Since the slogan of "taking function as the key link" was put forward by the reform movement of mathematics education in the early 20th century, function has been established as the core of mathematics teaching. This is not only because it is the important foundation of the whole mathematics system, but also because the function thinking method has become one of the main thinking methods of modern mathematics and can play a leading role in the design of mathematics curriculum. However, function has always been the most difficult thing for middle school students to learn. Some research and teaching practice show that function learning difficulties even accompany the whole mathematics learning process of many middle school students. This paper analyzes the difficulties of middle school students in learning functions, and puts forward some suggestions on the curriculum design of functions.
First, the analysis of learning difficulties function
In China's basic education curriculum reform facing 2 1 century, the design of mathematics curriculum highlights the main line of "function" and adopts spiral arrangement, but function is still the most difficult content for middle school students, resulting in the following three factors.
(A) the complexity of the function itself
Function is the most complicated in middle school mathematics, and it is the main factor that causes students' learning difficulties. A function contains two essential attributes (domain and corresponding rules) and more non-essential attributes (such as range, independent variable, dependent variable, set, etc.). ); In the definition of "variable theory" of junior middle school function, the word "Y is a function of X, denoted as y=f(x)" is an implicit expression with abstract symbols. Function involves "variable", the essence of which is the application of dialectics in mathematics; There are also many representations of functions, such as analytical method, list method, image method, arrow method and so on. Function has a complicated relationship with other contents; Wait a minute. These complexities of functions determine the inevitability of learning difficulties of functions, mainly in the following aspects.
1. Difficulties in understanding function variables
Variables are the building materials of all abstract things in mathematics, but it is not easy for students to understand the connotation of variables. The author made a survey on 300 junior high school students who have studied functions: Please point out the variables in the functional relationship L = 2π r between the circumference and radius of a circle. The survey results are as follows: 83 students think that L, π and R are variables (ask why, answer: all letters can be changed); There are 97 students who think that only R is a variable (ask why, answer: L is a function of R, π is pi, so only R is a variable); There are 59 students who think that only π is a variable (why, A: L is an independent variable, R is a dependent variable, and π can only be changed); 57 students think that l and r are variables; Four students didn't answer. Most students can't understand variables correctly. On the one hand, there are teaching reasons: in teaching practice, teachers often underestimate the difficulty for students to understand variables. On the other hand, looking at the content of middle school mathematics, it is basically constant mathematics before function learning, and it is normal for students to understand variables.
2. The difficulty of abstract function symbols
It is also difficult to accept the abstract representation of function symbols. In a middle school, after the teacher finished the definition of function, he gave the usual expression y=f(x). After class, many students asked the teacher: Is F and X multiplicative? Although students have learned the definition of function, some can even recite it, but they don't understand the true meaning of function. Some teachers think that we should not directly say "usually, we express the function of y as x as: y=f(x)" in teaching, but we can say that "f represents the corresponding relationship between independent variables and dependent variables. For any X in the definition domain (at this time, write' X' on the blackboard), we can correspond through the correspondence relation F (write' F ()' on the blackboard, and the X just now is enclosed in brackets). This kind of improvement can avoid students' illusion.
The author has done a survey, and more than 90% middle school students don't know whether the function refers to f, f(x) or y=f(x). Many students don't really understand what y=f(x) is after graduating from high school-the reason is that the symbol F is "hidden" and its specific content can't be reflected from the symbol-the thinking level of middle school students still lacks enough experience in establishing specific content for F.
3. Difficulties in the application of function images
Number and shape are two aspects of mathematics. In rectangular coordinate system, number and shape are unified, so it seems natural to study various properties of functions by image method. But for students, this is not the case. Although most students can make simple images, they often regard function images as something other than functions, and do not regard them as an organic part of functions. For example, students are not used to transforming functions into f (x) k, f (kx),
|f(x)|, f(|x|), f2(x), etc. Are related to graphic transformation (such as axial symmetry and central symmetry). It is not easy for middle school students to regard the image of the function as an organic part of the function. In fact, before learning functions, students basically study logarithms and shapes separately, and only need to think about logarithms or shapes in learning. Function requires the flexible conversion of thinking between symbolic language and graphic language, but the formation of middle school students' visualization consciousness (the idea of combining numbers and shapes) takes a long process.
(B) the level of thinking development of middle school students
The learning difficulty of function is related to the development level of middle school students' thinking, and the restriction of middle school students' mathematical thinking development level is its internal factor.
Students are required to construct a process that reflects the dynamic change process of "taking a function value for each specific value in the definition domain" according to a possible situation of the function. At the same time, we should condense the three components of the function: correspondence law, definition domain and value domain into an object to grasp, and understand the object as a whole, dynamic and concrete. At the same time, the dynamic process should be transformed into a static object, which can be static, moving or discrete. Psychological research shows that: [2] The thinking development level of junior high school students is gradually changing from concrete image thinking to formal logic thinking level, and the dialectical thinking development of senior high school students is gradually becoming the mainstream on the premise of continuing to improve the development of formal logic thinking. But dialectical thinking is the highest form of human thinking development, and the dialectical thinking of middle school students is basically in the early stage of formation and development. On the one hand, the dialectical thinking of middle school students is immature, and their thinking level basically stays in the category of formal logical thinking, and they can only understand things partially, statically and separately; On the other hand, the characteristics of functions are developing and changing, which are interrelated with many mathematical knowledge and belong to dialectical concepts. This contradiction constitutes the root of all cognitive obstacles in function learning.
(C) junior high school function convergence
In our country, the curriculum design of "Variable Theory" and "Correspondence Theory" of junior and senior high school functions has always been the external cause of the difficulty in learning functions. This design is reasonable, but on the other hand, it is easy to cause difficulties in students' cognitive cohesion.
First of all, it is necessary to explain to students why we should re-characterize functions and solve the compatibility problem between "variable theory" and "correspondence theory". Of course, it is not difficult to solve this problem simply, but because of the inherent defects of "variable theory" [3], students' thinking will be implanted in the teaching of junior high school functions, which will lead to preconceived misleading and will inevitably increase the difficulty of convergence. In the investigation, we found that expressing Y as a function of X in the Theory of Variables often makes students make a common mistake: Y is a function, so it is difficult to accept that the corresponding relationship F is a function in senior high school. In the process of changing students' thinking from "variable theory" to "correspondence theory", we need to make a big leap, abandon the saying that "Y changes with X (X is the independent variable and Y is the dependent variable)", abandon the non-essential thing of "change" and highlight the idea of "correspondence". This will inevitably increase the inadaptability of function learning in senior one.
Secondly, the "variable theory" is based on variables. The so-called "quantity" refers to measurable objects, such as length, distance and time. That is, the research scope is limited to real number sets. This not only affects the migration of function to a higher level of abstraction, but also prevents students from applying function ideas to various research objects.
Thirdly, although the "variable theory" has practical value in some occasions, in fact, in the life of junior high school students, the "variable theory" is not necessarily more natural and practical than the "correspondence theory". Because even if students can easily understand many problems related to "correspondence" in life with their own life experience, it is not so easy to understand "variables". In senior high school, the focus of function teaching is to pursue formalization and pay less attention to practical problems. This may be the reason why most middle school students can't apply functions to solve practical problems after learning them.
Second, the function of curriculum design recommendations
At present, the theoretical discussion of mathematical learning in cognitive psychology is still in the primary stage, and the theory that can be used to better explain functional learning has not been supported by mature practice. Therefore, on the one hand, the study of function learning difficulties needs to explore the psychological mechanism of its learning process and construct its teaching and learning strategies in teaching practice; On the other hand, the author believes that the reform of function curriculum design can not only eliminate the external factors of function learning difficulties, but also improve the quality of mathematics teaching and cultivate students' awareness of "using mathematics" and their ability to explore and innovate.
(A) the role of thinking in the whole curriculum system.
The so-called functional thinking refers to the thinking method of solving problems by using a special correspondence between things. It runs through many occasions of mathematical theory and practical problems, and is a tool for effectively expressing, processing, communicating and transmitting information, exploring the development law of things and predicting the development direction of things.
Functional relations exist widely in students' mathematics courses. Such as the correspondence between natural numbers, rational numbers and real numbers. Points on the sum axis; Algebraic operations, various algorithms, identity deformation, equations, inequalities, etc. Can be attributed to a functional relationship; Symmetry, similarity, translation and rotation transformation in geometry are the corresponding relationships from one atlas to another. The relationship between the size of various geometric figures and the perimeter, area and volume can be summed up as a functional relationship. For example, the problems of travel, mobility, proportion, value and pursuit in mathematical application problems can all be solved by function thought.
In a word, taking function thought as the soul of senior high school curriculum system can achieve high-level harmony and unity. This will also help teachers to re-create the whole textbook, help students form a good cognitive structure, cultivate students' mathematical ability and problem-solving ability, and improve the quality of mathematics teaching.
(B) Pay attention to the consistency and focus of functional curriculum design
The new curriculum of middle school mathematics in China still divides the function curriculum design into two stages. The first stage is in the third stage of compulsory education (junior high school). In the corresponding curriculum standard [4], only a few specific objectives of learning functions are put forward, which seems to leave more room for the compilation of teaching materials. However, almost all junior high school textbooks adopt the "variable theory". The second stage is arranged in the first grade of senior high school. In the corresponding curriculum standards, the requirement of "correspondence theory" is clearly put forward, that is, "use sets and corresponding languages to describe functions and realize the role of correspondence in describing the concept of functions", and in the teaching instructions and suggestions, it is pointed out that "teaching should help students understand the essence of functions from two aspects: actual background and definition. There are generally two ways to introduce the concept of function. One way is to learn mapping first, then function. Another method is to experience a special correspondence between sets of numbers through concrete examples, that is, functions. " And suggested "adopting the latter way". Under the guidance of curriculum standards, the latter method is adopted in the existing experimental textbooks of new curriculum in senior high schools. The author thinks that in the teaching suggestions of functions in curriculum standards, it is advocated that mapping should not be mentioned first, but should be introduced directly from correspondence through concrete examples. This diluted processing form provides an opportunity for the overall reform of function curriculum design.
In the international comparison and exchange of mathematics curriculum reform, we find that it is rare for junior high school and senior high school to adopt the curriculum design of "variable theory" and "correspondence theory" respectively. Developed countries generally adopt the form of desalination, and penetrate the corresponding ideas earlier through concrete examples. [5] For example, in the French mathematics curriculum, the fourth and fifth grades of primary school require students to know the functional relationship and its inverse correspondence between the values used in the decimal set; The sixth grade requires describing the situation with function correspondence diagram; Grades 7-9 use charts, analytical expressions and other ways to express functions and deal with problems, but no strict definition of functions is given. When entering the high school stage, we should implement the teaching of different subjects, and only pay attention to the formal teaching of functions in mathematics courses involving natural sciences. With the deepening and extension of function teaching, calculus is included in the mathematics curriculum in senior high school. The mathematics curriculum in Japan is also the initial concept of function from the fourth grade of primary school, and the overall design of function curriculum is similar to that in France. In American mathematics curriculum, the central theme of curriculum standards for grades five to eight is to study patterns and functions, with emphasis on the exploration of functions. Students are required to recognize, describe and summarize the model and establish a mathematical model to judge and explain the phenomena in the real world. All kinds of algebra textbooks above grade nine define "relationship" first, and then define functions as special relationships [5].
From the design of function curriculum in developed countries, we should dilute the form, adopt the strategy of "early" and "real", and pay attention to the consistency of function essence and the focus of learning stage.
(3) Strengthen the connection between functions and related disciplines and real life.
Function relation not only exists widely in students' mathematics courses, but also has close ties with other disciplines and students' real life. For example, the free fall in physics, the temperature when heating, the relationship between the speed and time of cell reproduction in biology, the accounting of production cost in economics, the improvement of production efficiency, etc., can all be attributed to functional relationships. Functional relationships are also closely related to students' real life, such as the corresponding relationship between height and weight and age, the relationship between telephone charges, utilities, taxi fees and time, the relationship between bank interest and deposit time, and so on.
All kinds of things in our living space are in a dynamic balance of mutual connection and restriction, which is a universal law of objective existence. In the course design of function, we should try our best to explore the connection with other disciplines, make use of the disciplines that students are familiar with and have realistic background, highlight the instrumental role of function thought, give full play to the role of function thought in solving practical problems, encourage and organize students to carry out investigation and study, learn to use the learned function knowledge to solve practical problems, and enhance students' interest and confidence in learning functions.
Attach importance to the role of modern educational technology such as computers (equipment)
In the design of function course, attaching importance to the role of modern educational technology such as computers can not only greatly enhance the intuition, improve students' interest in learning and teaching efficiency, but also help to improve the long-term dull and closed state of function teaching themes and methods. These functions are huge and multifaceted. For example, by generating images of various elementary functions on computers and graphic calculators, we can make comparative explanations and deepen our understanding of functions and their properties. Using computing technology, students can investigate the properties of various types of functions, including the positive and negative transformations and the changing rules of function images when the parameters in analytic functions change, and understand the connotation and essence of functions more deeply and comprehensively through different static and dynamic ways and different macro and micro perspectives, especially in the close relationship between mathematical facts and other disciplines and realistic backgrounds. You can also use computers to carry out experiments, guess and explore mathematical discovery activities, realize "mathematics teaching is the teaching of mathematical activities" and realize "re-creation" of function learning, so that students can personally experience the process of using function knowledge to establish models and explore laws, and cultivate students' scientific inquiry and innovation ability.