What Math To Learn Next After Calculus

Depending on your goals, either A) methods of proof, or B) linear algebra followed by probability & statistics.

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Sometimes math learners wonder what course they should tackle next after calculus.

Learning math through calculus feels like climbing the trunk of a tree and then reaching the edge where it starts branching out in all directions.

What branch do you climb next? You’ve got so many choices that it might feel overwhelming.

Here’s my take. What I recommend depends on whether you want to

• A) become an academic mathematician, i.e., major in math, do a PhD, and create/prove new mathematical theorems, or
• B) build math-heavy systems that solve real-life applied problems, e.g., rocket engineering or any kind of AI/ML/quant/algo-heavy software development.

For goal A (“academic mathematician”), I would recommend “methods of proof,” i.e., an introduction to what proofs are, how to write them, and common techniques for approaching various kinds of proofs.

Why? Because every year, countless aspiring math majors show up to university without a background in proof-writing and then get their ass handed to them by proof-based courses because proof-writing is not yet second nature to them.

(Coming into a Real Analysis or Abstract Algebra class without fluency in reading/writing proofs being second nature is like coming into a Shakespeare literary analysis class with shaky or nonexistent reading and writing skills. The content matter is challenging enough that it’s overwhelmingly difficult for most learners to pick up supporting skills on the fly.)

For goal B (“build math-heavy systems”), I would recommend linear algebra and then probability / statistics.

Why? Because those are two subjects that show up in basically every hardcore application of math.

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