2018 International Conference on Applied Mechanics, Mathematics, Modeling and Simulation (AMMMS 2018) ISBN: 978-1-60595-589-6
Study and Practice on Primary School Mathematical Modeling
Jin-jin NIU
1and Ji-ping NIU
2,*1
College of Educational Science, Hubei Normal University, Hubei Huangshi, China
2
College of Computer, Huanggang Normal University, Hubei Huanggang, China
*Corresponding author
Keywords: Mathematical model, Mathematical modeling, Elementary school mathematics.
Abstract. The mathematical modeling is one of the ten core concepts proposed by the Ministry of Education's “Compulsory Education Mathematics Curriculum Standards”. The cultivation of mathematical modeling ideas and methods can help students get clear mathematical thinking and gradually form a scientific method to understand the world to improve students' basic mathematical literacy and interest in learning. This paper expounds the connotation and the significance of elementary school mathematical modeling. Through the abstraction and simplification of practical problems, two cases of teaching modeling are given.
The Connotation and Significance of Mathematical Modeling
Mathematical Model Idea
The Ministry of Education's “Compulsory Education Mathematics Curriculum Standards”[1,2] clearly puts forward the requirement to focus on the development of students' model thinking. The model thinking is a basic understanding of the concept, structure and method of mathematics. The basic idea of the model is reflected in mathematical abstraction, mathematical reasoning and model thinking. The model idea refers to the method and strategy, the consciousness and the concept of solving a practical problem by analyzing and abstracting the problem, selecting the appropriate mathematical model, and solving the problem through the model. The essence of the model idea is to solve a problem first, and then establish a way of solving an entire similar type of problems. It is a very important mathematical thinking method, an important means to integrate mathematics into real life and let mathematics be used out of the classroom [3, 4].
Mathematical Modeling Idea
The whole process of establishing model is mathematical modeling. Mathematical modeling is a basic idea of learning mathematics, and an important idea in modern teaching, which runs through the whole teaching process. “Compulsory Education Mathematics Curriculum Standards” clearly states the importance of the mathematical modeling. At present, mathematical modeling has been widely used in high schools and universities, but it is rarely involved in primary schools. Therefore, experts nowadays focus on the application of mathematical modeling ideas in primary school to help primary school students establish models and seek solutions to solve mathematical problems during learning. It is of great significance to use the mathematical modeling method in the classroom teaching of elementary school mathematics. The cultivation and use of modeling can help students solve math problems and conduct a self-exploration study in the future, which can not only cultivate students' awareness of self-inquiry, but also enable students to have the ability and initiative to learn mathematics knowledge.
activities and teaching methods. During practical teaching, mathematical modeling ideas should be infiltrated through teaching.
The Significance of Mathematical Modeling Ideas in Teaching
Mathematical Modeling Ideas Conducive to the Development of Students' Thinking
As a leader, the teacher guide students' thinking activities. It is very important for the teacher to integrate mathematical modeling ideas into classroom teaching to effectively develop students' thinking ability. The process of establishing mathematical models is the way to develop students' thinking ability. By building models, students can not only improve their ability to solve problems, but also discover problems and build models. During the process of verifying the model later, the development of students' thinking is cultivated [5]. Through the whole modeling process, by and by, students’ thinking ability is promoted.
Mathematical Modeling Ideas Conducive to Problem Solution
Pupils' ability to solve problems is weak. Teachers should focus on cultivating students' problem-solving skills. Learn to build mathematical models in their daily learning, students can solve problems by analyzing some practical problems in real life. In the process of solving mathematical problems, students can find a learning method and then set up a new math problem. Many students have difficulties in solving problems. Researchers engaged in solving mathematical problems present several models in mathematical psychology, namely understanding models, schema models, process models, and strategy models. Study of these models can help students improve their problem-solving skills. The establishment of mathematical models in the process of solving problems is helpful for improving students’ ability of problem-solving [6].
Mathematical Modeling Ideas Conducive to Self-Exploration
The new curriculum standards require students to be the main body and teachers be the mainstay, and give full play to the initiative of students. Therefore, students' self-exploration ability is very important. It is necessary not only to cultivate students' independent inquiry ability in the classroom, but also to stimulate students’ ability to think independently and explore independently out of class. Teachers should gradually guide students to master the construction process of simple mathematical models in teaching to make students learn how to build mathematical models and improve students’ ability of using mathematical model to solve practical problems. By and by, students will develop a habit to think and solve problems using a mathematical modeling method, and they can apply mathematical modeling ideas not only in classroom but also in real life. Furthermore, this trained mathematical modeling consciousness will further conducive to students future learning and deepen their self-exploration ability.
The Application of Mathematical Modeling in Teaching
In the practical teaching of elementary school mathematics, the application of modeling ideas is to redesign traditional teaching content. Using modeling ideas to improve students' understanding ability and learning effect has a strong practical application value. The basic styles in mathematics (straight lines, curves...), the determination of the various variables in the model, and the relationships between the various variables can be used to identify their specific relationships in a model. Typical mathematical problems in elementary school mathematics, such as traveling distance problems, flow problems, restoration problems, tree planting problems, and profit and loss problems, can be solved through mathematical modeling, which can improve students' learning effects and interest [7, 8, 9].
Design Case 1: Travelling Distance Problem
Problem description: A and B are 480 kilometers apart. Car A starts from A and car B starts from B respectively at the same time. After 6 hours, they encounter. The process is shown in figure1. It’s known that Car A travels 45 kilometers per hour. Find the speed of Car B based on the above description.
(1) Creating a situation and guiding students to analyze the problem
Teachers guide and encourage students to analyze practical application cases through specific situational creations. Students are invited to demonstrate the encounter process to help them analyze the process of the travelling problem and obtain quantitative relationships.
Figure 1. An example of travelling distance problem. (2) Abstract summarization and model building
The teacher then guides the students to use the line graph to graphically describe the problem, abstract the mathematical expression of the travelling problem, and establish the corresponding mathematical model.
(3) Studying the model and solving the model
Analyze the known and unknown conditions in the mathematical expressions. Suppose Car B travels x km per hour, then the formula can be written as: (45 + x) × 6 = 480. Through calculation, the answer is: x = 35km.
(4) Generalization, summarization, and application
The teacher guides the students to analyze the problem, abstracts a general modeling method for the similar kind of traveling distance problems.
The model formula of the travelling problem: total speed × time = distance, which can be simplified as: S=vt.
Where S refers to the distance, v is the speed, and t is the time.
Design Case 2: Perimeter Problems
Problem description: “The Circumference of Rectangles and Squares” in the third-year mathematics textbook, uses two mathematical models: the perimeter of the rectangle = (length + width) × 2, and the circumference of the square = the length of the side × 4. The modeling process in teaching can be carried out as follows.
(1) Creating a situation and guiding students to analyze the problem
Create a specific situation that visually reflects the perimeter of the rectangle. By asking questions, we will guide students to think independently and quickly find the perimeter of the rectangle, highlighting the characteristics of the simplicity of the mathematical model.
(2) Self-exploration and building models
The teacher adopts the cooperative inquiry teaching method, let the students discuss the method of solving the rectangular circumference in groups, and let the students try to write down the calculation formula of the rectangular circumference. By this way, students gradually learn and master the method and the process of mathematical modeling.
(3) Model explanation
The third-grade students' understanding of the mathematical model of the calculation formula of the rectangular perimeter is relatively abstract. To further understand the model, the teacher brings specific practical examples into classroom teaching to help students summarize the calculation formula of the rectangular perimeter.
Guide students to use mathematical models to solve two daily problems: the circumference of the lawn and the perimeter of the playground, and then point out the characteristics of the mathematical model. In the end, students are able to quickly solve similar problems in daily life.
It can be found that different math problems have similar models. For example, many engineering problems, travelling problems, and work problems, they all have a common model, namely, total speed × time = distance, which can be summarized into a formula: (a1 + a2) × c = d. Teachers can ask students to build their own specific problems in real life into a model of travelling problems, engineering problems or other problems. This way, students’ learning interests, problem-solving ability as well as mathematical modeling ability can be enhanced.
The Way to Cultivate Students' Modeling Consciousness
Creating Situations to Stimulate Students' Interest in Mathematical Modeling
Based on the student's real life, the teacher transforms student's daily life problems into mathematical models, which makes it easier to stimulate students' interest. Teachers can set up corresponding scenarios according to the specific teaching content, so that students can clearly perceive the charm of mathematical modeling. Let students have a basic idea of mathematical model and develop interests in mathematical modeling.
Deepening the Concept of Mathematical Modeling
In the process of mathematical modeling, every problem needs to be carefully studied to help us understand the relationship between actual problems and mathematical models. During classroom mathematics teaching, when explaining the concept of knowledge and the method of solving problems, the modeling ideas should be fully incorporated to let students feel the power of mathematical modeling, think independently and explore on their own.
Deeply Understanding the Basic Characteristics of Mathematical Problems through Modeling
To strengthen the understanding of mathematical models, it is more important to understand its essential characteristics. During the process of modeling, teachers guide students to discover various quantitative relationships, the transformation and derivation of graphics, and so on, rather than just telling students the results of a mathematical problem. For example, graphics and geometry is just the mathematical model of basic geometric figure formed by lines, squares, and trapezoids in real life. If the model created is divorced from real life, then it loses its practical significance.
Using the Intuition of Mathematical Models to Enhance Students' Mathematical Problem-Solving Abilities
In the primary school mathematics classroom teaching, the mathematical model is used to transform the specific real problem into a mathematical problem, and the mathematical model is used to solve the practical problem. In this process, the usage of intuitive and concrete mathematical models can help students better understand mathematics and acquire the ability to solve mathematical problems.
Conclusions
mathematical modeling and integrating it into teaching can make students feel the charm of mathematics, the fun of mathematics learning, and develop interest in mathematics learning.
References
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