Mathematics will be true in 100, 10,000 years, or for all eternity [1]. What does this mean? You study things that don’t change so that when things change, you don’t have to. In your lifetime, many things will change, but mathematics will still be the foundation of everything in our world.
The foundation? Yes. Everything around us is mathematics. From the car you drive and the house you live in to the phone you use or the food you eat.
Math is involved in every industry, from engineering and accounting to medicine and airlines.
Pick any field and with enough math, you’ll be able to understand it and even contribute to it.
But those aren’t reasons to study mathematics. I learn mathematics for its own sake. I don’t know how it could be useful, but I know mathematics reveals deep truths about our world. But again, I learn math to learn math. That’s how you will enjoy it the most.
I didn’t start out with this mindset. Studying engineering introduced me to learning what math was really about.
You will always find people who will tell you, “Math is useless and theoretical.” They don’t get the point. Sure, you might not use calculus or differential equations when buying groceries. Understanding these subjects shifts how you see the world.
Paul Graham views mathematics as “a valuable source of metaphors for almost any kind of work.” He’s right. For instance, when I learned Calculus , I grasped concepts like happiness, wealth, or even the universe.
Alright, you want to learn, so let’s learn.
First, identify where you are at. How? Remind yourself of the last math class you took. It might be Algebra 2, Trigonometry, or Calculus. Whatever it may be. It’s ok if you’ve taken a class but forgot it or didn’t learn it well. It happened to me when I went back and taught myself trigonometry.
Now that you know where you are at. I will suggest a roadmap, a path to move forward.
Khan Academy!
The best thing about Khan Academy is the mastery approach where you have videos to learn. Then you have practice problems and quizzes so you can make sure you learn. This is why I recommend Khan Academy to start because of the built-in practice problems.
Start with Precalculus and move forward. If it’s too challenging, try to go back and see what you haven’t learned yet.
After Precalculus, this is when you can start learning Calculus [2]. Choose Calculus 1 and start making progress.
You will use Khan Academy to learn up to Calculus 2. Then, we will use different methods (unless Khan Academy develops more content for higher-level math).
If you remember nothing else from this essay, just remember to go to Khan Academy. I will tell you more resources in case you are curious but if you’re starting, Khan Academy is the best starting place.
Assuming you used Khan Academy, now everything won’t be in one place and will be up for you to decide the option that works best for you.
Here are my favorite resources I use to learn higher-level mathematics. Today, this is how most college students are learning math. Not from their professors but from these gifted minds.
Professor Leonard: Buff guy who teaches you math so you can understand it deeply. His lectures are thorough and will help you grasp concepts fully. He has full-length course videos to get started on your math journey. Take a look at his YouTube playlists with his courses.
From a Reddit user:
He's very good if you want the full and thorough Calculus experience. If you're just looking for how to accomplish a method or some algebra questions that pop up maybe give "patrickJMT" or "The Organic Chemistry Tutor" a try. Their videos are shorter and full of quick relevant examples.
Organic Chemistry Tutor: That brings us to the Organic Chemistry Tutor which I admire for his breadth of knowledge across STEM. Although his name is related to chemistry, he teaches almost everything.
His math videos are excellent, and like the Reddit user, I use them to review concepts and do practice problems with him. He has videos from basic math to some Calc III and differential equations.
Paul's Notes: A professor who shared his notes online for his college students. Now, his notes are used by millions of people.
These notes are organized by class with Algebra, all Calculus, and Differential Equations. His notes are useful because he does not assume that you know everything already. Instead, he breaks it down and explains it step-by-step.
I use Paul’s notes to review and learn concepts as well as do practice problems. I love this page because of how he walks through a problem.
This is a common problem. As long as you understand, you do many problems, and you are disciplined, you will know how to solve these problems. But solving does not mean you always get what’s going on visually or even where notation like dy/dx comes from.
That’s why I will recommend you use these resources.
3Blue1Brown: Grant is one of my favorite YouTubers. His math videos don’t just tell you, “Hey, here is how you do a problem.” He will give you such an intuition about what’s going on, why and how we do things, and where it all comes from. He’s a talented mathematician and programmer who uses the best animations to understand math. Start with the Essense of Calculus. His goal with his videos? “The goal here is to make calculus feel like something that you yourself could have discovered.”
Desmos: A graphing calculator, meaning you write an equation and you can see what it looks like. This is particularly helpful when you are not sure what is going on visually or are having trouble with a concept. Sometimes, when I graph a function or equation, I understand and can move forward.
Calculus Made Easy: You’ll be learning Calculus, and sometimes there will be a lot of jargon and complexity. If you ever get confused, open up the Calculus Made Easy book. This one is a classic to understand calculus because it quickly gets to the point.
Calculators: Wolfram Alpha and Symbolab are quick ways to check your solutions and verify your steps. Don’t rely on them too much though as you don’t want to get into bad habits [3].
Do more problems. Problems and problems. One of the most gifted mathematicians ever, John Von Neumann, said once, “Young man, in mathematics you don't understand things. You just get used to them."
Listen, many times I knew how to do the problems but I had no idea why it made sense. It could only make sense looking backward when I had more context. Let your hand do the work, and your brain will catch up.
How do you get used to these things?
Practice and practice.
Schaum's Outline of Calculus is the best place to practice problems. In each section, they briefly explain the concepts and include many problems with clear solutions.
To add to that, most of the people I know, including myself, learn through practice tests. It’s almost like we do the learning when we’re doing problems. Then, we go back and learn the material to fill in gaps of understanding.
For practice tests, go to the internet and type, “[class/concept you are learning practice exam].” If you’re learning Calc I, you may type, “Calculus 1 practice exams.” Make sure you, it also includes the solutions to check yourself. For example, I used this website for Calc 1.
“Yes, some people are brighter than others but I really believe that most people can really get to quite a good level in mathematics if they're prepared to deal with these more psychological issues of how to handle the situation of being stuck.” Andrew Willes
Math is difficult for everyone. The difference is those who are able to be patient and get unstuck. This is important to take seriously because you WILL get stuck but it’s ok.
If you think you’re bad at math is because you haven’t learned how to get unstuck. It’s something you can learn. This math professor gave a wonderful speech about how to get unstuck. She shares her journey from hating math to becoming a mathematics professor. I watch it every time, I get stuck on problems. Watch her speech.
What to do when you get stuck?
Take a break and walk around
Solve and research similar problems
Sleep and come back the next day
Explain why you don’t understand to someone (or to a rubber duck)
If nothing works so far, ask a math discussion forum on Reddit, Discord, or Math Overflow
Libraries usually have free online tutoring. For instance, here's Chicago Public Libray’s free online tutoring. I used Brainfuse a ton in high school
I believe in you
Being stuck is one of the reasons going to college makes sense because 1) you have a group of people learning the same thing, and 2) there are tutoring centers and office hours. However, with determination, you can find people learning math online or even invite some friends. And about the tutoring, your library’s tutoring will help you.
Pure beauty that teaches you how to think!
I hope this guide it’s useful and you get started learning math. Don’t worry about where, or how. You just have to get started.
If you have any questions or doubts, contact me. I’ll be more than happy to help you out.
Are you ready? Let’s see.
What is 2 + 2?
If the answer is 4, you are ready!
Oh and by the way, when the aliens come to visit us, mathematics will probably be the language to talk to them. This isn’t a joke. Gauss, one of the most gifted mathematicians ever, wanted to take forests and draw the Pythagorean theorem so that if aliens approach Earth, they can say, "Wow, they seem to be mathematicians."
Have fun, and let’s learn math!
Notes
[1] I don’t say, “It will never change” because who knows? We might invent new mathematics and find we were wrong all the time. We need to discover more math to solve more problems. Our current toolkit is incredibly lacking to make progress in science such as theoretical physics, and really a lot of problems we haven’t solved yet.
[2] In Khan Academy, you’ll see AP Calculus AB, Differential Calculus, etc. Just pick Calculus 1 as that’s the one with all the content you need to learn.
[3] Dr. Paul Abbott helped me see further about this point. Here are his comments:
“Calculators: Wolfram Alpha and Symbolab are quick ways to check your solutions and verify your steps. Don’t rely on them too much though as you don’t want to get into bad habits.”I disagree with your conclusion here about “bad habits”. Using Wolfram|Alpha or, better, #WolframLanguage, is a good habit—and, combined with (basic) math knowledge, you can go much further, learning and discovering new math along the way.The useful ability to do algebra and calculus by hand should not be over-rated—or overemphasised—as it is many math, science and engineering courses, partially because it is something that is easy to test. Instead, proper understanding of the underlying concepts is far more important—and then you can use any computational tool you like, optimising the steps using your acquired knowledgeFor example, hand computation of a Laplace transform, or a matrix inverse is, in general, just not something a human should be doing. Maybe once or twice—but an entire course devoted to such machinations is unproductive. But understanding what these operations do, and where they should be applied is important. Unfortunately, the emphasis of most high school and tertiary courses is tilted the wrong way.
He is right. I use Wolfram Alpha and Mathematica quite often to reach a further understanding. I don't do as well in some of my exams in classes like differential equations because it only tests memorization of the algorithms. When in reality, I understand and see the beauty of big ideas.
Other Helpful Resources and Inspiration
The Map Of Mathematics Video: If you’re thinking of studying math, this video will introduce you to the many interesting areas of math. Both what we’ve discovered and what we are yet to discover (or invent).
Learn Mathematics from START to FINISH Video: Video: A great roadmap to learn the topics any undergrad would learn in college.
List of Free Math Books: 77 free textbooks about pretty much any math topic.
Thoughts on why to study math:
“As for what to study, the key is that it needs to be difficult in ways that reshape your perspectives, like math, physics, or molecular biology.” - Alan Kay
"Mathematics is principally a tool to meditate, rather than to compute." - Nassim Taleb
Videos on Math History: Learning the history provides me additional context to understand who did what where, why, and how. You may be surprised but once you learned the history, you learn better. Andrew from moderndaymath is my go-to YouTube channel for math history. Start with Euler, then Maxwell, and let your curiosity run free.
Thanks to Azul Wells, Anthony Polanco, Henk Bruinsma, and Ismael Hozain for reading drafts of this essay.
If you’re into interesting ideas (like the one you just read), join my Weekly Memos., and I’ll send you new essays right when they come out.