Late to the Party, Still Can Do the Math: Dr. Katya Steiner’s ‘The Categorical Imperative’
# Late to the Party, Still Can Do the Math: Dr. Katya Steiner’s “The Categorical Imperative”
You forgot how to do fractions. You binged math videos at 2 a.m. and woke up thinking the binomial theorem was sorcery. Relax — the human brain is embarrassingly good at re-learning things it once tolerated. The real blocker is choice paralysis: the internet is a buffet, and you can’t gain arithmetic fluency by Instagramming the salad bar.
Below: a pragmatic, mildly opinionated roadmap for getting good (or deliciously addicted) again. I’ll sprinkle in some cross-disciplinary riffs — combinatorics, geometry, logic, and a cheeky nod to category theory — because math is less a tower of ivory and more an interlinked subway map. Bring a pencil.
## Starter kit: where to begin without drowning
– If you need arithmetic and fractions: Khan Academy. It’s boringly effective, free, and ruthless about repetition. Use the practice problems, not just the videos.
– For a gentle, paper-first reboot: CK-12 and OpenStax — free textbooks you can print and scribble in like a civilized person.
– Compact, rigorous intros: Active Calculus (single-variable) and “Linear Algebra Done Wrong” if you want linear algebra that treats you like an adult learner ready for proof.
– Bite-sized worked examples: PatrickJMT for mechanics; 3Blue1Brown and Mathologer once you want intuition and visuals.
– When you want to ask dumb questions without being shamed: Math StackExchange and r/learnmath. Be specific — a minimal, reproducible example is your friend.
Stop collecting tools; start using two. Pick one practice source (a book or Khan) and one visualization toy (Desmos or GeoGebra). WolframAlpha is a calculator, not a life coach. If you download every symbolic system, you will catalogue software, not skills.
## Visualizing (a+b)^3 without tears
If algebra reads like incantation, visualize it. Imagine a cube with side length a + b. Slice along each axis into lengths a and b. The big cube splits into 8 rectangular blocks, but grouped symmetrically they give us the expansion:
– One cube of side a → a^3.
– One cube of side b → b^3.
– Three slabs of size a × a × b → 3a^2b.
– Three slabs of size a × b × b → 3ab^2.
Sum the volumes: a^3 + 3a^2b + 3ab^2 + b^3. If you prefer counting: each factor (a + b)(a + b)(a + b) means choose a or b from each of three slots. The coefficients 1, 3, 3, 1 are just binomial counts. Geometry, combinatorics — same party, different outfits.
## Why the cross-disciplinary view matters
Math is not a sequence of isolated facts; it’s a network of perspectives. If you only memorize algebraic tricks, your understanding will wobble when you meet an unfamiliar problem. A few cross-section notes:
– Combinatorics: coefficients are often counting in disguise. Seeing algebraic identities as counting problems gives intuition and reduces memorization.
– Geometry: spatial reasoning rescues algebraic dread. Volumes and areas provide concrete anchors for polynomials and integrals.
– Linear algebra: teaches you to think in coordinates and transformations. Suddenly “solve this system” becomes “diagonalize the boring matrix.”
– Analysis (calculus): sharpens limits, approximation, and why continuity matters — the stuff that makes calculus usable rather than mystical.
– Logic & proof: formal tools (propositional and predicate logic) are the muscle memory for reasoning. You don’t need to be a proof artist, but learn to read a proof and sketch a short one yourself.
– Category theory (yes, the titular gag): it’s the language of structure. Where set theory counts members, category theory cares about relationships between structures. Think of it as the discipline that politely insists you see the forest instead of just the trees. The “categorical imperative” of learning math is: relate things; don’t hoard isolated facts.
Each discipline offers heuristics that fertilize the others. The binomial cube is geometry, counting, and algebra — a tiny interdisciplinary miracle.
## A modest strategy for not being overwhelmed
– Foundations first: spend 60–70% of your time on arithmetic fluency, algebraic manipulation, and equation-solving. Skip this and higher math is a house of cards.
– Problems > videos: videos inspire; problem sets teach. For 30 minutes of watching, do 60 minutes of doing.
– Micro-goals: 20–45 minute focused sessions on a single problem type. Then stop. This rhythm is mercifully effective.
– Spaced practice: revisit concepts across days. Memory consolidates slowly — that’s biology, not your failure.
And yes, if you find yourself delightfully addicted to solving integrals at 3 a.m., schedule it. Obsession is a useful engine until it forgets dinner and friends.
## Community, mentorship, and bad habits to avoid
Find a study buddy or an online forum. When you post for help, show what you tried and where you got stuck. The internet has brilliant teachers and many terrible ones — favor answers that show steps over magic results.
Bad habits to dodge: skipping algebraic practice, treating CAS outputs as “answers” without verification, and collecting resources like emotional security blankets.
## Practical shortcuts and when to escalate
– Desmos/GeoGebra: interactive graphs and geometry.
– WolframAlpha: good for checks, not for learning technique.
– LaTeX: spend an afternoon learning basic LaTeX; your notes will look like a pro’s and you’ll read math papers faster.
– Sage/Octave: useful only when you have an explicit computational need. Otherwise, resist the temptation.
## Balance and the small prescription
If you want to be dangerous at dinner parties, do ten problems, not ten YouTube videos. Teach someone something every week — explaining cements understanding faster than rewatching a lecture. If you’re over 30 and newly obsessed: excellent. The material doesn’t care your birth year. Your life does — so keep one social day a week.
## Closing: the Categorical Imperative of Learning
Here’s the categorical imperative I actually mean: relate. Relate a problem to a picture, a picture to a count, a count to an algebraic identity, and an identity to a proof. Structure beats memorization. Practice beats inspiration. Two tools beat a cluttered app drawer.
So here’s my smug-but-sincere challenge: do ten problems this week, one visual proof, and teach one tiny thing to someone. If you must binge, binge solutions — not videos.
What unexpected bridge between two math disciplines would change the way you see a problem you’ve been stuck on?
