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# Thoughts on the draft maths A Level content

So, before I had even read through the Government’s documents outlining the proposed changes to the A Level Mathematics and Further Mathematics courses I came across an excellent post by @srcav discussing the changes – I really need to learn to stay more up to date.

The Department for Education and Ofqual are currently in a period of consultation and seeking input on these proposed changes. The documents are available here, just scroll down the page to contribute to the consultation.

In this post I am going to outline my initial reactions to the changes and the proposed content upon reading the documents.

I think the linear model for maths, while perhaps being more challenging for the students, will be an improvement as it will emphasise the connected nature of the subject. I did like the element of choice in the previous A Levels, but prescribing 100% of the content of A Level Mathematics and 50% of A Level Further Mathematics should make it easier for university lecturers to understand where the students are if they decide to study mathematics at university. Like the new GCSEs there seems to be a greater emphasis on proof of various kinds. The lack of any content from the Decision modules surprises me and is a shame given the massive use of mathematics in Computer Science. I would have liked to have seen at least some discussion of algorithms and their computation cost, though perhaps this could be included in the numerical methods topics.

This draft document also says that the specifications must encourage students to

Read and comprehend articles concerning applications of mathematics and communicate their understanding.

This interests me; are we going to see some kind of mathematical comprehension element to the assessment of the new A Level courses?
The fact that 100% of the A Level is prescribed surely means that schools will be free to choose the exam boards for A Level Mathematics and A Level Further Mathematics independently. If there is a genuine difference in content between the boards for further mathematics I definitely think this is a good thing.

A Level Mathematics<:/span>

As mentioned above there is a greater emphasis on proof, and I like the explicit inclusion of the proof of the irrationality of $$\sqrt{2}$$ and the infinite number of primes. The omission of proof by induction is a surprise though (this is surely a mistake?!). The use of set theory notation is also mentioned as is the use of Venn diagrams – this must be a good thing for students going onto further study of mathematics.

I feel that it is a shame that no numerical methods are included in the AS content, but I am happy that one of the bits of knowledge in OT2 is for students to understand that many mathematical problems cannot be solved analytically. This is something that I did not appreciate fully when I was an A Level student, and as a numerical analyst by training believe is a key bit of knowledge.

I am intrigued by the requirement that students must

become familiar with one or more specific large data sets in advance of the final assessment (these data must be real and sufficiently rich to enable the statistical concepts in the specification to be explored.

This, I am sure, will place a greater load on the exam boards to provide meaningful examples for the students to explore in the statistical aspects of the course. Of course, this should hopefully also lead to these aspects being a bit less “dry” than the present S1 content.

Content A9 is nice as it explicitly links transformation of graphs to the transformations of the graph of the Normal probability function $$N(\mu,\sigma^2)$$ – this is something that in my experience, very few undergraduate and postgraduate students appreciate, let alone A Level students.

The trigonometrical content of the A level appears to have an increased focus on the geometrical proof of trigonometric identities – I guess this is in common with the new GCSE syllabus.

As a former numerical analyst I am sad to see no numerical methods included in the AS Level, however the inclusion of the Newton-Raphson method is encouraging as we could discuss some interesting properties of this method and its application. To me, it seems a shame that numerical differentiation isn’t included (though it is included in the Further Maths content) as this can be linked nicely to differentiation from first principles.

Those that know me, will know that one of my “pet niggles” with A Level is the explicit teaching of the quotient rule for differentiation so I am disappointed to see that this is included. I’m not sure I know any practicing mathematicians who would (if they had to differentiate by hand) use the quotient rule: Instead for example, they would consider $$f(x) = \frac{x^2 + 3}{(x+1)^3}$$ as $$f(x) = (x^2+3)(x+1)^{-3}$$ and use the product rule. The use of the quotient rule seems more error prone, I imagine due to the negative sign preceding one of the terms, than the product rule. The resulting expressions also tend to be more difficult to simplify.

In the mechanics component of the A Level Course it is encouraging to see the use of calculus for kinematics in 1 dimension. It’s a shame that currently this tends to be in M2 or higher and so isn’t often studied by pupils taking just the core A Level as M1,S1 and D1 are commonly chosen from to build up the “applied” component of the A Level. I feel that this will prepare students better for the applied aspects of university level mathematics.

Further Mathematics A Level:
The content for the 50% prescribed component further mathematics A Level seems pretty much the same as contained in FP1-FP3 currently.
I am pleased to see the inclusion of Viète’s formulae relating the roots of a polynomial to its coefficients as I think these are really neat results that can be explored in a nice way (Could the students derive these?).

The section on matrices I think is disappointingly standard. I had hoped that the content on matrices would be expanded due to their importance in computational and applied mathematics. I am always disappointed that the “matrix of minors” approach is taught to find the inverse of a 3 by 3 matrix – this is almost never used in practice and doesn’t enhance understanding in any real way, that I can see anyway. The row-reduction approach is easier, less prone to errors and can be used to build up to a practical method for solving linear systems of an arbitrary number of unknowns. The omission of eigenvalues and eigenvectors is also sad. I hoped these would have been prescribed as core content since some interesting results can be obtained.

I am pleased with the general emphasis (throughout both courses) of mathematical modelling and this is reflected in the prominence given to differential equations in the prescribed content for the Further Mathematics A Level. This should prepare students well for further mathematical study.

If the exam boards are prepared to be brave and have significant differences between them the 50% unprescribed content could allow students to study some more unusual and topical areas of mathematics such as mathematical biology, coding theory, cryptography. These topics could be introduced at the required level, though I guess this would put additional pressure on teachers who may not have studied these areas themselves.