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:

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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:

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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
  • links to other topics
  • 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.)

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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.

Maths in the Sticks 2

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Following on from the success of “Maths in the Sticks” last year, Maths in the Sticks 2 (#mathsticks) is now open for registration on Eventbrite. Please, however, read the important information below regarding the event.

Programme for the Day

Here is an outline for the day:

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This year’s opening speaker is Dr Dave Wood (@dave2809) from the University of Warwick. Dave is the director of undergraduate studies and a principal teaching fellow. As part of his research, he participates in the annual ‘European Study Groups with Industry’ and he will be sharing his experiences of mathematical modelling to solve industrial problems.

Hannah Lees (@LorHRL) and Rob Beckett (@RBeckett_Yd) are Underground Mathematics Champions and they will be giving us a hands-on look at their favourite resources from the Underground Mathematics website, with a focus on modelling and problem-solving.

The morning will be wrapped up with a session by Dan Whitehouse (@DanMWhitehouse), head of our Maths department here at Hurtwood House. Dan will be looking at possible approaches to planning as we prepare for teaching the new specifications in September.

After lunch, our focus moves to Statistics as Jane Annets, a trainer from Casio, will present a session describing the functionality of the forthcoming ‘Classwiz’ calculator and giving us a hands-on experience with graphics calculators so that we can evaluate the potential of these technologies.

After coffee, the rest of the afternoon has been set aside for some collaborative planning time. You might like to continue investigating potential resources and the approaches to planning that were discussed during the morning. Of course, there is also the opportunity to network with colleagues from other schools, share ideas and make new contacts which will be invaluable as we all begin the new syllabus later in the year!

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. I should be able to get a bottle or two of wine 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 around 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

Registration for this event uses Eventbrite and that will ‘go live’ on Monday 30th January.

Click here to be taken to the Eventbrite registration page

Places are limited to 36 but, if I have figured out Eventbrite correctly, we should be able to keep a waiting list in case people need to drop out. This is a free event, with places allocated on a first come, first served basis.

If you have any questions at all, then please contact me either through Twitter or using my school email:

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Click & Reveal

This week I faced a frustration which I’m sure can’t have been for the first time: I had a Word document to use as a “fill in the gaps” style handout, but I had already filled the gaps. I was looking for a way to display the document using the classroom projector, but to easily toggle the gaps between filled and blank.

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This situation must surely arise up and down the country quite frequently. I put the call out on Twitter and asked a few colleagues in school as well. It turns out there isn’t a straightforward solution (that I’ve come across so far).

The shortlist seemed to be:

  • Use interactive whiteboard software with white boxes covering gaps (downside: I need to create a lot of covering boxes and I don’t finish up with a nicely printable Word document. I also don’t have an IWB…)
  • Use Powerpoint and animations (downside: in addition to the above, without clever VBA coding, you can only reveal the gaps in the order in which you animated them)
  • Change the colour of the text in Word (downside: need to select all of the text beforehand to make it white, then go through and slowly change the colour of different pieces to reveal them)
  • Change the colour in Word using a style – this is the best so far, and the purpose of styles. If every gap is formatted to have the whole style then it’s relatively easy to change that style from white to blue for example. This could even be automated using some VBA and a keyboard shortcut. Thanks to @SamHartburn for help cracking this approach! (Downside: it’s an all-or-nothing reveal and hard to adapt to revealing one gap at a time.)

There must be another way

I’ve done a fair amount of programming in years gone by, and learned my fair share of HTML, CSS and Javascript. So in a free lesson this morning, I sat down to hack something together and, within an hour, I had a fully working prototype!

Here’s a (silent) video screencast showing how my mocked up page responds:

The Sorcery

Essentially, each gap (which is typically either a span or div) is given the class reveal. As the page loads, a Javascript onclick function is assigned to every document element with class reveal. The Javascript simply changes the CSS for that element: toggling the colour and toggling the mouse pointer style.

I also found some sample code online that styles the page to mimic an A4 page (in appearance and dimensions) and I include a Google font (PT Sans) as the closest match to the one I always use with my classes (Myriad Pro) – all my handouts have this same style!

So from now on, maybe I’ll switch to HTML over Word documents. We’ll see. Creating equations becomes tedious, and I have yet to fully get the print CSS correct for printing a blank version of the document. That will have to wait until I have another spare hour.. probably half term!

FP3 Vectors – Developing Deeper Understanding (Part 2)

This part picks up from Part 1 and looks at the ‘shortest distance’ scenarios in FP3 Vectors. I’m always stunned by the textbooks and formula books that promote such formulae as:

\text{distance} = \left| \dfrac{(\mathbf{a}-\mathbf{c})\cdot(\mathbf{b}\times\mathbf{d})}{|(\mathbf{b}\times\mathbf{d} )|} \right|

and

\text{distance} = \dfrac{|n_1\alpha+n_2\beta+n_3\gamma+d|}{\sqrt{n_1^2+n_2^2+n_3^2}}\,.

The former of these is for the distance between two skew lines, and the latter for the perpendicular distance from a point to a plane. Not only are these a hideous mass of letters but, even more concerning to me, they are both disguising the same mathematical principle. Let’s take a clearer look at things.

Understanding Components

Here’s my starter question before we consider the ‘shortest distance’ scenarios:

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This can take a little time but because the given statement is true, students can invest some time and thought into why it is so. After some short discussion, we are happy that dot and cross product can be used to find components. Moreover, it is only the ‘vertical’ component that we will need: sometimes found with dot product, sometimes with cross.

Similar Situations

Mathematics to me is all about identifying similarities. The formulae quoted at the start of this post are totally anathema to me. Consider these (beautiful…) diagrams:

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In every one of these situations, the ‘formula’ for finding the shortest distance is simply

\overrightarrow{AB}\cdot\hat{\mathbf{n}}

Much less to remember, much more understanding developed.

Two Other Cases

In the case of parallel lines or perpendicular distance from a point to a line, this approach breaks down: we don’t have a convenient unit normal vector. This is where the cross product alternative for finding a component comes in.

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Many Ways

Of course there are many ways to solve these problems. I simply like the above approach because it follows from a clear understanding of components and the geometry of the objects involved.

I wouldn’t often openly criticise a textbook but, for comparison, here is a worked example from the Edexcel/Pearson book [FP3, p128]:

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Of course it works. But. Yuck.

 

FP3 Vectors – Developing Deeper Understanding (Part 1)

I love teaching vectors in FP3 because I see such simplicity behind all of the geometric situations. In Edexcel’s module the chapter essentially breaks down into three sections:

  • Learning how to calculate a cross product, culminating in the triple product giving the volume of a parallelepiped.
  • Learning that the equation of a line (which was introduced in C4) can be written in several forms; and learning about the equation of a plane, again in several forms.
  • Finding points of intersection of lines and planes; angles between lines and planes; and shortest distances between points, lines and planes.

This post picks up on a lesson at the second stage of this journey. Part 2 looks at a lesson snapshot from stage three.

Productive Struggle

My understanding of the phrase productive struggle is to give students a task which they can’t launch straight into solving (with their current knowledge) but that they can chip away at and explore.

Each year, I look forward to this lesson:

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Occasionally near the beginning, I might need to clarify precisely what the question means but I very rarely give any indication about how to proceed. And I give my students a lot of time. Originally I thought I give them around 15 minutes but this week, my class had at least 20 before we shared some ideas on the board. I asked 4 particular students to write up what they had tried:

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 Fwd--8.jpg  Fwd--7.jpg

Each of these contributions had something interesting that I wanted to bring to the whole class discussion. The writing in black identified a geometric meaning to the problem. The student writing in blue worked from the principle that the cross product of a vector with itself is zero (ie the zero vector). The student writing in red took an algebraic approach and realised that the three equations do not have a unique solution, but she was able to identify a valid solution. The student in green worked along similar lines but was able to list multiple solutions and identify a pattern behind them. (The annotations in red and blue came later.)

Eventually we could have a full discussion about why (\mathbf{r}-\mathbf{a})\times \mathbf{d} = \mathbf{0} represents a line – from both an algebraic and a geometric perspective. Students are always quite surprised to see the shortest demonstration:

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Of course, a similar lesson could be built around \mathbf{r}\cdot \mathbf{n} = c but I’m not usually mean enough to make the students suffer twice in the same week!

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:

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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:

perimeter-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:

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