Well behaved …

Mea culpa.

In my time, I may have made a few things up. I’ve “created” a new meaning for the verb: “to pythagorize”, and I regularly tell my students of the “Forgotten Formula”

(The first: to pythagorize means to “to philosophize in the manner of the Pythagoreans”. In my classroom it means to use or apply Pythagoras’ theorem. For example, to find the magnitude of a vector I may tell my students to “pythagorize the i and j (and k if working in three dimensions) components.” I think it may be stretch to suggest we are philosophizing in the manner of the Pythagoreans when doing this.

The second is a bit of cod psychology. The most useful item not included on the A level formula sheet is, imho, the identity:

sec2x = tan2x + 1

yet my experience has taught me that students often grind to a halt in a question when remembering this identity would unlock the problem for them. Hence I have christened it “The Forgotten Formula”, have it labelled as such on my wall, and refer to it by this name. On more than one occasion, I have had to explain to my students that this is one of my little (many?) quirks and if they start talking of “the forgotten formula” with other teachers and mathematicians they will, at best, look back at them blankly. But if it helps my students remember, than I’m happy)

So it should have come as no surprise to me when introducing my students to “well behaved functions” they thought it was more of my make-believe (although they loved the concept, and I may have embellished my teacher talk with tales of mis-behaving functions having to spend Thursday lunchtime in detention.) So I had to convince them it was “a thing” and that we describe a function as well behaved if it is continuous and all of its derivatives are defined and continuous. For our purposes, in A level maths, we were exploring the sign change method to find a root between a pair of values, and discussing when this method may not work (if the function is not well-behaved!)

I’ve taught this class for two years, I hope I’ve managed to teach them something. I suspect that, in ten years time, they may not remember much maths, but probably will remember when we talked of well behaved functions!

Posted in Uncategorized | Leave a comment

Outstanding coursework!

Outstanding coursework

Weaning myself off school and education at the end of a busy half-term, I found myself scrolling through edu-twitter and reading of a suggestion that coursework may find its way back into GCSE maths. Being somewhat long in the tooth and grey in the hair, I remember “teaching” maths coursework and I don’t think I am unique amongst those of us that taught it in having very mixed feelings about it.

I absolutely loved some of the tasks and the way they encouraged mathematical thinking, discussion and communication were great. But there were a few big, BIG, negatives to GCSE maths coursework.

The first was the cheating. Lets not beat about the bush and talk of collaboration or discovery, there was a whole lot of cheating going on. Someone in the class would make a breakthrough in the task, or find a neat (aka algebraic – you needed algebra to score the big marks) way to solve the problem and within 5 minutes it had circulated the classroom twice and everyone had written down someone else’s work as their own solution.

Then there was the problem of getting students to hand in their work so it could be assessed, as my tale below demonstrates.

And lets not forget the additional workload on teachers who then had to mark and moderate (all unpaid and in their own time) piles of student coursework. Anecdotally, I know of more than one school who made the switch from GCSE to iGCSE maths simply so they wouldn’t have to mark any more coursework.

But on with today’s tale.

My memory dims, but I reckon that this must have been in the early 2000’s, a department meeting to check how coursework was progressing, who had handed it in (the school’s deadline had passed) who still had outstanding pieces of coursework, and what we were going to do to extract the work from those students. Our Head of Department directed us to phone the parents of everyone of our students who had not yet submitted their coursework.

So later that afternoon, I picked up the phone to Mrs Smith …

“Good afternoon, Mrs Smith, its Mr Mathsteacher from school. How are you, hope you are well ….

“Yes, I’m very well, thank you. Now I’m calling because Johnny has an outstanding piece of coursework”

There was a short pause, before Mrs Smith replied:

“Oh thank you so much for calling. That’s wonderful to hear …”

My heart sank. Johnny was what you might call a loveable rogue (well, his mum loved him) and she wasn’t used to getting positive phone calls from the school. But she did get a number of phone calls from the school. My ambiguous use of the word outstanding was (understandably)interpreted very differently by Mrs Smith from how we had been using it within the department.

“I’m so sorry Mrs Smith. When I said Johnny’s coursework was outstanding, I didn’t mean it was brilliant, I meant that he has not yet handed it in. If he doesn’t hand by a week on Thursday he will miss the deadline and score zero marks for this part of his GCSE”

Across the silent ether of the phoneline, I’m sure I could make out the sound of Mrs Smith’s crest falling. Another wound to bear from her disappointing son.

I explained what Johnny needed to do, and how it would help him secure his GCSE, we exchanged pleasantries and I hung up, feeling bad for my choice of words. I still feel bad over twenty years later and now I always, always, think of Johnny and Mrs Smith before I use the word “outstanding” in any context.

I can’t remember if he got a C or a B, but I do recall that Johnny did pass his GCSE maths – probably due to his mother ensuring he got his coursework done and handed in. There will be thousands of Mrs Smiths up and down the country ensuring that their Johnny hands in his coursework; there will be a significant number of Mrs Smith’s who will happily pay a “tutor” to do and submit Johnny’s coursework for him; and, sadly, there will be thousands of Johnny’s up and down the country who have no support in their coursework, other than from their teachers. And that’s not fair, and that’s (one reason) why maths coursework shouldn’t be re-introduced as an examined component of GCSE maths. (But there’s no reason why we shouldn’t expect students to tackle these rich investigative tasks as part of their mathematical journey.)

Posted in Stories, Teaching Tips | Tagged , , , | Leave a comment

The Professor and his protégé

Herr Schmidt was tired. Tired of the cold, tired of the class of forty-seven pre-teen children that sat before him and, he surprised himself to think, tired of his protégé. In fact, the more he pondered, the more he realised it was his protégé that was the root of all his problems. Life had been simple before young Carl was delivered to his door to be a deserving recipient of Herr Schmidt’s careful tutelage.

“Ah, the days before Carl” mused the moustachioed Schmidt to himself. Life was simple then. His forty-six eager students would listen attentively, with awe even, to his every word, and then he would pick up his chalk, write a few tricky sums on the blackboard, sit back, open his newspaper and enjoy a peaceful twenty minutes as the nearly noble young of the Duchy of Brunswick set about their arithmetic, struggling in silent, scholarly endeavour.

“Herr Schmidt, Herr Schmidt. I am finished.”  The professor’s reverie was once again broken, mere moments after it had begun. However, his immediate irritability gave way to a small, barely discernible, smile. An idea had entered the maestro’s mind, and not just any idea, a brilliant idea. “This will keep him busy for a fair while” he thought to himself. “Perhaps until luncheon – possibly beyond” and his mind turned to thoughts of wurst and kraut.

“Gauss,” he boomed a little too loudly, “Gauss, you are a talented scholar, of that there is no doubt. I have a task for you that is worthy of your consideration, but be warned, it may be a little too much for you just yet. You will need to focus and concentrate; it is not a simple problem and will take you some time. Do you think you can accept such a challenge?”

Young Carl Friedrich trembled with excitement. He knew Herr Schmidt was a good man, the best, most dedicated teacher in the Duchy, and he had a special problem for him. “Of course, sir. If you think I am worthy of such a challenge, I will dedicate my mind to it.”

“Very well, my boy” responded Herr Schmidt. “You are to sum all the counting numbers between one and one hundred. As I say, not a simple or swift task. Now go, sit, sit, sit.”

As the young protégé scuttled back to his desk, the professor picked up his newspaper, emitted a soft sigh of relief and began to think again of his luncheon to come, his eyes gently closing …

He was soon rudely shaken from his peace. “Five thousand and fifty, five thousand and fifty” came a tremulous cry from Carl Friedrich Gauss. “What do you mean boy” grumbled the professor, “what do you mean. You have merely made up an answer. Guesswork is not the hallmark of a scholar, young man.  You must re-evaluate your solution, and I must re-evaluate your suitability for this class. Five thousand and fifty indeed. There is no way you can have completed such a sum so swiftly.”

Gauss was confused. He had been set a task, a special task by the magnificent Herr Schmidt and he had solved that task, quickly and eloquently, and yet Herr Schmidt seemed displeased? And then in it dawned on him: it was a test. To gain the professor’s true esteem he must now explain his solution.

“Sir, you are a sage and wise teacher, and I am honoured that you set me such a task, but even more so that you demand I explain my solution.  As you know, sir, the trick is not just keep adding the numbers, one plus two is three, three plus three is six, plus four makes ten.  That’s what a simple Hanoverian would do” he whispered, with a shy smile.

 “No sir, I considered the counting numbers from one to one hundred, and then considered the same numbers but in reverse order, from one hundred descending to one.” The professor wasn’t sure where this was going, but his interest was piqued. “Continue boy, continue” he demanded.

“Well sir, then I paired up the numbers from the two lists: one with one hundred, two with ninety-nine, three with ninety-eight etc.  Each pair will sum to one hundred and one, and there will be one hundred pairs, so I multiplied one hundred and one by one hundred …”

“… giving ten thousand, one hundred” interrupted the professor “which is wrong.”

“But sir, we have two lists of the numbers from one to one hundred, so now we must halve your ten thousand, one hundred to give the answer of five thousand and fifty” announced a triumphant Gauss.

There was a pause. A pause that stretched into a silence. The other boys, sensing something afoot, looked up from their slates. A tension filled the cold, damp classroom. Breathes were held. Jonas Fischer, a particularly nervous boy, slid under his desk. And then the silence shattered.

“Well, what are you waiting for boy? Sit, sit” bellowed Herr Schmidt, pulling up a chair next to his desk, a smile breaking across the corners of his mouth. And together they sat, the professor and his protégé, checking his working, trying other examples and codifying a formula for the general case. For the first time in years, Herr Schmidt was enjoying himself, enjoying mathematics again, so much so, and so engrossed in the task, that he almost forgot to send the boys to lunch. Almost.

He particularly enjoyed his wurst for lunch that day.

Author’s note: this story is fiction. There is an unreliable anecdote that he discovered the formula for an arithmetic progression whilst at primary school, but it makes a great story, one that I often tell my students and, as it is National Story Telling week I thought I would put pen to paper, embellish my tale a little, and write it down.  I hope you enjoyed it, if not the tale, then the simplistic brilliance (not mine) of how to add up the integers from one to one hundred.

Posted in Stories | Tagged , | Leave a comment

Saving the best ’til last


As the year draws to a close, it throws up perhaps the best date of the year – if you write it in the American (month/day/year) format.

Today is the 31st December 2023, 31/12/23 to those of us in the UK, but across the pond today’s date is written as 12/31/23. Dispense with the slashes separating month/day/year and you are left with 123123 – a date which will not come around for another hundred years.

Whatever your resolutions and plans for the new year, I am sure you are already looking forward to the 4th February – 4/2/24 or 4224 – our first palindromic date of the new year (of course, our American cousins will need to wait until the 2nd of April …)

I hope that you had a great 2023, and wishing you an even better 2024.

Happy New Year!

Posted in dates | Tagged | Leave a comment

Heroes and Gifts

Presents for teachers

I don’t really “do” heroes, if I were to compile a list of my heroes it wouldn’t be long, but it would include Martin Lewis, of Money Saving Expert fame – he talks sense, knows his numbers and genuinely tries to make the world better place for everyone. He’s been in the news and across social media of late sharing his thoughts on gifts for teachers. (In essence, don’t feel obliged to buy them. If you really do want to give a gift to a teacher club together to give a class gift) As a teacher, I’m always grateful for any gifts I recieve, but certainly don’t expect them, or prioritise those that do give me gifts.

But this post is not about whether or not gifts should be given to teachers, but a true tale from my past when I was young teacher, a couple of years into my career.

Like pretty much all teachers, in addition to teaching my subject (maths) I was also the tutor to a number of students – they registered with me first thing each morning, I was a point of contact between parent and school and I did my best to look after the overall academic progress and pastoral needs of those in my tutor group. I had taken them on the year before, as fresh faced Year 7s, new into secondary education, and, for continuity, I had hung on to them for another year. So, by the end of the Autumn term of Year 8 I knew them pretty well, the (loveable) rogues from the well-behaved, the academic high fliers and those that found their studies more of challenge. I knew them, they knew me, and, on the whole we all got along well, most of the time.

As November turned to December, the days got shorter and the weather got colder, one of my “loveable rogues” approached me with his customary big beaming smile on his face.

“Sir” he said (things were still rather formal back then) “sir, I’ve asked my mum for five bottles of wine to bring in for Christmas gifts for teachers. I’m going to give one to you, and one to Miss Jones [his English teacher] ‘cos I like her. And I’m going to keep the other three for me.”

I was touched that Tom* felt I was worthy of a gift, and also a little touched but perhaps a bit more concerned that he was happy to share his plan with me. Anyway, later that day I picked up the phone:

“Good afternoon, Mrs Smith* I hope you are well. I’m Tom’s tutor at school. … Yes, I’m well, thank you, no nothing to worry about, no he’s not in trouble. I just wanted to give you a quick call to suggest that if you are sending Tom into school with any gifts for his teachers, chocolates may be a better choice than wine. … No, no problem at all. Merry Christmas.”

Later that week, I raised a wry smile as Tom – ever so slightly grudgingly – handed me a beautifully wrapped box of chocolates as my Christmas gift.

*not Tom’s real name. Although more than 25 years have passed since this story, and countless students have filed through my classroom, I still remember Tom – I’m not sure I ever taught him anything, but as a beginner teacher, Tom and his antics taught me a whole lot about my chosen profession.

Posted in Teaching Tips, Uncategorized | Leave a comment