First, experience
Take the study of the concept of centrosymmetric figure in Centrosymmetric as an example: a piece of cardboard-the midpoint O of the line segment AB is nailed to the blackboard with thumbtacks, so that students can demonstrate that the line segment AB rotates by the minimum angle around its midpoint O and coincides with the original line segment, that is, the position of point A turns to the position of point B, and the position of point B turns to the position of point A; Then demonstrate the parallelogram ABCD made of hard paper. After the hard paper of the parallelogram ABCD rotates a minimum angle around its diagonal intersection O, it overlaps with the original parallelogram, that is, the position of point A turns to the position of point C, the position of point C turns to the position of point A, the position of point B turns to the position of point D, and the position of point D turns to the position of point B. Similarly, it is rectangular, square, diamond and so on. Have this property. That is to say, the figure rotated by180 around a certain point coincides with the original figure, but isosceles triangle and regular triangle do not have this property, which leads to the definition of centrally symmetric figure.
Second, discrimination.
After a preliminary understanding of the concept, some concept judgment questions can be quoted appropriately for students to identify and compare, which is conducive to clarifying students' misunderstanding, enabling students to test their mastery and application ability of the concept in practice, and is conducive to accurately understanding the concept. The concept of negative number is given in descriptive language, such as, etc. Numbers preceded by a negative sign (except zero) are called negative numbers, and students are easily confused by the superficial phenomenon "-". At this time, after introducing letters to represent numbers, you can give some counterexamples, such as negative numbers? Is 3a necessarily less than 4a? Is 2+a necessarily greater than 2-a? Deepen the understanding of the concept of negative numbers. For another example, after learning the simplest concept of quadratic root, let students distinguish the following:,, and so on. What is the simplest quadratic radical? Which ones are not? Why? Through this practice, students' ability to make simple judgments by using concepts is cultivated, and the essential attributes of concepts are repeated in their minds every time students make judgments, so as to further understand new concepts.
Third, comparison.
It can only be identified by comparison. For concepts that are easy to be confused or difficult to understand, only after many times of comparative analysis and practice can we achieve the goal of correct understanding. Using comparative method can help students grasp the essence of concepts and correctly grasp the essence of concepts. For example: equality and equation, solution and solution of equation, factorization and algebraic multiplication, sum of squares and square of sum, etc. Students are often confused. Teaching can guide students to find out their similarities and differences and deepen their understanding of concepts. Equality and equality are two related and different concepts. Equation is a formula to express equation, which contains two kinds: one is an identity, such as 2+3=5, a+b=b+a, and the equation can always hold no matter what values A and B take; The other is a conditional equation, such as 3+x=7. Only when x=4 can the equation be established, otherwise it will not be established. An equation with unknowns like this is an equation. This shows that the equation must meet the conditions at the same time: ① contains unknowns, ② equation. Another example is the sum of squares and the square of sum. We can compare their operation order: the sum of squares is squared first and then summed, that is, A2+B2; The square of the sum is first summed and then squared, that is, (a+b)2. Factorization and multiplication of algebraic expressions can compare the operation results: Factorization is the multiplication of a polynomial into several factors, such as x2-y2 = (x+y) (x-y); Algebraic multiplication is to multiply several factors into polynomials, such as (x+y)(x-y)=x2-y2. Some difficult concepts can be revealed by comparison from difficult to easy, such as comparing the similarities between two algebraic expressions 12a2b2c and 8a3xy; Compare the similarities and differences between squares and regular pentagons; Wait a minute.
Fourth, analogy.
Sometimes, concepts can be better understood by analogy. Such as: fractions and fractions, inequalities and equations, similar triangles and congruent triangles, etc. After analogy, we can learn new things by reviewing the past and make up for each other, thus deepening the effect of conceptual understanding. For example, students learn the concept of trapezoid in a direct definition way, and can think about the related concepts such as parallelism, quadrilateral and the sides of quadrilateral in their own cognitive structure, and realize that trapezoid is a special original quadrilateral, so as to clarify its connotation and extension. By discussing various special cases of trapezoid, such as right-angled trapezoid and isosceles trapezoid, the essential attribute of trapezoid is further highlighted, which is different from some original concepts (such as parallelogram). It is integrated into the original concept (quadrilateral) system, and then examples are studied and problem-solving exercises are carried out, especially by letting students identify positive examples and counter examples (including some variant figures), so as to deepen their understanding of the trapezoidal concept and further consolidate it in the cognitive structure.
Verb (abbreviation of verb) variant
The purpose of changing the non-essential attributes of mathematical concepts is to give students a chance to experience the generalization process of concepts personally, to make the concepts mastered by students more accurate, stable and easy to migrate, to avoid treating non-essential attributes as essential attributes, and to let students better understand the essence of concepts. When learning the concept of triangle height, students are provided with some examples of different triangles with different shapes (acute triangle, right triangle and obtuse triangle) and positions. Through the thinking processing of these typical variants, students can abstract the definition of "triangle height". Therefore, students understand: ① the height of one side of a triangle is a vertical line that never starts from the vertex on that side, and the obtained vertical line segment is the height of that side; ② The height can be inside and outside the triangle, as long as the line segment from a vertex to the opposite side is perpendicular to the opposite side.
In short, students should understand the background and basic facts of the concept from their own situation, and can't use concept formation and concept assimilation in isolation, let alone separate concept acquisition from concept analysis. Otherwise, they can only know the appearance characteristics of the concept and learn the superficial knowledge of the concept, but they can't well understand the formation conditions (connotation), application scope (extension) and mathematical thinking methods contained in the concept. Therefore, in the stage of concept understanding, students should be helped to analyze the connotation and extension of the concept, understand the concept from both qualitative and quantitative aspects, and then analyze the concept itself layer by layer, and analyze and excavate the internal essence of the concept from the directions of similarity, correlation and opposition.