Stolyar pointed out in his book Mathematics Pedagogy: "Mathematics teaching is also the teaching of mathematics language". [1] Because mathematical language is a scientific language composed of mathematical symbols, mathematical terms and improved natural language, teachers should not directly use mathematical language as teaching language in mathematics teaching, but must transform it into a language that students can easily accept according to their knowledge base and psychological characteristics. In other words, mathematics teaching language is the combination of mathematics language and teaching language. Because the teaching language is always accompanied by every link in the teaching process, it is expressed in an interspersed way. Therefore, interspersed language has become the basic form of mathematics teaching language, and the art of mathematics teaching language is mainly the art of interspersed language in mathematics teaching.
As the name implies, the so-called interspersed language refers to the mathematics language and teaching language outside the explicit text in mathematics textbooks. Interpolation is not scripted. As far as every math class is concerned, interspersed language always occupies a large proportion. For example, according to its different functions and ways in the teaching process, there are guidance, explanation, question and answer, analysis, comment, transition, contingency, metaphor, humor, rendering and so on.
Being good at using interspersed language is a very important basic skill for math teachers. How can we make good use of interspersed language? "There are three sources in the process of teaching and education: science, skills and art" (Suhomlinski). Accordingly, the following three principles should be followed in the teaching process, according to its teaching content, language skills and language art.
First, scientific principles.
Classroom teaching is the unified expression of knowledge content and its language form, and the scientific nature of knowledge determines the scientific nature of language. Therefore, scientificity is the fundamental attribute of interspersed language in the teaching of various subjects. However, the scientific nature of language in mathematics teaching has its unique connotation.
(A) the three-dimensional nature of mathematical content
According to the concept of modern quality education, the so-called mathematics education is an education that takes mathematics knowledge (and its application) as a cultural medium, absorbs various nutrients from it and promotes students' self-growth. Among them, growth mainly refers to the growth of thinking mode and the spirit of seeking beauty. Because of the good growth of thought and spirit, people's behavior quality will naturally be improved [2].
Compared with mathematical knowledge, mathematical thought and beauty are "recessive" and "dominant". But the effectiveness of a lot of knowledge is short-lived, but the effectiveness of ideas is long-term, which can make people "benefit for life"; The form of knowledge is dull, and the form of beauty is vivid, which can make people's hearts "summoned". The history of mathematical development also shows that mathematical creation often comes from the breakthrough of old mathematical ideas or the creation of new mathematical ideas; Mathematical discovery often comes from the pleasure of thinking or the call of mathematical beauty. As mathematician George Polya said, "The perfect way of thinking is like the North Star, and many people find the right path through it." Poincare, a great French mathematician, pointed out: "The people who can make mathematical discoveries are those who have the ability to feel the beauty of order, harmony, symmetry, neatness and mystery in mathematics, and they are limited to this kind of people."
Therefore, as far as mathematics teaching is concerned, "knowledge is valuable, and the price of ideas is higher." If you want to create something, you can't throw it away. " The content of its interspersed language must reflect three dimensions: taking mathematical knowledge as the main body, and digging and displaying the mathematical thinking method and mathematical aesthetic factors reflected by it as the two wings. As the saying goes, "a bird can't fly without wings." Mathematics teaching with poor thinking methods and aesthetic factors is rigid and imperfect. The three-dimensional nature of the interspersed content is the guarantee to fully embody the language skills in mathematics teaching. Doing so can not only sow the seeds of knowledge and ideas in students' hearts, but also lead students into the gorgeous mathematics hall, so that students can naturally understand the mathematical thinking method and be influenced by the beauty of mathematics, thus fundamentally cultivating students' cognitive ability and creative ability.
(B) the duality of language paradigm
Stolyar pointed out: "Mathematics teaching is the teaching of mathematical thinking (activity)." In order to effectively cultivate students' thinking ability, mathematics teachers' language should first have a complete dialectical understanding of the concept of mathematical thinking. Mathematical thinking is an extremely complicated psychological phenomenon. As far as its composition is concerned, there are logical thinking and illogical thinking (that is, image thinking and intuitive thinking). As far as the types of reasoning are concerned, there are deductive reasoning (also called rational reasoning) and non-deductive reasoning (also called perceptual reasoning, including inductive reasoning and analogical reasoning). Their roles in mathematics research or teaching always complement each other. Poincare said: "Logic can be used for argument and intuition can be used for invention." In fact, logical deduction and illogical deduction are indispensable in mathematical thinking activities. Just as people explore in the fog, they not only have to use their eyes to identify the direction and seek the road, but also rely on their legs to reach their destination, so illogical deduction is like eyes, which plays a guiding and leading role; Logical reasoning is like legs. Without logical deduction, it is impossible to reach the destination. 〔3〕
However, for a long time, due to the "logical rigor" of mathematics and the characteristics of logical deduction presented by the textbook system structure, and even because there is a standardized procedure for logical deduction, teachers often focus on logical deduction, and even mistakenly think that "accuracy, rigor and logical requirements" is the only scientific paradigm of mathematics teaching language. In fact, this is a manifestation of ignoring or underestimating the duality of mathematical thinking.
"Language is the coat of thinking." The duality of mathematical thinking determines the duality of mathematical teaching language paradigm, that is, according to the age characteristics of students, we should not only pay attention to strict logical deduction, but also punctuate the non-logical language that can guide students to associate, imagine, guess, analogy, induction, epiphany and understanding, and strive for the perfect combination and high unity of logical deduction and non-logical deduction, so that students can fully understand and understand mathematics and actively discover and create mathematics.
Second, the technical principle "has three words, and it is better to be clever."
Speaking skills are eloquence, which reflects a person's expressive ability. In mathematics teaching, the skills of interspersed language are highlighted in the following two aspects.
(A) the order of language organization
Teaching is a process according to certain procedures, and teaching materials, students and teachers are the three elements of the teaching process. Therefore, if we carefully observe the process of mathematics teaching, we will find that it has been integrated into three teaching procedures, and thus presents three teaching clues.
The first is the logical order of the content of the textbook. That is, according to the arrangement system of teaching materials, the internal relationship between knowledge is systematically analyzed, and the basic concepts, basic principles and basic methods are logically connected in series with the teaching purpose and typical training as the center. This is a "main line" in teaching. With it, the lectures are clear, smooth, coherent and consistent.
The second is the teaching plan designed by the teacher. That is to say, according to students' cognitive rules, a "step-by-step" lecture scheme is designed step by step. This is a "diagonal line" in teaching. With it, where to speak, where to climb, where to break through, where to talk in detail, where to be brief, where to be urgent, where to slow down; Teachers know how to persuade, how to explain and extend, and how to undertake the turning point.
The third is the cognitive thinking procedure of students. That is, under the guidance of mathematical thinking method, students are guided to construct a new thinking activity program of mathematical cognitive structure through assimilation of old knowledge and adaptation of old knowledge by new knowledge. This is a "red line" in teaching. With this red line, we can grasp the essence of teaching materials, fully display and reveal the process of thinking activities, and make "mathematics teaching is the teaching of mathematical thinking activities" implemented. Whether a series of interspersed languages can be organically organized along the above three procedures, that is, three lines, to form a "language chain" with clear direction, clear thinking and clear internal logic is the touchstone to test the language ability of mathematics teachers.
(B) the wisdom of classroom infiltration
In classroom teaching, due to the diversity and complexity of students' intelligence factors and non-intelligence factors, students' information feedback presents diversity and randomness. Some stable factors (such as mathematics content and students' original knowledge level, etc. ) is predictable, while others are unpredictable. Therefore, teachers must start from students' feedback information at any time, use and play the function and role of interspersed language in time, carry out effective regulation and control, and make classroom teaching always in the best state.
The beauty of classroom infiltration is that teachers should be good at guessing and judging students' thinking trends, grasp and capture the opportunity of inspiration, and create angry situations in order to get enlightenment and make it happen; Secondly, students' reaction (answering questions, learning emotions, thinking expression, classroom discipline, etc.). ) even unexpected situations (unexpected problems, teachers' omissions, students' abnormal behavior, etc. We must make clever and timely adjustments, so as to turn boring into novelty, negative into positive, and promote the harmony of teaching.
Makarenko said that one of the necessary characteristics of educational skills is the ability to improvise. Interpolation of language in class requires thoughtful foresight and superb and extraordinary contingency skills.
Third, artistic principles.
Mathematics is a science, but mathematics teaching is an art. Comenius said: "Educating people is an art in art, and the language used in educating people is an artistic language." The language of mathematics teachers should pay special attention to artistry, and really use artistic language to give students beautiful enjoyment, spiritual pleasure and fruitful learning results, just like a playwright carefully chooses words in a script and an actor handles lines on the stage.
(A) vivid and intuitive image
Everything is tangible, and visualization is the dominant feature of art.
Although mathematics has a high degree of abstraction and strict logic, its content-space form and its quantitative relationship always exist in a certain form. Generally speaking, there are two kinds of images in mathematics: perceptual images (images that can be perceived only by human senses) and ideal images (images that exceed the limits that human senses can perceive and are produced through abstract thinking). In mathematics teaching, the application of visual language is based on the high abstraction of mathematics and students' love of thinking in images of form, sound, color and emotion, and it is also the intermediary between them. Of course, the so-called visualization in primary and secondary school teaching is mainly to directly train sensory perception. Teachers integrate teaching content with its image, and guide students to complete the transformation from vivid and intuitive to abstract thinking in concrete and sensible images.
Visual language is a language art that combines hearing and vision. It requires teachers to deeply feel, understand, imagine and experience the teaching content, display the image of the teaching content through appropriate metaphors and popular language, deepen understanding and memory with images, promote the development of students' abstract thinking with images, and obtain the artistic effect of teaching.
(B) the emotional appeal to the soul
"Emotion is the mother of all arts." Emotion is a hidden feature of art.
"Touching people don't worry about feelings first, then they don't worry about words, then they don't worry about voices, and then they don't worry about righteousness." This passage by Bai Juyi, a great poet in the Tang Dynasty, is concise and inspiring. It tells people that language, sound and semantics can trigger emotions, and emotions can touch the soul, that is, emotional language can not only affect students' senses, but also directly appeal to their hearts; Emotion, speech, sound and meaning are four key words to enhance teachers' language appeal.
"Emotion" means that teachers should use their passionate teaching emotions to stimulate students' active learning emotions; "Speech" means that teachers should use interesting, inspirational, suspenseful, humorous and literary language to stimulate students' interest in learning; "Sound" refers to the teaching content and teaching situation in which the teacher's language should be inserted with corresponding pronunciation and intonation, with clear pronunciation, clear pronunciation, accurate articulation and natural harmony; Relaxed, orderly and full of melody, moderate and rhythmic regulation, striving to maintain harmony with students' thinking, so that students' learning emotions are constantly encouraged; "Positive" means that teachers should integrate their inner feelings into the teaching content through research, so that the output teaching information is covered with emotional clothes and dyed with emotional colors.
From 65438 to 0990, Peter Salovi, a psychologist at Yale University in the United States, put forward the theory of emotional intelligence. Later, many experts believe that intelligence comes from emotions, and emotions dominate intelligence. Emotional intelligence is more important than IQ for people's success. The role of emotional language in mathematics teaching is not only the rendering of an atmosphere and the call to students' hearts, but also of great significance to promoting students' psychological activities and improving their intelligence level, which is irreplaceable by any other language.