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Mathematics
Algebra 2 Honors and Precalculus Honors are the two courses on the schedule right now, with coverage running from Algebra I through multivariable calculus and linear algebra. Most students arrive understanding more than their grade suggests.
Where the points actually go
Ask a student who just lost fifteen points what happened and you’ll almost always hear the same answer: silly mistakes. It’s worth taking that seriously and then not believing it, because silly and careless both describe a student who wasn’t paying attention, and that is rarely who is sitting there.
What’s usually happening is that nothing is being done the same way twice. They start one problem by drawing, the next by reaching for a formula, the next by rearranging whatever’s on the left. Sometimes the improvisation works. Under time pressure on a test, it stops working, and the results look random because they are.
So the useful question isn’t whether a student knows the chain rule. It’s what they do in the first fifteen seconds after reading a question they don’t immediately recognize. That’s the thing I teach, and it’s the thing that transfers to problems neither of us has seen.
The useful question isn’t whether they know the chain rule. It’s what they do in the first fifteen seconds.
What I cover
Foundations
Grades 8–10
Algebra I and II, geometry, and trigonometry. Linear and quadratic equations, polynomials, rational expressions, systems, exponentials and logarithms, triangle geometry, circles, coordinate geometry, the unit circle, identities, and graphing.
Gaps that open here stay open. Nearly every calculus student I work with who is struggling has an algebra problem underneath it.
Pre-college
Grades 10–12
Precalculus and AP Calculus AB/BC. Functions and their properties, analytic geometry, sequences and series, limits and continuity, differentiation and its applications, integration, differential equations, and the BC additions: series, parametric and polar.
AP Calculus rewards a settled approach more than knowing more calculus.
University
Early undergraduate
Multivariable calculus, linear algebra, and discrete mathematics. Vector functions, partial derivatives, multiple integrals, line and surface integrals, Green’s and Stokes’ theorems, vector spaces, linear transformations, eigenvalues and eigenvectors, logic and proof, combinatorics, and graph theory.
This is coursework I’m in the middle of rather than coursework I once took.
Competition
AMC, AIME
Number theory, combinatorics, olympiad-style geometry, and proof technique. Competition math is a different sport from coursework — it rewards being comfortable with a problem you have no method for, which is nearly the opposite of what school trains. Worth doing for students who enjoy it, and not worth forcing on students who don’t.
How a session runs
An hour, built around the first fifteen seconds.
Last week’s work, marked for method
Ten minutes. Not whether the answers were right — whether the same opening move got used twice.
The unit that’s currently blocking
Sometimes a gap two years back that Precalculus has only now exposed, in which case we go back and fix it, because continuing over the top of it doesn’t work.
A cold problem, narrated from the first read
Something the student hasn’t seen, so there’s nothing to pattern-match to. Explaining things at them feels productive and mostly produces students who can follow a solution but can’t start one. Then a short assignment, under thirty minutes.
Common questions
They’re in Precalculus but I think the problem is older than that.
You’re probably right, and it’s the most common shape of this. If the algebra is shaky we rebuild it, even mid-year, even though it feels like going backwards. Continuing on top of a foundation that doesn’t hold makes every subsequent unit harder, not easier.
How long before we see something?
How a student approaches a problem changes in two or three weeks and you can see it if you look at their work. Course grades take four to eight, because a grade averages in everything that happened before we started. If nothing has moved by week six, the plan is wrong and I’ll tell you so rather than wait for you to ask.
Do you just help with the homework that’s due?
We’ll use current assignments, since that’s what’s in front of the student and it’s where the real errors live. But finishing tonight’s problem set isn’t the goal, and a session that only does that has failed even if the homework gets turned in.
Do you allow calculators and AI tools?
Not while a concept is still being built, because the struggle is where the learning is. Once the mental model holds, both are fine and worth learning to use well — the exams allow calculators and university work assumes them.
None of this is only my opinion. The Common Core standards for mathematics treat conceptual understanding, procedural fluency, and application as three equal requirements rather than a ranking, and national NAEP results show the widest score gaps at the proficient and advanced levels — the levels that need reasoning rather than recall.