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## MATHEMATICS

Bacon said“Mathematics is the gateway and key to all sciences”.

Gauss stated“Mathematics is the queen of sciences and arithmetic is queen of all mathematics”.

All the above definitions emphasize mathematics as a tool specially suited for dealing with scientific concepts.

### MEANING OF MATHEMATICS

The word mathematics comes from the **Greek word ‘máthema’**, which, in the ancient Greek language means "that which is learned", "what one gets to know”.

In Latin, and in English until around 1700, the term mathematics more commonly meant "**astrology"** (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800.

**Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers.**

### BRIEF HISTORY OF MATHEMATICS

- Mathematics
**starts with counting**. It is not reasonable, however, to suggest that early counting was mathematics. - Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started.
- In
**Babylonia**mathematics developed from**2000 BC**. - Number problems such as that of the
**Pythagorean triples**(a,b,c) with a2+b2 = c2 were studied from at least**1700 BC**. - The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around
**450 BC.** - The major Greek progress in mathematics was from
**300 BC to 200 AD.**After this time, progress continued in Islamic countries. Mathematics flourished in particular in Iran, Syria, and India. - The most important mathematician of the 18th Century was
**Euler**who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry. - The
**19th Century saw rapid progress**. - The 19th Century saw the work of
**Galois**on equations and his insight into the path that mathematics would follow in studying fundamental operations **Maxwell**was to revolutionize the application of analysis to mathematical physics.- Statistical mechanics was developed by Maxwell, Boltzmann, and Gibbs.
**Fredholm's**work led to Hilbert and the development of functional analysis.

### NATURE AND CHARACTERISTICS OF MATHEMATICS

**Logical Sequence:**It is a subject in which the previous knowledge has a greater influence.**Abstractness:**Mathematics is abstract in the sense that mathematics does not deal with actual objects in much the same way as physics.**Mathematical Language and Symbolism:**Another most important characteristic of mathematics which distinguishes it from many other subjects is its peculiar language and symbolism.*Lindsay says, “Mathematics is the language of physical sciences and certainly no more marvelous language was ever created by the mind of man”.*

## MATHEMATICS AND ITS RELATIONSHIP WITH OTHER SUBJECTS

**Mathematics with Arts:** The arts and mathematics involve students understanding of relationships between time and space, rhythm, and line through the experience of these abstract concepts in various art forms and mathematical ideas.

**Mathematics with Civics and Citizenship:** The concepts developed in the study of mathematics are applicable to a range of civic and citizenship understandings.

**Mathematics in Geography:** There are changes in the fertility of the soil, changes in the distribution of forests, changes in ecology, etc., which have to be mathematically determined, in order to exercise desirable control over them.

**Mathematics with Communication:** Mathematics structure and working mathematically play essential roles in understanding natural and human worlds

**Mathematics with English:** Mathematical structure is strongly related to semantics syntax and language and to the use of propositions and quantifiers embedded in the principled argument in natural languages.

**Mathematics with Health and Physical education:** In health and physical education, mathematics provides tools and procedures which can be used to model

**Mathematics with Humanities-Economics:** Economics and mathematics are related through the use of mathematics to model a broad range of economic, political, and social phenomena.

**Mathematics with History:** The concepts and skills developed in mathematics support student understanding and interpretation of a range of historical sources and their presentation as evidence in demonstration historical understanding.

**Mathematics with Science:** In science students use measurement and number concepts particularly in data collection estimation of error analysis and modes of reporting.

**Mathematics in Biological Sciences:** Mathematical and computational methods have been able to complement experimental structural biology by adding motion to the molecular structure

**Mathematics in Chemistry:** Math is extremely important in physical chemistry especially advanced topics such as quantum or statistical mechanics.

## Values of Mathematics

Mathematics has got many educational values that determine the need of teaching the subject in schools. These values can be studied under the following heads:

**Practical Value:** Mathematics has great practical value. Everyone uses mathematics in every form of life. A common man sometimes can do without reading or writing but he cannot do without counting and calculating.

**Cultural Value:** Mathematics has got a great cultural value which is steadily increasing day by day. The progress of our civilization has been mainly due to the progress of various occupations such as agriculture, engineering, industry, medicine, navigation, railroad building, etc. These occupations build up the culture. Mathematics makes a direct or indirect contribution to the development of all occupations.

**Disciplinary Value:** Mathematics trains or disciplines the mind also. It develops thinking and reasoning power.

## OBJECTIVES OF TEACHING OF MATHEMATICS

- Teaching and learning of basic
**numeracy skills**to all pupils. - Teaching of
**practical**mathematics - Teaching of
**abstract**mathematical concepts at an early age - Teaching of
**selected areas**of mathematics - Teaching of
**advanced**mathematics - Teaching of
**heuristics and other problem-solving strategies**to solve non-routine problems.

## AIMS IN TEACHING MATHEMATICS

- Develop a good
**understanding of numbers**and the number system - Achieving a sound grasp of the properties of numbers
- Achieving a good
**understanding of place value and ordering** - Understanding the principles and practice of estimating rounding.
**Improving speed**- Achieving a good understanding of number operations and relationships.
- Achieving
**rapid mental recall of numbers**of facts - Maximizing the ability to undertake
**calculations**using pencil and paper methods. - To develop a good
**ability to solve the problem** - Developing the
**ability to make decisions**

## INSTRUCTIONAL OBJECTIVES

The instructional objectives are based on **Bloom's Taxonomy of objectives**. Bloom's Taxonomy was created in **1956** under the leadership of educational **psychologist Dr. Benjamin Bloom** in order to promote higher forms of thinking in education, such as **analyzing and evaluating concepts, processes, procedures, and principles, rather than just remembering facts (rote learning).**

## MICRO TEACHING

Micro Teaching is one of the recent innovations in the field of educational technology. It offers a new model for improving teaching.

**Microteaching is a method that enables teacher trainees to practice a skill by teaching a short lesson to a small number of pupils. Usually, a micro lesson of 5 to10 minutes is taught to four or five fellow students.**

Allen, D.W (1966) defined microteaching as“a scaled down teaching encounter in class size and class time”.

Passi,B.K(1976) writes that“the most important point in microteaching is that teaching is practiced in terms of definable, observable, measurable and controllable teaching skills”.

### PHASES OF MICRO TEACHING

**Clift (1976) described the following as the phases of micro-teaching.**

- Pre-active phase (knowledge acquisition phase)
- Interactive phase (skill acquisition phase)
- Post-active phase (Transfer phase)

## TEACHING SKILLS

Teaching skills are specific instructional activities and procedures that a teacher may use in the classroom.

According to Allen-A teaching skill is a group of teaching acts/ behaviors intended to facilitate pupils learning activity directly or indirectly.

### Some Teaching Skills

- The skill of Questioning
- The skill of Reinforcement
- The skill of probing Question
- The skill of explaining
- The skill of stimulus variation
- The skill of introducing a lesson
- The skill of illustrating with examples
- The skill of using blackboard
- The skill of silence and nonverbal cues
- The skill of using audio-visual aids
- The skill of recognizing attending behavior
- The skill of achieving closure

## LESSON PLAN

A lesson plan is a **teaching outline of the important point of a lesson arranged in the order **in which they are to be presented; It may include objectives, points to be made, questions to be asked, a reference to materials, assignments, etc.

**An ideal lesson plan must have the following essential elements;**

- Knowledge of Student’s entering behavior
- Knowledge of the Subject
- General Knowledge of other related Subjects
- Clarity of Objectives
- Division in Units
- Flexibility
- Knowledge of the Principles and Strategies of Teaching
- Time duration Sense
- Clarity about Previous Knowledge
- Knowledge of Class Level
- Use of Instructional material

## Strategies of Teaching Mathematics

Teaching mathematics needs to know multi-techniques, methods, and strategies, approaches that break the monotony of the teaching and sustain the interest of the learners in learning mathematics.

The main aim of mathematics education in schools is the mathematization of the child’s thought processes.

In the words of David Wheeler,it is “more useful to know how to mathematise than to know a lot of mathematics

## METHODS AND TECHNIQUES OF TEACHING MATHEMATICS

### INDUCTIVE METHOD:

The inductive method is to move from specific examples to generalization and the deductive method is to move from generalization to specific examples.

#### Merits of the inductive method:

- Scientific Method
- The content becomes crystal clear to students.
- Based on Actual Observation and Experimentation.
- Thinking is Logical
- Suitable for beginners
- Increases Pupil – Teacher Relationship
- Home Work is reduced

#### Demerits of the method

- Not suitable for all topics
- Time-Consuming Method
- Laborious Method
- Not Suitable for all types of students

### DEDUCTIVE METHOD

- It is a method of reasoning by which concrete applications or consequences are deducted from general principles or theorems are deduced from definitions and postulates.
- It is
**proceeding from Abstract to Concrete, General to Particular, and Formula to Example**s. - Students are given formula/rules/laws/principles directly.

#### Merits of this method

- Time-Saving Method
- Suitable for all topics
- Suitable to all Students
- Glorifies Memory
- Useful at Revision Stage
- Speed and efficiency
- Mostly Used at Higher Stage level

#### Demerits of this method

- Not a psychological Method
- No Originality and Creativity
- Blind Memorization
- Educationally Unsound
- Students are Passive Learners
- The reasoning is not clear

### ANALYTIC METHOD

**It proceeds from unknown to known,** ’Analysis’ means ‘breaking up’ of the problem in hand so that it ultimately gets connected with something obvious or already known.

It is the process of unfolding of the problem or of conducting its operation to know its hidden aspects.

### SYNTHETIC METHOD

- It is the
**opposite of the analytic method**. Here one**proceeds from known to unknown**. - In practice, synthesis is the complement of analysis.
- To synthesis is to place together things that are apart.
**It starts with something already known and connects that with the unknown part of the statement.****It starts with the data available or known and connects the same with the conclusion.**- It is the process of putting together known bits of information to reach the point where unknown information becomes obvious and true.

### PROBLEM-SOLVING METHOD

The problem-solving method is one, which involves the use of the process of problem-solving or reflective thinking or reasoning. The problem-solving method, as the name indicated, begins with the statement of a problem that challenges the students to find a solution.

Problem solving is a set of events in which human beings were rules to achieve some goals– Gagne.

#### Procedure for Problem-solving

- Identifying and defining the problem
- Analyzing the problem
- Formulating tentative hypothesis
- Testing the hypothesis
- Verifying of the result of checking the result

### LABORATORY METHOD

- The laboratory method is based on the maxim
**“learning by doing.”** - This is an activity method and it leads the students to discover mathematics facts.
- In it, we
**proceed from concrete to abstract.**

The laboratory method is a procedure for stimulating the activities of the students and to encourage them to make discoveries.

- This method
**needs a laboratory**in which equipment and other useful teaching aids related to mathematics are available. - For example, equipment’s related to geometry, mathematical model, chart, balance, various figures, and shapes made up of wood or hardboards, graph paper, etc.

#### The procedure of Laboratory method

- Aim of The Practical Work
- Provided materials and instruments
- Provide clear instructions
- Carry out the experiment
- Draw the conclusions

### PROJECT METHOD

The project method is of **American origin** and is an outcome of Dewey’s philosophy of pragmatism. However, this method is **developed and advocated by Dr. Kilpatrick.**

Project is a plan of action(oxford’s advanced learner’s dictionary).

A project is a problematic act carried to completion in its most natural setting– Stevenson.

#### Steps involved in Project Method

- Providing /creating the situations
- Proposing and choosing the project
- Planning the project
- Execution of the project
- Evaluation of the project
- Recording of the project

## MODERN TECHNIQUES OF MATHEMATICS TEACHING

### Brainstorming:

Brainstorming is a teaching strategy for releasing ingenuity and for enhancing critical thinking, especially in mathematics wherein higher-order thinking skills of students should be more developed.

**Benefits of Brainstorming in the Maths Classroom**

- Activates schema
- Helps set a baseline for learning
- Help identify misconceptions
- Helps guide teaching and differentiation
- Improve student's perception about their level of mathematical understanding

### Quiz Technique

Quick quizzes throughout the day can help teachers assess the effectiveness of their instruction, as well as student understanding of the concepts taught.

### Seminar Method

- The seminar method is the most
**modern and advanced method of teaching.** - It refers to a structured group discussion that usually follows a formal lecture or lectures often in the form of an essay or a paper presentation on a theme.
- A specific subject or topic is delivered as an article or report in the seminar.

### Discussion Technique

According to Cockcroft (1982),Mathematics teaching at all levels should include opportunities for discussion between teacher and pupils and between pupils themselves.

### Scenario building Technique

- Scenario building is a method of understanding and planning for outcomes of an uncertain
**future**. - In essence, it is a method for envisioning possible futures for complex systems to understand major drivers of future change.

## MODELS OF TEACHING MATHEMATICS

Undoubtedly teaching models play a vital role in arousing the interest of students. Hence it is the prime duty of mathematics teachers to adopt suitable models in their teaching.

**Some Teaching Models are:**

- Explicit Teacher Model
- The Path-Smoothing Model
- An Alternative - The Challenging Model
- Concept Attainment Model
- Advance Organizer Model
- The Jurisprudential Inquiry Model

## Classroom Interaction Analysis (Flander’s Interaction Analysis Category System (FIACS)

Flanders developed a system of interaction analysis to **study what is happening in a classroom when a teacher teaches.** It is known as Flanders Interaction Analysis Categories System (FIACS).

- Flanders and others
**developed this system at the University of Minnesota,**the U.S.A. between 1955 and 1960. - Flanders classified total verbal behavior into
**10 categories**. - Verbal behavior comprises teacher talk, student talk, and silence or confusion.

**There are ten categories mentioned in this analysis. They are**

- Teacher Talk – 7 categories
- Pupil Talk – 2 categories
- Silence or Confusion- 1 category

## AUDIO VISUAL AIDS

Audiovisual instruction simply means a supplementary device for waking to learn objective real and effective.

According to Kinder S. James:Audio visual aids are any device which can be used to make the learning experience more concrete, more realistic and more dynamic.

According to Burton:Audio visual aids are those sensory objects or images which initiate or stimulate and reinforce learning.

According to KP. Neeraja:An audio visual aid is an instructional device in which the message can be heard as well as seen.

### Essential qualities of Audio-Visual Aids

- It must be
**clear, clean, interesting, and cheap**. - It should be of a
**suitable size** - It must
**adequate, accurate,**giving up-to-date information - Must be
**relevant to the topic**being discussed - It must not be overcrowded with details
- It must illustrate the specific point being taught

### Importance of audio-visual aids:

- It helps in
**developing the perception**of learners. - It aids in the positive transfer of learning and training
- It
**facilitates in understanding**and comprehension - It
**provides reinforcement**to the learner - It
**increases the retention**of the learner

## CLASSIFICATION OF TEACHING AIDS

There are many aids available these days. We may classify these aids as follows

### Visual Aids

- The teaching aids which
**use the sense of vision**for teaching/learning are called Visual aids. **For example: -**actual objects, models, pictures, charts, maps, flashcards, flannel board, bulletin board, chalkboard, overhead projector, slides, etc.- Out of these blackboards and chalk are the commonest ones.

### Audio Aids

- The teaching aids that
**involve the sense of hearing**are called Audio aids. **For example -**radio, tape recorder, gramophone, etc.

### Audio-Visual Aids

- The teaching aids which involve the sense of
**vision as well as hearing**are called Audio-Visual aids. **For example: -**television, film projector, film strips, etc.

## MATHEMATICS TEACHER

A mathematics teacher of today is expected to possesses certain skills and qualities pertaining to the subject. Academic qualities and qualifications matter a lot in mathematics teaching.

**Mathematics teacher has to update the maths knowledge of the learners through mathematics club activities, mathematics fairs, field trips, Olympiad, and celebrating mathematics events.**

Math teachers educate students at all levels, from elementary school through high school

### QUALITIES OF GOOD MATHEMATICS TEACHER

**Knowledge**of Maths- Alternative strategies to solve mathematics problems
- Facilitating approach
- Good leadership
- Care and
**concern** - Understanding of information and communication
- Engaging
**Experienced**- Gives a lot of problems to practice with
**Helpful,**insightful, explains things well- Can teach a topic in different ways to help better understand something
- Organized, smart,
**Enthusiasm** - Able to fully answer questions & visibly demonstrate answer; able to make learning fun
**Patience,**intelligence- Explains new material in detail

## MATHEMATICS CLUB

- The Mathematics club plays an important role in creating interest in mathematics in schools.
- This helps the students in having an idea of the practical utility of mathematics in addition to creating their interest in Mathematics.
- It can serve a number of purposes.

**Importance of the Mathematics club**

- Useful in arousing and maintaining interest in Mathematics.
- Gifted students get an opportunity to satisfy their needs and interests by actively participating in the activities of mathematics clubs.
- Helpful in making proper utilization of leisure time.
- The students get an opportunity of mathematical hobbies, recreational mathematics, mathematical projects, mathematical games, mathematical discussions and debates, and mathematical innovations.
- Provides an opportunity to read mathematical literature.
- Provides an opportunity for leadership, cooperation, joint responsibility, active participation, and organizing programs.

**Activities of The Club**

- Arranging lecturers by renowned Mathematics Teachers or Scholars.
- Celebrating days and events pertaining to the history of Mathematics or men of Mathematics.
- Organizing Mathematical competitions.
- Organizing recreational activities in Mathematics.
- Preparing Mathematical aids and illustrations.
- Organizing Mathematical exhibitions or fairs.
- Mathematical articles for the school magazine.
- Organizing seminars and career courses relating to Mathematics.

## MATHEMATICS FAIRS

With a view to encourage, popularize and inculcate scientific temper among the children of the country, **NCERT** organizes a national level science exhibition every year where children showcase their talents in science and mathematics and their applications in different areas related to our everyday life.

## FIELD TRIP

Field trips are a time-honored tradition in most schools. The students love them. field trips give them a chance to get out of the classroom and experience something new.

**Advantages and Of Field Trips**

- Enhances the Curriculum
- New Learning Environment
- Team Building

**Disadvantages of Field Trips**

**Planning:**A disadvantage of field trips is that they take an incredible amount of planning.**Liability:**Field trips bring up a wide array of legal issues, most regarding liability.

## MATHEMATICS OLYMPIAD

The Mathematics Olympiad activity was undertaken by **National Board for Higher Mathematics (NBHM)** from 1986 onwards and is currently run in collaboration with the **Homi Bhabha Centre** for Science Education, Mumbai.

One main purpose of this activity is to support mathematical talent among high school students in the country

## MATHEMATICS LIBRARY

- A well-organized library is a source of attraction for its students.
**A mathematics library is the birthplace of future mathematicians.**- It inspires, stimulates, and equips them to follow the footprints of great mathematicians.

**Importance of Mathematics Library**

- The separate arrangement brings efficiency to the organization.
- Mathematics teacher remains in touch with the volumes and literature available in the library.
- It gives a
**sense of separate identity**and inculcates interest in the subject mathematics. - The student gets better
**facilities for reading books**and literature. - It helps in the activities of the mathematics club.
- It may help in
**nurturing gifted and potentials students**in mathematics.

## MATHEMATICS LABORATORY

**A mathematics laboratory may contain the following types of material and equipment’s:**

- Different types of pictures and charts
- Models
- Weighing and measuring instruments
- Drawing instruments
- The useful material concerning other subjects
- Surveying instrument
- Some modern equipment’s

**Advantages of Mathematics Laboratory**

- Help in
**creating an interest**in the students in the learning of mathematicians. - Help in
**making use of all the progressive methods**like inductive, analytic, laboratory, heuristic, and project methods in the teaching and learning of mathematics. - Help in the
**inculcation of scientific**, problem solving, and heuristic attitude among the students. - The
**theoretical concepts may be easily clarified**through practical demonstration. **Save the time and energy**of the teachers as well as students.**Help in training**the students for the practical application of mathematical facts and principles in their life.- Help in
**satisfying the creative and constructive urges of the students**.

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