Problem Solving and Technology

In our work on revamping the curriculum for the new specification we have been careful to make sure that we are considering the overarching themes of problem solving and use of technology. We are very keen to ensure that use of technology does not become ever more complicated and ‘interactive’ PowerPoint files that are demonstrated from the front of the class room with little chance for students to use and develop their own skills. We also want to introduce this aspect as early as possible to encourage students to think about technology as a vehicle for working on and solving problems when they get stuck. On Friday I met with Simon, Fiona Kitchen (from FMSP) and two colleagues from my department to discuss methods for this.

Our starting point was to use the worksheet “Problem Solving with Geogebra” from MEI’s scheme of work. We looked at solving the problems ourselves, trying to limit techniques to those that year 12 students were able to use. This proved rather difficult! After much wrestling (and a plea to twitter) we managed to create working models in Geogebra for the first three of the problems.

We had a lot of fun working on these problems but concluded that the level was a too high for students who are starting out in year 12. As such we will need to adapt to something closer to GCSE if we are going to introduce Geogebra in this way at the beginning of the course. One thing that struck me during the afternoon was that we persevered with the problems for a long period, around 2.5 hours. This was something that our students would have really struggled to achieve. The same morning one of my year 11 students, when confronted with a difficult question, said “It’s alright for you sir, you are good at maths and can do it easily.” I was unable to make her understand that I don’t find all maths easy and that I enjoy the struggle with harder problems. This perhaps sums up the major problem we have been fighting against with our A-level students over the last few years – the lack of resilience as soon as a problem gets complicated.

Hopefully this process of really concentrating on both problem solving and technology will help to improve this, which is certainly the focus of what we are looking at during this process. For now though here are the problems that we worked on.

Problem 1

With this problem I found it very easy to create a polynomial fixed by the points A, B and C. This initially created a point D that moved as A, B and C moved. To do this I used the measuring tool in Geogebra to calculate how far away the points were from the origin.

My second attempt used a division at the start of the polynomial to allow me to control point D as well. It was at this point that I realised the shortcomings of my method of measuring distances – when I moved the points to negative values Geogebra continued to measure them as positive. To complete the problem Simon showed me how to use just the x-ordinate of A etc. in the calculation. My final solution is at: https://www.geogebra.org/m/bZD6fARj

Problem 2

For this problem I started by drawing a circle centred on the origin with a point on the circumference fixed into the side AC. I then created another circle centred on C which connected to the first. I repeated this methodology to create a third circle centred on point B. The three circles could then be manipulated together until I had a solution that worked. This did not satisfy me – I wanted to be able to change the triangle and the circles to remain a solution to the puzzle.

My instincts for this puzzle were probably from spending time playing the mobile phone game Euclidea – maybe those hours were not completely wasted! I guessed that the points where the circles met on the edges of the triangle were those where the circle inscribed in the triangle also touched the sides of the triangle. My resulting solution can be found here: https://www.geogebra.org/m/jMSThZg8

Problem 3

Problem 3 caused me the most problems. For a long time I was able to either create a line that was perpendicular to the tangent from A or a line that passed through the point B but not both. After a long time trying (and a plea to Twitter), Simon came up with a solution that can be found at: https://www.geogebra.org/m/MFgcJaAd

Problem 4

We ran out of time before tackling problem 4 and I have not yet returned to it. I have some ideas about using a quadratic function with roots that are the x-ordinates of A and B and integrating it but have not progressed any further yet. I leave that one with you…

Thoughts on Large Data Sets

One of the thoughts that came out of my most recent meeting with Simon was that the choice of exam board will be influenced by the large data set. I had previously been of the opinion that I could leave the choice until January 2018, seeing if any more specimen/mock papers became available and analysing question types. However this would mean not spending as long familiarising students with the specific large data set for whichever exam board we choose. As a result of this I have downloaded the data sets for AQA, Edexcel and both specifications of OCR. I should point out that I am not a statistician, I have taught S1 once and try to avoid it if I can!

I have started to look at the data sets to see which is most useable, and which students will be able to best gain insight into for reproduction in their exams. We want to be revisiting the data constantly, so that students are really familiar with it. This means that portability is important as we will not always be able to access computer facilities.

AQA – Purchased quantities of household food & drink by Government Office Region and Country

The data given is split into 10 regions (under separate tabs), with the average amounts of various foods and drinks per person per week. There is also a tab with averages for the whole of England. Having spent some time in Excel playing around with the data it is possible to fit each region onto a single sheet of A3 paper (total of 11 sheets).

Looking at the questions in the specimen paper, students are expected to be able to recall information about the average amounts of certain food groups from different regions. This is something that could only be known by someone who has done extensive work with the data set before, and given the sheer scale of the data is unlikely to be something that you could repeat for all of the different food groups.

Later questions involving the data set give a small excerpt and ask questions about these. These are much more accessible to students who do not have as much familiarity, but will be easier for those who are aware of the context. For example there is question about the total amount of confectionery purchased, which does not state that it is based on averages.

Total Marks based on Large Data Set in AS Spec Paper: 9 (Out of 80 on paper 2, 160 across the AS)

OCR A – Method of Travel / Age Structure

The OCR A specification looks at the methods of travel to work, broken down into regions, taken from the national census in 2001 and 2011 (separated into two sheets). There is also data about the ages of the residents of the regions (2 further separate sheets). Each tab can be set to cover three A3 pages, so a total of 12 will be needed for a portable copy.

In the question pictured here it would be advantageous to be familiar with the data set, particularly for part (ii), as there are different codes for the authorities based on their type. If you knew this then you would know how to separate the authorities further and would merely have to explain this.

For the other question based on the data set (not pictured), a summary table has been created. It is not as obvious what the benefits to knowing that data are here, although general familiarity and having looked at possible summary statistics will help.

Total Marks based on Large Data Set in AS Spec Paper: 8 (Out of 75 on paper 1 and 150 across the AS)

OCR B (MEI) – Population data and Olympic success

The first thing to note here is that the MEI specification (OCR B) has taken a very different position to the other boards. There will be three different data sets that will be used in rotation. The data sets that will be used for ‘live’ specifications are not available yet.

The data set that is available for the specimen papers is far less ‘large’ than the others, reducing to two A3 sheets. The question included here really grabbed me as being interesting – what were the outliers in Sub-Saharan Africa? On inspection, the data that stood out was that from islands, rather than countries on the continent.

This data set seems much more manageable than the others, and over two years I would expect students to be able to become very familiar with it.

Total Marks based on Large Data Set in AS Spec Paper: 7 (Out of 70 on paper 2 and 140 across the AS)

Edexcel – Weather Data

Edexcel’s weather data consists of 5 weather stations in the UK and 3 from abroad, with readings from both 1987 and 2015. I have been able to fit the data for each station, for a single year, on one A3 sheet (total 16 sheets).

The questions based on this data set again seemed to not require much detailed knowledge of the readings. In the question shown here it is only the fact that there is one reading per day that will help with part (b).

Of course, as Edexcel has not been accredited yet, this may change.

Total Marks based on Large Data Set in AS Spec Paper: 11 (Out of 60 on paper 2 and 160 across the AS)

Summary

While the use of the data set will only form part of my decision on which exam board to use, I have found the process of sifting through the data sets, and the questions that relate to them, extremely useful. It has also shown me the benefits of this approach. In starting to look at the data sets it is already noticeable how the data is starting to feel familiar. I think that this will develop much more ownership of the data and make structuring easier. Now students know they are expected to know the data set, they are more likely to see the value in using it as part of exercises.

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.

Having a hard copy of the MEI SoW to hand (http://mei.org.uk/2017-sow) 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.

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…

Teaching Structure

Today I met with Simon and my head of department, Pete, again to continue our discussion with how to move forward towards the first teaching in September. The main talking point was what we need in place before we can really start creating a scheme of work.

One thing that has been confirmed since our initial discussion is that the school will want us to enter students for the AS examinations at the end of year 12. This will be school policy for all subjects and, as such, is non-negotiable.

The next decision we will need to make as a department is how we are going to divide up the teaching between two or three colleagues. We are very lucky in that all of our members of staff are confident to teach A-level and keen to do so, but any significant changes to the way we currently operate is likely to mean that at least one person will miss out next year. This of course assumes stability, something that at the moment seems likely but that we can never rely on.

In our current structure each module is taught by a different teacher, so that each class has three teachers. This means that each teacher is responsible equally for the attainment of students across the course. It also means that the content is neatly parcelled out, and there are not too many tricky decisions over when to teach content, although with the core modules some thought has been needed so that, for example, differentiation is reached in C1 before C2.

In my previous role at a different school I abolished the distinction between the two core modules and taught it as one block. Where one topic was included in both modules it was taught at the same time, allowing more time to be focussed exploring ideas around the area as a whole. This led to a better flow, with topics seeming less disjointed. The teaching was split between two colleagues, but taught in a linear fashion with them handing over at the end of each lesson. The applied unit was split across the year, dovetailing with the core content at appropriate times. It would make sense if we were to adopt this model to have each of the two teachers leading one of the applied sections.

A further model would be to have one teacher cover all of the core material and one covering the applied. This is probably my least preferred option given the new split in the applied material between statistics and mechanics. I have heard of this model being used successfully in other schools (although obviously with only one applied option being taught), but have also heard of complaints from the applied teacher about being seen as less of a priority than the “main” teacher covering core.

My instinct at the moment is that the two teachers sharing equally will be the most workable solution, so I will start building on this principle. The next thing to start thinking about will be the order in which to cover the content.

 

Initial Thoughts

From September 2017 new AS/A level Maths and Further Maths qualifications will be taught across England. This means that during the next 9 months mathematics departments will be need to think carefully about how they will implement the changes for their teachers and students.  Simon Clay (MEI TAM Coordinator) approached the mathematics teaching team at Tudor Grange Academy Redditch, to see if they were interested in working together to think through the issues which are raised by these forthcoming changes.  This blog will therefore document one department’s journey as they navigate preparing for the new A-levels through the eyes of the Curriculum Leader.

Over the last few years a number of teachers from the school have participated in professional development and other activities run by MEI, including the Teaching Advanced Mathematics (TAM) and Teaching Further Mathematics (TFM) courses. The school is part of the Tudor Grange Academy Trust and has 440 students on roll. Until recently it has been a high school, teaching students from years 9 to 13. This year it welcomed its first year 7 pupils. The maths department is made up of 6 teachers, all of whom teach some A-level maths.

 

Bruce’s thoughts:

In this initial blog I have recorded where we are as a department now and the initial thoughts I have for moving forward with the new A-level. This has been prompted by being approached by Simon Clay and MEI to work with them as we begin our journey towards first teaching in 2017.

As the Key Stage 5 coordinator for maths some key decisions need to be made over the next 6 months regarding the new A-level specifications. At this stage, with none of the specifications having been approved, I have not thought too hard about this. This is not a position I want to continue with as I am keen to start rebuilding our scheme of work while there is still time to do a good job with it.

We normally run two A-level classes in each year group plus a small further maths group and all members of the department teach at least one module. In our current scheme of work we have tried to include the use of activities which promote deeper thinking, such as those that are available through MEIs Integral website. However time pressures mean that these are rarely used. I see the change of course, coupled with the changes to structure, as an opportunity to totally rethink the way we are teaching and to promote this sort of activity further. This fits into the way the structure of the A-level is changing, with the new over-arching themes of problem solving and the increased use of technology – this is our opportunity to tear up what we have done before and start afresh.

 

AS levels, Specifications and Applied content

One possible barrier to a total rethink of the current scheme of work structure is that the school currently has a policy that students sitting the new specifications of A-level will all sit AS-level examinations at the end of year 12. If this policy continues it will impact on the amount of time we have available and the order in which we teach content.

This feeds into our need to think about which examination board we decide to use. We have offered Edexcel, which was the course that was offered when I joined the school last year. With the change to curriculum I thought it was unnecessary to make changes twice in quick succession. I have previous experience of teaching both AQA and MEI, but everybody else in the department has only ever used Edexcel. I have been reading specifications for all of the examination boards and have also attended an OCR Maths Network meeting. I am not in any rush to make a decision due to all of the content in A-level Maths being core, and again that specifications have not been finalised so we don’t know what the final examinations will look like yet.

Another consideration is how we split the teaching in the new courses. Historically we have run the A-level courses with a separate teacher for each of the modules. This has allowed everybody to build their own specialisms, but makes linking between content in the core modules more difficult. It does mean that we are in a position that we have somebody confident with all of the different areas required by decision to make all of the A-level content compulsory as we have offered mechanics and statistics in one or other of our courses previously. This is tempered by the huge change in the emphasis in the statistics module with the large data set.

 

In conclusion, the key questions that we need to address in the short term are:

– Do we continue to offer AS examinations for all students?

– Which examination specification should we use?

– How should we structure the course in light of the new linear format?

 

Bruce Hampton is the Curriculum Leader for Sixth Form Maths at Tudor Grange Academy Redditch. He has been teaching for 12 years, initially at a school in Birmingham, moving to Tudor Grange Academy Redditch in 2015. He completed the Teaching Further Maths course in 2014-15. He studied Mathematics at the University of Birmingham and has a Masters in Teaching and Learning from the University of Warwick. In his spare time he is a keen rugby fan and likes to get out on his bike.

 

Simon Clay is part of MEI’s Professional Development team where he is involved in a number of different aspects of MEI’s work including working with teachers on the Teaching Advanced Mathematics (TAM), Introduction to Mechanics and Head of Mathematics courses. Before joining MEI in 2012 he had taught Mathematics in schools and Sixth Form Colleges for 12 years, during which he had been a Head of Department for 5 years.