Conditional Probability

It seems, to me at least, that probability is one of ‘those topics’ that students find to be more difficult than others. Each year I have winced slightly as it looms on the horizon of the scheme of work and each year I’ve varied my approach to try and get them (and keep them) on board. This post focusses on conditional probability, now designated as an A-level (not AS) topic.

Prior Knowledge

Students should already have a fair amount of experience with tree diagrams and Venn diagrams. They should also have met the terms independent and mutually exclusive – I think that’s actually where the problems begin. Why are these concepts so challenging? Perhaps because one (independent) is a term borrowed from every day language, but in that standard usage it tends to mean ‘separate from’ which is actually closer to the meaning of mutually exclusive. Secondly, because we try to give students an intuitive and descriptive feeling for what independent means in statistics, they then lose sight of the fact that it has a precise, mathematical definition.

What would I do about this? If I could, I would separate the teaching of the two terms with as much time as possible. We could teach ‘mutually exclusive’ when we cover sampling techniques (the strata for a stratified sample should be mutually exclusive, for example). More contentiously, I wonder if we shouldn’t introduce ‘independent’ until the time we study conditional probability? Compare:

Definition 1: A and B are independent events if P(A\cap B)=P(A)\times P(B)

Definition 2: A and B are independent events if P(A\vert B)=P(A)

These are equivalent*, but I would argue the second gives a much better feeling for what statistical independence really means.

Conditional Probability

When teaching conditional probability this year, I demonstrated it separately in three contexts: two-way tables, Venn diagrams and tree diagrams. Prior to the discussion of conditional probability, of course I wanted to check that students brought their prior understanding to the surface – through a mixture of starter questions on the board (two-way tables; tree diagrams), or some mini-whiteboard work (Venn diagrams).

In each context I adopted the same approach: a probability question involving the typical ‘given that’ phrasing; a highlighted, restricted part of the diagram; and the formula for conditional probability. Each time, I described the highlighter approach as intuitive and the formula approach as ‘safer’ – not susceptible to misinterpretation. Each time, we also considered the reverse of the conditional statement, ie. we contrasted P(A\vert B) and P(B\vert A).

I split these three contexts over three separate lessons. This gave the students the opportunity to focus on each one fully, in isolation, and to also experience the recall of the formula for conditional probability each day. The examples below aren’t the exact ones that I used in my lessons, but should be illustrative enough.

In a two-way table

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Question from LearnZillion

Intuitively: \displaystyle\frac{203}{203+122}=\displaystyle\frac{203}{325}

More abstractly,

P(\text{survive} \vert \text{first})=\displaystyle\frac{P(\text{survive} \cap \text{first})}{P(\text{first})} = \displaystyle\frac{203/2201}{325/2201}=\displaystyle\frac{203}{325}

In a Venn diagram

venn.png

Image from ck12.org

To find P(A\vert B), we can think intuitively: \displaystyle\frac{0.3}{0.3+0.2}=0.6

More abstractly,

P(A\vert B)=\displaystyle\frac{P(A\cap B)}{P(B)}=\displaystyle\frac{0.3}{0.5}=0.6

In a tree diagram

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Given that a student can construct tree diagrams, what is the probability that they pass? (Image from stats.libretext.org)

Intuitively: just read 0.97 from the relevant branch.

More abstractly,

P(\text{pass} \vert \text{can construct})=\displaystyle\frac{P(\text{pass} \cap \text{can construct})}{P(\text{can construct})}=\displaystyle\frac{0.78\times0.97}{0.78}=0.97

Note: there there is a significant opportunity for confusion here: the calculation for P(\text{pass} \cap \text{can construct}) is the product 0.78\times 0.97, however this is unrelated to the product formula for independent events. Indeed, the events ‘able to construct’ and ‘pass’ are not independent here.

Results and Looking Ahead

We set weekly class tests in our department and thus get fairly rapid feedback about how well students have picked up new topics. I was pleased to see that my emphasis of using the slightly more abstract (but secure) approach given by the formula for conditional probability made an appearance in almost all of the students’ work. I was a little disheartened (but not completely surprised) by the proportion of incorrect answers to the very first question on the test:

1 (a) State what it means for events A and B to be independent.

(b) State what it means for events A and B to be mutually exclusive.

We still have 6 or 7 months to sort that out!

I’ll be returning to conditional probability at every available opportunity. Due to the ordering of our scheme of work, I am now moving on to discrete random variables and thus can include conditional questions there. This is also followed by the binomial distribution (which many of you may well have already taught in the AS year). I find it somewhat surprising that conditional probability has been designed to come after hypothesis testing as, within a hypothesis test, the probability we calculate is conditional on H0 being true. With the order of our own scheme of work, I can emphasise that conditional statement more heavily and hopefully the students will gain a better appreciation for the process we follow in this type of hypothesis test.

* Footnote

The definitions are not quite equivalent… The reason the first is the preferred definition for independent events is that the second cannot be used in the case where P(B)=0. In all other cases, they are equivalent!

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Visualising Coupled Systems of DEs

One of the new topics to have found its way into Further Maths is coupled systems of differential equations. The Pearson textbook begins with this example:

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As both x and y are functions of time, t, then there is a very appealing dynamic nature to the solutions of these equations.

One beautiful way to visualise the family of solutions is to use an online tool called ‘Field Play‘ which I believe was created by Andrei Kashcha (@anvaka). I’ve only been able to capture still images to post here, but the website shows a striking flow over time.

It’s not too hard to enter the equations you wish to visualise, especially once you have seen a couple of examples. The key points are:

    • v.x represents dx/dt, and v.y represents dy/dt
    • p.x represents x, and p.y represents y
    • If you have an integer coefficient (such as 2) then you must enter it as 2.0

Another nice feature of the site is that the settings you enter are encoded in the website link. Thus each link below goes directly to the visualisation of the equations I’ve entered.

Example 1: Bears and Fish

Using the equations from Pearson’s example above yields an image like this:

Example 2: Lotka-Volterra

A more sophisticated model for interacting predator and prey species is given by the Lotka-Volterra equations. (Note these are non-linear, because of the xy term, so A level students are not expected to analyse them!)

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Here, x represents the population of the prey species, and y the predator.

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I found it particularly helpful to project this animation onto my whiteboard so that I could annotate a few features. Note, for example, the location of the origin and labelling of the axes (we were discussing Foxes and Rabbits). The anticlockwise flow follows the typical storyline of alternating fluctuations in predator and prey numbers.

LK_annotated.jpg

Example 3: Random!

Another great feature of this site is the ‘Randomize’ button. It will generate its own random set of DEs and show you the flow. It can be great to do this several times in succession and notice some features:

  • These systems have ‘equilibrium points’ where there is no motion. These are points where (simultaneously) dx/dt=0 and dy/dt=0
  • There are limited types of these equilibrium points: nodes, saddles, spirals, stars etc, and they can either be stable (eg local trajectories spiralling ‘in’ to the equilibrium point) or unstable (spiralling out etc). [A classification based on eigenvalues can be found online in notes such as these.]
  • The pattern of flow surrounding a number of equilibrium points is strongly reminiscent of weather patterns and might motivate students to learn about Ed Lorenz‘s work on chaos theory.

lorenz.jpg

 

Teaching Proof (Part 1)

Proof is one area where the new specifications for A-level Maths are noticeably different from the legacy courses. Technically, proof was a part of the old courses but that box was often ticked by including a trigonometric identity question in exams. MEI gave it a little more prominence with questions like these from their C1 papers:

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MEI Core 1, January 2005

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MEI Core 1, June 2005

This year I have had to teach proof much more thoroughly than ever before, and it’s certainly not been a straightforward process! This blog post is a reflection on some of the issues that arose and ideas for future teaching.

What’s new?

Each board’s specification contains essentially the same wording when it comes to the content we need to be teaching to cover ‘Proof’ as a topic. Here I’ve copied Edexcel’s specification as it helpfully distinguishes between AS (in bold) and A2, and includes a little bit of exemplification.

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Edexcel GCE Mathematics 2017 specification

When I was planning my teaching for this topic, I knew we would need a preliminary discussion about mathematical statements and how they can either be true or false (on a rainy day you might want to disappear down the rabbit hole of incompleteness and Gödel’s work…). I then thought that direct proof by deduction could be done in the same lesson, saving proof by contradiction for a second lesson. Job done. I think it’s fair to use the Bushism that I misunderestimated the depth of this topic!

Lesson 1: warm-up

I wrote out a sheet of example statements for paired discussion:

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Mathematical prompts for discussion: true or false?

These prompts were designed to include a variety of concepts:

  • There are rules such as ‘odd+even=odd’ that students have known since junior school, and perhaps have never deeply questioned.
  • Some of the statements fail in particular instances (the top-left is false when p=2, for example) so we have the notion of a counterexample.
  • A finite set of cases for checking (top-right) so we have the notion of proof by exhaustion. Which we can then contrast with statements where there are infinitely many cases to check, and thus would need a different approach. [There was also the opportunity to digress and mention Goldbach’s conjecture.]
  • Some statements can be proved using a direct proof by deduction. These will be a nice way to introduce concepts such as writing odd integers in the form 2k+1, where k\in\mathbb{Z}
  • I included the converse of Pythagoras so that we could talk about the direction of an implication. Here, the converse is itself a true theorem but that is not always the case.
  • My other choices are clearly heading towards proof by contradiction.

Lesson 1: our first proofs

Of course I wanted to start simple. We were going to prove that the square of an odd integer is itself an odd integer. We discussed how that statement feels rather self-evident and I allowed the students to raise their “Do we really have to prove it? Isn’t it obvious?” concerns. And so we began…

Claim: If n is an odd integer then  is an odd integer.

Proof: If n is an odd integer, then n = 2k + 1, where k is an integer.

Ok, stop right there! “How do we know n=2k+1? Don’t we have to prove that, too?”. (Along with the usual lone voice of “what does integer mean again?”) We had to interrupt the proof and have an important discussion about definitions and their fundamental importance in maths. Fortunately, I’d prepared a handout with some key terms on it, and this was the ideal time to give that out.

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Now we could continue our proof

Then, n² = (2k+1)² = 4k² + 4k + 1 = 2 (2 + 2k) + 1,
which is odd, as 2 + 2is an integer.

Ok, stop right there! “Now we can just say that 2 + 2is an integer?! Shouldn’t we have to prove that?”. And, it’s a fair point: if I’m taking the trouble to prove something like ‘odd times odd is odd’ then shouldn’t I also prove that ‘integer times integer is integer’? So then we had to digress into axioms, which I can honestly say I was not prepared for. [For those of you who haven’t had the pleasure of this at university, the integers are essentially constructed to satisfy certain fundamental rules: the sum of two integers being an integer is such an axiom – it’s an assumption on which our arithmetic is built, not something to be proved from more basic principles.]

I gave Euclid’s “parallel postulate” as a classic example of a ‘take it or leave it’ axiom. Take it, and you can study Euclidean geometry and triangles have an interior angle sum of 180 degrees etc. Leave it, and you can study non-Euclidean geometry which is way cooler. However, if you don’t accept the ‘integer+integer is integer’ axiom then good luck to you on your lonely journey.

There was just enough time left in Lesson 1 to finish with a direct proof of: if is even, then n² is even, and that seemed to reassure students that these proofs really aren’t that bad.

Until you get to Lesson 2…

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:

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

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

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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]:

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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:

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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

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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:

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Example

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Section III (relationship analysis)

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

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Example

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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:

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Example

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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…

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Example

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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

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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?

Do share your own thoughts and experiences in the comments!