[Keywords:] curriculum resources, mathematics, open questions, principles, methods and precautions
I. The root of the problem
How to continuously develop and utilize mathematics curriculum resources, so that students can learn useful mathematics in colorful mathematics courses; Everyone can get the necessary mathematics; Different people learn different mathematics "? Writing effective mathematics open questions is a good way to develop mathematics curriculum resources. Because most or all of the conditions of mathematics open questions are different from those of closed questions, the questions are clear and the conclusions are clear. Moreover, open questions focus on asking questions from real life and production, and the result is often "uncertain future". In the process of solving problems, students need to explore, try, analyze and infer everywhere, and put forward new methods to solve problems, which is conducive to cultivating originality and divergence of thinking, and then forming innovative ability. This is the commanding height of quality education-what is needed to cultivate students' innovative thinking ability. At the same time, designing traditional math exercises into math exercises with open questions can also arouse students' interest, stimulate students' thinking and learning passion, and trigger classroom interaction between teachers and students, which is undoubtedly a great progress in building innovative math classrooms and teacher-student relations.
Second, the principles of writing open questions in mathematics
1. Reflect exploration and promote active learning. There must be many different solutions or possible solutions to the compiled open-ended problems, which requires more knowledge points and stronger comprehensiveness of mathematical open-ended problems. Therefore, the charm of challenging and interesting open-ended questions can attract students to study actively and happily, stimulate students' curiosity and thirst for knowledge, and stimulate students to carry forward the subjective spirit, take the initiative to participate and explore. In order to get more answers, students will learn to consult materials, study problems from different angles, and actively cooperate with others to share experiences, so that everyone can finally gain something, fully develop themselves and express themselves.
2. Reflect richness and promote the cultivation of thinking. Open questions should be rich and varied, which is conducive to cultivating students' good thinking quality and promoting students' profound, extensive and critical thinking. For example, students are required to fully understand the existing conditions in solving problems, comprehensively analyze the problems from different angles, and make correct judgments; Students are required to have no obvious solution to the same problem in teaching, associate vertically and horizontally, think more about one problem and solve more, and train students' divergent thinking; Students are required to find the simplest method from complex appearances and cultivate their logical thinking ability; There are redundant conditions, interference conditions or missing conditions in open questions, which require students to distinguish right from wrong, discard the false and retain the true, and cultivate students' thinking ability. Students' discovery of various conditions that make the conclusion valid is helpful to improve their illogical thinking abilities such as association and guessing, and their logical thinking abilities such as analysis, synthesis and abstract generalization. In the process of finding the optimal solution from various answers and the optimal conditions from various conditions, they can train their creative thinking ability and open up their thinking space.
3. Reflect education and promote personality development. Humanism education emphasizes human personality and fully develops human personality independence. One of the highest educational purposes of quality education is to shape a complete individual with harmonious personality and all-round development. The new curriculum standard also emphasizes paying attention to students' personality differences and implementing differentiated teaching meaningfully, so that every student can be fully developed. Open-ended questions can better meet the diverse learning requirements of all students. Students need to cultivate divergent thinking through a series of analysis, use what they have learned to reason and draw correct conclusions. It fully reflects the diversity of thinking, but also reflects the students' individualization of mathematics learning, thus comprehensively cultivating students' creative ability, making them have good mathematics quality and belief, and providing quality assurance for students to further study mathematics (even lifelong learning).
4. Embody creativity and promote teaching and learning. After the introduction of open-ended questions into the classroom, teachers are not the protagonists of teaching activities, but "writers" and "directors"; It is not the imparting of knowledge, but the designer, promoter, demonstrator, organizer and regulator of teaching content and teaching activities. Because students are both consumers and developers of curriculum resources, especially under the background that modern information technology is widely used in all aspects of teaching and life, students have diversified ways to obtain knowledge and information, and mutual communication and learning between students have become more and more frequent and important. They themselves become the developers of special curriculum resources, and their existing knowledge, experience, experiences and interests may become curriculum resources. At the same time, students' learning style has also undergone fundamental changes. In the process of independent, cooperative and inquiry learning, rich and colorful curriculum resources have been formed. Therefore, teachers should attach importance to students' active participation in the development and construction of resources, form a good situation for teachers and students to solve open problems together, and realize teaching and learning.
Third, the writing method of mathematics open questions
1. Weaken the conditions of the old topic and diversify its conclusions.
[Example: 1] A motorcade wants to transport 56 tons of goods from place A to place B. It is known that large trucks can transport 10 tons of goods at one time, and the freight is 200 yuan. The pickup truck can carry 4 tons of goods at a time, and the freight is 90 yuan; There are two big trucks at present. Q: How many minivans do you need to complete the freight at one time?
For this topic, if the conditions in the topic are hidden, it will be changed to: design several different car rental schemes, choose the one you think is the best, and how much is the freight? The disappearance of conditions makes the problem have many different results. Generally speaking, all students can design several different schemes, but the number and advantages and disadvantages of the schemes are in order, which reflects the thinking level of different students. Such a topic embodies the educational idea that "everyone can learn mathematics, everyone can learn useful mathematics, and different people can learn different mathematics".
2. Hide the conclusion of the old topic and let it point to diversification. For a mathematical problem, it is one of the main strategies to hide one or more conditions and find the important or optimal conditions of its conclusion.
[Example 2] As shown in the figure, D and E are two points on the BC side of the triangle ABC, and AD=AE. What other conditions need to be met to prove △ Abe △ ACD?
Answer: ∠ bad = ∠ CAE; ∠B =∠C； ∠BAE =∠CAD； EC = BDBE = CDAB=AC .
For another example, in the study of chord intersection theorem, teachers can let students observe the intersection of two chords in a circle, make appropriate auxiliary lines, and explore some conclusions (such as isometric and similar triangles). ), so that teachers can follow the students' ideas or explore by themselves, and thus get the chord intersection theorem; Further expansion: What if the intersection of two chords is outside the circle and one chord becomes tangent? For students to learn.
3. Explore various conclusions under given conditions. You can change one variable in a traditional math problem into another variable and adapt it into an open math problem.
[Example 3] CD is the height on the hypotenuse AB of Rt△ABC. Try to find out all kinds of relationships between the shape and size of graphics.
Please simplify:-first, then choose a number to make the original formula meaningful, and you like to use it instead of evaluation.
This is a novel and practical topic, which strengthens students' basic knowledge (algebraic simplification and its meaningful conditions) in the form of open questions.
4. Strengthen the conclusion of the proposition and find the sufficient conditions for its establishment (generally additional conditions are needed).
[Example 5] AB//CD is known to be an isosceles trapezoid, if the condition of "BC=AD" is added. Besides "BC=AD", what conditions can be added to make the trapezoid ABCD an isosceles trapezoid? (Write at least two kinds)
5. Compare the similarities and differences of some objects. Using the connection and difference of different knowledge to promote or analogy, compare some similar or identical mathematical propositions and methods, and deepen or popularize them, that is, analogy method. For some mathematical objects, such as geometric figures, numbers, formulas, solutions and so on. By comparing their similarities and differences or classifying them from different angles, we can often get open questions. For example, compare the similarities and differences between the following two monomials:12ab2c, 8a3xy. Similarities: they are all monomials; There are three letters; The coefficients are all positive integers; Both contain the letter a; The highest common factor is 4a3；; Are quintic polynomials. Difference: different letters; Different coefficients; Not in the same category; Although they all contain a, the number of letters a is different.
6. Design solutions to some practical problems or seek various solutions and conclusions in practical problems. According to some mathematical knowledge and methods, open questions of applied mathematics are designed in relevant situations. For example, the issue of interest payment or production profit is directly related to the knowledge of arithmetic progression and geometric series. The author and the students made up such an open math problem: ① There is a garden 4 meters long and 3 meters wide. Now we should open a garden in the garden, so that the area of the garden is half that of the garden. How to design? Give the pattern you designed and make relevant calculations. (2) An industrial and commercial enterprise needs to borrow money from a bank for technical transformation, and there are two schemes. Scheme 1: One-time loan is 6,543,800 yuan, and the profit in the first year is/kloc-0,000 yuan; In the future, the annual profit will increase by 30% over the previous year; Scheme 2: the annual loan100000 yuan, the profit in the first year100000 yuan, and the profit in subsequent years is 50000 yuan higher than that in the previous year. The loan term of both schemes is 10 year, and the principal and interest will be repaid in one lump sum at maturity. If the bank loan interest rate is calculated according to compound interest 10%, try to compare the advantages and disadvantages of the two schemes. If the loan term exceeds 10 years, please estimate the advantages and disadvantages of the two schemes. ③ Try several different schemes, divide the triangle ABC into five parts with equal areas, and point out which five parts have equal areas (keep the marks of division and necessary marks, and don't write).
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Four. Matters needing attention in writing open questions
When writing open mathematics questions, we should grasp the openness, suggestibility of language, development of questions and students' cognitive level, and control the openness and difficulty of questions.
1. Control the openness of the topic. Controlling the openness of topics is a key to the success of writing open topics. Ignoring or belittling this key and simply pursuing the diversity of answers are common reasons for the mistakes in writing open-ended questions. We can control the openness of the topic by limiting the range of answers and changing the values of parameters.
In order to participate in Beijing's bid to host the 2008 Olympic Games, a class of students won the task of making colorful flags. If both sides are 1, a (where a >；; 1), cut into three rectangular colored flags (no cloth left), so that the aspect ratio of each colored flag is the same as that of the precursor, draw the schematic diagrams of two different cutting methods, and write the corresponding value of A (without writing the calculation process).
This topic comes from junior high school mathematics open problem set 3. 1 topic "self-similar 3 division of rectangle" When we compiled this question, the original question was "self-similarity of a rectangle is 5 points", that is, "what is the aspect ratio of a rectangle?" In order to solve this problem, it is necessary to classify graphics in a rather complicated way. Obviously, this is a good question material, but it is not suitable for junior high school students. Therefore, we changed "5 points" to "3 points", which reduced the openness of the problem, but the problem-solving strategies and thinking methods used in solving problems remained basically unchanged, which was suitable for junior high school students.
2. Control the difficulty of the topic. In the open-ended topic, many people tend to only care about the difficulty of the open-ended topic itself, but lack sufficient understanding and attention to the control effect of the way of presenting the topic on the difficulty of the topic. Therefore, we can control the difficulty of the topic by changing the narrative way of the question, using the suggestion technology and changing the answering requirements.
There are only four points on the plane, and these four points have a unique property: the distance between every two points is only two lengths. For example, the square ABCD (as shown on the right) has AB = BC = CD = DA ≠ AC = BD. Please draw four other different figures with this unique attribute and mark the equal line segments.
This is adapted from an old question. The original question has six answers. Considering the knowledge background of junior high school students, it is necessary to illustrate the concept of "two distances and four points on the plane" involved in the question, so take the square answer as an example to help students understand the meaning of the question. In this way, not only the relatively easy-to-think square answer is left to the students, but also the difficult-to-think isosceles trapezoid answer is prompted: in the example given by the question, you can get the isosceles trapezoid answer by removing one point from five points (accepting this prompt also shows the students' mathematical level from one aspect). In addition, it is easier to ask students to find four out of a total of six answers than to find four out of five.
In a word, writing effective open questions is an economical and practical way to develop and utilize mathematics curriculum resources based on textbooks. How to further mobilize students to participate in the compilation work, how to feedback information in time and how to improve it in the process of using open questions, so as to establish different levels of open question resource database and achieve the goal of * * * *, we still need our efforts.
References:
[l] Liu Jian, Sun,. Interpretation of Mathematics Curriculum Standards for Full-time Compulsory Education (Experimental Draft) [M]. Beijing beijing Normal University Press 2002.
[2] Dai Zaiping. Open problem set of junior high school mathematics [M]. Shanghai: Shanghai Education Press, 2005.
[3] Paragraph. The connotation of curriculum resources and its effective development [J]. Curriculum? Textbooks? Teaching methods, 2003, (3).
[4] Liu Min. Design Strategies of Open Mathematics Problems [J]. Education and Research Forum, 2006, (4).
[5] party. Several writing methods of open mathematics problems [J]. Shandong Education, 2002, (1).
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