The Term So Far…

It seems amazing that we are already more (now much more than when I started writing this!) than half a term into teaching the new specifications of the A-level. When we planned this half term I was concerned that we had put far too much content in and that we would never get anywhere near completing it all. Remarkably this has not been the case and we finished exactly on schedule.

The way that we have set the scheme of work up has had massive benefits, this has been particularly evident in the linking of topics that were previously kept apart by being almost arbitrarily in different modules. We have found this is useful with the relationships between indices, surds, exponentials and logarithms. This week we have seen this again in teaching binomial expansion and the binomial distribution consecutively.

I am also much more confident that I know how students are coping with the course due to the changes we have made to our assessment structures. For each unit students have completed the online Integral assessment, for which they are provided feedback. This is an automated process where we copy the data from Integral into a spreadsheet which creates a sheet for each student, detailing which questions they have got correct and which they need to redo. Corrections are made on the back of this sheet so that we have a record that they are doing this.Eg1

We have also set students the chapter assessments from Integral, which unfortunately don’t mark themselves! These are marked and entered into another spreadsheet, which again provides individualised feedback for the students. This shows students their performance in relation to the rest of the class for each question, so that they can focus their efforts on areas that they need to. It also allows us to see any areas that need revisiting with the whole group.Eg2

Where there are significant issues arising from the assessment we are able to run intervention sessions either during form periods (fortunately as Head of Department I do not have a form group) or after school.

Another interesting aspect that we have incorporated into the scheme of work is reading and comprehension tasks. At the end of half term one, as a holiday task, we asked students to read an article from New Scientist and to answer questions based on this. Our aims with our development of our practice in the new course have always been to develop students as mathematicians, not just to allow them to answer the questions in the examination. The article in question looked at prime numbers, particularly into the question of whether there was a pattern in the way they appeared. Students were challenged to answer 6 questions, starting with information taken from the article, then progressing to independent research on the topic. Unfortunately none of the students managed the sixth…

  • Define “twin prime”.
  • Using the sieve method, which is the largest prime number that needs to be used such that all of the non-primes < 100 are sieved out?
  • When does sieve theory struggle?
  • Showing your calculations, write down a list of the first 6 Germain Primes (29 is the 7th).
  • Write a paragraph of no more than 200 words about either Goldbach’s Conjecture, Germain Primes or the Riemann Hypothesis.
  • Solve the Riemann Hypothesis, donating half of your winnings to Mr Hampton and Mr Davies.

The next challenge is to find a suitable article to set for reading over Christmas!




Scheme of work and developing a teaching plan

This post is contributed by Simon Clay who is part of the Teacher Support team at MEI.

Given that the changes to A level mathematics are significant, an overhaul of teaching schemes for the new two-year long qualification is not a trivial task.  During 2016-17 a number of members of staff at MEI developed a Scheme of Work for the new A levels with the aim of trying to produce something useful for as wide a range of audience as possible.  This result is this freely available SoW, accessible via the MEI website.

Some of the thinking behind the design of the SoW units was as follows:

– It aimed to break down the new A level content into manageable units.

– It needed to function as a starting point for discussions in departments and therefore needed to be editable.

– It needed to take seriously the changes in emphasis of the new A levels, including the three overarching themes – Mathematical argument, language and proof; Mathematical problem solving; Mathematical modelling.

– It needed to incorporate useful features such as ideas as to how the use of technology can permeate the teaching of A level mathematics, questions which promote mathematical thinking, etc.

– It needed to be both adaptable and useable in the classroom.

– It needed to exemplify, and give free access to, some high quality teaching resources which can be ‘picked up and used’ in any classroom.

Since its launch in March, we have been pleased with the way the SoW has been received.  A common request, however, was for the provision of a plan for how the units could be linked together in a cohesive way to ensure the content is covered in the time available.  We have therefore worked on producing a series of schedules which show how the units of the first year (or AS content) can be arranged depending on considerations or constraints a department may have e.g. two teachers sharing a group, one of whom teaches pure and mechanics while the other teaches pure and statistics.  (We have so far only tackled Year 1 content but Year 2 will follow in due course!)

The reason for a post in this blog is because Schedule E is as a result of the thinking and work done by Bruce and the team at TGA Redditch.  It has been my privilege to take part in the discussions in which this SoW Schedule has been developed.

Below is an image of Schedule E taken from and beneath this I describe the key features:

Image of 'Schedule E'

– The team wanted to begin the course with an emphasis on problem-solving and proof in order to set the culture of working in this way from lesson 1.  This means lesson 1 will contain no mathematics beyond GCSE and will instead focus on reasoning, language and proof.  Lesson 2 will look at indices but with an emphasis on reasoning and proof rather than subject content coverage.

– There was a strong desire to get the students working with and becoming familiar with the large data set (LDS) right from the start of the course.  Thus by the end of the first teaching week students will know about the LDS and have done some initial exploratory work using it.

– The team identified some units, namely ‘Problem-solving’ and ‘Graphs and transformations’ as being recurring themes which can be addressed in a number of different units throughout the course rather than taught as discrete topics.

– The team wanted to use a teaching model where the class is shared between two members of staff but essentially runs as a single series of lessons.  This will clearly involve a high level of collaboration between them but they are keen to dovetail their teaching so that the student experience is as coherent and seamless as possible.

– They wanted the applied units to be taught alongside the relevant pure unit so it is clear what mathematics is being applied.  It is hoped that this will also help with fitting in the content in the time available.

– They wanted technology to be used by teachers and students whenever possible and so in the first few weeks there are planned opportunities for this, in particular when analysing the LDS and exploring graphs of exponential functions .

– The school has made a central decision that all students need to be prepared and entered for AS level examinations at the end of Year 12. This means that although at points it would be nice to extend and cover Year 2 topics straightaway these will need to wait.

And now there are only a few weeks until the schedule can be implemented!

First Attempt at a Framework

On Friday I met with Simon and my head of department Pete to try and create an initial framework of topics for the scheme of work. The target was to have a loose order of topics to cover the first year of an AS course, changing from our previous structure of 3 teachers each teaching individual modules, to a linear structure that will probably be taught by two teachers.

One of the real benefits of moving to the linear scheme will be how much time it frees up compared to our old structure by removing some of the assessment. Previously we have tested students each half term in all three modules. This has been in the form of a one hour assessment based on past exam questions, starting off quite narrow and expanding as more content has been covered. By the time these assessments had been completed and feedback given we were looking at 6 hours of teaching time lost per half term. In a linear system I would anticipate that the assessment could be reduced to a single one hour paper initially, allowing us at least 4 hours more time each half term.

Using the AS topic headings from the freely available MEI SoW we began to organise the topics into a coherent order, focussing on pre-requisite knowledge, and links between topics.

workingHaving a hard copy of the MEI SoW to hand ( was useful as we moved topics around.  It is designed to be editable for any specification and allowed us to focus on the connections between mathematics topics

One of the striking things that came up in the conversation was how we had previously compartmentalised topics. Surds and Indices is a C1 topic, whereas Logarithms and Exponentials is a C2 topic. Yet they are different ways of looking at the same thing and surely if taught together would allow a much better understanding of where logarithms come from, something that I have always struggled to get students to see. As such we have decided that the first thing we will teach is logarithms and exponentials, while at the same time revising the surds and indices materials students should have met at GCSE. This means that students will be meeting something new straight away, hopefully catching interest, but also brings in a link to previous learning.

A provisional model is shown in the diagram below, pure units in green, statistics in blue and mechanics in orange.

first framework

The model we have come up with looks very heavily weighted to the first half term. However of the five pure elements, four should be revision from GCSE. Historically we have taught these as the first half term of C1, a third of our teaching time across the whole course. While thinking about the links in topic areas we discussed how some of the topics (see the right-hand columns of the grid above) might be better spread over the course, with pieces put into different topics to improve connections. An example would be that for transformation of graphs in the past we have taught completing the square early in the course and touched back to how this links to transformations much later. We feel that by expecting to make links with transformations at appropriate points throughout the course as it naturally arises the links should be much clearer and stronger for the students.

Coordinate geometry is another topic that we felt was better split across the year. Tangents and normal will fit in as an introduction to differentiation and circles has strong links to trigonometry.

With the statistics elements of the courses we decided that the large data set should be introduced as early as possible. This meant that we inserted data collection, which is largely about sampling, into the first block of topics. This also got me thinking – I had previously decided that I would not make the decision on which exam board to use until much later. However in order to introduce the large data set I need to have made the decision so that students are used to working with the relevant data.

Mechanics fits in very well with elements of the ‘pure’ maths, particularly with calculus and variable acceleration. This has always been something that I have felt is a missed opportunity in the teaching of A-level maths, it should create a connection and allows us to show the roots of these skills in a real life situations.

This of course is only a first attempt and will continue to evolve as we move forward. At our next meeting with Simon we are going to look at the individual content statements for each topic and to order those, whether within the current structure or moved to further emphasise links.

Beginning to Order

One of the first things I am considering when putting together my scheme of work is the order in which we will teach. It feels like I have performed this task repeatedly in the last few years. My first scheme of work included modular examinations in January, then those were removed so I shuffled C1 and C2 into a single ‘core’ unit. When I moved schools to my current position I wrote a new scheme of work, within which a teacher took control of a single module, meaning that each group had three teachers. Last year I had a slightly smaller job of transferring my existing scheme of work into a new format so that it was on the correct templates.

Should we teach the topics broadly in the order MEI have placed them in their scheme of work (albeit with the statistics and mechanics spread throughout) or re-arrange so that similar content is taught together. In the past, when teaching C1 and C2 as a single ‘core’ unit I have rearranged the content so that, for example, all of the differentiation is together. This has already been assumed in the scheme of work, but should integration be taught immediately after. Perhaps it would be even better to teach them at the same time? This would hopefully create a much better understanding of the inverse nature of differentiation and integration – students could differentiate a curve, then integrate to get back to it, using a point on the original to find c.

A different strategy would be to separate similar content, allowing more structured interleaving. As students come back to a topic, they revise the original content and then build on it. This has obvious benefits of seeing things more than once, but is likely to lead to too much time being spent going back over previously learned content and falling behind as a result. As time is already tight we cannot afford too much slippage.

On Friday I will be meeting Simon and my head of department to start to build up the scheme of work, beginning with this process of ordering topics. I have loosely grouped topics – we now need to come up with an answer to the questions posed above. I am ready with the topics on cards to move around, string to make links between the topics and a blank timeline…