The Integral Maths Ritangle Competition

There’s a new kid on the school mathematics competition block, and I am pretty excited about it.

The great A-Level resource site integralmaths have recently announced a new competition for Sixth Form students called Ritangle . As part of this competition 23 questions are going to be released on 23 consecutive days and you need the answer to all of them to be able to work out the 23 character string to be submitted. You can download the poster for the competition (shown below) from the IntegralMaths website screenshot-2016-11-01-22-10-01


Ritangle have released 5 preliminary practise questions, the last of which went live today: prelime_twitter

I like this question; it is fairly accessible but sufficiently challenging to be interesting to solve. My students loved solving the first 4 preliminary questions. I say they are for practise, but correct answers to the first 5 will unlock a clue for the main competition.

One aspect of this competition, that is different to other competitions, is the encouragement to use technology to help with solving the problems. Being a computer coder and someone who enjoys computational maths this excites me. I hope that during the competition I will be able to blog a couple of times about the use of technology. Of course, in the spirit of the competition I won’t be able to discuss methods to solve directly competition problems, but I will be able to talk about computational techniques in general.

I’m really excited by this competition, and am pleased that some of our A-Level students want to to take part. I would encourage you (if you are an A-Level Teacher) to see if your pupils want to take part. If the preliminary questions are anything to go by then there are going to be some great problems to solve.



After a very early drive to Kettering (via Leicester) I was ready for my busiest Mathsconf yet. As well as running a workshop on Geogebra, I was representing the East Midlands West MathsHub (@EM_MathsHub), and launching a new A-Level textbook for which I am one of the editors with Tarquin Group (@TarquinGroup). Because of this I didn’t manage to get to go to the opening or the speed dating segments sadly but I did manage to make the main sessions.

My descriptions below are really just the rough notes I made during the sessions (with a bit of neatening up) and so are my interpretation of the sessions – I may have interpreted things said differently to how the speakers intended.

The Genius of Siegfried Engelmann in Practice – Kris Boulton
Following Kris’ Research ED Maths and Science talk I bought and have started reading Siegfried Engelmann’s “Theory of Instruction: Principles and Applications“. This is a very interesting book, but I am finding it hard going.  Kris recommended another book by Engelmann and Carmine “Could John Stewart Mill Have Saved our Schools?” Engelmanns work is logico-empirical in the way in which it was devised. The logico aspect involves deconstructing the topic to a very fine level of detail to the point that it is logically impossible for a student to mis-understand what the teacher is trying to say. The empirical aspect then tests this in the classroom and makes small changes to the programme, gradually making it better and better.

The idea of ruling out incorrect inferences made by student by deploying examples and non-examples is, when you think about it, is a blindingly obvious way to clarify a concept in the mind of a student. I think

Kris split most of maths into Concepts, Fact and Process – but emphasised that there are some aspects, such as proof, that do not fit these characterisations. His session today forcussed mainly on concepts and processes. Engelmann splits concepts into five different types (names by Kris though)

  • Categorical – Is this a pentagon (the response is always “yes” or “no” in the initial questioning sequence)
  • Fuzagorical
  • Comparative – Did gradient go up or down? Does the speed get
  • Correlated Features – You understand the concept by correlating it with others.
  • Transformation

Kris explained that he would have originally thought of transformation as a process as opposed to a concept.

Kris considered an expression like \( 2^5 \times 2^4 = 2^9 \) and considered some inferences that pupils may have drawn if this was presented as an inquiry prompt:

  • Only works for the numbers shown.
  • Only works for single digit integers.
  • Only works for positive integers.
  • Only works for rational numbers.
  • Only works for numbers (not letters).
  • Only works for real numbers.

The inference made by any given student will be a result of that persons given prior knowledge. All too often I think I underestimate the range of inferences that can be made by pupils – this session reminded me that i need to try and think about this more.

For me the most useful aspect of this session was seeing how Kris structured a teaching sequence for indices laws. It was split up into an initial instructional sequence, an initial assessment sequence followed by an expansion sequence.

Initial Instructional Sequence

  • \(7^{10} \times 7^3 = 7^{13} \)
  • \(3^{10} \times 3^3 = 3^{13} \)
  • \(3^{8} \times 3^9 = 3^{17} \)

It was emphasised that the addition in the first example was deliberately trivial so as to not cause difficulties.

Initial Assessment Sequence

  • \(2^9 \times 2^{10} = 2^{[ \quad ]} \)
  • \(2^9 \times 2^{6} = [ \quad ]^{[ \quad ]} \)
  • \(12^9 \times 12^{6} = \)
  • \( 193^7 \times 193^14 = \)

Expansion Sequence

  • \(13^{50} \times [ \quad ]^{10} = 13^60 \)
  • \(97^{ [ \quad ]} \times 97^{[ \quad ]} =  \)
  • \(3^1 = 3 \) (true or false)
  • \( 3^8 \times 3 = [ \quad ] \)
  • \( 2^5 \times 2 \times 2^{10} =  \)
  • \( (-2)^5 \times (-2) \times (-2)^{10}\)
  • \( \left(-\frac{1}{2}\right)^5 \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2} \right)^10 = \)
  • \( (\sqrt{5})^7 \times (\sqrt{5})^3 =  \)
  • \( b^7 \times b^3 = \)
  • \( 17^{[ \quad ]} \times 17^{[ \quad ]} = 17^{10} \)
  • \(25^4 \times 25^{[ \quad ]} = 25^9 \times 25^3 \)
  • \( 502^{5m} \times 502^{3m} = \)
  • \( a^{5m} \times a^{3m} = \)
  • \( a^{5m} \times a^{3n} \)
  • \(a^m \times a^n \)

I need to look more into this approach I think; for one reason or another I haven’t had much time to read up on this stuff sadly.

Kris gave us a glimpse of the planning process behind a series of lessons on simultaneous equations and how topics are returned to in different lessons.

Sadly I couldn’t go to his and Bruno’s later session on planning more than a lesson.

It was fascinating to watch a video of Kris teaching a mixed attainment Year 9 class about solving simultaneous equations. The precision and economy of the language was evident, with very simple sentences used to ensure students understand the examples. He gave examples of equations which could be solved (i.e. They only have one unknown) and those that can’t (i.e. With more than one unknown), before doing an assessment sequence of questions similar to the example.

We then watched a video of him adding equations together, where a student asked “What if you had 6x and 4y?” Kris explained how he doesn’t take questions until the end, as this question will be answered in the sequence anyway. He treated this as a transformation and so chose numbers where it was obvious that they were adding. Hearing about the success rates of students adding equations where the x and y terms were put in a different order I found impressive.

This session gave me lots to think about, and I think I will be thinking about these for a long time to come.

What’s the Point of Maths? – James Valori

In this talk James gave 5 reasons for learning maths.

  1. So people can’t lie with statistics: James gave us three examples of where statistics have been used to mis-inform. 80% of dentists recommend colgate where only 80% of dentists had listed Colgate amongst other brands. The reporting of the statistics about head injuries hides the fact that there are vastly more people in cars than on bikes. He then talked about Simpson’s Paradox
  2. It Rules the World: As an example of this, James talked about a situation where many people witnessed a crime and each statement rinfluenced other people’s statements – this of course then related back to Google’s PageRank model.
  3. Understanding the world and modelling: James talked about modelling the situation “should I buy or rent a house?” And considered various factors Roth considering. I think this would make a great Core Maths question. He then mentioned Dan Meyer and the idea of removing information to motivate why you would want to model the filling of a tank. This is one of Dan’s Three Act Maths activities. 
  4. It teaches you how to think: I liked the activity of James Mason that he shared \(\int_0^2 (x-1) \mathrm{d}x \) where the idea was to remove the procedural aspect of showing this fact and to develop understanding and expose the creative side of mathematics.
  5. Because it’s there: Why wouldn’t you study it?!?

I really liked the JavaScript implementation of a Quincunx that James showed, I am hoping that he will send me the code. 

An Introduction to Geogebra for Beginners – Tom Bennison

I could hardly not turn up to my own workshop. I shall post details of this in a future blog post.

How Misconceptions Change Over Time – Craig Barton

This workshop was fascinating, if a little depressing. Craig presented 5 questions that have been answered more than 5000 times on his excellent website Diagnostic Questions by pupils of a range of ages. He considered groups of 10-11 year olds, 13-14 year olds and 15-16 year olds. For the questions about calculating the area of a parallelogram, sharing in a ratio and solving equations the eldest group did poorly. For the fraction question there was only a 6% increase in in the percentage of people getting in correct over 5 years of schooling.

Craig listed 5 key takeaway messages:

  1. The language may change, but the misconceptions remain the same. Students’ understanding of fractions does not improve much in 5 years.
  2. The more skills and concepts students encounter the harder it can be to retrieve the correct one. Depth of understanding is so important.
  3. The older children get, the less carefully they read the question. Maybe there is also an argument that the way primary school students are being taught ratio is leading to a deeper understanding.
  4. Rules (inverse, change side etc) without the depth of understanding behind them, cause problems. Two biggest causes of mistakes on DQ are: 1) Confuse related concepts, 2) Misremember of misapply rules.
  5. Older students struggle with long division if they have not practised it in a while.

I wholeheartedly agree with Craig’s “Big Takeaway”

Thanks again to LaSalle for organising another great conference.

I’d rather die! – tonight’s #mathsjournalclub chat

Tonight (11th July 2016) at 8pm we are meeting to discuss the 6th #mathsjournalclub article which is the famous article by Brown, Brown and Bibby “I Would Rather Die’: Attitudes Of 16-Year-Olds Towards Their Future Participation In Mathematics”

This article is now almost 10 years old, and despite the numbers of students studying A-Level mathematics having increased since the 2002 data talked about in the report the participation rates of UK students in 16-19 mathematics is still often highlighted as a concern for the Government, being considerably leass than many high performing jurisdictions. IMG_1343

As teachers, I am sure that we have all heard mathematics described as “boring” and “hard” and this paper makes unwelcome reading, particularly as the sample of schools were significantly above the national average in terms of GCSE performance.

A few things to think about during the discussion tonight:

  • A large dip in percentages considering further study is shown between those pupils who achieved an A and those who achieved a B. Do you think this is the same today?
  • Does your school take people on to an A-Level if they got a Grade B?
  • How can we, as teachers, teaching within the confines of the system, tackle the perception that mathematics is boring?
  • Hatred is a strong emotion – is this to be believed?
  • What kind of activities should we be using to enhance “enjoyment” of school mathematics?

MEI Conference Day 1

Session 1 – An Introduction to Probability Distributions, Terry Dawson

I decided to attend this session as I wouldn’t describe Statistics as one of my strengths.

I use Geogebra quite a bit but I have never used the probability calculator option before. It works well on the iPad edition as you can see below, and provides a really nice interface for visualising probabilities and calculating them.

There was an emphasis on linking the teaching of the binomial distribution to the teaching of the binomial expansion. I do this, but I like the idea of making it explicit to students by getting them to do a dice experiment and relate the outcomes to the binomial distribution. The experiment suggested was to roll a dice 4 times and count the number of fives that occur. these outcomes could then be tallied up and you could also construct a tree diagram. Then you could relate the branches of the tree diagram to the number of occurrences of \(p^4, p^3q, p^2q^2,pq^3,q^4\). One resource that I particularly liked from this session was a card sort linked to the following article from the Daily Mail:

  • “A study has found that 41 per cent of Britons admit to regularly using their phones, tablets and eReaders when sat on the loo.” “Smartphones were identified as the preferred gadget of choice, cited by 65 per cent of respondents”.


I think it is great that the card sort contains differing values of \(n\) and \(p\) taken from the information given.

Following looking at the Binomial distribution Terry talked about using data sets to obtain data that is approximately normally distributed. This is likely to be interesting in terms of the new A-Level, so I am going to do a bit of work on it and then blog at some point in the future.

Session 2 – Core Maths: a Teacher’s Perspective, David Phillip and Claire Phillips

At my school I am going to start teaching a Level 3 Core Maths next academic year and so I picked this workshop to get some ideas for classroom activities.

The initial activity of recording when popcorn pops in either a maker or a microwave was nice, but it’s hard to get accurate data due to issues with reaction times etc. Though, of course, this could lead on to nice discussions about the validity of recorded data.

I had a very interesting discussion with Mike Ollerton about whether some fermi style questions are more worthwhile than others. I am inclined to agree that students will get more out of some fermi style problems than others, but quite how to taxonomies this isn’t completely clear. Do you think some of the questions shown below are worth more (in a measure of learning) than others?img_1334-2

There were many other great resources shared during this session which will definitely be worth a look once MEI put all the materials up. I quite liked  this matching activity:image3

Session 3 – Teaching A-Level in Early Career, Cathy Smith

I found this session fascinating (as I do with Cathy Smith’s work in general) as in my experience there often isn’t much attention given to preparing ITE students to teach A-Level, and the demands of teaching A-Level are different to teaching lower school. I can definitely identify with some of these issues in A-Level teaching that were identified in Cathy’s research.

Screenshot 2016-07-02 00.03.39

This quote from one of her study participants was particularly intriguing to me

Screenshot 2016-07-02 00.04.08

I can definitely see where this come from, but I find it a little sad as if you do spend time on the preparation then you tend to get an awful lot out of the lessons.

As an added bonus from this session Cathy got all the attendees to work in groups on the card sort shown below.Screenshot 2016-07-02 00.14.08 Screenshot 2016-07-02 00.14.16

I really enjoyed doing this card sort and I think it would be a very valuable activity to do both with beginning teachers and A-Level students. Cathy made an interesting point about the language used in this task. She had re-written a task from nRich  but been more consistent in the descriptions of the processes. For example each process starts with a description, followed by the \(y\) variable and then the \(x\) variable. This highlighted to me that getting the right amount of structure in an activity is very important. Too little is just as bad as too much! Interestingly, many people seem to start with the bacteria ones first, possibly for reasons of similarity.

Plenary – All Change in Post-16 Mathematics, Charlie Stripp

Charlie Stripp, the Chief Executive of MEI gave the first plenary of the conference. He gave a fairly positive talk about the state of mathematics education in the UK and the influences of all the changes coming through.

The picture of A-Level participation is very positive, though it remains to be seen how this will change in the future.IMG_1346


Charlie said that we have a “duty to use technology in teaching mathematics” and I completely agree with this. He also mentioned that there was research evidence that students owning their own graphical calculators helps their mathematics learning. I haven’t heard of this before, if anyone knows where this statement comes from please let me know!

Session 4 – Further Pure with Technology, Richard Lissaman

Being a keen programmer I was looking forward to this session with Richard Lissaman looking at the MEI unit Further Pure with Technology. A typical programming question from FPT is shown below

Screenshot 2016-07-02 18.27.21

I think the idea of doing this kind of thing in A-Level as so much of real-life mathematics is computational, and I would love to do MEI maths so that I could do this unit.

I was quite impressed with the CAS capabilities of Geogebra. I’m not sure how robust it is, but for a free package Geogebra is very impressive.image4

I’m glad that they have been able to incorporate FPT into the draft specifications for A-Level 2017. It doesn’t seem to have changed too much apart from replacing the complex variables section with content on differential equations.

Sixth #mathsjournalclub Article Announced

Thank you to everyone who voted in the recent #mathsjournalclub poll. The winner with 33% of the vote was the (fairly famous) article “I Would Rather Die’: Attitudes Of 16-Year-Olds Towards Their Future Participation In Mathematics” – Margaret Brown, Peter Brown and Tamara Bibby 

We will chat about this on Monday 11th July t 8pm, I really hope you can join us.

As usual, about a week before I will post some points to think about.

A-Level 2017

So… I’m expecting to be busy with blogging in the next week or so. I will be commenting on and giving my feelings about all the specifications and specimen papers being released for the new A-Level Maths and Further Maths qualifications tomorrow (9th June 2016).

I’m currently enjoying the AQA webinar and I love what I am seeing so far.

I have set up a Dropbox Folder where I will put all specifications and associated resources from all boards as soon as possible so that I (and you) can access them in one place!

I’m excited for tomorrow to come!

Last Minute FP2 Revision

It’s my last lesson with my Year 13 Further Maths students this morning before their Edexcel FP2 exam.

One of the things that I am going to do with them is an increased difficulty version of the June 2012 paper. This is a very similar idea to my increased difficulty M1 paper I posted a while ago. Essentially, all that I have done is remove any unnecessary intermediate steps and diagrams.

Screenshot 2016-06-07 05.50.05

You can find the paper here.

#mathsjournalclub Poll for the Sixth Chat

As promised, here is a chance to vote on the next #mathsjournalclub article. The poll will be open until the 9th June and then we will discuss the chosen article on Monday 11th July at 8pm – this will be the last chat before the summer holiday.

The choices are as follows (thanks to Rob Beckett and Danny Brown for some suggestions):


Vote here!!

I look forward to finding out what article you decide 🙂

Complex Loci

Complex loci and transformations in the complex plane are probably my least favourite topics to teach in A-Level Further Maths. It feels very negative to say that, but I don’t think I am alone! I also think it is one of the hardest topics for students to get their heads around in the further maths A-Level.

Last week I posted on Twitter asking if anyone had a complex loci card sort and Hannah (@LorHRL) pointed me into the direction of this one on TES. However being a Mac user it wasn’t easy for me to open the Microsoft Publisher file I decided to make my own using Geogebra.

Download the card sort here to use with your classes.