Mathematics
​Introduction
Mathematics is just as important as every other STEM subject and is used in all the sciences. Many students develop a form of anxiety where they are scared to go forward with math. This can be remedied by learning the concepts, which gives the student intuition into the formulas and methods. Although this may require a bit more effort initially, it is incredibly beneficial in the long run, particularly when students encounter more advanced math topics.
We would like to provide an alternative means of educational enrichment by providing high-quality, accessible math resources for high school students. Below is a list of links to specific sections of resources, feel free to jump to any specific one!
Videos/Notes
Educational Channels
mathantics – Channel that focuses on topics from arithmetic to basic algebra. Explains math in a very intuitive and graphical manner. Perfect for those seeking to brush up on basics.
Khan Academy – Website with videos on pretty much all topics of grade school math from basic counting to Linear Algebra. Most of these topics come with review exercises. Highly recommended for math students of all levels.
TabletClass Math – Channel that focuses on topics from pre-algebra to beginner calculus (AP Calculus AB level). Contains lots of short review style videos, perfect for students seeking to study for an exam.
The Organic Chemistry Tutor – Despite the channel name being about Organic Chemistry, this channel has videos on a wide variety of math topics from algebra to calculus. Has lots of videos going in depth on a certain topic, perfect for students looking to brush up on certain topics.
Featured: The Organic Chemistry Tutor made a video full of precalculus questions and a step-by-step solution. This video is the perfect way to prepare your next precalculus exam. See below to watch.
Brian McLogan – Channel on topics from pre-algebra to beginner calculus (AP Calculus AB level). Has videos on many specific examples, and perfect for students looking for worked-through examples to practice specific math skills.
Featured: In this video, Brian McLogan explains three ways to solve a quadratic equation, which is factoring, AC method, and the quadratic formula. He brushed a bit about inverse operations as well. See below to watch.
Eddie Woo – Very good channel on topics from algebra to beginner calculus (AP Calculus AB level) that emphasizes looking at the intuition behind mathematical concepts through a graphical lens. Also includes various application videos to connect math to other science subjects.
Featured: Sir Eddie Woo explains why 0! = 1. He argues the same way why a^0 = 1. See below to watch.
PatrickJMT – Channel with videos ranging from algebra to calculus (AP Calculus AB + BC and multivariable calculus) to linear algebra to discrete math. Effectively contains all the information math students will need in high school. Contains videos explaining topics as well as many examples pertaining to those topics.
ukmathsteacher – Channel explaining topics on math relevant to the GCSE’s. Contains many worked out examples to previous exams as well.
Professor Leonard – Channel explaining topics from algebra to multivariable calculus and differential equations. These videos are geared towards those learning the topic for the first time. These videos are long and focus on the intuition in math, while containing many examples.
Hippocampus – Large website with math topics from Arithmetic to Calculus (AP Calculus AB and BC and Multivariable Calculus) to Statistics. Also contains practice questions and examples in a single source. Recommended for any math student.
MIT Math OpenCourseWare – Another large website that is a compilation of many MIT courses with most including video lectures, written notes, assignments, and exams. Highly recommended for high school students wanting to advance their math skills.
Brilliant – A website which allows you to visualize numerous topics and concepts ranging from fundamentals of mathematics to multivariable calculus and group theory. The website approaches problems through creative ways and allow the users to build an intuitive understanding of the topics. Would definitely recommend checking out for an interactive math experience.
EdX – A website that offers free online STEM courses by prestigious universities such as Harvard, MIT, Berkeley, and so on! If you are looking for a university that would fit you the best or just bored and need something productive to do in your free time, this is the perfect place for you to further your knowledge in the STEM field.
Coursera – An online and free source for you to learn all sorts of things related to STEM provided by top universities from all over the world, such as Stanford, MIT, Harvard, and many more. I would recommend Introduction to Mathematical Thinking by Stanford University at Coursera. You will learn number theory, real analysis, and mathematical language. It’s a great start for those who are looking to progress their math skills.
Dr. Trefor Bazett – A YouTube channel full of playlists including differential equations, calculus courses, and fun math theories. Loads of tips and series are embedded into this YouTube channel which will surely help you engage with math on a higher level.
Tom Rocks Maths – A YouTube channel that carries many videos on different theories, laws, programs, and problem solving questions. Lots of fun topics that’ll pick your brain and make you think about math on a deeper level.
Corbettmaths – A website to learn all kinds of mathematical concepts for free. For each topic listed on their website, there is a video explaining the concept, a worksheet for practice, and a textbook formatted practice sheet. After finishing the practice sheets, there is a link at the top of the page that provides the answers to all the practice sheets.
DoS (Domain of Science) – DoS creates videos about everything related to science. If you’re interested in Mathematics, I would highly recommend his vides where he created a map of mathematics. Check it out here!
Recreational Channels
3blue1brown – Variety of videos across many mathematical topics that enforces graphical intuition behind many complicated topics. Highly recommended for any student interested in math and science. Personal Recommendations/Favorites: “The Most Unexpected Answer to a Counting Puzzle“, “What they won’t teach you in calculus“, “Euler’s formula with introductory group theory“, and “Visualizing the Riemann hypothesis and analytic continuation“.
Mathologer – Channel that explains various abstract math concepts at an accessible level. Interesting and engaging content all around.
Numberphile – Another channel that also explains various abstract math concepts with layman’s terms. Perfect for anyone interested in pure math topics.
Flammable Maths – Entertaining channel that makes various skits as well as proofs on various topics. Most proofs are intended for higher level math students (calculus and above).
Vihart – A channel which create funny and laughable videos related to mathematics. She made numerous videos about the digits of pi. Truly the channel you’re looking for if you’re in love with mathematics and want to have fun!
Notes
Paul’s Online Notes – Written online notes on topics such as algebra, calculus (including multivariable calculus) and differential equations. Very thorough with many worked through examples as well as various other useful stuff such as cheat sheets, math tips, common errors, etc.
Wolfram MathWorld – Large database with many definitions to mathematical terms for reference.
Art of Problem Solving – Has many online classes on topics from pre-algebra to calculus to group theory. They also have classes focused on various competitions such as Math Olympiad and Physics Olympiad. Note that enrollment is not free.
MIT Open Courseware – Sets of high quality PDF calculus notes covering various units and topics that ensure an excellent learning experience. Published by one of the most prestigious universities in the world: Massachusetts Institute of Technology, commonly known as MIT.
Recommended Books and Textbooks
Personal Recommendations
Calculus of Several Variables (Serge Lang) – Written by a Yale professor, this is a well-written introductory multivariable calculus book for those just finishing calculus II or AP Calculus BC. This book takes a more pure math based approach of teaching and starts to expose the reader to proof-based concepts. Buy it now!
Linear Algebra Done Right (Sheldon Axler) – Being the perfect textbook for the second course of linear algebra, it is targeted towards undergraduate or graduate math majors in university or college. In a novel approach, this textbook helps the reader understand the structure of linear operators on finite-dimensional vector spaces which is the main goal of linear algebra. This San Francisco State University professor drives concepts and simplifies proof to the best of his abilities. Buy it now!
Challenge and Thrill of Pre-College Mathematics (V Krishnamurthy, C R Pranesachar) – The book does a very good job in allowing high school students to transition from school to college level mathematics seamlessly and building upon their understanding of different geometrical, trigonometric and algebraic concepts. What makes the book distinct is its detailed sections on geometry with some challenging problems. Employing distinct approaches to problems, the book is recommended for anyone wishing to explore fundamentals of geometry and related problems in depth. Buy it now!
Fundamentals of Differential Equations (R. Kent Nagle, Edward B. Saff, Arthur David S.) by Pearson – This book provides depth explanation from the introduction of differential to partial differential equations. It starts off by the basic theory, methodology, methods, and relevant applications of differential equations which are beneficial in the field of science and engineering. The perfect book for high school students learning to excel differentials at school. Buy it now!
How to Solve it (George Polya, New Foreword by John H. Conway) – Known as the father of problem solving, this Stanford professor researched complex mathematical physics and many other mathematical topics. His research includes number theory, numerical analysis, probability theory, combinatorics, geometry, and other complex analysis. His published papers of research and theory provides intense and deep results of the study of integral functions. Buy it now!
Advanced Calculus (Lynn Harold Loomis, Shlomo Sternberg) – Written by two professors from Harvard University who taught in the 1960s, Advanced Calculus remains a vital component of the Math 55 class taught at Harvard. Math 55 is known as one of the most rigorous undergraduate math courses in the United States and lasts two semesters. If interested in this textbook, it is recommended that the reader has knowledge about limit and continuity type arguments along with partial derivatives. Buy it now!
General Lists
List of Math Textbooks – Includes topics from basic algebra to topology, number theory, mathematical physics, and pretty much every other math topic.
Introduction to Probability by Charles Grinstead and J. Laurie Snell – This textbook teaches you about probability and and random variables. Before jumping into this book, you should already have a good understanding of multivariable calculus. In this book, you’ll learn about binomial, distribution functions, Poisson distribution, geometric, and hypergeometric. There’s problem sets available in the book for practice as well.
Online Research and Internship Opportunities
Pure math research is very hard to come by as much of the required knowledge often involves graduate-level classes. However, here is a list of some possible research projects that any high school student can get involved with. On the other hand, there are a bunch of applied math research projects that come in many different forms from bioinformatics to computer science to statistics. Many of these can be found on their respective pages.
Open Problem Garden – A collection of open and unsolved math problems for any high-achieving students to attempt.
Programs in Mathematics – A database full of all and any online research connected with math, updated whenever a new opportunity arises. There are also ways to be an editor of the database itself.
MathPrograms.org – This website will tell you all of the mathematical science related online events from all around the world! You can participate in online research activities, submit your research paper and publish it, collaborate on projects with other people, and even compete in math competitions for a chance to win prizes.
American Mathematical Society – This organization houses a variety of paid internships at universities all over the United States. The programs are developed for high school students and students in college. Additionally, many of the programs are rooted in research work.
Summer Programs
Summer programs are a great way to study math more in depth in a cooperative environment with other like-minded students. Some summer programs focus on education and others focus on research. Choose what is more relevant to you. This list of programs are mainly for high school students.
Personal Recommendations
Research Mentorship Program (RMP) – RMP is an intensive 6-week research program held at UC Santa Barbara (UCSB) which involves high-achieving students participating in university-level research in subjects from the humanities to math. The program includes multiple field trips and social events, creating a fun and collaborative working environment. The program accepts rising juniors and seniors and a limited number of rising sophomores. It is extremely competitive and costs $75 US Dollars, however it provides scholarship opportunities.
Research Science Institute (RSI) – A highly prestigious intensive 5-week research program held at MIT that involves students going through the research process. Research here is centered around science and engineering topics. The program culminates in written and oral presentations of their projects. This program is also free to all of those accepted.
Stanford University Mathematics Camp (SUMaC) – A prestigious and intensive 4-week math summer residential program at Stanford University for high school juniors and seniors. Daily lectures are taught by Stanford professors on topics such as group theory, number theory, and algebraic topology. After every lecture, students work on problem sets based on the lecture with help from the counselors if needed. Tuition for the program is $3,250 but students can apply for financial aid.
Program in Mathematics for Young Scientists (PROMYS) – An intensive 6-week math summer residential program at Boston University for students between 15 and 19 years old. PROMYS consists of exploring various topics in math with a focus on number theory. The program holds a strong emphasis on developing individual problem-solving skills and applying them to challenging problems. The cost of the program is $2,500 but students can apply for financial aid.
Mathcamp – An immersive 5-week math summer residential program at Champlain College that accepts students from ages 13-18. Students here are exposed to a wide range of math topics and work together with graduate students and researchers. Mathcamp is really what you make of it where you can either work on research projects are just enjoy the lectures and field trips.
IMC – The International Mathematics Competition is being held at the American University in Blagoevgrad, Bulgaria by the University of London for six days where students from universities around the world are allowed to participate. The problems that will need to be solved during the competition are from various different math topics and many past participants have gone out to be very successful in the math field.
AwesomeMath Summer Camp – This is a 3 weeks program for talented and gifted students in mathematics around the globe. AwesomeMath offers both in-person classes and online classes as well due to the rise of COVID-19 cases in recent years. Most students who enroll in this summer camp seek to improve their skills and prepare for mathematics competitions, such as AMC10/12, AIME, or USAMO. The program only held virtual classes for the summer of 2021, but they are looking to transition back to in person for the summer of 2022. The cost for the virtual program ranges from $875-$995 depending on how early the student enrolls in the course.
MathILY – MathILY is an intensive mathematics summer camp for high-school students from any parts of the world. It is a perfect summer camp for anyone of those who has curious mind for mathematics and is willing to commit for a 5 week program. However, because of the pandemic, MathILY will take place synchronously online in 2021. The cost of the program is $2150.
Idea Math Camp – Idea Math Camp invited students who enjoy solving difficult mathematical problems. In this summer camp, you’ll learn algebra, number theory, combinatorics, and geometry. Classes are online and relatively small in order to ensure the quality of education. This camp starts in July and costs $1245.
The Hampshire College Summer Studies in Mathematics – This is a summer camp dedicated to high school students seeking to further their mathematical studies and make connections within the field. It is a six-week program consisting of college-level math and class time of 4 hours per day (Monday-Saturday). Students will stay in the dorms at the Hampshire College in Massachusetts and the cost is $4913.
General Lists
American Math Society List – List focused on primarily math camps
Pathways to Science List – List focused on more general STEM camps
Competitions
Math Olympiads for Elementary and Middle Schools – Math contests for aspiring students from grades 4-8 that involves working in teams to solve challenging problems.
USAMTS – A proof-based math competition for high school students equivalent to the AMC 10/AMC 12. Students who perform well on this home exam may qualify for the AIME and USA(J)MO. The test has multiple rounds and students are given a whole month to work out their solutions.
International Math Olympiad – The premier math competition for high school students across the world. Qualification for this competition is competitive and rigorous and involves taking the AMC 10/12, then the AIME, then the USAMO for those in the US.
2021 International Math Contest – This contest is available for students across the world in grade 3 to grade 10. The contest lasts for 30 minutes and it is held online. The math questions are based on a national curricula and is developed by math professionals. If you are one of the qualifiers, you’ll be invited to the Russian Math Olympiad!
International Mathematics Competition (IMC) – The Chiu Chang Mathematics Education Foundation in China established the IMC in the early 2000s and it has grown into a multi-continental competition. The competition is directed towards Southeast Asia but is open to any country. Each country or city that is invited will send 2 teams (6 members per team) of promising individuals to compete in the competition. Host countries–including Bulgaria, China, Hong Kong, India, Indonesia, Macau, Malaysia, Nepal, Philippines, South Africa, South Korea and Thailand–are permitted to send 4 teams to the event.
Other Resources
LaTeX – A typesetting which is pretty much universal for typing any STEM-related document. Takes some time to get used to but is definitely worthwhile for any student interested in STEM to learn. Some links for those who are interested: Latex Tutorial, Overleaf (an online collaborative LaTeX editor).
Art of Problem Solving (AoPS)- An organization dedicated to teaching math to the world’s youth. AoPS offers online courses suited for middle and high school students and focusses on teaching its students a few critical concepts that can be applied to a variety of problems. Additionally, students have the opportunity to interact with others through an online discussion platform and ask for help from teachers specifically employed by AoPS.
Socratic Google – Powered by Google, it is a problem solving app that operates at a high level of mathematics. Ask Socratic a question and it will provide the best resources for you to further explore that subject and/or concept.
Desmos – A free online graphing calculator. If you don’t have a graphing calculator at home, this is the perfect website to visit! It has a scientific, graphing, four function, and matrix calculator in addition to geometry tools.
Arithmetic Game – A very simple online game which provide arithmetic questions for people to solve before the time runs out! The problems will not change until you find out the correct answer. Quite the simple, challenging, and fun game to spend your time.
GeoGebra – GeoGebra is a powerful tool for teaching and learning mathematics. It serves for many purposes, starting from scientific calculator for tests, graphing, to geometry display. GeoGebra is also perfect when dealing with elementary students, as it has a tidy and fun approach on the design! In addition, it works well for university project too.
Standardized Test Preparation
SAT Math – For those seeking to improve their existing knowledge in the math topics, I recommend Kaplan’s SAT Prep Plus and Barron’s SAT (both these books also include practice tests and non-math section topics). For those strong in the topics, I recommend simply just doing practice tests and math sections. Many of these sample tests can be found online at Khan Academy.
ACT Math – In terms of subject matter, the ACT Math section is highly similar to the SAT Math section. However, the ACT Math section involves solving more questions at a faster pace than the SAT Math. For those looking to improve their existing knowledge, I recommend Kaplan’s ACT Prep Plus (also includes prep and practice tests for non-math sections). For those strong in the topics, I recommend doing practice tests and math sections from the ACT Official Study Guide (also includes content for non-math sections).
AP Calculus AB & BC – In terms of content, AP Calculus BC incorporates a few additional topics that AB doesn’t such as more advanced integration methods, parametric equations and polar coordinates, and series. For both subject matter and practice, I recommend Barron’s AP Calculus (includes AB and BC) and Princeton Review’s AP Calculus (AB version, BC version). The differences between the two books are minimal but I would say Barron’s does a better job at explaining topics while Princeton explains basic concepts then does a lot of example problems. I also highly recommend doing past AP’s free response questions (AB version, BC version) for further practice.