# Learning paths Compare goals and step counts, then open a path to read one step at a time. Learning order is not a formal proof. 1 [2](/en/paths/?page=2#results) [3](/en/paths/?page=3#results)[Next →](/en/paths/?page=2#results) 59 results · Page 1 of 3 Learning path · 6 steps ## [Geometry Foundations to the Pythagorean Theorem](/en/paths/path-geometry-foundations/?from=%2Fen%2Fpaths%2F) Move from the language of points, lines, and planes through triangles, angles, length, and right triangles to the Pythagorean theorem. path-geometry-foundations Learning path · 3 steps ## [From Sets and Functions to the Tends-To Relation](/en/paths/path-function-to-limits/?from=%2Fen%2Fpaths%2F) Establish a foundational chain from set membership through total functions to convergence semantics. path-function-to-limits Learning path · 6 steps ## [Modern General Mathematics Literacy](/en/paths/path-modern-general-literacy/?from=%2Fen%2Fpaths%2F) Use six current objects to demonstrate a branching entry path into general mathematical literacy. path-modern-general-literacy Learning path · 4 steps ## [Entry Path to Formalized Mathematics](/en/paths/path-formalized-mathematics-entry/?from=%2Fen%2Fpaths%2F) Move from set and function semantics into precise Lean and Mathlib statements and content-addressed evidence objects. path-formalized-mathematics-entry Learning path · 4 steps ## [Proof and Evidence Literacy](/en/paths/path-proof-and-evidence-literacy/?from=%2Fen%2Fpaths%2F) Identify proof obligations before distinguishing human proof, finite computational evidence, and exact formal proof. path-proof-and-evidence-literacy Learning path · 5 steps ## [Foundational Mathematical Language](/en/paths/path-foundational-language-candidate/?from=%2Fen%2Fpaths%2F) Move from the concrete language of counting to sets, propositions, and logical connectives, establishing the vocabulary needed to read later mathematical objects. path-foundational-language-candidate Learning path · 11 steps ## [Wave-One Domain Foundations](/en/paths/path-domain-coverage-wave-one/?from=%2Fen%2Fpaths%2F) Branch from natural numbers, sets, and propositions into arithmetic, equations, probability, induction, factorials, and elementary combinatorial counting. path-domain-coverage-wave-one Learning path · 4 steps ## [Set Language toward Topology](/en/paths/path-set-language-to-topology-candidate/?from=%2Fen%2Fpaths%2F) Build the set-language prerequisites needed to read the definition of a topological space, progressing through sets, membership, subsets, and basic set operations. path-set-language-to-topology-candidate Learning path · 5 steps ## [From Sample Spaces to Events](/en/paths/path-probability-events-candidate/?from=%2Fen%2Fpaths%2F) Use set language and sample spaces to establish the precise boundary of an event as a set of outcomes. path-probability-events-candidate Learning path · 5 steps ## [From Natural Numbers to Absolute Value on the Real Line](/en/paths/path-number-system-extension-candidate/?from=%2Fen%2Fpaths%2F) Move from natural numbers used for counting to integers and rational numbers, enter the real-number system through completion language, and then express number-line distance with absolute value. path-number-system-extension-candidate Learning path · 16 steps ## [Seven-Round Foundation Expansion: Numbers, Functions, Counting, and Coordinates](/en/paths/path-pulse-foundations-20260930/?from=%2Fen%2Fpaths%2F) Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. path-pulse-foundations-20260930 Learning path · 17 steps ## [Seven-Round Algebra and Probability Foundations: Powers, Identities, Functions, and Finite Models](/en/paths/path-pulse-foundations-20261003/?from=%2Fen%2Fpaths%2F) Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. path-pulse-foundations-20261003 Learning path · 17 steps ## [Daily Seven-Round Cycle I: Quantifiers, Discrete Structures, Linear Systems, and Conditional Models](/en/paths/path-pulse-foundations-20261004/?from=%2Fen%2Fpaths%2F) Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. path-pulse-foundations-20261004 Learning path · 15 steps ## [Function Invertibility and Finite Probability Moments: Seven-Round Workflow](/en/paths/path-pulse-foundations-20261005/?from=%2Fen%2Fpaths%2F) Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. path-pulse-foundations-20261005 Learning path · 20 steps ## [Real Order, Average Rates, Coordinate Geometry, and Data Summaries: Seven-Round Workflow](/en/paths/path-pulse-foundations-20261006/?from=%2Fen%2Fpaths%2F) Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. path-pulse-foundations-20261006 Learning path · 17 steps ## [Matrices, Real 2×2 Linear Algebra, Labeled Triangles, and Coordinate Transformations: Seven-Round Workflow](/en/paths/path-pulse-foundations-20261007/?from=%2Fen%2Fpaths%2F) Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. path-pulse-foundations-20261007 Learning path · 18 steps ## [Real Limits, Continuity, Derivatives, and Polynomial Integration: Seven-Round Workflow](/en/paths/path-pulse-foundations-20261008/?from=%2Fen%2Fpaths%2F) Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. path-pulse-foundations-20261008 Learning path · 21 steps ## [Final Core80 Seeds and Five Rounds of Existing-Family Depth](/en/paths/path-pulse-foundations-20261009/?from=%2Fen%2Fpaths%2F) Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. path-pulse-foundations-20261009 Learning path · 17 steps ## [Cross-Domain Depth: Algebra, Sets, Distances, Finite Probability, and Quantifiers](/en/paths/path-pulse-foundations-20261010/?from=%2Fen%2Fpaths%2F) Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices. path-pulse-foundations-20261010 Learning path · 9 steps ## [Foundation Gaps: Natural Numbers, Expressions, Sets, and Finite Limit Models](/en/paths/path-pulse-foundations-20261011/?from=%2Fen%2Fpaths%2F) Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices. path-pulse-foundations-20261011 Learning path · 9 steps ## [Foundational Reasoning and Geometry: Induction, Counting, Typing, and Length](/en/paths/path-pulse-foundations-20261012/?from=%2Fen%2Fpaths%2F) Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices. path-pulse-foundations-20261012 Learning path · 14 steps ## [Finite Structures: Pythagoras, Number Systems, Inference, Events, and Labeled Triangles](/en/paths/path-pulse-foundations-20261013/?from=%2Fen%2Fpaths%2F) Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices. path-pulse-foundations-20261013 Learning path · 14 steps ## [Foundation Transforms: Angles, Number Lines, Equations, Factorials, and Circle Boundaries](/en/paths/path-pulse-foundations-20261014/?from=%2Fen%2Fpaths%2F) Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices. path-pulse-foundations-20261014 Learning path · 16 steps ## [Typed Data and Selection: Fractions, Ratios, Maps, Compositions, and Coordinates](/en/paths/path-pulse-foundations-20261015/?from=%2Fen%2Fpaths%2F) Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices. path-pulse-foundations-20261015 1 [2](/en/paths/?page=2#results) [3](/en/paths/?page=3#results)[Next →](/en/paths/?page=2#results) 59 results · Page 1 of 3