How to do well in elementary school calculation class in the context of the new curriculum

1. What are the problems in the teaching of calculation? What are the main problems?

There are four main problems in the current teaching of computing: the contradiction between creating a situation and reviewing the pad, the contradiction between the intuition of arithmetic and the abstraction of algorithms, the contradiction between the variety of algorithms and the optimization of algorithms, and the contradiction between the formation of skills and the solution of problems.

First, we will talk about the general aspects, and then we will talk about them in detail later. These four problems are more of a new problem after the curriculum reform

2, the original calculation of the teaching of the introduction of the way to review the pad, now more popular to create a situation, how to deal with the relationship between the pad and the situation, so that the boring calculation of the same can trigger the interest of the students?

The constructivist learning theory is that learning is always linked to a certain socio-cultural context, which is the "context", and learning in the actual context is conducive to the construction of meaning. Indeed, a good problem situation can effectively activate students' relevant experience, experience. The Compulsory Education Mathematics Curriculum Standards (Experimental Draft) also emphasizes that the teaching of arithmetic "should further cultivate the sense of number through solving practical problems and enhance students' understanding of the meaning of arithmetic," "should enable students to experience the process of abstracting quantitative relationships from practical problems and solving problems by applying what they have learned". ""Avoid separating arithmetic from its application"". However, nothing is absolute. Because the sources of mathematics, one is from the development needs of the real society outside mathematics; the other is from the internal contradictions of mathematics, that is, the development needs of mathematics itself. Both sources of mathematics may become the background of our teaching. For example, the teaching of "negative numbers" is seldom taught in elementary school in traditional textbooks, but now the curriculum standard stipulates that negative numbers should be introduced in elementary school. In real life, there are a large number of quantities with opposite meanings, which can be used as materials for revealing negative numbers; at the same time, from the perspective of mathematics itself, in order to solve the contradiction such as "2-3" is not enough to subtract, it is necessary to introduce a new kind of number, which is also a problematic situation that is easy for primary school students to perceive. Here, the choice of one of the two angles of introduction is desirable.

Nowadays, the teaching of calculation almost disappears in the traditional teaching of review padding, replaced by - the creation of the situation. At present, most of the general teaching process of computational teaching is: the teacher creates a scenario, students ask questions, independent thinking algorithms, feedback and exchange of algorithms, independent choice of algorithms. For this reason, many computation lessons either start with "shopping" or end with "shopping mall". Now the teaching of computing, it is difficult to see the common review pad in the past.

The other side of the question is, do we need to "review the pad" before teaching computation? In fact, the main purpose of the new lesson before the review pad, first, in order to activate the students' minds through the reproduction or re-recognition of the relevant old knowledge, and secondly, for the new knowledge of learning to disperse the difficult points. The former, as long as necessary, there is no excuse. The problem is the latter, there are some calculation teaching, often some teachers in order to make the teaching "smooth", designed some transitional, suggestive questions, and even artificially set up a narrow thinking channel, so that students do not need to explore or a little try, the conclusion came out.

Summary of the problem--

See, create a situation and review the pad is not a contradiction in terms, not all the teaching of computing must be from the life of the "prototype", the choice of how to introduce depends on the computing

3, how to deal with the relationship between algorithmic diversity and algorithmic optimization?

The Compulsory Education Mathematics Curriculum Standards (Experimental Draft) states in the "Basic Concept" that "due to the differences in the cultural environment, family backgrounds, and thinking styles of students, students' mathematical learning activities should be a lively, active, and individualized process. " In the "Content Standards" of the first section, it is said, "Oral calculation should be emphasized, estimation should be strengthened, and diversification of algorithms should be promoted." In the first paragraph of the "teaching suggestions" again pointed out that: "due to the students' different life background and thinking perspective, the methods used are inevitably varied, the teacher should respect the students' ideas, encourage students to think independently, and promote the diversity of calculation methods."

"Diversification of algorithms" was a buzzword at the beginning of the new curriculum reform.

At the beginning of the implementation of the math curriculum reform, we feel very new to "algorithmic diversification", and the teaching of computation has changed from the past "teaching materials selected algorithms, teachers explaining algorithms, students imitating algorithms, practicing algorithms to strengthen algorithms" mechanical mode, and there has been a very welcome change, "algorithmic diversification". The "diversification of algorithms" has become the most obvious feature of computation teaching.

〖Case〗"Two-digit minus one-digit subtraction" teaching fragment:

First of all, the teacher through the problem situation to show the example of 23-8.

Then, after the teacher's careful "guidance", there is a diversity of the teacher to explain the algorithm, students to imitate the algorithm, practice to strengthen the algorithm. ", a diversity of algorithms appeared, and the teacher spent nearly a class period to demonstrate (also demonstrated separately with animated courseware):

(1) 23-1-1-1-1 -1-1-1-1-1 = 15

(2) 23-3 = 20, 20-5 = 15

(3) 23 -10 = 13, 13 + 2 = 15

(4) 13 - 8 = 5, 10 + 5 = 15

(5) 10 - 8 = 2, 13 + 2 = 15

(6) 23 - 13 = 10. 10 + 5 = 15

(7) 23-5 = 18, 18-3 = 15

......

Finally, the teacher said "Use whatever algorithm you like to use in what kind of algorithm you like." （

After the class, I talked with the teacher, who said, "Nowadays, we must diversify the algorithms for teaching computation, and the more algorithms we have, the more we can embody the spirit of curriculum reform." I also asked the students who came up with the first algorithm in the classroom, "Are you really doing it this way?" The student said, "I don't want to use this stupid method! The teacher told me to say that before class." I asked several students in a row, but none of them used this method of subtracting 1 one by one. Then the next several algorithms (especially the sixth and seventh) really students come up with their own?

The above case reflects that a few teachers have a vague understanding of the basic contradiction between algorithmic diversity and algorithmic optimization in computation teaching. Algorithmic diversity should be an attitude, a process, algorithmic diversity is not the ultimate goal of teaching, can not unilaterally pursue the formalization. Teachers do not have to take great pains to "demand" diversified algorithms, nor do they have to deliberately guide students to seek "low-level thinking algorithms" in order to reflect diversification. Even if sometimes the algorithm of the textbook, but in the actual teaching of the students did not appear, that is, the students have gone beyond the "low level of thinking algorithm", the teacher can no longer show, there is no need to go back.

4, how to develop students' sense of number in computational mathematics?

Number sense is a good intuition of the relationship between numbers and counting. To develop students' number sense in computational teaching is to be able to grasp the relationship between the relative sizes of numbers in concrete situations; to express and communicate information in terms of equations and results of computations; to choose appropriate algorithms for solving problems; and to be able to estimate the results of computations and to explain the reasonableness of the results.

On the development of number sense in the teaching of calculation. I would like to say this much for now, this issue is rather abstract when unfolded.

5. What are the psychological factors affecting students' calculation? What countermeasures should be taken?

This question, I did a special survey and analysis 10 years ago.

The psychological factors affecting students' computation are: rough perception, attention disorders, memory reduction, fuzzy representation, emotional vulnerability, strong information interference, stereotyped side effects of thinking and so on.

Taking oral arithmetic as an example-

To perform oral arithmetic, first of all, the arithmetic formula consisting of data and symbols must be perceived through the students' sense organs. Elementary school students perceive things characterized by more general, rough, not specific, often only notice some isolated phenomena, can not see the connection and characteristics of things, and thus the impression left in the mind lacks wholeness. The oral math problem itself has no plot, monotonous external form, not easy to arouse interest. Therefore, students oral math, often only perceive the data, symbols of itself and less consideration of its meaning, similar, similar data or symbols are prone to perceptual distortion, resulting in errors. For example, some students often regard "+" as "×", "÷" as "+", and write

Note the dissonance.

Attention is the direction and concentration of mental activity on a certain object. The instability of attention and poor distribution ability is an important psychological factor in the production of oral math errors. Elementary school students' attention is not stable, not lasting, not easy to distribute, the scope of attention is not wide, easy to be attracted by irrelevant factors and the phenomenon of "distraction". In the process of oral arithmetic, it is necessary to pay attention to or allocate attention to different objects at the same time. Due to the attention of elementary school students to take into account the surface is not wide, requiring them to allocate their attention to two or more objects at the same time, they often lose sight of one or the other, lost three or four. For example, separate oral 6 × 8 and 48 + 7 oral math problems, most students can count accurately, while the two questions together, counting 6 × 8 + 7, students often get 45, forget to round and cause errors.

Memory reduction.

The purpose of memorization is not only the storage of information, but more importantly, the ability to accurately extract. Students store information in the process, due to physiological, time, review the amount of factors such as the impact of the stored information disappeared or temporarily interrupted, thus losing the beginning and forgetting the end, resulting in "forgetfulness error". In particular, the addition, subtraction, addition, subtraction, multiplication, division and other oral problems, instantaneous memory is large, such as oral 28 × 3, students are required to temporarily memorize the results of each step of the oral calculations, that is, 20 × 3 = 60, 8 × 3 = 24, and in the mind of the oral calculations of the 60 + 24 = 84. This type of oral calculation problems, the reason for the error, mainly due to the middle of the storage and extraction of the number of incomplete or forgetfulness.

Representation fuzzy--

Representation is the bridge of transition from perception to thought. From the form of arithmetic, the oral calculation of elementary school students from intuitive perception transition to representational arithmetic, and then to abstract arithmetic. From the thinking characteristics of elementary school students, their thinking with a great deal of concrete image, representations often become the virtue of their thinking. Especially in the lower grades, often due to the oral calculation method of the representation is not clear and produce errors. For example, some first-grade students oral 7 + 6, 8 + 5 and other progressive addition, the mind of the "decomposition" → "make up ten" → "merger" of the fuzzy appearance, can not imagine the The "make up ten method" process is vague in your mind and you can't visualize the specific process, so you make mistakes.

Emotional vulnerability--

When doing oral math, students want to get results quickly. Some students in doing oral math problems, due to the existence of the psychological rush, when the number of small, simple formula, easy to give birth to the "enemy" thought; and when the number of large, complex calculations, but also showed impatience, boredom. Oral math, some students often can not see the full details of the problem, carefully and patiently analyze, but also can not correctly and reasonably select the method of oral math, and then develop the topic is not clear to see the pen in a hurry, do not check the finished bad habits.

Strong information interference -- the visual and auditory perception of primary school students is selective, and the strength of the information received affects their thinking. Strengthened information in the student's mind to leave a deep impression, such as the number want to subtract 0, 0 and 1 in the calculation of the properties of 25 × 4 = 100, 125 × 8 = 1000 and so on. This kind of strong information comes to the eye first and tends to overshadow other information. Such as the oral calculation 15-15 ÷ 3, students do not not know "first multiply and divide, then add and subtract" order, but by the "same number of subtraction equals to 0" this strong information interference, some students first thought of 15-15 = 0, and ignored 15

The negative effect of stereotyped thinking--

Stereotyping is a kind of "inertia" of thinking, which is the result of a certain mental activity. This state of readiness can determine a certain tendency of similar subsequent activities.? It's a good idea to get a good deal on a new product. Tokyo Park The "bag of tricks" is the "bag of tricks" of the "bag of tricks". Piku freezing trekking heat scapula? Aobao? What is your breakfast? What's your name? What's your name? source halt toad thin H indistinct visit?40 ÷ 60, 450 ÷ 90, 360 ÷ 40 and other questions after the clip a 300-50, many students are often miscalculated as 300-50 = 6.

About the psychological factors interfering with the calculation, so much.

6, please talk about how to solve the arithmetic intuitive and abstract algorithm of the contradiction

There have been some teachers believe that the teaching of computation, there is no reasoning, as long as the students master the calculation method, repeated "rehearsal", you can achieve the correctness, proficiency requirements. As a result, many students are able to calculate according to the law of calculation, but because the calculation is not clear, the scope of knowledge transfer is extremely limited, and can not adapt to the calculation of the ever-changing variety of specific situations.

The algorithm is the theoretical basis of the four calculations, which is composed of mathematical concepts, properties, laws, and other components of the basic theoretical knowledge of mathematics. Algorithm is the implementation of the four rules of calculation of the basic procedures and methods, usually arithmetic guidance under the guidance of some of the perceived provisions. Arithmetic provides theoretical guidance for algorithms, and algorithms make algorithms concrete. Students in the process of learning to calculate clear arithmetic and algorithms, it is easy to flexible, easy to calculate, the diversity of calculations have the basis and possibility. It is inconceivable that a student who is not even clear about the principles and methods of basic calculations can not carry out calculations flexibly and easily. How can they have the ability to calculate diversity? Therefore, it is a very important topic to pay attention to arithmetic and algorithms in the teaching of calculation.

In the teaching we often see such a phenomenon: in the teaching aids demonstration, learning aids operation, picture control and other intuitive stimulation, the students through the combination of mathematical and physical way, the understanding of the arithmetic can be said to be very clear, but the good times do not last long, when the students are still lingering in the intuitive image of arithmetic, and then immediately faced with the abstract algorithms, and then the next calculations are the direct use of the abstract algorithm of simplification of the calculation.

So I think, in the arithmetic intuitive and abstract algorithms should be built between a bridge, paved a road, so that students in the full experience of the gradual completion of the action thinking image thinking abstract thinking development process.

In short, the teaching of computing needs to let students understand the theory of arithmetic in the intuition, but also needs to let students master the abstract law, but also needs to let students fully experience the transition and evolution of the abstract algorithm from the intuition of arithmetic to the transition and evolution of the process, so as to achieve a deeper understanding of the theory of arithmetic and the algorithm of the practical grasp.

7, the textbook of the curriculum reform clearly puts forward the "strengthening of estimation", how do you cultivate students' sense of estimation and estimation ability?

To reflect the requirements of the "standard" "strengthen estimation", we can focus on the following two aspects:

(1) Cultivate a sense of number is to lay a good foundation for estimation. Number sense is a good intuition of the relationship between numbers and counting. In estimation, number sense is mainly manifested in the ability to grasp the relationship between the relative sizes of numbers in specific situations, to choose appropriate algorithms for solving problems, and to explain the reasonableness of the results. Estimation can develop students' understanding of numbers and is important for the cultivation of number sense, and at the same time, a good sense of number is the necessary foundation for students to carry out estimation. In addition to strengthening the development of number sense in the number of knowledge, in the number of arithmetic process should be combined with specific calculations to develop students' number sense.

(2) In addition, we need to cultivate the habit of estimation. We often find that some students make some inexplicable mistakes in their calculations. In this regard, we should allow students to develop the habit of timely estimation and checking, every finished a topic, you can first estimate the value, and then compared with the actual calculation of the answer, timely detection of errors and corrections.

8, estimate 19 + 18, many students directly calculate 37, then the teacher how to do? How to deal with the relationship between estimation and accurate calculation in teaching?

Estimation is the ability to approximate or roughly estimate the process and result of an operation. Currently, much emphasis is placed on estimation in international mathematics education. With the rapid development of science and technology, there are a large number of facts that are impossible and do not require precise calculations. Numerous examples illustrate this - the number of times a person estimates and differentiates the product quotient in the course of a day's activities is far greater than the number of times he or she performs precise calculations.

And the ability to do exact calculations (both oral and written) is a necessary computational skill for students to develop in their teaching.

Estimation is mainly used in daily life when it is not possible or necessary to calculate precise results; actuarial calculations are based on the need to calculate the results accurately. Both have their own requirements in teaching and learning, and in elementary school it is mainly to cultivate the ability of students to calculate accurately, while allowing students to experience the need for estimation in specific situations.

9, now the textbook in the calculation of teaching are not appear in the calculation of the law, this, the teacher how to deal with?

The laws of mathematics reflect the relationship between several mathematical concepts. The law of calculation is a written expression of the rules of arithmetic, it is in the arithmetic guide to the implementation of the rules of the arithmetic process to make specific provisions, reflecting a standardized operating procedures.

One of the trends of the new curriculum reform is to downplay the form and focus on the essence. Therefore, the current teaching of computation has diluted the programmed narration of the arithmetic and computational laws, and strengthened the students' understanding of the arithmetic and mastery of the algorithm, and strengthened the students' experience in the computational process and active exploration.

For the calculation of the law that does not appear in the textbook, as long as the students understand the theory and master the algorithm on the line.

As for the narrative and generalization of the law of calculation, do not be too demanding, especially in the lower grades.

8, estimate 19 + 18, many students directly calculate 37, what should teachers do? How to deal with the relationship between estimation and accurate calculation in teaching?

Estimation is the ability to approximate or roughly estimate the process and result of an operation. Currently, much emphasis is placed on estimation in international mathematics education. With the rapid development of science and technology, there are a large number of facts that are impossible and do not require precise calculations. Numerous examples illustrate this - the number of times a person estimates and differentiates the product quotient in the course of a day's activities is far greater than the number of times he or she performs precise calculations.

And the ability to do exact calculations (both oral and written) is a necessary computational skill for students to develop in their teaching.

Estimation is mainly used in daily life when it is not possible or necessary to calculate the exact result; actuarial calculations are based on the need to calculate the exact result of the calculation. Both have their own requirements in teaching and learning, and in elementary school it is mainly to develop students' ability to calculate accurately, while allowing students to experience the need for estimation in specific situations.

9, now the textbook in the calculation of teaching are not appear in the calculation of the law, this, the teacher how to deal with?

Mathematical laws reflect the relationship between several mathematical concepts. The law of computation is a written expression of the rules of operation, it is under the guidance of the theory of arithmetic on the implementation of the rules of the operation process to make specific regulations, reflecting a standardized operating procedures.

One of the trends of the new curriculum reform is to downplay the form and focus on the essence. Therefore, the teaching of computation now dilutes the programmed narration of arithmetic and computational law, and strengthens the students' understanding of arithmetic and mastery of algorithms, and strengthens the students' experience of the computational process and active exploration.

For the calculation rules that do not appear in the textbook, as long as the students understand the theory of calculation and master the algorithm, it is enough.

As for recounting and summarizing the rules of calculation, don't be too demanding, especially in the lower grades.

10, calculation class, how to effectively improve the speed and accuracy of students' calculation, improve students' thinking ability?

Regarding the speed and accuracy of calculation, they are two important dimensions to measure the formation of students' calculation ability. The general trend of reform in the teaching of computing is to reduce the requirements for the rapidity of computation.

The author thought that for some basic oral arithmetic to allow students to achieve fast and correct to be. That is, in the primary stage of the oral content, two one-digit addition and its corresponding subtraction and table multiplication and its corresponding division is the four operations in the basic oral, commonly known as the "four ninety-nine table", which "four table" is the basis of all calculations, make sure that the student The "four tables" are the basis of all calculations, and it is important to make sure that the students achieve the degree of proficiency of the "off the top of their heads".

And for the written calculation, do not have to put forward too high speed requirements, the important thing is to let the students calculate correctly, and gradually improve the speed.

11: In the calculator into the classroom, students can usually use? How can we solve the conflict between modern teaching tools and written calculations? Introduce your experience to the group.

According to the Compulsory Education Mathematics Curriculum Standards (Experimental Draft), it is stated in the second semester that students should be able to perform complex operations with the help of a calculator, solve simple real-world problems, and explore simple mathematical rules. Therefore, some editions of textbooks introduce the teaching of calculators from Grade 4 onwards to help students perform calculations and explore patterns. Students can certainly use them in normal times as long as they are necessary. But it is also important to guide students to use the calculator reasonably and not to rely on it completely.

(1) Handle the relationship between written calculations and calculator operations. For primary school students, mastering some simple written calculations is the basic requirement for learning math, so it is necessary to play a good basic skills. And for some of the more complicated arithmetic, it can be replaced by a calculator.

(2) Cultivate the habit of exploring the laws of mathematics by using the calculator. In some textbooks, some topics are arranged for students to use calculators to explore the laws, so that students can use calculators to carry out activities such as calculations, observations, guessing and verifications, which have a great role in promoting the development of students' exploratory learning.

About the introduction of calculators into teaching, since I have not yet taught the fourth grade of the experimental textbook of the curriculum standard, I have not yet accumulated much experience in this area.

12. Do students have to practice more on calculations that are more difficult to master, such as those related to pi?

On the one hand, for students more difficult to grasp the knowledge of calculation, to strengthen the targeted practice, such as the calculation of pi can let the students through the calculation to remember some of the multiples of 3.14 6.28, 9.42, 12.56, 15.7, 18.84, etc.; on the other hand, for the calculation of the complexity of the content, to alleviate the students of the burden of complicated calculations, such as calculations related to the pi A calculator can be used to help calculate.

13, not long ago, you asked students in Beijing class vertical calculation, the whole ten separate line, such as 34 × 3, 11 × 5 of the vertical calculation process, respectively, as Figure 1, Figure 2. so that a better understanding of the arithmetic is certainly, but not so written can not be a good understanding of the arithmetic? I feel that you are complicating a simple problem, so I would like to hear your analysis of this design.

3 4 1 1

× 3 × 5

1 2 5

9 0 5 0

1 0 2 5 5

About this problem, please see a short article written by the author - "looks clumsy but is actually artisanal"

Teaching fragment (third grade "one digit by two digits "

Teacher: Students, looking at this picture, what math information do you know?

Student 1: There are two monkeys picking peaches.

Student 2: One monkey picked 14 peaches, and the other monkey picked 14 peaches.

Student 3: All 14 peaches were 10 in one basket and 4 in another.

Teacher: How many peaches did the two monkeys pick in one ****? How do you answer this question?

Student 1: 14 + 14.

Student 2: 14 × 2.

Student 3: 2 × 14.

Teacher: How did you calculate this question? You can discuss it among your table.

(Students talk to each other for discussion)

Teacher: Who will say how you came up with the result?

Student 1: I used 14 + 14 and got 28.

Student 2: I looked at the picture, a **** in the right basket is 8, a **** in the left basket is 20, and the total is 28.

Student 3: I thought of it by multiplication. 10 times 2 equals 20, 4 times 2 equals 8, 20 plus 8 equals 28.

Student 4: I thought of it differently. 14 is 2 sevens, multiplied by 2 it is 4 sevens, 4 sevens and 28.

Teacher: Oh, that's a good idea! (The whole class applauded for student 4)

Teacher (pointing to the screen): Just now a student said 4 times 2 equals 8, which part of the equation does he actually mean?

Student: It's the 8 peaches in the two baskets on the right side of the picture.

Teacher: So what is counting the peaches in the two baskets on the left?

Student: 10 times 2 equals 20.

Teacher: We just counted the ones in the first place and then the ones in the tenth place, what should we do next?

Student: add up.

Teacher: Yes, you have to add all the peaches in the right basket and the left basket to find out how many peaches there are in a ****.

(Teacher's step-by-step board is as follows:)

1 4

× 2

8......4×2 = 8

2 0......10×2 = 20

2 8 ......8 + 20 = 28

Teacher: An algorithm like this is what we call -

Students (in unison): using vertical form.