First, the ideological basis of "micro-inquiry" task design
According to the definition in American National Science Education Standards, inquiry refers to a series of activities, including observing, asking questions, collecting information, making research plans, evaluating existing conclusions based on experimental evidence, collecting and analyzing data, giving answers, explaining and predicting, and communicating results. Inquiry requires clear assumptions, critical and logical thinking, and other alternative explanations should be considered.
Inquiry learning is a learning method based on exploring the essence. It is a learning method according to the method and process of scientific inquiry. Through students' doing, thinking about how to do and even thinking about what to do, students can gain experience, understand and apply scientific research methods while mastering the subject content, and then form scientific research ability. Obviously, this is a more vital way of learning, especially for people's scientific research ability. However, the practice of classroom teaching proves that this kind of inquiry learning in a strict sense is not generally applicable to daily elementary mathematics learning, which is mainly reflected in the following aspects: First, it is difficult to balance time and income; Second, it is difficult to take into account the individual differences of students; Third, some learning contents are not suitable for inquiry learning. "Question" is the starting point of inquiry. No matter who puts forward the problems in mathematics classroom teaching under what circumstances, they need to be selected, oriented or optimized under the guidance of teachers, and finally form the task of learning and exploring. Through this process, we can simplify and refine the original inquiry questions, avoid the blindness of random inquiry, shorten the inefficient inquiry links as much as possible, and maintain the core spirit of inquiry, so as to ensure that mathematical literacy such as mathematical spirit, mathematical thinking and mathematical thinking methods can penetrate and fill the inquiry task and inquiry process. Teacher Tu Rongbao also advocates trying to explore throughout the teaching and insisting on one or two short explorations in each class. [1] This is the most "economical" teaching design. Because it is small and flexible, we might as well call this task "micro-inquiry" task.
Second, the educational value of mathematics "micro-inquiry" task
The task of "micro-inquiry" is the content and requirement of activities designed by teachers to cultivate students in mathematics classroom teaching. It is a teaching process in which teachers transform the "scientific form" of teaching content into "educational form", and it is an important carrier for students to transform knowledge into ability. It is obviously of higher educational development value for students to change the focus, difficulty, key or exercise form of each class appropriately and present it as a "micro-inquiry" task in integrable ware.
First of all, the task of "micro-inquiry" is conducive to stimulating students' deep learning. The task of "micro-inquiry" in mathematics classroom gives students something to do. Compared with the single way of listening or practicing, the multi-unit classroom structure has a sense of hierarchy and rhythm, which can better help students concentrate and keep their attention, concentrate on completing tasks and think positively. In task-driven learning, students can think and solve problems in their own way, adjust their learning progress according to their own pace, and seek help and communication according to their own needs. The inquiry task that teachers participate in and design is closer to the actual situation of specific students, which can be well controlled in the students' recent development area, which is easy to stimulate students' desire for knowledge and enthusiasm for inquiry, so that students can have more successful experiences. Learning in this task-driven, action-independent and good mood atmosphere, memory, understanding and application become the conscious needs of learning, and advanced thinking such as analysis, synthesis and evaluation can also be triggered naturally.
Secondly, the task of "micro-inquiry" is conducive to cultivating students' research methods and problem-solving ability. Because of the tiny task of micro-inquiry, we can design one or two inquiry links for students in each class, so that students can experience the inquiry process regularly. Research methods and problem-solving procedures generally belong to tacit knowledge at the methodological level. Only under the guidance of teachers' organizations, through regular independent exploration and practice, can we gradually master and develop habits. Critical spirit, questioning ability, rational reasoning and deductive reasoning ability, information collection and analysis ability, decision-making ability, etc. The internal inquiry process must be realized through constant personal experience. These "consciousness" can only be accumulated or crystallized through a lot of practical experience, and gradually form their own unique personal style and quality, which is irreplaceable by any other teaching method.
Thirdly, the task of "micro-inquiry" is conducive to improving learning ability and promoting the maintenance and internalization of learning results. The process of completing the task of "micro-inquiry" in mathematics is the process of students doing, thinking and speaking, that is, students' mathematical knowledge, skills, thoughts and activities are constructed through their own assimilation and adaptation process. The stimulus of active perception in the situation is easy to form a profound and comprehensive representation, and frequent thinking processing processes also enhance students' short-term memory ability and self-concept, and the long-term memory gained through experience is easier to maintain and internalize. Instead of being indoctrinated with rigid knowledge, students have established a widely connected cognitive structure system. With the increase of knowledge, such cognitive structure will be richer, more closely linked and present a snowball effect, thus laying a more adequate foundation and preparation for a new round of learning.
Third, do a good job in mathematics "micro-inquiry" characteristics of the task
1. is a well-planned scheme.
A successful teaching design is always inseparable from the support of system theory, learning theory, teaching theory and communication theory, and from the full grasp of teaching materials and accurate analysis of students. The task of "micro-inquiry" is the "finishing touch" in teaching design, which is by no means a product of whim and will not be produced out of thin air in the classroom. A good inquiry task is the product of logical thinking under the guidance of theory and a set of plans repeatedly conceived and rehearsed in teachers' minds. The task of "micro-inquiry" in mathematics is not only a problem, but also includes carefully designing or selecting situations, predicting the possible questions raised by students, predicting the possible thinking obstacles and possible reactions of students, and then taking corresponding measures for various possible feedbacks to consider how to inspire students to realize the transformation of their inquiry activities from scratch.
2. There is a just right question.
Inquiry often comes from doubt, and a good "micro-inquiry" task naturally needs a good question to "ignite". Just right, it is very close to students' reality, can induce students to think about relevant existing knowledge and experience, has strong sensory stimulation to students, can stimulate students' interest, fully attract students' attention, has a strong desire to explore, and is at the height of "one jump is enough". A good task always contains good questions, and the task execution process led by good questions is actually a spiral process of constantly solving problems and constantly generating new problems. 3. There is a lot of room for exploration.
A good task always contains rich activity resources, which can make students have something to do and be willing to think positively. A good "micro-inquiry" task can reflect the designer's value choice, and enable students to obtain sufficient qualitative and quantitative gains in knowledge acquisition, ability training, emotional attitude and so on, so that students can have internal needs and memorable reflection time after inquiry. A good task has certain complexity, and the direction of exploration is open. This is not an answer to choose or not. It is often necessary to divergent thinking before we can determine the exploration path and put forward our own original assumptions. The conclusion needs to be summarized, reasoned, induced and expressed by oneself.
4. Strong operability and wide participation.
A good "micro-inquiry" task is the design of taking local materials and teaching students in accordance with their aptitude, which is close to students' life and convenient for students to operate. It can make students enter the inquiry state unconsciously and avoid invalid activities well. There are appropriate resources and conditions for hands-on operation, and more importantly, there are designs to promote the internal operation of students' brains. Inquiry activity is not a performance of a few students, but can provide help or supervision, so that students of different levels can participate in inquiry activities; Stimulating students' thinking participation, emotional participation and value judgment participation is not only behavior participation, but also design and arrangement, which can obtain a certain degree of inquiry results.
Fourth, the design strategy of mathematics "micro-inquiry" task
1. Practical application strategy
The most common tasks in mathematics classroom teaching are classroom exercises and homework. These exercises or assignments may be transformed into "micro-inquiry" tasks as long as they are handled slightly. For example, after learning the elementary knowledge of triangles, we can design such a task: prepare some bamboo poles, protractor for teaching, meter ruler with millimeter scale and string grouping, take students to the river and let them measure the distance between two trees on both sides of the river; Take the students to the playground and ask them to determine the height of a tall building nearby. Such a task cannot be solved simply by applying formulas. Students need to determine the conditions required for reverse problem solving according to their own knowledge, make a solution, then measure, collect data and calculate the results. This not only contains the knowledge of solving general triangles, but also can be transformed into a more concise solution for solving special triangle problems with special angles if the scheme is well designed. This design not only gets rid of the boredom of directly applying formulas, but also cultivates students' practical ability and improves the level of mathematical thinking, which is of great significance to cultivate students' application consciousness, innovation consciousness and modeling thinking.
2. Analogy verification strategy
One of the important goals of mathematics learning is to train thinking with mathematical knowledge as the carrier. For learning different knowledge, we can often train and improve the same way of thinking. This is a process of acquiring knowledge and improving thinking methods and quality. Analogy is an important method of mathematical thinking and an effective means and strategy to explore conclusions through reasonable reasoning. For example, learning from arithmetic progression before learning from geometric progression requires students to infer and construct geometric progression's knowledge system based on arithmetic progression's knowledge and learning experience. Such a task, with a small slope and a foundation of inquiry, can not only consolidate the existing knowledge, but also stimulate students' curiosity and cultivate students' inquiry consciousness, thus enhancing students' ability of analysis, reasoning, judgment, choice and demonstration, especially enhancing students' initiative and self-confidence in learning and cultivating students' psychological habit of "re-creation".
3. Explore legal strategies
Middle school mathematics knowledge mainly appears in the form of concepts and laws, and formulas and laws are the focus of our mathematics classroom teaching. These mathematical laws exist objectively and can be explored in a certain way. At the same time, it is necessary to avoid the mathematical rules being mechanically accepted and becoming rigid formulas. If we package the ins and outs of the rules into a "micro-inquiry" task and let students discover the rules themselves, learning can achieve twice the result with half the effort. For example, after introducing the definition of combination number, students can explore the calculation method of combination number independently. Prompt students to use "professional" thinking strategy, first try less and try more step by step, and then verify and prove it through inductive reasoning. Another example is to give students a task: we have learned six trigonometric functions of an angle, so is there a certain relationship between the six trigonometric functions of the same angle? Consciously cultivate students' logical reasoning consciousness of "regression definition" This will not only enable students to memorize formulas, but also help to cultivate students' professional and generalized thinking methods, and help to cultivate students' awareness of looking at problems and the world with a connected and holistic concept.
4. System construction strategy
From the structural system of mathematical knowledge, although the specific contents of some knowledge are different, the structural system and learning procedures are completely similar. For example, the research function is to study images and properties; Learning analytic geometry is mainly to study the relationship and properties between graphs and equations. For the study of these similar knowledge, students can be guided to learn one or two examples first, and then the subsequent similar content is designed as an inquiry task, so that students can imitate the program and explore independently. For example, when learning conic curves, because students have experience in learning straight lines and circles and have a certain thinking method of analytic geometry, they can directly assign tasks to students: drawing graphics according to the specified requirements and methods, establishing their own coordinate system, determining the equations of graphics, and studying the properties of graphics. This reflects the typical mathematical education thought of "cultivating students' thinking methods and ways with different thinking materials", which is conducive to cultivating students' transformational thinking, that is, not only knowing how to do it, but also knowing how to do it, which has a subtle and deep-seated effect on promoting students' initiative and creativity in studying problems.
5. Grafting strategy
The strategy of replacing trees with flowers to design the task of "micro-inquiry" actually embodies the idea of "learning by doing". On the surface, we don't take mathematical knowledge as an inquiry question, but let students do a task and experience and discover mathematical laws or properties in the process of completing the task. For example, after introducing the definition of "1 radian", ask students to determine the radian number of a given angle. This seems to be an operational task, which essentially implies the essence of discovering the radian number formula. For another example, before learning solid geometry, students all know the geometry of cylinders, cones and frustums. Students can be arranged to make their own models of cylinders, cones and frustums with cardboard, and then make them according to the required size. According to the required size, its essence already contains the knowledge related to these geometric properties. Such a task can closely link mathematics with life, and it is also very simple to operate. It can not only enhance students' practical ability, but also skillfully solve the indoctrination teaching dilemma of "teacher's nanny teaching" and "students' passive acceptance"
6. Special research strategies
Some mathematical knowledge is relatively difficult and complicated, but it is the focus and difficulty of learning, and sometimes even the focus. No matter the size of this knowledge point, it needs to be studied specially. For example, the monotonicity definition of function contains logical ideas, requires formal expression, and students' foundation is weak. It is impossible to incorporate new knowledge into the original cognitive structure only by assimilation. If you simply introduce it to students, it is easy for students to swallow dates in their understanding and apply mechanical imitation, so they can't grasp the essential attributes of the definition. It is necessary for us to hold a special discussion on the definition of "monotonous increase (decrease)", asking students to try to express the definition in their own language, and then try to express the definition accurately in mathematical language, so as to give students sufficient time and space, create a platform for discussion, provide appropriate help, and urge them to experience thinking, discussion, discrimination, getting rid of the rough and getting rid of the essence, and gradually improve their understanding. By experiencing the process of knowledge generation, they can