Cambridge has just published Enrichment Mathematics for Secondary Students (22/01/26 – link fixed). Cambridge describes the set as “an inspiring and challenging new series for ambitious 7-10 maths students and teachers”, and I strongly agree (except for their use of the term “maths”). I don’t know of any other text, anywhere, that is as good at covering secondary pre-calculus curriculum material, and should-be-curriculum material, at such a deep level.
A disclaimer: the authors, David Treeby and Gareth Ainsworth, are ex-students and very good friends of mine. And a second disclaimer: I was an ad hoc proof-reader for various drafts of the text. So, the reader has cause to be wary of my opinion. But, I love this book. For many years, the drafts have been my first and pretty much only port of call whenever I was required or permitted to provide exercises for a secondary class; and that’s for any class, whether extension or pleb stuff. The book is elegant, with spare and mathematically coherent explanations, well-chosen examples and really excellent exercises, each section beginning with very routine bread and butter stuff, nicely done, and progressing naturally to very challenging problems.
Enrichment Mathematics has been in the works for many years, with David slowly formalising and polishing the notes he had prepared for the various extension and acceleration subjects that he taught. (The layout and the LateX and the graphics are all David.) Then, Gareth came on board, in particular contributing many deep and very good problems. The result, I believe, is a unique and really excellent text.
Below are a few excerpts from a late draft, and readers can check out Cambridge’s sample chapters and other details, here and here. (22/01/26 – links fixed.)
(30/11/25 As noted by commenter mat, the posted triangle excerpt had a typo (twice) in the proof. Gareth let us know that this typo had later been fixed for the published book(s); the corrected proof is now posted here.)
Looks lovely, thanks. Where can I buy it?
One minor point. I only ever use “evaluate” when asking to find a numerical answer. So I was a little surprised by Example 1 in 5C.
Thanks, tom. The sample links I included also lead to buying on Cambridge’s site.
Interesting comment about “value/evaluate”. That doesn’t bother me, although perhaps it might. (I was a bit more annoyed with “roots” of the equation in the Vieta theorem.)
It’s a first edition, basically home made, so there will be (i.e are) plenty of such judgment calls, and out and out inconsistencies/typos/errors. David and Gareth will definitely take any queries/flags on board for (hopefully) the next edition.
Perhaps “*express* the following as the difference of two squares” would be suitable?
Yes, arguable.
Sounds like it will be a wonderful resource. I just bought the first volume and ordered an inspection copy of the second. Thanks Marty.
I ordered both books directly from Cambridge.
Fantastic news! At last a really high quality textbook. Excellent contribution to the field and thanks very much to all the incredible hard work that must have gone into this. Have ordered a copy of each.
There seems to be a typo in the proof of Theorem 1.
Edit: it occurs twice.
Thanks, Mat. There will be plenty of such typos to find in the books, although, as it happens, not the one you noted. Gareth has told me that this typo was corrected before publishing. I’ll update the post with a corrected version soon.
Pity that these are not available in pdf formats. My shelves are pretty full.
I know the feeling…
In related news, Marty has a new scanner.
Is this designed as some sort of supplementary bank of questions or could a cohort work through these books and go all the way from 7-10 with nothing (or very little) else?
The text is definitely put together as the latter, but I have only had the opportunity to use it in the foremr manner, and it has worked very well. The sample chapters are a good indication of the chapters in general, and you can see the chapter lists.
It seems that the two volumes contain no statistics at all, and probability is not treated until near the end of volume 2.
Why should the book contain any statistics?
A good question. There are many answers. I think you (and many others) will choose answer 1 and move on, but for everyone else:
1. Statistics is not Mathematics. These books are about Mathematics. There are plenty of (really good) books about statistics available for anyone who cares to look (Tom Apostol’s Mathematical Statistics is my personal favourite, but it is long out-of-print).
2. Statistics is part of the Australian and Victorian curricula 7 to 10 and so a book that a school intends to use as the sole resource should probably cover the entire scope of said curriculum, but again, see point 1.
3. If I were offered two textbooks for a non-externally-assessed course and one was a depth book and the other a breadth, I know which one I would choose.
4. Probability, on the other hand is useful, important and perhaps worthy of greater treatment in a book for extending students. Maybe in the second edition(s)…?
What *Extension* can you and would you wish to do in stats?
Regarding the probability, for this book it all naturally belongs in one chapter. Given that, the chapter naturally belongs in Volume 2. One can do a little probability earlier but there is no necessity or any great benefit.
Having said that, the probability chapter is, I think, the clunkiest chapter of the book. It is very difficult to do this stuff well, and I think the probability needed more thought. But, the problems are still generally excellent.
A cynic could say that any proper treatment of statistics is an extension on what is currently available in most 7-10 textbooks. Beyond that, I have a few thoughts, but most of them require solid foundations in algebra and probability.
Regarding probability, I think there is a lot of extension work that can be done, but first and foremost, undoing some of the misunderstandings of existing curriculum. Tree diagrams come to mind, especially the idea that two events for different trials, when given the same letter designation somehow have different probabilities… and then we wonder why VCE Methods students do so poorly on conditional probability questions.
Of course good statistics is good, and an extension of the twaddle done year after year in the standard subjects. But statistics done properly comes later.
I agree with your second paragraph, on probability, but I don’t see how David and Gareth’s book fails to address it.
‘Mathematical Statistics’ – Tom Apostol , is this the exact title? I cannot seem to find any information on this book. Thank you
Anything by Apostol is going to be good, but I would imagine this is a university-level text.
It would be a university-level text but I am unable to source information on a publication by Tom Apostol titled: Mathematical Statistics. Listed below are titles authored by Tom Apostol. Maybe someone could reach out to Red Five and obtain the exact book title and ISBN.
Bibliography
Mathematical Analysis: A Modern Approach to Advanced Calculus, (1957) Addison-Wesley, ISBN 0-201-00288-4
Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9
Modular Functions and Dirichlet Series in Number Theory, (1990) Springer-Verlag, New York. ISBN 0-387-90185-X
Calculus, Volume 1, One-variable calculus, with an introduction to linear algebra, (1967) Wiley, ISBN 0-536-00005-0, ISBN 978-0-471-00005-1
Calculus, Volume 2, Multi-variable calculus and linear algebra with applications to differential equations and probability, (1969) Wiley, ISBN 0-471-00008-6
The Mechanical Universe: Mechanics and Heat, Advanced Edition ISBN 0-521-30432-6 (with Steven C. Frautschi, Richard P. Olenick, and David L. Goodstein)
New Horizons in Geometry ISBN 088385354X (with Mamikon Mnatsakanian)
Good Point. I think RF got either the title or the author wrong.
If I were extending students in stats, I’d basically get them doing project work – very simple – find bad examples out there (that could keep them busy for a very long time!). The other option would be to teach some data visualisation (which you get through stats anyway but would be accessible to Y7-10). Could fit nicely with a course on using Excel for data analysis – using Kaggle for sample data and getting them to do some real world stuff). Kids likely to be very interested and learning would be accessible easily for advanced students. Could do a great deal here – it’s not maths but is relevant and can be tied to maths without too much bother.
Whatever it’s worth, it’s not maths and trying to make it mathsy usually makes it worth less. And, it has nothing to do with a book like David’s and Gareth’s.
As maths teachers, we can’t be total purists and need to take the wins where we can. I begin each year by waving my phone around and talking about how maths is ever more important, especially since data is the new oil (or whatever the latest silly statement is) and of course lots of other relevant stuff like that without maths, we’d all be sitting on the ground without any buildings etc etc. It helps to reinforce in their minds that maths is important. Let’s not define our way out of an important way to relate maths to the real world and the stuff I’m talking about fits better into maths than any other subject. And if they leave with the conclusion after using Excel that no one in their right mind would do data on a CAS, all the better.
The point I was making is that it is entirely appropriate for the book NOT to contain stats as that is best done in other ways for advanced students.
I’m not saying that teachers can, or should, be total purists (although they definitely can, and definitely should, be more purist than they generally are). But I don’t think this stats application stuff, whatever its intrinsic worth, sells mathematics to anyone.
I can’t really see a way to properly present this without needing some sort of linear algebra. From my faint memory of the curriculum, there is some stuff which I think could work as an advanced extension:
– Bayesian analysis, posterior inference with basic loss functions
– Estimaors, sufficiency, the bias-variance tradeoff, and examples of Rao-Blackwell
– Hypothesis testing, Neyman-Pearson, p-values, power and size, including likelihood ratio tests
– Simple linear regressions, a proof that the LSE and MLE coincide
At the Y7-Y10 enrichment level, you’d probably be better off exploring areas less grounded in prerequisite theory.
Exactly. One *might* figure out a little that works without requiring too much prereqs, but it’d be a stretch, and hard, and all for unclear benefit.
The statistics taught in schools is largely based on exploratory data analysis (EDA) which was developed by John Tukey (1915-2000). It deals with *informal* methods for exploring data sets. EDA took off like wildfire; many universities were teaching their students about it; now it is just mainstream. But it has the added advantage is that the ideas are quite accessible even to school students. This is why statistics has become so popular in schools around the world.
In brief, it’s popular because it’s not mathematics but people can pretend that it is.
Are you saying that statistics isn’t maths or that exploratory data analysis isn’t maths? I agree with one of these things and vehemently disagree with the other.
Hi, Bell. I’m honestly not sure whether statistics is mathematics. I’m not even sure about probability. Sorting the foundations of P & S is very mathematical, and then the power of P & S comes through very solid mathematics, but they are nonetheless very strange beasts.
But, I think the question of what statistics is, is a red herring. The important question here is, what type of statistics, if any, should be taught in school, and when?
Statistics, at least in Victorian schools, appears to be of two types:
(i) low level data picturing, with a bit of simple parameters, in lower year levels;
(ii) the sample means and “hypothesis testing” and whatnot in VCE.
Either or both of these can be less or more real-worldly. I’m not quite sure what of this the term EDA is intended to include.
At the school level, both of these are mathematically trivial and thus, for the teaching of mathematics, largely worthless.
I can see some value in (i), coming under a rare, semi-sane use of the term “numeracy”. But there is just not much mathematics without a large effort, and which then is quite distinct from the “numeracy” aspect.
As for (ii), it is just nuts. The underlying mathematics is simply too hard to do properly at the school level, and so the VCE statistics is all fake, just garbage.
What *really* pisses me off with the teaching of “statistics” in school is not the statistics per se but, rather, the pretence that what is being taught is mathematics.
I reckon it might be interesting to have a discussion on the boundaries of what is worthwhile teaching in high school maths around the edges. I think that there could be some areas which are not strictly maths but are worth including at the edges. For example, while I think the ridiculous pseudocode and algorithmic thinking are out of place in VCE maths, at the same time in the late 70s, we learnt Fortran in maths to use on the school computer (yes – I am old – it only took punchcards). There wasn’t computer studies in those days. And then I think given data’s increasing role in the world, it wouldn’t hurt to educate students a bit about it. It’s not maths, but is closely related and maths is probably as good a place as any to bung it in.
Thanks, JJ. See my reply to bell, above.
It is of course worthwhile discussing what is worth teaching at school, and how. I’m largely a purist – quelle surprise – and my emphasis is on teaching kids to think clearly and solidly and deeply on difficult topics. I can appreciate, however, arguments for a more explicitly functional education.
I don’t want to get hung up too much on labels. If it is determined that X is worth teaching then X is worth teaching, and then how how X is categorised and under what banner it is taught are secondary questions. What I am railing against here is teaching X and pretending, by the use of an inappropriate banner, that one is teaching Y.
I simply do not believe that the teaching of “statistics” at school, of whatever type and of whatever value, has much of anything to do with the teaching of mathematics. I really wish people would stop pretending otherwise.
I note that the two volumes seem to make no mention of financial mathematics.
Either you’re talking about things like compound interest and marginal tax rates, which I see no point dedicating time to when they’re basic applications of the elementary algebra taught, or you’re referring to what I know to be ‘financial mathematics’, which is so far removed from the high school curriculum to be worth considering — things like stochastic processes, Ito calculus, and other technical concepts. Apologies if I’ve missed a joke.
I was referring to financial mathematics at school level. Financial mathematics has been part of the curriculum in some form or other since ancient times, and gets a lot of attention throughout high school these days. For extension problems at this level in financial mathematics, see chapter 1 of Kellison, S. G. (2009), “The theory of interest”, McGraw Hill.
Financial mathematics is not a topic; it is an application. You can give an example of an extension problem that you believe would improve David and Gareth’s book.
Jesus Christ, Terry. There is also no mention of the War of Independence. Read your Gardiner.
My books arrived today. They look great! Looking forward to sinking my teeth into ‘em tomorrow.
Hi Marty, I think I found an error with a problem (9A, 7 (b)).
I’m being brave posting this because it’s all at the edge of maths I understand but I can’t get an answer, I can’t graph it, and online calculators give no answers.
Do the authors accept emails about potential errors?
Thanks, Aaron. I’ll look at that particular question tomorrow (unless someone here gets to it first).
There’s certainly going to be errors and I know the authors would be very grateful to know of any. They are aware of this post and I will alert them to your comment (and they will likely see it anyway). So, for the future, you can just flag any possible errors in comments here and you can be sure that they will be checked and, if need be, noted for the future.
Hi Aaron. Yes, that is clearly an error: a stray “3” has appeared in the first term. So the intended equation to solve is,
Solving that equation gives the same answers as given in the text.
I’ve alerted the bosses. Thank you.
Thanks for that. It’s a stellar text. Please let the authors know that as well.
I agree, and done.
These books are gorgeous! I’ve been using them for a few days now to help me put together notes for a course I am teaching next year. The care and restraint that has gone into the text – so good. Section on rational root theorem is particularly beautiful. Everything is superbly presented too.
Well done David and Gareth and thank you for writing these books. Thanks Marty for getting me on to them.
Thank you, on behalf of the authors. Yes, they are beautiful books.
Hi Marty, and happy new year.
I think I may have found another error, unless I am misreading the argument. In the solution at the top of page 98 of Volume 1, we are instructed to subtract (1) from (2). Doing so gives (-9x = -9), which simplifies to (x = 1). That is consistent with the aim of the exercise (showing (0.1111… = 1). However, the text states that this subtraction yields (9x = 9), which appears to have the sign reversed.
Thanks very much, Aaron.
I don’t have the hard copies yet. (I’m waiting for Cambridge to realise all the work I did for David, and thus them, and gift me copies …). Could you please indicate the chapter and section and problem number?
In any case, it sounds like an error, albeit more on the nitpick side. (David and Gareth would still want to know, and thank you.) I’ll check it out once you pinpoint the question, and then I’ll see about setting up an errata list.
No worries. I hesitated to draw your attention to this because it felt nit-picky, but nothing ventured, nothing gained. The item in question is 4H, Example 1, Solution.
Thanks, Aaron. Obviously you were flagging it to be helpful, not as a gotcha, and it is appreciated. Yes, it’s a nitpick, and it’s unlikely to derail anyone for too long, but it is wrong and should be fixed for (hopefully) the next edition.
Anything you see, consider it worth mentioning/querying. One way or another, I’ll make sure soon that there’s an errata list visible to the public.
The links in the article no longer work.
Ugh! I’ll look.
Thanks, 404. Cambridge made a single page for both volumues, which makes sense. Links fixed now.
Hi Marty, I think I have come across another typo.
Volume 1, page 163, Example 2, solution should be (3x-1)(2x-2)(2x+2), they have (2x-1)(2x-2)(2x+2).
Thank you very much, Aaron. It is indeed an error (which originates in the previous line).
I alert David to all such errors. I’ve also ordered David to order Cambridge to set up an errata page. But no one much listens to my orders. (Hence the blog.) If Cambridge doesn’t do it, I’ll do it myself.
Probably of little consequence but in volume 2 on page 180, question 6i is the same as an example question.
And while I’m here, is it even possible to prove that a triangle has 180 degrees? Volume 1 offers a proof. I was looking at some stuff today and apparently that fact is equivalent to the parallel postulate. So is it just a definition that a triangle’s internal angles add up to 180 degrees? Just wondering about this because you go crazy when people “prove” 8^0 = 1, when it’s a definition.
Thank you very much, Myler. I’ve passed on your glitch-spot to David.
I think the postulate thing is fine, although the book could have been a little more explanatory. If you look at the beginning of Section 10B, there are statements about lines and angles, including a statement about “corresponding angles”, followed by a note indicating that “is” (better, “can be”) taken as an axiom; it is equivalent to the parallel postulate.
As it happens, David really likes the axiomatic Euclidean stuff and raves about the school text by Harold Jacobs (especially earlier editions). I am sure his own taste would have been to do the stuff in solid detail. But, and at risk of incurring the wrath of Ed Barbeau or the ghost of Tony Gardiner, the domination of modern mathematics by number means that axiomatic Euclidean geometry is pretty fringe these days.
Thanks, Marty. I’ve come across another mistake. Volume 1 page 43 question 13. “How many kilogram of food…”
Thanks, Myler. I’ve passed it on.
Volume 1, Page 46, question 6. “Connor walks into the same shop … How much does she spend?” Not sure this is a mistake, but I hadn’t heard Connor being used as a girl’s name before. A quick google says it’s very rare. I hope this isn’t annoying if I just comment every time I find a mistake, or if it’s better to hold it until I’ve got several to give at once. It’s a fantastic book by the way.
Absolutely, keep them coming, Myler. David appreciates *all* such corrections, and I pass them all on.
I’ll bet this has been picked up on, but at the start of the percentages chapter, “A percentage is a number expressed a fraction of 100”
No idea. But when I see something, I send it.
Volume 1 page 59 question 17. “There are 20% more boys that girls in a classroom.”
Keep it up and you’ll be coauthor of the next edition.
Volume 1, page 66, question 11. “Alice can paint the bathroom in 3 hours … Betty can paint the Bathroom in 4 hours …”
Thank you. Passed on.
Volume 1, page 71, question 2. “If she had walked at a speed of 5km/h, then she had have arrived 20 minutes earlier.”
Same page, question 5. “She walks for three hours and runs for the two hours.”
Page 72, not sure about this one but it made me double take. Question 10, “I can ride by bike three times as fast …” Should it be ‘my’ bike?
Relayed.
Volume 1 page 81, question 9b “How many ways can 31 can be written as the sum of four different prime numbers?”
Thank you. Relayed. (And don’t you have a life?)
Great way to spend my holidays.
Um, there is soccer and Wimbledon, you know. In any case, I know David is very appreciative.
Does it make a difference that question 17b and c in 4D (page 87) don’t say “exactly”?
I think the “exactly” is reasonably understood, but I’ve passed it on.
Volume 1 page 89 question 5. “What the smallest number of cards …”
And thank you.
Volume 1 page 90, questions 22-27. Inconsistent use of the Oxford comma. Feels like it’s splitting hairs, but I’ll put it here regardless.
No full stop at the end of question 30, same page; and 32 on page 91.
Volume 1 page 106 question 24b: Convert 1234_16 into a hexadecimal.
Thank you.
Noticed the book is a bit loose between writing numbers as numerals vs words. E.g. Page 123, question 10a “Let n be a positive integer. Show that if n is divisible by 3, then n^2 is divisible by three.” Can’t find your post where you talked about the difference….. Just looked for a bit and here it is. https://mathematicalcrap.com/2022/12/29/new-cur-18-to-be-two-or-not-to-be-2/
is switching style in the same sentence ok?
and while i’m on this page, in question 11b, does
mean n + (n/15) or (n^2)/15? I could check the answer and probably find out, and based on previous context it probably means the former, but I’ve never seen a mixed number written with pronumerals.
Thanks, Myler. No, it is not ok. I’ve noted that as well about the book, and point it out when I see it (and I’ve passed on this example).
There are three principles that guide this, I think, and a fourth overarching principle. First, if you’re counting something then you use the name: I have three cats. Second, if you’re using the number as a thing then you use the numeral: 3 is prime. Third, if you’re gonna do arithmetic, then use the numerals, even if they began as counts: if I have 3 cats and I get 2 more cats then I now have 5 cats. Of course this last example is a judgment call, since the arithmetic is trivial. But, word problems easily go from the numbers of people or things to involved arithmetic, and you want the key numbers to be highly visible. That brings the overarching principle: whatever the general rules, first and foremost be clear.
Obviously, the example you raised is plain silly: whatever the argument for the first 3 or three, it applies equally to the second three or 3. And, obviously both should be 3. The book is clumsy in this regard, and I’m happy to pass on whatever you find.
Volume 1, page 127, question 2c and 2k are identical.
Also I edited something into my previous comment, don’t know if you saw it. Thank you for passing these on.
Ah, I didn’t notice: I’ve passed on the extra query, and this one here as well. And thank *you*. I talked to David yesterday (about something else): he is bemused but also very appreciative.
Volume 1, page 144, question 7 says “… expressing your answers …” while the others say “… expressing your answer …”
Volume 1, page 180, question 19 “Her return journey takes 8 second longer.”
Some slightly awkward typesetting on page 176, question 11. The denominator of
is intruding on the exponent of 
.
Volume 1, page 182 question 5 says “find the value of x if …” which I think you’ve said before is poor writing, intead of just “find x if …” or “determine x if…”
“Value” appears a lot, so I don’t know why I have only just noticed it.
Thanks, Myler. All passed on.
I don’t necessarily mind “value” as a style thing, to give more visibility to the x. But it’s a judgment call, and is clearly overused. I think in proofreading I whacked a lot of the “value”s, and David is certainly alert to the redundancy issue.
Volume 1, Question 1, page 193, “Suppose p and q are the roots of quadratic …. ” I think it’s meant to say “… the quadratic…”
Volume 1, page 173, question 6g, should the denominator of the second fraction be
instead of 
? But the answer at the back fits with 
, so I’m not sure.
Huh. That’s a weird one. Maybe intended as a trick question (which i don’t like). Anyway, I’ve passed it on.
Volume 1, page 202, question 1, unless I’m missing something, should the angle be
instead of 110?
No, I think 110º is correct.
Is the 110 referring to the angle under theta? I’m confused.
Oh yeah. The diagram is screwed. The 110 in the diagram should be 70. I think I’ve already flagged this one.
Well the next one over says 85, but it can’t be, so I don’t know if it’s worth changing everything to be exactly to scale (or just changing the numbers). Sort of annoys me when these diagrams have some obvious “this can’t actually be right” issue, but when you sketch stuff out all that matters is the information actually written down: if it says 85, then just take it as 85.
Are they all screwed?
Let me get my protractor
Can’t find a protractor, but just looking at them, I reckon David must have fudged the numbers a bit just to make the tikz simpler. Like, 2b and 2c, they’ve got an angle of 115 thats bigger than an angle of 120. I’ve tried to make these to scale in latex, and it’s a bit of a nightmare for some of them (I’m just a novice though), just to get all the lines to meet up with these exact angles.
Thanks, Myler. I know that in general David cared about such things, so I’m not sure if this one’s a glitch, or that he just didn’t bother this time. (Obviously the 110-70 one was a separate stuff-up.)
Was just flicking through volume 2 just now and came across another error, page 221 question 11, the straight hand is listed as 2 4 4 5 6
Thank you.
Volume 1 page 211, “The icosahedron is has 20 equilateral triangle faces.”
Thank you.
Volume 1, page 262, question 9 and 10 has ‘half cylinders’ and ‘quarter cylinders’ without a hyphen, but question 16 does.
Page 266, volume 1, question 12, “A rectangle has dimension …”
Thank you, as always. I will pass them on.
Volume 1, page 289, question 5 ends with a colon, question 6 ends with a full stop.
Geez, you’ll drive him nuts. (and thank you.)
There’s probably heaps that aren’t consistent, just flicked over to 290 and questions 10 and 11 also end with a full stop, but probably of zero consequence, really.
Yes, of course. And don’t stress it. Substantive errors are of course more important. But you may as well note it if you have the energy.