Last semester, I listened to an open math class: the judgment of parallelogram (1). Teachers' positioning of teaching knowledge goals, ability goals and emotional goals is appropriate. The teaching method is to adopt the "goal-problem" teaching method, and strive to embody the teaching concept of "subject participation, independent inquiry, cooperation and exchange, and guided inquiry".
The following will briefly review the teaching process:
Teaching begins with the examination of questions: what are the important properties of parallelogram? Please review from three aspects: edge, angle and diagonal. Considering from the side: two groups of opposite sides are parallel, and two groups of opposite sides are equal; From the point of view; The two diagonal groups are equal respectively; Considering diagonally: the two diagonals are equally divided. Then, the teacher introduces the new lesson and does the following work with the students:
① Draw two parallel lines MN and PQ.
② Intercept BC and AD on MN and PQ lines to make BC=AD.
③ Question: Is the quadrilateral ABCD a parallelogram?
Taking students into the exploration of new knowledge, teachers guide students to write what they know and have verified, and prove it with the definitions of triangle congruence and parallelogram. When students find that the quadrilateral ABCD is a parallelogram, teachers introduce classroom teaching into key procedures and present them in the form of questions, requiring students to express the above findings as written propositions. In this way, the teaching objectives of this class have been initially realized. Then the teacher asked the students to explore the parallelogram judgment theorem 2 again, and threw out the question: "Are two groups of quadrilaterals with equal opposite sides parallelograms?" Students are required to rewrite the above proposition as known and verify it in symbolic language, and it is not difficult for students to prove the correctness of the proposition, thus obtaining the judgment theorem 2 of parallelogram. Looking back on this kind of discovery, we can draw a conclusion that there are three ways to judge a parallelogram: the definition of parallelogram, the judgment theorem of parallelogram 1, and the judgment theorem of parallelogram 2.
On one occasion, the teacher gave an example:
Example 1 It is known that the quadrilateral ABCD is the midpoint of the parallelogram.
Judge: Are quadrilateral AEFD and quadrilateral EFCB parallelograms?
Around the teaching focus, according to the teaching objectives, teachers and students cooperate, and then demonstrate. Then the teacher deepened the above questions and asked the following questions:
Example 2 It is known that the quadrilateral ABCD is a parallelogram, and e and f are the midpoint of AB and CD respectively. Is the quadrilateral EDFB a parallelogram? (individual students answer)
In Example 3, it is known that points E, H, F and G are the midpoints of sides AB, BC, CD and DA of parallelogram ABCD, and ED intersects with AH and GC at points A' and D' respectively, and BF intersects with AH and GC at points B' and C' respectively. It is found and proved that there are several parallelograms in the graph.
Example 4 It is known that the parallelogram ABCD, E and F are the midpoint of AD and BC respectively, and AG=CH. Prove that the quadrilateral GFHE is a parallelogram (the whole class does it on paper and individual students answer it).
These problems are simple and basic, and have deepened at different levels. Let students master the application of parallelogram judgment theorem 1 step by step and use parallelogram judgment theorem 1 flexibly, which not only expands students' thinking, but also enlivens the classroom atmosphere.
At the conclusion stage of the class, the teacher asked the students, "What methods have you learned to judge whether a quadrilateral is a parallelogram?" Let the students answer and make a summary to emphasize it. In the fun of teachers and students exploring and summing up knowledge, an open class is over.
Second, blow the yellow sand first and then cash it.
The almost monotonous narration in front obviously can't present wonderful classroom teaching. Although teaching is a regrettable art, it takes cash to blow away the yellow sand. A young teacher who has only been employed for more than two years can grasp the teaching materials more accurately, and after design-practice-redesign-practice, he has left us precious truths to ponder.
1. Analyzing and processing textbooks is the basic skill of teachers.
The content of the textbook "The Judgment of Parallelogram" (1) is the proof of two judgment theorems. The proof can be used as a basis for judging whether a quadrilateral is a parallelogram. From the perspective of learning task, it belongs to upper learning, which is a new method to prove whether a quadrilateral is a parallelogram by using the definition of parallelogram. According to the constructivist learning view, the interaction between new knowledge and knowledge in the original cognitive structure is mainly an adaptive process, that is, the existing cognitive structure is continuously developed and transformed to "accommodate" the new knowledge in the original knowledge framework. The value of mathematics in the process of human civilization is enormous, and geometry shows infinite charm with its graphic language, and parallelism is even more wonderful. The essence of parallelism is straight lines that never intersect in the same plane. A parallelogram that conforms to "two sets of quadrangles with parallel opposite sides" is the simplest figure with parallel characteristics in plane graphics. Unlike the study of geometry in ancient Greece, which was a strict axiom system and logical proof, China's ancient algebra focused on algorithms and made good use of area calculation, which was the most basic tool for our ancestors to study geometry. If teachers can master this level of teaching materials, they can guide students out of the misunderstanding of simply using triangular congruence method in teaching and give unique proofs with area method or parallel concept, which is of great benefit to cultivating the breadth and depth of students' thinking.
Therefore, studying the syllabus (or curriculum standards), analyzing the teaching materials and handling the teaching materials are the basic skills of teachers. Otherwise, it is not clear which content can be used as the basis for students to build a new knowledge structure and which content needs new input. Whether their interaction is "assimilation" or "adaptation"; Otherwise, in the limited classroom teaching time, it is difficult to highlight key points, break through difficulties and leave independent time and space for students.
2. Optimizing the teaching design that can embody modern ideas has a long way to go.
The teaching method of "full house irrigation" has been abandoned by more and more teachers; The teaching method of "asking questions in the whole class" is similar to heuristic method, but in fact, it is an accepting learning that teachers guide students and follow the teaching procedures designed by teachers in advance, so it has also been criticized by many people. Teachers adopt the teaching concept of "goal-problem" in teaching. It can be roughly divided into the following procedures: reviewing and laying a foundation-creating an environment to raise doubts-asking questions to guide exploration-solving problems-extending migration-consolidating summary. It is a successful teaching practice to try the "double-guided learning" mode of constructivist teaching view and connect all links naturally. Constructivist teaching view holds that the process of knowledge acquisition is not a simple process of "teachers imparting knowledge to students", but can only be actively constructed by students according to their existing knowledge and experience. In this construction process, students are the main body under the guidance of teachers and the active constructors of knowledge meaning. On the basis of the original cognitive structure, the leading role of teachers is to effectively organize students to explore and communicate and actively construct perfect cognition after bringing students into the "problem situation". Throughout this lesson, both the questions designed by teachers and the enlightening questions in the process of guiding students to explore pay attention to positioning the questions in the students' "cognitive nearest development zone", so the questions are oriented and progressive. "Problem is the heart of mathematics" is fully reflected in the lesson.
One of the cognitive goals of this course is to prove the proposition of words in plane geometry. Teachers creatively set the goal on the platform for students to participate in the process of proposition discovery. It is the nature of every student to guess and foresee. Grasping this psychological feature, the teaching design of "guessing before proving" effectively stimulates the sense of responsibility of students with mathematical learning difficulties and arouses their ability to actively perceive, explore and construct knowledge in class. This is a successful practice of the principle of teaching students in accordance with their aptitude.
3. Believe in students, so as to reflect the teacher's teaching philosophy of student development.
Judging from the teaching design of parallelogram judgment (1), teachers deliberately embody the concept of "guiding knowledge construction" and the practice of "seeking answers with students", which leaves a deep impression on people. Equally profound is that in the process of teaching, there is always such a trace that students are not regarded as equal individuals. These questions are carefully designed by the teacher. Half of the students answered the teacher's questions and were reminded by the teacher from time to time in the process of answering the questions, so that it is sometimes difficult to discover the students' real thinking process. Of course, "take small steps and ask more questions" is conducive to students' thinking and understanding of knowledge, and is conducive to understanding students' mastery of knowledge. However, in today's advocating the cultivation of innovative spirit and practical ability, we should pay more attention to the cultivation of students' problem consciousness. When asked about doubt, doubt comes from thinking. Teachers should create enough space and time for students to ask questions in class. The essence of target problem teaching method is to cultivate students' problem consciousness and their ability to find and ask questions in the process of solving problems. Unfortunately, the students in this class found and asked too few questions, especially after proving the judgment theorem 2 of parallelogram, they lacked corresponding questions and exercises. In the long run, students' problem consciousness will fade. In class, when exploring questions, teachers are impatient because of teaching plans and eager to give their own ideas, which is also a distrust of students. In this way, students will have thinking inertia.
Third, suggestions to improve teaching design
After proving that "two sets of quadrangles with equal opposite sides are parallelograms", two triangles are placed on the same plane, and a set of corresponding sides overlap each other. Must the resulting figure be a parallelogram? How to spell to get a parallelogram? Give full play to students' imagination and let them put two congruent triangles together, so as to guess whether all the corresponding sides must be parallelograms when two congruent triangles are put together. It is the application of parallelogram decision theorem 2.