Most people I have spoken to aren’t terribly keen on the current A Level textbooks. I think that the current Pearson Edexcel books are particularly poor, worse than the Edexcel books that I had when I was at A Level and definitely not as good as the Bostock and Chandler textbooks that I love.

I think the questions in the Edexcel books are quite samey, not necessarily challenging enough to stretch the most able and could also highlight links between topics better. The order that they tackle topics is also a bit weird, I can’t work out why the chapter on vectors is at the end of the M1 book for instance.

I was thinknig that the 100% prescribed A Level course (and 50% prescribed further maths) that is coming in provides a good opportunity to write an exam-board neutral text book / express what we want in a textbook.

Here is my list of things I would like to see in an A Level textbook:

- A book written in a logical order that highlights the links between topics.
- Rigorous derivations that highlight where methods come from.
- Sections that discuss the use of the mathematics in the real world.
- Plenty of exercises for pupils to practise (with worked solutions available for teachers).
- Scaffolded exercises to develop understanding and pupil’s confidence.
- Off syllabus sections to help prepare students for university level maths. For example, the Fundamental Theorem of Calculus is something that could be properly discussed at A Level and not just alluded to.
- Access to plenty of virtual manipulatives to demonstrate ideas. These should be accessible to any students that want to use them.
- Links to (free) mathematical software pacakages, especially for investigating data in the statistics sections.
- Explicit emphasis for multiple approaches to solving problems.
- Longer, more free-form questions to challenge students.
- Suggestions for self-study research projects (maybe ones that would be suitable for the EPQ).

I’m really interested to hear what other people would like to see in a text book.? What kind of things do they particularly like/dislike?

Please comment below or comment on twitter.

*Related*

The off syllabus stuff sounds great. Working open days at a uni which doesn’t require further maths, it was one of my key points that uni maths covers so many topics never seen before. STEP exams were the closest I got to uni style thinking during A level. Maybe a companion book, could go on for days!

Which STEP papers did you do?

Hmm I’m not sure, seems years ago now (it was)… I’m thinking it was I and II, as I had a friend applying for Cambs and it’s what I think they require. Much less structure to the questions, although I’m afraid I can’t offer much more insight than that as I haven’t picked one up since the 6 hours worth of exams for the two of them!

Funnily enough I replied to your tweet before I read your post – and said I don’t want exam board names on the cover and referred to Bostock & Chandler!

Bostock & Chandler – I always found so clear – the format with several worked examples then exercises is excellent and really helpful for self – study.

I think the balance has to be right with formal derivation – whilst we want to challenge the best Mathematicians, we want the subject accessible to students of varying ability. I find that sometimes formal derivation can be best after students have been using techniques. Try too much derivation at the beginning and they have nothing to hang it on to.

True about the derivation. Maybe clearly highlighted sections to differentiate :S I really like the exercises in Bostock and Chandler, I think they provide a good mix!

..and!

This text could get a bit fat given everything on your list! I’d cut out the off syllabus sections from the main text – maybe a separate text for that – in fact there’s some around already. Yes to IT to support – Desmos (free apps on android and iOS), WolframAlpha and virtual manipulatives as you say. I could write a whole blog post in reply to this – maybe I will!

Would be great if you do ðŸ˜‰ I love Desmos and WolframAlpha. Particularly good that you can now get Mathematica for free on the Raspberry Pi

I don’t know too much about the UK Level system, but I can say that one of the things that I like, but don’t see often in math texts is a distinction between “Exercises” and “Mathematical Exercises”. “Exercises” would be mostly mechanics and some application of theory. “Mathematical Exercises” would be mathematics — ie, working in a theoretical / abstract framework, dissecting proofs, exploring the assumptions underlying theorems, extending theorems, etc. Bhattacharya’s “Statistical Concepts and Methods” book from a long time ago does this, for example.

I visited a school in India this week as part of a school exchange and was asked to teach a coordinate geometry lesson to grade 12 (yr11) on coordinate geometry. I was given a textbook to look through so I could see the standard they were at while I planned my lesson. Question 1 in the chapter looked like this:

Find the area of the quadrilateral with vertices (-4,5), (-0,7), (5,-5), (-4,-2).

I can’t remember the exact vertices, but it didn’t really matter, because it wasn’t a rectangle anyway.

I liked this question because it wasn’t the usual ‘use this method when you see this’ type of question. Although it is not the most difficult question, it’s the sort where students can compare and contrast approaches. I’d like to see more of this type of question in textbooks.

Maybe an NRich textbook?…

I think we’re thinking similarly here. I generally think the answer is,simply “one that I’ve written”, there are so few good ones,about these days at any stage. I have recently looked at the new KS3 textbooks from pearson, and they seem excellent, so hopefully their new album level ones will be too.

Yep that is my thinking. Maybe we should just write one?…..

You and I have had many conversations about the standard of Modern A Level textbooks and I agree that they are not particularly helpful, particularly when considering what is expected of students beginning University studies.

I agree that the linearisation of the Mathematics A Level will certainly help to standardise the content, resulting in exam board-neutral textbooks. Also, I think constant reminders about connections between areas of mathematics studied can be highlighted in this setting, e.g. pointing out how useful calculus is in mechanics and statistics. The skill of seeing connections between degree-level modules is somewhat lacking in our undergraduates, so instilling this into them at A Level would be very helpful.

I agree that the use of scaffolded exercises to develop understanding would be useful, but I think these should be complemented with open-ended exercises that encourages students to think about using their known mathematics in unexpected ways or combine what they thought were self-contained topics (I guess this is what you were alluding to in your penultimate bullet point?). This will give them a flavour of what to expect in a mathematics degree.

For the off-syllabus sections, I think a good discussion of the need of proof and some general approaches for how to prove theorems would be very useful for students looking to pursue mathematics to a higher level. Many of our current first years do not consider this to be maths, saying “why are you just teaching us logic?”, to my dismay.

I think that the overall message of the textbook would be that there are plenty of ways of reading around the subject and exploring interesting applications of the mathematics the students are learning at A Level, rather than hundreds of basically the same exercise to hammer home each small point. In addition to learning the methods, they should have an appreciation for the wide reach of mathematics.

Hi Tom,

I welcome your suggestion about clearer links between topics in a textbook. As you’re aware, maths might be taught in chunks/modules but most of the time it’s not applied that way in practice. Not only can highlighting the links show the diversity of the techniques being applied but I think it may help to encourage interest via other, potentially unobvious or new, subject areas: mathematical biology and computational maths spring to mind ðŸ˜‰

During my Biology A-Level, we had a synoptic module in which questions relied on the student to use knowledge from the range of modules taught. A new textbook might be a good opportunity to introduce something along these lines in maths.

Derivations would be nice but you probably don’t want to scare students off so choosing the right amount is important.

Links to any sort of free mathematical software or experience of computation before coming to university would be great, otherwise its a bit of a shock!

From an industry point of view, well-rounded mathematicians/graduates are highly desired for solving real world problems and there’s no doubt that learning about a wide variety of topics in a well-structured and organised manner from as early as possible (without discouraging anyone!) is a good way forward.

I have been meaning to come comment here for ages, sorry for taking so long to get around to it. Mainly I agree with all the things you list so it’s hard to forget them and think up my own list. One thing that would be nice to have which you’ve not explicitly listed is having questions which require non-trivial work from multiple topics. I like to try and signpost, as much as possible, places where students will draw upon expertise from a variety of chapters of work that they’ve been studying to try and get them into the habit of appreciating everything is linked. Sure, this happens a lot whenever a more advanced question calls for differentiation etc… but it would be good to have examples which use other areas beyond merely using a method from another section to tackle a new idea in our current topic.

You also talk about free-form exercises… I have found more recently when teaching a computing element of a module to [university] students (for which it’s assumed they have very little prior knowledge) that exercises which are more free-form as you suggest, leaving them to their own devices to do stuff, rather than (i), (ii), (iii), (iv) step suggestions lead them to be more efficient learners as well. All too often if I try and be too helpful in presenting new ideas in computing (getting them to write a short script to do something useful, for example) they’ll get stuck on something I didn’t think they would, partly because of an artificial intermediate step I had to create. I think it’s so much more useful to merely describe the target and let them work out the details than do half the work for them and tell them to write a script to do “A” and then a script to do “B”… then stick them together and magically it does what we wanted. The drawback is you certainly need to work harder on expectation management — free-form problem solving is certainly more difficult, but a skill I’m sure you’d like them to develop.