# Arts Elective Modules (Mathematics) for B.Ed.

## OBJECTIVES:

This module aims at deepening the understanding of the concept of number. To achieve this, divisibility properties of integers will be studied, place value systems will be investigated and different number bases will be studied. The differences and common properties of different types of numbers (integers, rational, real and complex numbers) will be highlighted.
Numbers and their algebraic properties will play an important role in counting, solving equations and proving simple mathematical statements. This module will contribute towards a deep and robust understanding of number concepts for the student teacher.

## LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

• solve problems related to basic properties and to the representation of numbers of different types (integers, rational, real and complex numbers);
• solve equations;
• prove simple statements by using mathematical induction.

## MODULE CONTENT:

The number system hierarchy; key algebraic properties of addition and multiplication; subtraction and division as secondary operations. Integers; divisibility; elementary number theory.
Mathematical Induction; Counting.
Rational and irrational numbers; place value; finite and infinite recurring and non-recurring decimals; other number bases.
Complex numbers; quadratic and cubic equations.

## PRIME TEXT:

HUNTER, J.; MONK, D., Algebra and Number Systems (Blackie) 1971.

## OTHER RELEVANT TEXTS:

NIVEN, I.; ZUCKERMAN, H.S., An introduction to the theory of numbers (Wiley) 1980.

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## OBJECTIVES:

In this module, fundamental ideas for measuring length, area and volume are discussed. The close relationship between number concepts and measurement is at the core of this course. Mathematical modelling of problems based on real life situations will form an important part of this module. Mathematical tools such as limits, differentiation and integration will be studied so that they can be used to find numerical values of measures of length, area and volume.

## LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

• demonstrate knowledge and understanding of the main mathematical ideas involved in measuring lengths, areas and volumes;
• use the concept of function and apply differentiation and integration techniques to solve problems (for example min/max problems, surface area calculations).

## MODULE CONTENT:

measuring lengths; commensurability; relation to rational and irrational numbers;
limits of sequences and series;
functions; domain and range; sketching graphs;
functions as a tool to model reality;
limits of functions; rate of change; derivatives of functions; applications to optimisation problems;
measure of area and volume; integration;
Fundamental Theorem of Calculus.

## PRIME TEXT:

PRIESTLY, W.M., Calculus: An Historical Approach, New York, (Springer), 1979.

## OTHER RELEVANT TEXTS:

EDWARDS, C.H. Jr., The Historical Development of the Calculus, New York, (Springer) 1979.
HSIANG, W.Y., A Concise Introduction to Calculus, Singapore, (World Scientific) 1995.
GOWERS, T., Mathematics: a very short introduction, Oxford, (Oxford University Press) 2002.
WEISS, S., Elementary College Mathematics, Boston, (Prindle, Weber & Schmidt) 1997.

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## OBJECTIVES:

This module will provide an introduction to the theory of probability and to statistical techniques in a manner which will foster understanding of concepts and development of expertise in their applications.

## LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

• understand the background to statistical testing;
• apply this knowledge to statistical tests.

## MODULE CONTENT:

Description of sample data; graphical representation, measures of location and dispersion, mean and standard deviation.
Probability theory; applications.
Random variables; discrete and continuous, expectation and variance.
Probability distributions; binomial, Poisson and normal distributions.
Sampling theory; random sampling, sampling distributions.
Estimation; point and interval estimates, Student's t-distribution.
Hypothesis testing; error types.
Correlation and regression; least squares, errors.
Testing methods; chi square test, F-test, non parametric tests.

## PRIME TEXT:

HOEL, P., Elementary Statistics, (Wiley) 1976.

## OTHER RELEVANT TEXTS:

FRANK, H., Introduction to Probability and Statistics, (Wiley) 1974.
HOGG, R., CRAIG, A., Introduction to Mathematical Statistics, (Collier Macmillan) 1978.

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## OBJECTIVES:

This module aims at deepening the knowledge of classical Euclidean geometry. Another purpose of this module is to widen the perspective of the student so as to understand the place of Euclidean geometry within modern geometry. This module will contribute towards a deep understanding of basic geometric concepts and constructions for the student teacher.

## LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

• carry out and describe ruler-and-compass constructions with precision;
• use basic geometric facts to prove other geometric results;
• demonstrate understanding of geometry beyond classical Euclidean geometry.

## MODULE CONTENT:

• lines, angles, circles, area of geometric figures;
• geometric constructions;
• transformations, congruence, similarity;
• parallel lines, triangles, quadrilaterals, polygons;
• right triangles (Theorems of Thales and Pythagoras);
• special points in triangles;
• Euclidean geometry in 3D;
• Non-Euclidean geometry.

## PRIME TEXTS:

STILLWELL, J. (2005). The Four Pillars of Geometry. 1st edition. New York, NY: Springer.

LEVERSHA, G. (2008). Crossing the Bridge. UKMT.

## OTHER RELEVANT TEXTS:

BRUMFIELD, C. F., VANCE, I. E. (1970). Algebra and Geometry for Teachers, Addison-Wesley.

CLARK, D. M. (2012). Euclidean Geometry: A Guided Inquiry Approach, American Mathematical Society.

OSTERMANN, A., WANNER, G. (2012). Geometry by Its History, Springer.

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## OBJECTIVES:

In this module, basic problem solving techniques will be explained. The main emphasis is on understanding of the process of problem solving in its dynamic and cyclic nature. Therefore, a major component of this course will consist in giving the participants the opportunity to build up their own experience in solving mathematical problems. Most of the problems will not require mathematical knowledge beyond the school curriculum. One of the aims of this course is to enable the teacher student to provide support for the development of gifted children by exposing them to suitable interesting mathematical problems.

## LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

• tackle with confidence mathematical problems which require more than the straightforward application of a known procedure;
• apply basic problem solving strategies to unseen problems;
• generalise the results of a problem and suggest new problems;

## MODULE CONTENT:

• general techniques of mathematical problem solving;
• problem solving as a tool of mathematics instruction;
• solving problems;
• how to write a solution or a proof.

## PRIME TEXTS:

POLYA, G. (1981). Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving. Combined Edition. New York. John Wiley& Sons.

GAY, D. (1998). Geometry by Discovery. New York: John Wiley& Sons.

SCHOENFELD, A.H. (1985). Mathematical Problem Solving. London: Academic Press.

POLYA, G. (1973) How to sove it. Second Edition. Princeton, NJ: Princeton University Press.

## OTHER RELEVANT TEXTS:

GOWERS, T. (2002). Mathematics: a very short introduction. Oxford: Oxford University Press.

COURANT, R., ROBBINS, H. (1996). What Is Mathematics? An Elementary Approach to Ideas and Methods. Second Edition. New York: Oxford University Press.

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