Floor to Ceiling

Yesterday I posed a simple (ish) question on twitter:

“What is the nth term of the following sequence:

2, 20, 202, 2020, 20202, 202020, …?”

I love the question for a number of reasons that i’ll discuss here.

Firstly, on the surface it’s a really simple sequence. It’s very easy to describe it in English.

“Start with a two, then add a zero, then a two, then a zero, …”

Or, if you want to get a little more specific:

“follow the recurring sequence 202020 with n digits”

So far so good. But this is a great example of how something can use very few words as an easy instruction in English, but be arguably much more difficult to translate into mathematics.

Often it’s the other way around.

Consider the formula below:

2n2 + 7

Simple to write in mathematics, and say as “2 n squared plus 7”

but to describe it to someone less familiar with the terminology we’d say “take n, multiply it by itself, then multiply it by two, then add seven”.

In this case the mathematics is much simpler than the spoken English, if you’re fluent in both.

But how do we write mathematically :

“follow the recursive sequence 202020… with n digits”

You’d think it’d be straight forward… well, it depends on what tools you have at your disposal.

An initial approach

I began by ‘seeing’ the pattern 2, 20, 202, 2020, 20202, … as

2

2 x 10

2 x 100 + 2

2 x 1000 + 2 x 10

2 x 10000 + 2 x 100 + 2

(spoiler: there are better ways to see this pattern!)

Or, written utilising ‘n’:

inf series 1

This felt like progress. The pattern utilised ‘n’ and generated the sequence, but it’s not a neat ‘nth term’ just yet.

The issue I had now though was that the pattern wasn’t quite consistent enough with the tools I wanted to use. How could I define that if n is even, just do the thing you did for the term before, but if n is odd, you do one more thing? How do I tell the sequence when to stop? If n is odd then I stop at 2 x 10n-n but if n is even, then I stop at 2 x 10n-n+1

At this point I thought maybe I needed two nth terms, one for if n is odd, and one for if n is even. That wasn’t very satisfying though. Surely I could just merge the two things together…

I should probably have just decided to take an entirely different approach, but I was in it to win it now (or… lose ultimately).

Note:  on twitter, various incarnations of the above had begun to surface as others were also punching holes in this problem. I had a few people write solutions literally in words, with clauses for odd and even, and many others had latched onto variations of the 2 x 10n idea.

So, even though it felt messy, maybe this was just destined to be a messy nth term. I struggled on.

As I was continually adding things onto the sequence, I figured I’d need sigma. But how exactly would I use it to incorporate all the little clauses that were rearing their ugly heads?

sigma 1

Something like this maybe?

It should be noted that sigma is in my tool box. For a lot of people it might not be. Hence it’s a good time to mention that problems like these are great for reaching a point where you say ‘hang on, I think I need a new tool… is there a tool that does…”

I know there are some teachers who teach students along these lines, exploring a tool until you’ve exhausted it and discovered you *need* a new one, rather than just being given it.

I realised after a bit of thought that, using this model that I was kind of now blindly following because I’d invested in it, I’d need a second variable (maybe I don’t need it, but I decided I did). So I came up with this:

sigma 2

I felt like I was closer, and I was (barely) pleased that I was somehow managing to incorporate everything into one formula, rather than having an odd and an even… alas there were a few issues. Firstly, it doesn’t work for n=1. Secondly, it kind of scrappily works for n=2 (I think…), thirdly it doesn’t work at all once “n-2x+1” results in a negative number. I thought I could fix that last part with yet another clause:

sig3

But now it was scrappier than hell, and it still doesn’t really work, and I’m not convinced it’s even an ‘ok’ thing to do… I was basically laying out a flow diagram into a messy, erroneous blurg.

Time to change my approach and start over again. I felt like I’d hit a wall.

In the back of my mind when I noticed that odd and even ‘n’ seemed to need different rules, I imagined I’d possibly need to use either the ceiling or floor functions. Again, these are tools I’ve needed in the past, but may be unknown to many. ‘Floor’ rounds anything down to the nearest integer, and ‘ceiling’ rounds anything up to the nearest integer. They’re useful for many reasons, but it felt like they might be useful here to use some kind of division when ‘n’ is odd to get something more workable. Indeed, here is a submission from someone on twitter using the ceiling function:

C9W404_UAAAq6Uw.jpg_large

(It’s used for n/2).

Anyway, I left it alone for an hour or two, but it was on my mind. Then it struck me that 0.202020202020… is just 20/99. I hadn’t even been thinking at all along those lines. I’d got too obsessed with picturing each output as the sum of separate parts.

The rest fell into place very quickly.

floor soln2

hence the nth term is “simply”:

floor soln

I feel embarrassed to have even pursued the first line of enquiry (probably just a symptom of maths anxiety), which is silly because that’s what we all do: chase an idea until it stops working. I could probably have made it work eventually, but it would have been a mess. Below are some of the other solutions and almost solutions that people posted in answer to the tweet:

Reflections on 13 years of teaching…

I began my teaching career in 2004, and in two weeks I will be finishing my secondment in schools, effectively ending 13 years of teaching children. It feels appropriate to share a few thoughts and reflections on what I still genuinely feel is one of the most rewarding (and punishing) professions anyone can experience.

The Noble Profession

I did not start teaching because I wanted to help children. Nor did I start teaching because of a nagging dissatisfaction with being a small cog in a big corporate wheel, searching for something more meaningful. I joined because I needed a job, and at the time, teaching was offering a very attractive financial incentive – paying off my student loans over 10 years, and a ‘golden hello’ of about £5000 in my first year after qualifying. It felt like a better idea than continually applying unsuccessfully for graduate IT jobs that had seemingly dissipated after the millennium bug phenomenon.

Today the Government offers similar, but poorly structured incentives to join the profession. A large bursary that effectively salaries your training year (for shortage subjects). However, there are no strings attached. No conditions beyond ‘start training’, and no caveats should you decide to quit during your training, or after your training, or after a term of teaching. In fact, some trainees end up taking what amounts to a pay cut upon graduation.

I didn’t find my training year hard. Everyone always says it’s hard but it doesn’t have to be. It took up a lot of my energy, but the things I was being asked to do didn’t feel difficult, just time consuming. It bothers me that trainees now are often asked to do more than they need to, simply because people seemingly want them to find it as hard as they did. Encouragement seems in short supply.

A love of maths

I was not a fan of maths at all. I wasn’t even a maths teacher. I taught ICT (basically a tour of the Office Suite) which felt like a disappointment after studying programming and mathematics at University (I studied maths because it was valuable, not because I liked it). I was competing with a trainee teacher who had already established themselves in the school we applied to (he was working there), and the school ended up appointing us both because I could offer mathematics as well, despite having no formal training in teaching it. The first 5 years of my teaching career very much had maths as an aside. I was promoted quickly, and became the preposterously pretentiously titled ‘Director of ICT’ after about 4 years. Being promoted quickly was exciting and felt rewarding, but ultimately I became disillusioned, and then I quit.

I was not a good teacher

I don’t look back on my first 5 years of teaching with any particular pride over my style of pedagogy. I got on very well with my students, lessons were relaxed, behaviour was good – after I realised that being a total  asshole with no time for empathy (as instructed) didn’t work well for me at all. But in hindsight, I think people thought I was a good teacher because the students behaved and we were all having a good time. I have no doubt that that’s a key part of doing the job well, and without it you’ll probably struggle, but the way in which I actually taught stuff was pretty poor. I fell foul of many IT lessons that defaulted to ‘research this, present it on a powerpoint’ or ‘get on with your GNVQ coursework’. Worse still, the fact that I was being recognised as an ‘outstanding’ teacher felt counter intuitive. I wanted to get better. To this day I question the notion of what an ‘outstanding’ teacher is. I think a lot of schools still see it as a show. Kids engaged? Tick. Teacher enthused? Tick. Books look ok? Tick. This guy’s great.

Teaching maths felt harder, and more nuanced to me. I liked that, and I always finished a lesson, no matter how well it went, thinking ‘how could I do this better?’.

Working Abroad

As flattered as I should have felt at being promoted quickly, it left me feeling disillusioned and numb about the linearity of a teaching career. I sought a more significantly different edge to my career and applied for jobs that seemed above my station. To my surprise I ended up being an education consultant in the middle east, working with interpreters to help reshape the pedagogy of existing teachers in boys schools. It was a huge culture shock, and the most amazing experience of my life to date. It was here where I started to really invest in my own development. I began reading about the science of teaching, about the science of questioning, understanding, memory, the deeper questions around the purpose of school, and the history of education. Suddenly teaching became so much more interesting to me. It was a kind of awakening, a realisation and an appreciation of the precision, hidden depths and  craft of good teaching. I felt as though I had actively rejected my former teacher self, and the very notion of what was being labelled at the time as ‘outstanding’ (*spit*).I learnt to love mathematics as I worked alongside some of the most enthused and knowledgeable people I’ve had the privalege to meet, who showed me that making sense of mathematics is so much more important than just being able to do it.

I met so many incredible people from different walks of life, all of whom helped reshape my outlook on teaching, and life in general. I saw how valuable education is, and how little we appreciate it back home. I saw the imbalance of privilege – where I could be served in a coffee shop by someone with more qualifications than me, but with the ‘wrong’ colour passport, and I heard of what people go through to be able to give their children access to any kind of education. People who moved away from their families, their countries, just to earn enough to send home to give their children a chance. Children they would see grow up only in snapshots. I thought back to the children I had taught in the UK, utterly oblivious to the enormous liberties afforded to them simply because of where and when they were born. Indeed, I was one of them.

I returned to the UK with, for the first time, a genuine zeal for the profession, which has stayed with me since, and which I try to pass on to trainee teachers embarking on their own journeys.

What we do changes lives. Some will appreciate that, some won’t. Some children would have done just fine without you, some wouldn’t. Some may fail, most won’t. The tragedy of teaching is that you often don’t see just how much impact you had on people. The occasional run-in with a former pupil, or a heart-felt letter at the end of year 11, or a parent telling you how grateful they are for what you’ve done for their child. We cling to those moments, in a time where teaching is harder than it should be. You’ll find those cards, gifts and messages stuffed away in the desks of teachers, or pinned to their departmental noticeboards. Sometimes we need to revisit them to remind us what the hard battle is for.

I’ll miss the buzz of the classroom, the joy of seeing when a concept clicks, the wonder and intrigue of young minds, the eleven year old predicting a room is “about 20 metres tall”. I’ll miss the sixteen year old who is mortified that they just calculated a perimeter as a negative number, the belief reignited in someone who aced a test, the email from a student telling me maths ‘isnt shit with you’, and the nine year old who signed my leaving card on my first school placement “Goodbye Mr Southall, thank you for teaching me, you will always be suspicious.”

Bless them all.