This Wednesday was the Nottingham Regional Final of the UKMT Senior Team Maths Challenge and I was lucky enough to take my team of Sixth Formers who did really well (a very proud teacher here!).

I thought I would quickly share a couple of simple tricks for squaring numbers, which I am sure that most of you will know but I had never really had much of a need for things like this until I started preparing teams for this kind of thing.

The first one is for squaring numbers that end in 5. For any number ending in 5 take the digits before the five, call this \(n\) and then multiply by \(n+1\) before appending 25 to the result. For example for \(325^2\) work out \(32 \times 33 = 1056\) and so \(325^2 = 105625\). To show why this works just consider the following.

The last line of this is merely the product of \(y\) and \(y+1\) with 25 appended to the end.

Another trick makes use of algebraic identities to change the squaring of an arbitrary number into some easier squaring operations.

Consider \(a^2 = (a+b)(a-b) + b^2\). This can be used to help us work out some squares, for example:

This month the ATM have produced a special issue of their journal “Mathematics Teaching” focussed on assessment and made it freely available to all until the end of January 2016. People on Twitter seemed keen to have a special edition of #mathsjournalclub (in addition to our usual bi-monthly chats) looking solely at an article / articles from this issue.

So…. We will discuss two articles on Monday the 11th of January at 8pm. The articles with their abstracts are below and following this there is a link to the google poll to select our two favourite articles.

“The three most important things in teaching? Assessment, assessment, assessment!” by Ruth James – This piece describes how assessment becomes an integral part of classroom practice. The process begins with an anecdote from experience as an NQT and builds into a thoroughly documented approach to the demands of contemporary pupil assessment. The ideas and strategies have been developed in the real classroom,
in real time. The focus is Key Stage 1, but the issues relating to such things as fluency and mastery have wider appeal. By any yard-stick this is a considered professional approach to assessment in all its forms.

“Encouraging students’ formative assessment skills when working with non-routine, unstructured problems: Designed student responses” by Sheila Evans – Effective assessment is necessarily composed of numerous strands and strategies. Some strands will be more preeminent than others, but sound assessment depends on data derived from multiple sources. Designed student responses have the potential to gather new insights into student achievement and understanding. Here the case is well made, and students respond positively to the introduction of this type of assessment activity.

“Rich Task Assessments” by James Towner – Old habits endure because they offer a sense of
security. This security is often related to the investment of considerable time and effort into a past initiative, or directive. However, when change becomes inevitable, winning the arguments that are driven by the comfort of inertia can be difficult without the exposition of a clear vision for future ways of working. In terms of assessing and monitoring progress in mathematics through
Key Stage 3 here is a vision. The vision involves the collaboration of colleagues, progression maps, pathways, student involvement, and communicating with parents. This description shows the insights and experience
that have come together to build an assessment model that seeks to be both meaningful, and representative
of student understanding. Is it a model that can win the hearts and minds of all the stake-holders?

“Thinking outside the tick box” By Mark Pepper – The National Curriculum [2014] for mathematics
has a section headed ‘Solve Problems’. This article provides ideas that respond to the criteria set out in the Programmes of Study for Key Stage 3. The problems outlined relate to the ‘everyday’ and are not age or ‘level’ related. These problems have ‘worked’ for the author. They might well engage learners in other classrooms.

“Teaching and Assessing Mathematics in Primary Schools Assessment Without
Levels” by Michael Smith and Ray Huntsley – In schools and education the ‘one-size-fits-all’ maxim is an omnipresent source of tension between practitioners and policy makers. This tension might be interpreted as ‘all change’, but the reality is that ‘some change’ is often preferred to ‘no change’. Change requires time, and time is a resource that needs to be managed in schools with the same rigour that is applied to other costly resources. This is a model for assessment that has been developed as a whole school approach, having regard for ‘where the school is’. The time taken to read this piece will be well invested, but as the author modestly states; “What
I have attempted to do is take the limited guidance and make it work for the children I teach.” Teaching, learning, assessment – are surely all about children.

“Assessment: Beyond right and wrong” by Matt Lewis – This is an account of an on-going evidence-based project in a number of schools in London. The professionals involved seek to set the complex process of assessment of learning in mathematics in the context of its educational and social purposes. The article draws on examples
from recent work on approaches to the assessment of reasoning and problem-solving. The challenges are not underestimated given that we seem to be experiencing a culture that might be regarded as valuing measurement of performance regardless of the quality or meaningfulness of the data generated. The debates will continue, but the enduring need for statutory assessment should not dumb- down the process.

“A little knowledge …” by Mike Askew – The reported teaching of mathematics in Pacific-Rim countries can be relied upon to generate strong reactions from practitioners working in English classrooms.
The genesis of such reactions apart, there can be no substitute for first-hand experience. Here such first-hand experience, together with perhaps some pre-conceptions, reveals a strategy that is both powerful and simple. As
to be expected this piece mixes anecdote and pedagogy in equal measures, providing a counterbalance to some of the more strident reports published in popular media. There is also a suggestion that Arithmagons could become big in China.

“In conversation” by Mike Ollerton, Claire Denton & Jenni Black – Sadly, conversations about things mathematical, let alone assessment, are often a casualty of the ‘busy school
day’. This account showcases the power of a three-way conversation that focuses on reflection from innovative practice. Meaningful assessment is a challenge, particularly assessment formats that can be meaningful for both teacher and learner. Nobody claims the challenge is insignificant, but well considered responses do signpost a way forward. This signpost might indicate a direction
of travel that leads to improved communication and understanding of student achievement in mathematics.

“Making learning visible in mathematics with technology” by David Wright, Jill Clark & Lucy Tiplady – Innovation through research is necessarily a rigorous process that requires commitment, cooperation, and time. This report of an international project describes the progress observed thus far, and gives insights into some aspects of the work in progress. The technology is used as a device to focus on student responses to problem situations and to focus on assessment. The project sees assessment as learning, and seeks to enable learners
to benefit from assessment opportunities. A single tablet computer in the classroom can enable these ideas to
be developed in any classroom where the focus is on learning. Who knows, the image of a student response to a mathematics problem might become the new ‘selfie’?

“Damned if we do, damned if we don’t? Baseline testing our youngest learners” by Helen Williams & Sue Gifford – This piece presents a strong note of caution to current proposals that impinge on early years learning. The educational policy machine grinds inexorably to produce changes that, all too often, are not supported by evidence based practice or well reported to those likely to be most affected – schools, teachers, learners and parents. While the proposals are currently badged as non-statutory, who knows how things will morph with the passage of even
a short time. As the authors state with both conviction
and concern: “This is particularly harmful for mathematics learning which suffers relentlessly from negative
attitudes. In short, testing in this way gives a distorted message of what is of most value in education, leading to impoverished learning and risking the depression of future achievement.” Perhaps policy makers should arrange to visit a reception class some time soon.

“It’s easy to judge” by Ian Jones – Can you imagine assessment in mathematics
with no form of marking scheme? How about peer assessment based on a subjective assessment as opposed to any notion of correct v incorrect? This account of contemporary research suggests that using Comparative Judgement, teacher and peer assessments are statistically remarkably similar. Could this be the beginnings of an innovative approach to assessment in mathematics that has a place at all levels of achievement and performance? There is the opportunity to use
the dedicated website, and to be part of the research programme. Too good to be true? … perhaps. Maybe it’s a case of ‘only time will tell’.

“Year 7 – the problem with ‘expected levels’” by Naomi Harries
– Proposals to introduce a new SATs test for Year 7 students who do not reach national expectations in their Year 6 SATs are likely to affect 20% of students
in the cohort. In simple terms this will represent a new and significant challenge for teachers working with lower-attaining students. Here the author outlines an approach that might support the teaching and learning of mathematics for this group of learners. The response is positive in that it seeks to deploy strategies that target tasks and activities that are rich in assessment opportunities. The objective being to build a pattern
of assessment that can help to foster mathematical capabilities and student confidence. Transition from one phase of education to the next is often a cause for anxiety, without the prospect of further external assessment in Year 7.

“A ball bounces and – that’s all …” by Bob Burns – This is a ‘fly-on-the-wall’ account of a lesson and the initial interactions between teacher and students. This sparks a good deal of student-to-student interaction together with the occasional teacher intervention.
Not quite the same as a video-clip, but let the imagination create the scene.

“Young Children’s Mathematical Recording” by Janine Davenall – How children record their mathematics provides valuable insights into what they know, what they understand, and what they choose to record. What learners choose to record depends on what they feel able to record and
the confidence that it is meaningful. This account of observations made within a normal classroom situation demonstrates the spectrum of strategies that learners use to communicate an activity that might be described as mathematical story-telling. Interpreting the early emergent recording of mathematics, when well informed, can be a powerful classroom tool.

The Google Poll is located here. This one is slightly different to the usual one in that as we will discuss two articles there is a selection to make for first choice and then a second choice.

The scores will then be wighted (a first choice scores 100, a second choice scores 50) and the two articles with the most points will be discussed on the night.

Please make your choices by Tuesday 1st December so that we all have time to read the articles over Christmas.

This morning I made the journey down to the University of Leicester’s Fraser Noble Building for the latest ATM/MA East Midlands branch session.

Today it was John Mason – a big name in the Mathematics Education world. His session was entitled “Teaching More by Teaching Less: Getting Learners to Make Use of their Natural Powers” and I had a great time at this thought provoking session and I was lucky to sit with Pete (@MrMattock), Mel (@Just_Maths), Em (@EJMaths), Andrew (@ApApaget), Mary (@PardoeMary) and Andrew Price (@ColonelPrice) where we had many great conversations. It was also good to see a strong contingent from The University of Nottingham’s CRME in attendance.

For this post I am going to share some of the tasks we looked at and my thoughts on them, but I’m going to start with a few nice quotes from John Mason himself.

“Mathematics is working on problems.”

“Learning to read graphs is much more important than learning to draw graphs.”

“My job is not to choose questions kids find attractive, it’s to choose questions with the most potential for learning maths.”

Sadly I think these three highlighted thingd probably happen far too little in the classroom today with all of the current assessment and accountability pressures.

One thing from the sesssion that I liked straight away was the use of magic squares without numbers.I love using magic squares in lessons, but I admit I tend to use them as a way of practising procedues, I hadn’t thought of using them to “tease out” students’ reasoning. John pointed out that there was lots of research that students who have been deemed “low achievers” in mathematics can reason mathematically, it is their low level of numeracy that gets in the way of them demonstratng this in the classroom – so why not use tasks that don’t require numeracy. I firmly believe that improving numerarcy is important, but so is developing the mathematical reasoning skills and so I like the idea of using tasks that develop this without requiring a high level of nmeracy. Indeed, I suspect that developing reasoning will improve their numeracy anyway as the students will be able to reason more about the number system. Fo those of you who aren’t familiar with magic squares, they are arrays of numbers (commonly \(3 \times 3\) such that the column sums, row sums and leading diagonal sums are all the same. John suggested removing all the numbers from a magic square and using just mathematical reasoning to show certain properties. He started by asking us to reason out that the blue coloured squares in the magic square below add up to the same value as the red coloured squares.

Of course this is relatively clear, but you can develop this idea into some harder examples

Or even extend the idea to larger magic squares

I am definitely going to try this in the classroom.

This session has also inspired me to use more prompt photos and as students to formulate their own mathematical questions. Of course it is important for students to develop fluency, but I also think that to become a mathematician they need to be able to pose interesting mathematical questions.

John showed a vey cool applet that he has written using Cinderella. It contained a straight line of fixed length divided into sa blue section and a red section. The blue line segment was the perimeter of a square and the red line segment was the perimeter of a triangle and as the joining point moved from left to right the square gre and the triangle shrunk as you would expect. One interesting question to pose is “At what point is the perimeter of the square equal to that of the triangle”. Of course this is true when the join of the red and blue line segments is in the middle, but originally I (along with many others) made the mistake of thinking that we had to work with he ratio of the side lengths. The app allows you to plot the area and perimeter of the two shapes as the joining point moves, as shown in the video below.

This is a fantastic visualisation of something that I actually found quite counter-intuitive until you think about it in terms of the underlying maths. There are many other questions that you could pose about this visualisation.

We then looked at some more tasks around the topic of area and perimeter from this excellent sheet that John provided us with

I particularly like the “More or Less Perimeter and Area” and “Shape Signature” tasks and would quite like to work them into lessosn this year at some point. I really want to find the time to explore the questions posed by the “Shape Signature” task.

Towards the end of the session John talked about “Multi-Level Initiating of Tasks”: Between the three there is clearly an increase in the structure and scaffolding given to the student. At what point does this turn from beig a help to leading to a dependence on the part of the student and reduce their ability to reason mathematically?

All in all it was a very thought provoking session and I have taken away lots of things to ponder. John has kindly made all the materials from his talk available online here; in particular his PowerPoint slides are available.

I may be wrong, but it seems un-common to introduce the Poisson distribution in S2 by discussing its original application to real data. Before teaching in schools I developed an “Introduction to Quantitative Research Methods for Engineers” course and this was how I introduced the poisson distribution in the lecture notes. I think it is a great way to bring a bit of the history of mathematics into your teaching, and in my opinion certainly motivates the Poisson distribution more than the fairly awful way it is introduced in the Edexcel text book.

The distribution function for the Poisson distribution is as follows: If \( X \sim Po(\lambda) \) then \( P(X=x) = \frac{e^{-\lambda}\lambda^x}{x!} \)

In L. Bortkiewicz’s book of 1898, “The Law of Small Numbers” he discusses the number of fatalities in 10 corps of the Prussian cavalry that were the result of horse kicks. He has 200 years worth of data, and to this data he fits a Poisson distribution after calculating an appropriate rate parameter.

From the table it is clear that a Poisson distribution, with rate parameter \(0.61\) provides a pretty good fit (of curse the goodness of fit can be calculated!).

As an aside, tabulating Poisson probabilities is extremely tedious… You can save yourself a lot of time by using WolframAlpha, especially if you know some Mathematica / Wolfram Language syntax.For example to tabulate the values for a Poisson distribution woth rate parameter 2, for example you can do the following

In the above the command “N[Table[(Exp[-2]*2^x)/Factorial[x],{x,0,10}],4]” generates the following

Here I have used the Table command to tabulate between \(x=0\) and \(x=10\) the Poisson distribution function with parameter \(-2\). I have then wrapped this with the N command so that I get values to 4 decimal places output instead of the default exact expressions.

Whilst WolframAlpha’s natural language processing is good, using the syntax of the underlying language makes it a lot more powerful.

On Saturday 7th November Ed Southall (@solvemymaths) organised a morning’s conference. This was primarily aimed at his trainees but a few tickets were made available generally. It was definitely worth making the trip up the motorway in the awful rain to go to this and was great to catch up with Ed, Hannah (@MissRadders) and Beth (@MissBLilley) afterwards. Andrew (@ApApaget) and Megan (@MeganGuinan1) but I didn’t realise until the end.

The day started with Craig Barton (@mrbartonmaths) talking about his excellent Diagnostic Questions website. I confess that as yet I hsven’t really used this website to its full potential but it is one that I want to explore more. The GCSE Maths Essential Skills quizes are excellent (a question from one of these is shown below) as each of the multi-choice answers exposes a misconception on a common GCSE topic.

Craig’s weekly insight blogs are excellent and well worth a read, I found the one about writing algebraic expressionsparticularly interesting.

Another feature of this website that I like is that students have to explain their reasoning (admittedly a contentious issue at the moment) when they select an answer which is great information for the teacher to have. If they make a wrong selection they are shown a range of correct explanations from their peers.

After Craig I went to the lovely Beth’s session on using the history of maths to help teach maths.

I learnt a few fascinating things about the Greeks that I didn’t know, including an ingenious method to measure the radius of the moon.

Beth has written a great (first) blog post about her presentation here and has allowed her files to be made available either on TES or here.

Following Beth’s presentation I went to Laura Hadfield’s workshop “Making Differentiation Visible Throughout the Lesson”. For me there wasn’t much new here, but I was really interested in how everything in her lessons is traffic-lighted for differentiation. She doesn’t use the usual traffic light colours instead going for blue for “Practise”, green for “Mastery” and pink for “Apply”. I do something along these lines in some lessons, but certainly not in every lesson and it was interesting to hear of a department where this is a consistent practise. Like most things, I imagine it has more of an impact when consistently done. One comment that Laura made intrigued me – she said something like “there is no problem with using textbooks as long as the instructions aren’t ‘do question 1 to question 20’.” To me this is a problem with the textbook, not the instruction. If the textbook is well designed then working through a sequence of questions should develop fluency and understanding well. Admittedly there would have to be some teacher intervention but a good textbook would have pedagogically well designed questions.

To end the day we had the legend that is DonSteward present some excellent things. This was genuinely the best hour and a half of CPD that I have had.

Don opened with the following picture (I think the inquiry maths prompt has been inspired by this) as an investigation to do with Year 7 – “Do we get the same a nswer if we go either way round the diagram?”. Following a discussion students can then substitute some small numbers in and investigate what happens: Is there a pattern? Can this be generalised? What if we change the numbers/operations on the branches? I really liked his suggestion of how to introduce algebra into this task and it is definitely something I am going to try in the classroom. The idea being that when asked “how do we show this for all numbers?” many students will suggest trying a big number such as a million, but we can’t fit all those zeros into the circle and so we write ‘m’ and work through creating an algebraic expression. I think this is a fantastic way to introduce algebra in a non-threatening way. He then moved on to suggest that a student should think of a number (Beth must have been very excited to be included in Don’s presentation here!) but not say what it is, so we have to give it a letter, ‘b’ say. We then pass this through the operations deriving the same expressions, but as we don’t know what this letter stands for we have shown that the pattern always holds. Don has also written this activity up back in 2013 in his excellent post “Both Ways”.There was a large emphasis in Don’s session on visual representations for proofs, and Ed has posted the visual proof of this pattern in his write up of the session.

Don then talked about some great proofs that made use of the following diagram

I particularly liked the one for the arithmetic-geometric mean inequality (or two numbers) shown below:

Don then discussed some investigations involving relationships between the area and perimeter of rectangles – he has recently written about these on his blog here.

I was also pretty excited to see an alternative construction of an angle bisector that is is considerably easier than the one traditionally taught.

The session ended on a mathematical approach to Magic squares, but Ed has written an excellent post about this, so head over to solvemymaths.com for more details.

Overall I had a fantastic morning in Huddersfield, thank you Ed for organising this.

George Boole is credited with many things, including

Founding invariant theory.

Invention of boolean algebra.

Intorduction of mathematical probability.

Boole’s first mathematical interest was is the realm of mathematical analysis and differential equations. He published two textbooks that also included soem original research contributions concerning thr algebraic approach to differential equations. Broadly speaking we denote the derivative as the application of an operator \(D\). So that if \(D(f) = u \) then \(f = D^{-1}(u) = \int u \). For example we could have the following Differential equation (DE) \( \frac{\mathrm{d}^2f}{\mathrm{d}x^2} + 3\frac{\mathrm{d}f}{\mathrm{d}x} + 2f = \sin (x) \) can be written in operator notation as \( D^2(f) + 3D(f)+2f = \sin (x) \) or \( (D^2+3D+2)(f) = \sin (x) \). For his contributions in this field he was awarded The Royal Medal of the Royal Society.

In 1841 Boole published one of the very first papers on invariant theory, this paper is credited by Arthur Cayley in his paper of 1845 which is often credited as being the start of invariant theory.

Of course Boole is most famous for his contributions in the world of logic. He first pulished a book on logic in 1847 whilst he was still in Lincoln, before publishing again once he had moved to Cork.

I was interested to learn that Boole only allowed the operation of Union when the sets were disjoint, Evegeny then presented a nice table comparing modern notation with that of Boole.

Boole is also famous for his contributions to mathematical probability, most notably his name lives on in Boole’s inequality.

In conclusion “Boole made a giant step towards mathematics as a truly abstract discipline, causing a paradigm shift, giving mathematics enormous scope and potency” (Khukhro, 2015). Evegeny has made his PowerPoint slides available here.

The poll is now closed and the winner is…… “Contrasts in Mathematical Challenges in A-Level Mathematics and Further Mathematics, and Undergraduate Examinations” by Ellie Darlington, and published in the IMA Teaching Mathematics and Its Applications journal. This article achieved almost 50% of the votes, and I am really looking forward to discussing it.

We will be discussing this at 8pm on Monday the 7th of December. I know it will be the run up to Christmas, but I hope lots of you will still be able to make the discussion.

As usual, about a week before I will post some possible topics of conversation or things to think about. I hope you enjoy the article.