LEARN
Build understanding before you drill.
Use lessons, the prerequisite-aware Learning Map, the full curriculum, and your deep study plan from one organized starting point.
4learning destinations
PERSONAL MATHEMATICS ENVIRONMENT
Dashboard
Your adaptive study command center.
0day streak
0%overall mastery
0reviews due
STUDY
Learn a concept
Move from explanation to worked reasoning without leaving the existing lesson system.
ORIENT
See where the math connects
Choose a destination or inspect the complete prerequisite structure before starting.
Evidence boundary: opening a Learn destination never awards mastery. Learning evidence still comes from the existing lesson, practice, review, diagnostic, checkpoint, and exam systems.
PRACTICE
Choose the right kind of evidence.
Build a study session, target one skill, retrieve due material, recalibrate with diagnostics, or run a no-hint exam simulation.
5practice destinations
FOCUSED WORK
Practice and review
MEASURE
Diagnostics and exams
LABS
Explore mathematics by manipulating it.
Fourteen released lab destinations are grouped by mathematical role instead of competing for permanent space in the primary navigation.
14lab destinations
FOUNDATIONS
Number, algebra, and geometry
DATA & UNCERTAINTY
Statistics and probability
FUNCTIONS & CHANGE
Precalculus through advanced calculus
APPLIED & COMPUTATIONAL
Finance, linear algebra, numerical methods, and ML math
PROGRESS
Inspect evidence without inventing a story.
Use recorded-evidence analytics for the current learner state and attempt history for the underlying events.
2progress destinations
ANALYZE
Progress intelligence
TRACE
Attempt history
No synthetic trends: missing evidence remains missing evidence. These surfaces do not reconstruct historical mastery states that were never stored.
TOOLS
Use support when the mathematics calls for it.
Tutor, Notebook, and the Calculator Suite are grouped here as utilities rather than mixed with progress or curriculum navigation.
3core tools
EXPLAIN & RECORD
Tutor and Notebook
COMPUTE
Calculator Suite
Open the existing scientific, finance, graphing, and time calculators without creating another calculator product.
TODAY'S OBJECTIVE
Build mathematical fluency through deliberate practice.
Take the placement diagnostic so Math OS can establish your starting skill map.
FOCUS AREA
Calibrating…
--
Your highest-priority skill will appear here as evidence accumulates.
Retention model awaiting evidence.
ADAPTIVE COMMAND CENTER
Your next moves, ranked from current evidence.
Math OS reconciles mastery, confidence, retention, prerequisites, mistakes, checkpoints, and today’s goals before choosing what deserves attention.
PRIORITY STACK
Now / Next / Later
DOMAIN PULSE
Where attention is concentrating
SUPPORTING EVIDENCE
Detailed progress, queues, and goals
The command center above is the primary decision surface. These panels expose the underlying learner record in more detail.
TODAY'S PLAN
Today's plan
LEARNING SUMMARY
Current progress
DAILY TARGET
Session goals
0 / 20questions today
0 / 30active minutes today
MISTAKE PATTERN
Most common mistake
No misconception pattern yet. Wrong answers will be classified and accumulated here.
RETENTION
What needs retrieval
CHECKPOINTS
Skills ready to prove
MASTERY MAP
Highest-priority skills
PLACEMENT DIAGNOSTIC
Find your real starting point.
Math OS samples twelve foundation subskills twice. The second pass adapts upward or downward from your first response.
24questions
12subskills
0hints
CURRICULUM MAP
Your mathematics path, from foundations to advanced work.
Math OS organizes all 62 skills into prerequisite-aware domains so you can see what is ready, what is developing, and what should come next.
ROADMAP
Domains and prerequisites
RECOMMENDED LESSON
Calibrating…
DOMAIN READINESS
Curriculum readiness map
Internal readiness across the Math OS curriculum. These are learning signals, not official exam scores.
CURRICULUM
Lessons and mastery checkpoints
Start practice
Choose today’s plan, adaptive practice, a specific skill, or a word-problem session.
Word-problem sessions translate real situations into equations before solving.
SPACED RETRIEVAL
Your review queue is current.
RETENTION MODEL
Review signals
PRIORITY QUEUE
What needs retrieval next
MISTAKE INTELLIGENCE
Recurring error patterns
INTELLIGENT SESSION BUILDER
Choose review depth
Each plan uses the same evidence model with a different time budget. Review activity changes scheduling only through real question outcomes.
REVIEW SCHEDULE
Skill-by-skill retention
SKILLS
Mastery, retention, confidence, and prerequisites
LEARNING MAP 2.0
Prerequisites, bottlenecks, deep progress, and unlock paths
Navigate the real 62-skill prerequisite graph. Map signals are read-only views of existing learner evidence.
NUMBER SENSE
See how numbers are built, related, and scaled.
Explore place value, signed magnitude, factors, equivalent ratios, percentages, and powers of ten without leaving the core mastery system.
NUMBER SENSE MASTERY
Current signals
PLACE VALUE
Build the number from positional units
SIGNED NUMBER LINE
Magnitude and direction
FACTOR STRUCTURE
GCF, LCM, and primes
EQUIVALENT REPRESENTATIONS
Fraction → decimal → percent
SCIENTIFIC NOTATION
Powers-of-ten scale
ALGEBRA
Transform structure without breaking equivalence.
Build expressions, solve equations and inequalities, compare systems, factor quadratics, and connect standard form with vertex form.
ALGEBRA MASTERY
Current signals
EXPRESSION BUILDER
Distribute, combine, verify
Model k(x + b) + mx.
EQUATION SOLVER
Legal transformations
Solve ax + b = c.
SYSTEMS EXPLORER
Two equations, one intersection
a₁x + b₁y = c₁ and a₂x + b₂y = c₂.
FACTORING
Monic quadratic structure
Factor x² + bx + c when integer factors exist.
COMPLETING THE SQUARE
Standard form → vertex form
INEQUALITY EXPLORER
Track when the comparison reverses
VISUAL FRACTIONS
See the numerator over the denominator
Use the sliders to visualize a fraction as equal parts of a whole.
RATIO SCALING
Equivalent ratios visually
See how a ratio scales while staying equivalent.
Original
Scaled
EQUATION BALANCE
Visualize why solving equations works
Choose values for a two-step equation of the form ax + b = c. Math OS will build a balanced equation and show the solving path.
Left side
=
Right side
FUNCTION GRAPHER
Explore linear and quadratic functions
SYNCHRONIZED REPRESENTATION
Fraction · decimal · percent · number line
ALGEBRA TILES
Distribution as signed area and tile counts
GEOMETRY & COORDINATE MATHEMATICS
Connect formulas to pictures and coordinates.
Use the labs for visual understanding, then send the same topics into the mastery engine for practice and spaced review.
GEOMETRY MASTERY
Current signals
COORDINATE PLANE
Distance, midpoint, and slope
PYTHAGOREAN LAB
Right triangles
MEASUREMENT LAB
Area, perimeter, and volume
ANGLE RELATIONSHIPS
Complementary, supplementary, and triangles
TRANSFORMATION EXPLORER
Translate, reflect, and rotate a point
DYNAMIC TRIANGLE
Coordinates → sides → angles → area
CIRCLE EXPLORER
Radius controls the family
STATISTICS · PROBABILITY · DATA
Turn raw numbers into evidence.
Build statistical intuition with real calculations, visual distributions, probability experiments, and adaptive mastery practice.
STATISTICS MASTERY
Current signals
DATA LAB
Describe a dataset
Enter numbers separated by commas, spaces, or new lines. Math OS calculates center, spread, and a histogram from the same data.
DISTRIBUTION
FREQUENCY TABLE
PROBABILITY LAB
Theory versus simulation
Z-SCORE EXPLORER
Standardize an observation
A z-score tells you how many standard deviations an observation is above or below the mean.
REGRESSION & RESIDUALS
Fit the data, then inspect the misses
SAMPLING DISTRIBUTION · SHARED SIMULATION ENGINE
Repeated samples reveal the behavior of sample means
Define a population, draw many samples of equal size, and compare the observed distribution of sample means with its theoretical center and standard error.
Run repeated samples to compare observed variability with the theoretical standard error.
v0.35 · STATISTICAL INFERENCE & RESAMPLING
Move from descriptive samples to uncertainty-aware inference.
These modules build on the existing Sampling Distribution Explorer, Distribution Engine, and seeded Simulation Engine. They separate a statistic, its sampling variability, an interval procedure, and a hypothesis-test model.
Interpretation boundary: a confidence level describes the long-run behavior of an interval procedure. A p-value measures compatibility with a null model; it is not the probability that the null hypothesis is true.
CENTRAL LIMIT THEOREM
Sampling mean and standard error
Key distinction: σ describes individual observations; σ/√n describes the spread of repeated sample means. The bell curve shown is exact for normal populations and a CLT approximation when its conditions are appropriate.
MEAN CONFIDENCE INTERVAL
Known-σ z interval and margin of error
ONE-SAMPLE z TEST
Distance from a null model in standard-error units
p-value: under the null model, the probability of a test statistic at least as extreme in the direction(s) defined by the alternative. It is not P(H₀ is true).
PROPORTION INFERENCE
Wilson confidence interval + null-model z test
Approximation check: the z test relies on a sufficiently large null-model expected-success and expected-failure count; the Wilson interval is used for interval estimation because it behaves better than the simple Wald interval near boundaries.
POWER & SAMPLE SIZE
Detectable effects under a specified alternative
Tradeoff: for fixed α, effect size, and population variability, larger n generally raises power by reducing standard error.
SEEDED BOOTSTRAP
Resample the observed data with replacement
CONFIDENCE-INTERVAL COVERAGE
Repeat the procedure, not the interpretation of one interval
Long-run meaning: before sampling, a 95% procedure is designed to cover the fixed population parameter in about 95% of repeated samples under the model assumptions.
v0.36 · ADVANCED STATISTICAL INFERENCE
Extend inference beyond known-σ one-sample models.
This wave adds Student t methods, independent and paired comparisons, two-proportion inference, chi-square, ANOVA, and regression slope inference while preserving the existing Statistics Lab and shared engines.
Model discipline: every method has assumptions. The interface reports the mathematics, but learners must still check independence, sampling design, distributional shape, expected counts, and residual conditions where applicable.
UNKNOWN σ
One-sample Student t inference
Why t? replacing unknown σ with sample s adds uncertainty, producing heavier tails and a critical value that depends on degrees of freedom.
INDEPENDENT MEANS
Welch two-sample t inference
PAIRED DESIGNS
Analyze within-pair differences
Design matters: paired inference is about the distribution of within-pair differences, not two independent groups.
TWO PROPORTIONS
Difference in population proportions
Two standard errors: the interval uses unpooled sample proportions; the null test for equal proportions uses a pooled estimate.
CATEGORICAL COUNTS
Chi-square goodness of fit
Approximation condition: this implementation requires every expected count to be at least 5.
MULTIPLE MEANS
One-way ANOVA
Interpretation: a small ANOVA p-value indicates that the equal-means model is inconsistent with the observed between-group variation relative to within-group variation; it does not identify which group means differ.
REGRESSION INFERENCE
Slope uncertainty and t test
Inference conditions: slope inference relies on an appropriate linear model, independent observations, stable residual variance, and residual behavior suitable for t-based inference.
PROBABILITY · COMBINATORICS
Count possible worlds, then measure uncertainty.
Build from counting rules into conditional probability, Bayes’ theorem, expected value, and binomial models.
PROBABILITY MASTERY
Current signals
COUNTING EXPLORER
Permutations versus combinations
Change n and r to compare ordered selections with unordered groups.
CONDITIONAL PROBABILITY
Restrict the sample space
Use counts to compare conditional probabilities in both directions and check independence.
EXPECTED VALUE
Weight outcomes by probability
BINOMIAL DISTRIBUTION
Exactly k successes in n trials
SEEDED BERNOULLI QUICK CHECK
Theoretical probability vs long-run frequency
Run a reproducible Bernoulli experiment through the shared Simulation Engine.
MONTE CARLO · COIN & DICE
Watch experimental probability converge toward theory
Choose an experiment and compare a seeded simulation with its exact theoretical probability.
MONTE CARLO π
Approximate π through random area sampling
The circle occupies π/4 of its surrounding square. Random points estimate that area ratio.
DISTRIBUTION EXPLORER
Link parameters, theory, and sampled outcomes
Compare theoretical mean and spread with a reproducible simulated sample.
RANDOM WALK
From local step probability to a distribution of paths
For p = 0.5 the theoretical terminal mean remains at the starting value; changing p creates drift.
FINANCIAL MATHEMATICS
Model the value of money across time, cash flows, and portfolios.
Connect time value of money to compound growth, annuities, loans, bonds, investment returns, and weighted portfolio returns.
FINANCE MASTERY
Current signals
TIME VALUE OF MONEY
Present value → future value
Change the compounding assumptions and see how time and frequency change a lump-sum investment.
ANNUITY LAB
Repeated contributions
Model equal end-of-period deposits and separate contributions from investment growth.
LOAN AMORTIZATION
Payment and interest cost
BOND PRICING
Discount coupons and face value
RETURN LAB
Holding-period return and CAGR
PORTFOLIO MATH
Weighted expected return
Two-asset portfolio weights are normalized automatically so the relationship stays mathematically valid.
GROWTH COMPARISON
Simple · compound · continuous
RECURRING CONTRIBUTIONS
Contribution timing matters
INFLATION
Nominal vs real purchasing power
FEE DRAG
Gross vs net compounding
AMORTIZATION DETAIL
Payment schedule
v0.33 · SHARED CASH-FLOW ENGINE
Move every cash flow to a common point in time.
These modules share one validated financial core. A cash-flow timeline, annuity, loan, bond, or project valuation is a different curriculum experience built on the same time-value mathematics.
Model: cash flows are valued using the configured rate and timing convention. Interpretation: the result is a mathematical equivalence under those assumptions. Limitation: the calculation does not determine whether a real financial decision is appropriate.
CASH-FLOW TIMELINE
Choose a focal date and move every flow there
ANNUITY TIMING
Ordinary, due, and growing cash flows
Invariant: beginning-of-period payments have one additional period to compound. Growing annuities also change the payment stream itself, so payment growth and investment growth are distinct assumptions.
AMORTIZATION + EXTRA PAYMENT
Measure payoff acceleration, not just payment size
Uses the principal, APR, term, and frequency from the Loan Amortization panel above.
VARIABLE-RETURN GROWTH
Arithmetic average is not compound growth
Concept: arithmetic mean describes the average single-period return. Geometric mean / CAGR describes the constant compound rate that reproduces the same beginning-to-ending growth path.
CAPITAL BUDGETING
NPV, IRR, payback, and discounted payback
Calculation vs judgment: NPV and IRR summarize cash flows under a stated discount-rate model. They do not capture every operational, strategic, financing, or uncertainty consideration in a real project.
v0.34 · PORTFOLIO RISK ENGINE + FINANCIAL MONTE CARLO
Connect return, volatility, correlation, diversification, and uncertainty.
These experiences reuse the v0.32 seeded Simulation Engine and extend the v0.33 Finance architecture with a pure portfolio-risk core. Results are mathematical models under stated assumptions, not investment recommendations.
Model boundary: expected return, volatility, VaR, and Monte Carlo outcomes summarize assumptions or observed samples. They do not forecast a security, guarantee outcomes, or decide whether an investment is appropriate.
DIVERSIFICATION EXPLORER
Two assets: expected return, covariance, and portfolio volatility
Invariant: expected return is linear in normalized weights. Portfolio variance is not linear because covariance terms connect the assets. Lower correlation can reduce volatility without changing either asset's standalone volatility.
HISTORICAL TAIL RISK
VaR and Expected Shortfall from a return sample
Interpretation: historical VaR is a sample quantile, while Expected Shortfall averages the observations in the selected loss tail. Both inherit the limitations of the supplied sample.
SEQUENCE-OF-RETURNS RISK
Same returns, different order, different cash-flow outcome
Concept: with no intermediate cash flows, reversing multiplicative returns leaves the final product unchanged. Contributions or withdrawals make the path matter because different dollar balances experience each return.
MONTE CARLO WEALTH SIMULATION
Seeded paths, percentile fan, and goal probability
Simulation assumption: annual growth factors are sampled from a lognormal model calibrated to the entered arithmetic expected return and volatility. Reusing the same seed reproduces the same pseudo-random experiment exactly.
PRECALCULUS · TRIGONOMETRY
Understand functions before calculus asks you to change them.
Train transformations, polynomial behavior, exponential and logarithmic functions, right-triangle trigonometry, the unit circle, and sinusoidal graphs.
PRECALCULUS MASTERY
Current signals
FUNCTION TRANSFORMATIONS
Move, stretch, and reflect a parent function
UNIT CIRCLE
Connect angle, coordinates, sine, and cosine
EXPONENTIAL & LOGARITHMIC
Functions and their inverses
POLYNOMIAL BEHAVIOR
Degree, leading coefficient, and end behavior
TRIG GRAPH LAB
Amplitude, period, and vertical shift
FUNCTION MACHINE
Rule → input → table → output
TRIGONOMETRY
Connect triangle ratios, circle coordinates, and periodic functions.
Move between SOH-CAH-TOA, exact standard-angle values, the unit circle, and the full transformed sinusoid.
TRIG MASTERY
Current signals
RIGHT TRIANGLE
SOH · CAH · TOA
UNIT CIRCLE & EXACT VALUES
Coordinates and signs
SINUSOID TRANSFORMATIONS
Amplitude, period, phase shift, and midline
CALCULUS FOUNDATIONS
Study change and accumulation as visual ideas before memorizing rules.
Build intuition for limits, average and instantaneous rate of change, derivatives, tangent lines, and accumulated area.
CALCULUS MASTERY
Current signals
LIMIT EXPLORER
Approach a value from both sides
SECANT → TANGENT
Watch average rate become instantaneous rate
DERIVATIVE RULES
Power rule with a live slope
ACCUMULATION LAB
Approximate area with Riemann rectangles
CONVERGENCE TABLE
Watch approximation error shrink
ACCUMULATION / FTC
Accumulated amount and current rate
LINEAR ALGEBRA + VECTORS
See equations, vectors, and matrices as transformations of space.
Build intuition for vector magnitude and operations, dot products, matrices, systems of equations, determinants, and linear transformations.
LINEAR ALGEBRA MASTERY
Current signals
VECTOR LAB
Add vectors, measure length, and compare direction
MATRIX LAB
Operate on 2×2 matrices
SYSTEMS SOLVER
Connect two equations to one intersection
x +y =
x +y =
LINEAR TRANSFORMATION
Watch a matrix reshape the plane
v0.37 · LINEAR ALGEBRA DEPTH WAVE
Connect vector geometry, matrix structure, transformations, eigendirections, and approximation.
This release deepens the Linear Algebra view already present in Math OS. It reuses the seven existing canonical Linear Algebra skills and adds a shared mathematical engine rather than creating a second lab architecture.
Core idea: vectors describe quantities in coordinate form; matrices act on those vectors; row reduction solves constraints; determinants describe area scaling and invertibility; eigenvectors expose invariant directions; least squares handles systems that cannot be solved exactly.
DOT PRODUCT + PROJECTION
Decompose one vector relative to another
Orthogonal decomposition: u = projᵥ(u) + residual, and the residual is perpendicular to v.
DETERMINANT + INVERSE
When does a matrix undo itself?
Invertibility: det(A)=0 means the transformation collapses dimension, so no inverse can restore every input uniquely.
GAUSSIAN ELIMINATION
Row-reduce an augmented system to RREF
Classification: RREF can expose one unique solution, free variables with infinitely many solutions, or a contradictory row that makes the system inconsistent.
SPAN + BASIS
Do two vectors create the whole plane?
2D shortcut: two vectors form a basis for R² exactly when their determinant is nonzero.
EIGENDIRECTIONS
Find directions a matrix only stretches or flips
Meaning: an eigenvector keeps its line of direction under the transformation. The eigenvalue tells how strongly that direction is scaled.
LEAST SQUARES
Find the best linear approximation when Ax=b has no exact solution
Bridge to statistics: ordinary least-squares regression is also a linear-algebra problem. The best-fit coefficients solve the normal equations when the design matrix has full column rank.
MULTIVARIABLE + DIFFERENTIAL EQUATIONS
Extend calculus from one variable into fields, gradients, and evolving systems.
Build intuition for multivariable functions, partial derivatives, gradients, first-order differential equations, slope fields, growth/decay models, and Euler's method.
ADVANCED CALCULUS MASTERY
Current signals
MULTIVARIABLE FIELD
Explore f(x,y), partial derivatives, and the gradient
SLOPE FIELD
Read a differential equation as local direction
GROWTH & DECAY MODEL
Study y′ = ky as continuous change
EULER METHOD
Approximate a solution one tangent step at a time
v0.38 · MULTIVARIABLE CALCULUS, OPTIMIZATION & ODEs
Deepen the Advanced Calculus view with gradients, Hessians, optimization, double integrals, and higher-accuracy ODE solvers.
This release promotes the existing Advanced Calculus view into the shared LabShell. It reuses the seven canonical Advanced Calculus skills and the v0.37 Linear Algebra Engine rather than creating another calculus architecture.
Concept bridge: gradients are vectors, Hessians are matrices, optimization uses both, and numerical ODE methods approximate continuous change by controlled finite steps.
GRADIENT + DIRECTIONAL DERIVATIVE
How fast does f change in a chosen direction?
Tangent plane: the two partial derivatives are the local x- and y-slopes, so the gradient determines the first-order plane approximation.
HESSIAN + OPTIMIZATION
Classify a critical point and follow gradient descent
Second derivative test: the Hessian describes local curvature. For a nondegenerate 2D critical point, its determinant and fₓₓ distinguish a local minimum, maximum, or saddle.
DOUBLE INTEGRALS
Accumulate a scalar field across a rectangle
Iterated accumulation: on rectangular regions, polynomial terms can be integrated exactly by integrating powers independently in x and y.
LOGISTIC DIFFERENTIAL EQUATION
Growth slows as the state approaches carrying capacity
Autonomous model: the slope depends on the current state y. Equilibria occur where y′=0, including y=0 and y=K.
NUMERICAL ODE METHODS
Compare Euler, midpoint, and RK4 on y′=y
Accuracy hierarchy: for a smooth problem and the same moderate step size, midpoint generally improves on Euler, while classical RK4 is dramatically more accurate. Reducing h provides a convergence experiment rather than a guarantee for every ODE.
v0.39 · NUMERICAL METHODS & SCIENTIFIC COMPUTING
Approximate difficult mathematical problems while measuring error, stability, and convergence.
This lab connects Algebra, Calculus, Linear Algebra, Probability, and Scientific Computing. Every algorithm exposes its assumptions and an error signal rather than treating a numerical answer as automatically exact.
Core discipline: a numerical method is not just an answer generator. It is an algorithm with a stopping rule, an error model, convergence behavior, and sometimes stability or conditioning limits.
ROOT FINDING
Bisection · Newton · Secant
Tradeoff: bisection is slow but robust with a valid sign-changing bracket; Newton can be very fast but depends on derivative behavior and the starting point; secant avoids an explicit derivative but can still fail.
NUMERICAL DIFFERENTIATION
Forward vs central differences
Observed order: central difference should show approximately second-order truncation error before floating-point roundoff dominates at extremely small h.
NUMERICAL INTEGRATION
Trapezoid vs Simpson
Exact reference: supported presets have analytic antiderivatives, so the lab can measure quadrature error rather than only displaying an approximation.
INTERPOLATION
Lagrange / barycentric polynomial interpolation
Interpolation vs regression: interpolation passes through every supplied point; least squares instead minimizes residual error when exact interpolation is not the goal.
ITERATIVE LINEAR SOLVER
Jacobi iteration + conditioning
Conditioning: κ₂(A) measures sensitivity of the linear system to relative perturbations. Convergence of Jacobi is a separate algorithmic question and is not guaranteed for every invertible matrix.
MONTE CARLO INTEGRATION
Seeded stochastic quadrature
Stochastic error: Monte Carlo standard error typically shrinks like 1/√n. Reusing a seed reproduces the pseudo-random experiment exactly; it does not remove sampling uncertainty.
v0.40 · MACHINE LEARNING MATHEMATICS FOUNDATIONS
See machine-learning models as compositions of linear algebra, calculus, probability, and statistics.
This lab teaches the mathematics underneath common model families rather than turning Math OS into a model-training service. Every probability, loss, gradient, and projection is exposed as a mathematical object.
Core structure: feature vectors enter linear combinations; losses measure mismatch; gradients describe how parameters change the loss; optimization updates the parameters; probability transforms support classification; eigendirections support dimensionality reduction.
FEATURE VECTORS
Cosine similarity + standardization
Similarity is geometry: cosine similarity compares direction rather than magnitude. Standardization instead changes feature scale by centering and measuring values in sample-standard-deviation units.
LINEAR MODEL
MSE loss + gradient descent
Optimization: the MSE gradient points uphill in parameter space, so gradient descent subtracts a learning-rate-scaled gradient. The learning rate controls step size, not model complexity.
LOGISTIC CLASSIFICATION
Sigmoid probability + binary cross-entropy
Probability vs decision: logistic regression produces a probability under its model. A classification threshold is an additional decision rule, not part of the sigmoid itself.
MULTICLASS PROBABILITY
Softmax + categorical cross-entropy
Numerical stability: subtracting the largest logit before exponentiation leaves softmax probabilities unchanged while avoiding overflow.
REGULARIZATION
L1 / L2 parameter penalties
What regularization changes: the penalty changes the optimization objective. L2 grows quadratically with weight magnitude; L1 grows linearly and is nondifferentiable at zero, where a subgradient convention is required.
PCA
Covariance eigendirections + explained variance
PCA is variance geometry: after centering the data, the covariance matrix’s leading eigenvector points along the direction of greatest sample variance. PCA is not a supervised predictor.
PERSONAL NOTEBOOK
Capture what you want to remember
NOTEBOOK SIGNALS
Your knowledge base
SAVED NOTES
Personal reference
KNOWLEDGE BACKLINKS
Where this skill appears in your record
WORKED-EXAMPLE COMPARISON
Your recorded work vs reference
MISTAKE REPLAY
Replay the latest recorded divergence
FORMULA LIBRARY
Core formulas worth knowing
J
PRIVATE LEARNER PROFILE
Jacob
College math progression
Stored locally in this browser
LEARNING SNAPSHOT
Your Math OS record
IDENTITY & GOAL
Personalize the tutor
STUDY DEFAULTS
Set your normal workflow
Profile changes stay on this device unless you export your Math OS progress.
KeyboardTab moves through controls. Enter submits answers. Escape closes Calculator or confirmation dialogs. Alt+C opens Calculator.
CLOUD IDENTITY
Account & cross-device sync
Optional account sync keeps the browser as the immediate/offline store while copying your learner state to your private Supabase account.
SYNC MODEL
Local saves never wait for the network. Cloud writes are debounced, RLS-scoped to your account, and a lossless state snapshot is retained for recovery across devices.
Cloud sync is optional. Anonymous/local mode remains fully functional.
CURRICULUM PATH
Your progress by domain
PRIVATE TUTOR
Ask, explain, diagnose
This standalone tutor is rule-driven and uses your local Math OS data. It does not send your messages to an external model.
WORKED-EXAMPLE TRAINER
Study a complete solution, one step at a time
MISCONCEPTION REMEDIATION
Patterns worth fixing
PERT-STYLE DRILL
12-question mixed drill
Mixed arithmetic, fractions, proportional reasoning, and algebra. No hints. Designed for fast test-prep calibration.
FOUNDATION SIMULATION
30-question exam
A broader no-hint simulation using the current Math OS foundation graph. It is not an official PERT exam or score predictor.
EXAM HISTORY
Recent simulations
CURRENT LEARNER STATE
Authoritative evidence model
Recorded evidence window30 days
PROGRESS SNAPSHOT
What the platform actually recorded
PERIOD COMPARISON
Current window vs preceding window
EVIDENCE SOURCES
Where recorded work came from
EVIDENCE TREND
Recorded attempt volume over time
DOMAIN EVIDENCE
Current state plus selected-window performance
SKILL DRILLDOWN
Timestamped evidence without reconstructed mastery history
EVIDENCE COVERAGE
What is represented in this view
SESSION OUTCOMES
Completed logs matching these filters
ERROR PROFILE
Most common misconception signals
DAILY GOALS
Configure training targets
INTERNAL UI REFERENCE
Math OS design system
This internal reference keeps new interface work aligned to the same tokens, component grammar, mathematical presentation, and state language used throughout Math OS. It is for component QA and regression detection rather than learner content.
v0.28 canonical
TOKENS
Color roles
Base--surface-base
Primary surface--surface-primary
Raised surface--surface-raised
Primary accent--accent-primary
Success--success
Warning--warning
Danger--danger
TYPOGRAPHY
UI hierarchy
METADATA / EYEBROW
Section title
Card title
Body text is concise, readable, and deliberately lower contrast than headings.
Secondary metadata · 13px
BUTTONS
Action hierarchy
FIELDS
Form controls
PROGRESS
Evidence and progress
Mastery · 72%
72%Mastery
87%Confidence
5Reviews due
MATHEMATICS
Semantic math presentation
x = −b ± √(b² − 4ac)2a
Display mathematics receives dedicated spacing and uses the shared mathematical font rather than ordinary UI typography.
STATES
Feedback language
Correct. Independent retrieval recorded.
Hint. Identify the inverse operation before calculating.
Review. This skill is due for retrieval.
Check the sign. The magnitude is correct but the sign rule changed.
EMPTY / LOADING
System states
No review items due.
Math OS will add retrieval work when evidence becomes due.
Math OS will add retrieval work when evidence becomes due.
Preparing evidence summary…
DENSE DATA
Table / record treatment
| Skill | Mastery | Confidence | Review |
|---|---|---|---|
| Fraction operations | 72% | 87% | 3 days |
| Linear equations | 91% | 93% | 14 days |
ATTEMPT LOG
Recent work and reasoning signals
0 attempts