First, some basic ideas.
1. Problems and Mathematical Problems
According to the dictionary of psychology, a problem refers to "a task situation in which there are some obstacles to be overcome between a given state and a target state". Mathematical problems refer to situations that are characterized by intellectual challenges to people and cannot be solved by ready-made methods, programs or algorithms, or situations that cannot be solved by ready-made mathematical experience and methods. Mathematical problems have three remarkable characteristics: one is obstacle, the other is acceptability, and the third is inquiry.
2. Problem solving, application problems and application problems
(1) problem solving
Mathematics problems are generally divided into two categories, one is routine, that is, simple background, clear conditions, unique answers, solving common problems, mostly in exercises and exams. Another kind of unconventional problem, usually called "challenging" problem, has relatively complex scenarios, implicit conditions and open answers, and there is no ready-made solution to it.
There is no unified explanation as to what is problem solving. But in any case, there is a consistent view on how to understand problem solving: the so-called "problem solving" refers to solving "very expensive problems". The purpose is to cultivate students' inquiry consciousness and innovative spirit.
(2) Problem solving and application
Solving problems is not equal to applying problems. The difference between problem solving and application is that problem solving is the beginning of learning, not just application; Solving problems emphasizes close connection with practice and is open; The form of problem solving: asking questions, experiencing, constructing and forming innovative consciousness; Problem solving has room for communication and reflection. However, at the end of learning to solve application problems, the traces of artificial fabrication of application problems are obvious and closed; The teaching form of practical problems: finding types, remembering conclusions, setting formulas and forming "conditioned reflex"; "Condition+body type = question answer" constitutes the factor of application problem, and students do not need to reflect or reflect less in the process of solving problems.
(3) Problem solving and application
Based on the "gap" between "problem solving" and "application problem", some scholars put forward the term "application problem". Compared with "problem solving", "problem" in problem solving is a more practical problem, which is closely related to students' real life and often needs to consider many factors in real life, and has the characteristics of comprehensiveness and openness. As for the "problems" in practical problems, although it is advocated to conform to students' reality and strive to be open, on the whole, the problems have been simplified to a certain extent, and the background is relatively simple, and the quantitative relations involved are often familiar to students. Therefore, what students do is mainly to analyze the quantitative relationship between them and solve them with the knowledge and methods they have learned.
3. Method of solving problems
(1) A Four-stage Model for Solving Polynesian Mathematical Problems
As early as 1957, Paulia, a famous math teacher, made a detailed analysis and description of the process of solving mathematical problems, and his research formed the basis of the research on solving mathematical problems since the 1980s. Paulia divides solving mathematical problems into the following four stages:
The first stage: understanding the problem. What are you looking for? What information does this question give? Draw a schematic diagram.
The second stage: making a plan. Do you know similar problems? Do you know a simpler question? Can you rephrase the question? Try to solve a related problem, try to solve part of the problem.
The third stage: implementation plan. Execute the solution and check every step. Can you prove that every step is correct?
The fourth stage: review the answers. Check formulas and results. Can you get the answer in different ways? Can you apply this result to solve another problem?
(2) Singapore's four-stage mathematical problem-solving model.
The basic model of problem-solving is given in the appendix of Singapore's revised primary school mathematics syllabus in 2000, which requires primary school mathematics teachers to implement problem-solving teaching with reference to this model. The problem solving steps included in this model are as follows:
Understand the problem. Includes finding out the given information; Materialize this information; Organize this information; Contact this information.
Design plan (select strategy). Including: description and expression; Use charts and models; Make a systematic table; Looking for patterns; Take a step back and consider; Concepts before and after use; Guess; Make a hypothesis; Restate the problem in another way; Simplify the problem; Part of solving the problem.
Implement the plan. Including: using computing skills; Use geometric skills; Use logical reasoning.
Reflection. Comprises the following steps: checking the solution; Improve the method of use; Explore other methods; Extend this method to other problems.
(3) Problem-solving model in modern cognitive psychology.
In modern cognitive psychology, problem solving has always been an extremely active research field, and the problem solving models proposed by researchers emerge one after another. To sum up, the models related to mathematical problem solving can be summarized into five sub-processes:
The psychological essence of discovering consciousness problems is to detect the difference between the existing state and the expected state.
Define and characterize the problem-clearly define the nature of the problem, analyze the conditions needed to solve the problem and the existing conditions, and clarify the ultimate goal of solving the problem.
Determine the solution to the problem-including two basic processes: selecting the solution and determining the specific solution steps.
Implement the problem-solving plan-put the previously formulated problem-solving strategies and plans into practice to make the problem reach the target state.
Evaluate the result of problem solving-actively test and evaluate the process and result of problem solving, and judge whether the process of problem solving is reasonable and the result is correct.
4. Problem solving in Mathematics Curriculum Standard (20 1 1 Edition)
Whether it is the 200 1 version of the mathematics curriculum standard (experimental draft) or the 20 1 1 version of the mathematics curriculum standard, problem solving is regarded as a main line running through the mathematics curriculum in China.
(1) Solving problems is a way of thinking.
In the standard, problem solving is not only the content of the course, but also a concept throughout, which encourages students to experience the process of abstracting mathematical problems from the actual background-establishing mathematical models-solving models-explaining, applying, expanding analysis and solving problems.
(2) Solving problems is the goal.
The goal of process and method in the Mathematics Curriculum Standard (20 1 1 Edition) is divided into mathematical thinking and problem solving. The specific description of the problem solving goal is as follows:
Initially learn to find and ask questions from the perspective of mathematics, comprehensively use mathematical knowledge to solve simple practical problems, enhance application awareness and improve practical ability.
Get some basic methods to analyze and solve problems, experience the diversity of problem-solving methods, and cultivate innovative consciousness.
Learn to cooperate and communicate with others.
Initially form a sense of evaluation and reflection.
Among them, innovative consciousness and practical ability do not appear in other target parts of mathematics curriculum standards, but only in the problem-solving part.
(3) Solving problems is a requirement.
Verbs such as "experience, experience, exploration, attempt, expression, explanation, reflection, etc." mentioned in the mathematics curriculum standard are accompanied by problem solving, and should permeate every knowledge field and the whole process of mathematics teaching.
Second, the educational value of "problem solving"
Mathematics teaching in primary schools should take cultivating students' ability to solve problems as an important task and attach importance to the value of solving problems.
1. The ability to solve problems is an important symbol of students' mathematical literacy.
Pisa (programme for international student assessment developed by OECD) defines mathematics literacy as: "The ability to have knowledge, understand the position of mathematics in natural and social life, make mathematical judgments and participate in mathematical activities to meet the needs of individuals to become caring and thoughtful citizens in their current or future lives. Mathematical literacy includes several levels of using mathematical ability, from standard mathematical operation to mathematical thinking ability and observation ability. Students are also required to understand and apply a certain range of mathematical knowledge, such as probability, change rate, growth rate, space and shape, quantitative reasoning, uncertainty and membership. These include specific areas of mathematics courses, such as arithmetic, algebra and geometry. " Among the eight aspects of mathematical literacy in PISA design, at least three aspects are directly related to problem-solving ability. (1) Mathematical thinking. (2) Establish a model. (3) Questions and problem-solving skills.
2. The improvement of problem-solving consciousness makes students more aware of the value of mathematics.
In the process of analyzing and solving problems, students can experience the application of mathematics in reality, understand the mathematical problems around them, and then guide, understand and master the role of mathematical knowledge and ability. Some people call mathematical consciousness "seeing the world with the eyes of mathematicians". What others may not notice at all is really interesting mathematics for him. In others' eyes, it's not the math background. They can see mathematical problems from it and understand and analyze such problems with mathematical thinking. An important function of mathematics education is to cultivate students' mathematical consciousness and learn to see the world from a mathematical point of view.
3. Promote the understanding and mastery of the basic knowledge of mathematics.
The four areas of mathematics learning stipulated in the mathematics curriculum standards, especially the first three areas, all have specific learning contents, respective goals and tasks for specific knowledge and skills. However, through learning in various fields, it is consistent in cultivating students' awareness and ability to solve problems and cultivating students' emotions and attitudes. In the process of learning various content fields, solving problems should be regarded as an important task. At the same time, the improvement of problem-solving ability will also promote students' understanding and mastery of various fields.
4. Solving problems is an important way to cultivate students' innovative consciousness and practical ability.
The solution of mathematical problems can not directly depend on the existing knowledge and methods, but can only be realized by recombining the existing knowledge and methods and generating new strategies and methods. Therefore, the process of solving mathematical problems is an innovative process. This process urges students to seek new ways and methods, which can not only enable students to acquire initial innovation ability, but also enable students to develop innovative consciousness and innovative thinking habits from an early age, thus laying a good foundation for achieving higher-level innovation in the future.
Third, the analysis of common quantitative relations in "application problems"
Quantitative relationship analysis points to "application problems" in textbooks.
1. Basic quantitative relation
(1) Four operations:
Addition: Addendum+Addendum = sum and-one addend = another addend.
Subtraction: Minus-Minus = Minus-Minus = Minus+Minus = Minus.
Multiplication: factor × factor = product ÷ one factor = another factor.
Division: Divider/Divider = quotient dividend/quotient = divisor quotient × divisor = dividend
(2) Operating rules:
Additive commutative law A+B = B+A
Additive associative law A+B+C = (A+B)+C = A+(B+C) = (A+C)+B.
Multiplicative commutative law ab = ba
The law of multiplicative association ABC = (AB) C = A (BC) = (AC) B.
Multiplication and distribution law a (b+c) = ab+AC
(3) Basic nature:
The operational nature of subtraction: a-b-c = a-(b+c)
Operational nature of division: a ÷ b ÷ c = a ÷ (b× c)
Invariance of quotient: a ÷ b = (a× x) ÷ (b× x) = (a ÷ x) ÷ (b ÷ x) (x ≠ 0).
Basic attributes of a score:
The basic properties of the ratio: a: b = (a× x): (b× x) = (a ÷ x): (b ÷ x) (x ≠ 0).
The basic nature of proportion: AD = BC because a: b = c: d.
In the teaching of conventional application problems in China, the results have been very good. But courses and teaching are often taught for the sake of teaching. Compared with the teaching in Europe and America, there are still many shortcomings in raising questions, developing problems and handling application problems flexibly.