Regarding the thinking process, psychologists have studied the thinking process of solving problems from different aspects, among which the more representative theories are "four-stage theory of solving problems" and Will? Simon's information processing system theory, Paulia's four-link theory of problem solving, and Wallace, an American psychologist, have studied according to the self-reported experiences of many creative inventors and summed up four processes. Looking at the research of thinking process, we can see that the research results are rich and the exploration is very deep. However, on the whole, there are still some defects in the study of thinking process: the division of thinking process is too mechanized, stylized and incoherent. At the same time, students' thinking process of solving mathematical problems is different from that of mathematicians, and intuitive thinking is also different from mathematicians, which is a skilled stage of algorithmic thinking. The intuitive results of the two are different. The inventor's result is new knowledge unknown to mankind, and the student's intuitive result is re-invention and re-creation. However, the thinking process of both of them is a leap from the original thinking, which is an unconscious process. Moreover, intuitive thinking runs through every link of the thinking process, and every gap in the algorithmic thinking process is lubricated by intuitive thinking. Intuitive thinking is a lubricant, not after algorithmic thinking. Therefore, the thinking process of students' mathematical problem solving should be a systematic thinking process.
First, the systematic thinking process of solving mathematical problems
The thinking process of mathematical problem solving (hereinafter referred to as systematic thinking) is a systematic process in which the algorithmic thinking process and intuitive thinking process interact under the supervision of metacognition.
The algorithmic thinking process refers to the thinking process of solving mathematical problems, which is divided into four steps: preparation, exploration, clarity and construction. Among them, the preparation stage includes having a clear understanding of the problems faced, having the motivation to solve problems, and reading math problems; The exploration stage includes the expression of conditions or conclusions in mathematical language (algebraic language or graphic language), simplification (including factorization, general division, reduction, constant separation, combination of similar items, rationalization of numerator or denominator, reverse application of basic operation theorems and formulas, etc. ), equivalent transformation conditions and conclusions, trial and error, reverse thinking. The clear stage includes pattern recognition, recall and association, finding out the internal logical connection and gap between the conditions and conclusions of the problem and the intermediate steps and conditions (or conclusions) in the process of solving the problem, finding out the strategy to narrow the gap, determining the implementation strategy, evaluating the adopted strategy, deciding a standard and choosing the appropriate strategy. Construction stage: verify the conclusion, directionally control the thinking process, deepen and consolidate mathematics knowledge, improve mathematics methods, feel positive emotional experience and enhance mathematics learning interest.
Each stage of the above four steps may encounter failure and produce new problems. At this point, the process of metacognitive monitoring thinking has returned to the original place or environment for careful exploration. This cycle may go through many times, so the problem-solving process composed of these four basic stages is a nonlinear and dynamic cycle process. [2] Intuitive thinking is a highly compressed algorithmic thinking process. When the students' algorithmic thinking works well in the exploration and clarity stage and becomes subconscious, the problem information will be input into the brain for a short period of algorithmic thinking (sometimes even unconscious) and get the results instantly, which is the students' intuitive thinking process. When solving mathematical problems, we first go through the preparation stage of algorithmic thinking, and then metacognitive control thinking first carries out intuitive thinking according to the problem situation. If the problem can be solved correctly, metacognitive control thinking will enter the construction stage of algorithmic thinking, so as to verify the correctness of the results and analyze the mechanism of intuitive thinking, which will be stored as procedural knowledge in the cognitive structure as the trigger point of intuitive thinking in the future. Otherwise, metacognitive control thinking will turn to the exploration and clarification stage of algorithmic thinking, and finally enter the construction stage.
It can be seen that good and orderly algorithmic thinking is the basis of intuitive thinking, which can simplify algorithmic thinking. The thinking process of students in solving mathematical problems is systematic thinking. Systematic thinking has the following functions in solving mathematical problems:
Second, the function of systematic thinking.
1. Promote students' correct attribution
Many problem-solving subjects can't be correctly attributed to the low personal problem-solving ability and take effective measures, which leads to the problem-solving ability not being improved for a long time. Systematic thinking must be applied on the basis of a complete and orderly cognitive structure. If the cognitive structure is defective, systematic thinking will be hindered and stopped. At this point, the thinking subject can correctly attribute.
For example, a student who has just learned a topic can't do it in an hour or two. The class teacher of this student thinks that this is because of poor understanding ability and no understanding of the connotation and extension of the concept, so he has been guiding students to improve their understanding ability for a long time; Parents think it's because students' endowment is not high and they can't compensate. But the author explained the following question to him: "Let the variables X and Y satisfy the constraint conditions, then what is the maximum value of Z=3x-2y?" Students can understand the essential meaning of this question well: find the maximum and minimum intercept of a set of parallel lines Z=3x-2y by indicating the points in the area. However, the next day, students will not be able to test this type of question. When the author asks students to do sound exercises according to systematic thinking, it is found that students can successfully complete the preparation stage, but they can't do pattern recognition and recall association in the clear stage, and they can't remember doing such problems at all. When the author reminded me that I had talked about this kind of topic yesterday, the students could solve it smoothly immediately. By using systematic thinking to solve problems, the student found that the reason for his low problem-solving ability was his poor memory. Therefore, the effective guidance for this student should be to train memory. Therefore, systematic thinking can play a correct attribution role in solving mathematical problems.
2. Improve students' cognitive structure
The operation of systematic thinking needs the support of perfect knowledge structure. Without knowledge, systematic thinking will stop. At this time, students must take the initiative to make up for the lack of knowledge, and systematic thinking can continue to operate. Therefore, the process of systematic thinking is also the process of students' self-improvement and consolidation of knowledge structure, and it is also the process of students' active learning (or review), and its effect is higher than that of passive learning. Successful learning experience enhances students' confidence and interest in learning mathematics and improves their learning enthusiasm. Therefore, teachers should pay attention to cultivating students' systematic thinking in teaching. In order to cultivate students' systematic thinking, teachers should first reform the teaching methods of examples and highlight the role of systematic thinking.
Third, the teaching strategy of cultivating systematic thinking
The basic thinking of students' intuitive thinking is algorithmic thinking, so to cultivate students' systematic thinking, we must first cultivate algorithmic thinking and stimulate intuitive thinking with algorithmic thinking.
1. Change the way teachers explain examples.
When explaining examples, the teacher explains according to the process of systematic thinking, showing the running track of algorithmic thinking, the trigger of intuitive thinking and the adjustment process of metacognition, so that students can fully appreciate the effectiveness and importance of systematic thinking.
Example 1 Let the function f(x)=. If f(x)≥2 exists for any x∈R, find the value range of a. ..
The teacher explained the problem: this problem is to prove inequality, and the function contains two absolute values. According to the usual method of dealing with algebraic expressions with absolute values, the expression of f(x)=≥2 is transformed by zero-point division method.
Continuing to observe the conditions, we know that this problem is the third kind of inequality problem from "for any x∈R, there is f(x)≥2". The common solution is to find the minimum value g(a) of the function f(x), and then solve the inequality g(a)≥2. The relationship between the intermediate state "piecewise function f(x)" and "finding the minimum value g(a) of function f(x)" is analyzed. The common method to understand this kind of problem is the mirror image method, that is, draw an image with the piecewise function f(x)= and find the minimum value from the image. From the above analysis process, we can know that analyzing the relationship between intermediate state and intermediate state (conditions and conclusions) is the key to solve this problem; If the algorithm thinking is automated, we can immediately know that this problem is to find the maximum value of the piecewise function, and the solution method is the image method, which is the intuitive thinking of students. At this point, the teacher finished analyzing the problem. From the teacher's analysis of examples, students feel that there is a clear direction to solve problems with systematic thinking; Know what to think when thinking stops; Know what steps thinking has taken, what steps are missing and what steps are incomplete, so as to make up in time. In this way, students feel the effectiveness of systematic thinking and stimulate their desire to learn systematic thinking.
2. Cultivate students' metacognitive monitoring ability
When students have the psychological tendency of learning systematic thinking, questioning method (teachers ask questions in the early stage, students answer them, and students ask themselves in the later stage) is adopted to cultivate students' metacognitive monitoring ability. These questions are: can this topic be expressed in mathematical symbol (graphic) language? Whether the following simplification techniques can be used: equivalent substitution, merging similar items, factorization, general division, denominator rationalization, formula, common factor extraction, cross multiplication, separation of constants, etc. What kind of problems are you facing? What are the internal relations between conditions and conclusions, between conditions, between intermediate states and conditions and conclusions? What is the connection and difference between this problem and typical problems? Is there any way to eliminate this difference? Is the method of eliminating differences simple? Do you have anything simpler? Is this a new typical problem? Is the problem-solving method typical? Is it helpful to solve other problems? Students have good metacognitive monitoring ability, can control the operation of algorithmic thinking independently according to problems, correctly choose the key steps of algorithmic thinking, ignore unnecessary steps, and finally form intuitive thinking.
The author's four-year teaching practice has proved that the implementation of systematic thinking teaching strategy has improved students' mathematics scores, intuitive thinking and metacognitive ability, and enhanced students' interest in learning mathematics. Thus it is proved that the systematic thinking teaching strategy is reasonable in theory and effective and feasible in practice.
(Reference of this journal) (Editor)