Undergraduate Math Foundations LabPath
A first-year undergraduate math foundation that connects proof, linear algebra, discrete structures, multivariable thinking, numerical methods, and modeling.
Day 1
Teach the idea, run the model, defend the evidence.
Students start with a visual walkthrough, then use one clean control change to produce evidence they can explain, revise, defend, and transfer.
Mathematical Proof And Argument: Explore
Explore proof structure using mathematical proof and argument.
Open missionOne visible mathematical consequence per mission.
- Learn the structure with a visual walkthrough.
- Predict which control will change the model.
- Run one clean test and inspect the consequence.
- Explain the evidence in words, symbols, and context.
- Revise, defend, and transfer to a new situation.
Course depth with simulator-backed evidence.
Mathematical Proof And Argument: Explore
Invalid proof steps break under counterexample or missing premise checks.
Try itLinear Algebra As Transformation: Test
Matrix changes rotate, stretch, collapse, or preserve directions in visible space.
Try itMultivariable Surfaces And Gradients: Defend
Direction matters: the steepest change follows the gradient, not every visible path.
Try itDiscrete Structures And Algorithms: Explore
Small rule changes alter reachable states, runtime growth, and proof obligations.
Try itNumerical Methods And Error: Test
A method can converge, diverge, or become unstable depending on step size and assumptions.
Try itProbability, Stochastic Processes, And Risk: Defend
Individual paths vary, but long-run distributions and risk thresholds become visible.
Try it8 units with daily evidence loops.
Unit 1
Mathematical Proof And Argument
What makes a mathematical argument valid rather than persuasive-looking?
Mastery gate: Student constructs direct, contrapositive, contradiction, induction, and counterexample arguments.Unit 2
Linear Algebra As Transformation
How do matrices transform space and encode systems?
Mastery gate: Student interprets vectors, matrices, span, independence, eigenvectors, and systems as transformations.Unit 3
Multivariable Surfaces And Gradients
How do functions change across surfaces, contours, and directions?
Multivariable Surfaces And Gradients: ExploreExplore partial derivative using multivariable surfaces and gradients.Multivariable Surfaces And Gradients: TestTest gradient using multivariable surfaces and gradients.Multivariable Surfaces And Gradients: DefendDefend contour using multivariable surfaces and gradients.
Mastery gate: Student connects surfaces, contours, partial derivatives, gradients, and optimization.Unit 4
Differential Equations And Dynamical Systems
How do local change rules create global behavior?
Differential Equations And Dynamical Systems: ExploreExplore equilibrium using differential equations and dynamical systems.Differential Equations And Dynamical Systems: TestTest stability using differential equations and dynamical systems.Differential Equations And Dynamical Systems: DefendDefend phase portrait using differential equations and dynamical systems.
Mastery gate: Student analyzes slope fields, equilibria, stability, phase portraits, and model limits.Unit 5
Discrete Structures And Algorithms
How do graphs, recursion, and invariants power computation and reasoning?
Discrete Structures And Algorithms: ExploreExplore graph model using discrete structures and algorithms.Discrete Structures And Algorithms: TestTest recursion using discrete structures and algorithms.Discrete Structures And Algorithms: DefendDefend invariant using discrete structures and algorithms.
Mastery gate: Student uses graph models, recursion, complexity, invariants, and algorithm evidence.Unit 6
Numerical Methods And Error
How do approximations become trustworthy enough for real work?
Mastery gate: Student estimates error, convergence, stability, and algorithm limits.Unit 7
Probability, Stochastic Processes, And Risk
How do random processes create predictable long-run structures?
Probability, Stochastic Processes, And Risk: ExploreExplore stochastic process using probability, stochastic processes, and risk.Probability, Stochastic Processes, And Risk: TestTest expectation using probability, stochastic processes, and risk.Probability, Stochastic Processes, And Risk: DefendDefend risk using probability, stochastic processes, and risk.
Mastery gate: Student models stochastic transitions, expectation, variance, risk, and simulation evidence.Unit 8
Mathematical Modeling Capstone
How does a mathematician choose tools, defend assumptions, and communicate limits?
Mastery gate: Student builds a multi-tool model, validates assumptions, revises from evidence, and transfers responsibly.Evidence becomes action.
- Assign the next course mission to a class or misconception cluster.
- Inspect TeacherOS evidence by concept, representation, and transfer.
- Launch TeachProof practice for the hardest teaching move.
- Share readable parent/district evidence without reducing learning to completion.
Progress without reducing learning to completion.
- What Undergrad Math concept the student can explain
- Where the student is confusing structure, procedure, or interpretation
- The next best practice mission
- Undergrad Math mastery by unit
- Misconception resolution time