Christmas Calculated Colouring 2019

It is that time of year again…..

Here is this year’s A-Level calculated colouring – I hope your students enjoy it. The questions are of varying difficulty, some are very easy but some require a bit more thought. It should be accessible to Year 12 mathematicians.

Please download the file here

There will also be a Further Maths Calculated Colouring this year, featuring a beautiful illustration by a talented friend of mine – this should be going up on Wednesday night this week so look out for it.

A Level Books Teaching

Classic Maths Books 1 – Bostock and Chandler textbooks

This week I posted the following picture on Twitter:

A few people retweeted, favourited and commented how good they were for questions to stretch A Level students so I thought I would write a bit more here.
They were first published in 1982 and are still available from Amazon here and here.
I think I got my copies when I was doing my A Levels and asked my teacher if there were any other books he would recommend that weren’t the Edexcel ones.
I think they are still fantastic books, vastly superior to the Pearson published textbooks, though I think some students may find them a bit dry. They certainly assume more knowledge than the current books and the later questions are generally more involved (or it is less clear on how to start them). There are some really nice examples in the book, such as:

  • Find the complex roots of the equation \(2x^2 + 3x + 5 = 0 \). If these roots are \( \alpha \) and \( \beta \), confirm the relationships \( \alpha + \beta = – \frac{b}{a} \) and \( \alpha \beta = \frac{c}{a} \)

All these examples are well explained, and it is easy to follow them line by line.
One other aspect of them that I appreciate are the multiple choice questions at the end of each chapter; these are great for a quick test of understanding and I am using some of them in some QuickKey quizzes that I am trialling this half term.
I may make a blog post about a classic mathematics book a fairly regular thing… gives me an excuse to get some of the older books off my shelf again 🙂

A Level

Thoughts on A Level Mathematics and Making the Transition

Last night I read an excellent blog post by Stephen (@srcav) discussing the readiness of students for the current maths A Level. It turns out that this blog post was actually a response to another great post by Jo (@mathsjem) and I was going to comment on both posts, but then thought they would probably be a bit long for comments. So here are some of my thoughts…

The Importance of Algebra
As Jo points out it is possible to do well at GCSE without much of an understanding of algebraic manipulation. Algebra is, in my opinion, the bedrock of A Level, and students need to be able to perform algebraic manipulation correctly, confidently and quickly to do well at GCSE.
This appears to come as quite a shock to many students, who have got used to algebra questions being one of 4 main topics, and normally quite distinct from others. I have only been teaching at school level for a short time, however, to me a diagnostic algebra test at the beginning of Year 12 seems to be a very good predictor of A Level grades (far more so than GCSE grades) if the student doesn’t put work in to practice the algebra skills that they may be potentially lacking.

The choice of student’s applied modules seems to be important, and to me this choice means that an A Level grade is not really fit for being used by universities to compare students. M1 is a tougher module for a student to complete than either S1 or D1 and so an A Level where M1 and S1 have been chosen isn’t directly comparable to an A Level where the student has “chosen” (or pushed towards by a school looking for improved results) S1 and d!. The new 100% prescribed content of the new A Level should remedy this and enable the students to see maths as the connected subject that it is without the arbitrary distinctions placed on it.
Timing is also a problem for those students who are taking Maths and Further Maths. Ideally, a student could complete all of the standard A Level in Year 12 before studying Further Maths in Year 13. Where this isn’t possible students study Further alongside the standard A Level. Working on FP1 content whilst working on C1 content is obviously problematic as students don’t know all the techniques (polynomial division being one of them) that is expected for the Further Maths content. They also haven’t had the practice, or necessarily developed the mathematical maturity that is expected of them. This has obvious effects on confidence levels of the students, as well as creating time pressures when delivering the content.

Type of Exam Question
As Jo mentioned there is a large change in the wording of exam questions at A Level. I think that there should be greater parity between GCSE and A Level mathematics in this. I understand though, that this may go against some of the GCSE reforms to push questions towards simpler wording and grammatical constructs (as AQA definitely seem to be doing) so that students with weak language skills aren’t at a significant disadvantage in their mathematics exams. If we are taking students onto A Level who have achieved a grade B with less than 50% of the paper correct, it is entirely possible that students with weak language skills will end up taking A Level maths where understanding the exam questions could be problematic.
Students also sometimes find the use of more formal mathematical language and symbols confusing. This could be remedied by introducing some of this into the higher GCSE classes; for example sigma notation could be introduced when computing the mean of a set of numbers.

Managing the Transition
I believe that a mathematics A level is the most rewarding A Level a pupil could take, and so I, like all teachers I’m sure, want to make the transition from GCSE to A Level as manageable as possible.
In my school we set some summer work which is essentially revision of GCSE algebra, though it is clear that not all students have revised this sufficiently when they come to do their initial algebra assessment. This year I also sat the Further Maths students some harder work that was more open ended and of a problem solving nature. I wanted to stretch them and also get an idea of how their brains worked when faced with a challenge. These questions than formed the first lesson of the year. On the whole they were well received by the students with them saying that they enjoyed having a go at them even if they did not correctly solve them. This experience of struggling towards a solution is important. Many Further Mathematics students have probably never struggled with a maths question before and the increase in difficulty can sometimes be uncomfortable and hard for them to handle. Learning to fail and accepting that they will learn an awful lot in the process is an important thing to experience on the road to becoming a mathematician.
I’ve also found encouraging group work important too as this forces them to discuss the maths instead of working in isolation. Being able o discuss the questions and work through problems is important as there often isn’t enough time to talk through problems with all questions in class.

Off Syllabus Mathematics
Last year I was lucky enough to have some timetables time to work with the most gifted of the year 11 students, looking at mathematics that wasn’t on either the A Level or GCSE Syllabi. Due to my interests we looked at, among other things, floating point representation of numbers and round off errors, formulae for calculating Pi and iterative schemes for solving linear systems. Many of these topics are typically studied in Year 2 of an undergraduate degree, yet they were accessible to students who had just completed a GCSE course (of course some of the rigorous analysis was left out….). I am passionate about exposing students to maths outside of the syllabus as I think background knowledge often helps them see how to approach a question in a different light, or at the very least provides a motivation for studying what we do at school which is sometimes not explained enough.

A Final Niggle
One thing that I have always loved about maths is the “universal truth” that maths brings. A mathematical concept is always true no matter what, and something that you learn early on cannot be then shown to be rubbish. This is in contrast to the other sciences, for example in Chemistry when you learn about chemical bonding you seem to be forever told to forget what you have been told previously! For this reason it always pains me when teaching the quadratic formula that students begin to believe that getting a negative discriminant must mean they have gone wrong! I can’t see any reaso. Why a higher tier GCSE student couldn’t handle a question such as the following:


I certainly don’t understand why complex numbers are not in the standard A Level as they are so fundamental.

Anyway, I’m looking forward to joining in with Jo’s chat tomorrow and looking forward to hearing other people’s views.


Integration by a flowchart.

After years of dealing with any integrations that I had to do by thinking “let’s load Mathematica” I thought I should probably practice doing some by hand as I will be teaching Further Mathematics this year and will need to be able to integrate by hand.

As I was practicing a few I had a vague memory of a flow chart that my A Level teacher had given me (a long time ago) and thought it could be useful to try and produce something similar.

I had also just downloaded the iPad Grafio app so I decided that producing a flow chart for integration would be a good way to get used to Grafio. Overall a fairly easy app to use, sometimes a bit fiddly though to place content where you want.

The flow chart is shown below:


And a high resolution pdf can be downloaded at the following link.

Anyway, I would be very grateful if you commented and let me know what you thought of the flowchart. Do you work through a similar process when integrating by hand? Have I missed anything that you think is important?


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.

General comments:

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.

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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.

Final comments:

As is usual in a post of this nature I have expressed my own personal opinions and I would be very interested to discuss these with others. I would love it if you left any thoughts in the comment section below this page!