Learn
Nineteen skills behind the site’s tools, measured by an adaptive check and kept up by practice and spaced review. Separate from the courses: think of it as a quick gauge of where you stand.
Nineteen skills behind every tool on this site, each with a short lesson and a bank of questions, eighty in all. A short adaptive check finds your level, practice goes to what you haven’t mastered, and skills come back for review just as you’re likely to forget them. Your progress stays in this browser.
One short lesson (a plate) per skill: the idea, the formula, a worked example, a calculator to try, and three questions that count toward your mastery here. For full courses, go to the study home.
Start here
The first plate. Or take the check below, and this will point you to your weakest skill.
Six to twelve questions, each chosen to tell the most about your level given your answers so far. Right answers bring harder questions; wrong ones, easier.
Answer a question to start the estimate.
One question at a time. Skills due for review come first; otherwise the skill you have mastered least. Every explanation links to the tool that covers it.
One petal per skill. A petal grows with the chance you have mastered the skill and turns gold at 95%. A dot on a petal means it is due for review.
Three models from the research on learning, each small enough to run in your browser. Try them below.
Each question has a difficulty b, a sharpness a and a guessing floor c. The chance that someone of ability θ answers correctly is
P(θ) = c + (1 − c) / (1 + e−a(θ − b))
After each answer the check updates its estimate of θ (the average over a bell-shaped prior, weighted by how well each θ explains your answers) and asks next the question carrying the most information at that estimate: I(θ) = P′(θ)² / P(θ)(1 − P(θ)).
Each skill is either mastered or not, and only your answers hint at which. After an answer the chance of mastery is updated with Bayes’ rule, allowing for slips (a master answering wrong, 8%) and guesses (a beginner answering right, 20%), then for the chance you learned something on the way (15%).
after a right answer: P(L | ✓) = P(L)(1 − s) / [P(L)(1 − s) + (1 − P(L)) g]
Your starting mastery is individualised: the placement estimate shifts every skill’s prior, σ(logit prior + θ).
Memory fades on a curve. The chance you still recall a skill after Δ days is 2−Δ/h, where the half-life h grows with right answers and shrinks with wrong ones:
h = 2√right − √wrong − 0.5 × difficulty days
A skill comes back for review when the predicted chance of recall falls to 50%, which is exactly one half-life after you last practised it.
What these numbers are, and aren’t. The question parameters and model settings here are reasoned starting values, not calibrated from learners: calibrating item response theory needs hundreds of people answering the same questions, and the half-life weights in Settles and Meeder’s model were learned from millions of reviews. Treat your estimate as a guide to what to practise, not a grade.