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## Table of Contents

Introduction

Discussion

Result: Pedagogy for Higher Education

Realistic Method

Idealistic Method (Research Oriented Approach)

Conclusion

Open Problems

References

## Abstract

The aim of the article is to discuss an approach to teaching that generally a mathematics teacher follows in higher educational institutions of graduate and postgraduate students and of research scholars. It has been observed that all subjects including mathematics follow the same roots to develop. They all consist of three parts: assumptions, properties and applications, which brought them under the same umbrella of definition. In teaching too they follow the same steps to be explained to the students in order. Although there is no single best method available, an attempt has been made to propound one of the best idealistic method and a realistic inductive method. The article concluded with a short note that realistic inductive method is sufficient for graduate and postgraduate students while idealistic method is useful for research oriented students followed by the scope of further research in open problem section.

**Key Words:** Axiom, Theorem, Property, Teaching Methodology, Research.

**AMS Subject Classification:** 97C70, 97D40, 97D80

## Introduction

The action of a person who teaches is known as teaching. It is a process of transferring knowledge from teachers to students using different methods i. e., it is a process of imparting knowledge or skills. **Smith** defines it as certain tasks or activities the intention of which is to induce learning.

There is a difference of teaching in school, college and university level students. A school teacher basically helps students to learn. Whereas college and university teachers (or professors) conduct research and commonly teach undergraduate, professional and postgraduate courses in their fields of expertise. Professors may mentor and supervise graduate or postgraduate students also conducting research for dissertation. So we find the difference of teaching at these levels.

**Raghwan** (2018) says that the difference lies in the principles and methods of teaching, namely the pedagogy. He states that there is no pre-training programme available for university and college teachers prior to their taking up teaching as their profession. He stresses that the art and science of teaching, instruction and training school students is different from that of college students. In school, the emphasis is on making students learn facts and skills while college provides a learning environment where the student is required to think through and apply what s/he has learned.

He sees college lectures as in capsule form, presenting in a summarized form the main points of the topics of discourse. He believes that teachers in college deliver their lectures and expect the students to note the important points. They expect students to do self study, think and synthesize about different view point.

According to him the college teachers must have adequate scholarship and expertise in the area they have specialized at the master level. *There is no prescribed pedagogy to follow but they formulate their own pedagogy to create in the students a keen interest in the topics they lecture on and leave ample scope for the students to think, analyze, deduce and reach a conclusion based on their reasoning*. Therefore college teachers function as catalysts to make the students think for themselves. That is one of the reasons why college teaching has not insisted on pre-training.

As far as approach to teaching is concerned, formal teaching tasks include preparing lessons, giving lessons, and assessing students. **Weimer** states that “*when teachers think the best, most important way to improve their teaching is by developing their content of knowledge, they end up with sophisticated levels of knowledge, but they have only simplistic instructional methods to convey that material*”. He expresses that a love of the material and a willingness to convey that to students only enhances learning. He stresses on *what we teach and how we teach it are inextricably linked and very much dependent on one another*. Even though both are tightly linked, they are still separate. Development of one doesn’t automatically improve how the other functions.

If the method used to convey that knowledge are not sophisticated and up to the task, teaching may still be quite ineffective. It may not inspire and motivate students. It may not result in more and better student learning. *The best teachers do know their material, but they also know a lot about the process*. They have at their disposal a repertoire of instructional methods, strategies, and approaches—a repertoire that continually grows, just as their content knowledge develops. They never underestimate the power of the process to determine the outcome.

**Kelton** mentions that we can write up syllabi by simply handling all of the mandated material first. After that, go through each portion and decide if there is anything that you want to add or elaborate on i.e., *prepare a general list of the material to be covered*. **Raghwan** (2018) states that instead we lecture on topics, provide the *historical information*, identify *interdisciplinary connections* related to the topics, discuss *research* about those topics and leave the students to relate the lectures to the reference books. **Vaske** (1998) says that the one goal of education (teaching) is to develop students’ critical thinking. **Entwistle** (2008) quotes that “*the best practice is whatever helps students to engage more deeply with the subject and to become more actively responsible for their own learning*” is more important than others.

As far as mathematician’s approach is concerned, lecturing remains the prime delivery mode for teaching college students. **Shellard** et al (2002) states three components to effective mathematics instruction: Teaching for conceptual understanding, Developing children’s procedural literacy, and Promoting strategies competence through meaningful problem-solving investigations. **Protheroe** (2007) instructs to increasing abstract reasoning including thinking hypothetically, comprehensive cause and effect. He stresses the students to think deeply about the problems they are solving, reaching beyond the solutions and algorithms required to solve the problem. Thus encourage the learners to investigate further to generate new knowledge. He also suggested for interdisciplinary connections of different disciplines. **Barton** (2012) mentions that we lecture through three components: lectures and their development; student perspectives on mathematics; and interactions in the lecturing environment.

**Yadav** (2017) states that teaching completely depends on the basics required for the development of the subject. He propounded that *mathematics is the study of assumptions, its properties and applications*. He also claimed that every subject is the study of assumptions, its properties and applications. Therefore *all mathematical and non-mathematical subjects are the study of assumptions, its properties and applications* ’, whether it is sciences, arts, commerce, literature, etc. These three terms give the clue of teaching not only mathematics but other subjects also. He suggested that in teaching we must maintain the order of assumptions, properties and applications.

To the question raised by **Ronning** (2008) ‘ *What should we emphasize when we teach mathematics? What kind of understanding do we want the students to develop? What kind of mathematics, and how much, do all students need to know* ?’, **Yadav** (2017) stated that “*every chapter must be divided into three parts: assumptions, properties and applications*”. He stated that when we start teaching, we must mention “*what are the basic assumptions in the chapter*” keeping in view that definition is itself an assumption. What can we obtain from the assumptions and in last how and where can we apply these concepts? In this way students learn the definition, formulae and understand the basic structure of the chapter, which makes them perfect in application and for further research. Thus our motives become more and more successful in increasing the interest of mathematics among students.

## Discussion

As far as teaching in higher educational institutions is concerned, the content knowledge of the teacher and the contents of the lesson play an important role. The development of the contents of the lesson depends on the content knowledge the teacher and his interest in teaching and research or both.

A good teacher having interest in teaching without research experience would always focus on basics needed to understand the lessons, the assumptions, its properties and applications to complete the course of study required for graduate and postgraduate students. A very good teacher with research experience having interest in teaching would focus on the historical background of the lesson, basics needed to understand the lesson, its properties, its applications, its limitations, and current research going on in the related areas. But the best teacher having keen interest in teaching as well as in research would also focus on the philosophy and social values of the properties and results, interdisciplinary relations with different disciplines, future scope of research in the lesson, and would suggest the students to think critically to generate new knowledge including other subtopics as contents of the lessons.

Thus the content of the lessons depends on the interest of the teacher in the subject and their teaching and research experience. As far as the subtopic basics needed is concerned, it completely depends on the teacher’s experience of teaching whereas the inclusion of the research related subtopics and the philosophy of the results are due to the research experience of the teacher and his (her) deep knowledge in the lesson.

## Result: Pedagogy for Higher Education

From above we find that in preparing the lesson to teach college and university level students in higher educational institutions, we should collect the following important subtopics in order:

- Historic Background with its Development

- Basics Needed to Understand the Chapter (if any)

- Assumptions and Definitions

- Related Properties

- Applications in Real and Imaginary World Problems (if any)

- Interdisciplinary Connections of Different Disciplines (if any)

- Limitations of the Concepts (if any)

- Philosophy and Social Values of the Results and Properties (if any)

- Critical Thinking to Understand the Chapter (if any)

- Current Research Going on and

- Further Scope of Research to Generate New Knowledge

Thus in teaching we can include the above important subtopics before delivering lecture in class room in higher education institutions but in the era of completing course of study as first duty, it’s not always possible to follow the same. Therefore on the basis of the contents of the subtopics in preparing lectures, we can name them as follows:

## Realistic Method

This method consist only three subtopics: assumptions, properties, and applications. This is the more realistic method than others because under the constraint of limited time and period allowed for each subject, a teacher cannot discuss others subtopics. This method is useful for graduate and postgraduate students. In this case students are supposed to know the basics needed to understand the current lesson going on or to be started next.

Realistic method can be divided into two parts: Inductive Method and Deductive Method. *Inductive method* consist assumptions, properties and applications in order while *deductive method* consist these in reverse order as applications, properties and assumptions. The procedure discussed by *Yadav* (2017) is inductive. It would be better to call inductive method as realistic inductive method.

## Idealistic Method (Research Oriented Approach)

For research oriented students of doctorate and post doctorate work, a teacher needs deep knowledge of the subject and the chapter. In such case the subject is limited to a topic or a finite number of topics. Therefore a teacher can prepare the lectures on a particular chapter containing: Historic Background, Basics, Assumptions and Definitions, Properties, Applications, Interdisciplinary Connections, Limitations, Philosophy and Social Values, Critical Thinking, Current Research, and Further Scope of Research to Generate New Knowledge.

## Conclusion

As far as mathematician’s approach to teaching is concerned we conclude that the realistic inductive method containing only three subtopics assumptions, properties, and applications are sufficient for graduate and postgraduate students whereas idealistic method is more useful for research oriented students.

## Open Problems

All mathematical and non-mathematical subjects and its chapters can be studied (or taught) and its lessons can also be well prepared according to the subtopics contained in realistic inductive and idealistic methods, which opens the scope for further research.

## References

Barton, B., Ell, F., Miller, B. K., Thomas, M. (2012). Teaching Undergraduate Mathematics: Perspectives and Interactions. June.

Entwistle (2008). Teaching and Learning Research in Higher Education. A Paper Prepared for an International Symposium. Guelph, Ontario, Canada. April 25-26.

Kelton, S. (n.d.). An Introduction to Teaching Mathematics at the College Level.

Protheroe, N. (2007). What Does Good Math Instruction Look Like? Principal. 7(1). 51-54.

Raghavan, H. (2018). Learn to Teach: A Shared Process. University News: A Weekly Journal of Higher Education. Association of Indian Universities. 56 (4). 3-6, 22-28.

Ronning, F. (2008). Developing Knowledge in Mathematics by Generalising and Abstracting. Mathematics Newsletter: Ramanujan Mathematical Society. 17(4). 109-118.

Schleicher, D. & Lackmann, M. (2011). An Invitation to Mathematics from Competition to Research (Preface: What is Mathematics? by Gunter M. Ziegler). Springer, XIV. 220.

Shellard, E., & Moyer, P. S. (2002). What Principals Need to Know about Teaching Math. National Association of Elementary School Principals and Education Research Service.

Smith, B. O. (n.d.). Definitions of Teaching.

Vaske, J. M. (1998). Thesis: Defining, Teaching, and Evaluating Critical Thinking Skills in Adult Education. School of Education. Drake University.

Weimer, M. (n.d.). Editorial Page. Faculty Focus Special Report: Effective Strategies for Improving College Teaching and Learning. The Teaching Professor. Magna Publications, Inc. USA.

Weimer, M. (n.d.). Content Knowledge: A Barrier to Teacher Development, Faculty Focus Special Report: Effective Strategies for Improving College Teaching and Learning. The Teaching Professor. Magna Publications, Inc. USA.

Weimer, M. (n.d.). Finding the Best Method. Faculty Focus Special Report: Effective Strategies for Improving College Teaching and Learning. The Teaching Professor, Magna Publications, Inc. USA.

www.en.m.wikipedia.org/wiki/Teacher. Accessed on 29.07.2018

www.en.m.wikipedia.org/wiki/Professor. Accessed on 29.07.2018

Yadav, D. K. (2017). Exact Definition of Mathematics. International Research Journal of Mathematics, Engineering and IT. Associated Asia Research Foundation. 4(1). 34-42.

The Effective Mathematics Classroom. n.p.

- Quote paper
- Dr. Dharmendra Kumar Yadav (Author), 2018, Mathematician's Approach to Teaching. Pedagogy in Higher Education, Munich, GRIN Verlag, https://www.grin.com/document/440885

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