Tag Archives: maths tutoring

Teaching with multi-base : Escaping my own base-ten world first

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.

Derek Sivers

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-four Diene blocks from my set of blocks.

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. An task my tutees do and also 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.

Starting a new YouTube channel of tutoring videos

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.

Atul Rana Online Tutor YouTube Channel image

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.

Anyway do check out Atul Rana Online Tutor YouTube Channel

As I progress in this journey I will no doubt blog about more aspects of video production, including the software and hardware I use. Feedback in comments here or on any of the videos is very welcome.

Maths Conference Birmingham #mathsconf17

This blog post is a write up of my fourth maths conference. La Salle Education run the UK’s largest network of maths teachers’ professional development along with an online platform Complete Maths. Attended by around 400 maths teachers and a few tutors, the conferences are invaluable professional development, training and networking with fellow maths teaching professionals.

TLDR : All four workshops were phenomenal as standalone workshops. In the sequence I attended them, they compounded my learning even more.

Friday – Hotel tutoring and drinks

I have Friday tutees and Saturday is my busiest tutoring day. I wasn’t going to let that get in the way of going to #mathsconf17 though so I rescheduled many of my Saturday students well in advance. I took the train on Friday morning, checked into the hotel, set up my mobile online tutoring office (laptop + graphics tablet) and away I went tutoring until the evening.

Maths Tutors UK with Mr Corbett

Selfie time with Mr Corbett!

Fellow tutor friend and conference buddy Austin (@Lazyrunner78) arrived after a long evening of his own tutoring. We had a catch up and then made it to the bar late. The Friday drinks are always so welcoming, you can join anyone and instantly share your own enthusiasm of teaching amongst fellow teachers who understand the slightly crazy passion we have for teaching maths. I met some new teachers, both local to Birmingham and those who travelled. The friendly Mr Robert Smith was welcoming as always, introducing people to each other and keeping us all ticking socially.

Speed dating with some unexpected Science

The Saturday was buzzing and La Salle CEO Mark McCourt opened the conference with an intro. His story on how the conferences started and his passion of empowering and bringing together maths educators set the scene for the day. We kicked it all off with mathematical speed dating. This was missing at #mathsconf15 so it was good to see the return. A speed date is talking to a teacher for two minutes about your best teaching ideas then hearing the teacher for two minutes. That’s one speed date, next you find a random teacher and then rinse and repeat four more times. Mark reminded us all on how a mathematical speed date in Birmingham led to a wedding two years ago. Love that story!

Andrew Taylor from AQA at #mathsconf17

Andrew Taylor from AQA.

I shared my own ideas on using manipulatives to teach from a mixture of algebra tiles to the meaning of pi experiment. Amongst my dates I met a teacher from Birmingham who was retraining from being a Chemistry teacher to teach maths. As it was his first conference he felt a bit out of place amongst so many seasoned maths teachers. I reassured him that I felt even more out of place at my first maths conference as a private tutor but now I know the community is super supportive. Since we both also taught Science it was so easy to go straight into common themes between the two. His speciality was Chemistry so we had plenty to talk about that. I had my big aha moment right at the end of the day too on maths and Chemistry. More on that later.

Workshop 1 : Tech, Tech, Tech from Steep Roads to CGI Films

This is the second time I went to Douglas Butler’s (@douglasbutler1) talk, previously seeing him at #mathsconf10. This second helping was with a different flavour. He gave an overview of some of the items on this list.

  1. Top Google Earth Objects
  2. Top Large Data Sets on the Web
  3. Top Uses of Excel
  4. Top Problem Solving Ideas
  5. Top Twitterers to Follow
  6. Top Maths Blogs
  7. Top YouTube Channels
  8. Top Mathematics Entertainment
  9. Top Dynamic Software for KS3-4
  10. Top Dynamic Software for KS5

Maths cakes #mathsconf17

Maths cakes. Perfect for sugar rush.

I have installed Google Earth pro on my computer after seeing it in action at this workshop.  I use Google maps with tutees already to show them the similarity between New York’s grid layout and the x-y system. Google Earth Pro can do so so much more though. He showed the world’s largest equilateral triangle layout, parabolas, pentagon and the world’s steepest road. He also gave us all a hearing test. The airline industry is full of amazing data that can be used to show the perils of sampling data from a population. We also got a taster of Autograph and Excel. I am amazed by what those pieces of software can do. He finished off by making an animated version of the Starship enterprise from Star Trek to show 3D dynamic geometry in action, with music included!

Douglas tells great stories and delivers with such great enthusiasm that you are drawn into the world of maths he reveals with the help of simple technology. I’ve got such great ideas from this workshop which will no doubt help my online maths tutoring for KS3, GCSE and A Level students.

All that geometry and visual representation got me in the perfect mood for Singapore maths next.

Workshop 2 : Dyscalculia, Bar Modelling and the rise of Singapore

Dyscalculia and Singapore bar modelling are massive topics. I have been to day courses on both of them before. To deliver a concise idea of the two in one workshop was no small achievement by Judi Hornigold (@DyscalculiaInfo).

Counters are a powerful tool in learning maths.

Counters are a powerful tool in learning maths.

I have totally immersed myself into tutoring and understanding Dyscalculia after going to a day workshop on it 3 years ago. Judi told us how we can better define Dyscalculia so that we can then address it. She also discussed that in many cases Dyscalculia might appear to be the issue when in fact it is maths anxiety. Anxiety triggers a fight or flight reflex shutting students down to learning maths. Again, maths anxiety is a huge topic on its own.

So how can Singapore maths help? Students and teachers in Singapore had never heard of maths anxiety to her surprise. Judi went through a brief history of Singapore maths and then we got to the fun bit! Using counters, cuisenaire rods, Singapore strips (of paper) – Singapore strips sure got some chuckles in the room. We looked at the bar model method itself for a range of situations from number bonds, ratio questions, linear equations in counters to the idea of metacognition for students. Metacognition is about building into students how and when to recognise when a problem can be reduced down and then solved in a different way, rather than applying an algorithm on autopilot. A quick example is on finding 12.5% of a large number without using a calculator. If students recognise 12.5% instantly as one eighth then they can divide the number by 8 instead.

Singapore maths and bar modelling is changing lives for children. Judi had some amazing stories of students cracking things in maths. She had stories of students in tears of joy when they figured out concepts. I can relate to that as I had a Year 11 tutee who had battled with ratios all his life. It made sense to him after just half an hour when I used the bar model with him as the very first tutoring session I ever had with him. The utter delight and sigh of relief he had at understanding ratios is something I still remember so clearly.

What an inspiring, well thought workshop. Inspiring low motivation students was just about to be covered in workshop 3.

Workshop 3 : Re-visioning success and the marigolds of multiplication

Julia Smith (@tessmaths) is a motivational power house, absolutely no doubt about it. She works with some of the least motivated students, those who have retaken GCSE maths and in some cases, are still retaking. She has found many ways of motivating students and has some excellent methods on how to help them revise and pass their exams.

Re-vision workshop at the maths conference

Re-vision workshop in the school music room.

Julia started off the talk by clarifying that if students haven’t managed to figure a method out by the age of 15 and a high stakes retake exam is imminent, then it is time to re-visit the topic in a totally different way. If a method that works for them to give them the correct answer, then no matter how procedural or ‘quick fix’ the method seems, it is more than worth it to get the student to pass, gain confidence and go on to get a better paying career in life.

She broke down the word Re-vision into re and vision. I had never thought about it this way so this was very refreshing. We also discussed possible answers to the “I hate maths” line from demotivated students and a tea towel of her revision techniques was given to one of her favourite responses.

I am really torn when I have to teach to the test rather than teach for understanding. I will switch to teaching for the test in cases when I have to. To many of my students their dream might be to work in Veterinary Science, Sports Science, Nursing, Music Tech or something that requires that all important maths pass. I’ve got such students over the maths hurdle and it is truly satisfying.

Amongst her top tips was the idea of double marking past papers, one with the real mark and the other with what the mark could have been with all slip ups and silly errors were given. Getting students to visualise tough moments in exams and to work out strategies to overcome those tough moments and to continue. Her centrepiece was her toolbox, which amongst other methods uses the marigolds of multiplication. This helps students to instantly figure out the times tables of 6,7,8 and 9. It works and will get students out of jail when they most need it, I really liked it! The other technique that I learnt was Vedic multiplication using just line strokes and counting for long multiplication. Again, what a superb technique.

Maths Tutors UK Facebook group tutors

Maths Tutors UK unite.

She also stressed that the way to do maths is to do lots of it, the importance of good exam technique and plugging gaps in the nine basics. Corbettmaths revision cards were mentioned amongst mathsbot and a few other great revision resources.

What came across so well is Julia’s energy and a can-do attitude to get her students over that line. I will take a lot away from this workshop and have new found courage to help my GCSE retake tutees.

On to workshop 4. I was already primed for linear equations from the bar modelling workshop earlier in the day.

Workshop 4 : Atomising Linear Equations and an aha moment with Chemistry

Choosing this fourth workshop was a tough decision indeed. Between Jo Morgan’s workshop on solving quadratics, this one by Kris Boulton (@Kris_Boulton) and Pete Mattock’s one I had to pick just one. The title of this talk “How to solve linear equations, 100%, guaranteed” and a compelling description is what really sold it to me in the end. Perhaps the biggest motivating factor for me was that solving linear algebra equations is one of those pivotal key skills that once cracked, really gets students a firm grounding for algebra in general. I keep having to revisit it with some students.

Kris started off with Al Khwarizmi. This is  what I do when introducing linear algebra to students, so this struck a chord with me instantly. I ask students to google the origins of algebra and more on Al Khwarizmi’s book. We then talk about some of the words that come up, balancing, restoration, completion etc. Kris went into some detail about the appropriate use of the equivalent, equal signs and the word solve.

Atomising how to solve linear equations

Atomising how to solve linear equations

He has ‘atomised’ the process of solving one step linear equations in some very fine detail indeed, 17 steps in fact. Breaking and repairing equations was the sort of language I have not heard in this context, so it really gave me food for thought. These steps could be put into component process pretty much independent of each other.

  • Deciding
  • Simplifying
  • Breaking
  • Repairing

I was sitting next to Austin for this last talk of the day and we both tried to come to terms with the idea of breaking an equation. This careful ‘atomisation’ and the early Chemistry moment suddenly gave me a Eureka moment. At GCSE Chemistry students are given equations to balance. These are broken equations because atoms are quite literally in unbalanced numbers on both sides. Balancing equations is a nightmare topic in Chemistry and Kris’s talk has given me an idea on breaking the process down rather than teaching it as one big process from start to end.

There was a lot in this last talk of the day and by being forced to think in language I had previously not encountered I have taken a lot away from this workshop.

Fan moments, freebies and meeting other maths tutors

It is so refreshing to see more tutors turn up to these conferences. The Maths Tutors UK group has about five core members who attend these conferences and a new tutor local to the conference always joins in. It is vital that tutors get out there to such events as working in isolation has drawbacks.

CGP free books for teachers #mathsconf17

Free books from CGP!

The rest of the conference was all about goodies from CGP, maths cakes, selfies with the legend that is Mr Corbett (we were in a long queue of selfie takers!) and all round socialising.

In summary this was the best maths conference for me yet. On its own each workshop was perfection. By design or sheer coincidence the order in which the workshops followed one another complimented each other so well. Compounding at its best. Einstein wasn’t kidding when he said it is the eighth wonder of the world!

The positive, supportive, can-do energy of these conferences is what bring me back to them each time. Endless thanks to the La Salle team and Mark McCourt for making this all possible.