# Teaching Mechanics

Initially motivated by a brief discussion with one of my colleagues, towards the end of last term I ran a few Twitter polls to gauge different teachers’ conventions when it comes to teaching Mechanics. We’ll take a look at them one by one:

## Poll 1: Newton’s second law

The poll also attracted a thread of nearly 20 comments. Curiously, the majority opinion in the comments was the minority opinion in the final poll result (the second option)! Other comments made reference to the fact that we should of course label equations to indicate that we have resolved (ie applied Newton’s second law) on a particular object.

My personal opinion is the second option: R – mg = 0. It is a specific instance of Newton II where the acceleration happens to be zero. Why would we deal with the equilibrium case so differently and give students two (related but) different ways to tackle problems? [In fact, I sometimes wonder if we should teach some simple acceleration problems before looking at equilibrium as a special case?!]

Interestingly, all the comments that supported option 2 argued a similar point. The comments supporting option 1 didn’t really argue why it might help students. One commenter pointed out it could reduce sign errors and another liked developing students’ intuition around the balance of forces. (I have another worry here that students may forget that even if forces are ‘balanced’, the object could still be in motion.)

### My main point

• Neither is incorrect, but I would teach the “F=ma” approach for all situations, including equilibrium, so students only have to learn a single method. It also means they pay attention to the language used and deduce from ‘in equilibrium’ to that a=0.

## Poll 2: Acceleration Arrows

A very common student misconception in Mechanics is that the direction of acceleration matches the direction of movement. However, wherever possible, I think it makes sense to draw the acceleration arrow to match the direction of motion and this goes hand in hand with interpreting ‘deceleration’ to mean a negative acceleration.

If you show students (or teachers) the diagrams above without any context, our initial intuition would be to think of A as travelling upwards but slowing, and B as travelling downwards and getting faster. Of course, either of the diagrams could match either of those situations and we wouldn’t know which until we were given some information about the object’s velocity.

### My main point

• Neither is incorrect, but wherever possible I try to draw diagrams in a way that appeals to our intuitive sense of what is happening. (And never deduce the direction of motion from a force diagram!)

## Poll 3: Collisions

I’ll be honest here, and I admit that deliberately I included Option 1 as a bit of a red herring. Following on from the idea of intuition in the last poll, it’s perfectly reasonable to consider that the particles might travel in opposite directions after the collision, but in fact they don’t. Of course, we don’t know this for certain until we’ve solved the equations and we just get a good diagram drawn and then trust in maths to get the signs right for us at the end. I think one risk with Option 1 is if all students get the ‘speed of separation’ correct, but maybe I should have more faith.

I think collisions is the one area where I haven’t fully made up my mind and part of the purpose of this poll was to gauge a majority opinion and to see if that would make me reconsider my own view.

My approach to teaching this topic over the past few years has been to use Option 3: drawing all of the velocity arrows in a common direction, and then assigning them negative values if we know a particle is travelling in the opposite direction. My reasoning is that this helps get signs correct in the conservation of momentum equation and, moreover, I teach the restitution equation symbolically as

$e=\dfrac{V_B-V_A}{U_A-U_B}$

rather than using the phrasing ‘speed of separation’ and ‘speed of approach’.

I’m conflicted because this plays along the lines of giving students a single all-purpose method (as discussed with Newton II above) but somewhat goes against the idea of drawing a diagram that matches our intuition (as discussed with acceleration arrows above).

### My main-ish point

I really am torn here. I think I’ll continue to teach Option 3, with consistency winning over intuition. (To be honest, some days I’m just glad students draw diagrams at all!) I’m very willing to debate this one further though.

## In summary…

• Prioritise consistent approaches to solving a whole class of problems, reducing the number of methods students have to learn
• As far as possible, draw diagrams to match intuition (but don’t apply the converse and assume a diagrammed system is behaving as you intuit!)
• We all nag students to draw good diagrams and to label their equations. It took me a long time to figure out that I should isolate these skills in tests if I really want students to take notice. (Eg give the setup for a full mechanics problem, but only ask for a force diagram; or provide a force diagram and ask for a labelled system of equations (and for them to remain unsolved); or even, present the diagram and the equations and simply ask students to insert the appropriate labels.) In short, if you want them to develop a specific good habit, then test them on that specific good habit!

### A footnote on Newton II

Of course, we could argue that F=ma is a special case restricted motion in one-dimension and we should write F=ma and deal with vectors. But then this is a special case, assuming that a is constant so we should write F=m dv/dt. But then this is a special case that assumes the mass is constant, so we should write F=d(mv)/dt.

How do I reconcile this with my approach to consistency and teaching a single method for a whole class of problems? Well, up to (old spec) Mechanics 2 student don’t meet situations with varying mass etc so there’s no need for that extreme generality. It is always nice to discuss it in passing though, especially once they’ve learned about separable differential equations.

# Multiple Choice Questions in A-level Maths

There appears to have been a period roughly during the 1980s when multiple choice questions (MCQ) featured prominently in A-level exam papers. Precisely when or why they appeared and, indeed, when or why they disappeared again is a mystery to me at the moment. Moreover, the only textbooks I’m aware of that include MCQs are those authored by Bostock and Chandler. (There are also a couple of exam preparation books by Shipton and Plumpton that include them: Multiple Choice Tests in Advanced Mathematics, and Examinations in Mathematics.)

## In the new A-level

MCQs do get a cameo role in the new A-level assessments from AQA. Each of their Specimen papers includes a couple of MCQs worth a single mark each. Here’s an example from Paper 1:

In the document The Thinking Behind Great Assessment by Dan Rogan (Chief of Examiners), is the snippet

positioning MCQs as one of four features “at the heart of our aims for the qualification”. Later in the document comes some elaboration:

Aside from ease in assessment terms, I’m keen to focus on the pedagogical value of MCQs and the potential for their use in teaching. The above example feels similar to those offered on the Diagnostic Questions site although I would suggest different potential answers if I wanted to uncover misconceptions (for example, including 2 as an option to identify students who simply observe the coefficient of x).

## Diagnostic Questions

Indeed, here is such an MCQ on gradient from the Diagnostic Questions site [sign-in required when following that link]:

One of the excellent features of the Diagnostic Questions is the “insights” where you see students’ explanations for their answers. For example, this student offers their reasoning for (incorrectly) choosing option B:

## It’s an MCQ, Jim, but not as we know it

In the 1980s A-level papers, MCQs were a much more serious affair. For a start, there were five different types of MCQ. Even interpreting the instructions is no mean feat. I’ll describe each type and then include an example. [Photos from Plumpton & Shipton.]

### Section I (multiple choice)

There is a single correct answer among five options.

Example

### Section II (multiple completion)

Three responses are given (123) of which one or more are correct. The letter representing the student’s answer depends on which are correct:

Example

### Section III (relationship analysis)

These questions comprise two statements (1 and 2) and the student has to determine the logical relationship between them:

Example

### Section IV (data necessity)

A problem is followed by four pieces of information and the student must determine which (if any) piece of information could be omitted and the problem still be solvable:

Example

### Section V (data sufficiency)

These comprise a problem and two statements (1 and 2) in which data are given. The student has to decide if the given data are sufficient for solving the problem. Brace yourself…

Example

## Just a couple more variants…

In Core Maths for A-level, Bostock & Chandler simplify things down to three types:

• Type 1: exactly as Section I above
• Type 2: akin to Section II above but students simply write the letters corresponding to which items follow from the information in the question.
• Type 3: true/false

In their Further Pure Mathematics book, they have:

• Types 1 and 2 as in their other text
• Type 5 is true/false (as Type 3 in their other text)
• Type 3 corresponds to Section III above
• Type 4 is an amalgamation of Section IV and Section V: a problem is introduced followed by a number of pieces of information. Either all the information is needed (answer A); the total information given is insufficient (answer I); or some information can be omitted without affecting the solution of the problem (the letters for these items must then be specified)

## Use of MCQs in Teaching

I think there is good potential for the use of these more sophisticated MCQs in A-level teaching, although I fear students will either need simplified instructions (for example those used in Bostock & Chandler, rather than the London board papers of the 1980s) or significant training in how to respond. In particular, I agree with Plumpton & Shipton’s comments about Sections III – V:

These items enable coverage of topics which are difficult or unfair to examine by longer structured questions. Indeed, these more sophisticated item types are a far better test of mathematical understanding than some longer questions in which candidates may be applying a method or technique which they have learnt but not have properly understood.

Time to begin building a usable bank of these questions so that I can try them out next year!

# Maths in the Sticks 3

We are now delighted to confirm that “Maths in the Sticks” will be running again this year, on Sunday 22nd April 2018. Many of you will now be familiar with the routine: an engaging day of KS5 professional development with a hearty Sunday lunch! I am also very pleased that we have once again partnered with the Further Maths Support Programme to run this year’s event.

## Programme for the Day

All our speakers are now confirmed, though the precise timings of the day may be subject to change.

Our opening speaker this year will be Colin Wright (@colinthemathmo) with his hugely engaging talk on the mathematics of juggling. We are also pleased to welcome Daniel Griller (@puzzlecritic), author of the puzzle book Elastic Numbers, and Luciano Rila (@drtrapezio) from UCL who frequently runs sessions for both A level students and teachers. This year we also have Sue de Pomerai (@suedepom) from the FMSP who will be looking at some of the newer (or at least less familiar) content in the new Further Maths pure module.

## Sunday Roast

I have blogged before how CPD really shouldn’t be about the food, but hosting this event on a Sunday was a deliberate decision. Our school serves a roast lunch buffet-style, along with a salad bar and vegan options. I should be able to get a bottle of wine or two open, too!

## Hurtwood House

The event will take place at my school, Hurtwood House. Our postcode is RH5 6NU and, as you will see from Google Maps, we are in quite a remote location in the beautiful Surrey hills. Direct access to the school is only possible by car and, where possible, I would encourage people to drive and/or car share. However, so as not to exclude anyone, I can arrange for a minibus pick-up from Guildford train station at 9am. There is a box on the registration form where you can indicate if you would like this service. (Please check that suitable trains run on a Sunday for you to make it to Guildford before 9!) Of course, there will be a return minibus, departing Hurtwood House between 3 and 4pm at the end of the day.

Please also bear in mind that there is a flight of outdoor steps between the room we are using and the canteen. If you are keen to attend but are concerned about accessibility, then please contact me directly and we can see what arrangements might be possible.

## Registration

We typically have space for around 35 teachers each year. Please complete the registration form on Eventbrite to book your place.

## Job Vacancy – now closed

You might also be interested to know that we are currently advertising a position for a Maths Teacher to join us at Hurtwood. Full details are on our school website here, and the vacancy is also listed on the TES here. (Closing date 2nd February.)

# Water, water, everywhere

A particular issue on my mind recently is the sheer quantity of ‘in-house’ resources that departments create to either do away with textbooks, or at least supplement them with materials tailored to their schemes of work and students’ abilities. For some, photocopying costs must be paralleling the purchase price of textbooks!

I raised this issue in some recent tweets, wishing that there was a way to share this work and its products – in some parallel to the ‘open source’ software movement. The barriers to that are typically quality control; choosing the right platform for collaborative authoring; and keeping a generally consistent format throughout. None of these problems is necessarily insurmountable but the solutions come with the costs of inconvenience, learning curves, or simply time. (For examples of so-called open textbooks, take a look at Stitz and Zeagers’ Pre-calculus book or the ‘approved’ lists collated by the American Institute for Mathematics.)

## Nor any drop to drink?

The main product I envisage being most useful is simply a ‘problem book’ for A level Maths: minimal (if any) theory or worked examples, but masses of questions at different levels. Something along the lines of Drill, Exam Standard and Extension.

The more I think about this kind of project, the more I realise how many resources I’m surrounded by. I have textbooks of every style from every decade; past papers from every board for the past decade or more; papers from STEP, AEA and MAT exams which are great for extension problems; worksheets, packs of questions, and even more papers from Solomon, T. Madas, Delphis, Zigzag; the exercises and quizzes from Integral maths… The list feels almost infinite.

Las year I almost drowned my students during the summer revision period with huge packs of past papers from every source, supplemented with topic practice too!

## […] is this indeed, The light-house top I see?

Just the other day, I rediscovered the book Graded Exercises in Pure Mathematics which I’d forgotten I even owned. This is the closest approximation to what I think would be my ideal resource: chapters are focussed on each pure topic and the exercises are graded as Basic, Intermediate, Revision and Advanced – almost exactly the same gradings as I had considered.

But there’s a downside: published in 2001, this book has missed out on a couple of curriculum reforms and the (mis)ordering of the topics from our current perspective renders it almost as difficult to use as the usual cutting and sticking I do from every other book on my shelf.

## A speck, a mist, a shape, I wist!

I’ve been a big fan of Elmwood press‘s textbooks for a long while now: they contain some theory and examples but then great sets of exercises of increasing difficulty, exam standard questions and some review. I’ve been in contact with them and they have confirmed that new editions are being produced to match the 2017 specification.

I think I’ll hold out hope on these new editions, otherwise I’ll have a big project for myself in the coming academic year, creating what (for me at least) would be the ideal practice and problem book!

### Think, Pair, Share…

• Does your department systematically produce a large number of ‘custom’ resources for teaching A level Maths? (More than just the occasional photocopied exercise from another book?)
• Is most of your supplementary material geared towards giving students questions to work on? Or explaining aspects of theory in a way you prefer over the textbook approach for example?
• Do particular advantages come from having in-house resources, or might it be possible (over several iterations) to agree on a common ‘best practice’ resource?

# A Culture of Problem-Solving

There’s been quite a bit of puzzle sharing going around on Twitter lately and the timing fits well as our department is looking at how me might fully integrate problem-solving into our schemes of work for the new A level spec. Here are a couple of Twitter puzzles from recent weeks that you might like to try:

Also in the past couple of days, Jamie Frost (@DrFrostMaths) published this webpage in which he discusses a departmental shift-change to include more puzzles and problem-solving in their teaching, and the noticeable impact this has had on the pupils’ performance in UKMT maths challenges and olympiads.

With our A level students (particularly focussing on those doing Further Maths in the first instance) we are keen to give them much more exposure to mathematical puzzles and problems to solve. This will hopefully raise our students’ performance in the Senior Maths Challenge and also be of significant benefit to those who may need to take STEP papers for university entrance. Of course, another aspect of this is the building of a very positive culture of mathematics within our student community.

## Sources of Puzzles and Problems

Very briefly I’ll mention just a handful of sources of potential puzzles and problems that can be used, before dissecting a few problems much more deeply.

• Past maths challenge papers are available directly from the UKMT and they’re now also available online. (Jamie Frost is also integrating them into his online homework platform, available soon.)
• Cambridge have created a series of STEP Support assignments that help to bridge the gap between A level study and the demands of STEP questions.
• Books: there are innumerable books published with collections of mathematical problems but often it can be tricky to found those pitched at just the right level. One which I cannot recommend highly enough (and from which the problems below have been taken) is “A Moscow Math Circle” published by the American Mathematical Society. The introduction of this book also includes a very clear depiction of how mathematical circles are run in the Russian tradition.
• Magazines: I have recently, discovered a magazine called Quantum that was published in US in the 90s, based on the Russian publication Kvanta. All the past issues of Quantum are archived online, and many of the problems they contain are pretty demanding! (I’ve included samples at the very end of this post.)

## Homework for Part II

I’m breaking my original idea for this blog post into two separate sections. The second will focus on the difficulty with giving hints to students when they are stuck or have arrived at an incorrect answer. You might like to try the following problems (from A Moscow Math Circle) in advance of reading my thoughts on them in the next few days.

## Excerpts from Quantum

As mentioned earlier, Quantum magazine includes some quite challenging brain teasers (as they call them) and problems to solve. Here are the ones presented in Volume 7, No. 5 from May/June 1997.

# Silk from a SoW

As the exam boards compete to get us to sign up for delivering their flavour of the new A level Mathematics, it’s not easy to know on which criteria to judge and compare them. As the new content is fully prescribed, I think it will come down to the nature of assessment (number of papers, content of each, any quirks of style) and the support offered to teachers (additional teaching resources, schemes of work, online services, etc). Many departments are perhaps very comfortable with their current board and might only make a change if there is significant reason.

I’ve been fascinated to read Bruce Hampton’s Thoughts on A level Mathematics blog posts as he reflects on the process of comparing the exam boards from several perspectives. (Bruce is on Twitter as @bhampton271828.) He has already discussed issues such as problem-solving, planning a coherent scheme of work and a comparison of the large data sets that each exam board will use.

In this post I’m going to briefly look at what the boards offer in the way of schemes of work/resources to help with planning.

## AQA

This post is in fact motivated because of some AQA resources I saw recently (more on which in a moment). They have a dedicated webpage to collate resources specifically about planning, although bizarrely the kind of planning documents I’m looking for are actually on their Teaching Resources webpage.

One interesting approach is their “Route Map” which essentially uses Powerpoint as a way to organise a programme of work. There are initial slides (all editable) for Year 12 and Year 13, like this:

and each coloured rectangle jumps to another slide which (at the moment, at least) just lists subject content statements. I’d never thought of using Powerpoint to achieve such an interactive document, but I can see it has some merits.

The other planning documents that AQA will produce are ones they call “Teaching Guidance” (link is to a sample one for Differentiation). I’ve got to be honest: I think this document is rather disappointing. It summarises the content statements and then exemplifies them with a handful of examination-style questions. Apart from perhaps clarifying what is or is not examinable, I can only see this encouraging a teaching-to-the-test approach. There appears to be no thought about prior knowledge, links to other topics, common misconceptions etc.

## Edexcel

(At the time of writing, Edexcel is the only board currently awaiting accreditation but in spite of this, they are pushing ahead and publishing a number of draft documents.) Edexcel have a page for Teaching & Learning Materials which contains some suggested course organisation, documents that map between the old and new specs for Maths and Further Maths, and their schemes of work.

I’ll focus on the schemes of work and they seem pretty comprehensive. The introduction to these documents explains the type of content to be found within:

This is much more closely aligned with what I believe a scheme of work should be: a document that really focusses on important issues to consider when teaching each topic.

Edexcel are also providing a free online “interactive scheme of work” which, from watching the overview video, seems like a great tool. I wonder if it will be customisable to a deeper extent for departments to annotate teaching points etc.

## OCR

OCR’s resources are all linked directly from the qualification page. Of note here are the “Lesson element” samples such as this one on surds. This reminds me very much of the Standards Unit resources (still available online – I usually get them from mrbartonmaths): it comprises activities such as card sorts, ready to be used in class, but accompanied by teaching notes.

However, I can’t actually find anything resembling a scheme of work or even an example schedule of how the content could be organised in a school year.

## MEI (OCR B)

In a way, the page for this qualification on OCR’s site has even less than the OCR A page. There is of course reference to (and sample materials from) the Integral maths website which many people will know already. However, it’s on MEI’s own site that things get more interesting.

On their Schemes of Work page, MEI have divided the A level content into 43 units and shared a planning document for each one. (See for example, AS Differentiation.) The key elements of these documents appear to be:

• content statements from the specification
• commentary
• sample MEI resources (really selling Integral maths!)
• use of technology
• prerequisites
• prompts for mathematical thinking
• opportunities for proof
• common errors

Again, much of this is what I believe a scheme of work should be about.

## Some Summary

 Board + points – points AQA Interesting use of Powerpoint to collate a SoW, linking to other slides with more detail SoW only contains subject content items; Teaching guidance documents repeat content items and only demonstrate exam-style questions Edexcel SoW contains much more comprehensive information – detailed commentary; online SoW builder seems very interesting Fairly dry presentation: although the SoW is editable, it’s one long document; no references to uses of technology OCR “Lesson element” documents seem well thought-out and practical No scheme of work/planning documents that I can find MEI (OCR B) Separate documents for each of 43 ‘units’ of study; detailed commentary & opportunities for technology and proof It will be interesting to see how many of the referenced activities become ‘paywalled’ in the Integral resources site

# Click & Reveal (Part 2)

A couple of weeks ago, I wrote a post “Click & Reveal” describing a situation that I was trying to solve: for a number of my handouts, I’d like to be able to display them in class and – as and when I feel like it – click to reveal different parts of the working etc. In that original post, I mentioned a solution I hacked together using HTML, CSS and Javascript – not the ideal environment for writing maths handouts!

## PDF Sorcery

Well, I’m not one to quit on a problem like this and I’ve now come up with exactly what I want. Try downloading this PDF document and opening it in Adobe Reader. (The only limitation of this approach is that it does need Adobe Reader – not Mac Preview or Chrome’s built-in PDF viewer etc.)

Anything not coloured black in the PDF file is clickable! Click on any of the blue expressions to hide them. Click on the red ‘hide’ eye to blank all the expressions together. Click on the green ‘reveal’ eye to show all of the expressions.

Moreover, you can choose to print the document either filled or blank (and the eye icons don’t appear on the printout)! Students could even use the PDF file to test themselves in a similar way to flashcards.

The beauty of this approach is that the solutions can be revealed in any order, at any time (unlike say a Powerpoint where the order is pre-determined).

## LaTeX Sorcery

This solution works wholly in Latex (which is a drawback as I’d be surprised if many teachers know about Latex, let alone use it to produce documents). It uses the ocgx2 package which takes advantage of the Layers (and Javascript) functionality that Adobe Reader supports.

If you’d like to see behind the scenes, I’ve shared the source for this document on Overleaf.