The Categorical Imperative: Relearning Math as an Adult (Without the Rabbit Hole)

# Relearning Math as an Adult (without falling into a rabbit hole of videos and existential dread)

You graduated, got a job, and then life happened. Now you want to learn — or relearn — math. You open a browser and are assaulted by an army of YouTube channels, PDF textbooks, and graphing tools. Congratulations: you have the modern learner’s dilemma. The good news is math did not move. The bad news is the internet is a noisy bazaar. Consider this your polite, slightly snarky manifesto — equal parts practical roadmap and philosophical nudge from Dr. Katya Steiner, who believes the right categorical imperative for adult learners is: compose small, check often, and respect structure.

## Start with a mindset and a plan (your categorical imperative)

If you are returning after a decade or three, the real work is not the content but the method. Treat math like weightlifting: technique first, heavy stuff later. Fix the foundations — arithmetic, fractions, algebraic manipulation — and build deliberately. Schedule short, focused sessions. Mix reading with problems. Reward yourself for tiny wins (coffee, a smug playlist, telling a friend you did an integral and lived to tell the tale).

The categorical imperative here is: compose small, check often. In category-theory-ish terms, build small morphisms (tiny, verifiable steps) and compose them into bigger transformations that actually do something useful. This idea scales from arithmetic to abstract algebra: small, rigorous connectors beat grand gestures and flashy visuals when the goal is lasting competence.

## A suggested learning arc (not a ladder to climb in a weekend)

– Number sense and fractions: decimals, percents, estimation — the unsung heroes of everyday math.\
– Algebra and functions: manipulations, factoring, solving, graphs.\
– Geometry and trig: shapes, angles, and the unit circle.\
– Precalculus and limits: functional intuition, series as controlled approximations.\
– Calculus and linear algebra: one-variable calculus first; matrices and linear maps next.\
– Practice and proof: fluency with problems, and then learning to write arguments.

Yes, linear algebra and calculus are cousins who gossip. Linear algebra teaches you to think about structure; calculus teaches you how things change. Together they are the reason modern data science and physics work. You do not need both on day one.

## Cross-sections: what different branches of math will teach you about thinking

– Number sense teaches you to estimate and sanity-check. You will waste less time if you can tell when an answer is impossibly large.\
– Algebra teaches manipulative fluency: relocating terms, factoring, and seeing hidden structure. This is plumbing; not glamorous, but essential.\
– Geometry trains visual intuition and spatial decomposition. Geometry is where combinatorics and algebra meet a ruler and a ruler’s ego.\
– Calculus gives you local-to-global thinking: infinitesimals are tiny stories about change that add up to big ones.\
– Linear algebra is the art of compact representation and change of coordinates. Once you can think in vectors and linear maps, a lot of otherwise-difficult problems simplify into matrix juggling.\
– Proof theory and logic teach you to argue clearly. Here, the flavor of logic matters: classical proofs allow excluded middle and non-constructive existence; constructive logic insists you build the object you claim exists. Both are useful and both have places in the wild.

If you like unifying meta-theory, category theory is the philosopher’s vantage point: it says, “What matters is how structures map to each other.” That is a beautiful, compact lens that can help you prioritize: look for morphisms and composition, not infinite lists of formulas.

## Video teachers worth your binge time (but limit your subscriptions)

Pick one or two voices. Too many cooks ruin your cognitive stew.

– For step-by-step clarity: Khan-style lessons and short classroom playlists.\
– For intuition and visual delight: 3Blue1Brown — gorgeous for the why.\
– For worked examples: channels that solve specific problems — useful when you’re stuck and need a walkthrough.\
– For full-course depth: open university lecture series if you want the syllabus experience.

Remember: watch 20 minutes, then do 40 minutes of problems. The algorithm will thank you later.

## Paper over panic: books you can actually read

If you like paper, bless you. Print a book, underline like a Victorian, and make marginalia your new civic duty.

– Brush-up basics: approachable workbooks with lots of exercises.\
– Transition texts: friendly precalc and single-variable calculus books.\
– Proof-based math: clear introductions to proof and linear algebra that teach thinking, not just computing.

Free open textbooks are fine. They are frugal and often excellent.

## Tools that won’t make you lazy (if used properly)

– Desmos and GeoGebra: immediate visual feedback — use them to build intuition, not to replace thinking.\
– WolframAlpha: a brilliant, judgmental calculator — check your work, don’t outsource understanding.\
– Sage/Octave/Maxima: for programming experiments and numerical exploration.\
– LaTeX: annoying at first, useful later for clarity and résumé points.

## Explaining (a+b)^3 without algebraic trauma

Visualize a cube with side (a+b). Split each side into a and b. The big cube partitions into:

– one a-cube: a^3\
– three a^2b rectangular slabs (a by a by b): 3a^2b\
– three ab^2 slabs (a by b by b): 3ab^2\
– one b-cube: b^3

Count the blocks. The coefficients 1, 3, 3, 1 are just combinatorics wearing geometry as a hat. Pascal’s triangle is the same pattern, generalized. It’s bookkeeping with style.

## Short, ruthless study plan

– Weeks 1–4: arithmetic and fractions; 30–60 minutes daily of practice.\
– Month 2: algebra basics; do problems every session.\
– Month 3–6: geometry/trig and single-variable calculus; one concept at a time.\
– Keep a question log: write down confusions, find a short lecture, then solve three problems on that topic.

Pair watching with doing. The ratio is roughly 1:2 (watch:do) for durable learning.

## Final takeaway (and a little philosophy)

The internet tempts you with shiny explanations and existential wonder. Your job, dear adult learner, is to pick a couple of voices that click, do the grunt work, and adopt a compositional ethic: small steps, verifiable checks, and respect for structure. Whether you approach math for work, curiosity, or the pure pleasure of seeing patterns settle into place, the payoff is never the final answer alone — it is the moment a pattern stops being mysterious and becomes a tool.

So here is my parting, slightly bitchy question for you to chew on over coffee: once you can compose tiny verified steps into larger transformations, what real-world problem would you like to finally sit with, dismantle, and understand from the inside out?