If you want to learn something, teach it

The popular physicist, Richard Feynman, had an unparalleled way of communicating his subject.  His gift of teaching illuminated the lives of his students and, crucially, the non-expert public.  But it says a lot about the greatness of the man that he talks about teaching as a crucial method by which he himself learned his subject.  This beautiful side-effect of teaching is rarely mentioned; certainly, to the uninitiated (with perhaps children of their own that require some instant results), the teacher knows it all – he/she must be superhuman (or even Google itself!) so why must they need to learn, right?

I am an ‘Early Career Researcher’ (or ECR).  This title is self-explanatory (though I’ll carry on calling myself a Mathling on this page!).  As I have found out, some major firsts occur as a result of being an ECR:  first collaboration, first conference, first big seminar, first publication, and, as I will talk about here, first lecture course as a lecturer.  As well as being an achievement, this has been a learning opportunity that I will continue to treasure.

Some time before starting the lecture course, I was told by many experienced lecturers about how it takes over one’s life and how our research suffers as a result.  One academic mournfully wished me all the luck in the world – as if I’d just been diagnosed with a serious illness!  Ominous signs, but at least this advice prepared me for the worst.

Thankfully, this was one of those times where the reality was slightly less painful than feared, though it didn’t stop me from becoming a slave to lecturing!  But I wouldn’t change anything.  Far from being a hindrance, I look upon this stage of my life as vital to my research – I have learnt so much about my field of interest (complex systems and chaos) that I’d be much worse off in terms of subject knowledge had I not taken this challenge.  And so this brings me back to Feynman’s technique: if you want to learn something, teach it.


Feynman mentions his ‘teach to learn’ technique in his characteristically entertaining book, “Surely You’re Joking, Mr. Feynman!”

I will list what have I learned by teaching.  In doing so, I hope to show just why Feynman advocated this simple mode of learning.

  • That I love my subject even more!  Teaching chaos theory and nonlinear phenomena forced me to read up on beautiful examples such as the double pendulum and the background to Edward Lorenz’s iconic weather models.  I also finally dusted off my copy of James Gleick’s “Chaos” book, which tells the story of this subject in stunning fashion.  I now have a deeper appreciation for just why this topic is crucial to our lives, and I’m thankful that I get to make my tiny contribution towards the advancement of nonlinear dynamics.
  • Chaos doesn’t exist in 2 dimensions or lower (unless we live in a discrete world!):  a particular result that I hadn’t paid much attention to before.  If the system is based on continuous time, then it can be chaotic if the number of variables is 3 or more; otherwise, we can get chaotic behaviour in even one-dimensional models such as a discrete model of population growth (the one dimension simply recording the yearly population of some species).  Just think about what chaos mean if it exists in population models…
  • More research avenues opened up:  In thinking about this lecture course, I organically encountered topics that I think deserve further study. I also encountered countless examples of chaotic systems that I hadn’t encountered before – I am more of a mathling than I first thought!   My thirst is greater having also supervised various research topics in the form of final year dissertations (I intend to shed more light on these topics in subsequent blog entries).
  • That I need to timetable research and teaching and be STRICT with it:  I must have assigned 90% of my time to the teaching and 10% to my research.  (However, in light of the above, perhaps this number is more like 75-25).  In future, I must be strict in terms of how much time I assign to the teaching and research.  I hope that the experience and teaching material that I have now built up will allow me to lighten my teaching load.
  • That staff and students LOVE my lecturing style:  It is always rewarding to be thanked.  And I have had overwhelmingly positive feedback from both staff and students regarding my passion and dedication to my subject.  This is deeply fulfilling, and tells me that I should continue to teach wherever possible.

So, as you may have guessed, the intensity of teaching is why I have been quiet on this blog for what seems like an age (Barack Obama was president and the UK was still an EU member the last time I blogged!).  I have yet to strike the right balance between my teaching, writing, and research.  I guess that’s the flipside to a first – that we don’t quite have the experience to judge such matters.  But I’m much better off for it and feel a hunger for research that I don’t see being satisfied any time soon!

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This Complex System of Life

I recently read something that made me appreciate each and every one of us…

“…everything is connected with everything else. Stars, clouds, forests, oceans and human beings are interconnected components of a single system in which nothing can exist in isolation.

In such an interdependent universe human beings hold enormous responsibility; each individual is accountable, and every action has repercussions that reverberate far beyond the moment. Past, present and future form a continuum in which each generation inherits a world shaped by the actions of its forebears and endow human beings with an even more awesome task: they are the caretakers of the entire system, responsible for keeping the stars on their courses and the living world intact.”

This is an excerpt from David Suzuki’s 1999 book, The Sacred Balance: Rediscovering Our Place In Nature. (I haven’t read the book myself – I came across it while reading another fine article promoting veganism in this ever consuming society.  The reviews for the book look good, so it is now on my reading list.)


How amazing that we play a part in beauty like this. (The mountain is the Matterhorn in Switzerland.)

Suzuki’s quote reminded me of chaos theory – the idea that a small disturbance in a system can create radically unexpected consequences some time later.  Chaos theory is another aspect of mathematics that motivated me as an undergraduate – it says that, even in a deterministic system, things can quickly become unpredictable.  This is particularly true when we think of the weather – suppose today, a Thursday in the month of June, we experience the same sort of conditions – temperature, sunlight, cloud cover, precipitation, etc – as on some Thursday in June of last year.  Does this mean tomorrow we will get the same weather as the Friday of last year?  Probably not, right!  This is because despite the similarities in weather, the relatively small differences in temperature etc actually make a big difference.  It’s like two 100 metre  runners that start off so close but end up being seconds (a big time difference in the 100 metre) apart at the finish line – their minute difference in speeds was all-important.  We all know how annoyingly unpredictable the weather can be – a couple of days ahead is great to forecast, but any more than that and the chances of accurate forecasts diminish greatly.  Chaos theory has been described in a much more colourful way by James Gleick in his popular book, Chaos.   I highly recommend it – it’s non-technical and contains no equations!

Chaos theory is a fundamental part of the larger topic of Complex Systems, which I’m interested in.  A complex system is made up of similar individual objects that interact in very simple ways, yet produce an overall behaviour that is difficult to predict and fascinating to study.  Examples include traffic flow (think of the times you got stuck in a traffic jam!), viral outbreaks on a social network, crowd behaviour, and, of course, the weather.


The Butterfly Effect (popularised in this film of the same name) is a common phrase to describe the effect of chaos theory.  

So what has David Suzuki got to do with this?

Well, he’s made me realise that life itself is one big complex system.  As individual objects within it, we make subtle contributions.  But because of the complex nature, we don’t necessarily see the effects of our actions.  In fact, I’d go so far as to say we shouldn’t expect direct ’cause and effect’ outcomes, but expect surprises (both uplifting and sad), conflict, confusion, and chaos.  Indeed, take a look at the world today and notice how things go awry – wars, racial tension, environmental damage etc – and people (politicians and activists normally) offer ‘solutions’ even though ultimately there never seems to be an easy fix to the problems; this is symptomatic of a complex system.

But chaos theory does offer hope – Suzuki made me think that, no matter how ‘small’ our contribution is, this complex system of life ensures that we can make a difference.  In fact, we do make a difference.  So it also then serves to remind us to make that contribution count.  We should not waste the precious time that we have – because, one way or another, our actions will have repercussions on the whole of mankind, not just our own locality.



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“How we Apply Mathematics”

In October, I visited Caracas, Venezuela, where my brother works as Head of a school.  Since my father hasn’t lived up to being an embarrassing dad, my brother has taken it upon himself to step in and fully enact this role by raving about my achievements to everyone!  So, following another such instance, it fell upon me to give a talk to Years 10, 11, and Sixth Form at his school.

I gave the talk the title, “How We Apply Mathematics”, with the aim of presenting mathematics not often seen in school classrooms.  Thus, in a similar vein to this website, I tried to present what I find to be truly beautiful about mathematics, including fancy pictures, humour (definitely a work in progress!), and no equations.  It was an experiment of sorts, since I was flying against expectations and relied on my powers of speaking as much as the slides – things we don’t normally associate with mathematics in the class.  But this was the point.  The ultimate aim was to show how I find mathematics to be a flexible, creative subject, and the skills required to pursue it as a career are closely aligned with those applied by artists, painters, and musicians.

Having been introduced as the one thing my brother truly loved above all else (until his wife and beautiful daughter entered the world!), I delivered a talk for approximately 45 minutes – running through fractals, complex systems, cellular automata, my award-winning article on pendulum waves, and my current work of electricity forecasting.

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Pleasing to see my Venezuela visit was nicely reported in the school magazine!

It was pleasing to see the students were engaged – they asked interesting questions, such as, “How would you become a Professor?” and “What are you currently writing about?”  I loved the experience on the whole, and used it to more recently deliver a similar, shorter talk to a similar age group of students from around the country.  They had come to Oxford to learn more about the options that await them at university level and it was up to myself and two other researchers (from Law and Psychology) to show just how university can develop us as critically thinking, open-minded, and curious people.  We particularly talked about passion for our subject being the key element that made us pursue such a career.  Following this however, a tricky question was asked by a teacher: “How can you reconcile a passion for what you do with the large debt that students leave with following graduation?”  It is a valid, and very real, point that must be considered by a potential undergraduate.  While it wasn’t communicated to the audience, the speakers generally agreed that a university path is not for everyone – it really is more important that one follows one’s passions, and that may not necessarily lead to university.

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One of my slides from Oxford talk: Using a cellular automaton model to assess the popularity of a new smartphone app.

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Ebrahim’s Adventures in Wonderland

adventure noun. an unusual, exciting, and daring experience.

And so, isn’t adventure one of the foremost reasons for living?  To marvel at discovering new things and following new paths.  For some, the definition above is taken most literally – manifested in travel; but as children, we loved playing hide and seek, wandering through the woods, playing with Lego…  We tend to grow out of it because that’s supposedly ‘what adults do’, but should we?  As an ever-learning mathematician, my adventure is the mental journey I take – venturing through many obstacles and confusing avenues of proof to solve a difficult problem – as well as the broader step that is to take a risk in life – to follow your passion for enlightenment despite not knowing the map.  (See earlier post on Passion.)  So, adventure seems fundamental for a productive mathematician, which brings me to an eye-opening event that I recently attended.  Organised by the effervescent maths populariser, Marcus du Sautoy, it connected the narrative of literature with proof in mathematics; again, the underlying theme for both is adventure.  I urge you to read this extract from Marcus’ talk and view the video of the discussion, easily found online (which also involved the great Sir Roger Penrose, no less).

Like many, my first experience with Alice in Wonderland was as a child via the Disney animation.  I was fascinated by this most “unusual, exciting, and daring” adventure.  I first read the book only about 6 years ago so it fills me with quite some joy that fate has led me to now live just around the corner to Alice’s shop and Christ Church in Oxford.  For me, the main thing that struck me about this book is how Carroll forces us to to be a child and view things afresh; indeed, isn’t the childlike wonder a beautiful thing in itself?  To observe things as if you’ve never seen them before in your life enables one to appreciate where they are and to marvel at the minutest of details; perhaps, even, that difficult problem is reduced to one that is elementary as a result!  A recently famous example is that of Andre Geim, a supposedly ‘serious’ Manchester scientist, who discovered the “wonder material”, graphene, by playing around with Scotch tape!  Geim welcomed novel approach(es).  I think it is important to regularly step ‘outside of ourselves’ just to see how things really are, to not become too biased or influenced by the way things seem to be, and, to some extent, to shake off the shackles that training/education can trap us in.  After all, there is never only one road in life (otherwise we’d have all followed it and ended up in the same place), and the road itself may have to be created by us.  So, like Geim, the researcher is encouraged to “think outside the box”.  And what better way to think outside the box than to actually be outside the box?  In our case, that is by taking on the guise of a child.  Another character.

Graphene was discovered using Scotch tape, and earned the discoverers, Andre Geim and Konstantin Novoselov, the 2010 Nobel Prize in Physics

Graphene was discovered using Scotch tape, and earned the discoverers, Andre Geim and Konstantin Novoselov, the 2010 Nobel Prize in Physics

The genius of Lewis Carroll is thus that Alice takes the reader along with her down a rabbit hole of wonder – child, adult, or in between, there are elements that appeal to all in this book.  So, whilst it is labelled as a children’s book, it’s worthy of being a classic children’s book.  Indeed, I’m sure you can count hundreds of pieces of work (books, films, games, etc.) that are boxed off as ‘for children’ yet appeal to adults.  But let us be proud of that appeal and let us understand the importance of why they appeal to us. So I say yes, we have no choice but to grow up, but we should always retain that childlike wonder.

Alice, an iconic adventurer.

Alice, an iconic adventurer.

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Good things happen when you follow your passions

This is a quick update post to share the joy of winning a prize for an article I co-wrote with Andrew Irving (my fellow Bee).  The article, entitled “Pendulum Pattern Perception”, won the Early Career Mathematician Catherine Richards Prize 2014, awarded by the Insitute of Mathematics and its Applications (IMA).  You can read it in the latest issue of the IMA’s Mathematics Today magazine: http://www.ima.org.uk/activities/publications/mathematics_today.cfm.html

In their own words, “The Institute of Mathematics and its Applications (IMA) is the UK’s learned and professional society for mathematics and its applications. The IMA exists to support the advancement of mathematical knowledge and its applications and to promote and enhance mathematical culture in the United Kingdom and elsewhere, for the public good.”

In general, the IMA is an organisation whose publications and events form a regular part of our (i.e., mathematicians’) lives.  My PhD supervisor, Dave Broomhead, was himself a former editor of Mathematics Today.

For the above reasons, I feel extremely privileged and honoured to have been recognised by my peers for doing something I love.  As always, I am deeply indebted to Andrew, whose writing (as is evident in our article) never fails to delight, enchant and spark the reader’s imagination in precisely the way we intend.

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This blog entry is written in memory of David Broomhead*

One thing I forgot to mention in my previous post is probably the most important thing that has got me so far in life – passion.

I remember choosing to do a PhD after taking a year out to ‘assess my options’, which involved a couple of job interviews in the financial world which just weren’t for me – I knew because I felt out of place at the interview itself. I think a parting of the ways was best for both industry and person. I’m a firm believer in things happening for a reason, one door shutting for the opening of another and all those other clichés! Thus, I was led to where my heart pointed me – back to university and mathematics.

I knew I wanted to do a PhD, despite learning from various sources about the typical struggles and uncertainties that were involved. It actually seemed like most people didn’t have a good word to say about it – I can’t forget how a “PhD stall” at a jobs fair sternly told the audience to “not do a PhD unless you’re prepared for difficulties”. Whilst I’m all for ‘straight talk’, I don’t think this is how I’d promote a PhD!

Thus, passion it was that drove me on towards my goal – I scoured the internet for PhD positions at various universities, applied to one or two, but wasn’t thrilled by the project topics themselves. Eventually, I took the bold step of deciding to follow on from my final year project at undergraduate level – I contacted my supervisor, Dave, and simply asked if he would supervise a PhD project of my own choice even though I was unsure as to how exactly I might extend the ideas from undergraduate level.

The uncertainty of research direction was compounded by a major drop in financial help – I would be paid less than half what I would be paid if I chose to do a council (EPSRC usually) funded PhD. In addition, such projects would be well-structured with targets and deadlines, almost guaranteeing a successful thesis in the allocated 3/4 year period. On the other hand, neither I nor Dave had a definitive timeframe in mind for our project!

So how did I get through this? There are actually many reasons, many of which I have stated and others are below, but I think overarching these is passion. I loved maths, and particularly the research topic that I chose. Moreover, Dave was always encouraging of any ideas I came forward with (the nice guy that he was!), and this made life so much easier – I think he understood that a researcher needs time to be creative and to simply think of different directions and solutions. Another supervisor may have (quite understandably) wanted results every fortnight or so, but such a pressurised environment was not my way of working and I’m really very fortunate to have had a supervisor who agreed. I never got to ask him but maybe Dave saw my enthusiasm for the subject and realised it was best to let me work in my own way to see my ideas take shape naturally.

There were many stages during that first year where I asked myself why I was going through such a struggle – both financially and mentally. I used to compare myself with my peers who were all in well-paid jobs, and this did not help – I advise against comparing because the grass really does always seem greener on the other side. I got through it by first writing down reminders for why I wanted to do this – “I enjoy maths”, “freedom to pursue own ideas”, … If one word were to form an umbrella over these, then that would be (yes, you’ve guessed it) passion. Another aspect that helped me was my family – I needed them both for moral support but also to put a roof over my head, for which I am still grateful. This meant that I could focus solely on the work. The bonus effect of all of this was that I developed my organisational skills from the start – it forced me to plan my weeks quite meticulously to ensure I made the most of my time at home, on the train, at the office.

There were other obstacles, including (sadly) Dave falling ill halfway through, meaning I didn’t see him again until after I actually passed the viva. Wow, I hadn’t really thought of how serious this could’ve been! However, I now think that I managed to overcome this because of the autonomy that Dave allowed me to have during the early parts – by now, I was naturally thinking of ideas, deciding which to pursue, when to type up, etc. In essence, I was supervising myself! This has proved to be a valuable skill as I grow towards becoming a more independent researcher in the ‘publish or perish’ world that I’m in now. So, again, adversity created opportunity. In the end, I was forced to submit by the deadline (which was now extended) by my replacement supervisor, Mark – another nice guy but one without whose insistence on my handing in chapter drafts I would have failed. I am eternally grateful to Mark. I’m also very fortunate that I’ve had such good people around me, including my fellow PhD students – although we couldn’t really help each other in terms of the work, the conversations about life were lessons in themselves.

In hindsight I would not have changed anything – the difficulties that I went through were there to make me a better researcher and better person. I really am glad to have gone through everything that I did during the intense first year, the calming middle period and the frantic writing up stage – it has made me who I am now.

There is a message here – if the circumstances permit, then wholeheartedly pursue your passion and never look back or sideways! Sure, things WILL be difficult at times, but what’s the point of life if it’s plain-sailing? It’s MEANT to be about ups and downs. Besides, pursuing your passion also ensures that you’ve been brave enough to give it a go – you would have no regrets. (The following is a favourite TED talk of mine, humourously telling us to have no regrets: http://www.ted.com/talks/larry_smith_why_you_will_fail_to_have_a_great_career) Having said that, the great thing about passion is that, not only does it produce the success, it also eases the mind during the low periods – it reminds you why you are here. It worked for me and I can safely say that it’s the best decision I’ve ever made. Pursue your passion 🙂

*Sad note: Only a couple of days after I first drafted this entry, Dave Broomhead, my supervisor, sadly passed away.  This was unexpected and so, I have unwittingly paid tribute to him in this post, though I don’t think it conveys exactly how much of an influence Dave was (and is) to me, not only in how I do research, but also in how I live life. His smile and welcoming aura showed that pursuing your passion should not come at the expense of happiness, enjoyment and loved ones around you – involving your family and friends in helping you achieve your passion makes it all worthwhile.

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Come with me and you’ll be in a world of pure imagination

Maths is beautiful.  If you’re thinking this is a cliche, then you are precisely the person I wish to address with this post!

I feel like mathematics is traditionally misunderstood.  In fact, being a ‘mathematician’, I am often misunderstood myself.  Yes, I was bright in school and did really well in my exams, eventually getting a PhD.  But that road was a big struggle (which deserves a blog post itself) – maths was and is tough for me.  In terms of my personality, I’m anti-social at times and am a bit of an introvert – traits you might say are linked to being a geek!  I don’t like labels though – they are too confining.

Often, I would rather play football than be doing maths – I’m actually rubbish at mental arithmetic and, indeed, sport is where I’ve made some of my best friends.  I’ve always been quite sporty and arty and I hated maths in school – just like you did!  Exams were not a doddle for me either because of the strict time limits that are imposed.  I think this is why a research career is suited to me – I get the time to think about problems and find innovative solutions.  I guess this also brings out the best from my character, which is somewhat closed-off from society at times.

Why am I stereotyped?  Maybe because I tell people I do maths and this instantly brings back nightmare visions of numbers, algebra, long multiplication, yawning in class and spectacles (I don’t even wear those).  But I think my perseverance in the subject (by taking A levels and going further with it) opened up this beautiful world to me and showed me what maths can actually do.  Maybe perseverance could have the same effect on you too – in anything that you pursue – you just have to give it a chance!  Generally, I have a thirst to understand the world and how it works – from the large-scale to the microscopic, I feel a responsibility to fulfil the gift of reflection and wonder that we humans are endowed with.

I guess schools could do more to show the beauty in maths (I have indeed tried to take my fractal work into schools; but only as an ‘outsider’, not a teacher).  In reality, maths encompasses so much more and is not like the maths we see in school.  I like the analogy from this video , which says that school maths is like taking art lessons but not being shown the great works of art.  There are many great works of maths and I think this should serve as an inspiration to mathematical education.

But inspiration doesn’t have to be restricted to traditionally mathematical domains.  Amongst so many other works of art (which I hope to delve into in later posts), Willy Wonka’s Chocolate Factory is a favourite of mine.  It is creative, imaginative, inventive and more – all important traits of a mathematician.

Willy Wonka's Chocolate Factory fantastically displays the appealing traits of a mathematician.

Willy Wonka’s Chocolate Factory fantastically displays the appealing traits of a mathematician.

From my experience, problem-solving is maths.  Finding imaginative ways of doing old things is maths.  Being creative is maths.  Thinking logically is maths.  Contemplating is maths.  Wondering is maths.  How is this all possible?  I guess that is my challenge on this website – I must try to win you over and show you this fantastic world that’s discovered with an imagination fuelled by maths.  I hope you like it, I think you will…

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