# Complex Numbers, Pythagoras, and Protractors

Honestly, you could hardly make it up. In the last week or ten days you may have spotted some or all of these:

***   Simon Jenkins in The Guardian pontificates about maths teaching.   Among his assertions is that a primary school complains that a child he knows hasn’t mastered complex numbers.   It’s not clear how much Jenkins knows about complex numbers but it doesn’t seem to extend to knowing the difference between complex numbers and fractions, or possibly decimals.

***   Then there’s a YouTube clip of Piers Morgan smugly telling us that he understands Pythagoras Theorem and can recite it as 3.147….

***   And yesterday I read that a golfer is being investigated for using his protractor – creatively, he’s apparently invented a way of using his protractor to measure distances rather than the usual angles. We then get a helpful clue that his protractor is also called a “compass” – we also learn that sailors have been using them a long time. And in further clarification we learn that the device has a third manifestation as a pair of “split dividers”.   Actually, it was pretty obvious to all of us that the protractor / compass was in fact a pair of dividers all along – but not a single member in the newspaper’s editorial and production team knows what a protractor looks like, let alone what one does.

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# A Year Of Non-French

At the middle school I taught in, the top year (Y8) timetable had a unique subject called non-French.  Pupils were selected for this elite group using the sole criterion that the French department didn’t want to be bothered with them.  As you’ll imagine, this didn’t exactly enhance the children’s self-image; things were made worse still by the fact that nobody had the faintest idea of what the lessons should involve and that the group was taken reluctantly by whichever teacher had a free slot on their timetable that year.

I have to confess that the French department and the Head weren’t far-sighted educational thinkers, and neither of them saw anything wrong with putting a subject on the timetable for which the only guidance was that it wasn’t French.  So one year I decided I’d have a go with the group, and I must say we had a lot of fun.

Rule 1 was no text-books, and above all no worksheets.  Rule 2 was that we’d actually do something positively rather than because it would fill up a lesson or two.  So we read and performed comedy scripts such as the classic Tony Hancock Blood Donor.  Then we spent a month of lessons exploring some number magic tricks, leading up to a nerve-wracking and rather brilliant performance by the group in which they stunned the audience (i.e. the non-non-French remainder of the year group) by a variety of magical effects and amazing predictions.

I thought of them when I spotted a tweet forwarded by Simon Gregg about measurement using a trundle wheel.  One fine morning in early summer we decided to help the groundsman by measuring out the 100-metre track.  They organised themselves in half a dozen groups and remarkably, every group chose a different method.  One group used strides, one pigeon steps, one a metre stick, another made themselves a 10-metre length of string, ….  Even more remarkably, I was flabbergasted that the various methods all agreed within two or three metres – and were in close agreement with the official measurement determined by the school’s trundle wheel.

(Which is why the tweet intrigued me.  Simon’s correspondent reported that his trundle wheel measured differently from A to B than from B to A – and that their groundsman says that’s invariably the case!)

Another big topic we did was to make a simulation game of the development of railways across our local Chilterns area.  The pupils worked in teams, building routes which avoided hills and rivers to connect revenue-generating towns.  We looked at costs, and scheduling, and created a wonderful map of hexagons; the map filled the entire wall of the classroom.  One day a visitor came and was a little snooty that we’d played a game to model the process.  “Wouldn’t it be better to study what really happened?”, she said.  Which was what we’d done the very day before, so the non-Frenchers proudly demonstrated how one of their routes had followed the GWR track out to Reading and the West, while another had followed the Great Central route to Aylesbury and beyond, and another had taken the route out through Luton and Stevenage.  I did enjoy that moment!

As the climax of the year the group created and published a comic magazine called Creeps!, sales of which (enhanced by the donation of a huge box of Monster Munch by Walkers Crisps) were large enough to necessitate a reprint.

There was one spin-off that was nicely gratifying.  Upsetting the French department was always fun, and they were quite aggrieved by the whole business.  They’d been quite happy to slough off their discards, but didn’t much enjoy it when they faced a queue of their students asking if they could drop French and do non-French instead.

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# Problems with Gabriel’s Problem

Here’s an activity I’d like some help with. It seemed it would chime pretty well with my Envelope puzzles (see for example, https://established1962.wordpress.com/2015/04/   and also  https://established1962.wordpress.com/2017/03/03/a-wow-conversation-with-amy/  )  Things started well enough, but as the session went on I couldn’t find the right way to take it further.

We started with me sketching out a 2×2 grid. I asked Chaz and Jo to put the digit cards 1 to 4 into the squares. I then wrote in the products of the rows and columns.

The next stage was to give them another grid, this time showing the products only. Could they place the digit cards?

Of course they were happy to do this, and we were quickly ready to move on.

I think I missed a good line of development here. Chaz and Jo have good knowledge of the multiplication tables, and could have handled a situation where I used other combinations of digit cards. They would have been able to work out the cards used to make this grid:

Instead, I moved to a 2×3 grid, with the numbers 1 to 6. This worked fine. There were lots of possibilities, some decent practice in handling two- and three-factor multiplications, and plenty of decisions to make.

So, after several of these, and with twenty minutes to go, we simply had to take things a bit further forward. It was pretty obvious that we had to tackle 3×3 grids, using cards from 1 to 9. They were very happy to kick me out of the classroom while they composed puzzles for me to solve. Well, I’m glad I tried it this way round instead of me setting them problems, because the 3×3 grid turned out to be a different beast entirely.

Chaz’ problem made me think for a while, but would have been too much of a challenge for my Y6s, though I’m pretty sure some of their more confident classmates could have wrestled it out.

But I found Jo’s to be a complete brute. I gave it 15 minutes of effort, but there were too many Sudoku-type multiplicities and I had no clue which one to follow. Later I did have another look and found a more successful line of attack, but it was definitely more difficult than Chaz’ puzzle.

I was really disappointed at the way this had developed. It had begun with a nice gentle starting point accessible to all, and when I’d thought I’d turn up the challenge dial one just more notch I found it had jumped from strength 3 to about strength 10.

Afterwards I spent some time trying to figure out how I could move from the rather successful 2×3 grid and introduce a new challenge without making it so intimidating. In fact I spent most of the weekend trying, and rejecting, ideas. I’ll offer a couple of the better ones next time, but in the meantime I’d welcome any suggestions you might have.

And who’s Gabriel?   This was my attempt at using a simpler version of the very similar NRICH puzzle called Gabriel’s Problem. The NRICH version is targeted at secondary pupils, and is located at   https://nrich.maths.org/11750   The challenges set by Jo and Chaz fit very well with the NRICH puzzle.

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# One Week’s Photographs

When I was about ten I’d be given a roll of film to take on holiday.  Deciding which scenes to use my exposures on was a tricky business.  Using them too early meant I lost any opportunity to take a special souvenir late in the week, but what if it rained solidly in the last few days?  In practice it didn’t matter much, because the resulting pictures were always pathetic – blurred, dull grey, with the carefully composed subject barely visible in the middle of the landscape.

Now of course, we carry with us the chance to capture almost anything, instantly, and of good quality.  Here are some pictures from the last seven days.

I went to a meeting at the Royal Society on Friday, and have the badge to prove it:

Now don’t imagine the Royal Society and I are closely connected, but once a year they kindly host a meeting of the Joint Primary Group of the Mathematical Association and the Association of Teachers of Mathematics, and give us a splendid buffet lunch.

Considerably more attractive to my mind than that offered to the Fellows themselves – if you’re an FRS you clearly take very seriously indeed the idea of fish as food for the brain:

A couple of days earlier I was in Wetherspoon’s.  Can anyone possibly explain why an Americano – which Wetherspoon’s defines as black without milk or cream – contains six times as many calories as black coffee, and more than filter coffee with milk?  (The staff certainly couldn’t, and I don’t blame them):

Here’s another I don’t understand (particularly as by my count, the bus only has 28 seats):

On one of the cooler, damper days last week, this one made me smile:

And on one of the sunnier, warmer days here’s the canal.  One of the delightful things about canals is that you get these lovely tranquil scenes when you’re in a shopping street with dozens of cars trundling past, and with the supermarket 100m away to your left, and the railway station 100m away on your right.

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# Valid and Reliable?

When it comes to assessment I guess there’s no name bigger than Dylan Wiliam. Dylan’s advice has been sought, and subsequently ignored, by governments for at least thirty years. His website says “Dylan Wiliam is Emeritus Professor of Educational Assessment at University College London …. he has taught in inner-city schools, directed a large-scale testing programme, … and pursued a research programme focused on supporting teachers to develop their use of assessment in support of learning …. Over the last 15 years, his academic work has focused on the use of assessment to support learning (sometimes called formative assessment). He now works with groups of teachers all over the world on developing formative assessment practices.”

I can’t recall how we met, but I do know that when in 1980 I launched a postal games magazine built around games of interest to mathematics teachers Dylan was an early subscriber. No doubt I used this connection when we invited him to speak to an audience of a couple of hundred 15-yearolds in Hertfordshire. Hatfield is about 25 miles up the A1, and Dylan arrived on an impressive-looking motor-cycle. Even more impressive looking was Dylan himself.

It’s not quite so obvious in today’s photographs, but the Dylan Wiliam of thirty years ago in full motorbike gear with rich black hair and beard was a hugely glamorous figure. Few of us could even dream of making so dramatic an entrance.   Without a moment’s thought, and quite unconsciously, the female half of the audience reacted as one. The hall filled with pheromones; every bosom swelled, and every skirt was hitched a couple of inches. Simultaneously, every one of the boys shrank and instantly became one of a hundred little brothers. You can guess which half of the audience mobbed Dylan at the end of the afternoon.

I was reminded of this by receiving an invitation to a webinar by Dylan entitled “There is no such thing as a valid test …. All those who produce tests claim that their tests are valid and reliable. Unfortunately, such claims are meaningless, because tests cannot be valid, and they cannot be reliable. A failure to understand why such claims are incorrect means that the substantial power of assessment to support learning is often lost. ….

Now clearly I don’t know what Dylan will be saying, but it does give me a nudge to write once again about Edmond Holmes, the remarkable Chief Inspector of Elementary Schools over a hundred years ago. In a blistering denunciation of how the authorities rely upon tests – and this was before the First World War – Holmes pointed out that tests:

• Assess the recall of information and rote methods rather than understanding,
• Label children with a number,
• Dominate the curriculum and force schools to concentrate on pupils attaining their target grades,
• Demotivate those given lower scores,
• Fail to identify other abilities,
• Force all schools to accept a standard approach,
• By requiring standards to be attainable by all schools consequently lower aspiration of all those which are above average,
• And, in his own words, “value the results of education for their measureableness, … ignore those results which are too intrinsically valuable to be measured”.

Over the last twenty years or so we’ve had to make these same points again and again, but it’s rather remarkable to think that they were being made by the Chief Inspector at the very beginning of the twentieth century.

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# The Car Park

I never thought for one moment just how much geometry is involved when marking out the car park at the supermarket.

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# Two Chief Inspectors – Edmond Holmes and Chris Woodhead

It’s approaching a quarter of a century since Chris Woodhead became Her Majesty’s Chief Inspector of Schools. I couldn’t tell you a thing about any of his immediate predecessors; I don’t even know their names and probably never did. Woodhead, however, happily raised the profile of the position and used it to push for a back to basics programme based on traditional teaching methods (even though in his own career as a teacher and lecturer he’d advocated a much more liberal approach). As HMCI he saw little need to be seen as a friend of teachers, and asserted thousands of them were incompetent. I’m rather proud of the fact that during his time it was made clear to me I’d not be welcome as an HMI; my own approach to inspecting was perceived as being far too soft on teachers to fit the Woodhead ethos.

It was Woodhead’s period which saw the introduction of SATs throughout primary and secondary schools.   There were plenty of objections from the profession, both to the concept and the implementation. Nevertheless governments of different hues followed a tradition dating back to Palmerston in 1860. They consulted teachers, thanked them politely – and went ahead anyway, and the basic system has now been in place for more than twenty years.

Many of those objections had been foreseen years earlier, by one of Woodhead’s predecessors, who would have been appalled.

He railed against the assumption that the ability to recall factual information could be used as a simple yardstick which measures knowledge and understanding:

“…. it is quite easy to frame an examination which will ascertain, with some approach to accuracy, the amount of information that is floating on the surface of the child’s mind; and it is also easy to tabulate the results of such an examination, — to find a numerical equivalent ….”

He recognised that a school’s need to demonstrate good results in tests and examinations tends to dominate its curriculum and squeeze out everything else, particularly anything relating to independent thinking:

“In a school which is ridden by the examination incubus …. suppression of the child’s natural activities becomes the central feature of the teacher’s programme. In such a school the child is not allowed to do anything which the teacher can possibly do for him. He has to think what his teacher tells him to think, to feel what his teacher tells him to feel …. As an educator, the teacher must do his best to reduce the child to the level of a wire-pulled puppet.”

He knew that tests labelled children and whether accurate or not, that everyone, teacher and child alike, came to believe the label:

“…. the child who is low in his class is apt to accept the verdict of the class-list as final, and to regard himself as a failure …. there are many kinds of capacity which a formal examination fails to discover …. he not unnaturally acquiesces …. ends by becoming the failure which he has been taught to believe himself to be.”

He pointed out that focussing upon rules and algorithms in a mechanical manner meant that, because they were instructed that the first number in a subtraction question should be placed on the top line, children might perform a subtraction question presented as “From 95 take 57” but be unable to handle one presented as “take 57 from 95”. Likewise, using learned rules didn’t stop children giving:

“an entirely nonsensical answer to a simple arithmetical problem, – to say, for example, as I have known half a class of boys say, that a room is five shillings and sixpence wide.”

All these points seem at least as true today, not least that we value what we can measure, rather than measuring what is valuable:

“And in proportion as we tend to value the results of education for their measureableness, so we tend to undervalue and at last to ignore those results which are too intrinsically valuable to be measured.”

………

It’s impossible to imagine Woodhead or any of his successors making these statements, so just when were they made? All the “he’s” and “his’s” gave you a clue it was a good time ago. I’ve written before how lazy and uninformed it is to imagine that anything remotely liberal or child-centred began in trendy college departments in the 1960s. The quotes were all made long before 1960; they all come from “What Is and What Might Be”, by Edmond Holmes, HMCI from 1905 till 1911, which makes them more than a century old.

Holmes had been an inspector since 1875, enforcing, with much reservation, the Payment By Results system which concentrated upon using mechanical methods to teach a “3 Rs” curriculum. He soon recognised its inadequacies, and in his Report for 1878-79 he was bold enough to comment – even though he was still sufficiently junior that he was still in his twenties –

I do not reproach them [the teachers] …. I only wonder that it is not more striking and more disastrous than it is. Circumstances are against the teacher from first to last.

In “What Is and What Might Be”, written after his retirement, Holmes returned to this theme with a vengeance:

“No one knows better than I do that the elementary teachers of this country are the victims of a vicious conception of education which has behind it twenty centuries of tradition and prescription, and the malign influence of which was intensified in their case by thirty years or more of Code despotism and ‘payment by results’ …. which seems to have been devised for the express purpose of arresting growth and strangling life, which bound us all, myself included, with links of iron, and which had many zealous agents, of whom I, alas! was one.”

You can read “What Is and What Might Be” for yourself via the The Project Gutenberg ebook version at     http://www.gutenberg.org/ebooks/20555

………

Three months back I wrote about the remarkable Harriet Finlay Johnson and her school at Sompting.   When Holmes learned of her work he became her biggest fan. At Sompting he found everything he’d been looking for in a school for thirty years; he visited the school time and time again and featured it extensively in “What Is and What Might Be”, using the codename Utopia.

Harriet had a considerable flair for publicising what she was doing at Sompting, and with Holmes’ encouragement approving stories regularly appeared in local and national papers. She went further, and at Holmes’ suggestion wrote a book called “The Dramatic Method of Teaching”. This received wide publicity, even as far as Japan; it also went to an edition in the United States. It too is easily to find online:

https://archive.org/details/dramaticmethodof00finlrich

………

Unfortunately, totally unconnected events meant they both entered retirement in 1910, and I plan to cover these in a later post.

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# A Place In The Sun

Here’s something I’ve never noticed before.  We live at number 6, and our kitchen caddy was emptied this morning.  It was a cool night, but by the time I collected the caddy the sun had warmed things up a bit.  Not quite enough, however, to complete the job – the light-coloured 6 had reflected the warmth back again rather than absorbing it, and the underside of the lid still had the condensation visible.

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# Tilly Roth

I’ve no idea what made me think of her after so long, but I saw yesterday was the funeral of my friend Tilly who has died, at the age of 94. I think she was my friend, though for much of the time she would probably have admitted it only through gritted teeth. Our relationship was like no other in my life; several of us joined the newly-expanded Hertfordshire maths advisory team at the same time, and there was an interesting couple of weeks while we all sized each other up and began establishing our working and personal relationships. There were six of us, and we both stood out. I did because I was the only male, and Tilly because no-one had ever met anyone like her before.

Apart from her, we’d all come from within the county, so most of us knew each other vaguely and we all shared common experiences and practices.   Tilly, however, was a fully paid-up veteran of the famously progressive and leftish Inner London Education Authority, so although all the others in our team were comfortably mainstream feminists Tilly was far more committed and often aggressive with it. Perhaps this was aggravated by a bitter divorce, but we didn’t have many meetings without her complaining about male-dominated curricula, male role-models in textbooks, males in senior positions. On one occasion we were experimenting with a new robot toy controlled by the computer. I’d learned long before then to take a back seat in such situations; the device was called something like a pupil-instructed-performer, abbreviated to PIP. This was meat and drink to Tilly. “Why do robots always have to be given boys’ names?”, she grumbled. For perhaps the only time in my life I came up with a response I couldn’t have bettered given a week’s preparation. “Because women give them orders and they do as they’re told, Tilly” I snapped. Even Tilly had to give a faint smile at that.

This was after a couple of years when we all knew each other pretty well.   By then the team had grown and sucked in another couple of women; we were all on the best of terms, and they’d appointed themselves – Tilly included (she’s second from the left) of a fan club of my wife in sympathy for her having to put up with me on a full-time basis.

She was an inspired appointment. She challenged and refreshed us on a daily basis and gave us a perspective no-one else could. She pointed out that even the most deprived schools in Hertfordshire – perhaps the richest county in England – tended to have ample playgrounds and playing fields and easy access to countryside. She was a resource for us all whose commitment to equality moved on our thinking faster and better than all other bodies put together.

Because we all liked and respected each other we could argue furiously without giving or taking grudges. Once Tilly accused me of knowing nothing about board games. I couldn’t have been on much safer ground here, since board games were my hobby as well as a professional interest. I didn’t see any reason to make a big thing of it to the whole team, so I waited till we had a moment together at the coffee machine and told her I’d been invited to go to Rotterdam as the Guest of Honour at a European games convention in a few weeks, and would she like me to bring a bottle of genever back for her?

Every now and then life throws a coincidence at you no author would dare to suggest. Tilly was Dutch, and older than the rest of us, and it was Rotterdam where she’d been for much of the Second World War. Her obituary   https://www.familynotices24.co.uk/com/view/4543029/tilly-friederichroth   mentions she was in the Dutch Resistance.  Once when she came for a meal she mentioned to our son how she’d travel round Rotterdam with Resistance messages rolled up inside the handlebars of her bike.  I’d spent the war as a toddler; Tilly had spent it in near-starvation under a vicious occupying-power. I’ve written about a number of people who’ve been heroes of mine in a professional sense, but Tilly is the only one who needed to call upon heroism on a daily and continual basis. She made our team stronger and better than it would otherwise have been, and I’m proud to have known her and worked with her.

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