Mathematics, the science of calculation and logical reasoning, has played a decisive role in building up every civilization. Mathematics, like every thing else that man has created, exists to fulfill not only certain needs and desires but also helps in faster development in this digital age. So it also occupies an important place in the curriculum. The book written in simple language may be found useful by its users.
Chaman Lal Banga, M.A., M.Ed., NET & J.R.F. (Education) has been associated with teaching profession for the last seven years. He taught to B.Ed. students in Himachal Pradesh and is presently working with Department of Education, Government of Himachal Pradesh at Chamba, HP.
Preface
SECTION I
1. Meaning, Nature and the Scope of Mathematics
Introduction/1; Definitions of Mathematics/1; Meaning of Mathematics/2; Nature of Mathematics/3; Elements/5; Axioms or Postulates, Propositions and Theorems/5; Scope of Mathematics/11; Basic Mathematics/11; Applied Mathematics/11; Modern Mathematics/12
2. History of Mathematics
Introduction/16; History of Metric System/19; The Babylonians/20; Writing and Base 60 System /20; Multiplication and Division /20; Pythagorean Triples and Problems/20; Egyptians and Romans/21; Number System/21; Egyptians and Romans/21; Multiplication/21; The Notation System /21; Contribution to Mathematics by India/22; Zero and the Place-Value Notation /22; Vedic Mathematics and Arithmetical Operations/24; Geometry of the Sulba Sutras /25; Jaina Contribution to Fundamentals of Numbers /26; The Anonymous Bakshali Manuscript/26; Astronomy/27; Classical Contribution to Indeterminate Equations and Algebra /27; Indian Trigonometry/29; Kerala Contribution to Infinite Series and Calculus/30; Modern Contribution/30
3. Contribution of Indian Mathematicians (Bhaskaracharya, Aryabhatta, Lilavati, Ramanujan)
Bhaskaracharya II/34; Lilavati (A Book on Arithmetic)/36; Bhaskara's Contributions to Mathematics - An Overview/39; Preliminary Concept of Mathematical Analysis /39; Arithmetic/40; Definitions/40; Algebra /40; Positive and Negative numbers/41; Trigonometry/41; Calculus /41; Astronomy /42; Aryabhatta Ist/43; Aryabhatta's Contribution to Mathematics/44; Aryabhatiya/44; Place Value System and Zero/45; Pi (p) as Irrational/45; Mensuration and Trigonometry/46; From "ardha-jya" to "sine"/46; Indeterminate Equations/46; Astronomy/47; Motions of the Solar System/47; Eclipses/48; Sidereal Periods/48; Legacy/48; Aryabhatta Numeration/49; Its Basis/50; Srinivasa Ramanujan/50; Life Sketch/50; Contacting English Mathematicians/52; Contribution of Ramanujan to Mathematics/53
4. Contributions of Euclid, Pythagoras, Rene Descartes
Euclid/56; The Elements/56; Other Works /57; Pythagoras/61; Life Sketch/61; Musical theories and investigations/62; Quotations by Pythagoras/62; Rene Descartes/63; Life Sketch/64; Philosophical Work/65; Mathematical Legacy/65; Descartes' Greatest Contribution to Mathematics /66; Descartes Observations in Detail/66
5. Educational Values of Mathematics
6. Aims and Objectives of Teaching Mathematics in Terms of Instruction and Behaviour
What is Aim or Goal?/75; What an Objective is?/75; Examples of Aims/76; Difference between Aims and Objectives/76; Classification of Aims of Teaching of Mathematics/77; Educational Objectives/77; General Aims of Teaching Mathematics/78; Taxonomy of Instructional Objectives/78; Behavioural Objectives/79; Taxonomy of Educational Objectives/79; Need for Writing Objectives in Behavioural Terms/79; Methods of Writing Objectives in Behavioural Terms/79; Robert Mager's Approach/79; A List of Associated Action Verbs for the Cognitive Domain/82; List of Action Verbs for Affective Domain/82; List of Action Verbs for Psychomotor Objectives/82; Objectives at the Entire School Stage/83; Knowledge and Understanding Objectives/83; Skill Objectives/83; Application Objectives/83; Attitude Objectives/84; Appreciation and Interest Objectives/84; Objectives of Mathematics at the Secondary Stage/84; Formulation of Objectives/85; Writing Objectives in Behavioural Terms/86; Creativity Objectives/87; Different Methods of Writing Objectives in Behavioural Terms/87
7. Relation of Mathematics with Other Subjects
The Relation of Mathematics with Other Subjects/90; Relation of Mathematics with Physical and Biological Sciences/90; Relation between Mathematics and Physics/90; Relation between Mathematics and Chemistry/91; Relation between Mathematics and Biology/91; Relation between Mathematics and Engineering/92; Relation between Mathematics and Agriculture/93; Relation between Mathematics and Social Sciences/93; Relation between Mathematics and Economics/93; Relation between Mathematics and Psychology/94; Relation between Mathematics and Philosophy/95; Relation between Mathematics and Logic/96; Relation between Mathematics and Geography/96; Relation between Mathematics and History/97; Relation between Mathematics and Fine Arts/97; Relation between Mathematics and Drawing/97; Relation between Mathematics and Languages/97; 16. Relation between Mathematics and Commerce/98
8. Mathematics in School Curriculum and in Everyday Life
Mathematics in Everyday Life/99
SECTION II
9. Methods of Teaching Mathematics
What is Method?/102; Lecture Method /103; Demonstration Method/104; Inductive-Deductive Method/104; Inductive Method/104; Deductive Method/108; Analytic-Synthetic Method/109; Synthetic Method/112; Laboratory Method/115; Project Method/118; Lecture-cum-Demonstration Method /120; Heuristic Method/120; Problem-Solving Method/122
10. Curriculum in Mathematics
Rationale of Curriculum Development/126; Meaning of Curriculum/126; Definition of Curriculum/127; Principles of Curriculum Construction in Mathematics/127; Need for Planning of Curriculum in Mathematics/132; Preparing the Syllabi according to the Age Groups of Children/132; The Dynamic Approach to Curriculum/132; The Teacher's Concern in Curriculum Development/132; Objectives of Curriculum/133; Organisation of Curriculum/133; Place of Learning Experiences in the Mathematics Curriculum/134; Basis of Curriculum Development in Mathematics/134; Existing Syllabi of Mathematics/138; Primary Level/138; High and Higher Secondary Level/138; Critical Evaluation of a Syllabus/141; Suggestions for Improvement in the Existing Mathematics Curriculum/143; Critical Study of Mathematics/143; Syllabus of 10+2 Class (CBSE/PSEB)/143; Conclusion/147
11. Mathematics Library and Laboratory
Mathematics Library/149; Need and Importance of Mathematics Library/149; Library Facilities in the School/150; Need of a Separate Mathematics Library/151; How to Use Mathematics Library/151; Material for Mathematics Library/152; Mathematics Laboratory/154; Need and Importance of Mathematics Laboratory/154; Equipment of the Mathematical Laboratory/155; Summary of Effective Use of Blackboard Writing/156
12. Mathematics Club
Need and Importance of Mathematics Club/161; Organisation of Mathematics Club/162; Office Bearers of Mathematics Club/162; Activity of the Club/163
SECTION III
13. Unit Plan and Lesson Plan in Mathematics
Unit Planning/165; Definitions of Unit Plan/165; Types of Unit Plan/165; James Michael Suggests Two Basic Types of Unit Plan/165; Major and Minor Units of Teaching/166; Teaching Unit Plan: Guidelines/166; Elements of a Unit Plan/167; Morrison's Steps in Unit Planning/167; Essential Characteristics of Unit Plan/167; Merits of a Teaching Unit Plan/168; Limitations of Unit Plan/168; Principles Involved in Unit Planning/168; Example of Unit Plan/168; A Sample Unit Plan/169; Steps of a Unit Plan/170; Lesson Planning /171; Definitions of Lesson Plan/171; Need and Importance of Lesson Plan/171; Principles of Lesson Planning/172; Another Classification of Principles of Lesson Planning/172; Levels of Lesson Planning/172; Types of Lessons/172; Types/172; Features of a Lesson Plan/172; Different Approaches of a Lesson Plan/172; Herbertian Steps of Lesson Planning/173; The Project Approach of Lesson Planning/174; Morrison's Approach of Lesson Planning/174; Characteristics of a Good Lesson Plan/174; Demerits of Lesson Plan/175; Sample Lesson Plan/175
14. Teaching Aids in Mathematics
Meaning of Teaching Aids/179; Definition of Teaching Aids/179; Place of Teaching Aids in Mathematics/180; Need of Teaching Aids in Mathematics/180; Importance of Teaching Aids/180; Suggestions for Effective Use of Teaching Aids/181; Effective Use of Teaching Aids/181; Principles of the Teaching Aids/183; Classification of Teaching Aids/184; Important Characteristics of Audio-Visual Aids/185; Important Teaching Aids Used in Teaching-Learning Process/185
15. Mathematics Teacher
Qualities and Characteristics of a Mathematics Teacher/192; Qualities in a Mathematics Teacher/193
SECTION IV
16. Techniques of Teaching
Techniques of Teaching Mathematics/196; Self-Study/196; Drill Work /197; Supervised Study /199; Home Work/201; Oral Work/203; Written Work/204; Group Work or Group Study/204; Review/205
17. Textbooks of Mathematics
Meaning of Textbook/207; Need and Importance of Textbook/207; Importance for the Teachers/207; Importance for the Students/208; Importance for Maintaining Uniformity of Standard/208; Characteristics of a Good Textbook/209
18. Teaching of Modern Mathematics
Understanding Modern Mathematics/212; Essentials of Modern Mathematics/212; Objectives of Teaching Modern Mathematics/212; Understanding Modern Mathematics/212; Classification of Sets/215; Venn Diagram /218
19. Gifted and Backwards Children in Mathematics (Enhancement Programmes)
Gifted Children/220; Definition of Giftedness/220; Identification of Gifted in Children in Mathematics/220; Enrichment Programme for Gifted Students/221; Suggestions for Teaching the Gifted Children/221; Backward Child/222; Meaning /222; Causes of Backwardness/222; Remedies for Overcoming Backwardness/225; Backward Students in Mathematics/226; Identification of Backward Students/226
SECTION V
20. Evaluation in Mathematics
Evaluation-Meaning/228; Principles of Evaluation/228; Definitions of Evaluation/229; Purpose of Student Evaluation/229; Evaluation Process/230; Comprehensive Evaluation /231; Purposes of Evaluation in Mathematics/231; What is the difference between Examination and Evaluation/231; Procedure for Test Construction/231; Planning the Test/232; Purposes of Educational Tests/232; Types of Tests/233; Tools and Techniques of Evaluation/233; Testing Procedure/234; Self-reporting Tests/234; Observational Tests/234; Projective Tests/234; Characteristics of Good Evaluation Programme/234; Situations of Evaluation/235; Preparation of Evaluation Programme/235
21. Teaching of Mathematics - Shortcomings and Remedial Measures
Defects and Possible Remedies/237
22. Creating and Maintaining Interest in Mathematics
Tree of Knowledge/241
SECTION VI
23. Teaching of Arithmetic
Aims of Teaching Arithmetic/246; Major Objectives of Teaching Arithmetic/247; General Rules for Teaching of Arithmetic/247; Methods of Arithmetic Teaching/247; Teaching of Some Important Topics in Arithmetic/247
24. Teaching of Algebra
Objectives of Teaching Algebra/257; Importance of Teaching Algebra/257; How to Introduce Algebra to the Students/258; Methods of Teaching of Algebra/258; Teaching of Different Topics of Algebra/259; Teaching of Addition in Algebra /259; Teaching of Substraction in Algebra/259; Teaching of Multiplication in Algebra/260; Teaching of Division in Algebra /260; Teaching of linear equations/261; Teaching of Factorization of Algebraic Expressions/261; Teaching of Linear Equations in Two Variables/262; Teaching of Method of Completing the Square/263; Method of quadratic formula /263
25. Teaching of Geometry
Stages of Teaching Geometry/265; Types of Geometry/265; Demonstrative Geometry /265; Practical Geometry /266; Historical Background of Geometry/266; Aspects of Teaching of Geometry/266; Need and Importance of Teaching Geometry/266; Different Stages of Teaching Geometry/267; Objectives of Geometry Teaching/267; Teaching of Some Important Geometrical Topics/267
26. Teaching of Trigonometry
Method of Approach/271; Trigonometric Formulas and Relationships/271; Teaching the Functions of Special Angles/272; Functions of the General Angles/273; The Trigonometric Ratio/273; Teaching of Solution of Triangle/274; Fundamental Identities Used in Trigonometry/276