Non-linearity of Sine

While at a meeting yesterday the following question came up due to a student “Why isn’t \(\mathrm{sin}(a+b) = \mathrm{sin}(a) + \mathrm{sin}(b) \sin\)“. My immediate response would be to say “because \(\sin(x)\) isn’t a linear function”, but this isn’t a terribly satisfactory response since the likelihood is that the student doesn’t have a deep understanding of the difference between linear and non-linear functions.

On reflection I think the best explanation is to ask the student to sketch the graph of \(\sin(x)\) and ask them to consider \(\sin(a+b)\) using the graph. singraph

It should become clear from the graph that \(\mathrm{sin}(a+b) \) can only be equal to \(\mathrm{sin}(a) + \mathrm{sin}(b)\) when either one of \(a\) or \(b\) is \(360^\circ\) (in fact any multiplicity of the periodicity of \(\sin(x)\).

Another approach could be to ask them to consider a sequence of non-linear functions such as \(f(x)=x^2, f(x) = x^3+3,  f(x) = 2^x+6\) and ask them to compute \(f(a),f(b),f(a+b)\). This I hope would get rid of the expectation that f(a+b) = f(a)+f(b). The geometric proofs of the true formulae for \(\sin(a+b)\) could be a nice way to close the discussion.

I wonder if this is a consequence of an over-reliance on linear functions for examples of substitution etc in lower school.


Thoughts on the A-Level Reforms

As usual Jo Morgan (@mathsjem) has beaten me to it with a great post setting out some of her thoughts concerning the A-Level reforms for Mathematics.

Here are some of my thoughts in a less ordered structure than Jo’s.

  • Teaching Structure and Specialisms Jo has outlined a good structure for teaching the content for the standard A-Level that allows people to teach according to their current specialisms. In theory, this would enable a smoother transition to the new A-Level, however, my concern is that this approach risks diluting the idea that maths is an interconnected subject (something that I think is an issue at the moment). To me, an ideal structure would blend the applied content in with the pure content, for example vectors could be introduced in conjunction with mechanics. I appreciate that this could cause issues with staffing though.
  • Teacher CPD There will be a high demand for CPD for current A-Level teachers, and I think this will be especially true for teachers of Further Maths. I am sure the FMSP and MEI will have various things on offer, but due to the scope of the changes and the difficulties of releasing staff for cover I think much CPD will have to be delivered in school, maybe through events led by your local MathsHub.
  • Student Calibre In her post Jo highlighted a concern about students not being able to “drop” the subject after an AS year. This is a very real concern for me! To some extent this may be mitigated by the increased demands of the new GCSE, but currently you can get an A at GCSE and not succeed at A-Level. We perhaps need to come up with more rigorous entry criteria (especially for Further Maths) and monitor students closely in the first few weeks of Year 12. Of course, this may affect student numbers which moves me on to my next thought.
  • Student Numbers Recently the numbers for A-Level Mathematics and A-Level Further Mathematics have risen (the work of the FMSP has been instrumental in this in my opinion), the A-Level reforms present a very real threat to this trend. Mathematics is already seen as a “hard” and “demanding” A-Level, and with the removal of many of the topics and modules that are perceived to be easier this perception will probably increase. The changes to the funding formula are likely to make students taking 4 A-Levels seem less attractive to Sixth Forms and Colleges (financially they don’t gain much from it) and this is bound to impact the numbers for Further Mathematics.
  • Decision Maths As the 100 prescribed content for A-Level includes only Pure, Statistics and Mechanics those A-Level teachers who currently only teach D1 and D2 will either have to up-skill or stop teaching A-Level. There is some scope for decision maths to be included in Further Mathematics specifications, but it is quite clear that the current content of D1 and D2 is not valued by higher education, so I suspect that if any of this content makes it on to specifications it will be mainly stuff that is currently examined in the second Decision module at A-Level. There is some scope for Decision maths teachers to move over and teach Core Mathematics instead as there is some decision content in the optional papers.
  • What will MEI do? At the moment MEI offer a radically different (and more innovative) offer for A-Level and A-Level Further Mathematics, with modules such as “Numerical Methods” and their “Further Pure with Technology”. I am looking forward to seeing their approach to the reforms.
  • Text Books I suspect that many schools will spend a significant mount of money changing the textbooks they use to match their choice of new specification. I’m not keen on many of the current textbooks, so it will be interesting to see what publishers come up with, or whether they just re-jig their current offering. To me this seems an ideal time for an exam board neutral text book thought from the ground up to be a guide to advanced level maths as opposed to a “test performance” factory. Once specifications and SAMs are released I am hoping to put together something along these lines that anyone is welcome to use, though whether I will have time to produce a full textbook remains to be seen. I also like the possibility of a digital textbook including virtual manipulatives, demonstrations and videos (for example of concepts in mechanics) but this would be a massive undertaking!
  • Problem Solving Seeing how exam boards approach the requirement for more problem solving in assessment is perhaps the most interesting thing about the A-Level reforms. To have a genuine problem solving question in an exam I think will be hard to achieve (and if they manage it possibly not a “fair” examination question) and will make the marking significantly more involved. The use of comparative judgement technology could perhaps be used to mark this style of questions. Basically I am very curious about this.
  • Large Data Sets The meaning of this still hasn’t really been clarified by Ofqual. Will we have some kind of pre-release? Will students need access to statistics packages for examinations? There is a lot of uncertainty over this.
  • Readiness to Teach It seems unlikely that exam boards will all get accredited with their first submission to Ofqual and so there is the possibility that accredited specifications may not be out until the beginning of 2017. This doesn’t leave much time for schools to make decisions about which to go with.
  • Performance Measures With the proportion of A-Level students continuing with maths becoming a performance measure will we be pressured to accept more students on to A-Level course regardless of ability? Will many schools implement a compulsory AS Maths or Core Maths decision?

I will add to my above thoughts if an when I think of other things to say.

I am looking forward to discussing the A-Level reforms with the other attendees of Stuart’s (@sxpmaths) “Maths in the Sticks” event next month. Preparing for the reforms will also be key content at the “East Midlands KS5 Mathematics Conference 2016” that I am co-organising. Thank you for mentioning this Jo, I hope many people can spare the day to attend during the summer break!


Approximating the Binomial Distribution

As part of the Edexcel S2 course students study the Poisson distribution (which I have written about in two posts here and here before) and also it’s application as an approximation to the Binomial distribution.

I find that people, most definitely including myself, don’t have an intuitive understanding of how good this approximation is and so I have made a small GeoGebra applet that allows you to explore this for varying values of \(n\) and \(p\). I have embedded it in a webpage on my website and there is a screen shot below.Screenshot 2016-03-27 22.37.42

I’ve found this applet also useful for just exploring the shape of the distributions as you can get a real “feel” for the effect of the probability parameter \(p\).

If you would like to, you can also download the original GeoGebra file.


From Fibonacci to da Vinci – an IMA Talk

On Wednesday last week there was an IMA East Midlands Branch talk at Derby University. This talk was organised in conjunction with the British Society for the History of Mathematics (BSHM) and was delivered by Fenny Smith. Fenny has delivered lectures for Gresham College and is an expert in Italian Renaissance algebra. This talk was full of little bits of fascinating information and in this post I hope to share some of them.

It was fascinating to see the accounts from a florentine company’s French branch. 

  To us – used to Arabic numerals – it was hard to see where the numbers in the accounts were as they were written in Roman numerals. I wasn’t aware of the use of the numeral J to mean the same as I but to also denote the end of a number.  Fenny explained that one advantage of our modern number system is the fact that you can check your work, or indeed the work of someone else. If you were using an abacus, the only way you could check your work is to re-do the calculation. This in part explains the adoption of our modern number system.

What we now know as Arabic numerals were developed India, and by the 7th Century were being used by astronomers and mathematicians. They arrived in Baghdad in the late 8th century when Hindu scholars brought their astronomical tables with them.In Baghdad Caliph al-Mansur had a centre of learning, and he didn’t really care where you came from if you had something to contribute you were welcome. We owe a lot of our knowledge of Greek Maths, Philosophy and astronomy to al-Mansur who translated and preserved many Greek works. 

Following this the Hindu-Arabic Numerals travelled westwards through the Arabic world towards North Africa and Spain. 

 In 976 two monks wrote a large manuscript called the Codex Vigilanus and in this they wrote about the new 9 Indo-Arabic numbers. At this point they were seen as curiosities as opposed to something of practical use.
In around 1200 Carmen de Algorismo by Alexander de Villedieu, a popular copy of Dixit Algorizmi (one of the books by Muhammad bin MĆ«sā al-KhwārizmÄ«, who is often credited with the introduction of Algebra), was published introducing the Arabic numbers to the French. This book was written  written in Latin, in rhyming couplets!! This seems pretty impressive to me.

At first there was much resistance and suspicion concerning the new numerals. Fenny explained the suspicion that people had of the concept of zero is to do with our Christian faith where God was said to have created everything, so how could he have created nothing? Conversely in Hinduism nothing is seen as a kind of “nirvana” and so there was no issue with the idea of zero as a placeholder.

 The famous engraving Melencolia I by Albrecht DĂŒrer made an appearance during the talk as a demonstration of the development of the numerals too which was interesting.  


Another interesting piece of information from this talk is that Gelosia multiplication was so named because he grid structure resembled Venetian window shutters.

It was also interesting to be reminded of Egyptian fractions, which were all unit fractions. An advantage of this representation of fractions that had passed me by, is that it is very easy to compare fractions and decide which is bigger. For example, \(\frac{4}{5} = \frac{1}{2} + \frac{1}{4} + \frac{1}{20}\) and \(\frac{3}{4} = \frac{1}{2} + \frac{1}{4}\) and so it is immediately obvious that \(\frac{4}{5}\) is the larger fraction.

I found learning about the Commercial Revolution was very interesting – a lot of the banking concepts we take for granted now had their origins during this period of time. If you even get a chance to go to a similar talk by Fenny I would encourage you to do so.
I have a recording of the talk, but I have been asked to not share it widely so you will need a password to view the video.


Poisson and a PGCE Student

I am really enjoying having a great PGCE student (@MrPullenMaths) in some of my Year 12 Further Maths lessons. (He isn’t massively active on Twitter at the moment so please follow him and welcome him!) He has recently been teaching the Poisson distribution and on Monday this week discussed using the Poisson distribution as an approximation to the binomial distribution. He knows that I like to delve a bit further (and maybe more rigorously) into the maths with my classes and so he looked at the proof of the validity of this approximation. I would probably have avoided this as it is quite technical, however it went really well and I was particularly impressed as this wasn’t in his lesson plan and he ad-libbed the proof. He asked for assistance a couple of times, but I actually think this improved the presentation as it became more of a discussion with everyone in the room than just a “work through of the proof”. A shot of the board is shown below: 

 In future I think there are a couple of aspects that can be improved, just by tweaking the layout to emphasise a few points:  

 This lesson made me realise that even though I have very high expectations of my students I am sometimes guilty of limiting the mathematics I expose them too. 

Finally, give @MrPullenMaths a follow on Twitter and encourage him to be more active.


Hypothesis Tests and Edexcel

Recently I have been covering hypothesis tests with both my Year 12 and Year 13 further maths groups and I noticed differing wordings in exam mark schemes (Edexcel S2).

I’ve always taught that you cannot say “we accept \(H_0\)” as you can’t prove that \(H_0\) is correct, and that you should say something like “There is insufficient evidence to reject \(H_0\)“. However, as the example below from January 2013 shows they seem to be happy with you saying “accept \(H_0\)“:

January 2013

Screenshot 2016-03-12 20.02.02Screenshot 2016-03-12 20.02.20

But strangely they don’t always use this language:

May 2011

Screenshot 2016-03-12 19.48.14

Screenshot 2016-03-12 19.52.57

I’m interested – how do you teach it?


Pi Day 2016

Following on from last years Pi day post, I thought I would briefly post about another series method for calculating pi.

I came across this cool infinite product involving the primes a few years ago, and was recently reminded of it when I saw Matt Parker (@standupmaths) in Nottingham recently.

This infinite product is due to the genius Ramanujan and was published in the Journal of the Indian Mathematical Society in 1913.Screenshot 2016-03-13 21.02.35The product I am interested in is numbered (5) in the article above, namely \(\left(1+\frac{1}{2^2} \right) \left(1+\frac{1}{3^2} \right)\left(1+\frac{1}{5^2} \right)\left(1+\frac{1}{7^2} \right) \cdots = \frac{15}{\pi^2}\). If \(P_n\) denotes the partial product up to the \(n\)th prime then we can say that \(P_n \approx \frac{15}{\pi^2}\) and so \(\pi \approx \sqrt{\frac{15}{P_n}}\).

I was interested in how quickly this converged so I coded it up in MATLAB (in an easy to understand style, not a way to maximise speed)Screenshot 2016-03-13 21.14.37

In the above bit of code the MATLAB function \(\texttt{primes}\) produces a list of all prime numbers below the integer n. Thus, the code doesn’t directly compute the product up to the \(n\)th prime in the sense that you might expect. I then called this function from a script file to generate a list of outputs.Screenshot 2016-03-13 21.20.41

With this we obtain the following table of results:

Screenshot 2016-03-13 21.21.09

Broadly speaking we see that for an increase in an order of magnitude of the number of primes we get an extra correct decimal digit of pi (it’s too late on a Sunday when I am writing this for me to do a proper convergence analysis). I believe the issues in the last three lines are likely due to the limitations of using floating point arithmetic, but I have not had a chance to try this method out using a variable precision arithmetic library.

If you have MATLAB and want to try it out for your self the files are available from my website at prime_pi.m and prime_product1.m.


Using Software and Games to Inspire and Motivate A-Level Mathematicians

This is a joint blog post by me and Jasmina Lazic, my co-presenter at #mathsconf6 (see my write up of the day) on Saturday 5th March 2016.

Me and Jasmina first interacted over twitter, and over time the idea of leading a workshop at a #mathsconf took root. Collaborating over email and a couple of WebEx calls we came up with a presentation and running order for our 50 minute session before meeting for the first time on the morning of the conference. Neither of us had presented with someone we hadn’t met before and it was an enjoyable experience, despite the issues with wifi at the beginning of our session.

In this post we intend to write a bit about the content of our presentation in case you couldn’t make it. The Prezi is embedded below:

[prezi id=”” align=center width=600 lock_to_path=1]


We started by introducing ourselves and our motivation for using MATLAB. I have used Matlab for almost 12 years, having started using it during my first year at university. I immediately took to MATLAB as it was so easy to use, and in fact emailed Cleve Moler (the creator of Matlab) who then sent me a copy of the original MATLAB to try out – I actually found it quite cool to plot functions in a terminal window using asterisks. MATLAB has been developed significantly since that first edition (which was focussed on matrix computations – hence the name MATrix LABoratory) and now comes with many tool boxes which enable you to quickly prototype sophisticated applications. Since completing a PhD in computational applied mathematics where I programmed predominantly in Fortran I have become a secondary maths teacher and try to bring technology, including MATLAB into my A-Level classes.

Jasmina was a maths tacher in Belgrade before completing a mathematics PhD and researching global optimisation and heuristic design. Since 2011 she has worked at The Mathworks, the developers of MATLAB, and is the education core team lead with them and an application engineer.


Sudoku is often held up as an example of a puzzle that mathematicians are particularly good at – infact you need no mathematical knowledge to solve sudoku problems, which require only logical thinking. There are many heuristic methods used to solve Sudoku puzzles by hand (see this book for some). A common way in is to “look for pairs” like in the picture below: There is a 5 in the 4th row and in the 6th row, we know that there must be a 5 in the 5th row; since there is already a 5 in box 4 and box 6 we can insert a 5 in indicated position.

   I must confess that I have never solved a Sudoku by hand (in fact the above is as far as I have ever gone with a Sudoku) and when I was given a book of Sudoku puzzles as a present my first thought was to write a program to solve them so I would never have to waste my time solving one by hand.

One way of solving Sudoku is to implement a recursive back-tracking procedure. Recursion is a fairly basic computer science technique where a function can call itself. The classical simple example of recursion is in the computation of the factorial function since \(n! = n \times (n-1)! \). This definition of the factorial function can be implemented in Matlab in the following way, not the call to itself in the body of the function.

Screenshot 2016-03-12 22.02.40

Having familiarised ourselves with the concept of recursion, lets return to solving Sudoku. An algorithm can be explained as follows:

  • Read in the blank grid.
  • Compute candidates for empty cells by applying the rules of sudoku.
  • Fill in all singletons
  • Exit if there are no candidates for any cells.
  • Fill in a tentative value for an empty cell.
  • Call the program recursively until the puzzle is solved.

As an example consider the simpler “Shikoku” puzzle below: at each step there is a suitable singleton we can select. Thus, to solve this puzzle there is no need to fill in any tentative values and the solution is easily computed:

Sometimes there aren’t any clear singletons to choose and we have to tentatively choose a number to fill in and then recompute all the possibilities, such as the example shown below.


These examples motivate a few questions to think about

  1. How can we optimally choose singletons in order to compute the solution in the minimum number of steps?
  2. For a Shikoku, what is the minimal number of clues required for a solution to exist.

In the example above, once we have filled in one tentative value all further steps have at least one singleton we can choose from. Of course Sudoku is harder than Shidoku, and often the selection of an entry results in there being no candidates a few steps later, hence the need to go back, choose another entry and restart the solution process from that step. This can be seen nicely in the video below

The video above has been created using the Sudoku code described by Cleve Moler in his “Experiments with Matlab” book. If you are interested in learning more about the recursive backtracking approach to solving Sudoku I would encourage you to read the appropriate chapter and look at the MATLAB code.

Following this, Jasmina described another approach where a Sudoku problem was converted into a linear programming problem and then the power of MATLAB’s Optimization Toolbox was employed to solve the resulting problem which has linear objective functions, linear constraints and the constraint that some variables must be integers.

To represent Sudoku as a binary integer program the 9-by-9 grid of clues (which cells have entries) and answers into a 9-by-9-by-9 cube of binary numbers.  With reference to the picture below, a clue of 5 in position (3,4) in the Sudoku problem becomes a 1 at level 5 in the cube at coordinate (3,4).Screenshot 2016-03-12 23.14.54

Of course at a solution we must have exactly one 1 in the stack at each grid coordinate. We can represent this mathematically by letting \(x(i,j,k)\) represent the value of the \((i,j)\)th entry at level \(k\) then we must have \(\sum_{k=1}^{9}x(i,j,k) = 1 \). We can represent the other rules of sudoku similarly, and then use a ‘black–box’ integer programming solver to solve the resulting problem. This Mathworks blogpost explains the more technical details if you are interested, it also goes on to describe an extension to Hyper-Sudoku.

The cool webcam solver we demonstrated makes use of the MATLAB Image Processing and Computer Vision toolboxes, and is similar to the demonstration shown in this video. You can also download the code at this link.

The Monty-Hall Problem

The Monty-Hall Problem is a classic problem in probability that causes a lot of confusion. This Numberphile video explains the problem very well:

This problem is often used as an exercise in the Statistics S1 module, for example, this question taken from the current Edexcel S1 text book.Screenshot 2016-03-04 21.45.15

Before looking at the mathematically correct approach to this problem making use of conditional probabilities and Baye’s theorem we discussed the naive approach resulting in the diagram below Screenshot 2016-03-09 22.44.33

Of course without the probabilities at the end of each branch being shown this diagram backs up the a naive intuition that it makes no difference to switch – there would be a 50% chance of winning if you stay or switch.

Working through the mathematics, we instead see that the probability of winning if you switch is actually \(2/3\).Screenshot 2016-03-09 22.51.36

Probability is notoriously counter-intuitive and the Monty-Hall Problem is a fantastic demonstration of this. Even when shown the mathematics, many people fail to be convinced. The aim of this part of our workshop was to demonstrate how easy it is to use MATLAB to generate some simulations of the situation and quickly calculate the proportion of those where you win by switching.

Jasmina live coded the following simulation in a couple of minutes during the session – I’m always impressed by live coding as it is so easy for things to go wrong.

This demonstration showed how easy (and quick) it is create a MATLAB file that demonstrates the Monty Hall problem and hopefully convinces even those sceptical that it is indeed better to switch.

Screenshot 2016-03-12 23.38.32

We hope our session gave you lots to think about, please don’t hesitate to contact us if you have any questions.


East Midlands West MathsHub Subject Network Meeting

Today was this half term’s Subject Leader Nework Meeting at the University of Nottingham for the East Midlands West MathsHub (@EM_MathsHub). It felt pretty strange being in A32 again; I spent a lot of time in their when I was on my School Direct course.The agenda for today was the following

  • News from Ofsted
  • Feedback and Marking
  • What is a good GCSE?
  • New A-Level content
  • Maths Hub Secondary steering group

This was a much more info filled session than sometimes – no maths to do at the beginning ;(

Below are some notes, more for my benefit than for anyone else…

News from Ofsted

Matilde talked about the presentation from Jane Jones that happened at last week’s National Maths Hubs Forum. A survey by Ofsted highlighted that many “schools’ principal focus is the new GCSE and that too many don’t seem to realise that getting KS3 right should boost performance at GCSE” – to me this was to be expected. The massive change in style of GCSEs has definitely spooked many (if not all) schools and it is no surprise that there will be a large focus on the new GCSE. Ofsted have also highlighted that so far the new national curriculum has had no significant impact on teaching quality – again this doesn’t seem to be a surprise as it hasn’t been around long. A fact from Matilde that I found interesting is that the introduction of new national curriculum in Singapore in 1980 didn’t show improvements in teaching and results for 10 years! She also highlighted that none of the current secondary textbooks develop reasoning well; “worded” application questions at the end of a chapter doesn’t develop this.

Ofsted have highlighted that the recruitment and retention of suitably qualified staff and subject leaders is a challenge. Personally I am quite worried about the lack of subject specialists – I wouldn’t want my daughter taught by non-specialists (or at least people who haven’t gone through subject conversion courses) in any subject. Maths is an subject full of connections, I think that a certain level of exposure to maths or a love of exploring maths is important to be able to express this nature of the subject well to students.

It’s always good to hear views from John Mason and his view that activities are not good activities unless you can make them deeper I think is great. 

The following quote from Bruno Reddy was highlighted:

“Ofsted don’t have a preferred lesson style, marking approach, differentiation approach, pupil grouping arrangement, textbook, lesson activity, assessment system or curriculum. So long as you can evidence how your school’s choices on all of these impacts your students’ learning, there is no need to “do it for Ofsted”.”

I enjoyed the reminder to re-read the summary that Bruno Reddy (@MrReddyMaths wrote following last years #ofstedmaths chat which is available here.


We discussed the challenge of marking and how Ofsted will check that teachers mark in line with a schools policy, and how this potentially may not be best suited to the teaching of mathematics. 

New GCSE Update:

In the league tables 5 will count as a ‘good grade’, but that for the first two years, grade 4s won’t need to resist. This post from Mel (@Just_Maths) was mentioned, and I find her timeline particularly useful.  

 New A-Level

This very useful document from The Further Mathematics Support Programme was shared which is well worth a read, and if you haven’t already take a look at their A-Level 2017 page.

Planning for the Future

Interesting discussions about CPD need for local departments in the coming years.



After a quick McDonald’s breakfast treat (the Suasage and Bacon sandwich is back!) I begun the early drive down to Peterborough for #mathsconf6. It was a fantastic day, and it was lovely to catch up with many friends again and meet. This post aims to provide a bit of a summary of the day… This time I have tried something new and wrote bits of this during the day (and then fleshed it out later) instead of writing lots of notes and then typing up a post. It has turned into a bit of a mammoth post!

The Opening Address – Mark McCourt – La Salle 

As always an amusing introduction to the event where he made the point that between Year 1 and Year 11 there are only 320 mathematical concepts that students need to master and they have to do this in around 1600 hours of maths teaching. I have to agree with him – surely we should be expecting everyone to manage this?! However, for some reason this currently doesn’t happen.

Andrew Taylor – AQA

Andrew started by sharing a few of the things he hates about assessment (this isn’t all of them, I was stupidly distracted making some last minute presentation changes):

  • Getting the grade is in the way of students getting a deep understanding of maths. “Surely we can find a way to get above that”.
  • Testing for the sake of putting numbers on a spreadsheet.

I really liked how he expressed a desire that kids grow up with a love of maths and that they appreciate the power of maths.

This speech I found really interesting as it wasn’t a conventional exams and assessment update;  it was a much more personal view of what he values in education and assessment. He mentioned the Cockroft Report as a a large influence on him, and I was interested to hear that Cockroft had expressed a concern about students taking exams where they can only achieve a third of the marks – this is certainly still an issue today. The SMP books also got a mention – I wish these books were still being published 🙁 Andrew emphasised that the most important key to success is “to go back to great teaching” and it is this that will get good results. The issue of jumping around tiers to obtain extra marks by “gaming” the system, for example, is going to be a lot harder to do with the new GCSEs and so we need good teaching to ensure everyone achieves.

A great quote from Cockroft is the one shown below on testing:

 Andrew then talked about what AQA does to help, giving the three points below.

  • To get the correct assessment you must have the curriculum expertise; and this is why they have an expert panel meeting of people involved in maths teaching and maths education.
  • Listening to people who use AQA and responding to provide the information that they require.
  • They work hard to gain the trust of people using the assessments and providing the results. AQA strive for consistency in the questions / papers they produce.

Speed Dating Segment:

Despite best laid plans I hadn’t got my thing to share and instead I talked about the upcoming KS5 CPD day I’m organising in August. I managed to pick up some great ideas from others though, which I have summarised below.

Megan Guinan – Megan first shared the Tarquin protractors; I’ve not seen these before and it is cool that they are flexible.

 She then talked about how to fold a Kite out of A4 paper and shared some excellent pictures of display work produced by her students. The link she shared for instructions for this is very good.

Deb Friis – Shared the MEI Introducing Trigonometry material – I particularly like the Aqua Ferris wheel idea for plotting sin graphs above and below the \(\) x-\(\) axis.

Jennifer Shepherd – I really really liked this idea for a constantly changing display that Jennifer shared

 It shows a progression of a topic through the Key Stages really well, and nicely allows students to contribute to a display that is easily kept fresh. Since the event Jennifer has tweeted a picture of her display,which I have included below.  

 Emma Brooksbank – Quick and easy marking: At then end of every lesson she puts a question on the board which students answer on a post-it note which she then collects, marks and returns. The idea of returning the post-it notes is simple but something I don’t think I have ever done.

Philippa Winstanley – Problem of the Week: All year 10s do this to develop their problem solving skills. She shared a lovely bookmark of problem solving strategies that she gives each pair. I wish I had taken a photo of this…

Following the sped dating it was on to the workshops.

Workshop 1 – How to Get The Most From Your Data

As I am a massive data geek I was really looking forward to Amir (@workedgechaos) and Ben’s (@MrBenWard) session

In answer to the question “What is data?” They provided the following 3 points (I particularly like the third one), before saying that it is not a panacea to all your problems and in fact often creates more questions than answer.

  •  The result of recording the results and attainment of students.
  •  A source of information to inform decisions about your department.
  • A means of providing objectivity around often subjective arguments.

Ben made a very good point that we should be “measuring what we value” and not just creating value in “what we measure”. This post by Tom Sherrington (@headguruteacher) about the following diagram is well worth a read and I am thankful for them highlighting this as I had missed it on Twitter.

Most maths teachers I am sure see themselves as being pretty data savvy but Amir talked about the fact that being data savvy isn’t just about collecting data it is about using it effectively. I really liked how they included the concept of “live data” being used in the classroom such as QuickKey, diagnostic questions etc which can be used to inform (and modify) classroom teaching instantly. I’m guilty of not including this in what I see as data, and to be honest I don’t know why. I should make more use of this kind of data, and I actually think that recording this data could give me some interesting insights over a longer time frame of say a few years – definitely something for me to consider…..

The use of a spreadsheet containing information about staff interested me as I think many teachers would find this controversial and be reluctant to have this used in a department. I feel that as long as this was used in a non-judgemental way (which Amir emphasised that it should be) it could be really powerful in terms of quality assuring opinions on colleagues, identifying each teacher’s strengths that could be shared with the department as part of some CPD and identifying any support that is needed to ensure a department of all staff are happy. Ultimately, I believe that being happy in the workplace has a large impact on performance (both of teachers and of their students) and using data can help to ensure happiness.

I also thought Ben’s comment about having a department meeting to complete data entry together was fantastic as this would definitely open up the department in terms of sharing their planning and assessment of students. The importance of teachers valuing their data I think is also key: if staff value their data then it is more likely to have an impact on their teaching and change aspects of their teaching. From a personal point of view, at the moment it is easy to not value the data that we are entering for our Year 10s, for example, as we don’t have much knowledge of how the new GCSE is graded. However, we should keep in mind that across the year cohort this data is still powerful.

I am a big fan of data, but it is certainly important to remember that the data cannot tell the whole story as every pupil is an individual and so the use of data needs to be supplemented with professional judgement and knowledge of the pupils which can only come with devoting time  to get to know them. But, being devils advocate here – could we just not collect more data?! (I think I may write more about this over the coming weeks once my thoughts have crystallised a bit).

Over the next few weeks I will certainly be thinking about and reflecting on these 3 questions:

  1. What do you need your data to do in your context?
  2. What gaps do you need to fill with your use of data?
  3. What will you do on Monday to begin to address those gaps?

Workshop 2 – Using Software and Games to Inspire and Motivate A-Level Mathematicians: The Case Studies of Sudoku and the Monty Hall Problem

I was co-presenting a workshop in the second slot with Jasmina Lazic. This was an interesting experience as I hadn’t presented a workshop with someone that I hadn’t met face-to-face before today but I think (aside from the technical issues at the beginning of the presentation) it went pretty smoothly. We will write a blog post about the session soon, but here are the slides:

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Workshop 3 – Classroom Culture

Bruno Reddy (@MrReddyMaths) led this workshop and  was very prepared and had his PowerPoint slides ready to download before his session. Bruno emphasised that to achieve the fantastic King Solomon Academy results the culture of the department (and school) was key.

The use of clapping as a way of saying well done and getting the class’ attention if they have been working in pairs was something that I had seen Bruno talk about before and I have used successfully in the classroom so it was nice to be reminded of this.

Bruno articulates culture as being what you stand for and how your actions articulate this. The emphasis on culture being from the beginning (or even before) was important, Bruno used some photos of KSA to ask us what we can spot about the culture from the word go. I was particularly drawn to the notices/banners on the ceiling as a way of making some of the cultural beliefs of the school evident. Bruno provided three Questions for reflection:

  • What culture is evident in your prospectus?
  • What culture is evident from open evenings?
  • What culture is evident from your Year 6 days.

I drove back from the conference this evening thinking about these questions, along with many other things!

The use of clicking to provide feedback to the teacher (and their peers) definitely made sense as it is certainly true that students don’t often nod along as you teach them. To me, this would feel strange at first but it is surprising how quickly it begins to feel normal.

Bruno talked about 3 goals for lesson one at the beginning of the year:

  1. Routineering: Clear entry routines for all classes so that the expectations are there from the beginning. A task present for them to do as soon as they get in. A good entry routine reminds pupils of the high expectations in the classroom. I particularly liked this slide:  The chance to see Bruno model establishing a new entry routine was fantastic – I am going to try this on Monday with one of my classes and I’m looking forward to seeing how it goes. As the routine begins to slip, he explained how it is necessary to bring this to the classes attention and explain how this would be sanction worthy. Bruno talked about Do Now tasks should give the pupils a chance to achieve and do well to be effective.
  2. Visioneering – “You have to go big with their dreams and win hearts and minds”I think this is an aspect I would struggle with as to be honest I’m not very good with acting but thinking a head to the beginning of next year I want to give it a go and am going to speak to my head teacher about making a results day video.
  3. Expectation Setting  – Bruno set out two non-negotiables: “100%” and “Every Second Counts”. The “100%” expectation expresses that there is only one acceptable percentage of students following an instruction and that is 100%. I’ve found it hard to stick to this recently but after this session I am feeling re-invigorated 😉 Bruno’s advice to use sanctions for situations where there is a black/white divide is great in my opinion; implementing a sanction for something like an instruction to talk quietly is much harder to do

Workshop 4 – Ask the Exam Board

Eddie Wilde (OCR), Graham Cumming (Edexcels) and Andrew Taylor (AQA) panel discussion. Below I try to present a summary of some of the answers given to a few questions. Unfortunately I felt that about half of this session was wasted as they were answering questions that have either been (very) publicly answered before or the answers have been in published materials, but it was interesting to hear a more human take on them.

  1. What do the new grades look like? Will there be grade descriptors? GC – “we do not know what the boundaries look like, we can model them to an extent. We know roughly that 4 <-> C as Ofqual have stated earlier. They know that the ramping of difficulty will be different to current papers. Foundation papers up to Grade B, Higher to start at Grade C. AT – If we put a 9-1 grading on last years cohort we could see how the grades fall out, but it is a completely different system so it is an unknown. We are required to have 50% of the higher exam targeting grade 7 and above but it is hard to see how this will translate to student performance. EW – Individual questions will not have grades attached, it doesn’t happen now and it won’t in the future.
  2. How are we supposed to assess progress with current students? EW – For a strong Grade C candidate you are looking at a 4/5. All boards have ways to assess current students, eg Edexcel’s Check In Tests. But they will not translate into grades on the 1-9 scale. GC – All boards to have mock papers written for October, papers designed for students close to the end of Year 11. I particularly liked this quote from Graham “Hannah’s Sweets will be Question 1 on all our papers now”.  AT – The ordering of pupils using old style assessments may not necessarily carry over to the same rank order in 2017 due to the different style of assessments.
  3. Can we have some guidance on what tier we should enter students for? AT – This is a common question at the moment. 10% of students getting a grade on  higher tier on the current specification ended up with a U when matched up with the new style of assessments (it is currently around 1% to 4.5%).  EW – If genuine C/D borderline students he would put them in for foundation as they could expect to be able to do about 2/3 of the paper, but he doesn’t want to tell you what to do. GC – If you want to use a current paper, look at the Results Plus data and chop off the first 20 marks to give an indication of how they would do on a new 80 mark paper. This would be optimistic though due to the changing of style.
  4. How many forms can questions about bounds and limits take? GC – “The possibility for bounds questions is of course unbounded”. Questions on this topic can be very simple but can be top grade questions. The higher grade questions are likely to be similar to current questions. Edexcel are trying to cover every style of question through their practice papers that they are releasing. One of the assessment objectives is to talk about assumptions, if a question was demanding more interpretation or analysis it would occur later in the paper. EW – Questions are not designed to trip students up.
  5. Trial and improvement, in or out? EW– Trial and Improvement is a perfectly valid technique but it is no longer a required technique in the new specifications so students could use this unless the question directs students to use another technique, eg iteration. AT – Trial and improvement is part of iterative methods which is only on higher tier content, and so questions that now involve trial and improvement must be set at a level to appear on the latter half of the paper  (unlike the current “draw a trial and improvement table for this question).
  6. Truncation – one question from OCR, are there others? Is truncation not as important as others? EW – the fact that something appears in a SAM doesn’t indicate an order of priority in terms of content.
  7. Will there be an alternative maths qualification for those students who would not achieve at GCSE standard? EW – Four is the new “good” for the first two years with the expectation that it will change to 5. Only the top third of those students who currently get a grade C are expected to get a 5. AT – Entry Level Certificate has been re-developed in parallel with the new GCSE specification and will be there to be used. We don’t know if students will be evenly distributed in the grade 5, it is an arithmetic thing. Grade 4 and 5 are comparable across the tiers and so parity will be achieved through common questions etc. GC – Edexcel have the Level 1 awards in Number and Measure alongside Level 1 qualifications in Functional maths.
  8. What are your views on the up coming changes to the new A Level? AT – we don’t have enough time. There are pros and cons to both modular and linear approaches. One of the pros to the modular approach is the increase in uptake for A Level maths and further maths in particular. Ofqual are very close to providing final guidance on assessment conditions.
  9. What can teachers be doing differently? AT – Teach well and concentrate on that. When a team of teachers are coming together before teaching a topic go to the spec and decide how to teach so it is not a walk in the park for top sets or how to bring it down to a lower level for lower sets.
  10. How do I promote a love of maths? EW – easy, enjoy it yourself and promote this to the kids. GC – show kids that they can love it. AT – Teach the subject and not how to pass exams.

Other Fun Aspects of the Day 

The Tweet Up was enjoyed by many and it was great to catch up with Kyle McDonald (@jk_mcd) from Wellingon College and be told about their Harkness resources for A-Level. Sample versions of these are available on their website and further information can be obtained from Aidan Sproat (@AidanSproat).

I was also pretty excited to meet the Corbett Maths (@corbettmaths) in the lunch break.

It was lovely to finally meet Naveen Rizvi (@naveenfrizvi) and have a decent conversation with her in the bar after the event. I also enjoyed chatting with Neil Turner (@Mr_Neil_Turner) about various teaching related topics.

I ended the day with a lovely catch up with Jo Morgan (@mathsjem) before dropping her off t the station and driving home. Jo has just published her 50th “MathGems” post and Julia (@tessmaths) organised the most amazing cake for her!

I also got possibly the best conference freebie ever today courtesy of OCR:

All in all a fantastic day and I’m looking forward to the next one in Leeds.