Resources

Textbooks, Technical Manuals, Websites and Tools on Anything Higher Mathematics.

Criteria for Inclusion (Books)

The web is full of materials in higher mathematics, but finding high-quality materials of the right level proves to be a much tougher task. Over the years of reading textbooks and other educational materials, We’ve come to discover that great textbooks tend to share the following characteristics:

 

  • Compactness: As a general rule of thumb, people tend to successfully make through a book if it has around 300 pages or less. Remember, we are not talking about novels or reference books here — learners get cognitively exhausted from attempting to absorb many concepts at the same time.
  • Concise, Personal Style of Writing: Readers tend to be more engaged when they feel the authors are talking to them, and learners prefer having materials delivered to them in simple words, in few paragraphs.
  • Pedagogical Minimality: While it doesn’t always show, experienced authors tend to take great care in choosing words and avoiding extraneous concepts. They would present the proofs in intuitive and easily reachable ways and leave the tiny details and future explorations to the readers. Most often than not, instead of feeding the readers with complete infos, these authors merely assist the readers in discovering mathematical insights on their own.

 

 

Tips on Reading Educational Materials

While not all textbooks/manuals are of the same quality, every one of them is limited by its presentation of materials (which is linear by design). Consequently, one is better off reading several books covering similar topics at the same time, as this allows one to have access the same concepts from different perspectives.

Simply put, we usually don’t nail a subject by reading one single textbook.

Also, don’t skip the index! It is a powerful tool for gauging and reviewing what we have learned. This is especially true when the book does not contain any other reviewing devices such as keyword sections and summaries. Indeed, leveraging the index to elaborate on what you’ve learned throughout the book can sometimes mean the difference between a 50% retention rate, and 90% retention rate.

Online Math Learning Platforms

Specialized, in-depth learning platforms for university/college math courses and standardized test preparation, with integrated modules based on textual / audiovisual materials.

Khan Academy

You know this one right? Originally a no-name guy posting math tutorials on YouTube, Salmon Khan would later be discovered by Bill Gates, subsequently receiving the financial backing from several private foundations to start a non-profit organisation — with the goal of providing free, world-class education to the world.
While much of the educational contents on Khan Academy are primarily built to supplement applied-math-based curriculum with a learning philosophy skewed towards video-based learning, many of their topics — especially the ones on Probability & Statistics, Calculus, Differential Equations and Linear Algebra, nevertheless remains substantially valuable to  millions of college/university math students worldwide. In addition, preparatory modules on standardized admission tests such as SAT and GMAT are offered as well — always at zero cost to the students and their parents.

Magoosh

Back in the old days, offline standardized test preparation courses can cost up to several hundreds of dollars. With the advent of Internet and its associated technologies, access to affordable test preparation is now possible to most computer users within a few clicks.
Built as an online testing platform with the goal of providing a robust test preparation experience, Magoosh now offers test materials for the ACT, SAT, GRE, GMAT (among others) in the form of timed practice questions and video lessons. Available both on the Web or via its mobile app, Magoosh’s testing platform also allows for its users to track their own progress, create customized practice sessions and receive assistance from the support team if needed. While admittedly not a replacement for students requiring one-to-one instructions to master the background materials, for those who fare well with self-study as a primary test preparation strategy, Magoosh can be a great asset in terms of boosting their test performance and increasing the chance of admission to their college(s) of choice.

Online Communities for Higher Mathematics

While there exist many math communities online, a vast majority of them targets students under mainstream K-12 curriculums. For those looking for specialized learning platforms where people gather together for in-depth discussions on higher mathematics, the following sites can help.

Quora

Quote touts itself as a website with “the best answer to any question“. While predominantly a general question-and-answer community, with the advent of its LaTeX integration via MathJax in May 2015, Quora is increasing becoming a popular site for university math students to ask course-related questions (among others), and for higher-mathematics enthusiasts to answer them — usually with a surprisingly-fast response rate.

Brilliant.org

A community website devoted to the intellectually-gifted, Brilliant.org aims to provide a social learning platform whereby students of math-related disciplines (e.g., mathematics, physics, quantitative finance, computer science) get to practice their problem-solving skills through the access to its ever-growing repertoire of user-generated problems — the latter of which are classified by their subtopic and tailored to an individual’s level of proficiency. Furthermore, Brilliant.org users also have the options to create their own problem sets and wikis for others, making it a formidable tool of accelerated learning for anything mathematics and beyond.

Art of Problem Solving (AoPS)

The Swiss knife of Math Olympiad training hubs, the Art of Problem Solving — originally founded by Richard Rusczyk in 2003 — went from a textbook publisher into a mathematical-education powerhouse featuring accredited online school, textbook series, customized online learning system (Alcumus) and community forum for high-performing math students all around the world, with additional resources including YouTube channel, collections from math competitions, games and a lot more!

Math StackExchange

A specialized question-and-answer website for math students and professionals of all levels, Math StackExchange provides a relatively-clean platform with LaTeX integration where topics from different branches of higher mathematics (e.g., number theory, calculus, abstract algebra, probability theory, combinatorics, group theory, analysis, graph theory) can thrive. Although possibly not intended for the general public, it’s a favorite place among the technically inclined.

Online Tools for  Scientific Computing

Graphing calculator, data analyzer, derivative/integral calculator and the like.

Desmos

A powerful HTML5 online graphing calculator, Desmos establishes itself as a free web/mobile application with a deceptively-simple user interface. From graphing equations and inequalities to creating tables and regression models, Desmos has all kinds of functionalities in its toolbox (as summarized in its 12-page user manual). If you want to learn all the stuffs you can do with Desmos, then this definitive guide can definitely get things going, and you haven’t heard of Desmos Art, then you are probably missing out!

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