Keywords: mathematical view; History of mathematics; Logarithm; plural
In teaching, students often ask such questions: "Teacher, why am I just not interested in mathematics?" "Teacher, what's the use of learning these concepts, theorems and formulas in the future?" What's more, he asked, "Teacher, why are you forcing me to learn math?"? I won't do math research in the future. " ……
Indeed, at present, many students think that mathematics is a boring and difficult subject because they can't figure it out; Because you don't understand mathematics, you think that mathematics is a game in which concepts and rules fall from the sky; Because you don't realize the value of mathematics, you think that mathematics is a subject with no practical significance, and learning mathematics is only to cope with exams; Because I don't understand the thought and spirit of mathematics, I think that "being able to recite concepts, use formulas, prove theorems and do problems" is the highest standard for learning mathematics well. ...
These phenomena show that the problems in students' minds can no longer be taken lightly, so the author has carried out relevant research.
First, the analysis of the current situation of senior high school students' mathematics view
Senior high school students' view of mathematics mainly refers to students' belief in mathematics itself, in mathematics learning and in themselves. [1] Because individuals have different knowledge backgrounds, or accept different philosophical concepts, or are influenced by different teachers, plus their own practical experience, they gradually produce and form their own different understandings and experiences in the process of mathematics learning.
(1) belief in mathematics itself
In the process of learning mathematics, students have different feelings and understandings of mathematics itself. Through the investigation of 6 14 senior high school students, it is found that about 52.5% of them "never thought about what mathematics is"; 24.9% people "thought about what mathematics is, but don't know what it is"; 7.8% people "have heard the teacher say what mathematics is"; 14.8% people "thought about what mathematics is, so they know what it is". But in their eyes, mathematics is mainly related to numbers and graphics; It is a discipline composed of concepts, formulas, theorems, rules and symbols. It is a subject with strong skills and methodology, but it is not easy to master; It is a subject about calculating and solving problems; It is a subject to explore the relationship between spatial form and quantity. ...
(2) Belief in mathematics learning
Davis et al.' survey (Li Shizhen 200 1, 2 17-222) shows that students hold different views on mathematics learning in the process of learning. The author found that high school students' mathematics learning beliefs mainly include:
(1) Learning mathematics means doing problems;
② Learning mathematics is to get good grades in the exam;
③ Learning mathematics mainly depends on memorizing, imitating and formulating formulas;
④ Learning mathematics is to cultivate a person's ability of calculation and thinking; Solid geometry mainly cultivates a person's logical reasoning ability and spatial imagination ability;
⑤ Learning mathematics means learning to use what you have learned to solve problems in real life.
(3) Belief in learning mathematics.
There are obvious differences in students' beliefs in learning mathematics. In the investigation, we found that:
(1) Full of self-confidence-some people are interested in mathematics and think that they have certain talents and advantages in mathematics, and they are confident and capable of learning mathematics well.
(2) Plain and confident-some people are interested in mathematics in general, and think that they don't have much talent and advantages in mathematics, but they can still meet the basic requirements as long as they work hard and study hard.
Lack of self-confidence-some people are not interested in mathematics and think that they have no talent for learning mathematics at all and have no ability to learn mathematics well. They often say that they have been poor at math since primary school, which shows that they can't learn math well.
(4) Types of mathematical views
According to the investigation and analysis, senior high school students' view of mathematics can be summarized as follows:
① Dynamic view of mathematics. In the eyes of students, mathematics is knowledge in the process of constant change and development, so there may be shortcomings and mistakes, and only through constant trial, correction and improvement will it be gradually improved. So learning mathematics is also a step-by-step and continuous improvement process. Be tolerant of your own confusion and mistakes, and know that only by taking a positive attitude can you learn math well.
② Static absolutist mathematics view. They regard mathematical knowledge as a collection of truths that have remained unchanged for thousands of years. It is a highly rigorous and extremely abstract knowledge system. So they emphasize acceptance and memory, imitation and training, and advocate that practice makes perfect; Or think that their memory ability is not good, their abstract ability is poor, and mathematics learning is bound to be difficult.
③ The mathematical view of instrumentalism. They think that learning mathematics is to learn the methods and skills to deal with and solve various (mathematical) problems. Therefore, they pay more attention to practical problems, advocate the close combination of mathematics and life, and pay more attention to accumulating materials related to mathematics.
④ View of mathematical culture. They believe that mathematics is a human culture that has a certain relationship with social nature, class consciousness and national spirit, and it is a specific knowledge system that reflects people's thinking methods, aesthetic consciousness and cultural values. Of course, this concept is less found and accepted among students.
The above concepts reflect students' knowledge and understanding of mathematics itself and their understanding and judgment of the value of mathematics from different angles. Of course, some ideas have a positive effect on students' learning, while others obviously have a negative effect.
Second, analyze the influence of mathematics view on mathematics learning.
At present, there is still a lack of accurate data analysis on the influence of mathematical concepts on students' mathematical learning. However, according to historical data and current research, students' view of mathematics has a considerable influence on their learning methods and achievements. Schoenfeld's research shows that the development of students' thoughts has become an important factor in the process of mathematics learning, and there is an obvious correlation between mathematics belief and mathematics achievement. [2] Carlson's research found that some ubiquitous and lasting mathematical concepts played a decisive role in their later learning. [3] Zheng Yuxin pointed out that for students, the importance of thinking lies in that mathematics learning not only refers to the learning of knowledge and the improvement of ability, but also a process of forming opinions, beliefs and attitudes, which will have an important impact on their future mathematics learning and even their life. [4]
In fact, for individuals, a correct view of mathematics can control their own factors and make them actively participate in learning activities. If a student does not have a certain mathematical concept, then he will be a person who lacks initiative, has weak subjective consciousness and can only passively act according to instructions; If students' views on mathematics are inconsistent with those contained in the curriculum, then this concept may become an obstacle to their study; If a student faces a mathematical situation and doesn't realize that it is related to mathematics, then he won't start to deal with it by mathematical methods; If students regard mathematics as a collection of concepts, theorems and symbols unrelated to social production and practice, they will inevitably adopt a static and passive attitude to accept "mathematical truth" in the learning process; If students regard mathematics as the product of mathematicians' imagination and free creation, then a stereotype that is far away from society, objective, strict and highly abstract will occupy the sky in their hearts, which will inevitably lead them to have little interest, little meaning or too difficult psychology and stay away from them. If students regard mathematics as gymnastics of thinking and think that learning mathematics needs to use their brains repeatedly, then mathematics seems to be a yardstick to measure whether a person is smart or not. When they feel depressed because they can't solve math problems, they will feel that their intelligence is not as good as others and they are pessimistic and disappointed. If students think that mathematics learning is calculation and problem solving, then in their eyes, mathematics is closely related to formulas, formulas and formulas, or that mathematics is an activity of giving a bunch of numbers and then finding answers through formulas, then they will inevitably lose interest in lengthy and complicated calculations and endless sea of questions; If students think that math learning is to imitate the thinking of mathematicians or math teachers with superior intelligence, they will often lose confidence and sigh. Practice has proved that students' view of mathematics really affects their learning attitude and interest, their choice of cognitive materials, their choice of cognitive methods and their evaluation of learning results. (Li Shizhen, 200 1, 2 1 1) For groups, the concept of mathematics can control various forces among individuals and make them actively participate in social construction activities. Learning is a social construction activity, and there are various forms of communication between teachers and students, teachers and students, students and families, and students and society. In these activities, on the one hand, mathematical concepts provide the basic principles of activities, thus adjusting the behavior of subjects and determining the degree and scope of communication. On the other hand, through the communication, exchange and collision of individual mathematical viewpoints, all subjects gradually reach * * * knowledge and form a joint force. Although there are individual differences in the views on mathematics in the same group, there is always a dominant view of mathematics at work, and it is this dominant view of mathematics that makes the learning objectives, learning methods and evaluation standards of the whole class tend to be consistent, thus ensuring the smooth progress of learning activities. On the contrary, if the mathematical concepts among students, teachers and students, and students and textbooks are often in conflict, contradiction and conflict, and there is no bond to maintain them, there will be a state of "gathering and dispersing", and learning activities will be difficult to carry out effectively.
Thirdly, the experimental exploration of the influence of mathematics history on senior high school students' mathematics view.
1, experimental purpose
As early as 1876, H. G. Zeuthen, a famous Danish mathematician and historian of mathematics, emphasized that "by studying the history of mathematics, students will not only gain a sense of history, but also look at mathematics from a brand-new perspective, thus having a sharper understanding and appreciation of mathematics." [5] 1977, American scholars McBride and Rollins found that the history of mathematics is very effective in improving students' enthusiasm for learning mathematics [6]. Wilson and Shawort pointed out that students and teachers should think about "Who does mathematics", "How does mathematics do it" and "What is mathematics", so that students can understand the extensive relationship between mathematics and other disciplines, mathematics and society. It can broaden the view on the essence of mathematics [7]. J. Fauvel, a British historian of mathematics, summed up 20 reasons for applying the history of mathematics to mathematics teaching, one of which is that the history of mathematics can change students' views on mathematics [8]. Brugell pointed out that the historical knowledge about how the mathematical concept developed helps students to understand the concept, and pointed out that mathematics was created by human beings in a specific historical period, not eternal [9].
Since the establishment of 1972 "International Research Group on the Relationship between Mathematics History and Mathematics Education" (HPM), more and more European and American scholars have done a lot of research on the relationship between mathematics history and mathematics education. There are also some scholars in China who pay more attention to the relationship between the history of mathematics and mathematics education. However, it is rare at home and abroad whether the history of mathematics can change students' view of mathematics and thus affect their mathematics learning. Inspired by history, this paper intends to further explore whether the history of mathematics has an impact on senior high school students' mathematics view on the basis of previous research results.
2. Determination of the theme
Experimental class: Class 4, Grade 03 Preparatory Course, Sugaogong Campus; Control class: Class 3, Grade 03, Su Campus. The experimental class and the control class are randomly selected. The mathematics teaching of the two classes is undertaken by the author alone.
3. Experimental process
(1) Pre-test. Test the math scores of students in two classes, and the results are shown in Table 3.
A questionnaire survey was conducted on students' views on mathematics in two classes (see appendix 1), and the results are shown in table 4.
⑵ experimental method
(1) Introduce the relevant history in combination with the teaching content.
During the one-year teaching process, the experimental class introduces at least one knowledge about the history of mathematics every week, while the control class changes to solving problems and exercises.
② Select some contents to test and compare.
Experiment 1: Logarithmic Concept
When learning the concept of logarithm, the two classes adopt different teaching methods: first, organize teaching according to the textbook system; In addition, combined with the reading material "A Brief History of Logarithm and Exponential Development", it answered students' various questions, and also triggered an unexpected logarithmic class [10]. The results of the after-class test (see Appendix 2) are as follows:
Table 1 pre-and post-test statistics of logarithmic concept learning in two classes
The results show that after learning A Brief History of Logarithm Development, the difficulty of learning logarithms in the control class is obviously reduced, the interest in learning logarithms is obviously improved, the purpose of learning logarithms is more clear, and the process of generating logarithms is more clear.
Experiment 2: Complex number concept
The teaching organization of these two classes is different. The control class organizes teaching according to the content and system of the textbook. Starting with the development of complex numbers, organize teaching in experimental classes. The findings (see Appendix 3) are as follows:
Table 2 Statistics of Complex Concept Learning Test in Two Classes
The results show that the experimental class is easier to accept imaginary numbers than the control class, and the proportion of imaginary numbers as meaningful real numbers is greater than that of the control class. The proportion of number system as dynamic development is higher than that of control class.
We also learned in after-class communication that the introduction of historical process makes students understand the concept of logarithm more comprehensively, accurately and profoundly.
① Complex numbers are constructed in a certain way. The generation of complex numbers is a process of organizing and systematizing mathematical contents based on the principle that operations can be performed without restriction [1 1]. This is a way for human beings to construct the number system, and it is also one of the ways for students to construct the cognitive structure of the number system.
The emergence of complex numbers is a historical development process. Through the analysis of the development process of complex number, students realize that complex number is the product of the joint efforts of several generations; It is a process from scratch, from doubt to acceptance, from vagueness to clarity, from one-sidedness to perfection; It develops with the development of society and mathematics itself. Complex number is produced after the supplement and popularization of real number theory. This is the inevitable result of the accumulation of inherent achievements in mathematics itself, which leads to a new abstract stage and rises to a new general concept [12].
(3) imaginary number is not mysterious, absolute authority. Judging from the "growth" process of the concept of imaginary number, even mathematicians have gradually deepened their understanding. At first, people were skeptical and unacceptable about imaginary numbers. Leibniz called imaginary number "a strange creation of ideal world", "a wonderful protector of God, an amphibian almost between existence and non-existence" [13]. However, although Euler also believed that imaginary numbers only existed in imagination. It was not until Hamilton established complex numbers on the basis of real number theory and strengthened the application of complex numbers in physics and other fields that people really began to accept imaginary numbers. This is consistent with students' lack of understanding of its practical application and their psychological obstacles in understanding and accepting concepts. However, the presentation of history helps students to dispel their mysterious mentality and authoritative psychology and reduce the feeling of being excluded.
The emergence and development of complex numbers is a breakthrough in people's ideas. Students think that such an equation has no real solution. Since there is no real number solution, why discuss it? Since negative numbers can't be squared, why should we admit that they are meaningful? This is a psychological contradiction, a cognitive conflict and a closed concept. Dialectics tells us that nothing in the world is completely unchanged and will not develop anyway. No matter how accurate the definition is, any mathematical concept will develop with the development of science. People's understanding of things always spirals up. Through the investigation of history, we realize that the introduction of imaginary numbers is a kind of creation, an invention and a breakthrough in thinking.
⑤ Differentiate the ancients' view of mathematics and promote the formation of students' view of mathematics.
When learning solid geometry, let the students discuss Euclid's view of mathematics. When learning analytic geometry, let students discuss Descartes' view of mathematics and the birth of analytic geometry.
(3) Post-test: After the end of a school year, two classes were tested and investigated by questionnaires (see appendix 1), and the results are as follows:
Table 3 Statistics of the results of the initial and final exams of the two classes
Note: (1) There is no significant difference between the experimental class and the control class.
⑵ The experimental class and the control class have final grades, so we can't think that the history of mathematics has no influence on students' grades.
Table 4 Questionnaire survey statistics at the beginning and end of two classes
The results show that the introduction of mathematics history has obviously improved the students' interest in mathematics learning in the experimental class; It has strengthened students' mathematics learning motivation and changed their mathematics concept. Let students know more about the essence of mathematics and promote the improvement of mathematics scores.
4 conclusion
Through a year's investigation, it is found that the history of mathematics can change students' view of mathematics to a certain extent, thus affecting mathematics learning.
Through the understanding of history, students can psychologically shorten the time to accept an idea.
Through the analysis of history, let students accept that mathematics is the result of human social activities.
③ The history of mathematics is helpful to cultivate students' dynamic view of mathematics.
The history of mathematics helps to cultivate students' ideas of creation and invention.
⑤ The history of mathematics helps to cultivate students' mathematical cultural values.
The history of mathematics helps students to understand the process of formalization, abstraction and precision of mathematics.
⑦ The history of mathematics helps to change teachers' view of mathematics, thus affecting students' view of mathematics.
5 Some suggestions
Based on the research of this paper, I suggest: attach great importance to the cultivation of students' mathematical outlook; Seriously deal with the relationship between the history of mathematics and mathematics textbooks; Organize the compilation of appropriate historical materials; Seriously organize the training of mathematics history for in-service teachers; Vigorously carry out HPM research.