Maths Solver-
What a proof actually is, and why mathematicians won’t let it go
A proof is not a “convincing argument.” It is the difference between believing something and knowing it. A worked example, three styles of proof, and why even non-mathematicians benefit from learning to write one.
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How SAT Math scoring actually works on the digital test
The two-module adaptive structure, what your score on Module 1 does to Module 2, and why the “raw to scaled” conversion is more interesting than the College Board lets on.
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The week before a maths exam: what I would and would not do
The case against last-minute revision, what to actually do with the seven days, and the morning-of routine I have given to every Year 11 I have ever taught.
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Simple vs compound interest, and why the difference is bigger than you think
Linear growth vs exponential growth, with a year-by-year table for $1000 at 5% over thirty years. The number that comes out at the end is genuinely surprising.
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Pi Day, and the things people say about pi that aren’t true
No, pi was not discovered in 1988. No, the digits are not random. And no, you do not need to memorise more than four of them to be a working scientist. What pi actually is, in 1500 words.
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Graphing calculator vs Desmos vs phone app: what to actually buy
A TI-84 costs £100 and Desmos is free. Here is when each one is the right choice, what your exam will let you bring, and the calculator habits that quietly cost students marks.
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AP Calculus AB or BC? An honest comparison from a former TA
BC is not just “harder AB.” It is AB plus a coherent block of new material that some students will love and others will find pointless. How to decide which one to take.
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Why 0! is 1 (and not 0, like everyone first thinks)
A two-line answer is “because the recursion forces it,” but the better answer is “because the empty arrangement is still an arrangement.” Both, with diagrams.
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What dx and dy actually mean (a calculus student’s guide)
Leibniz wrote dy/dx as if it were a fraction; Newton thought it was nonsense. Both were partly right. What the symbol really represents, and when you can get away with treating it like a fraction.
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Why we have imaginary numbers (it wasn’t to solve quadratics)
The story you usually hear — “mathematicians invented i because $x^2 + 1 = 0$ has no solution” — is wrong. The real history is more interesting and explains why complex numbers turned out to be so useful.
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What is the number e, explained without calculus
Most explanations of $e$ start with limits and derivatives, which makes the topic feel like it requires calculus to understand. It does not. Compound interest is enough.
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How I actually learn a new maths topic, in five steps
The method I started using in graduate school after wasting too many hours rereading textbooks at 1 AM. It works for everything from algebra to differential equations.
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Mental maths tricks that actually work, and the ones that don’t
The percentage swap, multiplying by eleven, squaring numbers ending in five — the tricks worth the ten minutes to learn. Plus the “Vedic maths” ones I would skip.
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How to read a maths textbook (it isn’t how you read a novel)
Forward, then backward, then forward again, with a pen. Why textbooks make sense on the third pass and not the first, and the trick that makes worked examples count for double.
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Why maths feels hard, and what that actually means
Most students who say they “can’t do” algebra have a gap two topics back, not a problem with algebra. The dependency tree of school maths and how to find your own missing piece.
Blog
Notes from two maths teachers. Half of these are answers to questions students keep asking us; the rest are things we wish someone had told us when we were learning the same material.