How Frontier works, and how to learn with it

Everything Frontier teaches lives in one graph: 371 topics across 7 courses (Arithmetic & Pre-Algebra, Algebra, Functions & Precalculus, Calculus, Linear Algebra, Probability & Statistics, Mathematics for Machine Learning), split into 813 knowledge points. A knowledge point is one taught step, with its own worked example and practice.

Arrows in the graph say what you need first. A topic only opens once you know everything it builds on, so you're never asked to stand on ground you haven't built. A second set of links records how much of each simpler skill an advanced topic exercises. That one is what lets hard work count as review of easy work (section 4).

Problems aren't stored, they're generated: every knowledge point has a generator with exact answers, so practice never runs out and you'll rarely see the same question twice. Answers are checked for meaning rather than formatting, so 1/2, 0.5 and 2/4 are the same number unless a question asks for a particular form, like lowest terms.

In the appThe Map shows the whole graph, coloured by course and by how well you know each topic. Select a topic to see what it needs and what it leads to.

Your knowledge frontier is every topic whose prerequisites you've mastered. New lessons only come from there: new enough to be worth learning, close enough to what you know to be doable. A lesson is a few knowledge points, each the same small loop.

1. 1Why it matters. A short reason first. Some topics open with a question to predict, or sliders to play with, before any rules.
2. 2A worked example. Revealed one step at a time. Try each step on paper before you show it.
3. 3Practice. Fresh problems of the same kind. Two right in a row and you move to the next knowledge point.
4. 4Done. The topic joins what you know, and the lesson tells you what it just opened up.

If two in a row stops being possible within five questions, the lesson pauses. It comes back after a break with fresh problems, and you work on other things meanwhile, which genuinely helps. If the same step trips you up twice, you'll first get reviews of the key earlier skills it relies on.

Memory fades on a curve, and it fades a little slower each time you successfully recall something. Frontier keeps an estimate of your memory for every topic you know. It halves over a set interval, and when it reaches the review line (50% unless you change it), the topic comes back. Each review done on time roughly doubles the interval: about 1½ days, then 3, 6, 12, 24, 48 and onwards.

Why wait until it's fuzzy? Recalling something that has become a little hard strengthens it far more than rereading it while it's fresh. That's also why a review can feel harder than the lesson did: it's meant to. Reviews done early count for less, for the same reason.

A review is three fresh questions on one topic, often including one that asks you to find the mistake in a worked solution, explain why a step works, or predict a result. Three right passes; two right passes if the last one was right; otherwise a fourth question decides.

In the appSettings → Memory strength moves the review line: Efficient (40%), Standard (50%), Strong (60%) or Exam mode (70%). Higher means more reviews and crisper recall. The Today page's Memory card shows what's fading fastest.

Advanced math uses simpler math. When you practise multiplying and dividing fractions, you're also simplifying fractions and factoring numbers. Frontier knows this: each advanced topic records what share of each simpler skill it exercises, and practising it passes that much credit down as an automatic partial review.

So when reviews come due, Frontier first looks for a new lesson whose practice would cover them, and offers you the lesson instead. Over its lifetime, most topics need only a handful of explicit reviews. Credit flows down, and trouble flows up: missing a simple skill nudges the topics built on it, so they come back sooner. Topics you find hard refuse this free credit and always get real reviews.

The idea is known as fractional implicit repetition, and the sources below describe it. Frontier's model, weights and numbers are its own.

The list on the Today page is rebuilt from your memory model whenever things change:

1. 1Find what's due. Every topic at or below its review line.
2. 2Cover it with new lessons. Prefer lessons whose practice reviews the most due topics at once.
3. 3Review the rest. Anything left becomes a review, at most three at a time, so new lessons keep coming.
4. 4Fill up with lessons. Up to five tasks: shallow topics before deep ones, ones that unlock a lot, and different from each other so similar ideas don't blur together.
5. 5Mix in the rest. A timed quiz about every 150 XP and a multistep problem about every 100 XP.

Every task says why it's there. Pick any one; they're all worth doing. Coming back after four or more days with a pile of reviews? A single welcome-back check replaces them: one question per topic, chosen so that each right answer also refreshes the simpler topics underneath it.

In the appHalf-done lessons wait in a Continue card at the top. On the Map, today's tasks wear their numbers from the list.

Practising one kind of problem at a time feels productive but doesn't teach you to recognise which method a problem needs. Mixing them does: in a classroom trial, interleaved practice beat blocked practice by a wide margin (an effect size of 0.83).

Quizzes are timed (about two and a half minutes a question), closed-book and mixed: mostly topics you've learned since the last quiz, plus a few older ones that are fading. Expect to miss some; around 80% is a good score, and every missed topic gets a review straight away. Multistep problems chain three or more topics the way real problems do. Understanding checks ask you to find the wrong step in a worked solution, explain why something works, or predict before you calculate. Explaining your reasoning is one of the better studied ways to learn (a meta-analysis put the effect at about 0.55).

• Placement. An adaptive test (12 to 45 questions, depending on how far your path goes) asks the question that teaches it the most each time. A right answer credits everything underneath; a wrong one casts doubt on what's built on top. Topics credited on thin evidence are marked as such, and turn back into lessons if you struggle with them.
• How fast you learn each topic. A perfect lesson starts with a longer first interval; topics you find hard get shorter ones and more explicit practice.
• How fast you forget. Once you've answered 30 review questions, Frontier compares how your answers hold up over time with what its model expected, and stretches or shrinks your review gaps to match. It only moves when the evidence is clear, and you can switch it off.
• Your goals. Pick a target course, set a focus topic to fast-track everything it needs, or switch to maintenance mode (reviews only) for a busy week.

One XP is about one minute of focused work, so XP measures effort, not luck. A lesson is worth about three XP per knowledge point, scaled by how many you got right, with a bonus for a perfect run; a review is worth four. Set a daily goal for each day of the week, including rest days (a streak skips them). Frontier forecasts when you'll finish your path at that pace.

Rushing costs. A wrong answer in under four seconds counts as rushed, and from the third one in a task each costs an XP. Guessing fast teaches nothing, and the XP stays honest.

When the AI tutor is switched on, it works as a ladder of hints: a nudge, then the next step, then one worked step. It never gives the final answer before you submit. After you answer, you can ask “why?” about any step of the solution, or “where did I go wrong?”. The checker, not the AI, decides what's right. A nudge keeps full credit; once a step is shown, that answer counts as practice instead.

The ladder is deliberate. In a study with nearly a thousand students, open access to a chatbot raised practice scores but lowered later exam scores; a tutor limited to hints avoided the harm. “Try a similar one” gives you a fresh problem of the same kind, checked the same way, whenever you want another go.

Frontier is built for a phone first: big answer boxes, a math keyboard, and a row of keys for fractions, powers and roots. It follows your device into dark mode, and it works without a connection. Open it once online and today's tasks are saved on the device; your answers sync when you're back, and a task already finished on another device isn't counted twice. Signing out clears that saved copy from the device.

Frontier is a personal project: a free math notebook for working from the basics up to the mathematics of machine learning, built on the most efficient method we know of and made kinder to use. Three ideas guide it.

• Keep what works. Mastery learning on a knowledge graph, spaced and implicit review, interleaving, and lots of problems over lots of reading. Justin Skycak's writing about Math Academy inspired the idea; Frontier is an independent project with its own curriculum, problems and design, and isn't affiliated with it.
• Add what was missing. The “why” behind each topic (intuition first, examples step by step, explanations on demand), a map you can see, a reason for every task, settings instead of rigid rules, a gentler return after a break, a phone-first design with dark mode, and offline use.
• Stay honest. The constants in the memory model are educated guesses, so Frontier measures them against your own answers. Every generator is tested on dozens of random problems, and every lesson is played start to finish by automated tests. We don't claim results we haven't measured.

1. 1A little, most days. Twenty to forty minutes beats a weekend marathon. Choose a goal you can keep, and use rest days.
2. 2Keep paper next to you. Work every step on paper and type only the answer. Math is learned through the pen.
3. 3Try before you look. In a worked example, attempt each step before revealing it. The effort is the learning.
4. 4Slow is fast. A careful right answer is worth more than three quick guesses, and rushed misses cost XP.
5. 5Ask for the smallest hint first. A nudge keeps full credit. Save shown steps for when you're truly stuck.
6. 6Hard reviews are working. If a review feels easy, it came early. Fuzzy is the point.
7. 7A paused lesson isn't a failure. It comes back with fresh problems after a break. Do something else meanwhile.
8. 8Take quizzes cold. No notes, no peeking. Around 80% means the quiz is finding what to strengthen.
9. 9Use the map. See why a topic matters and where it leads. Set a focus topic when you have a goal.
10. 10Back after a break? Start with the welcome-back check. It clears the backlog quickly.
11. 11Busy week? Maintenance mode keeps what you know without new lessons.