Hong FANG

College of Basic Sciences, Tianjin Agricultural University, Tianjin 300384, China

Abstract For young teachers, the ability to transform teaching materials and the ability to control teaching is a basic teaching regulation and control ability. Through a case teaching of function limits in the mathematics course, this article studied how to explore deep into the content of textbooks, use the textbooks in a creative and proper manner, help students better understand and master the content of teaching, improve students ’autonomous learning ability, and promote comprehensive improvement of students’ knowledge, ability and caliber.

Key words Transformation ability of teaching materials, Teaching regulation and control ability, Case teaching

1 Introduction

In theOpinionsoftheMinistryofEducationontheImplementationofFirst-classUndergraduateCourses, it stated that course is the core element of talent cultivation, while the quality of the course directly determines the quality of talent cultivation. Course teaching is not simply imparting knowledge, but a combination of knowledge, ability and quality. Teachers’ teaching should not just copy the textbooks. Instead, teachers should go deep exploration of the content of the textbooks, to refine the teaching content that meets the students’ cognitive level, and through the teaching of the course content, let the students understand the cutting-edge and era of the subject development, improve their comprehensive ability and advanced thinking to solve complex problems[1]. The main field of first-class undergraduate course construction is classroom teaching. The design of course teaching activities should conform to the direction of student-centered course teaching reform, focus on stimulating students’ learning interest and learning potential. In the teaching process, teachers should not instill knowledge into students, but students actively construct knowledge. Besides, teachers’ control and treatment of teaching contents, regulation of teaching process and organization and management of classroom teaching activities are particularly important.

2 Problems to be solved in the case teaching

Understanding and mastering the language of function limits is a difficult point for students.ε-δFor the students’ transition from the intuitive definition of the limit to the abstract concept, there should be a bridge. How can the teacher extend the abstract concept from the specific problem reflects the teacher’s ability to transform the teaching materials and the processing content of teaching contents. Besides, the process of demonstrating mathematical formulas, theorems, and concepts contains important methods of thinking. Teachers should not simply impart knowledge in accordance with the original textbooks. They should demonstrate their thinking processes and allow students to master new knowledge, learn to actively establish a knowledge system, and cultivate the spirit of actively seeking new knowledge.

3 Teaching mode of the case teaching

3.1 Encouraging students to learn with questionsAs regards the understanding of the concept language of functional limitsε-δ, teachers may set the following questions for students to preview. (i) What is the relationship betweenεandδin functional limit concept? How does it differ from the relationship betweenεandNin the definition of the sequence limit? (ii) What is the difference between the geometric shape when the function limit exists and the geometric shape when the sequence limit exists? (iii) Understanding of certain sentences (or formulas) in the definition of function limits.

3.2 Teaching and learning process of the case teaching

3.2.1Exploring new knowledge. We have learned how to describe the two infinite approximation processesx→∞ andf(x)→Awith precise mathematical language and formulas. How should we define the limit processesx→x0andf(x)→A? Starting from specific problems, this lesson would guide students to summarize, understand, conclude the concept of function limits during the learning process.

Students: through observing the picture of functionf(x), it found that whenxis closer to 3, the corresponding function valuef(x) is closer to 6; whenxis infinitely closer to 3, the corresponding function valuef(x) is infinitely closer to 6, but when the functionf(x) is undefined atx=3? How to understand this question?

Teacher: the limit of functionf(x) limit whenxis close to 3 refers to seeking variation trend off(x) whenxis infinitely close to 3, rather than to calculating the function value of the functionf(x) whenx=3. Therefore, it is not connected with the functionf(x) atx=3. This indicates that it is only required that the functionf(x) is defined in a punctured neighbourhood ofx=3.

Students: we can measure the degree ofxapproaching 3 using the distance formula |x-3|, and measure the degree off(x) approaching to 6 using |f(x)-6|, what is the connection between the two?

Students: in the case (1), as long as taking the radiusδ<0.01 of the punctured neighborhood ofx0=3, it is able to satisfy the requirement; in the case of (2), as long as taking the radiusδ<0.001 of the punctured neighborhood ofx0=3, it is able to satisfy the requirement.

Teacher: very good, please think that if given an arbitrarily small positive number, can we find the positive numberδ?

After inspiring students to think, it could be concluded that as the given positive number decreases, a positive numberδcan always be found to satisfy the condition; that is, given a positive numberεno matter how small, a positive numberδalways exists, so when 0<|x-3|<δ, the inequality |f(x)-6|<εholds. This is the essence of functionf(x) being infinitely close to 6 whenx→3. Finally, the teacher guides the students to give a precise definition of the limit of the function.

3.2.2Further exploration of teaching contents. (i) After giving the definition ofε-δof the function limit, ask students to continue their research in small groups: inε-δlanguage, a positive numberεis used to describe the closeness off(x) andA, and a positive numberδis used to describe the closeness ofxandx0. Give aε, we can find aδ, then is the value ofδonly value? (ii) How many cases doesx→x0in the function limit include? After careful discussions, students reached the conclusions that: The value ofδis not only, because the selection ofδdepends onε;x→x0includes two cases,xcan approach tox0from the left side ofx0, and it can approach tox0from the right side ofx0. Therefore, it should be divided into left and right limit forms, which is different from the sequence limit.

3.2.3Application examples:

Analysis: through the previous learning process, it was found thatδdepends onε, that is, given a positive numberε, then find a positive numberδ, so the "backward inference method" can be used in the proof. In the solution of specific problems, in order to make the calculation of the positive numberδsimple, the technique of simplification or application of |f(x)-A|<εcan be converted into a functionf(|x-x0|) about |x-x0|, and then the inequalityg(|x-x0|<ε) can be solved to find the positive numberδmeeting the conditions.

Students: the amplification method has an influence on the selection of positive numberδ.

Teacher: this question examines the students’ understanding ofx→x0in the concept of function limits. Please give answers after the group discussion.

4 Teaching introspection

In this paper, we discussed how to guide the teaching content in the teaching materials step by step through the teacher, so that students can approach the essence of the problem they are seeking step by step, and experience the process of knowledge construction. The teacher’s teaching is not simply imparting the content of the teaching materials into the students. Instead, teachers teach students efficient learning methods, ponder over the essential way of thinking through phenomena, and guide students to autonomously analyze, summarize, understand, and learn how to solve abstract problems[2]. Teachers provide exercises that define the limits of the proof function to test students’ ability to use what they have learned to solve specific problems. We hope that through this form of teaching, students can interact with teachers in the classroom, dare to question, and dare to express their views, and cultivate the independent thinking ability, dare to criticize and question, and have innovative ability and innovative thinking. The overall construction of first-class undergraduates should attach importance to "making courses better, teachers stronger, students busier, management stricter, and results more effective", so as to establish a high level talent cultivation system. Therefore, it is recommended to improve teaching ability of teachers, and better guide their teaching work. Teachers should make effort to comprehensively improve students’ knowledge, ability, and caliber.