Fish don’t know they’re in water. If you tried to explain it, they’d say, “Water? What’s water?” They’re so surrounded by it that it’s impossible to see. They can’t see it until they jump outside of it.
Sitting in a park in central London on a warm, sunny June 2018 day I knew something in me had shifted forever. My head was full of swirling mathematical thoughts processing just what had happened during those last hours. Before this day, I had no interest in working with multi-base. It seemed impractical and if I am honest, an unnecessary indulgence. But something in my brain was broken and unresolved after I was unable to do an exercise at that CPD workshop. And I had to get to the bottom of it. So I began a journey of curiosity, frustration and wonder into the world of multi-base.
That CPD day was delivered by La Salle Education CEO Mark McCourt and was titled “Making Maths Memorable”. I had learnt many things that day, including the split attention effect, non-examples and the careful use of silence when presenting visual information. The workshop wasn’t a multi-base workshop as such but it had clearly piqued my interest. Over the years I started adapting ideas from the workshop to my seemingly alien world of online tutoring. My teaching was to be transformed.
It took me several months to do anything with multi-base after that workshop, perhaps because I had no starting point or representation to grip the idea. Until I heard about numbers, numerals and digits in a podcast by Mark again. It turned out I didn’t know abut this either. How could I convey these early, basic ideas to my tutees if I didn’t understand them myself? There was a big, gaping blind spot in my teaching staring back right at me. It existed because none of these things are tested in the current English maths curriculum or any of the other systems I had encountered. And I’ve tutored in over 25 countries! So if it isn’t tested for in any curriculum then is it worth learning?
Well obviously yes, because mathematicians are curious and seek enlightenment. I knew that understanding number in depth held an important key, not just for my lowest attaining pupils, who I felt would benefit most from the knowledge at first, but for all pupils. I looked up various definitions and started exploring the world of numbers. Having grown up in Libya and India I was already familiar with modern Eastern Arabic numerals and Hindi Devanagari numerals.
If you want to master something, teach it. A great way to learn is to teach.
Richard Feynman
I found a Year 7 tutee to test my ideas and understanding with multi-base. I had tutored her since Year 5 and her parents were open minded on my teaching of ideas beyond the curriculum. She was a diligent, curios and bubbly learner. She was honest and clear live when teaching : “I don’t get this”, “What do you mean when you say numeral?” etc., This batting to and fro was what I needed to tweak and refine my delivery in real-time. This is a fairly routine aspect of tutoring, a conversation and constant running of experiments to gauge where the tutee is at.
With my help she made an odometer type counter on our digital writing platform Bitpaper by programming the numerals in steps of 1. Pressing the forward or rewind button (Undo and Redo) would get the odometer to count up or down in various bases. Much like the counter of fuel at a petrol station does in base-ten. An odometer counting up is all I could think of and I didn’t come back to multi-base again for a few more months. Unknown to me there was a bigger issue I had to resolve first.
An immediate problem I faced was that the number system in base-ten was so deeply ingrained in my mental programming that it was difficult to think outside of it. Whenever I saw 14 written in various bases, I read and saw it as fourteen. I needed numerals from another language altogether to break the association of base-ten with the way digits are combined to make everyday numerals. I found the perfect bridge both for me and tutee by using Hindi (or Devanagari) numerals.
It is somewhat embarrassing to admit that despite having grown up in Libya, Yemen and India, I had no idea that the modern numerals we use today are Hindu-Arabic numerals. It is never too late to learn of course and it sure makes for good stories with tutees. Incidentally, Libya has a lot of Roman monuments preserved immaculately, so I was surrounded by a blend of ancient Roman numerals, Hindi numerals and Eastern Arabic numerals during my childhood days in Tripoli. It felt amazing to look back at something so familiar and find deeper meaning through these numerals.
Base-three Diene blocks from my set.
Over the next two years I attended various other La Salle CPD days in London on multiple representations. Each one had a mention of numbers, digits and numerals and counting in different bases. I read books, found videos and podcasts too. I started to see how the area model could be used to understand number systems in other bases. In particular the use of Dienes blocks which Zoltan Dienes used in various other bases. I even got hold of an incomplete set of physical wooden sets of multi-base blocks, which looked very cool. My understanding was starting to deepen, but I didn’t feel confident enough to start teaching with virtual multi-base blocks, the only ones I could use online with my tutees. Base-ten blocks were no problem at all and I was using them before anywyay.
In all this time, I started to fine tune my skills in teaching through various representations. Particularly the use of the Rekenrek, algebra tiles, Cuisenaire rods and two sided counters. After seeing a back to back session on the use of two sided counters by Jonathan Hall (aka mathsbot) and Bernie Westacott, I started to realise how incredibly effective two sided counters could be for teaching so many mathematical ideas.
Fast forward to March 2020; a global pandemic happened and life turned upside down. I was fortunate that I kept tutoring online as I had done for all these years and that time was well spent in exploring virtual manipulatives to teach students in far flung corners of the globe. This experience of operating in the virtual 2-D world of online tutoring was about to pay dividends in how I could understand and teach multi-base. Not just to my tutees but as CPD later, both receiving and delivering it.
#MathsConf23, like many events, went virtual a few weeks into pandemic. The “Explode your mind with exploding dots: A global phenomenon” presentation was given by James Tanton who radiated a teaching life force and infectious enthusiasm. By then I had already been using dots/counters/rekenreks, so I got this representation immediately. For the first time, multi-base started to make clear sense and a whole new universe revealed itself to me. I was breaking out of the shackles of base-ten.
I then started helping my tutees prize this association apart. Disrupting someone’s worldview is no easy task but my tutees trust me. Besides, younger tutees had not lived with base-ten for as long as I had, so they were fairly quick to grip multi-base. Nonetheless, I took an incredible amount of care and caution to make sure that tutees do not get muddled up. Always starting from an open ended exploration of numbers, digits and numerals before presenting clear cut definitions. Regularly reminding them that a numeral is the written code and representation of number, whereas number is the thing itself, the idea.
Once familiar with binary with Hindu-Arabic numerals, I encourage tutees to make up their own digit symbols. Here we have a comb (one) and pumpkin (zero) from a tutee.
I sidestepped working in multi-base with Hindu-Arabic numerals with my tutees and started using a mix of ancient Egyptian and Roman numerals first. Roman numerals turned out to be a great bridge into this world for my tutees as they were already familiar with them. And in this last academic year, I have thoroughly explored exploding dots as my go to representation for multi-base, both with whole numbers and whole + fractional numbers.
I have now emerged on to the other side; now when I see 14, I see a numeral that is one-four. A symbol that could represent various other numbers depending on base choice. I even get the joke: “There are 10 types of people in this world, those who understand binary and those who don’t”! And since September 2020 I have been covering a range of multi-base ideas with tutees, from long addition algorithms to division. It is only a few months on but I am already seeing them develop robust and flexible generalising skills. They are becoming more mathematical and some of them are already comfortable working in base-x.
I have a lot more to write about teaching multi-base and will do so in a series of blog posts, particularly on the idea of place value. Once you are enlightened about place value, it is impossible to teach place value in just base-ten. Because teaching it in just base-ten does not feel like teaching place value at all. Education and CPD is the way out of this, just like it has been for me. Place value in other bases has also been mentioned by Charlotte on her blog post and there are some really great definitions of various related ideas on Mark’s blog post too.
Exploring ancient Egyptian numerals. A task my tutees do and one I set for my #MathsConf25 workshop.
So how to get started on multi-base (radix)? You can of course look it up online, in various books or take CPD. The exploding dots website is a great place to get started on it. I presented a workshop at #MathsConf25 titled “An introduction to Multi-base” which was specifically designed for complete novices with my fresh viewpoint. I really encourage maths teachers, tutors and pupils to explore the world of multi-base. There is something profound missing if you don’t understand it.
Disrupting one’s own existing worldview and frame of reference is no easy process. But as teachers we know this better than anyone else. An incredible journey awaits for you if you haven’t explored this world yet and want to dive into it.
Everything changed the day after I came back from the weekend #MathsConf22 trip to Manchester on 15th March 2020. The seriousness of the current pandemic truly hit home. It would still be another week before the government would stop politely asking people not to go out but to enforce a lockdown. I’ve been in my London flat ever since that weekend and have barely ventured a mile away from it during my daily walks. Avoiding the high street and tube station. It is bizarre as a Londoner not to have used the tube station for two and a half months.
In the immediate aftermath of lockdown I witnessed tutor colleagues lose their jobs, Bitpaper go from bust to survival and I threw myself into training friends to learn how to work online.
My March 2020 step count went down and screen time went up.
The constant attention to news, social media and the lack of being able to go out enough made for a cocktail of restlessness and being on edge. I have no garden in my small London flat so each outing was invaluable respite. As the clocks changed and spring arrived, the earlier sunrises further reduced sleep hours.
Nonetheless, given everything I am still extremely fortunate to not have had more serious family and work related stress. I was already working from home and in that aspect business continued for me as normal. Indeed, one of the few constants I was able to provide my tutees was our same tutoring slot at the same time of the week. That wasn’t to last long as like many tutor colleagues I too lost tutees for a myriad of reasons. I was having daily conversations with parents and realised the pressure that families are under. It made total sense that for some families tutoring would not be a priority at this point in time.
I was disappointed to see fake news that the tutoring industry was booming, a more detailed and comprehensive Sutton Trust report confirming that in the UK the amount of tutoring taking place was down overall. There are various global Facebook groups of tutors with thousands of tutors in them. We get a far more accurate and nuanced picture of the industry through hundreds of daily conversations between tutors. Almost every news story on the tutoring industry has an agenda.
Transferring my online teaching skills – Two to tango?
My job to transfer my skills had begun back on the train from Manchester to London already. I saw a Facebook message from my Argentine Tango dance instructor that they were offering free online trial lessons. The thing that was their life and works purely in the physical realm was seemingly going to be gone for months. I knew I held a part of the solution as I had pushed online tutoring a long way. I had chosen to stop tutoring in person altogether in 2016, so working from home and teaching was now the norm for me. While still on the train I got my instructor to install zoom and we made a video call while I was on a train going at more than 100 miles per hour.
After 2 months of running several group and 1-to-1 classes online my dance instructors are one of the few who have quickly adapted to the art of teaching online and done it well. I have a lot more to write about this but what they achieved is a truly rare event. There have been a few other exceptions, those tutors that were already adaptable and creative in their tutoring. But on the whole mass attempts to deliver online tutoring have been sub standard to say the least. Live online teaching is a highly sophisticated form of teaching requiring its own separate pedagogy and intersection of skills. Like learning anything worthwhile it requires consistent, deliberate effort and constant improving. Those who tutor well online, tutor all or mostly all online.
One of the most effective ways how online learning can work was for me to actually just live stream examples of this teaching. This happened in a CPD context and is mentioned later on this blog post.
Bitpaper or bust – further consequences of the rapid shift to online
The whiteboard company I work for called Bitpaper was facing an emergency. Tutors who previously either dabbled in online tutoring or were complete newbies were forced into online tutoring by the pressure to save their livelihoods. I am glad that the perceived quality of Bitpaper from existing users meant that so many people chose Bitpaper. The product was still in beta mode and therefore free at that stage. Nothing comes for free though as there were server costs amongst the other costs that included mine and other people’s time.
Bitpaper is a central part of my online tutoring set up.
With a huge increase in user base, server and AV platform costs were escalating out of control. This was running the company and the team into the ground very quick. The levels of stress placed on the team and me as the social media community and PR person was immense. Plans to commercialise were brought forward by a few months and it was now imminent or we’d go bust. We made an announcement to commercialise which went down well with most but there was an inevitable backlash by a minority. After a live stream to a big facebook group to fully explain our reasons we could breathe easy. We commercialised and I went out of emergency mode to focus on other aspects of the product. I remember sleeping well that night for the first time in several days.
Livestreaming
I had numerous messages and requests from tutors on training them to become online tutors. I was way too busy with my own tutoring, life and Bitpaper to do this. I had already prepared detailed guides on online tutoring a long while back. I can only think of maybe 10 tutors at most who are expert at tutoring maths online. Along with me, their voices got drowned in providing what is the correct and expert advice on online forums.
I offered free training to those who wanted it but on the condition that this training is live streamed so that others can see this too and a recording is kept. Not many signed up to this as one might imagine. But the few who did made it totally worthwhile. I then extended this to live streams with other experienced online tutors, something that had been on my to do list for a year already. Our voices were being drowned by constant panic and terrible mass novice to novice advice. The video platform was far better to present this expertise.
I now knew that live streaming was going to be the most valuable use of my time and expertise. Previous experience with tutors showed me that only a small minority of tutors put in the effort to take CPD or read books. The few that do value CPD turn up to meetups and events anyway. The next step for me was logical, I turned to the mainstream school teaching community which does actually care about CPD and has had well evolved networks formed over the years. I approached La Salle Education CEO Mark McCourt to live stream on twitter as I had known him from attending several CPD days and maths conferences. This would leverage on an already well formed network on twitter in a novel and more human way.
#MathsChat Live streams on twitter and facebook.
The #MathsChatLive live streams were born and took right off. I have learned a lot from this level of live streaming. They have around a thousand viewers on each live and more that view the recording later. In many ways all my experience of tutoring online, making tutees comfortable in a virtual environment, holding the space and using tools to teach online was preparing me for this. We are only a few live streams in and I’m keen for this to evolve and get better with time. I will write a separate blog post on multi live streaming for CPD as it has been a fascinating learning experience.
And then the rest of my life moved online too
On 1st March 2020 in a studio under a church on a sunny Sunday I took my first ever singing grading exam. Grade 3, Trinity Rock and Pop. I had been taking singing lessons consistently for about 2 years, every week in fact. It was all online. I had convinced my singing teacher to teach me online 3 years ago and we had found a way for it to work for me. The only part of the entire grading that took place in person was the actual grading exam itself. I am glad to say I passed with a distinction. Bring on grade 4.
But ultimately we exist in the real world and not in a virtual space. I take two dance classes a week and a pilates class as well. The reason I was doing this was to get out of my home and get out! With even these moving online, everything I do now is online. Now that it is no bad thing as I have made my room a practice space for dance classes and pilates, something I never thought was possible. I have also learned to live stream music performances from my room and have made an art form of that in itself. This will all come in useful even after lockdown.
With Gigabit internet, 5G, the internet of all things, video walls, autonomous cars, bio technology, data driven algorithms etc. all developing very fast we should expect a different world in the coming years. The American civil war propelled the use of the telegraph to the mainstream when it was a niche product for the railways initially. And so this global pandemic has also moved the use of video calls, cloud computing etc. more to the mainstream I hope. And we can truly now ask, which appointments need our physical presence and which do not? Something that I have tested for years now. I have formed fairly sophisticated relationships with tutor colleagues and families in other parts of the world that I have never met in person. A lot is possible online.
Facebook live streaming music performances. It is how I first learnt to live stream in 2014.
All the technology aside, I keep in mind the seriousness of the global pandemic we are in now and that there have been lives lost. I have a civic duty in playing the small part asked of me to minimise risks. And it has been fascinating to see what that means for my tutees in other countries and my extended family in India.
The peak of the academic year ends for me after around the second week of June. By then the exhaustion has really kicked in and I’ve been running on adrenaline with total commitment to helping my students. The demand on my hours is highest in the build up to exams as existing tutees (some who I have been working for many years) need that last minute support, reassurance and specific troubleshooting with exam questions or technique in general.
At a family music festival in early June
Things then gradually slow down but never really stop during the summer until I decide to block a week or two off and go on holiday. I am still to do this and definitely need to do take some time off completely from tutoring to refresh and re energise in doing the thing I enjoy.
Every academic year is unique and different. This year the highlights for me have been:
Taking Sundays off every week
At the start of this academic year back in September 2018 I made the decision to not tutor on Sundays. This is something I had done almost every year previously with tutoring 7 days a week being the norm. This seemed like a tough decision at the time but it has been the single best decision I made last year. Having one day off a week meant cramming my Saturdays as a result. Previously I’d like to keep Saturdays and Sundays light but now I felt I needed a full day off entirely. That one day off a week, spending time with family, doing music and relaxing has been priceless for my well being.
Taking time off for CPD and the value it provides
At #MathsConf19 in Penistone (Sheffield) in June 2019
The biggest cost to me for taking CPD is taking the actual time off tutoring. Lost tutoring hours is lost income and disruption to the regular tutoring timetable. There are other costs like train, hotel, food etc. when travelling to conferences. The cost is well more than worth it, some of it helping reduce the tax bill a little and the rest is all about increased confidence and finding a community of teachers. CPD is a long term investment and like many things in life, taking a hit in the short term is necessary to play the long game. Besides, many teachers who deliver CPD often do so at their expense and two events I went to were free which I am grateful for.
There is absolutely no doubt that I have learnt more about teaching maths and developed more as a maths tutor this year than in any other year. I went to 3 maths conferences, #MathsConf17 near the start of the school year, #MathsConf18 just before Easter and then #MathsConf19 as a treat after exams. Last summer I attended a workshop on 11+ exam entry prep and a La Salle one on ‘Making maths memorable’. Continuing with the La Salle ones again with two phenomenal workshops in the autumn term (Multiple Representations and CPAL). This summer so far I’ve been to a making maths videos afternoon, Maths Teachers Network day and a Dyscalculia conference.
Using manipulatives for teaching maths.
I am always excited to tell parents of tutees about all the new ways I learn about educating their child. And I have new self belief that I am becoming a much better online maths tutor for primary, Dyscalculia, GCSE, IGCSE and A Level.
CPD is not just about attending courses though, I am reading books (more on that below) and engaging in conversations with teachers and tutors on twitter + Facebook all the time. It is invaluable to learn from other teachers and to articulate what is on one’s mind.
Books, books, books, a microphone and a chair
These are the books I read during the last academic year. Some directly related to tutoring and some on general knowledge.
Sapiens: A Brief History of Humankind by Yuval Noah Harari
Why We Sleep by Matthew Walker
Homo Deus: A Brief History of Tomorrow by Yuval Noah Harari
Factfulness: Why Things Are Better Than You Think by Hans Rosling
Dyscalculia: from Science to Education by Brian Butterworth
Last year I spent all summer reading Craig Barton’s book ‘How I wish I’d taught maths‘ and this summer so far I have been getting well into Mark McCourt’s ‘Teaching for Mastery‘. I have a massive backlog of books after that still. I am addicted to reading about education.
Book reading this summer.
Being a sound and music nerd I also bought a shiny new USB microphone to improve my online tutoring sound quality. The Rode NT USB gives crystal clear sound to my tutees. I can’t believe it took me so long to buy a high quality external mic!
Last June I bought the most important piece of hardware of all, a proper fully adjustable desk chair. Previously I was starting to get back pain and other back problems that I don’t even know of. Long days of online tutoring at home on rigid chairs was not good for my back. The new chair, together with taking plenty of standing breaks has helped my back recover back to normal this year.
Tutoring community of those showing up and an award!
I am so glad to see tutors showing up to the London Tutors meetups that I organise. And those who turn up to the maths conferences and CPD events. It is refreshing to see the same group of tutors regularly engaging in meeting each other and going to CPD events. The tutoring communities are relatively young and it is thanks to these that I found such a wide world of educators and teaching CPD.
The Profs Tutors Summer Party 2019
At the end of year summer tutors party I was ecstatic to receive an award from The Profs for being an ambassador for them. In helping promote them and the work I have done with online tutoring communities. I am very grateful for the award and will treasure the trophy and speech that was given during the award.
Maturing as a SEN and teaching adults tutor
With a few years of teaching SEN students (Dyscalculia is really my specialism but it is often comorbid with other issues), I feel like I am starting to mature in teaching in this area. By tutoring in this area I am developing real sensitivity, good pedagogy, excellent communication skills and most of all thinking outside the box with constant innovation in online tutoring technology. There is always plenty to learn though so this maturity process has only really just started.
A community of EdTech maths teachers and tutors.
I also started teaching adult students this year on a regular basis. I always believed that maths can be learnt at any age and I now have my own proof of this from various case studies. I really look forward to developing more into an online maths tutor for adults as well.
BitPaper and TheWayUp! game
Work hasn’t been all tutoring though, I continued to keep pace with the rapid new developments and features being rolled out by BitPaper, the digital interactive paper I use for my tutoring. My job in the team has been to communicate, interact and get feedback from an online community of tutors.
Tutoring using BitPaper
TheWayUp! was another project this year that I was involved in. This required an entirely new way of thinking about digital PR. Much more planned, strategic and with a team involved. I learnt a ton about digital PR.
Both BitPaper and TheWayUp! game meant I was working with a group of people in a team. This has been really refreshing to me as a solo tutor.
Time to relax
My workload is the lowest now with just 2hrs of tutoring daily and all of Saturdays and Sundays off. I’ve been catching up with friends, going on day outs with family, doing a lot more music, more CPD courses and ticking away with reading books too. As I relax I can also ponder on some of the longer term things I want to do in life. And the summer is now the perfect opportunity for it all.
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.
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’.
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
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.
