Someone described the learning process of primary school mathematics as follows:
Simple and complex.
Memory comprehension.
Intuitive abstraction.
Passive and proactive.
It is also said that students should achieve the goal of learning mathematics in six years: understanding ideas, accumulating experience, and developing thinking.
These statements are true!
However, in order to achieve the goal, in the long learning process of six years, it is a big test for the "teaching" of teachers and the "learning" of students. If the teachers are unprofessional and have no teaching methods, and the students have no rules and habits, in fact, many children have a huge gap in the harvest in the course of six years of learning. The "Lao Lin family" is obsessed with exploring appropriate ways of "teaching" and "learning", just to shorten this "gap" and shorten the ......
Take the process of "intuition to abstraction" as an example, it is related to several core concepts mentioned in the new curriculum standards: number sense, symbol awareness, spatial concept, geometric intuition, and model thinking. One of the more relevant and more commonly mentioned is: "geometric intuition".
Compulsory Education Mathematics Curriculum Standards (2011 Edition) points out that geometric intuition mainly refers to the use of graphics to describe and analyze problems. With the help of geometric intuition, complex mathematical problems can be made concise, visual, and helpful to explore the ideas for solving problems. In simple terms,Geometric intuition is to rely on and use graphics for mathematical thinking and imagination, which is essentially a kind of thinking power developed through graphics.
How can we develop this kind of thinking?
The most straightforward and effective way to do it: draw!
Attaching importance to cultivating students' drawing ability is a common topic in primary school mathematics. There are many experiences and discourses in this area, butThe cultivation of students' drawing ability is a long process, and it cannot be achieved by a good public class or a few cases or experiences once in a while. We often find that many students in the upper grades still do not use drawings to think about how to solve problems, and students have neither the awareness nor the ability to do so.
Why? Think about it, actually.
The literacy of drawing ability cultivation in the textbook (Renjiao Edition) is relatively fragmented, and if we do not systematically grasp and target penetration, it will be difficult for students' drawing ability to be effectively improved.
How to develop students' drawing skills?
Let's go firstSort it out
What does "drawing" mean? What does "graph" mean?
AboutThe action of "drawing".It generally refers to the use of one or more actions by students in various drawing operations: such as "circle", "paint", "draw", "connect", "draw", "mark", etc.
AboutThe type of Graph, we divide it like this:
1.Concept map. Refers to a diagram that uses numbers, letters, symbols, or graphics to represent the essential properties of a mathematical concept. Students' comprehension of words in primary school is not rich or rigorous enough, and it is more abstract to understand the concepts described in words, and the use of drawings can improve students' understanding of the concepts. For example, drawing straight lines, rays, drawing various shapes, drawing parallel and perpendicular, drawing fractions, etc.
2.Diagram. Refers to the use of numbers, letters, symbols, or graphs to represent mathematical information, graphic features, quantitative relationships, or diagrams of thinking paths. In this context, it refers to the common lattice diagrams, relationship diagrams, analysis diagrams, simple plane diagrams and flow charts in primary school mathematics, such as line diagrams that represent collocation methods, flow charts that arrange time reasonably, relationship diagrams between cylinders and cones of equal base and equal height, and combination diagrams that use two identical triangles to form a parallelogram.
3.Segment diagram. It refers to a kind of diagram that is composed of several line segments together to represent the quantitative relationship in practical problems in life, so as to help analyze the meaning of the problem and solve the problem. (Note: Because line diagrams are more important in primary school mathematics, they are divided into a separate category).
Let's give it a try:
1. Normal infiltration
As a front-line teacher, I know very well the importance of students having the ability to draw, but it is not possible to achieve it overnight if you want students to consciously pick up the thinking tool of "drawing". We must infiltrate the drawing ability training from the daily drip teaching, and constantly summarize the drawing experience every time we use it.
Case:
[After-class practice for the first volume of the second grade "Preliminary Understanding of Multiplication".
[After-class practice for the first volume of the second grade "Multiplication in the Table (1)".
2. Intensive training
On the basis of normal infiltration, two special drawing exercises are customized according to the learning content of each textbook: basic practice and improvement practice. Basic exercises are used to awaken the cognition of drawing, try to draw operations, and pave the way for the learning of relevant knowledge during the semester, and are generally used at the beginning of the semester. The practice is used to consolidate the drawing operation, improve the drawing skills, and strengthen the geometric intuitive, and is generally used during the final review.
Case 1: People's Education Edition of Primary School Mathematics
The first volume of the second grade is the basic exercise of "drawing mathematics".
(Compilation: Zhong Lanzhi, Dongcheng Eighth Primary School, Dongguan City).
Design Intent + Practice Reflection:
The first question is the application of the knowledge of length units, which requires you to measure the length of a known line segment and then draw a line segment of the same length. In daily teaching, some students tend to miss the endpoints when drawing line segments. This question reinforces both the method of measuring line segments and the method of drawing line segments of a specified length. The practice of this question is good, the students have a good grasp of the method, and the drawing is accurate.
Question 2 is the application of the knowledge of right angles, which requires students to add a line segment to the figure to create different numbers of right angles, which not only consolidates the understanding of right angles, but also consolidates the method of judging (drawing) right angles. In daily teaching, some students do not use the right angles of the triangular ruler to draw corners in a standardized manner, and use any side of the ruler to draw right angles at will. In the first practice, some students made mistakes in reviewing the questions, and the original right angles were also counted, ignoring the key word "increase"; After the teacher's guidance, the students carried out a second exercise, some students did not use the right angle of the triangular ruler to add the line segment, the increase is an obtuse angle or an acute angle, after the explanation again, the student can successfully increase the corresponding right angle.
Question 3 is to consolidate the knowledge of finding how many more numbers are than a number, and help students further understand the important role of drawing in solving problems. In daily teaching, students are often too lazy to use drawing analysis to understand the meaning of the topic. The practice was good, and the students were able to draw a picture to represent the meaning of "Class 2 (2) gets 3 more sides than Class 2 (1)", and understood that Class 2 (2) was divided into two parts, one part was the same number as Class 2 (1) and the other part was more than Class 2 (1).
Title 4 is designed to consolidate the understanding of angles, and the exercise gives vertices and requires you to draw acute, right, and obtuse angles respectively. The practice was good, the drawing was accurate, and only 2 students did not draw corners with known vertices.
Questions 5 and 6 are to guide students to analyze the meaning of the question with the help of drawing, and further clarify the problems and differences of "several additions" and "several additions", as well as the methods of solving the problems. In teaching, students tend to confuse "several additions" with "several additions", especially when solving problems, it is easy to calculate excess information as useful information. This problem is well practiced, and with the help of drawing and analyzing the meaning of the question, students can distinguish the difference between "several additions" and "several additions".
Case 2: People's Education Edition of Primary School Mathematics
Second gradeThe upper volume of "Drawing Mathematics" improves the exercises
(Compilation: Zhong Lanzhi, Dongcheng Eighth Primary School, Dongguan City).
Design Intent + Practice Reflection:
This "Drawing Mathematics" improvement exercise in Year 2 is designed to consolidate the knowledge of the semester, train the drawing operation and improve the drawing skills.
Question 1 is about the extended application of measurement knowledge, requiring students to read the question carefully, can find the key word of the question "incomplete ruler", and notice that the ruler does not start from the scale 0 (the ruler diagram deliberately does not show a "broken" pattern), you need to use the method of counting large grids or the method of calculation: ( = 4 cm, find 4 cm, and then color the band. Most of the students completed well.
Question 2 is about the deepening of the knowledge of drawing and counting angles, and Question 1 requires students to complete the right angles on the known vertices and line segments, and mark the right angle symbols. Question 2 asks for a line to be added to question 1 to make it 3 corners. In the first question, some students omitted right-angle symbols. In the second question, some students use different ways to turn into three corners.
Question 3 is the deepening of the knowledge of observing objects, which requires students to clearly know how many faces can be seen and which faces cannot be seen when observing the combined cube from the front, side, and above. The practice is not ideal. When observing the combined figures, it is difficult for the students to imagine the shape of the objects they observe, so the teacher uses the real object to pose and guide the students to understand the question. Some students made mistakes when they were observing, finding the wrong place to observe. After many observations, students can basically find out which side is not visible.
Question 4 is the extended application of drawing right angles and counting right angles, requiring students to use a right-angled edge of the triangular ruler to coincide with an edge in the figure, add a line segment, and increase the required right angle. With the experience of the basic practice of "drawing mathematics", the students practiced well.
Question 5 is to understand the problems related to "multiplication and subtraction" in real life with the help of drawing, and students are asked to draw a corresponding formation (schematic diagram) according to the meaning of the question and the equation. The practice is good, and the students can basically use simple symbols to represent 6 groups, 6 in each group and 2 in the 7th group, and express their thoughts intuitively.
Question 6 is an exercise in answering the "is enough" question with the help of drawing analysis, asking students to find the total number of wheels needed, then compare the size of the existing 20 wheels with the total number of wheels needed, and finally make suggestions if there is not enough. Some students used the wrong number to compare, some students did not make suggestions, some students suggested to assemble only 4 bicycles and 4 tricycles, and some students suggested buying 2 more wheels.
Question 7 is the application of mathematical wide-angle collocation knowledge. It is difficult to ask students to draw a picture to analyze the handshake problem, and reverse thinking to analyze the number of people shaking hands. The practice was not ideal, and the students were able to analyze the handshake method and understand that the handshake problem was a simple combination and had nothing to do with the order. In order to guide students to understand this question, I used the basic question of "3 people shake hands, every two people shake hands, a total of 3 times", and guide students to practice, try to add 1 person, and deduce that adding 1 person will increase 3 times.
Our insistence:
It's not just the textbook that requires it
If there is no requirement in the textbook, you can also draw
It's not just a matter of asking for a drawing in class
I also draw when I practice after class
Long-term attention, careful selection of materials, patient guidance
Carefully customized, insisted on training, and improved
interest - interest is cultivated by a little understanding. Experience – Experience is something that has been tried and earned. Foundation - Foundation is built by knowledge. develop - development is brought about by reflection.