"Mathematics Curriculum Standard" points out that mathematics curriculum "should not only consider the characteristics of mathematics itself, but also follow students' psychological laws of learning mathematics, emphasizing that starting from students' existing life experience … mathematics teaching activities must be based on students' cognitive development level and existing life experience". ① In the teacher's action education based on lesson examples, we design origami activities to let students practice, explore independently and cooperate with each other, which enriches students' learning methods and teachers' teaching methods. In this process, students find the pleasure of learning, and teachers have a new understanding of the way of mathematics teaching and learning.
First, design the background of origami activities.
"Triangle midline" has always been a classic content in junior high school geometry textbooks, and many open classes have chosen this content. However, in a large number of lectures and teaching, we find it difficult to prove the nature of the triangle midline. Only a few excellent students can do it independently in class, and most students face difficulties in proving it. How to effectively solve this teaching difficulty is the starting point of our lesson study. As we all know, using the methods of "operation", "observation", "conjecture" and "analysis" is an important way to learn geometry. From this, we thought of redesigning the teaching process of "triangle midline" according to students' existing life experience and mathematical foundation. Let students explore and explain the midline characteristics of triangles from the study of "graphic characteristics in origami"
On the one hand, the origami activity itself can arouse many good memories of students, such as origami airplanes, paper sailboats, paper cranes, treasure gourds and so on. On the other hand, origami is an effective operation activity. Students can feel the geometric properties of graphics through their own operations, and use the movement of graphics to find and analyze problems. Moreover, origami itself also carries many important things.
We can extract a more general geometric method, which is of great value to cultivate students' interest in learning, curiosity and exploration spirit.
Second, the teaching objectives.
1. In the case of origami, we can comprehensively use the properties of angular bisector and line segment perpendicular, as well as some properties and judgments related to triangles and quadrilaterals.
2. Establish a variety of connections between some activities in the life world (paper-cutting and origami games) and the geometric world to stimulate the interest in learning geometry.
3. Establish the connection between geometry and real life problems, and cultivate mathematical thinking modes (association, analogy and intuitive thinking).
4. Experiencing the process of mathematics learning: observing, exploring, guessing and verifying, and experiencing the general laws of scientific discovery.
Third, the teaching process.
1. Create a situation.
Teacher: Students, have you ever played origami? Origami planes, paper boats, paper gourds, paper cranes and so on are all very interesting. The paper that we contact most in our daily life is rectangular. If you fold one corner of such a piece of paper, you will get a right triangle (teacher's demonstration). How do you fold an isosceles triangle from a rectangular piece of paper? Please give students a discount.
The students think of the old origami. )
Ask questions.
(1) Import Problem-Fold a right triangle into a rectangle.
Teacher: We already know that rectangular paper can be folded into a right triangle. Now consider the problem in the opposite direction, that is, can a right-angled triangular piece of paper be folded into a rectangle?
Students observe, try and discuss origami in groups, explore folding methods and express their findings. )
Teacher: (Physical projection) We unfold the paper, draw creases and mark letters (as shown in figure 1). Looking back at the origami process, what did you find? (Teacher's Tip: Pay attention to the relationship between the position and length of the line segment in the picture. Is there an isosceles triangle in the picture? Which triangles are congruent? )
A
British natural gas company
Figure 1
Student: (The teacher summarizes and writes on the blackboard) 1EF = GB = GC = BC/2. EG=AF=FC=AC/2。 Therefore, EF‖BC, EG‖AC.
② Fold divides triangle ABC into four congruent right triangles and two isosceles triangles.
③ Connect EC, AE=BE=EC=AB/2, ∠ A+∠ B = 90.
Teacher: By observing our paper (figure 1), we know that E is the midpoint of AB, and we have found three points, two of which have been proved before. Today we will explain them again through origami. Please make a crease through the middle points G and F, and think about the relationship between this crease GF and the hypotenuse AB. Can it be one side of a rectangle?
(2) General problem-folding any triangle into a rectangle.
Teacher: Now, let's consider a more general question, that is, can a piece of paper with triangles be folded into a rectangle? Please give students a discount.
The students tried to fold a rectangle with any triangle. Teacher's guidance during the tour: Students can recall how it was folded just now. At the end of the activity, students demonstrate the folding process of transforming the high line into two right triangles on the projector. )
Teacher: We open the paper and flatten it, draw all the creases and mark them with letters (as shown in Figure 2). From the origami activity just now, what is the relationship between the position, shape and quantity of line segments, angles and triangles in this figure? Ask each group of students to discuss and publish the results of the group discussion.
A
H C
Figure 2
(The teacher sums up and writes the results of the students' discussion on the blackboard. )
① About the midpoint: AE=BE=AB/2, AF = CF = AC/2.bg = DG = BD/2.ch = DH = CD/2; ② The median line on the hypotenuse: DE=AB/2, DF = AC/2; ③ About neutral line: EF=BC/2, GE=AD/2. FH=AD/2 .
3. guess.
Teacher: Under what conditions do you think one line segment is half the length of another line segment?
Students found: ① the midpoint of the line segment; ② the midline on the hypotenuse of the right triangle; (3) The midpoint lines on both sides of the triangle.
Teacher: Actually, we have found the midpoints E and F on both sides of △ABC. We call the line segment connecting the midpoints on both sides of the triangle the midline of the triangle. Now, what is the relationship between the middle line and the third side?
Students put forward a guess: the middle line of a triangle is parallel to the third side, which is equal to half of it. )
4. Explain the conclusion.
Teacher: Just now, everyone guessed the nature of the triangle midline. Now, can you verify this property and explain it?
(Students origami, using origami to compare the length of each side and the size of each corner. )
Teacher: Group discussion. How to verify? How to explain it? Teacher's guidance during the lecture tour: Your explanation should convince others that you are correct. Who wants to come here (podium) to explain it to everyone! Do you have any questions?
Students explain and argue with each other. Explain ①∠A+∠B+∠C= 180 on the physical projector. ; ② The quadrilateral EFHG is a rectangle. )
Teacher: Together, we discovered the nature of the midline of the triangle: the midline of the triangle is parallel, equal to half of the third side, and verified and explained it with origami. We will further prove and apply this property in the future.
5. Exchange experiences.
Teacher: What did you learn in this class? What have you learned? What did you find? What experience do you have? Are there any questions or puzzles?
Student 1: This class taught me that there is also mathematics in origami, and I feel that there is mathematics everywhere in my life. Observe and think more in the future.
Health 2: When I fold a rectangle with a right triangle, it is different from other students in the group. After comparison, I found that the folded rectangle is not as big as other students. I folded it several times and found that the area is the largest, which is half of the triangle area.
S3: I think it is very convenient to compare the relationship between line segments and angles with origami. For example, you can compare two identical numbers at the same time ... it's easy to guess while doing, so you should use this method more in the future when learning geometry.
Teacher: Students, in the origami operation, we find the relationship through observation, form a guess, prove our guess and draw a conclusion. This is an important way for people to discover new knowledge.
6. assign homework.
Teacher: Today's homework after class is to fold the graph with a square piece of paper, and explore the operation according to the worksheet to find out the problems.
Fourth, teaching and research after teaching activities.
From the above process, we can see that the process of teaching activities has gone through six links: creating situations, asking questions, guessing, explaining conclusions, exchanging experiences and assigning homework. In the subsequent teaching and research activities, teachers discussed the following issues, which caused us to think more.
1. About activity teaching.
Activity-based teaching method mainly emphasizes that students start from their existing life experience and learn in the process of hands-on activities, thus completing the active construction of knowledge. However, the occurrence of mathematical inquiry activities is different from scientific inquiry activities. Manipulation and operation of specific physical materials (origami activities) are only "external activities", while substantive mathematical inquiry often happens in students' minds-the teacher's task is to let students experience the process of "intuition-perceptual knowledge-rational thinking", and at the same time experience and feel the happiness and challenges in the process of mathematical discovery (from guessing to explanation/proof). The course "Graphics in origami" undoubtedly pays attention to students' learning of process knowledge and enhances students' emotional experience of mathematics learning process. Bruner also pointed out: "When we teach a subject, we don't want students to become a small library of the subject, but to participate in the process of acquiring knowledge. Learning is a process, not a result. " It can be seen that it is more important for students to "learn to learn" than "learn what" in activities.
2。 On the design of problem situation.
Dewey's "Five Steps of Teaching" embodies his educational thought of "learning by doing", which is embodied in the following aspects: teachers should prepare a real situation of applying experience for students in teaching-a situation related to students' real life experience; At the same time, some hints are given to make students interested in understanding a problem. This lesson "Folding Triangle into Rectangle" takes the real problem in origami situation as the thinking stimulus to stimulate students to learn geometric properties. Teachers don't provide ready-made textbooks for students, but hope that students can participate in activities, inspire and guide students to "naturally" generate methods from their own life experiences and origami activities (in fact, effectively use students' existing life experiences) to deal with problems arising in origami situations, consider things that have not been recognized before, and make experiences really grow and form new experiences. Moreover, there is a lot of tacit knowledge in situational practice, so the key to effective activity teaching lies in dealing with four relationships between explicit knowledge and tacit knowledge learning-namely, the organic combination of verbal communication, internalization, dominance and tacit understanding; . (4) On this basis, effectively carry out knowledge inheritance and innovation. ⑤
3. Talking about the cultivation of students' mathematical thinking.
One of the characteristics of mathematics is its high abstraction. For example, abstract concepts and abstract relationships, but they all have a lot of realistic backgrounds. This course pays attention to this feature in teaching design, trying to reflect the realistic background of mathematical facts, and selecting situations closely related to students' living world from it, so that the abstract process of students' thinking seems to happen naturally. In this way, students feel fresh mathematics, not just its cold beauty. Another feature of mathematics is rigor, which is manifested in strict logic and accurate calculation. This rigorous process reflects the gradual deepening of human understanding. In the class, we also pay attention to students' cognitive characteristics, and transition from the rigorous process of "intuitive geometry" to "proof geometry" to explanatory geometry, which requires students to "convince others that you are correct". The thinking mode of inspiring proof and refutation. At the same time, it embodies an attitude of gradually pursuing rigor.
Cheng. In the problem-solving activities of lesson plan design, some mathematicians' common thinking methods are reflected: (1) thinking about the inverse (opposite direction) problem of the problem, thus putting forward new problems (for example, from "folding a triangular problem with ordinary rectangular paper" to "folding a rectangular problem with triangular paper"); (2) From the special case of the general problem (right triangle is folded into a rectangle), find the solution; (3) The idea of transforming a general problem (turning a general triangle into a rectangle) into a solved problem (turning a right triangle into a rectangle); (4) The idea of inductive classification (summarizing and classifying many relationships found in origami); (5) Looking for the idea of invariance from change (the relationship between the length of changing line segments in origami).
4. About the development process of activity class.
How to develop students' mathematics activities in activity class? This depends on many factors, including the characteristics of teachers, students' foundation, content level, method application and situational introduction, etc. There is no doubt that students' active exploration and attempt is the core of activity class, and how teachers guide them here is very critical. When designing teacher guidance activities, we have experienced such problems as "verifying theorems (reviewing) or discovering problems (mathematics)", "taking the organization of knowledge structure as the main route or taking progressive cognitive activities as the main clue" and "how to perceive tacit knowledge in activities". Design teaching, practice several times, it's easy. For example, in the design of the first draft, the teacher intends to "review the theorems of line segment perpendicular bisector, angle bisector and hypotenuse midline of right triangle". I learned it through origami activities this semester, including 30. The right triangle property theorem of the angle also explores the triangle midline theorem that has not been learned before. " This is actually to verify the theorem through origami. Origami activities "run through" the review and discovery of theorems, and the capacity of the classroom is naturally not small. However, in the later study of * * *, everyone realized that verifying the learned geometric theorem through origami operation would lose the meaning of operation and occupy more classroom time. The focus of teaching should be "students discover new knowledge through origami operation, and provide students with more opportunities and time to ask questions, try, explore, discuss, communicate and summarize." Encourage students to open their minds, form innovative ways of thinking and look at problems in active exploration, and gain knowledge through this. "For another example, after the first class, the teacher reflected:" In the past, we paid attention to geometric argumentation and the rigor of logical reasoning, and paid little attention to how knowledge was generated; Origami activity is operational geometry, so it is difficult for teachers and students to adapt to explore, discover and verify from the perspective of origami. "After carefully observing the classroom video, everyone thinks it is necessary to create a plot closely related to students' real life experience, activate students' original experience, and reflect step by step to apply what they have learned. For another example, in a later parallel class, we found that students relaxed their thinking in origami activities, and there were many attempts and results, which better reflected students' subjectivity, but the direction of operation and attempt was not clear enough and the depth was also lacking. Careful observation of classroom video makes us understand the role of another subject in teaching. How teachers guide and guide students to extract key questions and useful knowledge from various attempts and results is not only very important in teaching design, but also the embodiment of teachers' and students' wisdom in teaching practice, and also related to the level of "tacit knowledge".
In addition, when designing activity-based teaching, teachers realize that if the pace of inquiry is small, it seems that students will be led into a "trap"; If the pace of inquiry is great, students' inquiry activities will be greatly hindered or even will not happen. So, how to master the rhythm of inquiry? Our understanding is that the steps of exploring and trying must be suitable for students' reality. Let students face moderate difficulties, stimulate their interest in exploration and thinking, and get some gains and a sense of accomplishment from this process of overcoming difficulties. Danshe
Planning the problem should not be too difficult, otherwise students will wander too much in front of the problem and waste a lot of precious time. At the beginning of the activity, the pace of exploration and attempt is smaller, so that more students have the opportunity to invest and participate. As students are familiar with the environment, situations and problems, they can step up their exploration and try, and constantly increase creative factors.