Mathematics Atlas | The Living Map of Mathematics at BTT

Build meaning before speed. Begin with observation and concrete representation; do not assume that one age implies one ability.

1. R01.01 Early quantity and comparison.
2. R01.02 Place value and operations.
3. R01.03 Multiplicative thinking.
4. R01.04 Fractions and decimals.
5. R01.05 Ratio, percentage and rate.
6. R01.06 Measurement and geometry.
7. R01.07 Data and mathematical language.
8. R01.08 Primary progression and transfer.

Completion test: Explain a relationship, solve a changed example and check it without copying.

Entry identified; coverage audit pending: Primary worked learning hub · Primary journey.

Separate the school route from the mathematical work and the examination environment. Entry: Primary number, language and proportional reasoning, with earlier dependencies repaired when needed.

1. R02.01 PSLE conceptual integration.
2. R02.02 PSLE question and paper architecture.
3. R02.03 Primary-to-Secondary transition.
4. R02.04 Lower-secondary algebra and graphs.
5. R02.05 Secondary geometry and data.
6. R02.06 G1, G2 and G3 pathways.
7. R02.07 Upper-secondary synthesis.
8. R02.08 Secondary assessment and transfer.

Completion test: Select the correct cohort and subject route, then complete a mixed problem with a check.

Entry identified; coverage audit pending: Curriculum overview · SEC subject routes · Secondary worked repair guides.

Use the existing A-Math diagnostic library and synthesis guides before commissioning overlapping explanations. Entry: secure algebra, functions, graph interpretation and mathematical reading.

1. R03.01 A-Math entry and prerequisites.
2. R03.02 A-Math algebraic synthesis.
3. R03.03 A-Math functions and trigonometry.
4. R03.04 A-Math calculus and geometry.
5. R03.05 A-Math examination independence.
6. R03.06 JC pathways and course scope.
7. R03.07 JC synthesis and modelling.
8. R03.08 University transition.

Completion test: Choose and justify a method across topics; state restrictions and verify the result.

Entry identified; coverage audit pending: Additional Mathematics directory · A-Math synthesis guides · JC Mathematics.

Map different routes without implying identical content or current class availability. Entry: the learner’s actual programme, prior knowledge and purpose.

1. R04.01 Integrated Programme routes.
2. R04.02 IB Mathematics routes.
3. R04.03 IGCSE and international routes.
4. R04.04 Polytechnic and technical mathematics.
5. R04.05 University service mathematics.
6. R04.06 Career mathematics.
7. R04.07 Adult numeracy and quantitative judgement.
8. R04.08 Re-entry and self-directed study.

Completion test: Choose a bounded learning route and demonstrate its intended practical or academic task.

Entry identified; coverage audit pending: Mathematics pathways · Adult tutorial.

School representation switching is not university representation theory. Begin with familiarity with the quantities or objects being represented.

1. R05.01 Mathematical language.
2. R05.02 Notation and symbol roles.
3. R05.03 Diagrams and bar models.
4. R05.04 Tables and graphs.
5. R05.05 Equations and inequalities.
6. R05.06 Coordinate and vector representations.
7. R05.07 Representation switching.
8. R05.08 Abstraction and modelling.

Completion test: Translate one relationship into two forms and explain what each form hides or reveals.

Entry identified; coverage audit pending: Representation Switching library.

Separate examples, conjectures, proofs and computational evidence. Entry: mathematical reading and elementary algebra.

1. R06.01 Statements and quantifiers.
2. R06.02 Implication and equivalence.
3. R06.03 Sets, relations and functions.
4. R06.04 Direct proof and counterexamples.
5. R06.05 Contradiction and contrapositive.
6. R06.06 Induction and recursion.
7. R06.07 Axioms, models and formal systems.
8. R06.08 Foundational limits.

Completion test: Produce a valid argument and identify why a tempting invalid converse or example fails.

Entry identified; coverage audit pending: Representation and verification corridor. This starting route is not a complete foundations course.

The common foundation underneath school and applied mathematics. Entry: concrete quantity and mathematical language.

1. R07.01 Number systems and magnitude.
2. R07.02 Arithmetic structure.
3. R07.03 Fractions, decimals and percentages.
4. R07.04 Divisibility and prime structure.
5. R07.05 Ratio, rate and proportion.
6. R07.06 Units and dimensional reasoning.
7. R07.07 Estimation, rounding and bounds.
8. R07.08 Financial and everyday arithmetic.

Completion test: Calculate reliably, interpret units and test whether the answer could be reasonable.

Entry identified; coverage audit pending: Number and place value · Ratio, rate and proportion.

Treat school topics as connected foundations, not miniature substitutes for every advanced theory. Entry: R07 number and proportion, with R05 representations.

1. R08.01 Expressions and algebraic transformations.
2. R08.02 Equations and inequalities.
3. R08.03 Polynomials and factorisation.
4. R08.04 Functions and graphs.
5. R08.05 Exponentials and logarithms.
6. R08.06 Sequences and recurrence.
7. R08.07 Trigonometric ratios and functions.
8. R08.08 Coordinate geometry and vectors.

Completion test: Choose a useful form, preserve equality or domain, and verify by another representation.

Entry identified; coverage audit pending: Algebra · Functions and graphs · Trigonometry.

A wrong answer is evidence; the earliest unstable step determines the repair. Entry: a learner attempt or a clearly specified task.

1. R09.01 Learner state and task intake.
2. R09.02 Prerequisite tracing.
3. R09.03 Diagnostic probes.
4. R09.04 Misconception and error analysis.
5. R09.05 Repair design.
6. R09.06 Independent checking.
7. R09.07 Transfer and delayed retrieval.
8. R09.08 Return and correction records.

Completion test: Distinguish competing causes, test a repair and verify transfer on a new task.

Entry identified; coverage audit pending: Mathematics diagnosis · Mathematical Lab · Diagnostic probe bank.

Retain the Engineer, Tutor and Tutorial series as distinct perspectives. Entry: a stated learner stage and evidence of current capability. Engineering terms are an explanatory model, not literal physical measurements of a child.

1. R10.01 Development from early years to adulthood.
2. R10.02 Capacity, load and reserve.
3. R10.03 Fluency and connected understanding.
4. R10.04 Tutor judgement and intervention.
5. R10.05 Tutorial design.
6. R10.06 Small-group orchestration.
7. R10.07 Independence and handover.
8. R10.08 Learning evidence and research limits.

Completion test: Describe what changed in independent capability rather than report attendance alone.

Entry identified; coverage audit pending: Engineer series · Tutor series · Tutorial series.

Practice should change a specified capability; assessment should reveal it. Entry: a bounded learning goal and the appropriate programme specification.

1. R11.01 Worked examples and fading.
2. R11.02 Retrieval and spacing.
3. R11.03 Variation and mixed practice.
4. R11.04 Question design and diagnostic assessment.
5. R11.05 Marking and mathematical communication.
6. R11.06 Timing and paper navigation.
7. R11.07 Error logs and feedback loops.
8. R11.08 Exam readiness and maintenance.

Completion test: Recover knowledge in mixed conditions, show defensible working and use feedback to change the next task.

Entry identified; coverage audit pending: Examination craft · Evidence and validation.

Tools support mathematical agency; they do not supply automatic correctness. Entry: the mathematical task, access needs and tool constraints.

1. R12.01 Calculators and calculator state.
2. R12.02 Dynamic geometry and graphing.
3. R12.03 Spreadsheets, code and computer algebra.
4. R12.04 Adaptive practice and diagnostic tools.
5. R12.05 AI assistance and verification.
6. R12.06 Accessible notation and alternative formats.
7. R12.07 Inclusive lesson and task design.
8. R12.08 Tool selection, privacy and evaluation.

Completion test: Use a suitable tool, verify its output and explain what remains the learner’s responsibility.

Entry identified; coverage audit pending: Mathematics technology atlas · Accessible mathematics technology.

Study finite structures through explicit counting rules, constructions and invariants. Entry: R06 proof basics and R07 exact arithmetic.

1. R13.01 Counting principles and bijections.
2. R13.02 Permutations and combinations.
3. R13.03 Recurrences and generating functions.
4. R13.04 Graph structure and connectivity.
5. R13.05 Trees, paths and networks.
6. R13.06 Matchings, colourings and flows.
7. R13.07 Extremal and probabilistic methods.
8. R13.08 Designs, enumerative structures and discrete geometry.

Completion test: Give a counting argument or graph algorithm and test it on a boundary case.

Illustrative entry identified; coverage audit pending: Graph connectivity in a local example. This application does not replace the theory.

An algorithm needs a correctness argument, a cost model and a stated domain. Entry: R06 logic, R13 discrete structures and elementary programming concepts.

1. R14.01 Algorithm specification and invariants.
2. R14.02 Asymptotic analysis.
3. R14.03 Searching, sorting and data structures.
4. R14.04 Greedy methods and dynamic programming.
5. R14.05 Graph and network algorithms.
6. R14.06 Randomised and approximation algorithms.
7. R14.07 Computability and reductions.
8. R14.08 Complexity classes and limits.

Completion test: Trace an algorithm, justify an invariant and distinguish runtime from mathematical correctness. Unresolved complexity questions remain labelled unresolved.

Entry identified; coverage audit pending: Mathematical computing.

Approximate answers need error, conditioning and stability information. Entry: R08 algebra and functions, with calculus or linear algebra when the method requires it.

1. R15.01 Floating-point arithmetic and round-off.
2. R15.02 Conditioning and numerical stability.
3. R15.03 Root-finding and nonlinear systems.
4. R15.04 Numerical linear algebra.
5. R15.05 Interpolation and approximation.
6. R15.06 Quadrature and numerical differentiation.
7. R15.07 Numerical differential equations.
8. R15.08 Simulation, reproducibility and verification.

Completion test: Compute an approximation, estimate its error and explain how a small input change affects it.

Applied entry identified; coverage audit pending: Applied algorithms gateway. Reconcile its numerical-method articles before commissioning a parallel library.

Keep this educational and mathematical. A toy construction is not a secure deployed system. Entry: R13 discrete mathematics, R19 modular arithmetic and R29 probability as needed.

1. R16.01 Information and entropy.
2. R16.02 Compression and source coding.
3. R16.03 Error-detecting and error-correcting codes.
4. R16.04 Finite fields and algebraic coding.
5. R16.05 Cryptographic mathematical foundations.
6. R16.06 Public-key concepts and protocols.
7. R16.07 Communication limits and decoding.
8. R16.08 Security claims and mathematical boundaries.

Completion test: Explain a mathematical guarantee and identify the assumptions outside that guarantee.

Prerequisite entry identified; coverage audit pending: Computational number theory. This is not certification of cryptographic safety.

Make vectors, maps, bases and inner products explicit before using matrix shortcuts. Entry: R08 algebra and R06 definitions and proof for the abstract route.

1. R17.01 Vector spaces, span and independence.
2. R17.02 Linear maps and matrix representations.
3. R17.03 Systems, rank and null spaces.
4. R17.04 Determinants and eigenstructure.
5. R17.05 Inner products and orthogonality.
6. R17.06 Spectral decompositions and singular values.
7. R17.07 Tensor, symmetric and exterior constructions.
8. R17.08 Applications and numerical connections.

Completion test: Describe a linear map in two bases and justify a decomposition with its conditions.

Entry identified; coverage audit pending: Vectors and coordinates · Matrices and operators example.

Use a common structural language without flattening distinct theories into one analogy. Entry: R06 proof and sets; R17 linear algebra for module and representation routes.

1. R18.01 Groups and group actions.
2. R18.02 Rings, ideals and polynomial algebra.
3. R18.03 Fields and Galois theory.
4. R18.04 Modules and associative algebras.
5. R18.05 Order, lattices and universal algebra.
6. R18.06 Non-associative algebra.
7. R18.07 Categories, functors and naturality.
8. R18.08 Homological algebra and K-theoretic viewpoints.

Completion test: Verify a structure’s axioms and explain the role of a homomorphism, quotient or invariant.

Illustrative entry identified; coverage audit pending: Groups and linear actions. Broader algebraic coverage still needs reconciliation.

Connect integer structure to algorithms, proofs, certificates and computational limits. Entry: R07 exact arithmetic; R06 proof and R18 algebra for advanced branches.

1. R19.01 Divisibility, Euclid and Bezout identities.
2. R19.02 Congruences, inverses and CRT.
3. R19.03 Primes, sieves and primality.
4. R19.04 Factorisation and discrete logarithms.
5. R19.05 Quadratic residues and reciprocity.
6. R19.06 Diophantine equations and approximation.
7. R19.07 Algebraic and analytic number theory.
8. R19.08 Elliptic curves and arithmetic geometry.

Completion test: Solve a bounded integer problem and verify the proposed answer independently.

Entry identified; coverage audit pending: Computational number theory entry · Continuation.

This is higher mathematics, distinct from changing between a school diagram and equation. Entry: R17 linear algebra and R18 groups, with analysis for continuous groups.

1. R20.01 Representations and equivalence.
2. R20.02 Invariant subspaces and reducibility.
3. R20.03 Characters and orthogonality.
4. R20.04 Restriction, induction and reciprocity.
5. R20.05 Tensor products and duality.
6. R20.06 Lie groups, Lie algebras and weights.
7. R20.07 Modular and non-semisimple representations.
8. R20.08 Applications across mathematics and physics.

Completion test: Translate a symmetry action into linear structure and justify a decomposition or explain its failure.

Entry identified; coverage audit pending: Representation theory entry · Continuation.

A drawing represents conditions; it does not create them. Entry: R08 coordinate geometry and R06 proof.

1. R21.01 Euclidean geometry and constructions.
2. R21.02 Similarity and transformations.
3. R21.03 Coordinate and analytic geometry.
4. R21.04 Affine and projective geometry.
5. R21.05 Convexity and polytopes.
6. R21.06 Discrete and computational geometry.
7. R21.07 Non-Euclidean geometries.
8. R21.08 Geometric problem solving.

Completion test: Prove a geometric relation or construct a counterexample independently of a suggestive picture.

Entry identified; coverage audit pending: Geometry and measurement · Constructions and loci.

Study continuity and global structure without assuming distance is always the primary object. Entry: R06 sets and proof, with R25 analysis for many introductory routes.

Completion test: Use a definition correctly and distinguish a topological property from a geometric measurement.

Batch coverage published; audit and bridge review pending. Use the Mathematics Hub topology guide batches for the live learning route.

Use coordinates locally while preserving geometric meaning globally. Entry: R17 linear algebra, R25 calculus and R22 topology.

Completion test: Calculate a local geometric quantity and explain its coordinate-independent meaning.

Batch coverage published; audit and bridge review pending. Use the Mathematics Hub differential-geometry guide batches for the live learning route.

Keep algebraic equations, geometric spaces and arithmetic questions connected but distinct. Entry: R18 commutative algebra, with R21 geometry and R22 topology as appropriate.

Completion test: Translate a small polynomial example between its algebraic and geometric descriptions.

Batch coverage published; audit and bridge review pending. Use the Mathematics Hub algebraic-geometry guide batches for the live learning route.

Limits explain when familiar calculations are justified and when they fail. Entry: R08 algebra and functions, with R06 proof for rigorous analysis.

Completion test: Control a limit or integral with the relevant assumptions and an instructive counterexample.

Batch coverage published; audit and bridge review pending. Use the Mathematics Hub calculus and real-analysis guide batches for the live learning route.

Use complex structure deliberately rather than transferring every real rule automatically. Entry: R25 calculus and analysis, with complex algebra.

Completion test: Apply an analytic method with its domain, singularities and contour assumptions stated.

Batch coverage published; audit and bridge review pending. Use the Mathematics Hub complex-analysis guide batches for the live learning route.

Move from finite vectors to function spaces while recording the additional analytic conditions. Entry: R17 linear algebra and R25 real analysis, with R26 when needed.

Completion test: Choose a function space, state the operator domain and justify a transform or convergence claim.

Batch coverage published; audit and bridge review pending. Use the Mathematics Hub functional/harmonic/operator analysis guide batches for the live learning route.

Solving, proving existence, predicting stability and approximating a trajectory are different jobs. Entry: R25 calculus, with R17 and R27 for advanced routes.

1. R28.01 Ordinary differential equations.
2. R28.02 Systems and phase portraits.
3. R28.03 Partial differential equations.
4. R28.04 Boundary and initial-value problems.
5. R28.05 Difference, delay and functional equations.
6. R28.06 Stability and bifurcation.
7. R28.07 Chaos and ergodic viewpoints.
8. R28.08 Analytic-numerical comparison.

Completion test: Specify a model, initial or boundary conditions and the meaning of its solution.

Illustrative entry identified; coverage audit pending: Quantum dynamics example. The broader differential-equation estate remains to be reconciled.

Probability begins with a model and sample space, not a confidence-sounding number. Entry: R07 fractions and proportion, with R25 and R06 for advanced routes.

1. R29.01 Sample spaces and probability laws.
2. R29.02 Conditional probability and independence.
3. R29.03 Random variables and distributions.
4. R29.04 Expectation, variance and transforms.
5. R29.05 Limit theorems and concentration.
6. R29.06 Markov chains and stochastic processes.
7. R29.07 Martingales, Brownian motion and stochastic calculus.
8. R29.08 Simulation and rare events.

Completion test: Build a probability model, check its conditions and interpret its uncertainty honestly.

Entry identified; coverage audit pending: Probability knowledge object · Sample spaces and independence.

Data analysis must preserve how the observations were produced. Entry: R29 probability, with R17 and R25 for more advanced methods.

1. R30.01 Data collection and exploratory analysis.
2. R30.02 Sampling and estimation.
3. R30.03 Intervals, tests and uncertainty.
4. R30.04 Regression and generalised models.
5. R30.05 Experimental design and causal reasoning.
6. R30.06 Time series, spatial and multilevel models.
7. R30.07 Bayesian and computational inference.
8. R30.08 Robustness, reproducibility and communication.

Completion test: State what the data support, quantify uncertainty and identify a plausible source of bias.

Entry identified; coverage audit pending: Statistics and data · Data and decisions.

An optimum is meaningful relative to an objective, feasible set and model. Entry: R08 functions, with R17, R25 and R13 for the chosen method.

1. R31.01 Objectives, constraints and modelling.
2. R31.02 Linear programming and duality.
3. R31.03 Convex optimisation.
4. R31.04 Nonlinear optimisation and constrained conditions.
5. R31.05 Discrete and integer optimisation.
6. R31.06 Calculus of variations and optimal control.
7. R31.07 Stochastic and robust optimisation.
8. R31.08 Operations research in practice.

Completion test: Formulate a problem, find or approximate a solution and check feasibility and optimality conditions.

Entry identified; coverage audit pending: Functions, optimisation and systems.

Distinguish descriptions of behaviour from prescriptions about what people should do. Entry: R29 probability and R31 optimisation where required.

1. R32.01 Decision theory and utility.
2. R32.02 Strategic games and equilibria.
3. R32.03 Cooperation, bargaining and allocation.
4. R32.04 Social choice and voting.
5. R32.05 Mechanism design and matching.
6. R32.06 Mathematical economics.
7. R32.07 Behavioural and bounded-rationality models.
8. R32.08 Decision under uncertainty.

Completion test: Explain a decision or equilibrium under explicit information, objective and behaviour assumptions.

Illustrative entry identified; coverage audit pending: Market allocation example.

A physical result needs a correct model and a return to measurable quantities. Entry: R17 vectors, R25 calculus and R28 differential equations.

1. R33.01 Kinematics and reference frames.
2. R33.02 Forces, momentum and energy.
3. R33.03 Lagrangian and Hamiltonian viewpoints.
4. R33.04 Rigid-body and multibody dynamics.
5. R33.05 Vibrations and stability.
6. R33.06 Signals and system models.
7. R33.07 Feedback and control.
8. R33.08 Engineering validation.

Completion test: Derive a bounded model, check dimensions and test a conservation or limiting case.

Entry identified; coverage audit pending: Calculus and kinematics.

The equations and their physical validity are separate matters to verify. Entry: R28 equations and R23 geometry where required.

1. R34.01 Continuum assumptions and conservation laws.
2. R34.02 Elasticity and solid mechanics.
3. R34.03 Fluid dynamics and transport.
4. R34.04 Waves, acoustics and electromagnetism.
5. R34.05 Heat and thermodynamics.
6. R34.06 Relativity and geometric physics.
7. R34.07 Multiscale and coupled systems.
8. R34.08 Boundary models and numerical validation.

Completion test: State the continuum assumptions, boundary data and the interpretation of the computed field.

Mapped; entry-point reconciliation pending. Existing applied articles must be checked before a missing-coverage decision is made.

Higher mathematics and enrichment, not a school syllabus shortcut. Entry: R17 complex linear algebra, R29 probability, and R27–R28 for deeper routes.

1. R35.01 States, amplitudes and measurement.
2. R35.02 Operators, spectra and evolution.
3. R35.03 Tensor products and entanglement.
4. R35.04 Density operators, channels and generalised measurements.
5. R35.05 Quantum information and entropy.
6. R35.06 Quantum algorithms.
7. R35.07 Symmetry, stabilisers and error correction.
8. R35.08 Statistical mechanics and many-body bridges.

Completion test: Compute a bounded example and state which mathematical and physical assumptions support it.

Entry identified; coverage audit pending: Quantum mathematics entry · Quantum algorithms and correction route.

Educational models, not personal medical advice or substitutes for domain evidence. Entry: R29 uncertainty, R28 dynamics and the relevant scientific context.

1. R36.01 Population and ecological dynamics.
2. R36.02 Epidemiology and health modelling.
3. R36.03 Biological networks and systems.
4. R36.04 Environmental and climate modelling.
5. R36.05 Geophysical and Earth-system mathematics.
6. R36.06 Astronomy and celestial mechanics.
7. R36.07 Inverse problems and parameter inference.
8. R36.08 Scientific validation and ethics.

Completion test: Fit or derive a bounded model, inspect assumptions and identify what observations could refute it.

Illustrative entry identified; coverage audit pending: Sampling and uncertainty in the rainforest.

Reuse BTT’s finance and banking algorithms library. Map the mathematics without giving investment recommendations. Entry: R07 financial arithmetic; R29–R31 and R15 for quantitative models.

1. R37.01 Cash flows, interest and amortisation.
2. R37.02 Discounting, curves and fixed income.
3. R37.03 Risk, probability and portfolios.
4. R37.04 Derivatives and stochastic models.
5. R37.05 Banking balance-sheet mathematics.
6. R37.06 Payments and market mechanisms.
7. R37.07 Numerical pricing and calibration.
8. R37.08 Model validation and governance.

Completion test: Explain a financial calculation, its assumptions and a scenario in which its interpretation fails.

Entry identified; coverage audit pending: Finance and banking algorithms gateway.

Use mathematics to inspect systems, not decorate technological claims. Entry: R17 linear algebra, R29–R31 uncertainty and optimisation, and R14 algorithms.

1. R38.01 Mathematics of machine learning.
2. R38.02 Data pipelines and measurement.
3. R38.03 Computer graphics and vision.
4. R38.04 Signal processing and communications.
5. R38.05 Networks, queues and logistics.
6. R38.06 Manufacturing, reliability and quality.
7. R38.07 Robotics and autonomous systems.
8. R38.08 Systems validation and human consequences.

Completion test: Identify the objective, data, constraints and validation needed for a system-level claim.

Entry identified; coverage audit pending: Mathematical computing route · Applied systems gateway.

Use the world as a source of mathematical questions without inventing observations or cultural claims. Entry: a concrete question, a suitable representation and sources for contextual facts.

1. R39.01 Everyday quantity and household decisions.
2. R39.02 Place, maps and navigation.
3. R39.03 Architecture and public space.
4. R39.04 Nature and local field inquiry.
5. R39.05 Art, pattern and design.
6. R39.06 Music, games and shared rules.
7. R39.07 History and philosophy of mathematics.
8. R39.08 Mathematics communication and citizenship.

Completion test: Return a mathematical result to the original situation with its limitations intact.

Entry identified; coverage audit pending: Mathematics of Bukit Timah Hill · Batik and mathematical pattern.

A research question can be explored rigorously without presenting a conjecture as a theorem. Entry: the relevant subject prerequisites and a precisely stated question.

1. R40.01 Problem formulation and literature mapping.
2. R40.02 Olympiad and inventive problem solving.
3. R40.03 Conjectures and counterexamples.
4. R40.04 Proof checking and formal verification.
5. R40.05 Computational experimentation.
6. R40.06 Cross-disciplinary bridges.
7. R40.07 Open problems and frontier surveys.
8. R40.08 Research communication and correction.

Completion test: Produce a reproducible argument, counterexample or bounded result, including what remains unknown.

Entry identified; coverage audit pending: Evidence and validation harness · Mathematical Lab. Their presence is not a claim of independent research certification.