How to effectively evaluate primary school mathematics?

Instructional design (ID for short), also known as instructional system design, is a special design activity aimed at teaching system and solving teaching problems. It is a systematic process to analyze the problems and needs in teaching, design solutions, try out solutions, evaluate the trial results, and improve the design on the basis of evaluation by using modern learning and teaching psychology, communication, teaching media theory and other related theories and technologies. Teaching design is not only a science, but also an art. As a science, it must follow certain educational and teaching laws. As an art, it needs to integrate many personal experiences of designers, re-create according to the characteristics of teaching materials and students, and flexibly and skillfully use the methods and strategies of teaching design. Then, how to design primary school mathematics teaching, so that it not only has the general nature of design, but also follows the basic laws of teaching, thus fully embodying the educational wisdom of teaching designers?

R Mager, a famous American expert on instructional design, pointed out that instructional design consists of three basic problems in turn. The first is "where am I going", that is, the formulation of teaching objectives; Then there is "how do I get there", including the analysis of learners' initial state, the analysis and organization of teaching content, the choice of teaching methods and teaching media; Finally, there is "how do I judge where I am", which is the evaluation of teaching. Teaching design is an organic whole composed of goal design, analysis and design of various elements to achieve goals, and evaluation of teaching effect. Therefore, in order to carry out effective primary school mathematics teaching design, we must focus on the above three basic problems.

First, determine the appropriate teaching objectives

Teaching objectives are not only the starting point of teaching activities, but also the preset possible results. The goal of primary school mathematics teaching includes not only the requirements of knowledge and skills, but also the requirements of mathematical thinking, problem solving and students' feelings and attitudes towards mathematics. Different understanding of the goal will lead to different teaching designs, thus forming different levels of classroom teaching. For example, the same course "Orientation" has formed two different levels of teaching design because two teachers have set different teaching objectives.

A teacher's teaching goal of "determining position" is: "master the method of determining position with' number pairs' and be able to determine the position of objects with' number pairs' on grid paper." Based on this goal, the teacher gave each student a card with the first column and the first line written on it, and asked the students to stand in front with the card and then find the corresponding position according to the requirements on the card. Under the guidance of the teacher, the students reported how to find a position and finally achieved the teaching goal. Judging from the goal determination and teaching process design of this class, the cognitive teaching goal is the main body. Although the teaching design is simple, taking into account students' original knowledge base and life experience, it causes students' single cognitive development, but lacks good emotional experience and opportunities to solve practical problems with knowledge.

Another teacher's teaching goal of "determining the position" is: "Make students explore the method of determining the position in specific situations and tell the position of an object; Ask students to use "number pairs" to determine the position of objects on square paper; Let students feel the close connection between mathematics and life in specific situations, discover and solve mathematical problems independently, gain successful experience from them, and establish confidence in learning mathematics. " Under the guidance of this goal, the teacher first asked the students to try to describe a classmate's position in the class with the simplest mathematical method, and then classified and compared the different representations of this classmate, on this basis, the similarities and differences of different representations were obtained-all of them used "the third group and the second group" to describe the classmate's position in the class. At this time, the teacher pointed out that in fact, the position of this classmate can also be expressed by (3,2), which is mathematically called "number pair". After the teachers and students studied the reading and writing method of "number pair", the teacher designed a game activity-the teacher pointed to a student and asked the student to tell his position with "number pair", and other students judged right and wrong; The teacher said, "Count right". Please sit in the corresponding position and stand up. The other students use gestures to judge right and wrong. Finally, the teacher also designed an interesting egg-beating game, in which "number pairs" representing each student's position were input into the computer, and the students stopped at random. Lucky students go to the front, correctly use the "number pair" to tell the position of the golden egg or silver egg they want to break on the grid paper, and then they can break the egg. After clicking, a word of blessing will appear on the computer. Through this teaching design, students not only feel the simplicity and uniqueness of using "number pairs" to determine the position of objects, but also realize that mathematics is closely related to life. In this process, students not only master knowledge, but also enjoy success and experience happiness.

Through the comparison of the above two teaching designs, we really feel that to determine the appropriate teaching objectives, we must correctly handle the relationship between curriculum standards, teaching materials and students' level, and pay attention to different levels of cognition, emotion and motor skills. Bloom takes explicit behavior of learners as the basic point of goal classification and complexity of behavior as the basis of goal classification, and puts forward six-level classification of educational goals in cognitive field-knowledge, understanding, application, analysis, synthesis and evaluation. 1964, Crasworth and others put forward the classification of affective teaching objectives, and divided them into five levels according to the internalization degree of values: acceptance, attention, response, value, organization of values, and personality of values or value system. Simpson divided motor skills into perception, orientation, guided response, mechanized action, complex explicit response, adaptation and creation. The classification of the goals of the three educators provides a basic basis for us to determine the teaching goals. When designing primary school mathematics teaching, we should consider these three target areas as a whole, and treat higher-level goals as the theme and fundamental purpose that affects the content. Only in this way can we determine the appropriate teaching objectives.

Second, a reasonable analysis and organization of teaching elements

(A) Analysis of students' situation

Students are the main body of learning. If we want to carry out targeted teaching design, we must analyze the learning situation, focusing on the initial ability of learners, the background knowledge and skills that have been formed and the way of thinking of learners.

1. Diagnosis of learner's initial ability

Gagne's classification of learning results and his thinking on learning conditions provide theoretical basis and basic ideas for the diagnosis of learners' initial ability. Gagne divided learning achievements into five categories: intellectual skills, cognitive strategies, verbal information, motor skills and attitudes. According to the complexity of intellectual skills's study, he divided it into several subcategories, namely, discrimination, concepts, rules and advanced rules (problem solving). Discrimination is the basis of concept learning, concept is the basis of rule learning, and applying a few simple rules is the basis of solving problems and obtaining advanced rules. For example, in the lesson of "area of triangle", students need to sum up the formula for calculating the area of triangle through experiments and use the formula to solve simple practical problems. This content belongs to the category of rule learning, and the premise of rule learning is to acquire the ability to use related concepts. Area of triangle = base × height ÷2. This formula includes seven concepts: triangle, area, equality, bottom, height, multiplication and division. If you don't master any of these seven concepts, there is no way to learn rules. At the same time, students must master the strategies of "cutting", "spelling" and "transforming", otherwise the formula for calculating the triangle area cannot be derived independently. Therefore, accurate diagnosis of learners' initial ability is the basic premise of effective teaching design.

2. Analysis of learners' background knowledge

When learning mathematics knowledge, students always have to get in touch with the background knowledge, understand the knowledge with relevant knowledge, including the knowledge gained in formal and informal learning, and reconstruct new knowledge. Primary school mathematics teachers' analysis of students' background knowledge includes not only the analysis of students' existing old knowledge that is beneficial to the acquisition of new knowledge, but also the analysis of background knowledge that is unfavorable to the acquisition of new knowledge.

According to the different background knowledge of students, a teacher made three different teaching designs for the course Prime Numbers and Composite Numbers.

Design 1: In the activity of "sending teachers to the countryside", according to the background knowledge of natural numbers, classification, odd numbers, even numbers, divisors and so on that the students in rural central schools have mastered, first, let the students classify the class numbers according to odd numbers and even numbers ── 1~ 16. Then ask the students to find all the divisors of each number from 2 to 16 and divide them into two categories according to their characteristics. On this basis, let students try to sum up the characteristics of these two numbers, and then under the constant questioning of the teacher, teachers and students can sum up what is a prime number and what is a composite number.

Design 2: In the inter-school exchange activities, according to the background knowledge that the students in the county experimental primary school have mastered, first, let the students classify the class size according to odd and even numbers-1~ 59. Then ask the students to find all the divisors of 1~59 and classify them according to their characteristics (they should be divided into three categories). On the basis of classification, let the students reveal the concepts of prime numbers and composite numbers through independent attempts to summarize, discuss and exchange, report and debate, and make it clear that 1 is neither prime nor composite numbers.

Design 3: At the "Provincial Excellent Teachers' Teaching Achievement Reporting Meeting", according to the fact that about one-third of the students in the class have learned the concepts of prime numbers and composite numbers through different channels (although they know the concepts, they have not really understood them), the teachers let the students read the textbooks and understand the concepts of prime numbers and composite numbers, so that all the students can really understand the connotation and extension of prime numbers and composite numbers under the argument between teachers and students.

Through the analysis of three different teaching designs of Prime Number and Composite Number, we realize that the correct analysis of learners' background knowledge is an important basis for effective teaching design.

3. How do learners think?

Ed Rabinowicz said in his book Thinking, Learning and Teaching: "As teachers, we educate our children. Since teaching children, we must understand how children think and how children learn ... Maybe we just think we know them. " Indeed, many times we think we know students, but we don't. Many primary school math teachers pay more attention to how to teach when designing teaching, but rarely consider how students learn and how students think. A teacher designed the lesson "cuboid and cube volume": First, review unit of volume, show the cube blocks corresponding to 1 cubic centimeter, 1 cubic decimeter and 1 cubic meter, and then ask students to estimate the approximate volume of a larger cuboid. Next, let the students put all kinds of cuboids and cubes of different sizes together and record the data. On this basis, let the students sum up the formula for calculating the cuboid volume independently. In actual teaching, students did not estimate the volume of this larger cuboid according to the designer's idea, but said that this rectangular body is about 30 cm, 25 cm, 50 cm, 20 cm, 30 cm, 40 cm wide and 40 cm, 50 cm high. In the process of recording data, the length, width, height and volume of the cuboid are also not recorded according to the designer's idea, but the number of small wooden blocks is directly recorded. The main reason for the difference between teaching design and actual teaching is that designers lack basic judgment on how students think. Therefore, when designing teaching, primary school mathematics teachers should not only diagnose learners' starting ability, analyze their background knowledge, but also pay attention to how students think. In addition, the analysis of students' learning attitude and interest is also very important to realize the teaching goal, and it is also an important content in teaching design.

(B) the organization of teaching content

Organizing teaching content is an important work of teaching design. The teaching content is to solve the problem of "what to teach and what to learn" according to the specific teaching objectives. Therefore, first of all, we should analyze the writing characteristics of the textbook and understand the editor's intention; Secondly, we should grasp the position and function of teaching content in the whole teaching system; Thirdly, it is necessary to analyze the key points and difficulties in teaching, and effectively highlight the key points and break through the difficulties through appropriate content. A teacher organized the teaching content of the lesson "Compare with One-Average" like this:

At the beginning of class, boys and girls are divided into three groups (5 boys in each group and 4 girls in each group) to have a glass ball competition, and the recorder of each group records the results of the competition. According to the total number of balls in each group, the male and female champion groups are judged. Then choose the final champion from the boys and girls champion group. Because the number of boys and girls in the champion group is not equal, it is unfair to determine the final winner according to the total number of balls pinched, which leads to the problem of finding the average. The teacher showed two sets of statistical charts of ball clamping. After teachers and students jointly explore the method of finding the average value and understand the meaning of the average value, let students solve three practical problems-finding the average temperature, finding the average height of five students and finding the average weekly water consumption of students.

The reason why the teaching content is organized in this way is because the teacher first carefully analyzed the teaching materials. In previous textbooks, students have mastered the methods of collecting and sorting out data, can express statistical results with statistical charts and tables, and can ask and solve problems according to statistical charts. The teaching content of this unit is based on students' existing knowledge and experience, using the information in the statistical chart to understand the meaning of the average and explore the method of finding the average. In order to let students know the characteristics of the average, the textbook discusses which group of students has strong overall strength according to the statistical chart, and leads to the concept of the average, so that students can realize the necessity of learning the average and understand its significance. In order to make the students really realize the necessity of learning the average, the teacher did not ask the students to compare the basketball throwing situation between the two groups, but organized the students to play glass ball games in groups on the spot to arouse their enthusiasm for participation. It is easy to solve the problem when determining the champions of men's and women's groups according to the total number of balls pinched, but whether the final champion can be determined according to the total number of balls pinched will cause students' thinking conflict, which will lead to the problem of finding the average. In order to let the students explore the method of average value independently, the teacher prepared a statistical table for the students on the number of catches of male and female champions. Let the students explore the method of finding the average value through observation. In order to better understand the meaning of the average and master the method of finding the average, the teacher finally arranged three simple practical problems for students to solve independently.

(C) the choice of teaching methods

Whether the teaching goal can be achieved depends largely on the choice of teaching methods. We should not only choose teaching methods according to teaching objectives, teaching contents, teachers' personal characteristics and students' age characteristics, but also mobilize students' learning enthusiasm to the maximum extent, so as to truly highlight students' dominant position. Let's take the lesson "Compare with each other-average" as an example. The teaching objectives of this lesson are determined as follows: 1. With rich examples and statistics as the background, make students understand the necessity of average, understand the meaning of average and master the method of average; 2. Cultivate students' ability to use what they have learned reasonably and flexibly to solve simple practical problems; 3. Understand the application of averages in real life, let students understand the close relationship between mathematics knowledge and daily life, infiltrate corresponding ideas, and improve students' interest in learning mathematics. In order to achieve the above teaching objectives, teachers should organize students to hold glass ball competitions when designing teaching. Because students take part in the competition themselves, they are very active, and their enthusiasm for participation is effectively stimulated through practical operation. By allowing students to decide the final champion group for boys and girls, students' thinking conflicts are aroused, and the intrinsic motivation of students' active learning is stimulated, so that students can truly feel that it is fair to decide the final champion by the average number of boys and girls in each group, thus understanding the necessity of seeking the average number. Next, by observing the statistical chart made by the teacher according to the results of the live competition, let the students think about how to determine the champion group when the number of participants is different. Teachers choose to let students explore independently, understand the meaning of "average" and master the method of seeking "average". In order to understand students' ability to use knowledge to solve simple practical problems, the teacher designed three practical problems for students to solve independently. In the process of solving problems, students not only learn to use knowledge, but also realize the practical value of mathematics, which stimulates students' enthusiasm for learning mathematics. By using this teaching method to carry out students' learning activities, students' dominant position is highlighted to the maximum extent, and students' subjectivity is fully exerted.

Third, the correct evaluation of teaching effect

Whether the teaching objectives put forward in the teaching design are achieved or not needs to evaluate the teaching effect. The main purpose of evaluation is to understand students' mathematics learning process, paying attention to both the results of students' learning and their learning process; We should not only pay attention to students' learning level, but also pay attention to students' emotions and attitudes in mathematics activities. A teacher made the following teaching effect evaluation design in the design of Statistics.

Question 1: What do you think of this class?

Please cooperate with the field survey to see how many students in this class are happy, happy and unhappy. Make the data obtained from the investigation into statistical tables and charts, and put forward corresponding mathematical questions and answer them according to the statistical tables and charts. In addition, please interview unhappy students to find out the reasons for their unhappiness and help them study and live happily.

This kind of question design can not only make all students experience the whole process of data collection and arrangement, but also try to make statistical charts according to the collected data, answer questions about mathematics according to the statistical charts, learn to read statistical charts, understand students' learning experience in this process, and provide basic basis for improving teaching.

This kind of question is very challenging and requires a certain degree of creativity when answering. Designing such questions when evaluating the teaching effect can not only examine students' understanding of statistical knowledge, but more importantly, examine whether students have statistical consciousness, creativity and imagination, as well as their understanding of practical problems.

The evaluation methods of teaching effect should be varied, including classroom application exercises, classroom observation, student interviews, homework analysis and so on. Through the comprehensive evaluation of teaching effect, we can understand the basic situation of students in knowledge and skills, mathematical thinking, problem solving, emotional attitude and so on, and provide a more scientific basis for further improving teaching design.

Teaching design is a systematic project, including the determination of teaching objectives, the analysis and organization of teaching elements, and the evaluation of teaching effect. The holistic view of the system holds that only when all the components in the whole system are harmoniously unified and coordinated can the overall optimization be realized. Therefore, when designing mathematics teaching in primary schools, we should not only master the characteristics, functions, design methods and strategies of each subsystem, but also deeply understand the relationship and mutual constraints between each subsystem and gain insight into the relationship between each subsystem and the overall teaching goal. Only in this way can we take the overall situation into consideration, focus on the overall situation, and start from a small place to optimize the overall design of primary school mathematics teaching.