After reading a famous book carefully, I believe everyone must understand a lot. Why not write down your reading notes? So how to write reading notes is more infectious? The following is a sample essay of junior high school reading notes I compiled for you, hoping to help you.
Reading Notes for Junior High School Students in Mathematics Monographs 1 After reading What is Mathematics, I was deeply influenced by the content and felt deeply, and I had a shocking feeling about the evolution of mathematics. I think I must write down this feeling in my notes so that I can forget to remember it later. Why write it down with a pen? I don't need to say more. I think everyone knows the secret.
Now, we will start with a series of axioms, from the generation of natural numbers to the perfection of real number theory. Perhaps there will be a new understanding of the "scientific nature" of mathematics.
Natural number is the most natural number in mathematics, which is used to describe the number of objects, and then some abstractions are the number of elements in a set. In the early days of human civilization, people have naturally used natural numbers. It can be said that natural numbers are naturally generated, and others are slowly expanded and evolved from natural numbers. Mathematician Kroneck once said that God created natural numbers, and everything else is human labor. (Godmadethenaturalnumbers Alsace Wolfman) )。
With the development of some mathematical theories, we urgently hope to have a mathematical description of natural numbers themselves. Logically, what is a natural number? There have been many attempts to describe natural numbers by mathematical methods in history. Mathematician GiuseppePeano put forward a series of axioms for constructing the arithmetic system of natural numbers, which are called Piano's axioms. Peano's axiom holds that natural numbers are a series of symbols that satisfy the following five conditions:
1.0 is a natural number;
2. Every natural number A has a subsequent natural number, denoted as S (a);
3. There is no natural number with a successor value of 0;
4. Different natural numbers have different successors. That is, if a≠b, then s (a) ≠ s (b);
5. If a set of natural numbers S contains 0, and the successors of each number in the set are still in the set S, then all natural numbers are in the set S (this ensures the correctness of mathematical induction).
Figuratively speaking, these five axioms stipulate that natural numbers are a one-way ordered linked list starting from 0. The addition and multiplication of natural numbers can be simply defined by recursive method, that is, for any natural number A, there are:
a+0=a
a+S(b)=S(a+b)
A 0=0
a S(b)=a+(a b)
Other operations can be defined by addition and multiplication. For example, subtraction is the inverse of addition and division is the inverse of multiplication. "a≤b" means that there is a natural number C that makes A+C = B. The basic properties of exchange law, binding rate and distribution rate can also be derived from the above definitions.
After Peano's axiom was put forward, most people thought it was enough to define the operation of natural numbers, but Poincare and others began to question the compatibility of Peano's arithmetic system: Is it possible to draw such an absurd conclusion that 0= 1 from these definitions through a series of strict mathematical deduction? If two contradictory propositions can be deduced from a series of axioms, we say that this axiom system is incompatible. The second of Hilbert's 23 questions is whether Peano's arithmetic system is compatible. This issue is still controversial.
In the history of mathematics development, introducing the concept of negative number is a major breakthrough. We want to be a
(a-b)+(c-d)=(a+c)-(b+d)
(a-b) (c-d)=(ac+bd)-(ad+bc)
We can naturally extend the above rules to
Another problem encountered in life is "not enough points" and "not enough points". Three people share six cakes, and one person has two cakes; But what if three people share five cakes? At this point, a number that exists between two adjacent integers must be generated. In order to better express this problem, we use a symbol a/b to indicate that consumers in unit B share the goods in unit A equally. One step that really plays a decisive role in the development of mathematics is to regard the symbol a/b composed of two numbers as a number and define a set of operation rules that it abides by. With the help of the life experience of "dividing cakes", we can see that for integers A, B and C, there are (ac)/(bc)=a/b, and (A/B)+(C/D) = (AD+BC)/(BD), (A/B) (C/. This definition is necessary for new numbers used to measure length, volume and mass. But in the history of mathematics, it took mathematicians a long time to realize that logically, the operation rules of new symbols are just our definitions, which cannot be "proved", and there is no reason for us to do so. Just as we define the factorial of 0 as 1, this is only to make the permutation number A(n, n) still meaningful and conform to the original algorithm, but we can never "prove" 0! = 1 Come on. In fact, we can completely define (a/b)+(c/d)=(a+c)/(b+d), and still meet the basic arithmetic laws. Although in our view, the result of this definition is ridiculous, there is no provision to force us not to define it like this. As long as it does not conflict with the original axiom and definition, this definition is allowed. It is just a new arithmetic system that is not suitable for measuring most physical quantities in the world and is not recognized and used by us.
We call all numbers with the shape of a/b rational numbers rational numbers. The appearance of rational numbers makes the whole number system more complete, and the four operations are "closed" within the scope of rational numbers, that is to say, the results of addition, subtraction, multiplication and division between rational numbers are still rational numbers and can be carried out without restrictions. From this point of view, it seems unlikely that we can get another number "beyond rational number".
When our number system is extended to rational numbers, the whole number system has undergone essential changes, which makes us believe that the expansion of the number system has come to an end. We say that rational numbers are "dense" on the number axis, and there are other rational numbers (such as their arithmetic mean) between any two rational numbers. In fact, no matter how small the interval on the number axis is, we can always find a rational number (when the denominator m is large enough, there is always a moment when 1/m is less than the interval length, and at least a rational number with the denominator m will appear in this interval). This makes people take it for granted that rational numbers have completely covered the whole number axis, and all numbers can be expressed in the form of A/B.
Incredibly, such numbers can't cover the whole number axis; Besides the a/b-shaped numbers on the number axis, there are other numbers! This is one of the most important discoveries of early Greek mathematics. At that time, the ancient Greeks proved that there was no a/b, so its square was exactly equal to 2. The number that equals 2 after squaring is not without (this number can be found by dichotomy), but it cannot be expressed by the ratio of two integers. In the present words, the root number 2 is not a rational number. The root number 2 is not an imaginary number without practical significance. Geometrically, it is equal to the diagonal length of the unit square. Our existing figures cannot express such a simple physical quantity as the diagonal length of a unit square! Therefore, it is necessary for us to expand our number system again to include all possible quantities. We call all numbers that can be written as integers or the ratio of integers "rational numbers", while other numbers on the number axis are called "irrational numbers". Together, it is a "real number", representing every point on the number axis.
In fact, constructing an irrational number is far less complicated. We can easily construct an irrational number to explain the existence of irrational numbers. 0.12345678911213141465438 ... is obviously an irrational number. Considering the process of expanding rational number into decimal form by trial division, because the value of remainder is limited in many cases, the remainder divided at a certain moment will inevitably be repeated with the previous one, so the result must be a cyclic decimal; Chamberowne constant is obviously not a cyclic decimal (no matter what you claim its cyclic segment is, I can construct a long enough number string so that a number in your cyclic segment does not appear in the string at all, obviously this string will appear in Chamberowne constant indefinitely). This example shows that there are still a large number of irrational numbers on the number axis, and the numbers with root signs only account for a negligible part of irrational numbers. This example also tells us that not all irrational numbers can be used to test people's memory and geek level like pi.
In the algorithm of defining irrational numbers, we once again encounter the problem we faced when introducing natural numbers at the beginning of this paper: what is irrational number? How to define the operation of irrational numbers? This problem has puzzled mathematicians for a long time. /kloc-In the middle of the 9th century, the German mathematician RichardDedekind proposed Dedekind segmentation, which skillfully defined the operation of irrational numbers and further improved the theory of real numbers.
Up to now, we have defined a new number with ordered number pairs, and defined the equivalence relationship and algorithm between ordered number pairs. However, the existence of Champernowne constant, a dumb irrational number, makes the hope that this method can continue to be used to define irrational numbers rather slim. Dedekind does not use an array of two or more rational numbers to define irrational numbers, but uses division of all rational numbers. We divide all rational numbers into two sets A and B, so that each element in A is smaller than all elements in B ... Obviously, there are only three rational number divisions that satisfy this condition:
1. 1.A has the largest element ax, for example, A is defined as all rational numbers less than or equal to 1, and B is all rational numbers greater than 1.
There is a minimum element bx in 2.2.B For example, a is defined as all rational numbers less than 1, and b is all rational numbers greater than or equal to 1.
3.3. There is no maximum element in A, and there is no minimum element in B. For example, A consists of 0, all negative rational numbers and all positive rational numbers with squares less than 2, and B consists of all positive rational numbers with squares greater than 2. Every time this happens, we say that this division describes an irrational number.
4.4. Note that it is impossible to "A has the largest element ax and B has the smallest element bx", which violates the density of rational numbers. Ax and bx are rational numbers, there must be other rational numbers between them, and these rational numbers do not belong to set a or set b, so they are not a division.
Why does every situation 3 describe a definite irrational number? Actually, it's very vivid. Because there is no largest element in a, we can take out more and more numbers from a endlessly; Similarly, we can get smaller and smaller numbers from B, and the numbers on both sides will get closer and closer, and the interval between them will get smaller and smaller. Its limit is a point on the number axis, which is greater than all the numbers in A and less than all the numbers in B. But the sets A and B already contain all rational numbers, so this limit must be irrational. So in essence, the essence of Dedekind segmentation is to approximate an irrational number with a series of rational numbers.
Now we can naturally define the operation of irrational numbers. We write the Dedekind corresponding to an irrational number as (a, b), then the result of adding two irrational numbers (a, b) and (c, d) is (p, q), where the elements in set P are obtained by adding each element in a and each element in c, and the remaining rational numbers belong to set Q. We can also define the multiplication of irrational numbers in a similar way. In addition, we can quickly verify that after the introduction of irrational numbers, our operations still meet the basic laws such as commutation law and combination rate, so I won't say much here.
Reading Notes on Junior High School Mathematics Monographs 2 I recently read Mathematical Thinking and Elementary School Mathematics, and I was deeply touched. The book says: Only by revealing the thinking method hidden behind the content of mathematics knowledge can we really "live", "understand" and "deepen" the mathematics class. This means that teachers should show students "living" mathematics research work through their own teaching activities, not dead mathematics knowledge; Teachers should also help students really understand the relevant teaching content, instead of swallowing dates raw and memorizing them; In teaching, teachers should not only let students master specific mathematical knowledge, but also help students deeply understand and gradually master the internal thinking methods.
Primary school students learn mathematics, which means that in the process of mastering basic knowledge, they constantly form mathematical ability and mathematical literacy, and obtain methods to think and look at problems from multiple angles, so as to think and solve problems mathematically. Mastering basic knowledge is the way, and the acquisition of multi-angle thinking mode is the ultimate goal. French educator Dostoevsky said: "A bad teacher gives up the truth, and a good teacher teaches people to discover the truth." Students' learning mathematics is an activity, an experience and a process, which cannot be said, but only participation and experience. Therefore, teachers should change the learning style centered on book knowledge and teaching, teach students the initiative of learning, and make students get the real feeling of knowledge in the operation experience, which is the driving force for students to form a correct understanding and turn it into ability. As the striking motto on the wall of the Washington Children's Museum says, "What you do, you will lose your muscles."
On weekdays, in the face of teachers' questions, if they are simple questions, many students respond. Once there is a thoughtful and profound question, only a few students tentatively raise their hands. Most students choose silence. What's more, sometimes the classroom is quiet, really, students dare not even go out of the atmosphere ... at this time, my heart began to tremble. How does a child with a full face of excitement come to the classroom to ask questions during recess? Where does the student's thinking come from? The answer is the teacher's enlightenment and training. As teachers, we often focus on making students master ready-made things and memorize them by rote. Over time, students never have to think, and gradually develop to be unable to think, and finally they are unwilling to think when they encounter problems. This will happen.
Our teacher should do two things in class:
First, teach students a certain range of knowledge.
Second, make students smarter and smarter.
And many of our teachers often ignore the second point, thinking that students are born smart when they master knowledge, but in fact, a curious, studious and diligent student is really smart. Then this kind of cleverness lies in the teacher's enlightenment and cultivation. Now the classroom attaches importance to group cooperative learning and students' hands-on operation ability. In fact, these practices are all about cultivating students' thinking ability.
Mathematics teaching is the teaching of mathematics activities and the process of interactive development between teachers and students. Teachers are the organizers, guides and participants of students' mathematical activities, and the enlighteners of students' mathematical wisdom. In the eyes of wise teachers, we should not only pay attention to whether students have mastered a certain knowledge, but also pay attention to the significance of the whole teaching process to students' growth and its influence on students' life. Be a wise teacher, focus on the future, enlighten students' thinking, cultivate students' mathematical wisdom, let students learn to learn and promote lifelong development.
;