Monthly Archives: June 2019

Maths Conference Sheffield (Penistone) #MathsConf19

This blog post is a write up of my sixth maths conference. #MathsConf19 was held at Penistone Grammar School near Sheffield on 22 June 2019. Run by La Salle Education, the conferences are attended by around 400 maths teachers, trainers, publishers, suppliers, academics, tutors and others involved in maths education.

TLDR ; Both real and virtual double sided counters are very versatile, the term radius is a relatively new word to the circle party. Just a few alternative methods for constructions brings the topic alive.

Penistone Grammar School

A beautiful summer’s day for #MathsConf19 at the idyllic location of Penistone. Picture by @LaSalleEd

Pre-conference Friday night socialising

The Friday night pre-conference drinks are an invaluable opportunity for informal CPD in itself. Teachers have so many things they want to share and bounce their thoughts off others. This is the part I enjoy so much as a one man band online maths tutoring business who doesn’t get the opportunity to do much of this in person. Twitter is useful for these things but there really is no substitute to meeting in person.

I got to check in with teachers with the new A Levels for example and how teaching the first full first cohort has been. I was so impressed to meet a couple of teachers who teach everything from further A level maths to Year 7 students, from top to bottom sets. A lot of skill and versatility is needed for this which I need as well as a tutor. I am looking to teach further maths in the future so I asked some questions on the various modules for that.

Some of us were also doing maths games and puzzles. I was playing Albert’s insomnia bought in by Drew Foster. A game using mental maths and the order of operations. The beer, chatter, games and socialising continued through the evening. Unlike Bristol I took an early night after the bar closed this time. Thankfully there was no Atul’s insomnia after playing Albert’s insomnia and I was in good spirits for the following day of conferencing.

Introduction, twitter and a MacMillan award

La Salle CEO Mark McCourt kicked things off with an introduction to the maths conference. AQA maths head Andrew Taylor also gave a short talk with a “guess the year this question was set” slides showing how certain stylistic elements of questions go in an out of fashion from the 1940s to date. Mark also mentioned that there are about 300,000 maths teachers in the country and encouraged us to tweet about the event so others can get involved with the network and get out to know each other. I couldn’t agree more on the immense power gained from meeting and learning from other teachers. La Salle truly excel at creating this community; online and in person through these events. And you really can’t go wrong if the entire conference title is a hashtag itself!

Mark was pleasantly surprised by an announcement from the audience to receive the 2019 Douglas MacMillan award. All arranged and nominated for by Julia Smith. He always doubles the amount (with some generous rounding up) raised on the day from raffle ticket sales. Mark also has a new book out “Teaching for Mastery” which I really look forward to getting into. I’ve been to three of his full Complete Maths CPD days and continue to learn from his vast understanding of maths teaching.

Speed dating and some new ideas for teaching

Next up was speed dating, 4 ‘dates’ where each delegate gets 120 seconds to share their favourite teaching idea with another delegate. 120 seconds to share all my life’s knowledge on maths teaching and my greatest hits of ideas. This was going to be pretty difficult I thought. Coming out of it I learnt a lot from these dates about maths teaching; from goalless problem solving to a highly atomised approach in teaching some topics. I talked mainly about ‘backwards fading in example-problem pairs’ and the ‘pretest effect’ that I have been trialling out with some good success.

Workshop 1 : Double sided counters

This workshop was delivered by Jonathan Hall aka mathsbot. He has created a very rich resource of online manipulatives that I very highly recommend using. Double sided counters have been late to this manipulatives party for me as I still haven’t started using these with tutees. So this workshop would serve as the perfect intro to using them.

Double sided counters workshop.

It was certainly a lot more than just an intro. Jonathan showed how this simple and one of the cheapest manipulatives can be used to explain numbers, probability, algebra and proof. Each delegate had their own set of manipulatives to play with. To start off with we were given a hotel problem with 12 closed doors to try out in our heads. It was apparent very quickly that this would be pretty hard to do mentally. As soon as the counters came in, it was easy to solve the problem with the yellow side as a ‘door open’ and red as ‘door closed’.

Quadratic sequences using double sided counters.

Students can explore patterns using counters. Eventually coming to their own conclusions on the general formula of a pattern. Presentation slide by @studymaths

We then looked at sequences. Now I have seen these on 13+ papers a lot in picture form but there really is something else about having the actual counters in physical form and to actually build the patterns with your hands. There is something satisfying about the process of building the patterns by hand and there is no doubt this very act leads to richer understanding. We looked at a couple of sequence examples and while both examples were for quadratic sequences, the counters work very well with linear sequences as well. We were then shown some great examples of visual proof and probability questions using Venn diagrams. Everyone had an A4 sheet in which to make a Venn diagram and place the counters. Each application eventually leading to a generalised form where a total of n counters can be used. Probability being finished off by looking at a Simpson’s Paradox example case.

I was really impressed to see the counters being used for factorisation and finding the mean. In this example we had three separate groups of red and yellow counters (first row on image) then redistribute it all to get three identical rows of 2 yellows and 3 reds in each row, i.e 3(2y + 3r). The last row in the image showing elegantly how the mean is simply two yellows and three reds 2y + 3.

The first row of three separate groups rearranged to show both how factorising and finding the mean of yellow and red counters. Presentation slide by @studymaths

The presentation wrapped up showing the many uses of double sided counters. These being; Directed number, Ratio, Sequences and nth term, Proof, Averages, Collecting like terms, Factorising, Venn Diagrams, Probability, Tree Diagrams, Factors, Multiples and Primes, Square and Triangle numbers, Long Division and Modelling Problems.

I’ve already got myself a set of the counters and can’t wait to use these in my teaching.

Workshop 2 : Ratio and Proportion

The next talk was by David McEwan who is the Curriculum manager of Maths at AQA. Ratio, proportion, scaling, fractions, percentages are all of course linked topics. #MathsConf18 gave me a real appreciation of the idea of ‘scaling from unity’ so I was really looking forward to this particular workshop. Each one of us had a list of specification extracts and exam questions to accompany the workshop too.

We kicked things off by an open ended discussion on how one could define ratio (see image).

An open ended discussion on what ratio means.

David also mentioned that ratio and proportion appears in some form or other mostly on Foundation or Higher-Foundation content. Analysing the June 2018 series he mentioned that ratio and proportion questions appear almost at the start of the paper and are evenly distributed towards almost the end. And the proportion of proportion questions? Roughly 25% in Foundation and 20% in Higher. The pun here is unavoidable and bought some chuckles around the room.

David showed the equivalence of fractions with ratios leading on to equality of ratios. Finally linking it all up with a really neat cross multiplication method suitable for all ratio-equivalence calculations.

Using bar modelling as well each percentage problem could be solved using this cross multiplication technique once the problem was set up the right way. Find the percentage, finding the number, percentage changes, reverse percentages could all be done using the bar model. I really liked the idea of going for one consistent representation and following it through.

A neat bar modelling and ratio cross multiplying method that can be used to solve various types of percentage problems with the same consistent representation.

We were also shown some slides to remind us that an introduction to trigonometry is all about ratios as well and that students can essentially be introduced to trigonometry at earlier ages when introduced to right angled similar triangles. Also discussed were ratio tables showing the conversion factors for area and volume scaling and a few other concepts that showed the same thread of proportional relationships.  It was really good to get such a clear reminder of this.

Workshop 3 : The Evolution of Vocabulary in Maths Education

Next up was Jo Morgan with a talk dedicated to the use of words in maths and how words change, evolve or fade out of use through time.

Words change in general over time because..

  • They become obsolete (e.g ‘cassette’)
  • Go out of fashion (‘groovy’ or that 90s word ‘naff’)
  • They get superseded by newer ways of speaking (‘telephone’ becomes just ‘phone’)

I was very relieved to hear that “thrice” was once indeed a word. I used it when I lived in India and other countries. I stopped using the word in Year 11 when I arrived in the UK as my classmates told me that no such word exists. It must have been faded out here in the UK by that time. And apparently “twice” is on its way out now too. Being gradually replaced by “two times”. The words ‘Evenly even’ (divisible by 2 and then 2 again) and ‘evenly odd’ (divisible by 2 just the once) were also mentioned.

Jo Morgan discusses “Evenly even” and “Evenly odd” numbers.

Jo then moved on to use of some words in the context of solving and simplifying equations. Transposition: “The act of transferring something to a different place.” and ‘concinnation‘ (simplifying in an equation) make a regular appearance. And so do terms such as ‘destroying‘, ‘clearing the fractions‘ and a verb in its own right ‘to vinculate‘.

The word ‘concinnation’ made me think of the word ‘concatenate‘ (link things together in a chain or series) that I vaguely remember using in computing. The ‘concatenate’ command is used to stitch up two or more files into one big one using the MS DOS command prompt.

On to circle geometry next. It is hard to believe now but the word radius is one of the youngest words to be used in circles and has only joined the circles party relatively recently. Mathematicians managed for a very long time without the word and using ‘semi-diameter‘ was enough. The earliest reference to radius as a mathematical term in English is Hobbes writing in 1656.

After that we got into some quadrilateral language. Rhombus “So called from the Greek word Rhombos, which signifies the Fish called a Turbot, and the Quarrels of Glass in a Window.”  Rhomboids was also mentioned and discussed as what we call the modern parallelogram. And interestingly oblong is the old word used for a rectangle. The new oblong is a lot different to the old one in that way.

Jo finished off the workshop with a look at Welsh mathematician Robert Recorde‘s contribution to maths. His book The Grounde of Artes was written with a lovely tutor and student narrative with Recorde doing some tutoring to his imaginary student and the student responding back. Encouraging the scholar with “well said”. Good tutoring practice has remained unchanged all these centuries then!

The meaning of equals in the original language by Robert Recorde and its English translation. Presentation slide by @mathsjem

The meaning of equals in the original language by Robert Recorde and its English translation. Presentation slide by @mathsjem

One cannot mention Robert Recorde without referring to his most well known contribution, the use of the equals sign = After a little training on how to translate old English we were given the original text to translate to see if we could spot the mention of the equals sign. Recorde also invented new English mathematical words with many not surviving common usage today. Language is something that changes through time and perhaps in a 100 years some of the maths terms we use today will be obsolete too.

I really enjoyed this workshop, it flowed very well, was paced just right and left me with curiosity to go and explore more.

Workshop 4 : No gimmicks learning and teaching using Algebra tiles

This workshop was delivered by Bernie Westacott who I recently found out about after his video podcast with Craig Barton on manipulatives. I very highly recommend watching that video series. Bernie has an incredible depth of knowledge in the use of manipulatives and in particular getting the teaching for young children absolutely correct the first time round. Not only that but introduces algebra right at the start when children first start their maths journey without using the notations yet. I got to meet him the week before for the first time at another workshop in London and this week he had a packed audience ready to get into virtual manipulatives.

A packed hall for Bernie Westacott’s presentation. Picture by @LaSalleEd

The workshop was based around the use of virtual manipulatives app brainingcamp. We spent some time exploring the use of double sided counters and then algebra tiles. Bernie also uses real counters when teaching young children. Incredibly enough he does that without using any symbols or written work, yet he can start getting children to understand the ‘rules of negative numbers’ and even simple simultaneous equations. Young children are perfectly comfortable with the ‘upside down’ world of negative numbers for instance once they have had a play with the counters.

Algebra tiles in action on the Brainingcamp app.

Like Jonathan Hall he also started off with the field axiom of mathematics on the idea of there being an ‘additive inverse’ rather than ‘takeaway’ for the idea of subtraction. He stressed that there is no such thing as ‘takeaway’ at all. The app is a great way to show the additive inverse, the zero pairs can be greyed out when brought close to each other which is pretty neat. These zero pairs can also be used in teaching Chemistry as the positive and negative charges can be used to model electrons, protons etc. I use coloured dots in chemistry teaching as well. But that’s a seperate blog post altogether.

Bernie showed us very elegantly with the counters how a negative of a negative gets back to a positive. What it means to add a negative to a positive and to a negative. And the moment that got the biggest ahhh moment was a demonstration of how multiplying a negative with another negative gives a positive. The clarity and evidence given by this representation using the field axiom idea is irrefutable

And here you have it. Why multiplying two negative integers gives a positive integer. Very straightforward in the context of the ‘additive inverse’ field axiom.

There was a little demo of Alge disks then, which seem to be a halfway house between algebra tiles and place value counters. The difference being that instead of numbers the counters have x and y labels on them. Factorising using these disks seemed to tie in very well with the factorising I had seen earlier in Jonathan Hall’s workshop.

We then moved on to algebra tiles themselves. The tiles can be used for a number of things and I have been using them for nearly two years now. Though I only use them for showing the area model and how they can be used to factorise quadratic equations. There’s loads you can do with them, including zero pairs that disappear when merged together

Finally Bernie stressed the point made at the introduction once more that these representations are only there for students to slowly learn and get a feel and sense for what the abstract version of such representations should lead to. And that with time the use of manipulatives need to be faded out of use. They can of course always be bought back as and when necessary on a topic per topic basis in the non linear journey of learning maths as and when required. Which is exactly what I do as a tutor. Bernie now also has a video channel that I recommend watching.

Workshop 5 : Yes But Constructions

The final workshop of the event was delivered by Ed Southall, author of the books ‘Yes, but why?’ and ‘Geometry snacks’ fame. Constructions as a topic is really interesting to me, having done lots of constructions during my Mechanical Engineering degree. In first year drawings are all done on paper with proper equipment before moving on to CAD after that. And subsequently practical sheet metal requires the use of constructions with good equipment.

Ed Southall discusses other ways of bisecting a line.

Constructions for teaching school students is none of that however, it is mostly wobbly compasses, broken pencil leads and nothing ever quite lining up. And teaching it online is a pain as well with the document camera kinda getting in the way. Mathspad and Bitpaper help me though and are usually enough. But I just get the bare minimum done that way.

A sensible order of teaching constructions @solvemymaths

Before starting any construction work whatsoever it is important to make sure the very hardware students will use is in reasonable working order, fastening the compass screw tight so it is not wobbly and making sure the pencil is not very sharp. Keeping it a little blunt makes the lines a little thicker and gives scope for covering up a little when things don’t match. Just getting used to joining up two points into a line requires practice and fluency (this always seems to have some degree of randomness as the pencil may not follow the ruler track as we think it does) and getting used to drawing circles of various diameters.

We then moved to perpendicular bisectors, bisecting it the classic way. But making sure to draw the full circles so the symmetry and context behind it all is clear to see. In fact drawing full circles instead of arcs is always recommended. Except for when your line is at the bottom of the page, then what? Enter alternative forms of bisecting a line.

Another way of creating the perpendicular bisector of a line.

Next up was angle bisection “The Don” method and another one. We also did an exercise with circles and lines, eventually leading to something looking very pleasing to the eye in an islamic art type style.

“The Don” method of angle bisection

And I learnt about a special type of triangle called a Reuleaux triangle. I finally know what the shape of my guitar plectrum is called and why it rolls so nicely!

Reuleaux Triangle. Good design for guitar plectrums as well.

There was also drawing an incircle of a square, incircle of a triangle and a circumcircle of a triangle (see video).

While teaching and leaving a class to do the constructions Ed suggested having gifs of constructions on a loop so students can look at them if they missed a particular step during the presentation so they can go back to it and see the whole sequence. He does this very well indeed on his own twitter account with the gifs which I highly recommend looking at.

Overall another great workshop with loads of great ideas to take away and implement.

A superb experience from the Friday to conference day

The workshops and the entire day is very carefully planned to bring maximum benefit to the delegates and also to make sure as many teachers get to know each other as possible through the various tea breaks, lunch, tweetup event, exhibition, speed dating etc. Penistone was not an easy location to get to particularly for those like me who don’t drive but once you got there it was difficult not to be wowed by the idyllic location and the spacious school layout which made the day feel so much more relaxed despite so much going on.

I say it every time but quite genuinely this was again my most favourite maths conference. I learn so much from everyone, not just the workshops but from every conversation with a maths teacher. With so many new things to try out and full of inspiration I am ready and refreshed for some light summer tutoring followed by a brand new academic year.

Maths Conference Bristol #MathsConf18

This blog post is a write up of my fifth maths conference held on 9th March 2019 in Bristol. Run by La Salle Education, the conferences are attended by around 400 maths teachers, trainers, suppliers, academics, tutors and just about anyone who is passionately into maths education. The maths teaching ecosystem is very diverse indeed and I learn so much from fellow professionals who live and breathe maths teaching. In this post I cover my thoughts straight after the conference with some reflection on how I have used what I learnt nearly 3 months ago.

TLDR ; I got a deeper appreciation of the idea of unity, the unit, one.

Pre-conference Friday drinks

As on previous occasions I travelled up to conference city on the Friday afternoon to tutor my Friday evening students online from the hotel room. After tutoring I headed to the Friday night pre-conference drinks; one of my absolute favourite parts of the whole experience. Conference day on Saturday is an intense day so I really like the Friday to relax into it all over some drinks. At the bar I got a chance to check in and catch up with some of the teachers I’ve got to know through La Salle and I also made some new connections. Tutoring is an isolated profession and it is so useful to share experiences with other maths teachers.

When the bar closed there was only one thing to do, go to another bar. And like Manchester #mathsconf15, the last men standing ended up exploring the nightclub scene. An epic time was had dancing to 1990s tunes.

Onwards to conference day itself.

Workshop 1 : Rekenreks rock, the new manipulative on the block

This workshop was delivered by Amy How who has recently become the ambassador of this invaluable manipulative. I learned about this manipulative recently from the Bernie Westacott video podcast with Craig Barton and from Mark McCourt’s workshops on CPAL and Multiple representations. Since I work with a few Dyscalculia students I am always seeking new ways of developing number sense for students. The Rekenrek turned out to be a tool that can be used for far more than developing number sense alone.


The Rekenrek is an invaluable manipulative. This is a 10 row version.

I used the Rekenrek for about two months previous to this conference for number bond work, doubles and subtraction but nothing more than that so far. So frankly I was blown away by how much more is possible with this manipulative from Amy’s workshop. Starting from the very simple idea:

  • Build it
  • Say it
  • Write it

The building part is moving the red and white beads, saying affirms the language and writing it gets it into symbolic form. Amy showed us how to use the beads to show the times tables in action through various arrangements. The patterns that started to emerge by exploring the six times tables got some aha moments in the room.

The 100 bead rekenrek can be used to build fluency for the number bonds of 50, 100, rounding and working out the times tables. We moved to the idea that a 10-row 100 bead rekenrek could also just represent one. Meaning that the same manipulative can also be used to embed the idea of fractions, decimals and proportional thinking. It is so clear and obvious when two fifths of 30 is explored by selecting two rows (of 10 each) out of five. There is an elegance and clarity to using this manipulative and while I initially made the mistake of thinking they are suitable for just number bond work, I have started using these with older students for fractions, decimals, percentages and ratio work.

One other small but important thing I implemented straightaway after this workshop was the idea of “one finger one push” to move the beads. Previously if students were to count the beads out by touching them and moving them one by one then I would let them do that. I realised that for one of my tutees who counts in clusters of one, this had to be corrected immediately. 3 months on and I have used this manipulative with various tutees now, including a superb virtual version of it on the mathsbot site.

Amy’s workshop had primed me with the idea that we can make something ‘one’ and then work from there. This theme of one or the unit then ended up repeating in nearly all the workshops I attended.

Workshop 2 : Cuisenaire rods, Metallica and the one

This workshop was given by Drew Foster who is a big fan of puzzles, manipulatives and hosts #BrewEdPreston. I got a set of original Cuisenaire rods back in 2015 after my Dyscalculia training with Patricia Babtie. For a good three years I used them mostly for number bond and number sense work. I have become more drawn in and fascinated tutoring number bonds over the last 3 years, there is something very profound about them. A few months ago I got some training on how to use Cuisenaire rods from the La Salle CPAL and Multiple representations CPD courses which has accelerated my use of these to no end. I am starting to find so many ways in which you can use these, with students at various stages of their maths learning journey. They are so useful for secondary maths as well; from arithmetic sequences, linear equations, ratios, percentages, area scaling, volume scaling and surface area to volume ratio. I am merely at the tip of the iceberg in terms of their full application.

Drew gave each table a set of Cuisenaires to play around with and move about. He stressed the idea that we should try and see things from the children’s point of view. You have to move these around physically first until stumbling into the right answer through the right combination of the rods. I was making the mistake initially of trying to do things in my heads and then move the rods about. We did a bunch of basic exercises at first from setting up equations in colour, to staircases and a pyramid.

The moments that wowed us all were how a simple ‘up and down staircase’ which looks like a Stats distribution and is linked to square numbers as well. We were also shown how Quadratic factorisation could be shown using the rods. Although I have used Algebra tiles for quadratic factorisation I never linked it using it with cuisenaire rods.

Cuisenaire rods.

We were shown the video below which is the ‘cuisenaire rods way of the zen’. All to the soundtrack of Enter Sandman by Metallica, my head was bobbing along for that one for sure or was that from the Macarena a few hours earlier at Popworld?

The fact that you can make one anything you like and then the other rods take on a different value is very profound. A couple of tutor colleagues who were at the workshop have now bought their own cuisenaire sets, remembering not to buy any sets with graduations or marks on them which Drew stressed are not the right type of rods. They must have no writing or markings on them. The beauty of these rods is that by keeping them free of any markings you can make them whatever value you want them to take. I was so inspired by this workshop to use this manipulative more and a few months on I have innovated their use in online tutoring ever more. I intend to write more about this in future blog posts.

Workshop 3 : Creativity and Curiosity, there’s more to maths than convergence

Even if creativity is not assessed it is important

As a tutor I often feel the pressure in helping a student prepare for an imminent high stakes exam, particularly for Year 11, 13 or 11+/13+ prep. Every tutoring hour has to count, not just the teaching itself but in supporting the tutee, communicating how the system works to the parents and showing the tutee how to revise by themselves in a structured way. Teaching for a test or exam is very ‘convergent’ in that way. There is a right answer to be got as quickly as possible.

Andrew Sharpe’s presentation reminded us why we really got into maths and how creative it can be. We started off with a number grid puzzle and a coordinate grid exercise. It was ok to fumble about and come to the solution, and in that fumbling process is the joy of discovery.  Trial, improvement, iterative processes are all part and parcel of the overall learning process in maths. He encouraged us to make up our own puzzles and for students to do so as well.

An exercise in divergent thinking that I remember was of the ‘Alternative Uses Test’ where 10 alternative uses of a brick should be brainstormed. Highly successful people are creative thinkers and problem solving itself is a creative endeavour. Some students like to think divergently rather than convergently, which is certainly something I have encountered as well. Andrew gave plenty of examples using number maths and geometry for the Alternative uses.

This workshop has inspired me to bring that creativity to more of my students. I do some fairly divergent teaching to those who are homeschooled but there is no reason to have elements of it with other tutees. During preparation for an exam gears have to be switched completely but there’s always some scope for creative and explorative thinking. Most importantly this presentation helps me give myself permission that divergent teaching is all part and parcel of overall teaching and in getting the best out of students. Andrew’s presentation for this workshop with some very useful extra resources are on here.

Workshop 4 : Unit Conversions, metre rules and following the multiplicative arrow

Jo Morgan’s in depth series of presentations are invaluable and I often refer to her presentation notes on the other in depth series, all available on her site here. For this topic on unit conversions Jo started off by mentioning that while unit conversions is not a big topic like solving quadratics or angles, it is a topic that often carries a lot of easy marks and students miss out on these.

She mentioned that units are covered first in Year 3 and then consistently again every year until Year 6. In fact students see mixed units like 1 kg 200 grams very early on in their maths journey and often year 6 students can know their millilitre <–> litre conversions better than Year 11 students. I often teach Year 6/7 students back to back with Year 11 ones and have observed this too. Somewhere in the middle years students don’t see unit conversions as often and lose that fluency in this topic. Jo showed us several examples of GCSE students losing fairly straightforward marks on unit conversions, both in the higher and foundation tier from examiner reports and example scripts (AQA and Edexcel boards). Quite often students were losing marks in just getting the simple decision on whether to multiply or divide the conversion factor.

The presentation then followed the structure of following this strategy in converting units.

  • Step 1: Be fluent in multiplying and dividing by powers of 10
  • Step 2 : Memorise the conversions
  • Step 3 : Perform the conversions

Step 1 is a self evident prerequisite and so Jo covered the other two steps.

Step 2 is all about memorising and recalling the conversion factors. There are surprisingly few conversions to be memorised for GCSE maths in the full range of conversions and all these are given below from one of the slide presentations. I have highlighted in purple the ones that need to be memorised.

GCSE maths unit conversions (from presentation by Jo Morgan)

This then lead to the discussion and history on why the adoption of the metric system. One interesting fact that I didn’t know was that the prefixes for bigger units (kilo, hecto, deca etc.) are Greek and the smaller ones are Latin (centi, deci, milli etc.). Definitely very useful to know and a great point of discussion for A Level Physics students who need to be aware of the fuller spectrum of units.

To remember units using manipulatives Jo mentioned that an actual metre rule is essential to show to students and water bottles also help. I really like bottles of water in addition to how useful they are to get a feel for volumes. They have lots of lovely Chemistry stuff on the labels, including the ions present, the charge on them, the exact type of plastic (polymerisation) and the concentration of the ions (a compound unit in chemistry that GCSE students also need to know). Lots of Chemistry and maths in water bottles.

Step 3 is carrying out the strategy once students have memorised the actual conversions. There are a number of methods possible, from ratio tables to Don Steward type grids and for calculator papers it can all be done on the new Classwiz calculators. This is very useful to know as I have a few IGCSE students who would benefit from getting the new calculators as both their papers are calculators.

The method that really appealed to me was ratio tables. I have seen them appear on twitter but only during the talk I realised how invaluable a technique it is, a very simple layout with the arrows representing the direction of multiplication. Going against the arrow means to divide. This will be very handy indeed for many of my students. The method that was very popular amongst maths teachers was “Last man standing”. It is a neat method using the idea that (100cm/1m) can be expressed as 1 and then the units can be ‘cancelled down’. I’ll refer to the excellent interpretation of Last Man Standing by Mr. Bracewell and Jo’s presentation slides referenced earlier.

This presentation gave me plenty of thought on another interpretation of the unit as being one and how you can use it convert between units. I tutor the Sciences at GCSE as well and dimensional analysis is very useful indeed there. Ratio tables to convert between units is something I certainly want to implement and last man standing is a useful new technique to have in the teaching toolbox.

Nearly three months on the thing I have used most is the use of ratio tables with my two GCSE retake tutees, particularly the idea of ‘going against the arrow’ to divide. Teaching students for an imminent exam when time is short does require such shortcuts which I feel can be justified for the long term benefits it will bring to the student’s future chances. As for what division sense actually is and how to convey the idea to students over the long term, that was covered in the next workshop.

Workshop 5 : Time to revisit…Division, the beast of division tackled

This was the first time I attended one of Pete Mattock’s talks having chatted to him before about the use of Algebra tiles and other manipulatives. In the preview blog post for this workshop he mentioned that “Professor Emeritus in the department of education at the University of Oxford calls division ‘The Dragon’. Those pupils who slay ‘The Dragon’ tend to go on to do well in mathematics; whilst those who don’t tend to struggle from that point on.”

Cuisenaire rods and counters

There’s certainly a lot of truth in this. Making sense of division has many important implications further down the line, including unit conversions which I had just seen on the workshop before. By being able to make sense of the division process from the very outset using the appropriate concrete and pictorial methods as support, a long term framework can be built up for pupils to understand the process of division. Concrete and pictorial methods eventually being faded out. I am currently reading Pete’s book ‘Visible Maths’ that looks at a variety of topics from different representations of whole numbers, powers and roots, the laws of arithmetic to algebraic manipulations. Division being just one of the topics in that book that was presented during this workshop.

Double sided counters and cuisenaire rods were the two manipulatives used during this workshop with delegates having ample opportunity to move the manipulatives around on each table. Counters being used to show the discrete view of division. 12 ÷ 3 can be shown as 12 being put into groups with 3 counters in each group (creating 4 groups) or 12 being shared into 3 shares (with 4 counters in each share).

Left : 3 counters in each group (creating 4 groups) Right : 12 being shared into 3 ‘shares’

Using the other side of the counters can then be used to show division where negative integers are involved. The red side of the counter represents -1 while the yellow side represents +1. Putting a red and yellow counter together creates a zero pair. Having a line of 12 reds represents -12 which are then put into groups containing three -1 counters, representing -3 together. Thus creating 4 groups.

Red counters for demonstrating division with negative counters (see blog post)

300 ÷ 20 can be demonstrated by making the 20 as 1 (in the same way as shown in Drew’s earlier workshop). Once the cuisenaire train of 20 is 1 then it is easy to make a multiplicative comparison with 300. 15 multiples of the ‘1’ unit.

And how to tackle the beast of division itself? Division and fractions. Again a multiplicative comparison is used to compare part to “whole”. The part being assigned the value of 1. Pete talks about fraction division in his podcast with Mr Barton, which is an excellent and more comprehensive description of the idea.

We finished off with a practical context where a simple speed, distance, time context could be visualised using the idea of multiplicative comparison. Speed is the distance travelled in unit time, i.e the distance traveled in one hour, one minute, one second etc. The example used was finding the time when 75 miles are covered at an average speed of 20 miles per hour. Once again 20 is made into the unit, i.e one. There are 3 of those complete units of 1 hour time ‘blocks’ and a remainder. The remainder being 15 out of 20 = ¾.

75 miles covered at an average speed of 20 miles per hour. 20 is called the unit, i.e one. There are 3 of those complete units of 1 hour time ‘blocks’ and a remainder. The remainder (not drawn in) being 15 out of 20 = ¾

The idea that really stuck with me, from Pete, Jo, Amy and Drew’s workshops being that the ‘unit’ is of profound significance. Exactly what that means was summed up by a tutee of mine in the last days of GCSE exam preparation with the quote at the end of this blog post.

Summing up the whole experience, this was by far my most favourite MathsConf so far. I am grateful to all those teachers who put in so much effort to run these workshops for the rest of us. And big thanks to the La Salle team and visionary Mark McCourt for putting these together.

Nearly 3 months after this conference and just days before the final IGCSE paper, my retake tutee exclaimed with delight in that light bulb ‘aha’ moment that we teachers all live to experience.

Once you know one, you know everything!

Mission accomplished, so many things fell into place for my tutee in that moment of realisation. I can’t wait for #MathsConf19.