Course title: Foundation of Mathematics Full marks: 100
Course No.: Math Ed. 416 Pass marks: 35
Nature of the course: Theory Periods per week: 6
Level: B.Ed. Total periods: 150
Year: First Time per period: 55 minutes
1. Course Description
This is a specialization course offered to the students majoring Mathematics Education in B.Ed. The main aim of this course is to develop an in-depth understanding of different aspects of mathematics and statistics. The first five units of the courses deal with symbolic logic, infinite sets, number theory, graph theory and linear programming, and the last four units cover correlation and regression, probability distributions, sampling distributions and test of hypothesis
2. General Objectives
The general objectives of this course are as follows:
· To enable the students in demonstrating the ability of giving judgments with the help of logic.
· To develop ability in students to prove the characteristic properties of infinite sets and fundamental theorem of arithmetic.
· To make the students able in investigating and establishing important properties in arithmetic.
· To provide in depth knowledge to the students regarding classification of graphs, proving simple properties of these graphs and discussing their applications.
· To enable the students in solving problems on linear programming.
· To impart practical knowledge and skills in deriving properties of correlation and regression and applying them to solve problems.
· To make the students familiar with different discrete and continuous probability distributions.
· To make the students able to use sampling distribution and estimation, and use test of hypothesis in research work.
3. Specific Objectives and Contents
|
Specific
Objectives |
Contents |
|
Identify
tautology and contradiction. Describe
the features of different methods of examples. Determine
the validity of arguments. |
Unit I: Symbolic logic (8) 1.1
Conditional 1.2
Bi-conditionals 1.3
Algebra of propositions 1.4
Negation of compound statements 1.5
More connectives and their truth values and
truth tables 1.6
Tautology and contradiction 1.7
Validity of arguments 1.8
Use of Euler diagram 1.9
Deductive proof and their application. |
|
Describe
countable and uncountable sets with examples. Prove
the characteristic properties of infinite sets. Prove
Cantor's theorem and cardinality. |
Unit II: Infinite sets
(9) 2.1
Denumerable sets 2.2
Countable sets 2.3
Uncountable sets 2.4
Cardinality of infinite sets 2.5
Cardinal arithmetic 2.6
Cantor's theorem 2.7
Schoeder Bernstein theorem 2.8
Continuum hypothesis. |
|
State
the divisibility theorem of integers with examples. Prove
the properties of primes and unique factorization theorem. Describe
congruence modulo and apply it for divisibility tests. State
and prove Fermat's little theorem, |
Unit III: Number Theory (20) 3.1 Divisibility
theory 3.1.1 Division algorithm 3.1.2 Euclidean algorithm 3.1.3 Diphantine equation 3.2 Primes
and their distributions 3.2.1 Unique factorization theorem 3.2.2 Goldbach's conjecture 3.3 Theory
of congruence 3.3.1 Properties of congruence 3.3.2 Divisibility tests 3.3.3 Linear congruence 3.3.4 Fermat's little theorem and Wilson's theorem 3.3.5 Euler's theorem |
|
-Describe
traversability of Eulerian and Hamiltonian graphs and apply them. - Prove
the properties of trees, planar graphs and directed graphs. -Apply the
concepts of graph theory to solve different problems. |
Unit IV: Graph Theory (30) 4.1 Basic concepts 4.1.1 Complete graph 4.1.2 Bipartite graphs 4.1.3 Connectivity 4.1.3 Sub graphs 4.1.4 Metric representation of graphs 4.1.5 Isomorphic graphs 4.2 Travers ability 4.2.1 Eulerian & Hamiltonian graphs 4.2.2 Properties and applications 4.3 Trees 4.3.1 Properties of trees 4.3.2 Spanning tree and minimal spanning
trees 4.3.3 Rooted and binary tree 4.4 Planar graph 4.4.1 Properties & theorems of planar
graphs 4.4.2 Graph coloring 4.5 Directed graphs 4.5.1 Diagraph 4.5.2 Connectivity 4.5.3 Traversability 4.5.4 Tournament 4.5.5 Traffic flows. |
|
Solve
the linear programming problems using simplex methods. Use
duality principle for minimization problem. Apply
the properties of basic feasible solutions. |
Unit V: Linear Programming (8) 5.1
Formulation of linear programming problem in
three or more variables; simplex method 5.2
Dual problems 5.3
Basic feasible solution and application of
LPP. |
|
Explain
the concepts of correlation and regression. Derive
the properties of correlation and regression. Apply
correlation and regression to solve problems. |
Unit VI: Correlation and regression (6) 6.1
Correlation 6.1.1
Pearson's
correlation 6.1.2
Rank
correlation 6.1.3
Probable
error and properties of correlation 6.2
Regression 6.2.1
Equation
of regression 6.2.2
Angle
between regression lines 6.2.3
Properties
6.2.4
Standard
error of estimate. |
|
-Describe probability distribution and distribution function of discrete and continuous variables. -Derive mean and variance of binomial distribution and normal distribution. -Use binomial table and normal table. -Prove Chebyshev's theorem. - Solve problems on binomial and normal distribution. |
Unit: VII Probability Distributions (15) 7.1
Axioms
of probability, some theorems on probability, Baye's theorem. 7.2
Discrete
random variables: probability distribution, cumulative distribution,
mathematical expectation, moments, mean and variance; uniform distribution
and binomial distribution:– mean and variance, binomial probability table,
recurrence relation. 7.3
Continuous
random variables: probability density, cumulative distributions, mean and
variance, Chebyshev's theorem and laws of large numbers; Normal distribution
– properties, mean and variance, area under standard normal curve, normal
approximation to binomial distribution. |
|
·
Apply
the concepts of standard error of the mean and central limit theorem. ·
Estimate
the population mean for large and small samples. |
Unit: VIII Sampling distribution and Estimation (20) 8.1
Parameter and statistics 8.1.1 Sampling distribution of mean, variance
and chi- Square, 8.1.2 Standard error of statistics (concept
only), 8.1.3 Central limit theorem (concept
only). 8.2 Estimation of parameters 8.2.1
Confidence interval for mean (difference between means) 8.2.2 variance |
|
·
Describe
the basic concepts underlying hypothesis test. ·
State
the steps in hypothesis tests. ·
Test the
significance of difference between two means for large samples. ·
Test the
significance of difference between two means for small samples. ·
Apply
chi-square significance test of independence. ·
Apply
significance test for correction coefficients. |
Unit IX: Test of Hypothesis (34) 9.1 Basic concepts 9.1.1 Null hypothesis 9.1.2 Alternative hypothesis 9.1.3 One - tailed and two - tailed tests 9.1.4 Type I and Type II errors 9.1.5 Level of significance 9.1.6 Critical region 9.1.7 Value 9.1.8 Test statistics 9.1.9 Steps in hypothesis testing. 9.2 Z-test: difference between two means of
large samples with unknown population variance. 9.3
T-test: difference between two means of
small samples with unknown common variance. 9.4
Chi-square test: significance test of
independence. 9.5
Significance test for correlation
coefficient. |
Note: The figures in the parentheses indicate
the approximate periods for the respective units.
4. Instructional Techniques
Because of the theoretical nature of the course, teacher-cantered instructional techniques will be mostly used in teaching learning process. The teacher will adopt the following methods/techniques.
4.1 General Instructional Techniques
· Lecture and illustration.
· Discussion
· Demonstration.
4.2 Specific Instructional Techniques
· Inquiry and question answer.
· Individual and Group work/project
· Report writing and classroom presentation
5. Evaluation
Students will be evaluated on the
basis of the written classroom test in between and at the end of the academic
session, the classroom participation, presentation of the reports and other
practical activities. The scores obtained will be used only for the feedback
purposes. The Office of the Controller of Examinations will conduct the annual
examination to evaluate student’s performance. The types, number and marks of
the subjective and objective questions will be as follows.
|
Types of questions |
Total questions to be asked |
Number of questions to be answered and marks allocated |
Total marks |
|
Group A: Multiple choice items |
20 questions |
20 x 1 mark |
20 |
|
Group B: Short answer questions |
8 with 3 'or' questions |
8 x 7 marks |
56 |
|
Group C: Long answer questions |
2 with 1 'or' question |
2 x 12 marks |
24 |
6.
Recommended Books and References
Recommended
Books
Freund, J. E. (2006). Modern elementary
statistics.
Maharjan, H. B. & Sharma, L.
N. (2008). An introduction to graph theory.
Maskey, S. M. (1998). First
course in graph theory.
Maskey, S. M. (2005). Introduction
to modern mathematics. Vol I & II.
Pandit, R. (2008). Elementary
modern mathematics.
Pandit, R. (2008). Mathematical statistics.
References
Crawshaw, J. & Chambers,
J. (2002). Advanced level statistics.
Freund, J. E. (2001). Mathematical statistics.
Garrett, H. E. & Woodworth,
R.S. (2000). Statistics in psychology and education.
Koshy, T. (2005). Elementary number theory with applications.
Pokharel, T. R. (2062). Fundamentals of number theory with
application.
Upadhaya, M. P. (2000). Introduction
to linear programming.
West, D. B. (2002). Introduction
to graph theory 2nd edition. Pearson Education
Wilson, R. J. (2002). Introduction to graph theory (4th ed. , 2nd Indian reprint).
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