- #1

Howers

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This thread is intended to give high school students the necessary information to cope with university level mathematics. I find the “who wants to be a mathematician” thread too convoluted to be of use when looking for books, and that it serves college students more than k-12. Nonetheless, it is a good read and contains a wealth of information for those with time and patience. I find a lot of people asking for high school resources, so hopefully this thread will serve as an oasis of sorts. Most importantly, I hope this advice exposes students to the beauty of math which has been all too absent in the modern curriculum. May it bring hours of joy and discovery.

Before I begin, I will first mention a little about mathematics education and how it has changed over the last few years. Traditionally, math has mostly been taught as a set of disjoint algorithms. Students were taught to memorize a bunch of procedures and to apply them to certain problems. It was criticized on the basis that students had weak conceptual understanding of material and relied heavily on rote memory. Traditional math was replaced by “new math” for a brief period in the 60s, partly because US was falling behind with Russia in the sciences. Books were more proof based and followed an axiomatic approach. Most modern math professors raved the new approach, and books from that era are considered ‘classics’. This new approach was quickly abandoned however, due to the fact that theory often obscured the main point and students had weak intuition of the material they knew. Also, it was deemed highly difficult for all but the ablest students. It was replaced by ‘reform mathematics’ in the 1980s, which sought out to connect mathematics to the real world and focus on applications and problem solving. This is largely what we are settled with today, mixed with the traditional approach. It is criticized as being dumbed-down math, and is part of the reason students experience difficulty transitioning to university mathematics.

In my opinion, I think students need both “new math” and “reform math”. No matter what math professors will want you to believe, mathematics has its roots in applications and solving problems; they create new branches of math and provide motivation for most subjects. Despite what engineers and scientists will tell you, a rigorous approach to math is absolutely essential as intuition is often wrong and limited; theory is more than “useless math” or an “art form”, it validates the tools of science and provides insight that intuition alone never could. Conceptual understanding is rarely needed by non-mathematicians, although at high school level I think everyone should have a strong background in elementary mathematics. Most scientists and engineers will succeed with reform math, as it was catered to their needs.

In what follows, the books are listed in the correct order of progression. In each grade, two subjects should be tackled simultaneously instead of one. Ie. Work through algebra and geometry at the same time, instead of first doing all of algebra and then all of geometry. Books are designated by two categories: skill and theory. Skill consists of books that are largely algorithm based, where you apply skills and techniques to solve problems; they are your cookbooks with recipes on how to solve questions. The material is often presented with a geometric/intuitive approach, and is of most use to the sciences and applied mathematics. This is the approach you’re used to in high school, so you can ignore these sections if you’ve been to class. Theory will give you the nitty gritty details of the contents, with a rigorous backbone and an abstract approach. These are your slick, black and white, no colors/photos booklets. You will be taught to discover mathematical methods on your own, and get practice at proofs. The questions will usually be highly difficult, requiring a lot of patience and determination. Solutions are not usually provided; you will have to gain confidence and get them yourself. These books will mostly be useless and annoying to most scientists, and are geared more towards mathematicians or those who want to perfect their gears. Snobbish mathematicians will tell you this is the only way to learn true math. You can ignore them if you’re in physics, but if you are taking pure math in university you had better heed their word. Lastly a few words of advice: do as many problems as time permits. Do not get obsessive compulsive about not getting every single question in a book. A good effort on half the questions is much better than rushing through all with minimal effort. More is merrier, but don’t let one question bog you down too long. A good rule is if you can’t solve a question in one hour, then move on. If a section is too difficult at first, take a break from it and return later. Do a section every two days, as this will aid in remembering the material. Where exercises occur is when a section ends, for those books that are not numbered by sections. Different people have different abilities, so that some may advance faster or slower relative to others their age. Background and interests will also affect how you progress. Generally, try to spend at least 10 hours on math each week. To a secondary school student, this list can take anywhere from 2-4 years to finish. College students brushing up can probably complete this in less than 6 months. Everybody finds math difficult at first. With practice and patience, it gets easier with time.

Addition, subtraction, multiplication, division, ratio, proportion, area, volume, decimals.

You should know how to multiply 2 digit by 2 digit numbers in your head. Learn to divide and do percents/decimals as well. Mostly memorization here, practice about 5 of these a day. (Ie. Do 4/7 and 67x92 in your head). If these take longer than 1min each… you need practice!

See Singapore math books for this. (ie. https://www.amazon.com/dp/9810184948/?tag=pfamazon01-20)

Arithmetic laws (distribute law, etc), variables, equations, number line, proportion, graphs, y=mx+b, substitution/elimination, exponents and their laws, quadratics, polynomials, fractions, division, basic number sequences.

https://www.amazon.com/dp/0716710471/?tag=pfamazon01-20

https://www.amazon.com/dp/0817636773/?tag=pfamazon01-20

Introductory proofs, from lines to planes, triangles, circles, quadrilaterals, platonic solids, constructions.

http://www.amazon.com/dp/071671745X/?tag=pfamazon01-20

An alternative, especially if you can't find an older edition of Jacobs, is the following:

https://www.amazon.com/dp/0977985202/?tag=pfamazon01-20

https://www.amazon.com/dp/0977985210/?tag=pfamazon01-20

Functions, graphs, analytic geometry, logarithms, trigonometry, conic sections, inequalities, sequences, series, vectors, matrices, dot and cross products (use Stewart in calculus as a source for more on vectors and 3d space)

https://www.amazon.com/dp/0132256886/?tag=pfamazon01-20

https://www.amazon.com/dp/0817635327/?tag=pfamazon01-20

https://www.amazon.com/dp/0817635335/?tag=pfamazon01-20

https://www.amazon.com/dp/0817641610/?tag=pfamazon01-20

FREE: http://www.mathlogarithms.com/

https://www.amazon.com/dp/0883856034/?tag=pfamazon01-20

Triangles, right angle triangles, trig ratios, unit circle, ptolemy’s theorem, pythagoras’theorem, trigonometric functions, graphs, inverse trig functions.

https://www.amazon.com/dp/0817639144/?tag=pfamazon01-20

Symbolic logic, truth tables, negation, conditional statement, equivalent statements, quantifiers, sets, unions, intersections.

https://www.amazon.com/dp/0471406473/?tag=pfamazon01-20

https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

Combinations, permutations, binomial theorem, induction, number theory, primes.

https://www.amazon.com/dp/0883856158/?tag=pfamazon01-20

https://www.amazon.com/dp/0883856018/?tag=pfamazon01-20

Limits, derivatives, integrals, taylors formula, analytic geometry. Partial derivatives, multiple integrals, vector calculus, vector analysis, change of variables & coordinate systems.

https://www.amazon.com/dp/053439339X/?tag=pfamazon01-20

Your first brush of calculus should be with a book like Stewart. This is how calculus was first conceived by Newton and Leibniz. Modern books will rob you of that experience, which I think is wrong. It’s best to experience calculus as infitesimals at first, even if it means upgrading to epsilon-delta formalism later. The modern approach is too slow, and you will need this math in all your science courses from freshman year. There will be plenty of time for you at university to develop the theory of calculus, where they won’t cover this stuff. I don’t recommend Spivak, Courant, or Apostol as first exposure… save them for undergrad.

Transformations, isoperimetric theorems, modern circles and triangles.

https://www.amazon.com/dp/0883856190/?tag=pfamazon01-20

https://www.amazon.com/dp/0883856042/?tag=pfamazon01-20

https://www.amazon.com/dp/0883856085/?tag=pfamazon01-20

These books don’t really belong to a curriculum, but are good to know if you intend to compete in Olympiads or want to have a general knowledge of math. Most should be accessible once you finish Grade 11 section.

Compete in Olympiads! These problem books are excellent preparation and sharpen your problem solving skills. Once all of the above is mastered, test your knowledge with these.

https://www.amazon.com/dp/0387982191/?tag=pfamazon01-20

https://www.amazon.com/dp/052154677X/?tag=pfamazon01-20

https://www.amazon.com/dp/0387242996/?tag=pfamazon01-20

https://www.amazon.com/dp/0486277097/?tag=pfamazon01-20

https://www.amazon.com/dp/0486284247/?tag=pfamazon01-20

https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

Classical books by originators. Not really recommended unless you have the time. Modern approach is a lot better.

https://www.amazon.com/dp/1888009195/?tag=pfamazon01-20

https://www.amazon.com/dp/0486420841/?tag=pfamazon01-20

https://www.amazon.com/dp/0486600688/?tag=pfamazon01-20

https://www.amazon.com/dp/0520088174/?tag=pfamazon01-20

https://www.amazon.com/dp/0300094736/?tag=pfamazon01-20

Curriculum similar to mine, but you need to apply (and pay) to use resources. I much prefer the books I listed to theirs.

http://epgy.stanford.edu/courses/math/secd.html

A thread on these forums by a professional mathematician. Gives solid advice on how to cope with undergraduate mathematics, and more or less provides a scattered list of university level books.

https://www.physicsforums.com/showthread.php?t=122924

A list of standard undergraduate books for pure mathematics. This is mostly university level stuff, but you can look into Stage 1 for some free resources.

http://hk.mathphy.googlepages.com/puremath.htm#1calculus

Before I begin, I will first mention a little about mathematics education and how it has changed over the last few years. Traditionally, math has mostly been taught as a set of disjoint algorithms. Students were taught to memorize a bunch of procedures and to apply them to certain problems. It was criticized on the basis that students had weak conceptual understanding of material and relied heavily on rote memory. Traditional math was replaced by “new math” for a brief period in the 60s, partly because US was falling behind with Russia in the sciences. Books were more proof based and followed an axiomatic approach. Most modern math professors raved the new approach, and books from that era are considered ‘classics’. This new approach was quickly abandoned however, due to the fact that theory often obscured the main point and students had weak intuition of the material they knew. Also, it was deemed highly difficult for all but the ablest students. It was replaced by ‘reform mathematics’ in the 1980s, which sought out to connect mathematics to the real world and focus on applications and problem solving. This is largely what we are settled with today, mixed with the traditional approach. It is criticized as being dumbed-down math, and is part of the reason students experience difficulty transitioning to university mathematics.

In my opinion, I think students need both “new math” and “reform math”. No matter what math professors will want you to believe, mathematics has its roots in applications and solving problems; they create new branches of math and provide motivation for most subjects. Despite what engineers and scientists will tell you, a rigorous approach to math is absolutely essential as intuition is often wrong and limited; theory is more than “useless math” or an “art form”, it validates the tools of science and provides insight that intuition alone never could. Conceptual understanding is rarely needed by non-mathematicians, although at high school level I think everyone should have a strong background in elementary mathematics. Most scientists and engineers will succeed with reform math, as it was catered to their needs.

In what follows, the books are listed in the correct order of progression. In each grade, two subjects should be tackled simultaneously instead of one. Ie. Work through algebra and geometry at the same time, instead of first doing all of algebra and then all of geometry. Books are designated by two categories: skill and theory. Skill consists of books that are largely algorithm based, where you apply skills and techniques to solve problems; they are your cookbooks with recipes on how to solve questions. The material is often presented with a geometric/intuitive approach, and is of most use to the sciences and applied mathematics. This is the approach you’re used to in high school, so you can ignore these sections if you’ve been to class. Theory will give you the nitty gritty details of the contents, with a rigorous backbone and an abstract approach. These are your slick, black and white, no colors/photos booklets. You will be taught to discover mathematical methods on your own, and get practice at proofs. The questions will usually be highly difficult, requiring a lot of patience and determination. Solutions are not usually provided; you will have to gain confidence and get them yourself. These books will mostly be useless and annoying to most scientists, and are geared more towards mathematicians or those who want to perfect their gears. Snobbish mathematicians will tell you this is the only way to learn true math. You can ignore them if you’re in physics, but if you are taking pure math in university you had better heed their word. Lastly a few words of advice: do as many problems as time permits. Do not get obsessive compulsive about not getting every single question in a book. A good effort on half the questions is much better than rushing through all with minimal effort. More is merrier, but don’t let one question bog you down too long. A good rule is if you can’t solve a question in one hour, then move on. If a section is too difficult at first, take a break from it and return later. Do a section every two days, as this will aid in remembering the material. Where exercises occur is when a section ends, for those books that are not numbered by sections. Different people have different abilities, so that some may advance faster or slower relative to others their age. Background and interests will also affect how you progress. Generally, try to spend at least 10 hours on math each week. To a secondary school student, this list can take anywhere from 2-4 years to finish. College students brushing up can probably complete this in less than 6 months. Everybody finds math difficult at first. With practice and patience, it gets easier with time.

__K-8__**Arithmetic**Addition, subtraction, multiplication, division, ratio, proportion, area, volume, decimals.

You should know how to multiply 2 digit by 2 digit numbers in your head. Learn to divide and do percents/decimals as well. Mostly memorization here, practice about 5 of these a day. (Ie. Do 4/7 and 67x92 in your head). If these take longer than 1min each… you need practice!

See Singapore math books for this. (ie. https://www.amazon.com/dp/9810184948/?tag=pfamazon01-20)

__Grades 9-10__**Algebra I**Arithmetic laws (distribute law, etc), variables, equations, number line, proportion, graphs, y=mx+b, substitution/elimination, exponents and their laws, quadratics, polynomials, fractions, division, basic number sequences.

__Skill:__*Elementary Algebra, Jacobs*https://www.amazon.com/dp/0716710471/?tag=pfamazon01-20

Theory:Theory:

*Algebra, Gelfand*https://www.amazon.com/dp/0817636773/?tag=pfamazon01-20

Geometry I (aka Euclidean/plane geometry)Geometry I (aka Euclidean/plane geometry)

Introductory proofs, from lines to planes, triangles, circles, quadrilaterals, platonic solids, constructions.

*Geometry, Jacobs*(1st or 2nd edition = theory, later = dumbed down )http://www.amazon.com/dp/071671745X/?tag=pfamazon01-20

An alternative, especially if you can't find an older edition of Jacobs, is the following:

*Kiselev's Geometry, Kiselev*https://www.amazon.com/dp/0977985202/?tag=pfamazon01-20

https://www.amazon.com/dp/0977985210/?tag=pfamazon01-20

__Grade 11__**Algebra II (aka intermediate algebra + precalculus)**Functions, graphs, analytic geometry, logarithms, trigonometry, conic sections, inequalities, sequences, series, vectors, matrices, dot and cross products (use Stewart in calculus as a source for more on vectors and 3d space)

__Skill:__*Precalculus, Sullivan*https://www.amazon.com/dp/0132256886/?tag=pfamazon01-20

__Theory:__*Functions and Graphs, Gelfand*https://www.amazon.com/dp/0817635327/?tag=pfamazon01-20

*(This book is optional; you will need trigonometry for it.)*

The Method of Coordinates, GelfandThe Method of Coordinates, Gelfand

https://www.amazon.com/dp/0817635335/?tag=pfamazon01-20

*Lines and Curves, Gutenmacher*https://www.amazon.com/dp/0817641610/?tag=pfamazon01-20

*Logarithms,*FREE: http://www.mathlogarithms.com/

*Introduction to Inequalities, Bellman*https://www.amazon.com/dp/0883856034/?tag=pfamazon01-20

Trigonometry (aka Geometry II)Trigonometry (aka Geometry II)

Triangles, right angle triangles, trig ratios, unit circle, ptolemy’s theorem, pythagoras’theorem, trigonometric functions, graphs, inverse trig functions.

__Skill:__included in algebra II curriculum, in Sullivan’s Precalculus book.__Theory:__*Trigonometry, Gelfand*https://www.amazon.com/dp/0817639144/?tag=pfamazon01-20

__Grade 12__**Proofs & Logic**Symbolic logic, truth tables, negation, conditional statement, equivalent statements, quantifiers, sets, unions, intersections.

__Skill:__*How to Read and do Proofs, Solow*https://www.amazon.com/dp/0471406473/?tag=pfamazon01-20

__Theory:__*How to Prove it, Velleman*https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

**Discreet Math**Combinations, permutations, binomial theorem, induction, number theory, primes.

*Mathematics of Choice, Niven*https://www.amazon.com/dp/0883856158/?tag=pfamazon01-20

*Numbers: Rational and Irrational, Niven*https://www.amazon.com/dp/0883856018/?tag=pfamazon01-20

**Calculus**Limits, derivatives, integrals, taylors formula, analytic geometry. Partial derivatives, multiple integrals, vector calculus, vector analysis, change of variables & coordinate systems.

__Skill:__*Stewart, Calculus*(ANY edition will do... older are better & cheaper)https://www.amazon.com/dp/053439339X/?tag=pfamazon01-20

Your first brush of calculus should be with a book like Stewart. This is how calculus was first conceived by Newton and Leibniz. Modern books will rob you of that experience, which I think is wrong. It’s best to experience calculus as infitesimals at first, even if it means upgrading to epsilon-delta formalism later. The modern approach is too slow, and you will need this math in all your science courses from freshman year. There will be plenty of time for you at university to develop the theory of calculus, where they won’t cover this stuff. I don’t recommend Spivak, Courant, or Apostol as first exposure… save them for undergrad.

**Geometry III (optional)**Transformations, isoperimetric theorems, modern circles and triangles.

*Geometry Revisited, Coxeter*https://www.amazon.com/dp/0883856190/?tag=pfamazon01-20

*Geometric Inequalities, Kazarinoff*https://www.amazon.com/dp/0883856042/?tag=pfamazon01-20

*Geometric Transformation Series, Yaglom*https://www.amazon.com/dp/0883856085/?tag=pfamazon01-20

__Beyond__These books don’t really belong to a curriculum, but are good to know if you intend to compete in Olympiads or want to have a general knowledge of math. Most should be accessible once you finish Grade 11 section.

**Problem Books**Compete in Olympiads! These problem books are excellent preparation and sharpen your problem solving skills. Once all of the above is mastered, test your knowledge with these.

*Problem Solving, Engel*https://www.amazon.com/dp/0387982191/?tag=pfamazon01-20

Cauchy-Schwartz Inequalities, SteeleCauchy-Schwartz Inequalities, Steele

https://www.amazon.com/dp/052154677X/?tag=pfamazon01-20

IMO Compedium, DjukicIMO Compedium, Djukic

https://www.amazon.com/dp/0387242996/?tag=pfamazon01-20

*USSR Olympiad Book, Shkalrsky*https://www.amazon.com/dp/0486277097/?tag=pfamazon01-20

General Audience

General Audience

*Concepts of Modern Math, Stewart*https://www.amazon.com/dp/0486284247/?tag=pfamazon01-20

What is Mathematics?, CourantWhat is Mathematics?, Courant

https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

**Originals**Classical books by originators. Not really recommended unless you have the time. Modern approach is a lot better.

*Elements, Euclid*https://www.amazon.com/dp/1888009195/?tag=pfamazon01-20

*Works, Archimedes*https://www.amazon.com/dp/0486420841/?tag=pfamazon01-20

*Geometry, Descartes*https://www.amazon.com/dp/0486600688/?tag=pfamazon01-20

*Principia, Newton*https://www.amazon.com/dp/0520088174/?tag=pfamazon01-20

Disquisitiones Arithmeticae, GaussDisquisitiones Arithmeticae, Gauss

https://www.amazon.com/dp/0300094736/?tag=pfamazon01-20

OTHER RESOURCESOTHER RESOURCES

*Stanford’s Education for Gifted,*Curriculum similar to mine, but you need to apply (and pay) to use resources. I much prefer the books I listed to theirs.

http://epgy.stanford.edu/courses/math/secd.html

*Who Wants to be a Mathematician?*A thread on these forums by a professional mathematician. Gives solid advice on how to cope with undergraduate mathematics, and more or less provides a scattered list of university level books.

https://www.physicsforums.com/showthread.php?t=122924

How to Become a Pure MathematicianHow to Become a Pure Mathematician

A list of standard undergraduate books for pure mathematics. This is mostly university level stuff, but you can look into Stage 1 for some free resources.

http://hk.mathphy.googlepages.com/puremath.htm#1calculus

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