I think that extension work should take what there is for syllabus and extend the work. Obvious? If so, then why do so many people treat extension work as an excuse to teach what is in next year’s syllabus? I say that attitude is wrong. Wrong at so many levels: not least, it makes the gap between the competent and the brilliant all the bigger, because when whatever it is that officially belongs in next year’s syllabus is eventually covered there are two new problems—those that understood first time round continue to be bored, and those that didn’t ‘get it’ have a whole new baggage of unlearning to do before whatever faults in understanding there are are found and fixed. And, of course, the competent are made to feel far from competent, which is not at all useful.

It would be far better, methinks, to develop other skills, especially those of integrated study, cross-curricular links and those that require the use of what has been learned from the curriculum. To this end, I’d far rather see (and hear about) problems being presented in context (i.e. not as Maths), so that they are identified, puzzled over, so that solvable sub-problems are found and that the skills of recording and presentation are given value. This may be read as ‘project work’ of which I am something of a fan, but investigations in maths can take anything from minutes to weeks—I have taken a GCSE investigation and a Y9 class has, to their minds, ‘finished it’ within a double lesson; this because they drove the investigation while I recorded on the many classroom boards what it was they had discovered. They might say in later weeks I had pushed, but at the time the sense of collective adventure (“We don’t know where this is going”) was the driving force in the room. On a really good day we had several bits of homework (one session, several problems, not everyone doing the same single thing) set by the class for them to report back on later in the week.

In particular, what would be useful, I suggest, would be to extend any maths problem to *earlier* in the problem-solving routine. So instead of finding 15% of £56 one looks at the problems attached to applying VAT (at 15%), including how to collect that tax. Then we can also look at such a problem as if sales manager, business owner, bookkeeper, accountant or chancellor. We could explore what happens with percentages when it is not clear what it is that is 100% and look at why/how these confusions occur (a language issue mostly). Thus we might extend maths from where we need the maths back towards where we try to model a wider problem. This, I say, is the missing skill, seeing where a little maths would be useful. I think of this as 'the hidden problem'.

Another issue, and a general one, is the perennial one of precision. So many reach seciondary school saying that they like maths because the answers are exact. But the world is not like that and in practice we need answers that are usable (and memorable, rough & ready). So we must not only learn to use precision wisely, we must be able to approximate, to estimate with speed, and to recognise the stability of an answer. Stability is when a small change in one place only resutls in small changes elsewhere.

What I offer in this collection is very little of that modelling and prject work, as there are many resources available. What I put here is stuff I’ve written that falls outside a current syllabus but which can (easily) be accepted by students as worthy, interesting and motivating. Much of this content was, once upon a time, in a syllabus for the age-group you have in mind (I’m thinking 11-14). Some of what I put here falls into a general category of the hidden problem: here’s a problem described in words; you need to find the maths within it and solve that; you also need to take that result and return it to the context as a considered action or recommendation. The idea is to persuade pupils into the habit of using their skills, to educate them into the confidence to explore an open or closed problem and hopefully to gain enjoyment from so doing.

So what I’ve written is for the student who is exploring on their own and perhaps for the teacher looking for something to take to a class. Most of what I’ve written is in the form of a worksheet that I could (and often did) use with classes in a suitable frame of mind, at any age from 11 to 16. And I did: for much of this material the difference between the ages is the speed of acquisition of idea and the recognition of available skills. Sometimes the older students had much bigger problems rediscovering what it is they know (knew at the time of reading) that would be useful—sometimes the less knowledgeable can find their way more easily.

DJS20150327

At 2017, when re-creating this collection, I shuffled it into an order that might be construed as ‘difficulty’. That means that I have put them in an order I think makes sense to me with my classes (those I taught). That will not necessarily agree with anyone else’s such ranking. A side-effect of creating the section is that the section entitled Lower School now has stuff I recognise as being within core syllabus for Y8-Y11 and what I put here is the remainder.

DJS20171127

I moved the stuff that is end of term-level fun into a sub-group (of seven, at the moment);

I can see that perhaps this is the greatest contribution I can make; extension work and maths that is simply fun. As in fun to find out, as in messing around with numbers….

- End of term and fun (7)
- Climate change (4)
- Borderline Statistics
- Saving time and fuel
- Metering
- Find the Number
- Roman Numerals
- Horrible paint
- Going to Cambridge
- in whom we trust
- Natural Units
- More Weird units
- More Weird Units Ride Again
- Referendums
- Traveller’s Arithmetic
- Pollution
- Rabbits
- Long Decimal Reciprocals
- Reasonable Questions
- Running and Walking
- Perfect Numbers
- Index: Xⁿ starts with n
- Shopping
- Sewage sludge
- Stopping
- Filling the tank
- Vaccine effectiveness
- Tiling the Bathroom
- Time zones, sunset, etc.
- Trams and Runners
- Trespass of your social distancing bubble
- Viruses are very small
- Upper Middle School - athletics (Mech)
- the N-dimensional cube
- Logs: Who cares?
- a^b^c explored
- Fuel bills
- Extra over