If you examine closely the syllabus in H2 Maths, you will notice that a lot of topics are incomplete. Some of the missing portions take up a major component in the chapter and as a result, students find themselves not being able to fully understand the topic. The following will discuss all the missing materials that are supposed to be covered. The use of GC has reduced a lot of tedious work but the consequence is that students dont understand all the theoretical aspects in the topic.
(1) In Systems of Linear Equations, students are taught to use the GC to solve simultaneous equations in 3 or 4 unknowns using the POLY SIMUL EQUATION SOLVER in the GC. Students just need to enter the matrix elements into the GC and they can immediately obtain the results. The theory behind all these which is Gaussian Elimination is not explained at all. As a result, students will not fully appreciate and understand how the 3 outcomes(Unique Solution, Infinite No. of Solutions, No Solution) is obtained. This will in turn affect their understanding of the analysis in Intersection of 3 Planes in the topic Vectors.
(2) In the topic Recurrence Relations, students are taught how to find limit of sequences given a recurrence relation. Then questions on analysing the recurrence relation are asked. Most of them involve using a graphical or algebraic approach to study the behaviour of the sequence(increasing or decreasing) in certain intervals. However, a lot of JCs do not even bother to teach this topic properly and students are confused on whether is this topic important in the exam.
(3) In Mathematical Induction, students learn how to prove statements involving series and sequences. But they dont learn how to prove questions on divisibility which is an important concept in Number Theory and also induction involving nth order derivatives.
(4) In 1st Order Differential Equations(DE), students only study Separable DE without learning Method of Integrating Factor for 1st Order Linear Equations. They only need to know to to use a given substitution to convert a non-separable DE to a separable one. This topic turns out to be one of the easiest among all the Pure Maths topics. This topic doesnt really teach students any important concepts except how to separate the variables to different sides of the equations. 2nd Order DE is no longer in the syllabus and as a result students who intend to study Maths, Physics, Engineering do not have a strong foundation in DE which is very important in the abovementioned disciplines.
(5) In Complex Numbers, students learn de Moivre’s Theorem but they dont see how powerful it is in evaluating trigonometric series or integrating powers of trigonometric functions. Loci in Argand diagrams only involves sketching and finding maximum or minimum values of moduli or arguments. Transformation in complex plane is not taught at all.
(6) In Vectors involving Skew Lines, students learn how to distinguish skew lines from intersecting lines. But they dont study how to find the shortest distance between the two skew lines. Furthermore, intersection of 3 planes is not taught in details. Most of the schools only teach them how to use the GC to find the solutions. A lot of student face difficulties in solving questions involving 3 planes but GC is not allowed due to nature of question.
(7) In Binomial and Poisson Distributions, theory on Discrete Random Variables is omitted and students have no idea what is the definition of expectation and variance in statistics. They only blindly memorise the formulae for E(X) and Var(X) in both Binomial and Poisson Distributions. The same goes with Normal Distribution where theory on Continuous Random Variables is not taught at all. In fact most students dont even understand the difference between a Discrete and Continuous Random Variable.
(8) In Sampling Theory, students learn how to find Unbiased Point Estimates of Population Mean and Variance but Confidence Interval is omitted from the syllabus. The same happens in Hypothesis Testing where students only study hypothesis testing involving population mean but not population proportion. 2 sample tests is also not taught at all.
The above only lists out the major missing stuff in H2 Maths. There are some other minor issues which are not mentioned above.
The ‘A’ level H2 Maths was introduced in 2006 after MOE decided to remove Further Maths from the syllabus. The use of graphic calculators is allowed in exams to solve questions that cannot be done analytically. Most of them involve graphing techniques, solutions of equations, system of linear equations and finding point of intersection of 2 graphs and evaluation of definite integrals. The main function of the graphic calculator is still used in Statistics where students can use all the built in functions to work out probabilities for various distribution and hypothesis testing. However we can see from all the past year ‘A’ level exams that almost all the questions in Pure Maths section do not require them to use the graphic calculator. In fact some questions even prohibit students from using graphic calculators to work out the answers. We can see that when the questions specify “answers in exact value”, “without use of graphic calculators”, “by algebraic method” etc.
MOE should seriously take a closer look at the H2 maths syllabus and review it after 7 years has passed by. The standard of the H2 Maths is so much lower than the old Further Maths. A lot of topics which are very important had been taken away. Some students who have the passion and aptitude in Maths do not have the chance to study those topics anymore. Examples are further applications of integration, evaluation of integrals by reduction formula, mathematical induction involving divisibility, 2nd order differential equations, application of de Moivre’s Theorem in proving trigonometric series, linear algebra. Moreover the Mechanics option in Further Maths serves as a foundation for students who intend to study physics or engineering in university. The ‘A’ level Physics has a different focus compared to the Mechanics in Further Maths. It is more theoretical in nature and the syllabus has certain limitations.
In fact, the subject Further Maths is still being offered in UK but not in UCLES Singapore ‘A’ level exams. MOE should consider whether should Singapore students be taking the same exam as in UK so that all students get a chance to study Further Maths. The level of difficulty in H2 Maths is still considered as average compared to Further Maths. Most questions are about the same standard as the C Maths in the past. Moreover, the H3 Maths syllabus has got not much direct relations to H2 Maths. Some of the topics in H3 Maths are too advanced and only taught in advanced undergraduate maths courses in university.