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.
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.
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.
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.
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.
“It is estimated that up to six months will be saved in teaching time by the introduction of decimal money”
British Pathe news video clip – ‘Decimalisation (1970)’
On 15th February 1971, 50 years ago today, something changed in Britain forever. Day to day life would never be the same again. On Decimal Day the monetary system went from mixed-base to base-ten or decimal currency. Bringing to an end a monetary base system that had been in use for over 1000 years.
Deci comes from Latin ‘Decimus’ (tenth), closely linked to Decem’ (ten). By switching to a base-ten currency, 100 new pence became equal to 1 pound sterling. Gone were all the other exchanges.
The history of the pound sterling goes back to the Romans. The £ symbol is an ornate version of L, for libra, worth one pound mass of silver. Libra is abbreviated as lb which we still use to measure weights. The shilling (s) came from a type of Roman coin called the solidus and the penny was abbreviated with its Latin name denarius (d).
So the old system was abbreviated as £sd (or Lsd for librum, solidus, denarius) or LSD for those groovy 1960s times. The main exchanges were:
Although the preparation had been going on for years, the currency transition was tricky. Prices were initially shown both in old and new forms. People had to get used to the new pence and other new coins. There were many benefits proposed by the government for going decimal, from reducing teaching time at schools to the use of fewer accountants for businesses.
Even though our monetary system was decimalised all that time back, we use mixed-base every single day today in fact. Time.
1000 milliseconds to one second (base-ten) 60 seconds to 1 minute (base-sixty) 60 minutes to 1 hour (base-sixty) 12 hous or 24 hour clock
And these days pupils have to understand binary and hex numbers for use in computing.
50 years ago is not that far back and there are many adults who will remember Decimal Day. If you were one of them, I would love to hear your experience of it all. So do drop a comment here on this blog post.
Just over 3 years ago I attended my first ever maths conference #MathsConf10 in London. That event was to change the course of my teaching life forever. I plugged into a network of passionate maths teachers and went deep down the rabbit hole of CPD; from books, articles, journals, CPD videos, podcasts, twitter discussions, conferences, the La Salle CPD days, webinars to hosting #MathsChatLives and more. Back in June 2017 I could not have dreamt that I would one day be presenting at a MathsConf. I felt so out of place even amongst tutors since I was the oddball one teaching entirely online. None of that mattered once I plugged into a network where everyone had the love of teaching maths in common.
I attended the mathsconfs religiously over the last 3 years and got to know the regulars. With the La Salle CPD days I started finding parallels with my previous experiences of CPD in industry and in academia. I knew that presenting to peers is an important aspect of professional self development, personal growth and accountability. Imperial College had a great ethos of communicating Science and I had already presented at Tribology conferences, both in poster and presentation format. That journey ended for me in 2005 however when I left university academia. So to go back to powerpoint after all these years was a daunting and exciting prospect.
When #MathsConf23 went virtual, it was brand new territory and I was prompted to put in a workshop proposal. I had run some live online CPD courses on Zoom for tutors before (Indie Tutors) and of course #MathsChatLive but that was very different to a full maths CPD workshop. I was both excited and nervous at the prospect of presenting a workshop. Excited, because the virtual teaching environment is now my natural teaching environment. Nervous, because of imposter syndrome! All my teaching is one on one and therefore I rarely ever talk for more than 5 mins before interacting live with a tutee. So I didn’t have the nerve to put in a proposal for #MathsConf23 where I’d be talking for an hour. Instead, I took the smaller steps of presenting for 5 mins first to the #LockdownStaffRoom group of maths teachers, then at the Seneca maths conference, and then as a panelist on the Tutors’ Association webinar. I gained more confidence after these short presentations. So finally, I submitted #MathsConf24 proposal for a full one hour presentation.
But what would I present on? I didn’t want to talk about online tutoring like I had done for the mini talks. Instead, I wanted to focus on specific aspects of mathematics teaching itself. I had 14 years of teaching behind me and felt that I should cover things I have learnt in all these years. The imposter syndrome inner voice said “you have nothing new to add, everything you have to say has been said before and what if the stuff you are saying is wrong?” and the other voice said “that might be true but every teacher interprets and executes the teaching of the same idea in a unique way, and therefore what you have to add to the canon of knowledge will be valuable to someone. And you now have accountability from your peers. This is the opportunity and feedback you have been waiting for all these years”.
I decided to present on how to help the lowest attaining pupils. The ones that have made me totally rethink how I teach. And have made me a more patient and understanding human being. In the last 3 years I have learnt a lot about using multiple representations which has been an eye opening and exciting journey. This quest to help my pupils has sent me down the path of learning how to teach the very earliest of maths ideas. I had to learn to teach everything from skip counting, number bonds to the meaning of “fraction-ness” and so much more. And this was hard, really hard. The earlier the maths idea or concept, paradoxically, the harder it seemed to be able to teach it well to someone who didn’t grip it. I had never thought about what a number means. Or what the difference is between numbers, numerals or digits. But once I started understanding these concepts more (thanks to CPD), I was also able to help my pupils make meaning of it. I felt enlightened in a way I had never before. And this journey is what I wanted to share. Along with practical ideas of course. The engineer’s mindset has never left me, practical, pragmatic actions that yield results. Some that I had hacked up along as a result of CPD and real world implementation.
So I started preparing my presentation. After a little battle with Powerpoint, I thought I’d try out Google Slides instead. I just couldn’t do it as I couldn’t figure an easy way of writing live on my presentation using a digital pen or to move manipulatives about on either PowerPoint or Slides. After some consideration I decided to use BitPaper for the presentation itself. BitPaper is not a presentation software as it has only one font and no option for any text formatting other than changing size and colour. But that was enough for me. And in some ways I found that restriction quite liberating with such limited choice.
The more I prepared it, the more I realised that my presentation will resemble my day to day teaching canvas itself. This gave me a great sense of relief, as snippets of my day to day online maths tutoring, from content relevant to primary teaching, secondary, GCSE etc. was what I wanted to share. I even started to copy and paste actual teaching examples from tutoring sessions into my presentation. Planning a whole hour teaching session was also something new I had to learn. And if I am honest I have never prepared a one hour lesson in my life ever, as it is not needed in tutoring. But I had prepared research presentations before, so that was the frame I was taking. Still, I just couldn’t run a presentation with any live and real interaction, so I put my thinking cap on to design elements of this. I had seen some La Salle CPD webinars that had neat interactivity with chats and polls, so that stimulated me for more ideas.
I started putting little snippets of text as ideas on the notes app of my phone and then to the presentation file itself. Every time I got an idea I felt I wanted to present I’d make a short note of it. Like so many teachers/tutors I’ve had the busiest of terms ever in my teaching career so far and I didn’t get a good enough stretch of time to work properly on the presentation. Two weeks before the big day arrived, I barely even had a slide ready. The publicity for the conference was going full steam ahead by then and my workshop would occasionally be plugged.
It was time for me to knuckle down and deliver. I started throwing all my ideas down now. At first I thought I wouldn’t have enough material for a full hour talk. But it was clear that if anything, I now had too much material with one week to go before the workshop. So I now started trimming some ideas down. A couple of teachers on twitter also tweeted they had cut their presentations down so I knew I was on the right track.
In fact I was still finalising slides the night before the conference and I was still feeling bogged down by it all. I had not even done a run through of the talk to see if it ran on time. Though I did a technical check with my tutor colleague Paul the evening before. What helped tremendously was being on a live zoom group social call in the #LockdownStaffRoom with some of the maths teachers I’ve got to know so well, including my La Salle presentation host being there as well. The company of other maths teachers face to face virtually made the finalising of slides such a pleasurable and motivating experience. I realised that I’m presenting to colleagues, who just like me live and breathe maths teaching. And have taken a day off from their busy lives to present and be part of the conference. This breathed new life into me and I finished the whole presentation off. I did one last test run after the #LockdownStaffRoom social. The presentation ran to 50mins, plenty of time to spare for the real thing. I had built in some real time interaction with the audience as well using twitter, which I was really glad about. But would this actually work out on the day?
My presentation was in the first period for 9:30 am out of the six on the day. I was mightily happy about the prospect of getting it out of the way first thing and not feeling nervous all day instead. I did some pre flight checks on all my tech; microphone level, camera image and lighting, screenshare window of presentation, chat window with La Salle, chat window with audience and twitter browser tab. All ready to go.
With great energy my host Jonny Hall aka mathsbot introduced me and I started it all off by welcoming everyone to the mathsconf and thanking everyone for taking time off on their weekend to hear me live. After about 5 seconds of nerves right at the very start I was in the flow of it all. Like a teaching session with a tutee it flowed very smoothly and I was well into my element. From performing as a musician I know when a singer records in the sterile and lonely environment of a studio they have to put themselves in the headspace of singing that song to just one person. Live at a venue is different and the energy for the same song needs to be dialled up to fill the room. So in my head I was presenting to just one person, the delegate on the other side of the screen. This made me feel more relaxed as well as I wanted every delegate to feel that I was talking to them individually on their side of the screen. Just like I do in my day to day tutoring.
The audience chat was very useful to keep an eye on as I got to know which ideas were resonating and leading to peer to peer discussions. I asked some questions on ideas behind numbers to get everyone thinking and was delighted to see answers coming. The exercise that I really wanted to test on on scale using twitter was one based on the use of Egyptian Hieratic numerals as part of the sections on numerosity (numberness), numbers, digits and numerals. This is one of my absolute favorite exercises that I do nearly with all my tutees of any age. It is one thing doing these with tutees so I was curious to know how it would play out with fellow teachers. As part of the section on numbers, digits, numerals, I went to explaining how the numberness of a number can be drawn out using Egyptian Hieratic symbols live using an example. I asked delegates to draw their own versions out, take a picture of that on their smartphone and post it live to twitter.
There was a 20 second delay in the live video being streamed to delegates so I had to wait a little and not talk at all during this short time window. I initially made the mistake of switching to the twitter tab too fast and not giving enough time for delegates to see the screen. The chat window feedback was instant. I corrected and went back to the slide with the task and stayed on it. After about a minute, something beautiful happened. My twitter timeline was flooded with answers. Handwritten with pen and paper. Each one with its own unique personality. There was something really human and organic seeing delegates handwriting appear in real time. I took two of those examples from twitter and snip-pasted them live into my presentation. So two of my delegates now became part of the presentation! My gamble had paid off and I felt real connection with delegates from that point onwards. It was now two way real time communication.
I went through a bunch of things on the presentation, sometimes writing out explanations using my digital pen, moving virtual manipulatives and of course talking in the context of the slides. The presentation itself would be a blog post of its own but for anyone interested, the video recording is available to buy as part of the La Salle Teacher CPD college #MathsConf24. I concluded my presentation, looked at chat questions and answered as many as I could. Jonny Hall my host also asked a question on behalf of a delegate and we had a little bit of a chat at the end. And then I wrapped it all up. Phew, that was a good one and I was glad that was out. Onwards to taking part in the rest of the conference which was so stimulating.
So my first workshop was fourteen years in the making. I would possibly be the first career tutor to present at a conference like this. Hopefully paving the way for many other of my tutor colleagues who I know are also keen to present. I really recommend tutors to take the plunge as La Salle are really open minded with these things. A few other teachers also made their debut for a maths conference presentation so I wasn’t alone. I have much still to learn about teaching mathematics. So being able to share whatever I have learnt so far with my peers is a superb opportunity. The beginning of a journey of accountability, enthusiasm, personal and professional growth.
In the middle of my short holiday at the Peak District an article appeared on the famous US newspaper on the 6th of August 2020. The article was titled ‘Everybody Is a Live-Streamer in Covid-19 Era‘ and I featured on the opening paragraph for that story. The article was published in print of the Wall Street Journal as well.
“Soon after the coronavirus outbreak was declared a global pandemic, Atul Rana started using the video app Zoom to keep up with peers in his field. The high-school math tutor found the hourlong conversations so enlightening, he began broadcasting them live once a week on Twitter, Facebook and YouTube.”
Sarah E Needleman – Wall Street Journal, 6th August 2020
I had been contacted by author Sarah earlier and this seemed like a pretty cool story to be part of. Little did I know that I was a part of a new type of media creator in a family of live-streamers, a niche of sorts, people who stream about fitness training, video games, music and more. Which in itself has given me more ideas to try out for other things I do.
I already mentioned in a previous post that one of the highest leverage activities I thought I could engage in as soon as the Covid-19 pandemic went global was to live stream with UK based maths teachers. I was an early adopter to live streaming, starting from streaming live music gigs in my room from 2014 on YouTube, to 2016 on Facebook, to education live streaming on my Facebook Page and eventually Indie Tutors to help other tutors.
The need for a community of passionate maths teachers to communicate and share their thoughts has always been there. And there is a really rich and diverse community of UK maths teachers and tutors on twitter. The pandemic brought to an end of face to face conferences and meetups for a few months. I felt that loss as well. I felt I had to now rise up to a new role. To bring my expertise as an online maths and science tutor in holding an online space for tutees, to now holding it for these passionate maths teachers and to learn from them.
Unlike the earlier live streams, these were multi streamed simultaneously on twitter, Facebook and YouTube. I invented a new way of maths CPD online as more and more people gave me ideas. One great idea was to take questions live on twitter and discuss them on a panel.
Anyway, enough of my self congratulatory post here 🙂 I really like the idea that now anyone can run their own TV show online, not just mainstream or any other media. And over the coming months I hope to train up others to livestream like I do using OBS, restream and social media, so this becomes more of the norm and a new thing.
Over the last 2 years I have been mentoring Paul Morgan, an A Level Chemistry tutor on his move to tutoring Chemistry online. With Paul being based in Burnley and me in London, we are still yet to meet in person. We have been meeting on a video call every week which serves as part staffroom and part CPD. We discuss a range of things from the business side of tutoring, teaching techniques online and more recently I have been his test pupil for the Chemistry teaching videos he records while we are on a call. So in effect we are on a video call, an interactive canvas and online video recording studio.
I teach up to GCSE Chemistry and Paul being an A Level Chemistry specialist, I have learnt a lot from him. Particularly how what I am doing at GCSE and earlier stages links up with what comes up at A level. Always keeping a long term view of learning Chemistry and making sure that things are taught correctly the first time round in the learning journey. I did A Level Chemistry and did research in a multi-disciplinary Tribology lab at Imperial College. So Chemistry has always been of interest to me.
One of the great things about using interactive online whiteboard Bitpaper (disclaimer : I work for them as their social media rep) is that you can build a whole bunch of filled shapes. Both the tutor and tutee can move these shapes around. For maths teaching this has been excellent since I have made my own two coloured counters, algebra tiles and cuisenaire rods. A tutee can be thousands of miles away from me yet both tutor and tutee can move the manipulatives around on screen as if we they were on the same table.
I have started to use virtual counters in teaching Chemistry as well now. In Chemistry students may use ball and stick manipulatives at school to understand the basic formation of molecules, the relative arrangement of atoms in space in that molecule, bond breaking and bond breaking etc. Quite a lot of that movement and playing with things can be replicated online.
We have created a series of videos to show exactly this. This first video is the very first one that introduces virtual manipulatives. Some things to note:
The atoms sizes are relatively in proportion to each other approximately.
The subscript symbolism of what the molecule represents with a pictorial representation of the molecule needs to be explicitly practiced by students.
Embedding what monoatomic and diatomic means using a pictorial representation.
Common misconceptions are addressed using non-examples. For example, while Methane’s molecular formula is written with Carbon first and Hydrogen with a subscript of 4, that does not mean that the molecular representation has a carbon atom, followed by four Hydrogen ones in a row.
The whole process of forming a molecule from atoms is then animated by the tutee once they have manually formed the molecules. This is done while I keep silence. Silence from the tutor is key to reduce cognitive load on the tutees while they discover and make these molecules.
I have tested this with a number of pupils now online and Paul has started using it with his tutees as well. It has been really useful for me to consult Paul and together we are creating this new way of working with manipulatives for Chemistry when teaching online. Please do let us know in comments or on twitter/Facebook what you think of this.
26 November 2005 was a dark night for my family. I was in Budapest where my dad was on a posting and had lost his short struggle for life after a sudden brain hemorrhage. I had no idea what had just hit us as he was a perfectly healthy and fit person. In that state of shock it was inconceivable that he had left our lives forever.
As the eldest sibling of my family I took the call from embassy staff 36 hours earlier that he was in hospital. I left London with just a rucksack of some clothes. My brother and sister who I lived with at the time were at the flat in London. We were no strangers to talking to mum and dad on webcam every week, something we had done since around 1999. Dad liked calling us on video on MSN messenger. We were all university students at the time and were used to chat programs on those old desktop computers. Dad would always try to catch us online and call us on video two or three times a week. He was never to be online again on our MSN list.
For those last 36 hours I had held myself together for my mum and my siblings. I knew dad was in far more a serious condition than they all realised while me, mum and embassy staff were visiting hospital. I held hope, great hope that he will recover and that he would be in ICU for a few more days and then be out. He wasn’t even there for 2 days. At midnight we went to see dad as he had lost his last struggle for life. My mother was inconsolable at the hospital. I called my siblings on the phone as my brother and sister tried to console her. I was a student on a roaming pay as you go phone and my £50 top up ran out in minutes.
I reached our home in Budapest and started the MSN video call with my younger brother and sister. 2005 webcams were grainy but still we were able to see each other, see the sadness in our eyes, some of us cried and some were still in shock. This was the era in which the term 24/7 internet was coined as broadband was still in its infancy. It was about 2am and we decided to try and get some sleep. I told my brother in London to just keep the video call going, whether or not we were sleeping. I don’t know why I did this at the time, I needed to know that even if we were not talking to each other live that we were connected together.
I couldn’t sleep, the reality that my dad had just died hours ago and that I was the one now who had to inform his brothers and sister in rural India dawned on me. None of them had phones, they were too poor and mobile phones in India were expensive at the time.
I frantically looked around for phone numbers of relatives with a landline or mobile phones. I couldn’t do it, every time I saw a piece of paper with my dad’s writing I realised the writing is of my now dead father, and that I will never see any new writing from him again. Shock finally turned into mourning and I started wailing out loud, I turned into a screaming 5 year old again. My mum woke up and to my surprise even my brother who was on a live link to the MSN call woke up in London. We talked for a while and I calmed down. I then located the phone numbers and called various relatives in India all night. How can anyone ever be prepared to call up to tell the news of a brother’s death? You can’t. I did my best at the time and talked to them as long as I could. They didn’t know what to do either, mourn or console me? They consoled me and my mum who were at the flat in Budapest.
2 weeks later I travelled to India and met my family in India after 12 years of not having gone to India. In a suburban poor part of outer Delhi, my mother’s brother, my uncle found us four a small room. We were lucky to have running water near that room but my uncle’s flat didn’t. I helped my aunt in those next 3 months to bring buckets of water in the morning to her flat. It became a morning routine but never a chore as the temple blared out holy songs and India woke to life with that life energy of India that is hard to put in words. I even got used to cold showers in December. My aunt’s food was delicious, the Indian sun, Hindu philosophy and a sense of community in that suburb was nourishing to me.
My uncle’s grandchildren were three boys, the eldest of them was 13. The boys were amazed by the the tales of London and the Western world. They were inspired by what my dad had done for me and my siblings. Tales of how total dedication to education got my dad a school scholarship and eventually a job in the Indian diplomatic service. The prodigal son had escaped poverty, travelled the world and represented his nation. If he could do it, then so could they all.
I travelled to wrap up my dad’s paperwork, pension, bank accounts etc. from suburban Delhi to central Delhi for the next 3 months. It was such a trip, a tiny minibus that rattled through the rough suburban dusty roads with loud Bollywood music blaring. I had just finished my PhD and was in debt myself, so the 5 Rupees (~6 pence) journey suited me fine. I couldn’t afford a car or taxi either and was far from being a middle class Delhiite. So I ditched my GAP flares, bought clothes from local markets and just fit in with everyone else there.
The nearest cyber cafe with a webcam was about 45 mins bus ride from the suburb I lived in. I went to the cyber cafe to chat to my brother, sister who were now back in London after two weeks in India. I also got to keep in touch with my friends from Imperial College and catch up on the local student news, MSN was just about to be replaced by MySpace and Facebook. My nephews started to come along to the cyber cafe with me. I put two of my older nephews into an IT skills and English speaking course three mornings a week as that part of Delhi had more resources. The eldest ended up going to university, got a degree and is now working in an accountancy firm. He and his brothers got my uncle’s family new luxuries. 15 years later they have running water, plenty of appliances, a bigger house, a motorbike and more.
In those moments when I had left that video call on and continuously running shortly after my dad’s death, I felt that my flat in London was joined to my dad’s flat in Budapest virtually. It is as if the walls and boundaries had disappeared, as if that the room in London just became an extension to the room in Budapest. They were one. We could see each other and talk to each other. The quality of call didn’t matter, the fact that there was a live connection put us in a virtual space. That space is perceived and enhanced in our minds.
This idea of interconnected virtual spaces has never left me since that night in 2005. I started tutoring in 2006, fast forward to 2012 and I started tutoring online. Many of the things I learnt back then on how to stay connected with my family is what I now apply in my teaching. A vast part of that is forming very real and human connections with my tutees and their families. The tech has always been secondary, I just used whatever tech I had at the time to do the teaching I wanted to do. I turned the limitation of the tech into a strength, a pedagogical tool in itself. I made ‘remoteness’ a powerful tool. The engineer and problem solver in me has just learnt to adapt and innovate with whatever tools I have at any given moment of time.
Over the next weeks and months, I will try to open out from my head what these tools are. What it might mean to hold a space for someone online, to get them to articulate, to articulate myself, to value the silence and realise that perhaps the most important communication is happening in that silence between the talking. Once I have my own clarity on this, I will be able to convey it to the wider world.
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.
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.
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.
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.
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.
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.
Video killed the mp3 star – my first video channel circa 2007
Back in 2007 I started a video channel on YouTube for my band followed by my own channel in 2008. Phone cameras were pretty rubbish back then, so for any decent videos you needed a video camcorder (remember those?!). I bought a mini DV camcorder and along with my grainy phone videos I started uploading videos.
My band used to be filmed nearly every gig so I had tons of footage to look at, upload, edit etc., It was a fascinating learning curve on video editing, compressing and uploading. But the real learning came after the video was already uploaded. From what people were commenting on, likes, dislikes and analytics data.
I found that mostly no one cared about my band’s music and it wasn’t ‘viral’ enough. I was surprised to find I was getting more views on my personal YouTube channel than my band’s channel. I was posting acoustic cover versions of songs and was one of the first of the video blogger generation, holiday videos mixed with concert outings and even random ramblings.
Video feedback – harsh, instant and very useful
Through YouTube I realised I could in effect ‘beta test’ my voice with cover versions of songs. This would help me gauge for how good I really was as a singer. The feedback was nearly instant and very clear. People respond to a good voice and a well played cover version. The harsh truth I realised is that most of my covers bombed but a few kept getting hundred or so views every month, not bad going but definitely nowhere near some of the top video song covers out there. I’ve been taking one on one singing lessons for a year now so at some point I will revive the song covers.
Starting my tutoring video channel at long last
After a little dabbling in tutoring videos over the years (I made about 3), I finally committed myself to making videos on a dedicated channel. Through my tutoring video channel I hope to present some of the teaching methods I have learnt over the years. Now that might not be unique but it is my own take on things, my voice and my style of communicating. Much like posting the music videos, I feel like I will find my audience, whoever that might consist of.
At the moment I have been sending the videos to other tutors, teachers and also to parents of tutees. In future, as I have more videos I can use them as a resource with my current tutees who are used to my voice, video and specific way of explaining things. Any method they want to know again in the way I taught it to them….well, the videos will all be there.
My tutoring YouTube channel is born
Making maths video course and finding other tutors keen on videos
Earlier this summer I went to a ‘Making maths videos workshop’ delivered by educ8all, which I must say has been really helpful in the whole process. The most important thing was finding another tutor who is keen on making videos. Taking action is what I value and only by taking action will I become a better ‘YouTuber’. The one single act of having an hour appointment with another tutor every week at the same time to work just on video has set the wheels in motion.
Currently I am making videos with Catherine and Paul who both jumped in with the idea of video. Working with both of them individually makes the videos more like a live tutoring session, keeping things more bouncy and with a real person on the other side it is just much easier to articulate. I find solo videos harder to do. But with the wheels set in motion that too becomes easier to do now.
Things I have learnt so far about making tutoring videos
It turns out that there is a LOT to making videos. Here are just a few aspects I am learning about:
Video planning and storyboarding – freestyling videos is actually pretty hard. Scripting them and having a plan is far better
Storyboarding – How one ‘scene’ flows into the next. E.g two tutors talking, cut, then to the whiteboard
Various scenes – There turns out to be many possibilities on what can be filmed. Eg. two tutors talking, switching to a second camera to show manipulatives, writing together on a digital paper canvas with both tutor headshot videos showing, sharing computer screen and other ‘scenes’
Looking presentable and natural on camera
Having a presentable background – i.e removing visual clutter and getting a nice background colour. A real background, none of those fake green screen ones that I have seen in too many tutoring videos
Lighting – Bad lighting can destroy a video. I mostly film in natural light and add some additional soft lighting. I have done this already for live online tutoring so this was easy to implement
Sound – Great sound quality is an under-rated part of getting a good quality video. I have a Rode NT USB microphone and a mic stand to close mic me. Laptop mics are terrible and they also pick up tapping sounds, best avoided
Delivery and video presentation – The energy has to be solid in the video intro. I felt stiff and robotic during the first few videos but now I am starting to relax and be more ‘myself’ on camera
Editing video – Smooth transition between intro and main body of video, fade out as we both say bye, overall flow of video should be good
Rendering – Compresses the edited video size down. Takes a surprisingly long time
Uploading and Publicising – Video must be described and tagged appropriately. And then I publicise the video on social media
Feedback – Comments, likes and video analytics along with any feedback by talking to others then helps me figure out how things are working, or not working
Reducing unnecessary cognitive load for the video viewer
Both myself and Catherine have carefully thought out how the video should work. We always want to make sure the videos are short, to the point but still feel organic and lively. We have taken on board the lessons learnt on the video training course. I.e not making rookie mistakes such as writing and talking at the same time all the time (a little bit is ok). This diverts the attention with the audience having to focus on the dynamic writing and also listening at the same time, split attention effect etc. We have kept in mind cognitive load theory and hope to make very clear, well flowing videos.
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.
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
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.
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.
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.
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.
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.
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 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!
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.