The common propositions in mathematics curriculum are true propositions that reflect the basic facts of mathematics and have certain cognitive and practical functions. They constitute the core of middle school mathematics knowledge structure, and their main forms are formulas, theorems, principles and rules.
Proposition learning is based on concept learning and is a more advanced learning type than concept learning. The main learning process includes proposition acquisition, proposition proof and proposition application.
According to the relationship between the concepts in the proposition and the related knowledge in the original cognitive structure, the learning of mathematical proposition can be divided into three forms: upper learning, lower learning and parallel learning. No matter what form of learning, "when a learner can properly and correctly apply a proposition to some different conditions, he learns this proposition."
Proposition learning is the ability to respond to a series of conditions (stimuli) with a series of actions (reactions), which has outstanding value in promoting students to develop effective and coherent behavior ability. Therefore, proposition teaching has received special attention in mathematics education.
Case: counting principle
Counting problem is one of the important objects in mathematical research. Counting principle is not only the initial knowledge of combinatorial mathematics, but also the basic knowledge of learning probability statistics, calculus and other mathematical contents, and it is in the most basic and important position in the mathematical knowledge system. Because the two basic principles of counting are closely related to real life and are also common methods to deal with daily problems, it is easy for students to find and summarize the basic contents of the principles from personal experience, so the focus of the teaching process is to guide students to have a simple and intuitive understanding, develop the ability of mathematical expression and application, and establish a clear and systematic knowledge structure.
Although the content of this lesson is not difficult, it belongs to a higher level of law study. "Principle" is the most complicated mathematical object, which is the order of concepts plus the relationship between these concepts. Constructing a principle requires students to "respond to a series of conditions with a series of actions". Therefore, the learning psychology of this course is more complicated, and the cognitive process has to go through intuitive experience, pre-scientific concepts, mathematical concepts, relational judgment, principle creation, algorithm formation, situational application and other activities, which constitutes a difficult point of this course.
It is not easy to teach such a seemingly simple but complicated class well, which is a test for teachers in many aspects. First, accurately grasp the essence of mathematical knowledge and curriculum requirements, so that students can master the essence and application of principles in a given time; Second, accurately grasp students' cognitive characteristics, design classroom activities that adapt to students' psychology, and let students exercise their mathematical thinking and develop their mathematical modeling, mathematical expression and mathematical communication skills.
The two core knowledge points of this lesson are classified counting principle and step-by-step counting principle. The principle of classification and counting is also called addition principle, and its knowledge base is: completing one thing, classification and addition. The principle of step-by-step counting is also called multiplication principle, and its knowledge base is: completing one thing, step by step, multiplication. Learning these two principles is to clarify the following questions in understanding:
What is the meaning of one thing, how to complete one thing, whether it is completed by classification or step by step, how many ways or steps are there to complete one thing, and why addition is used for classification and multiplication is used step by step. These problems are the cognitive basis for a comprehensive understanding of the principle, an important aspect of correctly grasping the connotation of the principle, and the basic content that must be considered when applying the principle. If these problems are overcome, the two principles will be basically mastered. Therefore, in teaching, teachers should guide students to understand the knowledge base of the principle, constantly strengthen students' understanding of "completing one thing", "step by step", "several types" and "several steps", grasp the essential relationship of the principle, and establish a clear and stable algorithm model, so as to correctly judge and apply the corresponding principle in solving practical problems.
"Finish one thing" is an abstract word. In teaching, students should distinguish it with examples, so as to grasp it intuitively and have a clear point of view, and lay a unified understanding foundation for the following discussion.
"Classification" and "step by step" are the only criteria to distinguish the two principles, and they are also the basis for applying the two principles and the selection algorithm. When discussing "completing one thing", the method of completion will be involved. Further analysis of the relationship between the completed methods and this matter will show that some methods can complete this matter independently, while others can only be partially completed. When students realize this, they have grasped the essence of "classification" and "step by step", followed by fine processing and mathematical treatment of "classification" and "step by step".
To sum up, the core concepts of this lesson are classified counting and step-by-step counting, and the key to mastering the principle is to understand "completing one thing", master the standards of classification and step-by-step, correctly select the algorithm, and correctly calculate the number of steps and classes. The main idea running through this lesson is the formation and application of mathematical model, which realizes the thinking method of analysis, synthesis, induction and generalization and the general idea of "controlling complexity with simplicity" Therefore, the teaching design should adopt the inquiry mode of "question situation guiding inquiry-induction and generalization" to guide students to analyze typical cases, summarize the same characteristics, further summarize the essential characteristics, and finally deepen their understanding of concepts and thinking methods through application examples.