problems; The generalization process of problem solving methods and laws. For example, in the concept teaching of equations, the traditional method is to give the definition of equations first, and then give some formulas to let students distinguish which equations are. The practice of inquiry learning is to give some formulas first, and then let students observe and find out some similar characteristics. For example, some formulas are equations, some formulas are algebraic, and some contain unknowns, so we call this equation with unknowns equations.

3.2 Constructive inquiry refers to guiding students to establish knowledge systems and networks and form a good cognitive structure on the basis of understanding mathematical knowledge. This process should be completed by students themselves, which is conducive to deepening their understanding of the knowledge system they have learned and laying a good foundation for cultivating innovative thinking. For example, when learning the content of "triangle internal angle sum", the most common way for students to operate in teaching is to measure each internal angle of a triangle with a protractor at the teacher's prompt or request, and then add them up, so that the conclusion that "triangle internal angle sum is 180" is over-sampling. With the understanding of modern intuitive teaching, we can't help asking such a question: besides drawing this conclusion, students can't help asking. If students know the sum of the internal angles of other polygons such as quadrangles and pentagons. Can students only use hands-on measurement? In fact, teachers can provide materials (triangles of different sizes and types) without asking or prompting, and students can actively solve their own problems. In this way, students will not only use the angle measurement method (the simplest general method), but also use the cutting and spelling method (a special reduction method in mathematics). In this way, students not only get the desired conclusion, but also master the most commonly used thinking method in mathematical understanding-induction. After learning the knowledge of quadrilateral internal angles, it can be completely transformed into the understanding of the sum of two triangles and pentagonal internal angles, or it can be completely transformed into three triangles.

3.3 The purpose of applied research is to enhance students' application consciousness, cultivate innovative consciousness and improve their research ability in the development of knowledge; Teachers should actively guide students to get in touch with reality and understand the society, so that they can study in a more open environment and effectively improve their ability to analyze and solve practical problems. For example, "Survey of Middle School Students' Myopia Rate" and "Internet Cafe Cost".

To carry out inquiry activities, we should start with the contents of teaching materials and teaching facilities; We should proceed from the actual situation of students' ability, teach students in accordance with their aptitude and adjust measures to local conditions. Inquiry activities should be carried out in a planned way, combining students' psychological characteristics and cognitive level; Teachers' guidance should also gradually transition from more to less to students' independent inquiry.

4. The implementation of inquiry learning in junior high school

4. 1 Implementation organization form

Inquiry learning has many organizational forms, including group cooperative inquiry, individual independent inquiry and cooperative inquiry in classes and grades.

In the inquiry learning of comprehensive practical activity class, more research groups should be formed to carry out inquiry activities in the form of group cooperation. The research group can be composed of students freely, or students with similar interests across classes and grades. The research group is generally composed of 3-6 people. Students choose their own leaders and hire adults with certain expertise (such as teachers in our school) as tutors. In the process of inquiry, the members of the research group have their own advantages and disadvantages, and they cooperate with each other and complement each other.

When an individual explores independently, the teacher usually puts forward a comprehensive exploration topic to the whole class, and then each student decides on a specific topic and carries out exploration activities independently, which will be completed in a period of time.

In the form of collective inquiry, the whole class needs to collect information and carry out inquiry activities around the same inquiry theme through division of labor and cooperation. Through several classroom discussions, information and ideas can be shared and thinking collided, thus promoting students to deepen their exploration on the original basis.

Take the form of group cooperative inquiry and class collective inquiry, emphasize the active participation of everyone in the group on the basis of individual independent thinking and serious exercise, and avoid the phenomenon that some people are busy, some people are idle, a few people do it and most people watch it. Take the form of individual independent inquiry, guide students to communicate and discuss with others frequently and actively, and learn to enjoy information and resources.

4.2 General process of implementation

The development of research-based learning in junior middle schools is generally divided into three stages: the stage of entering the problem situation, the stage of practical experience and the stage of summary, expression and communication. In the process of learning, these three stages cross each other and advance each other.

(1) enters the problem situation stage.

At this stage, first of all, we should take various forms, create problem situations and clarify the task of inquiry. Generally speaking, you can tell stories, hold lectures and organize tours. The purpose is to pave the way for background knowledge, activate students' original knowledge reserve, provide exploration scope and induce exploration motivation. At the same time, under the guidance of teachers, students should learn to find problems and analyze and think from multiple angles, set up inquiry learning groups, invite instructors inside and outside the school to provide help and participate in inquiry activities; We should actively explore, search for relevant information, enter the state of exploring problems, summarize the specific topics to be explored, and form the most basic goals and ideas.

(2) Practical experience and problem-solving stage

After determining the problems that need to be explored and solved, students should enter the specific problem-solving process, form certain concepts, attitudes and master certain methods through practical experience.

At this stage, the contents of students' practice and experience include: actively collecting and processing information in open situations, group cooperation and various forms of interpersonal communication, solving practical problems with a scientific attitude, understanding the environment from a certain angle and discovering themselves. In the process of solving practical problems, students often encounter various difficulties. The characteristics of junior high school students' quick interest stimulation and change easily make their inquiry activities unsustainable, which requires teachers' timely care, guidance and supervision.

(3) Summary, expression and communication stage

Students should complete the task of research-based learning project from beginning to end and strive to achieve the expected goal. However, it is normal to get unsatisfactory results after hard work, which does not mean failure in learning. At this stage, students should organize and process their own or group's harvest through practice and experience, and form written materials and oral report materials. Students share their achievements with their classmates through communication and discussion, which is an indispensable part of research-based learning. In communication and discussion, students should learn to understand and tolerate, learn to analyze objectively and think dialectically, and be brave and good at defending themselves.

4.3 Teachers' guidance in implementation

In inquiry learning, students are active learners, which does not mean that the role of teachers can be ignored. Whether teachers can use promoting guidance skills is of decisive significance to the development and effect of research-based learning.

(1) In view of the fact that junior middle school students have a weak knowledge base of culture and science and lack methods and experience in solving practical problems, in the initial stage of inquiry learning, students can be given some basic training with examples to help them master the skills of using reference books (such as indexes, abstracts, encyclopedias, etc.). ), using audio-visual media, taking notes, conducting interviews and sorting out classified materials, so as to facilitate students to enter the inquiry smoothly. (2) In the process of implementing inquiry learning, teachers should keep abreast of students' inquiry activities and give guidance, explanation and supervision in a targeted manner; Organize flexible and diverse exchange and discussion activities to promote students' self-education and help them maintain and further improve their enthusiasm for learning; Give individual counseling to groups with special difficulties, or create necessary conditions, or help adjust the inquiry plan. In the implementation of research-based learning, teachers should change their roles from imparting knowledge to organizers, directors and participants of students' learning. (3) When carrying out inquiry learning, teachers should pay attention to the concern, understanding and participation of parents and social stakeholders, develop valuable educational resources inside and outside the school, and provide good conditions for students to carry out inquiry learning. (4) In the process of implementing inquiry learning, students should be instructed to write inquiry diaries, record the inquiry situation in time, and truly record their personal experiences, so as to provide a basis for future summary and evaluation.

5. Design examples of inquiry learning

Every theorem and conclusion in mathematics was discovered by predecessors through hard exploration. Even a general proposition, a conjecture, the process of putting forward embodies the wisdom of mathematicians. The traditional practice is often to give a ready-made conclusion and then copy the ready-made proof. Doing so puts students in a passive position, and students always have doubts: How did this theorem come about? How did you come up with this proof? Junior high school mathematics inquiry learning is to change the passive situation of this kind of learning, eliminate students' psychological doubts, let students actively participate in inquiry, try to find out and become the masters of learning.

For example, the teaching of circular power theorem in mathematics interest classes in grade three:

Question: Any straight line passing through a certain point P intersects with a circle O with a known radius R at two points A and B, and it is known that PO = d, so how to find the value of PA PB?

(1) Draw graphs as required (see Figure 1, Figure 2 and Figure 3) (cultivate divergent thinking)

(2) It may be better to study the case that point P is outside the circle first, which seems difficult to start with, and we can start with a special case. (This process cultivates students' inquiry ability)

Let's see that P is outside the circle, and the straight line passing through point P passes through the center O, as shown in Figure 4. If PO = d and AO = BO = r, we can get PA PB = (d- r)(d+ r) = d2- r2 r) = d2-r2.

I see that the straight line at point P is tangent to circle O at point A, as shown in Figure 5. At this time, point A and point B coincide, and Pa Pb = PA2. According to Pythagorean Theorem, PA PB = PA2 = d2- r2.

Now can you guess the value of PA PB? Try to prove your guess.

As shown in Figure 7, let OM⊥AB have a vertical foot of m, and according to the vertical diameter theorem, point M is the midpoint of AB.

Let AM = BM = x, then pa Pb = (pm-x) (pm+x) = PM2-x2.

Obviously, AM2+ OM2 = r2, PM2+ OM2 = PO2, so there is x2 = r2- OM2.

Therefore, papb = PM2-(R2-om2) = PM2+om2-R2 = D2-R2. Confirmed.

My guess is correct.

The situation that point P is in a circle can be solved similarly; As for the point p, PO = r = d on the circle, it is easy to know that the conclusion is correct.

(3) Students get the theorem of circular power through induction (to cultivate inductive thinking ability)

(4) Can the circular power theorem be proved in other ways? (Cultivate students' innovative ability and the ability to flexibly use mathematical thinking methods)

After group discussion, some students suggested that it can be proved by triangle method.

First, the case where point P is outside the circle is studied, as shown in Figure 8.

Let ∠APO =α, PA = x 1, PB = x2.

△PAO contains x12+D2-2dx1cos α = R2, that is, x12-2dx/cos α+D2-R2 = 0.

In △PBO, x22+ d2- 2dx2cosα= r2, that is, x22-2dx2cosα+ d2-r2 = 0.

So x 1 and x2 are the two roots of the equation x2-2dxcosα+ d2-r2 = 0. According to Vieta theorem, x 1x2 = d2- r2. That is, PA PB = d2- r2 (fixed value).

When point P is in the circle, as shown in Figure 9, let ∠BPO =α, Pa = x 1, Pb = x2.

△PAO contains x12+D2-2dx1cos (л-α) = R2, that is, x12-2dx1cos α+D2-R2 = 0;

In △PBO, x22+ d2- 2dx2cosα= r2, that is, x22-2dx2cosα+ d2-r2 = 0.

Therefore, -x 1 and x2 are two roots of the equation x2-2dxcosα+ d2-r2 = 0. According to Vieta's theorem, (-x 1)x2=d2- r2. That is, PA PB = r2-d2 (fixed value).

If point P is on the circle, then PO = r = d, and PA PB = d2- r2 = 0 (fixed value).

Therefore, regardless of the position of point P, there is PAPB = | D2-R2 ||| (fixed value).

(5) Can you summarize this proposition? (cultivate innovative ability)

After students' group exploration, we can get: intersection theorem, secant theorem, secant theorem. At this point, students' creativity has been fully exerted.

(6) Summary: each group summarizes the problems or mathematical thinking methods that should be paid attention to, and find a representative to speak. Teachers should encourage or improve it.

The whole process is mainly group activities, and teachers give appropriate guidance and explanations at key points, so that students can experience the whole process of scientific exploration. At the same time, it greatly mobilized the enthusiasm of independent learning and inquiry. We have reason to believe that this experience is a thousand times more effective than solving some exercises.

6. Take the pulse for "research study"

6. 1 obvious advantages

(1) The dominant curriculum in the school has become a hidden curriculum, and the classroom has extended from indoor to off-campus. Brand-new learning methods will enable more and more students to study more actively and create a good humanistic environment for their healthy growth. (2) The evaluation of educational results will be far away from the formal theory on paper, and will no longer talk about "heroes" by scores and pursue students' new development in inquiry. I only hope that they will pursue and think at every step, and learn and grow through participation, discovery and experience. (3) "Inquiry learning" will comprehensively cultivate students' pioneering thinking and experience self-confidence, and fully tap the potential of students' sustainable development.

6.2 Potential crisis

(1) "inquiry learning" course is defined as the category of non-exam-oriented education, and it enjoys the treatment of "minor course" for life, but it is dusty because it has been shelved. (2) Over-reliance on "inquiry learning" as a panacea for cultivating students ignores well-trained "basic knowledge and skills", negates the role of "receptive learning" and downplays the psychological shaping of students' personality, will and quality.

In a word, how to carry out "research study" is a realistic topic for every educator. We should constantly try, communicate, discuss and summarize, so as to gradually improve it and make our due contribution to the development of education.

refer to

1. Hu. Ideas on Implementing Inquiry Learning in Schools, Shanghai Educational Science Exploration,No. 1, 2000.

2. Zhang Hua On the Development of "Inquiry Learning" Course in Basic Education 200 1.5

3. Zhao manjun. A guide to research study for senior high school students. Liaoning Normal University Press.

4. Shan Wenhai. Inquiry learning starts with questions. Mathematical bulletin 2003.4

5. Jiang Peijin. Some thoughts on developing "research study" activities. Mathematical bulletin 2003.5

6. Wang Guangming. Several problems about research-based learning. Mathematical bulletin 2003.5