# Playing by the Rules

This post has been motivated by a discussion that’s surfaced on Twitter in recent days of which this is a taste:

Is it wrong to have a game that doesn’t abide by the standard order of operations? Is it likely to teach children misconceptions that will need to be unpicked in maths class?

## Why we have rules and conventions…

Consider the game of chess. If two people decide to play a game of chess, they both know exactly where they stand: white begins, the possible moves are clearly defined, turns alternate, the objective is clear – there really is no room for disagreement. But there are alternatives that can be played: suicide chess where you try to lose your pieces as quickly as possible (I prefer that to chess as I have a short attention span); or any of the many variants such as those listed on Wikipedia.

To me mathematics is in many ways like a board game. We have the pieces (usually set up in terms of definitions) and we have moves (the logic of deduction, theorems and the use of operators and functions etc). We need to agree on a system that is consistent within itself (as far as possible – I don’t want to get into set-based paradoxes or Gödel’s theorems).

In A level Maths and Further Maths, we teach the dot and cross product for vectors. Why do we have two different ways to multiply vectors? Why are they calculated in the way they do? Why don’t we just multiply component-wise? My usual explanation is that it is down to mathematicians to define multiplication and to do so in a way that has longevity/purpose. There is an essence of natural selection in the history of maths: some ideas just don’t have enough in them to last more than a cursory exploration. I’m curious to look back into the history of matrices sometime and see how that multiplication process developed.

## Order of Operations

You might disagree, but I believe the order of operations (BIDMAS, PEMDAS, PIMDAS, and many other variations) exists so that around the globe we all interpret expressions in a consistent manner. Parentheses (and brackets and braces) are a useful marker for prioritising the evaluation of different parts of an expression. But why didn’t we all agree to “left-to-right” evaluation instead? That could still be consistent around the world if we wanted it to be. Simple four-function calculators still work that way (and, in fact, we do use that when an expression is simply say a mix of addition and subtraction). Or why not reverse Polish notation which handles operations in a manner that programmers would call a stack?

Let’s compare them by finding the perimeter of this rectangle:

• Standard Order of Operations: $2\times(3+7)$
• Left-to-right evaluation: $3+7\times 2$
• Reverse Polish: $3\,\,7\,+\,2\,\times$

## Survival of the Fittest

I think it’s clear why RPN didn’t survive in the mainstream: although it’s easiest for a computer program to evaluate, it places a huge demand on the user to enter the input correctly. (As a distraction, there’s a cute discussion on Stackexchange about how to handle the quadratic formula in RPN!)

I think the reason the ‘standard order’ generally wins out over left-to-right evaluation is the flexibility it allows, without introducing ambiguity. Following the standard order, commutativity of operations can be used fully:

$2\times(3+7) = 2\times(7+3) = (7+3)\times 2 = (3+7)\times 2$

whereas in left-to-right evaluation, only these two would evaluate the perimeter correctly:

$3+7\times 2 = 7+3\times 2$

and once parentheses are introduced to LTR evaluation, it just feels like a restricted version of the standard order.

## Right or Wrong?

No, just different. Of course, however, young children could receive mixed messages. Are these best handled by only exposing them to the standard order of operations, or  is it better to open up the discussion a little and see why we have the standard order?

I’ll leave all the die-hard BIDMAS fans with this ‘brain training’ from the Daily Mail:

“Multiply by itself twice”… is that $x\times x \times x$ or $(x^2)^2$? 🙂

# A2 Trigonometry (Parts 2 & 3)

Yes, I’m beginning with Part 2. At some point I will try and write about my approach to sec, cosec, cot and the inverse trig functions but following on from some recent Twitter comments, it seems many of us are around the sin(A+B) and Rsin(x+a) mark right now.

My approach this year has made a much bigger deal about the motivation for these formulae and techniques. Here’s a brief run-down of the stages we went through.

# Part 2: sin(A+B)

## Stage 1 – Motivation

A number of statements written on the board. Students to discuss in pairs if each is true or false. Typically something like this:

 1.) $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$ 2.) $(a+b)^2=a^2+b^2$ 3.) $\ln(a+b)=\ln(a)+\ln(b)$ 4.) $e^{a+b}=e^a + e^b$ 5.) $\frac{5}{a+b}=\frac5a+\frac5b$ 6.) $5(a+b)=5a+5b$ 7.) $\sin(a+b)=\sin(a) + \sin(b)$ 8.) $(a+b)^{-1}=a^{-1}+b^{-1}$

Purpose: this gives the opportunity to air a few of the typical misconceptions that always arise again near the beginning of C3 (but hopefully not so much later in the course…) But it also emphasises that most functions cannot simply be ‘expanded’ (mathematically speaking, they are not linear).

Discussion: hopefully students will correctly identify true/false. It’s nice to make improvements to several of them, such as:

• 2.) $(a+b)^2=a^2+2ab+b^2$
• 3.) $\ln(a)+\ln(b)=\ln(ab)$
• 4.) $e^{a+b}=e^a e^b$

Most students are wise enough to not trust number 7, and thus we have some initial motivation for wondering how to ‘expand’ $sin(a+b)$.

## Stage 2 – A Proof

I’ve seen various diagrams and constructions (Cut the Knot has several pages dedicated to them) but my current favourite is this one:

The students are to label all the remaining sides in the diagram (I usually model how to proceed by using say $\cos\beta$ as an example. The worksheet I use has all the necessary labels underneath the diagram to guide them, too. Finally, the sheet poses:

Note, unless I have particularly high-flying students, I don’t make much of a fuss about the construction only really working if $\alpha+\beta<90^\circ$.

## Stage 3 – Examples and Non-Example

Of course, it’s tradition to find an exact value for $\sin 75^\circ$ at this stage, or perhaps $\latex \cos 15^\circ$, but some calculators now provide these values directly which is a shame.

I would typically demonstrate some equation that requires one or both of the new identities, but then I think a non-example is also crucial here. Something like:

Solve $\sin(x+\frac{\pi}{3}) = \frac12$.

Just because we can expand, doesn’t mean we always should!

That’s more than enough for one lesson. The next would begin with this slide

and then move on to the double-angle formulae.

# Part 3: Harmonic Form

For this lesson, I place perhaps an even greater emphasis on motivation. There are two initial activities that I use.

## Stage 0 – The Settler

I make quite a big deal about some things being “just a C2 question” – trying to make the point that C3 isn’t such a big leap as they might think, and that they really should be able to solve C2 questions by now…

Clearly the middle equation causes the problem and they’ll be clamouring to know how to solve it!

## Stage 1 – The Motivation

I then give the students a sheet that begins like this:

There are a couple more parts further down, but you get the idea. Given time to work in pairs and discuss what they are able to do, the students realise that filling in the column for $f(x)$ is relatively straightforward (and essentially C2 work, but it gives the opportunity to get a bit of help too) whereas the column for $g(x)$ is nigh on impossible.

It’s great to see what they then try: some will plug a few values in their calculator, using a bit of trial and error to find the domain of g. Others will try and guess what the sum (or rather, difference) of a sine and cosine graph might look like. Basically though, they’re stumped.

We use Desmos to do the big reveal. Type in f(x) to check all their answers to the left column. Then type in g(x) and. Wait. What? It’s the same function? *general surprise*

## Stage 2 – Backwards and Forwards

Some then realise that they could ‘expand’ f(x) using the cos(A+B) identity and we do this on the board. The stage is then set for, effectively, the reverse of this process.

## Stage 3 – So, it’s not a big deal

The whole point of harmonic form is to make very difficult questions (the right column) very easy (the left column)! They love it.

## Stage 4 – Practice

All of the above, perhaps with the inclusion of just another example or question is usually plenty for the first (hour-long) lesson.

I don’t have much more to add, except this idea which occurred to me for no particular reason. I make use of the classic Solomon worksheets relatively often (maybe on per week, ish) but this is the first time I’ve annotated one before issuing it to students:

I’ll try and get a higher resolution pic at some point, but the idea was to highlight the purpose (or distinguishing feature) of each question [done in red ink] and then recommend what to complete, such as “just do one of these” [done in green ink]. I’m sure the big piece of paper and colour-copying played its part in engaging the students, but they responded to it very well: for the able ones, it gave them clear guidance about what was worthwhile to complete; for the weaker ones, they had the detail of the question emphasised and felt more secure in tackling them.

# The Organised(?) Teacher

For some reason, my colleagues are often surprised when I say that I’m disorganised. I think people expect mathematicians to be organised as a consequence of the logic of the subject. It doesn’t take a mathematician (or perhaps it does) to see the fault in this. I can think through a problem, come up with a strategy and follow it through sequentially to a conclusion. But if you tell me the problem when I’m on my way to a lesson and then ask me about it later in the day, I’ll more than likely say: “Ah yes, I forgot about that. What was it again?” I’m forgetful, I procrastinate, I’m easily-distracted and I’m disorganised when it comes to managing tasks. But memory, focus, organisation and record-keeping are crucial for teachers, so I thought I’d take some time to share my strategies (or trade secrets, perhaps).

## The Year Plan

I know we have a scheme of work. But it’s in a shared network folder which isn’t easy to access outside of school. If I made an electronic copy it wouldn’t get updated. If I printed it, it would use too much paper and I would leave it in the wrong place. Each year, I now produce a “student year plan” for the courses I teach, which briefly summarise the topics on a week-by-week basis. Here’s an example:

Every student has a copy to put in their folder so they never have to ask which topic is next, or which textbook to bring. I have a copy pinned above my desk so even if by Wednesday I have forgotten what we’re doing that week, it’s right in front of me. It’s also saved on Google Drive so I can access it from anywhere.

Do we stick to it? Well, no. Of course there is a natural ebb and flow to teaching. Right now, we’re ahead of the game with M2 so Statics will shift from November to October. That doesn’t mean I’ll re-print the whole thing – we’ll all just annotate our own copies. Come November, I might try and start teaching Statics again but I’ve told the students to remind me not to…

## The Week Plan

I’ve never been able to plan a whole week’s lessons in advance. Every day I like to see how the class have found that day’s material and what they would benefit from as a next step. However, we do have some weekly structure set by the school: students are tested every week and they are graded every week. Thus to maintain a routine, I allocate Thursday as test day for all my groups so they can have their tests returned and discussed on Fridays. Grades are then entered on our system by Monday morning.

A corner of one of my whiteboards is always set aside to record the topics that will be covered in that week’s test for each group. This reminds the students permanently that tests are on Thursdays and stops them giving me a headache asking “What’s this week’s test going to be on?” three times a day. But again, it’s there for me as much as for the students: sometimes I have to look there on a Wednesday afternoon as I ask myself “what do I need to put on this week’s test?”.

Our grading system is excellent: students are graded in every subject, every week, for both achievement and effort. This is all recorded online and students, tutors, parents and teachers can all see the grades each week. And I get a relatively friendly nudge at 10:30am if this hasn’t been done…

## Planner or Markbook?

And planners? They just don’t seem to work for me. I’ve tried the standard teacher’s planner product. I’ve tried creating my own custom version, printed at the beginning of term. I’ve tried an old-fashioned simple markbook. All the same problems come up: it’s never in the place I need it and I fail abysmally at keeping it up to date. It becomes another chore in the week to retrospectively fill in all that week’s gaps, by which point 90% of the information is redundant. I’ve even tried the app iDoceo: a “powerful teaching assistant for teachers”, which some people swear by. I’ve tried it twice or three times now. I get as far as adding in students and blocking out term dates/holidays and adding our timetable structure. Then I don’t have my iPad with me or I don’t have an internet connection and it all goes out the window before the first full week of teaching. I’ve tried Moodle too: you can record test scores, attendance and share documents with the class. This time, sadly, it was the clunkiness of the gradebook that let me down and, although I could access it anywhere through a browser, the sheer number of clicks needed to get through to any useful information also put me off.

## New Year, New Plan

This year, I’ve tried to rationalise everything: I need convenient access and I need to store some key student data. It turns out Google Sheets might be the solution that works for me: quick to access, available anywhere and I’ve learned about cell comments.

I spent quite a while tinkering and experimenting with the layout and merged cells, but I like the simplicity of what I’ve set up: attendance is taken care of with the 5 small boxes each week; and underneath these are the student’s test score and weekly grade. At the top I can also quickly see the main topic, and with conditional formatting I get the ‘at a glance’ view of strengths and weaknesses. I have one spreadsheet per class and one tab per half term to reduce scrolling. As for cell comments, this is what I always felt was lacking but I Googled for them out of desperation and they do exist! If you notice, for example, the cell saying ABSENT. It has an orange mark in the upper right corner. This indicates I’ve added a comment, and by hovering the mouse on that cell it will pop-up giving more details about the absence.

## Other Records

I should have a record of teaching and for me this is covered by 3 things: the year plan, the scheme of work and the Google Slides presentations that I build up over the course of a topic. See for example these ones on M2 Moments or C3 Trigonometry II – depending on when you view them there may be more slides. Of course, I can also easily share these with students via the same direct links. If I make any alterations, I don’t have to re-send the whole presentation or re-upload it to a VLE.

I should have a record of prep. I think I’m in a grey area right now, if I’m honest. As you may well have read from my posts here and here, I’m running prep in a different way from my colleagues. The students themselves have the full record of all the prep they’ve completed, along with my feedback. They’re all tackling different combinations of questions, so I could never hope to record that myself. If they lose their orange book, this record is gone. But they have still done the prep and received feedback, and I think that’s the most important part of the system.

## All The Other Jobs

As you might be able to imagine, I am also hopeless at keeping on top of any kind of “to do” list. The problem isn’t recording jobs to be done, it’s getting in the habit of checking that list and acting on it. Again, I’ve tried apps, post-its, and even the old-fashioned note on the back of my hand. (I currently have ‘Q6’ written there in red, but I haven’t the faintest clue why. I’m sure it’ll come back to me…)

The habit of checking is always the downfall to the systems I try. For my boarding duties it’s simple: I have the rota stuck on my fridge door. I only need to check it when I’m at home, and I always see it when I’m at home.

This term, I’m trying out a system called the Bullet Journal. I’ve bought a Leuchtturm notebook, so I’m literally invested in the system, and I took the time to mark out calendar pages and the beginnings of my to-do list. The essence is that tasks are listed, then either completed or rescheduled. Each day you add to and/or cross tasks out from the list. It turns out I’m not alone as @dazmck, @MathematicQuinn and @MrDraperMaths are also using this system. However, it does rely on my keeping this notebook at my side throughout the day. I chose an orange one – my favourite colour and hard to lose. Only time will tell if this helps!

## Basically, I Cheat

The secret to my organisation is really relying on other people. I give the students full information so they know what’s happening on a weekly basis. All teachers love the starter task of “Tell me what we learned last lesson”. It’s rare that I have to rely on that to know what I’m doing, but it has happened… I keep minimal records in an easy-to-access online place and I drop in to see my HoD when I have that feeling that I should be doing something, just to check if I really should be doing something.

What works for you?

# Purposeful Prep, Part II

This post follows on from Purposeful Prep and Manageable Marking that I wrote last week. In that post I explained briefly the idea that I’m trialling with all of my classes: maths homework every night, self-checked by the students, handed in for comments and feedback, perhaps leading to a one-to-one discussion as well.

## Aims – My point of view

My main aims were to ensure that students were working on mathematics much more frequently outside of lessons (i.e., every evening) and that they were ultimately completing a greater quantity of work that is directly relevant to them. Here is an enlarged image of the expectations sheet, much easier to read than last week’s (highlighting added by the student):

I’m also trying to keep my marking load well under control. By my nature, I’m relatively disorganised and hopeless at settling down to work on jobs. Face-to-face, I have all the time in the world for my students. Sat at my desk, I could win a gold medal for procrastination.

Ultimately, I decided that the best value I can add for my students is to provide specific feeback, advice and corrections on questions that they have identified as problematic. I could never keep on top of every class’s work, when every student is potentially tackling different questions and topics!

I frequently chat very openly with my students about how we organise our week, about the nature of class tasks and work, and, now, about how homework can be most effective. Some people see marking as a significant part of a teacher’s role, but I have to disagree. The marking isn’t anywhere near as important as expert feedback. (Expert in any sense: efficiency of method, clarity of presentation, accuracy of algebraic work, precision of written responses, providing a structured strategy when students are stuck etc.)

## The Routine

“Orange books, please” is now my opening sentence in every lesson. They’re all handed in – I count them explicitly as they reach me. If no homework was done, I note it in their book. If they failed to check their work, I comment on that – then return the book and expect the student to check the work during the lesson so that I can review it again.

At some stage in every lesson, the students are busy working on tasks and I have the time to look over each book, carefully taking in:

• the quantity of work they have done
• the source of the work and it’s approximate level of difficulty
• the topic of the work, and if it is similar to past work or something new/more advanced
• the presentation of the work: legibility, clarity of algebraic work etc
• the problems that the student has highlighted (and often they will have written in corrections in a different colour, which I can review for correctness)
• the wording of written responses, especially in statistics (many of my students are EAL and, moreover, I don’t expect students to be able to mark such questions simply from comments in a mark scheme for example)

This sounds like a lot but the whole process is incredibly efficient. I can add my annotations, closing comments, recommendations for possible future work etc within a few minutes. Where necessary, I can also chat one-to-one with the student to explain a point or, if necessary, fix a time for them to come back later in the afternoon.

Indeed, much of what I write is common across many books and so today I took some time preparing stickers that I can use (somewhat sparingly..  I don’t want to go sticker crazy):

These are just a first iteration and, as with everything I do, I will ask students for their honest thoughts on using them. It is likely that I will adapt some after another week of seeing what works. Of course, I will still be writing comments too!

## The Guilt

After mentioning this trial approach with another colleague in the department, she has also adopted it (taking care to use red books instead of orange, as we have a number of students in common!) We have both seen a wonderful shift in balance: the students are completing more independent work than ever before, and our ‘marking’ (for want of a better label) is taking less time than ever before. It all happens within lesson time.

It is a strange feeling to be free during a free lesson, let alone not having to take a pile of marking home for the weekend. We are almost creating odd jobs to do just so that we don’t appear idle. (Hence printing sheets of coloured sticks today, for example!) Of course, there are term reports, leavers’ references, forecast grades etc all to be done, so we’re not that idle really.

## Sustainability in the Future

This approach really does seem ideal for this summer term: there is no new subject knowledge to be taught this term as the students prepare for AS and A2 exams. As I mentioned in a tweet earlier this evening, it is possible that everyone of my students completes different classwork and different homework from all the others. But, equally, I can set a common homework task if I believe they would all benefit from it.

The big question on my mind now is to what extent I can continue this approach next year, during the Autumn and Spring terms when we are teaching the subject. I have plenty of time to mull that one over!

### A Note on Preparing the Stickers

The stickers I’m using come on A4 sheets with 65 per sheet. I ordered them online a long time ago, but have never made great use of them. Avery provide template documents for all manner of sticker sheets, so I downloaded one of those to help with the layout.

Within each sticker, I created a two-column table. The left column contains an emoji (which I copy and paste from iemoji.com) and the right column contains the text. Print them on a colour printer and voila.

I’ve shared my sheet on Dropbox – feel free to play with it.

# Purposeful Prep and Manageable Marking

This term, I’m trying a new approach to homework with my classes. I only teach AS/A level and so this term is fully focussed on preparing for the exams. I think it is nigh on impossible to prescribe a homework that fits all the students in any one class: in my A2 Maths class, many of the students also do Further Maths and they have little difficulty with Core 3 and 4, whereas others do struggle with those modules; in my AS class most are confident on the core but are still getting to grips with statistics but, equally, there are a couple of students in the opposite situation.

The other risk at this time of year is setting (and thus marking) a vast number of past papers which, for me and my disorganised ways, is generally unmanageable. Also, how long should I allow the students to work on a paper? Overnight? Over several days? Some will fly through a C1. Others will need 2 or 3 days to complete a paper. Others still will leave it til the last minute and then see if they have enough time or not.

So, the experiment is to go old school and use exercise books. I bought a pack of A4 squared books from Amazon (our college doesn’t usually use books) and, in true KS3 fashion, printed an A5 sheet of expectations for them each to glue in.

The most important aspects from my point of view are that:

• the students do some maths outside of lessons every day (we teach them every day, so can monitor this on a daily basis)
• the students choose the work they are completing (most are very good at this, and occasionally my feedback will include specific follow-up questions)
• the students check their work against numerical solutions or mark schemes etc, highlighting problems that they were not able to resolve themselves

During a typical hour-long lesson, I will collect all the books at the beginning then, when during time when they are all working in class, I will scan through each one adding in comments and chatting to the students on a one-to-one basis. The books are always returned to students by the end of the lesson so they can work in them that evening. So far, I’ve not had problems checking every book during class time. I have, however, had to arrange some follow-up one-to-one support to spend enough time picking up on some students’ difficulties.

I have to say, the students are responding very positively to the process. They know they are spending more time on maths than they perhaps otherwise would, but they see the value in choosing appropriate work and getting very specific feedback from me. We do have a grading system that can penalise students on a weekly basis if they are not meeting our expectations but, so far, I have not had to use this as a punitive measure.

And the best bit? All my ‘marking’ (which isn’t marking it all, but giving feedback and advice) is done during class time.

# A Case For Grammar

This blog post is a little off my usual path: I typically stick to morsels of maths and anecdotes from my classroom. However, a post on Facebook and Twitter caught my attention this morning because of the flurry of supportive comments that followed it. The post is this one from Michael Rosen, linking to his latest blog post about the requirements of teaching grammar:

To try and pre-emptively subdue some of the storms that can arise, let me be clear about the position I’m writing from: I’m a mathematician / maths teacher (another long-running discussion involved in adopting those labels for myself..). I’ve never taught 10 year olds. I know nothing about creative writing. However, I am fascinated by grammar and languages.

## The Argument

Having read further posts and this piece in the Guardian, also by Michael Rosen, I now better appreciate the point that he is making: schools are being given very prescriptive guidelines about grammatical points and, not only must pupils understand the terminology, but it appears that the quality of their writing will, in part, be judged on the variety of constructions that it contains. With the best intentions in the world, shoe-horning a fronted adverbial into writing (as I think I have succeeded in doing here) has nothing to do with creativity and the ability to express oneself. Even as one devoid of literary creativity, I can appreciate the point that just shoving in constructions to ‘show off’ can be counter-productive to producing interesting writing. It reminds me of my GCSE Spanish written paper: the aim of the game there was to demonstrate my ability to use as many grammatical structures as possible. What did I do? I memorised a sentence that included an imperfect subjunctive (and, weirdly, I still remember it to this day)…

Had I not eaten the fish, I wouldn’t have become ill. Whack that in to your writing and you’re well on track for an A*. I almost certainly couldn’t have adapted it to other situations, let alone use it in the speaking exam.

So far, I completely agree with Michael, as best I understand the situation: the curriculum-listed grammatical rules are overbearing (to the point of defying convention in some cases) and the insistence on their inclusion in written work is a poor measure of ‘good writing’.

## The Digressions

Continuing the discussion a little further in a brief exchange of Tweets, Michael and I raised a few views that I think are worth their own air time.

### Classification

As I stated from the outset, I have no experience of teaching 10 year olds. The phrases ‘fronted adverbial’ and ‘subordinate clause’ seem unnecessarily advanced for that age – I don’t think I met subordinate clauses until Year 9, and I had never heard of fronted adverbials prior to this debate! However, the idea of following a classification, based (hopefully) on precise definitions, is an important abstract skill. That classifications can overlap is also important to recognise. As maths teachers we are familiar with how students are uncomfortable with statements such as “a square is a rectangle”. If it meets the criteria, then it gains the label. Of course, all of these arguments boil down to the definition of the term involved, and I’m under no illusion that grammatical terms must be incredibly difficult to specify to the same precision as mathematical definitions.

• Is Guildford a city?
• Is a platypus a mammal?
• How can 141ft Mount Wycheproof be classed as a mountain?

I’ve never thought too deeply around this issue, but I’m intrigued to know how disciplines other than mathematics formalise their definitions.

### Purity

I likened my appreciation for grammar to my appreciation for pure mathematics. I can (and did for four years!) study mathematics from a wholly ‘pure’ point of view, without any care for whether or not it has found applications in ‘the real world’. Pure mathematics, to me, is all about formalism and abstraction. Group theory, for example, abstracts the essence of a huge variety of seemingly disparate mathematical operations. Whether group theory itself has practical applications (I believe it does in physics and chemistry) is of no concern to me. The introduction to Ian Stewart’s Concepts of Modern Mathematics cites a number of examples of applied maths catching up approximately one century after developments in pure mathematics.

Years ago, I read a neat little book called Logic, by Wilfrid Hodges. As someone who had grown up with little appreciation for literature, and even less grasp of how to produce a piece of creative writing, a formal deconstruction of sentences and language was incredibly appealing. I had also been learning computer programming around the same time and appreciated how a computer would need that kind of logic to approach understanding human languages. I was fascinated how (structural) ambiguity could be represented by a kind of tree diagram:

How he got in my pajamas, I’ll never know. (Groucho Marx, image source.)

### Languages and Linguistics

I think it is an unarguable truth that learning other languages necessitates an understanding of the principles of grammar. (From a mathematical perspective, this is the abstraction and terminology that allows us to see commonalities and differences in languages’ structures, and that allows us to check if what we are writing is grammatically correct.) I love learning how other languages ‘tick’: Spanish has a huge number of tenses (not least of all the imperfect subjunctive I threw in earlier); German has a wonderful art of incorporating adjectival phrases and nesting clauses to a depth paralleling Inception; in Catalan you can uncover patterns behind words that have morphed from French to Spanish; Chinese has almost no verb conjugation or number/gender agreement.

For me, the culmination of this study in abstraction is the Linguistics Olympiad. It’s rare that I can find a few students as interested as I am, but a couple of years ago I did get a few to take part!

### Etymology

In Year 9, we began our study of Chaucer with, I think, part of the prologue of the Canterbury Tales and then the Miller’s Tale. I was fascinated by the Middle English and how we were given some insight into making sense of it. I hadn’t even heard the word ‘etymology’ before, but I was hooked.

Do you know the origin of the word helicopter? Here’s a strange-looking clue:

## The Short Story

I would agree with Michael: drafting a curriculum of overzealous grammatical rules and forcing their use in compositions for them to be deemed ‘good’ is wholly inappropriate for creative writing.

I would disagree with Michael: there is a beauty (and purpose) in the study of grammar and languages from a point of view of abstraction, and it is one that captivated my own interest in English at a time when I felt lost in the creative side of reading and writing. At what age and rate to introduce this formalism, I cannot say.

# Harkness at Wellington

Now that Torchwood is a distant memory, presumably there won’t be too many readers here by mistake. The ‘Harkness’ in the title refers to the Harkness Method of teaching that developed at Phillips Exeter Academy in the US. To cut a long story short, a benefactor (by the name of Harkness) donated sufficient funds for the college to develop an approach to teaching that he thought ideal. In its purest form, the classroom contains an oval table with a dozen chairs around it. But it’s not so much about the table.

## Maths the Harkness Way

To quote from Phillips Exeter’s own page, students taught in this way

are exposed to problem solving in a very student-centered, discussion-based classroom. Students are held accountable for attempting solutions to homework problems and the class as a whole decides on correct solutions.

How does this come about? The homework that students complete comprises a set of problems that are to be tackled in advance of a lesson. Then, during the lesson time itself, they present their methods/solutions and a discussion ensues where misconceptions are tackled, misunderstandings are explained and, for example, the efficiency of different methods is compared. The teacher’s role in this, to grossly oversimplify things, is to guide the discussion, direct the students’ questioning to other students, and to tease out interesting ideas that are worth discussion.

A few people that I’ve talked to about this approach have thought that it’s just “flipped learning” by another name, but I would disagree. Flipped learning, as I understand it, still takes the linear topic-by-topic approach and still explains to students (albeit by video etc) “this is how you do X, Y and Z”. Then class time is spent with the students doing X, Y and Z with support and direction from their teacher.

In contrast, through the Harkness approach, the students are developing their own sense of mathematics through the tackling of problems. Moreover, several themes are developed concurrently.

## Teaching Resources

Clearly, a significant factor in the success of such an approach lies in the problem sets themselves. Phillips Exeter have for a long time freely shared their resources, aligned to the American curriculum. At Wellington, Aidan Sproat (@aidansproat) along with colleagues, has worked hard to create appropriate problem sets for the core modules of the UK A level curriculum. Aidan is more than willing to share these resources and has set up a web page where you can access samples, or contact him directly for the full versions.

## The Lesson Routine – Beginning

The first thing that struck me is how the students are fully comfortable with the routine and are busy as soon as they enter the classroom. They cooperate in terms of clearing the whiteboards (the classrooms I saw each had a good number of whiteboards – small and large – around the walls) and sharing out pens and board rubbers. Individually, they would elect one of the problems from the set they had worked on prior to the lesson and wrote up their work on a board. I didn’t time the whole process, but I would estimate no more than 10 minutes were taken up of the hour lesson. That 10 minutes is invaluable to the teacher for surveying the work going up on the boards, taking a register, recording which students have contributed solutions etc. Moreover, the class organise themselves well in terms of having notes, pens and calculators out from the very start.

The first time you see one of the problem sets, you will be struck by the variety of topics it covers. The second lesson I observed was a lower sixth Further Maths class. Their problem set comprised 7 questions covering:

1. The chain rule for functions of the form $(ax+b)^n$
2. Creating functions with a given domain and range
3. The chain rule for functions of the form $e^{f(x)}$ (and also including $x^e$ as a function to be differentiated)
4. Exact trig values for an obtuse angle, given $\sin\alpha$
5. Parametric equations that had to be explored, essentially to determine why they were not defined on part of the specified domain
6. The magnitude of 2d vectors (given a definition to work from)
7. A proof of the $\cos(A+B)$ and $\sin(A+B)$ results using matrix multiplication (of matrices representing rotations)

(Note: Aidan explained to me that following the staggered changes to A level subjects, Wellington have opted for no exams in the lower sixth in any subject. This frees up a considerable amount of teaching time! Thus, students in that year will make good progress on A2 study before the summer.)

## The Lesson Routine – Middle

Once students were settled and solutions on display, the first order of business was asking the students which problems they wanted to discuss. These were listed on the projector (a Microsoft Surface and a projector with a wireless HDMI adapter makes for a very efficient set up!) and then Aidan added his own request to discuss the second problem. A student had a quick question she wanted to ask about number 4 and that was duly added to the list, too.

The majority of the class time was then spent with a very active discussion on the part of the students: whoever had written the solution explained their approach, and then questions were directed between students. The teacher’s role in this is a delicate one: orchestrating the discussion without jumping in too quickly; taking care to ensure the mathematical accuracy of points discussed (but again through questioning more than telling).

One of the most interesting discussions (from my point of view) was the exploration of the parametric equations. I’m not doing the original question justice here, but its mathematical essence was:

Explain why $x(t)=2t+3,\, y(t)=\sqrt{(2t-1)(t-1)}$ is not defined on the whole interval $t\in[0,5]$.

Points that came up included: is it possible to square root 0? Can we form a Cartesian equation for this curve? I tried every number from 0 to 5 but didn’t find any problem (by which they mean they tried every integer – a nice aspect of the problem); It came to light nicely that the values of $t$ for which it was not defined corresponded with the values of $x$ for which the Cartesian equation was not defined. Moreover, the skill of sketching $y=\sqrt{f(x)}$ from the graph of $y=f(x)$ also arose naturally.

Question 3 was also fascinating for me. The way $e^x$ is introduced to the students is a whole blog post in itself. But by the time they reached this problem set, the students were piecing together knowledge developed over previous weeks so that they could combine their understanding of the chain rule with the fact that $\frac{d}{dx}e^x=e^x$ to differentiate functions such as $y=e^{x^2+3}$ without any explicit instruction.

## The Lesson Routine – Ending

In the remainder of the lesson time, the students were invited to complete a few problems from the ‘review’ section of their problem sets. These are designed to give them extra practice: to reinforce what they have already covered and to prepare for a test which takes place once a fortnight. During this time, the teacher dealt one-to-one with the student’s question about problem 4 but, again, through a careful process of questioning and relating the problem to earlier work that had been tackled.

## So, does it work?

Desperate as we are to simply ask ‘does it work?’, that’s not really a well-defined question. Even to ask if it is a successful approach to teaching the subject, you need to define some success criteria. Mathematics is renowned for being a ‘chalk and talk’ subject: “Here’s a topic, here’s more or less how it works, here are several examples of all the main variations that can come up. Now you try a few.” In my teaching, I try hard to include ‘warm up’ activities that prompt students to retrieve prior knowledge and push them to try and tackle a new problem before the spoiler of the main part of the lesson. Going all out for a full Harkness approach is a brave move and I cannot imagine how much time and effort has gone in to the planning of the problem sets. As I alluded to above, for example, the development of $e^x$ as a function is incredibly well-crafted and it takes place through individual problems over the span of many sets.

What this approach certainly does achieve, and what I particularly like about it, is the community of students discuss mathematics together and they are able to explain their ideas (and indeed their questions) very coherently. Questions came up occasionally such as “do we need to learn this?” (formula for $\cos(A+B)$) or “should we copy this?” (a student’s protracted and not completely correct working on a problem) but, in fact, the students demonstrated very good independent and discerning study skills.

What about the student that doesn’t do the homework? To be honest, that’s not a mathematics teaching issue. That’s a pastoral concern.  If anything, this Harkness approach makes it very clear to students that to study mathematics, they must come to class having made the best attempts they can on the full problem set. To prepare for their fortnightly tests, they must ensure they are confident with all the problems and they can use the review sections of problems to practise. As in every department, additional support is available outside of lesson time.

## Am I a Convert?

As I already mentioned, I like to think I have a significant amount of problem-solving in my teaching already. However, I am fascinated by this approach and I think many of my students would get a lot out of being taught in this way. It’s early days for me: after observing it ‘in action’, I have been able to answer many questions I initially had but only for them to be replaced with new thoughts and questions.

In July, Wellington are holding a training course led by instructors from the Exeter Mathematics Institute. I was initially surprised by the fact that this runs over four days, but with so much involved in the development of the material and so many subtle details in how to lead the classroom, I think it will be an intensive but fascinating experience. I can’t wait to learn more!