Abstract

1. Mathematics Curricula at Higher Secondary Level: A Review and ProposalsbyM. Etezaz IbrahimSir Syed University of Engineering and Technology , Karachi, Pakistan.

2. Abstract • This study reviews the mathematics curricula in Pakistan at Higher Secondary level and presents proposals for improvement. • A review of mathematics curricula both at secondary and higher secondary level shows that some very important topics are missing though they are very necessary from the applications point of view. • All these curricula are already very heavy and cannot bear more topics to be added in them. Instead, it is proposed here to add one/two more papers of mathematics at the higher secondary level for the students of all disciplines including biology, arts and commerce.

3. Some of the topics to be added may be the following. • 1.Data analysis (interpretation and forecasting techniques) and Fundamentals of Statistics. • 2. Computational and Numerical techniques • 3. Discrete Mathematical Structures. • 4. Geometry of Solid Bodies. • 5. Observation and Monitoring of the Skies • 6. Dynamics of Physical and Financial System. • 7. Use of Mathematical Soft Wares. • .

4. Introduction • It is a bitter fact that knowledge of mathematics of students entering at BS level is constantly deteriorating. • This declination is a consequence of the decline of the knowledge of mathematics at SSC level and HSC level. • This is because the recent developments in mathematics are not incorporated accordingly

5. There are two different courses of mathematics at secondary level • (i) General Mathematics course which is compulsory for general group ( for students who do not select science group). In future they study social sciences and commerce. • (ii)Mathematics for science students ( for the students who select engineering sciences, medical sciences and computer sciences ). In future for their higher secondary studies they opt Sciences like physics, chemistry (and now biology too).these subjects demand mathematics to perform better evaluation, analysis, forecasts and measurements.

6. Many social science subjects such as political science, history and population sciences involve more delicate mathematics for their evaluation, analysis, forecasts and measurements. • It is the need of the time that the tradition of a light course of mathematic for social science students and a stronger course for science students should be discarded • A harmony among the courses of mathematics for all the subjects at higher secondary level is required • In this perspective it seems necessary to introduce mathematics at higher secondary level in all the disciplines.

7. In the past four decades computational sciences are developed enormously in theory and practice. In addition, computational facilities are far more developed and easily available. • Computational techniques are also improved in nature and utility. Modeling and simulation of natural and anthropogenic phenomena are now easy to handle. Different software and associative libraries are frequently available. But all these require a larger and most developed knowledge of mathematics. • Mathematics has changed a lot in its basic structure and applicability. Now mathematics has advanced from crisp structures to fuzzy structures. Randomness can be explored in more detail now.

8. Nonlinearity now can be utilized with more perfection than in the past for the analysis of phenomena. Discreteness now can be studied at the least levels. • Fractals geometry and methods now needed more and in the comprehension of a larger number of minds. So the basic knowledge of mathematics needed now need larger platforms. • Therefore it is an urgent need to add more contents in our existing courses of mathematics at higher secondary level. The current courses are already overloaded. • Thus it is needed to increase the number of papers of mathematics at higher secondary level. The existing number of papers at higher secondary level is two. It is therefore proposed to add more papers of mathematics at higher secondary level.

9. Instead, it is proposed here to add one/two more papers of mathematics at the higher secondary level for the students of all disciplines including biology, arts and commerce. • In these courses. This study indicate the weaknesses of mathematics courses at the HSC and SSC level and proposed to add additional content of mathematics in both these courses at appropriate levels • This is because the recent developments in mathematics are not incorporated accordingly. • Reviewing the secondary level mathematics courses it can be easily observed that there are two different courses of mathematics.

10. Comparison of Mathematics Courses at SSC Level

11. Following observation can be made at a glance. 1. Sets are not included in GM 2. Contents of the theory of numbers included in GM are insufficient. • The contents of chapter on matrices must be the same in GM and M courses. • The contents of the chapter on statistics should be same in both the courses as it is equally needed for the students of natural and social sciences.

12. 5. Comparing the geometry portions of GM and M it is obvious that the contents of geometry included in GM are deficient in demonstrated geometry and the geometry of circles. Demonstrative geometry provides a basis to logical reasoning, deductions and comprehensive presentations. These skills are necessary for the students of social sciences also. It is strange to observe that why the geometry of circles is not considered to be important for social sciences students. Now it is well known that simple geometrical elements like lines, triangles, squares and circles can be used to study patterns involved in social life, politics, population, geography and fine arts. Rotation, translation, and reflection are some very simple but important mathematical tools involving symmetry, interlacing, gridding and self similarity that can be utilized to study social structure and distribution

13. 6. Trigonometry is an important area of mathematics which is the basis for various analytical tools such as Fourier analysis. These analyses are equally important for social scientists and natural scientists. So trigonometry should be included in the course of mathematics for arts students also. It is further suggested that in addition to plane trigonometry concepts of spherical trigonometry should also be provided to all the students.

14. Mathematics Courses at Higher Secondary School (HSC) Level

16. We have mentioned above that in the past four decades computational facilities are very much increased and mathematics itself has changed greatly in its structural foundation and applicability. Consequently, it has increased the applicability of those areas of mathematics which were considered hard from the applications point of view. • Fuzzy sets have replaced crisp sets, nonlinearity has widened its scope, discrete structures have gained popularity instead of analytical structures, randomness has surpassed determinism, non Euclidean (fractal) geometry has overcome the Euclidean geometry and simulation and modeling a large software and computational support. • These applications are not limited to natural sciences. Social sciences like history, political science and education now utilize strong analytical techniques and forecasting methods. So social science subjects cannot be left deficient with mathematics.

17. In view of the above mentioned contents of mathematics courses at HSC level and a review of the more applicable areas of recently developed mathematics it can be easily observed that the courses are deficient. It is now necessary to include additional topics in these contents. • Some such topics may be data analysis (interpretation and forecasting techniques), statistical techniques, computational and numerical methods, discrete mathematics, geometry of solid bodies and fractals, astronomy, dynamics and some most necessary softwares. • Astronomy is necessary because “the future is in skies”. Dynamics because physical principals of dynamics are equally useful and applicable on other systems such as monitory and financial systems and many forecasting techniques are developed using dynamical systems. • Finally, it is to point out that the same mathematics courses should be studied by all the students irrespective of their disciplines (mathematics or biology, social sciences and arts). To balance the marking plan mathematics students can be provided with papers covering topics from social and biological sciences like philosophy and genetics.

18. Some images using Geometry

19. Some images (Architecture)

20. Some images (Nature)

21. Some images of Geometrical patterns A family of geometric tiles , derived from the octagram which through different combinations can generate a designs.

22. Some images of Geometrical patterns in Sindh Concentric Circles are prominent in Chaukundi Graveyard

23. Some images of Geometrical patterns in Sindhdifferent Geometrical Patterns are prominent in Chaukundi Graveyard

24. Conclusion • The following conclusions are made in view of the above discussion. • In general, the content of mathematics courses at SSC and HSC level are deficient in the sense that they don’t make any mention of recent advancement in mathematics. • Contents of General Mathematics (GM) courses at SSC level are deficient in providing a variety of topics • GM courses at SSC level are in itself deficient in contents • HSC courses in mathematics are deficient in contents from the point of view the modern developments in mathematics. • At HSC level mathematics is not included in the disciplines of social sciences and arts.

25. Recommendations • Following recommendations are made here. • The same course of mathematics should be provided to all the groups at SSC level. • In addition to idea of real number the idea of complex numbers should also be introduced at secondary level. • The concept of Fractals should also be introduced at secondary level. • Two additional papers of mathematics should be included at HSC level. • Theses content of these two additional papers can be sorted out from the following topics.

26. Data analysis (interpretation and forecasting techniques) and Fundamentals of Statistics, Computational and Numerical techniques, Discrete Mathematical Structures, Geometry of Solid Bodies, Observation and Monitoring of the Skies and meteorological systems , Dynamics of Physical and Financial Systems, Use of Mathematical Softwares • At HSC level mathematics must be compulsory for all discipline (Sciences including biology group, social Sciences, and arts) it is expected that the proposal s here will get due consideration by educationist and educational department for the perusal of the encasement and up gradation of mathematics and mathematics education.

27. References • (2015) Mathematics for class IX and X, Sindh text book, Jamshoro • (2014) General Mathematics for class IX and X, Sindh text book, Jamshoro • (2016) Mathematics for class XI, Sindh text book, Jamshoro • (2015) Mathematics for class XII , Sindh text book, Jamshoro • John R. Durbin, (1979). “Modern Algebra”, United States of America • D. K. Washburn and D. W. Crowe (1998), “Symmetries of Culture” University of Washington Press • A.V. Shubnikov (1974), “Symmetry in Science and Arts” Plenum Press