A little diversion

‘Teach me to divide’ she said

One wet afternoon when rain had stopped all play.

‘I can share OK to ten, but I need to learn the long way.’


Whatever for, my innocent young friend, for don’t we have a calculator as our foil,

And did not Ada Lovelace work such miracles, to take away such toil?


Learn algebra by all means

Seek patterns and relationships

For life is never really what it seems.


But better still, learn a poem, or a few

So that on some moonlit night

You might recite a verse or two,

To some young lover, arm in arm with you.



I have had great difficulty in embarking on this little piece; how can I write about criticism without being critical, when I find being critical so difficult?  Of course, I come home from work and complain, and from lots of other places and complain, but in the safety of my own dinner table and in the knowledge that my grumblings will not leave the room.  What is it that prevents me from sharing my expert knowledge with colleagues and other professionals – is it simply my own diffidence, or do we lack the culture that might encourage robust criticism of one another?  As teachers we have ingrained habits and attitudes, the way we always do things, and often assume other methods are either inferior or simply wrong.  (Actually, they may be better ways of doing things, but we lack the skills of doing it that way.)  And to compound this we may be doing things in the way we were taught ourselves – we who succeeded in the education system now relying on the methods that brought us success.

By way of example I will take one of my pet hates – the taking of the mouse.  As a learner myself at a local university I was following the teacher’s instructions to load a programme, but getting somewhat lost I asked for assistance. The teacher came over, took hold of the mouse, and proceeded to do the thing for me.  ‘Get off my mouse!’ I shouted, and explained that I was doing the learning, not her, and it required me do to the thing.   More recently a variation on the situation happened.  I was in the process of interviewing a prospective learner. I had given her a set of simple instructions to follow to undertake an online diagnostic test and asked a colleague the password to get on the local system.  My colleague immediately took the learner’s mouse, logged her in, selected the options, and set her off on the test. Inside I’m shouting ‘get off!’ but I can’t say it, then or later, for whatever reasons I still can’t fathom.  From the first meeting my aim is to empower my learning, and when I stand back to do so, and someone else jumps in, I get rather irate inside, but keep it inside.

Going back to the same university teacher, on a course for teachers, she had us cut out a triangle, tear off the corners, and arrange them to make a ‘straight line’, thus proving, she said, that the angles of a triangle add up to 180 degrees.  ‘No we haven’t!’ I cried ‘you have demonstrated, rather roughly, that it works for this specific triangle, and in no way proved the general rule.’  I’m not confident that she grasped the difference, but why is it we find it so easy in a hierarchical order? I’m always happy to criticise someone teaching me, and someone I’m teaching, but never fellow teachers.

Online forums and resources seem to exasperate my feelings.  One of my managers often suggests that I look at something that has been recommended to her at some training session, or that she has read about, and I’m quite happy to write a page of thoughtful criticism.  When I see things that I think are a waste of money, considering the huge amount of free resources available, then I simply ignore them.  But sometimes I see free things that so offend my pedagogical beliefs I want to offer that criticism. And often those things have been willingly shared by other teachers who, like me, spend hours preparing resources they are glad to share.  If it is something that is based on a sound idea I’m happy to steal the idea and build upon it in my own way, and I hope people do the same with me.  But what if the principles are at odds with what I passionately believe in?

Should we keep it bottled up, being the best place for it, or should we share our thoughts in the hope that all of us can learn from one another, and that the listener or the reader will hopefully appreciate the place that criticism is coming from – a deeply held love of learning?

Rich Questions

Rich Questions

Many years ago, in the early days of Inset and ‘Baker Days’, when people still knew who Baker was, a colleague gave a demonstration of a simple resource, an apple, and how it could be used to enhance his teaching of modern languages.  I’ve never forgotten it, as the epitome of a crass and thoughtless teacher showing off his knowledge without a care for our bored minds and beleaguered bums.  He spoke without pause, and we probably forgot every word he said.  I’m entirely in favour of simplicity, and love a simple question to start things off, and watch the threads take off in a variety of directions; in fact, the simpler the better.  My group on Friday afternoon are returning learners of assorted ages, nationalities, and existing skills.  I’m confident all of them have a wide range of mathematical knowledge, and my job is to get them back into critical thinking mode before term starts in September.  We’ve covered some algebra and some ‘shape & space’, and lots of incidental number stuff in the first couple of weeks, and this week returned to shape.

Me – ‘Draw something with an area of 18 square centimetres.’

I’d supplied centimetre square paper, rulers and pencils, and the rest was up to them.

After a few moments:

A – ‘I’ve got them all – rectangles 1 x 18, 2 x 9, and 3 x 6 – that’s all there is.’

C- ‘What about 4 x 5? 4 + 5 + 4 + 5 = 18?’

A – ‘No, you are confusing area with perimeter – area is the inside, perimeter is round the outside.’

(A demonstrates to C on C’s drawings the difference between the two concepts.)

Me – ‘Are you sure this is all?’

Another pause…

C – ‘Can we use halves?’

Me – ‘Did I say we couldn’t?’

C – ‘In which case we could have 4 x 4.5, 1.5 x 12…’ and goes on to list numerous other cases. By this time, C has moved entirely from the concrete to the abstract, and has ceased to draw rectangles.

Meanwhile, B is thinking…

B – ‘Can we have a square?’ And answering herself: ‘No, because the side would be the square root of 18, and that isn’t a ‘real number’’. She might not have said real, but something like it, indicating it wasn’t something we could measure with a ruler.  I actually take part for a moment, stop asking questions and open a calculator on the interactive board to show that root 18 is a damned tricky number, if you want to catch hold of it with a calculator.

Me – ‘You might like to come back to that one.’

B- ‘What about triangles? Can we have an equilateral triangle?’

Me – ‘That might require some serious mathematics, but there are lots of other types of triangles that you might look at.’

C then produces a number of right-angled triangles, and demonstrates to B how she has halved assorted rectangles of 36 square centimetres to produce triangles of 18 square centimetres.

Meanwhile, A is ordering her thoughts around the number of variations on the original question.

Me – ‘Are there an infinite number of shapes with that area?’

A – ‘Yes.  I’m thinking of a splodge, and that splodge can be any shape at all that covers 18 squares.’

Me – ‘A splodge – I like that.  Sticking with rectangles for a moment, are there an infinite number of rectangles?’

A – ‘Yes there are.  If I have an area of 18, and a given length, I can divide to find the width.’ We are now into re-arranging formulas, by intuition.

Me – ‘What did we refer to when we were looking at patterns last week, and general terms?’

C – ‘A variable.’

So now I’m inclined to open a spreadsheet on the screen, and A demonstrates confidence and fluency with Excel.  I put in some headings, and she explains the formula for the ‘Area’ column.

A – ‘It is the contents of the cell A2 multiplied by the contents of the cell B2.’ Spoken like a pro, and delivered with more clarity than a good many IT teachers could muster.  We pause whilst A explains the concept of spreadsheets to B and C. She then does all the clever bits, like copying down formula, and creating a new sheet for doing it backwards, with area fixed and length as the variable.  After which A and I get into a technical discussion where she initially sees ‘infinity’ as going on forever, and I’m hoping for her to see the infinite number of cases bounded by the ones she has.  In other words, she can see rectangles stretching forever and getting thinner, and I’m seeing an infinite set of widths, say, between 4cm and 5cm.  But time is pressing.

Back to D, who by now has moved from right-angled triangles to isosceles and scalene triangles, all the while rediscovering her knowledge and naming those shapes.

M- ‘What about the square – is there any way to construct it, without having to measure root 18, and thinking of what you have done to find the triangles?’

D – after a very short reflection: ‘Yes, if you draw a square of 36, then cut off the corners to create… What’s it called?’

A – ‘A rhombus?’

D – ‘Yes, a rhombus,  a square that’s half of the square, another square inside it, tilted…’

Me – ‘Come up and draw it!’

She does, and we have to break off shortly, with the briefest recap of what has ensued, what has been revisited, learnt, expanded upon.  It’s show time, and C and B have a suitable array of rectangles and rectilinear shapes to share.


And an hour was far too short. What could we have done with two, and what might we do next term when they start for real:

Do all the shapes have the same perimeter, and if not, which has the least?  Is there a maximum?

Will we find that parallelograms are easy to compute, and are trapeziums just as easy?

Are there any simple regular shapes that have an area of 18 sq cm, apart from a square (which wasn’t that simple at all), or do they all behave like equilateral triangles and need skills we might not learn in a few months?

Will we get to circles, and rearrange that formula that Maggie Thatcher said every pupil should know, to find a circle of 18 sq cm, and has it the smallest perimeter? (Cherry pie delicious, apple pies are too.)

And can we generate the formulas for the ones we’ve found, producing simple elegant statements with symbols and operators?

And could we use those formulas to generate variations, as we rearrange them according to known variables?

I’m not entirely pleased with myself, although I’m pleased that everyone has shown learning, and that both A and D have demonstrated true mathematical thinking and insight. B was not confident that he’d contributed, but the others reassured him that he’d be supported in future.  I think I was indulging, and could have possibly tried to keep things simple at this early stage. And next term I will.  But there again, challenging is good.  And I had fun, and remember, that’s the important bit, I’m the teacher!

Teacher Lust

Teacher Lust

I’m about to make a plea for the laid-back teacher.  We’re damn lucky if we get a job in the first place, risk little chance of promotion, and have a constant battle with learners who think they ought to be ‘taught’.  And yet it is us laid-back pedagogues who put the emphasis on learning and not teaching, the ones who don’t believe we know everything, believe that our way is the only way of doing something, and we are the ones able to refrain from imposing our will and our constructs on the mind and actions of the learners.  It has been nearly twenty years since I came across the construct ‘teacher lust’, and well over a hundred years since it was defined by Mary Boole, and yet still teachers act and behave as though they must be in control, and that enthusiasm equates with dictating everything a learner does. A definition from Tyminski says:
‘Examples of enacted teacher lust can include imposing mathematical knowledge or structure; directing and/or limiting student solution paths and strategies; or telling information in a manner that reduces the level of the task.’
How many of us have not been on the receiving end of this approach, and how many teachers are stuck in this mode? Furthermore the prevailing political will is not at odds with most teachers, who believe that not only should learners have no freedom of experiment and discovery, but that the teaching time allowed is far too precious to admit such concepts.
Consider the first job on my search of the TES: ‘We wish to appoint for September 2012 an enthusiastic, well-qualified, motivated and dynamic teacher of Mathematics.’  I can accept the ‘well qualified’ bit, since I’ve worked with more than a few damned ignorant teachers whose understanding of mathematical concepts was so limited they could only deliver learning in a force-fed manner, any deviation from the script by the learners sent them into freefall.  But what about enthusiastic? ‘showing lively interest; extremely keen’ (Chambers).  Why can’t I just show an interest, and be keen, instead of hurrying things along towards some great ‘challenge’ for the bright ones, and a few simple tasks for the slower ones. And remember, there’s sod all wrong with being a bit slow – quite a virtue in many ways. Motivated?  Surely my task is to motivate the learners, not overwhelm them with my own motivation.  And as for dynamic, ‘full of energy, enthusiasm and new ideas.’  Bit of redundancy of words there – yet more enthusiasm.  What about quiet, reflective, and encouraging teachers?

Where does it all lead?  New teachers do what their teachers did, and that is to stand at the front and teach. So many write resources and produce activities that are designed to direct the learning, and then keep the learner happy.  And all this makes more work, for the teacher feels the need to mark everything.  But there is hope – with experience and age comes wisdom.  Like all good 12 step programmes to combat addiction, there are ways to overcome lust.  Tyminski again:

‘As instructors become more conscious of their actions in the classroom and better understand the outcomes, they are more resistant to the influence of teacher lust and have more pedagogical options at their disposal to use to combat its pull.’
And finally, remember the learners, who do appreciate good teachers: @colin28 you are laid back! The very reason we all love you x  (and I love you too, SL!)

References: don’t be silly, use Google and do a bit of research yourself!

How to teach mathematics

There are a few rules to being a teacher, and most of them involve trying to avoid being the centre of attention; rules like never answering a question, and never doing anything yourself if one or more of your learners can do it for you, perhaps not so well as you think you can, but nevertheless, learn to let go.  Some of the difficult rules involve feedback; don’t tell the learner what they should have done, but lead them to a solution by asking questions – never forget Socrates, and if you have to be critical, as I do when giving formative feedback to budding teachers, suggest what they need to do to improve the assignment or write-up.  So when I find myself awfully critical of fellow professionals I find myself at the point of wanting to tell them what is wrong, and accepting I need to say what I might have done, or what I do do, which I think might be better.  Today is my first attempt at that, and I hope to continue with suggestions for teaching mathematics and the topics of the National Curriculum.

So what got me grumpy in the first place?  A number of posts on the TES twitter feed have led me to ‘superb’ resources that I found to be mind-numbingly boring.  The first time I replied, and got a nice response accepting my point, but I was told that young, new teachers loved ready-made materials that engaged learners in the classroom, and kept them busy and happy.  All very well, but has it anything to do with learning?  The latest one was described as ‘A very high quality substitution code breaker activity to help with algebra.’  No it isn’t.  A code breaker suggests some thought needed to break the code, not a set of simple calculations using letter substitution to reveal some ‘hidden’ message.  If you want the learner to break a code, make them do some work, make them think, not simple swapping of letters for numbers to find new letters to make a catch-phrase.

We need to go back and ask ourselves what we understand by algebra.  Or more appropriately, what does the National Curriculum understand by algebra?  I well remember the NC and the need for all children of all ages to ‘do’ algebra.  It was a complete disaster because primary school teachers, by and large, had no grasp of algebra themselves, and hence no idea how to teach it.  So in 1992 a task group was set up to review algebra teaching and I recall being part of that discussion.  Algebra, at all key stages, is about moving from the concrete to the abstract, for learners to make generalisations and to use symbols other than numerals to define mathematical relationships; to look for patterns, to describe and define those patterns, to tabulate patterns and to turn them into pictures using Cartesian co-ordinates, and to arrive at generalisations using symbols, usually letters.

If I get encouraged by the realisation that someone is reading this, and I hope someone does, I hope to continue and explain how I introduce adult learners to algebra.  I teach in FE, and have to cover the National Curriculum mathematics in less than eleven months, and nothing like the eleven years that school teachers have to develop the subject, so I don’t have time for mind-numbing tedium.  But I do have grown-ups who can be trusted to stay on task, and to bring their own experiences. So I’ll start with one simple example, and if it flies, I’ll develop it over the next few months.  My example is teaching a sculptor, a learner who needs to gain a GCSE in mathematics in order to commence a PGCE.  In a crash course in the month before her exam she asked me to explain a formula question on a past paper.  ‘What do you understand by a formula?’ I innocently asked.  She thought for a moment and replied  ‘Like A+B?’.  ‘No!’ I’m screaming inside, but keep that inside.  I looked around her studio (we were working at her place) and saw the model she is working on for the Wedgewood company.  ‘How do you price a sculpture?’.  The answers started to flow: ‘Materials, labour…’, which we were able to break down to a simple linear relationship, being M + T x R, where T and R represented Time and Rate of pay.  Simple, and so common a relationship in so many exam questions: electricity bills, phone bills, taxi costs, anything of the form y=mx + c, which is pretty much all that is required to pass the hurdle of gaining a ‘C’ in mathematics.  I was able to draw on the knowledge and experience of the learner, but then kids have enthusiasm and intelligence and a desire to learn, which makes teaching such an easy and enjoyable job.  So don’t make formulas to be mindless repetition of sums, but make them abstract representations of real-life, or imagined, situations.

Next I could explain what I did this week with a packet of 1000 ‘matches’, which cost me £1.99 and worth every penny, and provided a lovely photographic record of patterns in number, and how I will use the hand-shaking lemma in September when the next cohort arrives.  Will I be tempted to produce some easy repetitive worksheets?  Who knows, it might depend on the classes I get!  All I need is some feedback.