Learning The Language Of Mathematics By Jamison Pdf

PEDAGOGY OF MATHEMATICS | Teaching of Mathematics | Pedagogy of Maths | Teaching Of Maths

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.

LearningClassesOnline B.Ed Notes

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)

Sample Micro Teaching B.Ed Lesson Plans For Maths (All Skills)

S. No.	Name Of The Skill	Link
1	Writing Instructional Objectives In Behavioral Terms Lesson Plan	Click
2	Introducing A Lesson	Click
3	Blackboard Writing Skill Lesson Plan	Click
4	Fluency In Questioning Micro Lesson Plan	Click
5	Probing Questioning Lesson Plan	Click
6	Stimulus Variation Skill Lesson Plan for Micro Teaching	Click
7	Explaining Skill Lesson Plan	Click
8	Demonstration Skill Lesson Plan	Click
9	Reinforcement Skill Of Microteaching Lesson Plan	Click
10	Achieving Closure Skill Lesson Plan	Click

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 Examples.
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.

Author Remarks:

PEDAGOGY OF MATHEMATICS Is A Subject Taught In B.Ed And In Some Other Teaching Courses Also. On This Page, You Will Find Teaching of Mathematics Short Examination Notes And Downloadable Free PDF Book In English Medium For B.Ed First Year And Second Year and Semester 1, 2, 3, 4, 5, and 6. Here We Have Covered Some of The Main Topics and Important MCQ Questions of Pedagogy of Maths ( Teaching of Maths) Which Will Really Help in Your Exam Preparation and Also You Can Make Your Assignment Report and File for BEd Very Easily with The Help of These Notes. These Notes and Free PDF Book on Pedagogy of Math ( Teaching of Math) Subject Will Be Helpful for All the Students and Teachers of Any College or University. We Have Also Suggested Some of the Best Reference Books and Study Material PDF for PEDAGOGY OF MATHEMATICS That you can Also Go Through. Students and Teachers Preparing for All The Teaching Exams Like CTET, TET, UPTET, HTET Can Also Learn With The Notes Provided Above.

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Further References:

What Is Mathematics? [In Simple Words]
What Is The Definition Of Mathematics?
Characteristics And Features Of Mathematics
Aims Of Teaching Mathematics
What Is The Main Focus Of Mathematics Education? | Objectives Of Teaching Mathematics
9 Reasons Why Teaching Math Is Important? | Importance of Mathematics Teaching
Brief History Of Mathematics | Evolution, And Development Of Mathematics Over The Years
Relation Of Mathematics With Other Subjects