nculwell.github.io

Grade school math

High school math subjects

The standard US textbooks used by schools are pretty widely derided, although I do give links to some of them near the bottom of this document. I’ve focused on other alternatives here. I begin with Pre-Algebra (basically a one-year overview of all mathematics that is expected to be mastered before beginning algebra) because, although it’s not usually considered high school level, it is the first course that has many adult-style textbooks dedicated to it.

Series that cover all subjects

See below for more information about these.

Pre-algebra

Introductory algebra

Continuing algebra and trigonometry

Encompassing the portion of the US curriculum referred to by the terms Algebra 2, Trigonometry and Precalculus.

Geometry

Might work

Sources for texts and resources

UCSMP

http://ucsmp.uchicago.edu/about/overview/

CME

Connected Mathematics (CMP)

Seems to be the most widely used constructivist mathematics series for middle schools.

Constructivist mathematics approaches math by asking the students to solve problems that allow them to “construct” mathematical concepts for themselves. Research generally indicates that this is a good way to do things, because thinking about things is what makes you learn them (as opposed to repeating what you’re told to do, which is far less effective and thus requires more hours of studying to achieve the same results). The big caveat is that the constructivist approach requires a competent teacher to supervise learning and make sure students don’t get off track: the teacher’s job largely becomes one of recognizing and correcting misconceptions.

These features make constructivist books inappropriate as a primary resource for self study (because you have no teacher to correct your misconceptions), but they might still be useful in combination with another text as a source of problems.

For a look at the thinking behind constructivist mathematics instruction, Concept-rich Mathematics Instruction (ISBN 978-1416603597, Amazon) is a useful overview.

OpenStax

The OpenStax project offers free, high-quality textbooks for a variety of subjects. Here’s the link to the math section:

https://openstax.org/subjects/math

Print copies have been available on Amazon but many are on longer in print and there aren’t a lot of used copies floating around.

Currently, they offer textbooks for Pre-algebra, College Algebra, Algebra-Trigonometry, Precalculus and Calculus I-III. Notably, there is no Geometry text.

As is typical with such sequences, College Algebra, Algebra and Trigonometry and Precalculus are essentially the same book with a slightly different beginning and ending point for each. Thus, there’s no point in owning all of them.

As far as I can tell, College Algebra contains a strict subset of the chapters of Algebra and Trigonometry, and Precalculus is the same as Algebra-Trigonometry except that it omits the first two chapters (“Prerequisites” and “Equations and Inequalities”), it has slightly different coverage of trigonometry (one section removed that I noticed) and it adds a new chapter called “Introduction to Calculus”.

For self-study purposes, Algebra and Trigonometry seems to be the most desirable to have, since the material in the “Introduction to Calculus” chapter from Precalculus can be found in any calculus textbook.

Note that these books have two PDF versions: the “OP” version that is linked to from the main math books page, and the “Print Edition” version that can be found in a link on the details page. I’m not sure what the difference is, but I’ve noticed that the OP version is a larger file, whereas the Print Edition contains the student answer key (solutions for odd-numbered exercises).

Since the OpenStax website is annoying to use, I provide some direct links here, although these might be out-of-date in the future:

Sheldon Axler

Sheldon Axler is a university professor who has written a number of math textbooks for undergraduates. These offer a no-nonsense presentation for an adult audience. Axler doesn’t try to entertain you (much), but he does offer a lot of helpful tips about how to think about math as you learn it. Reading his books gives me the sense that he’s taught a lot of undergraduates in his time and he has a pretty good idea of what he can and can’t expect them to figure out on their own. He is also careful to give you an idea about why he’s telling you something, instead of taking the “it will make sense when you get there” attitude that many math professors adopt in writing their books.

These two books overlap quite a bit; I’m not sure if there is any substantive difference between them besides the sequencing of topics. His other book at this level, College Algebra, is superfluous because it is entirely contained within Algebra and Trigonometry (and it’s only slightly cheaper).

I suggest buying a decent-quality copy of whichever edition is cheapest. As of this writing, this one is the cheapest option to buy used:

Note that the covers of these paperback books are not especially strong and they tend to curl up.

The Art of Problem Solving

I gather this series began as two books by Richard Rusczyk, The Art of Problem Solving volumes 1 and 2, which cover the pre-algebra curriculum and are aimed at Mathcounts contestants. However, the series has been expanded to include textbooks from pre-algebra through all of high school mathematics. People rave about them, so they’re probably pretty good. I’ve only looked at Introduction to Algebra, but I found it interesting and thorough in its explanations.

The main high school curriculum subjects covered here (Prealgebra, Introduction to Algebra, Intermediate Algebra, Introduction to Geometry, Precalculus), aren’t usually given a challenging treatment in the other available books, so these make very attractive alternatives.

I don’t know if there’s as much need for the calculus book, as students already have lots of different options for that (as detailed on my other books page), but what I like about it judging from the samples available online is that it gives nice, detailed walkthroughs of more complex problems than you see in the worked examples of a typical textbook.

Since there is often a lot of overlap between courses on “Algebra II” and “Precalculus”, I compared the contents of Intermediate Algebra and Precalculus. They are mostly different, with Intermediate Algebra focusing polynomials, other functions, and series, whereas Precalculus is almost entirely devoted to trigonometry and an introduction to linear algebra.

AoPS also has a set of books called Beast Academy which is planned to cover grades 2-5, although currently only grades 3 and 4 are complete. I have no idea if those are any good.

These can be found here: https://www.artofproblemsolving.com/

The Art of Problem Solving website also offers online courses that go with these books.

The Russian texts

Gelfand

Israel M. Gelfand (Гельфанд) wrote a series of books to teach fundamental mathematics to grade school students as part of a correspondence program. The resulting series of books covers from algebra through precalculus:

Rutgers still runs the Extended Gelfand Correspondence Program in Mathematics (http://www.egcpm.com/), which allows you to have your exercises graded. You could just do them by yourself of course, but no answers are provided. The program accepts students of ages 13-17.

Gelfand’s books seem like a pretty interesting way to introduce high-school material to motivated students. They aren’t as dry as most American textbooks, though they lack color and gloss. However, I’m not sure if they really go into enough depth in some areas, so you might need to supplement them with a book like Axler’s.

Kiselev

Andrei P. Kiselev (Киселёв, also written Kiselyov) wrote a book called Geometry, which was the standard text in Russia for many decades (and was revised many times during that period). It is available in English in two volumes, Planimetry and Stereometry, which were translated and adapted for the USA by Alexander Givental.

Volume I: Planimetry covers lines, circles, similarity, regular polygons and areas. Volume II: Stereometry covers figures in three dimensions: lines and planes, polyhedra and round solids; then it wraps up with an introduction to vectors and then an overview of the history of geometry and a brief introduction to non-Euclidean geometry. Volume II in particular goes well beyond what high school students in the US are expected to learn from a geometry course, and it gets very dense.

Others

Yakolev, High School Mathematics. (Worldcat) Dorofeev, Elementary Mathematics Vygodsky, Mathematical Handbook: Elementary Mathematics Govorov, Problems in Mathematics: with Hints and Solutions Bukhovtsev, Problems In Elementary Physics

Harold Jacobs

Jacobs books combine entertaining commentary and careful exposition. He aims to convince people who “don’t like math” that really it’s an interesting subject. These books are copiously illustrated, with lots of well-chosen images that help illustrate and reinforce the mathematical content.

Elementary Algebra and Geometry are fairly standard secondary school curricula for those subjects. A Human Endeavor is a “topics” type book that would be appropriate for anyone at the pre-algebra level on up who isn’t already familiar with the topics. The topics it presents aren’t particularly advanced, but they are important ones, and many of them (e.g. statistics, topology) are not covered in a standard high school curriculum.

A number of the topics covered in A Human Endeavor (e.g. symmetry) are ones that are typically introduced in the pre-algebra curriculum, then ignored for so long that students wonder why they were ever brought up until they re-emerge in an abstract algebra course a decade later for any students whose studies make it that far. But that’s an entirely different discussion.

Barbeau

A quote from the introduction of Barbeau’s Polynomials:

This book is not a textbook. Nor is its topic being particularly recommended for inclusion, indiscriminately, into the school curriculum. However, it should convey some of the breadth and depth found close to the traditional school and college curricula, and encourage the reader not only to follow up on some of the historical and technical references, but to pull out pen and paper to tackle some problems of special interest. Some of the mathematics will be difficult, but I believe that it will all be accessible.

The intended audience consists of students at both high school and college who wish to go beyond the usual curriculum, as well as teachers who wish to broaden their mathematical experience and discover possible material for use with their regular or enriched students. In particular, I am concerned about two groups of students. There are those who romp through the school curriculum in mathematics while they have yet to complete other subjects. A standard response to this situation is to accelerate them, either into calculus or into college prematurely.

While this is undoubtedly appropriate for some, my experience is that very often such acceleration is counterproductive and leads to an unsettled academic experience. Then there are those who get caught up in contest activity. It is now possible to spend much of the spring semester preparing for and writing contests, and this may have some value. However, there are some for whom contests are not congenial and others who emphasize the short-term goal of solving problems and winning contests at the expense of proper mathematical growth.

What seems to be needed is a mathematical enrichment which starts with school mathematics, broadens it and yet is sufficiently down-to-earth that the student can explore it in an elementary way with pencil and paper or calculator.

This book covers algebra, but it covers it much more deeply than typical high school courses. It would be appropriate, as Barbeau suggests, as enrichment reading for a high school student who is too advanced for the usual curriculum but has not been accelerated through it. It would help to remedy one of the failings of the typical high school curriculum, which is that algebra is typically not taught very thoroughly, then when it comes time to learn calculus, students’ algebra skills are weak and their calculus skills suffer for it.

Many of the problems are taken from various math contests.

As an example of what you get in this book, it includes methods for solving cubic equations, which, while not really hard, are more involved than methods for quadratic equations and aren’t normally taught to high school students. In addition to being useful (cubic equations do come up and not knowing how to solve one is embarrassing), I find these techniques to be very instructive as examples of how you could go about solving problems.

Barbeau has another book, Pell’s Equation, that is similar in spirit, and (unsurprisingly) explores Pell’s equation.

Old books in the public domain

Leonhard Euler wrote an introduction to algebra called (in English) Elements of Algebra:

These are apparently still really popular in India. Since they are in the public domain, you can find them available online for free. PDFs are available on archive.org, probably elsewhere as well. They are still in print, though not always as cheap as you’d hope given that the copyrights have expired. Be warned that since they’re old, the writing style isn’t what you’d expect from a contemporary book.

Elementary Algebra for Schools is probably unnecessary these days. It covers roughly the equivalent of Algebra I and maybe Algebra II in US high schools, but modern textbooks do a better job with the material.

On the other hand, Higher Algebra is still a very useful book because much of the material that it covers is ignored by modern curricula. Some of what it covers is similar to Barbeau’s Polynomials (e.g. Hall and Knight cover Cardano’s method for solving cubic equations). Other topics in here, like continued fractions, are rarely discussed in modern textbooks.

The famous Indian mathematician Ramanujan read Hall and Knight’s Higher Algebra and cited it in his notebooks, and this association may account for its continuing fame in India, as Ramanujan is somewhat of a national hero.

Loney’s Plane Trigonometry has a lot of information, but I’m not sure how useful it is anymore as a textbook. This may be just a personal bias of mine, but I don’t find trigonometry interesting enough to dedicate this much time and energy to all by itself. A book like Gelfand’s or Axler’s gives you the basics you need, and the rest is generally covered elsewhere (e.g. in calculus and analysis) when it’s needed. If you really want to know a lot about trig, though, Loney might be what you want.

Hall and Knight and Loney are available from Indian publishers for fairly cheap. Try searching AbeBooks to buy them if you’re in the USA. I can’t vouch for the quality of those cheap editions, but I’ve heard they’re better than you might expect (unlike the terrible Indian editions published by companies like Wiley).

Chrystal’s Introduction to Algebra and Algebra: an Elementary Text-book form an elementary and advanced pair similar to Hall and Knight’s Elementary Algebra for Schools and Higher Algebra, and similarly, the elementary one isn’t of much interest. Chrystal’s advanced text, Algebra: an Elementary Text-book (misleadingly titled), begins with the basic principles of algebra but it is not really a beginner’s book, and it progresses quickly to a great variety of topics that are rarely taught these days, or are only taught at higher levels of algebra and analysis. Chrystal’s books offer a lot of challenging problems.

I don’t know if anyone still uses these other old books or not, but some people like them.

Newer classic texts, not yet in the public domain

Schaum’s Outlines

Schaum’s has several of their “outlines” which are appropriate for this level. Schaum’s outlines are full of worked problems, which you can use to study or to practice. (The answers are right there, though, so you have to cover them up if you want to use them for practice.) The outlines also provide some explanation of the topics, so in theory you could even use them as textbooks, though that isn’t really what they’re designed for.

Others

Free

More-or-less standard curriculum

Enrichment

Problem books, problem solving, math competition prep

AMS Mathematical World series

From the AMS website: “This accessible series brings the beauty and wonder of mathematics to the advanced high school student, the mathematics teacher, the scientist or engineer, and the lay reader with a strong interest in mathematics. Mathematical World features well-written, challenging expository works that illustrate the fascination and usefulness of mathematics.”

Here’s a link to the whole series on the AMS website: http://bookstore.ams.org/MAWRLD

Some titles that have caught my attention:

Cambridge reading list (pre-university)

http://www.maths.cam.ac.uk/undergrad/admissions/readinglist.pdf

Sequences from major publishers

I don’t know if these are good, but they are affordable.