Author: tombennison

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.The 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)

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.

With this we obtain the following table of results:

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.

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=”http://prezi.com/j497g5f6_vep/?utm_campaign=share&utm_medium=copy” align=center width=600 lock_to_path=1]

Introduction

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

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.

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).

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.

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 

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 \(2/3\).

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.

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

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.

Marking:

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:

[prezi id=”http://prezi.com/j497g5f6_vep/?utm_campaign=share&utm_medium=copy” align=center width=600 lock_to_path=1]

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.

 

We are coming up to the end of @staffrms #29daysof writing and for this post I have stolen an idea from Stephen Connor and am going to answer a few questions about my education.

What were you like at school? 

Overall I was pretty good at school and did well academically. Looking back, as a teacher, I don’t think I would have enjoyed teaching me! If I didn’t particularly respect my teacher I think I possibly made that clear but never did anything outrageous enough to really get in to trouble. I was also pretty stubborn, and if I thought something was stupid I made it quite clear. 

Primary, secondary, college or uni – which was your favourite?

To be honest it is hard to pick here as I enjoyed all of them for different reasons. I guess overall my time in Sixth Form was probably my favourite. First time I got to only do subjects that interested me (specifically I wasn’t forced to do PE!) and I got to spend lots of time in friends. I can’t remember that time being too stressful work wise and it was a good experience to be involved with organising the Sixth Form prom.

Which of your teachers do you still admire?

From primary school it has to be Miss Carpenter, I really enjoyed Year 2 as there was none of the stupid “let’s listen to a boring story while children tie up the teachers hair with rubber bands” rubbishy that I endured the previous year. Miss Carpenter was also the first teacher to really stretch me with maths, providing me with extra questions. I still admire my Year 6 teacher, Mr Massey and it was sad that he died so young. I think my love of computing technology is in a large part down to him as I was allowed to be an “IT Monitor”, which essentially meant I got to spend a lot of time outside the classroom helping other teachers with things like printing and fixing basic issues with their computer – I don’t think that kind of thing would be possible now.

From my secondary schooling there are a couple that stand out: 

 The first is Miss Dodwell, who I had for English from Year 9 through to Year 11. I had never been particularly keen on English as a subject before, always doing the bare minimum in terms of work but I always really enjoyed Miss Dodwell’s lessons (though I’m not sure if I let it be known that I did). I certainly don’t think I would have done as well with my GCSEs if I hadn’t have had her, and her approach to teaching has had a pretty big impact on how I want to be as a teacher. My classes now often find it strange when I say that I really liked my English GCSE classes and will quite happily discuss with them one of their current books. 

The second has to be my A-Level maths teacher, Adrian Green. His lessons went a long way in leading me to choose maths at university instead of physics or engineering. I really respected how he always found the time to help with maths despite being a deputy head. He was a great A-Level maths teacher and my current teaching is very much inspired by his approach.

Which lessons do you still remember?

From Primary I remember a ridiculous craft session (in Year 1 I think) where I had to use pasta shapes to make a picture of a beach and a sandcastle. I wanted to use the same shapes for the beach and the castle but was told that I wasn’t allowed as you couldn’t make out the castle so they had to be different pasta shapes. I pointed out that it was unlikely that the sand castle would be made of different sand than was present on the beach but that point fell on deaf ears. This picture went straight in the bin as soon as I got in…

At Clarendon I can vividly remember GCSE drama lessons in the run up to performances and spending lots of time with Justin and Luke up in the Tech room on the Zero 88 Illusion 120 lighting desk and sound desk – lots of fun times were had doing tech with these guys. 

For Sixth Form I had to take three of my subjects at Clarendon, and two of them at John of Gaunt. Chemistry was one of the ones that I took at John of Gaunt and I can remember doing lots of cool experiments with Mr Treble.

What are the main differences between the classrooms you were taught in and those you work in now?

I guess one of the biggest differences is how rare blackboards are now, quite a few of my classrooms at secondary had blackboards and no digital projectors. I can remember in primary school it being pretty exciting when we got an overhead projector for the hall – how times have changed.

The rise of the Internet and the fact that these days you can get a reasonable mobile data connection almost anywhere mean that collaboration with other teachers (or people in general) that are miles from where you are is a very real possibility.In the past I have always used something like Skype for this purpose, but that really isn’t very suited to the job. Yesterday I had a web conference with Jasmina Lazic, my #mathsconf6 co-presenter. We will be meeting in person for the first time on that Saturday morning and so all of the preparation for the session and communication beforehand has been through emails. The Mathworks (the makers of MATLAB) must have a corporate contract for Cisco’s WebEx product, and for me a WebEx meeting was a revelation! 
Cisco’s WebEx  software enables sharing of files, sharing of screen etc which actually allows proper useful meetings to take place. I also couldn’t guarantee that the software would work on my school’s computers and so as a back up I installed the iPad WebEx app and I was very impressed with it. Using the app I joined the meeting smoothly, and you can even share files and screens from your iPad too which I didn’t expect to be able to do.  
If you have never used it before, I would encourage you to check it out. The free accounts allow you to host meeting containing up to 3 people.

Today one of my colleagues asked me how this followed:  Thinking back my response (in front of the class) was something along the lines of “that’s a fairly standard trick used when integrating”. This troubles me as it makes it seem that it has come out of nowhere by some kind of magical process and probably adds to the feeling that mathematics is a just a collection of rules to be followed.

A better description, in my mind would be something like the following: “When faced with an integral that you can’t do straight away, try to manipulate it to something that you can do by either ‘adding 0’ or ‘multiplying by 1’. That is you do something that changes the form of the integrand but which leaves it equivalent to what you started with. In this case you ‘add 0’ by adding 4 and then subtracting 4:”  

 Do you think that avoiding the use of the word “trick” is important? 

In my experience professional mathematicians use this phrase to mean a “commonly used technique” and so it doesn’t have the connotations that the phrase can have. 

Should we avoid the phrase? Or try to push for a “maths specific interpretation”?

A lot of you probably know that for a while I have had the idea of organising a CPD conference aimed at KS5 mathematics. I feel that this is an area that isn’t served as well as lower Key Stages (of course MEI, FMSP do fantastic work in this area!) and with the imminent changes to A-Level mathematics there is a need for some collaboration in my opinion.

With this in mind Tom Wicks (the University of Nottingham’s School of Mathematical Sciences’ teaching officer) and Ria Symonds (the FMSP Derbyshire and Nottinghamshire coordinator) and myself have decided to organise a day conference towards the beginning of the summer holiday. We intend for this day to provide some great CPD opportunities for teachers and plenty of time for networking.

The day will take place on the University of Nottingham’s beautiful University Park campus and we have tried to keep cost to a minimum as we realise that many teachers will be paying this themselves.

Nottingham is well placed for people travelling from further afield  and if you come by train you can now jump on the tram to the University which is very convenient.

The workshops announced today are just a taster of what will be on offer – we intend to have 4 workshop slots throughout the day with a minimum of 3 options for each workshop. Our closing plenary is going to be delivered by the mathematician and author David Acheson. He regularly takes part in the Maths Inspiration lecture series and if you haven’t seen him speak before we promise that you will enjoy it!

Tickets and further information can be found on our Eventbrite page and a webpage for the conference with further details will be going live within the next week.

We very much hope you can make the time to attend!

Back in 2012 I was asked to take part in some videos promoting PhDs in the maths department at Nottingham. I didn’t realise it was online until some people in one of my classes found it…

So for your enjoyment here is the awful video!!!

https://www.youtube.com/watch?v=8bHkDRR0PdU

 

On Tuesday 23rd of February I am hosting the NCETM’s #mathscpdchat on the topic of “Proving: how do pupils learn to do it?”. This post is a collection of some thoughts and opinions that I have had over the last week or so….

A month or so ago I watched this fantastic video from the Museum of Math in New York on the subject of “Proofs from the BOOK”, if you have an hour or so to spare it is well worth a watch:

To me the concept of proof is what sets mathematics apart from all the other sciences and gives me more faith in anything proved mathematically than a theory from another branch of science. As such I think it is crucial for students to be exposed to proof as early as possible (even if it is just by giving them exposure to the question “How can you be sure this will always work?”). To reduce mathematics to various procedures for performing calculations of one sort or another is a depressing idea.

Typically geometric proofs are see as a nice way to introduce proof, for example this proof of the expansion for \((a+b)^2 = a^2+2ab+b^2\) 

Personally I question whether geometric proofs are the best way to introduce proof to students as sometimes they struggle to see what the diagram is showing, don’t necessarily appreciate that some diagrams are only sketches etc.

I thought it would be interesting to see what some of my old colleagues (3 still in academia, one in teaching and one in industry) thought about teaching proof at secondary level – this is what they had to say.

Person 1:

“Yeah. I’m definitely for the idea but it’s about mathematical rigour and I’m not sure a lot of students would appreciate it at that age. At an advanced level, proofs are associated with theorems and methods. I think there’s very few topics you could apply it to, which make it accessible. Pythagoras’ theorem is one that comes to mind or other geometry related topics (angles, shape and area) are probably the most obvious and intuitive topics – but how do you make it interesting. I think it’s one of many advanced concepts that becomes harder to integrate into the early curriculum. Maybe with operations on fractions too?”

 

Person 2:

“I think it is a good idea to teach proof at secondary school level (although maybe KS3 is too early). The problem is at Uni level and higher proofs are important – if students haven’t learnt and understood them from an earlier age they do struggle. However, students who don’t go onto Maths (or other subjects where proofs are useful) probably won’t need/understand proofs – and you risk not engaging them. I’m not sure how to aim proofs at lower levels.”

Person 3:

“Good question, and in an ideal world, of course we should be teaching mathematical proof alongside everything else. Two problems: our “pure” idea of mathematics in terms of proof being the final word seems to conflict with others’ views of mathematics in terms of methods and formulae used to work out solutions to “real-world questions” – such as voltage in a circuit. So would people accept the idea that maths is actually all about proof, rather than bare calculations? The other is that proof does rely quite a bit on logic, and so you then have the question of how to introduce rigorous logic at KS2 or KS3. In particular, why proof by contradiction or counterexample works at all.”

Person 4:

“Regarding teaching proof to secondary school children, I think it’s important to introduce the idea to children at a much younger age than A Level.  I’m not sure how much better the new A Level is, but it’s barely covered there, especially without Further Maths. I think this serves two purposes. It better prepares those students that go on to study maths at undergraduate level, which is the obvious benefit. The second advantage is that younger students will be exposed to another side of maths that is different and hopefully more interesting than just calculating things and doing sums. Other subjects, such as English, history, human geography start to get analytical from around Year 9 onwards and this typically engages students rather than putting them off, so why should maths be any different?

I think the way to do it is perhaps introduce the ideas first in the current way, such as Pythagoras’ Theorem, triangle inequality etc. and get them used to using the result so they are fairly convinced it works. Then motivate the question “how can we show that this works in ALL cases” to get them used to the idea of general ideas. Some may attempt to just do lots of examples, but will hopefully realise that no list is exhaustive enough and they need to try something else.”

Person 5:

“Proofs are fundamental throughout the whole of mathematics and they are what make mathematics so different to others sciences. Theories in other sciences are based on our best knowledge at a certain point in time, they can be backed up with evidence, but could be disproved in light of new experiences. Mathematical theorems, on the other hand, once proved, are fact. As an example, consider Newton’s laws of motion, for 200 years these were assumed to be fundamentally true, until Einstein (and others) showed that they did not hold at small scales and high speeds. In contrast, Pythagoras’ theorem is as valid today as it was 2,500 years ago in ancient Greece. For those wishing to study mathematics past A-Level, understanding how to prove things is essential; indeed, a large part of a professional mathematician’s work is taken up in the quest for new proofs. Exposure to as many proofs as possible will leave students with the building blocks to develop their own novel proofs. For those inquisitive students, who do not with to pursue mathematics further, attempting to prove a theorem can lead to better understanding of that theorem, as well as teaching the student the crucial skill of logical thought. Of course, for some students, proving theorems is not required, a firm grip on how to use the theorem is the most important thing, and this is OK too. By teaching students why we need proofs and how useful they can be, hopefully we can avoid situations similar to the one I faced when lecturing engineering students. I was told that ‘I did too many proofs’, when, in fact, I had done none. It has been a while since I was at school and I cannot recall exactly what exposure to proofs I had. I have this vague idea that I may have been proving things, but I didn’t know it. I am not convinced that developing proofs amongst KS3 students is required, except for those with a definite aptitude for the subject. For older students, I think it is vital they are told why proofs are so important in mathematics. If a theorem is given, and its proof is feasible, then I think it should be explored in class. Geometry and number theory seem to me the most accessible areas where theorems could be proved. Later, trigonometric theorems could be proved, as well as those fundamental to Calculus. Students should also be encouraged to come up with their own proofs, after all, there are often multiple ways to prove something.”

I think the quotes above throw up some very interesting ideas that could be discussed in our twitter chat, for example:

• Should all students of mathematics be exposed to proof?
• How should we develop the logical thinking required to prove something?
• Does not teaching proof disadvantage children?
• Do we need to reformulate the idea of “what maths is”  for many people to understand the importance of proof?
• How do we make developing proofs engaging?

Last week I also ran a poll on twitter asking the question “When you teach ‘angles in a triangle’, do you demonstrate it or prove it? If you prove it please reply with how.” 61 people voted and the results are shown below.

To be honest I was a bit surprised by the outcome as I expected a significantly higher proportion of respondents to say that they proved it. Of course a demonstration of the result by either tearing the corners of a triangle and sticking them on a straight line or folding of the triangle is popular and a nice active way for students to visualise why the rule works. But one of the proofs is a classic application of the angle rules for parallel lines, as nicely demonstrated by Dawn (@mrsdenyer) on the back of an envelope, and so I expected a higher percentage of teachers to go through the formal proof with their students. 

Some time ago Professor Smudge (@ProfSmudge) shared a file of proof questions that were used with year 10s in the “Longitudinal Proof Project” research study and I can remember being particularly struck by question 4:

This is a really nice collection of likely responses you would get in the classroom if you asked this question – Eric’s is particularly interesting as I can imagine many students being convinced by this because of its algebraic nature, despite being nonsense. I haven’t managed to find the time to read through all the reports from this research project and the subsequent “Proof Materials Project” which are all helpfully available online on the mathsmedicine website.

Finally, for tonight, Mark Greenaway (@suffolkmaths) shared with me a fantastic page on proof from his sufmolkmaths website which lists many activities designed to promote students thinking about proof, including this one from Mark Dawes about the prime numbers. Showing that all primes (greater than 3) must be of the form \(6n\pm1\) is a particularly nice thing to discuss with students as you can develop their thinking from an intuitive grasp of the result to a proof involving modulo arithmetic.

Anyway, enough of my rambling – I am looking forward to the discussion tomorrow.