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Use of AI in Mathematical Education

Use of AI in Mathematical Education Research-Oriented Table of Contents Part I — What Exactly Is Being Represented? 1. Mathematical knowledge as a carrier 1.1 Standard representation: curriculum as ordered topics; hidden assumptions: monotone progression, stable prerequisites, one dominant decomposition. 1.2 Knowledge graphs: nodes = concepts, edges = prerequisite/implication/dependence. 1.3 Learning spaces: admissible knowledge states K ⊆ 2 Q K\subseteq 2^Q , allowing multiple legal paths through the same domain. 1.4 Hypergraph representations: prerequisites may be conjunctive, disjunctive, compensatory, or context-dependent. 1.5 Concept lattices and Galois structures: concepts represented through shared attribute closure rather than curricular sequence. 1.6 Equivalence boundary: sequence ≃ \simeq DAG only when prerequisite relation is effectively total or near-total; DAG ≄ \not\simeq learning space when alternative acquisition paths matter. 1.7 Degenerate cases: one-concept domain...
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Semantic Cloud 2

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  Semantic Cloud Research-Oriented Table of Contents 0. The Entire Primitive Alphabet 0.1 ḓ := THIS | NOT-THAT 0.2 Why distinction precedes object, identity, relation, context, space, probability and semantics 0.3 Orientation already implicit in THIS | NOT-THAT 0.4 Negation/complement: what exactly is NOT-THAT ? 0.5 Self-application: can distinction distinguish distinctions? 0.6 Closure problem: C l o s u r e ( ḓ ) Closure(ḓ) 0.7 Minimality criterion: p p is primitive only if p ∉ C l o s u r e ( ḓ ) p\notin Closure(ḓ) I. First-Order Consequences of Distinction 1.1 Difference from distinction 1.2 Boundary: THIS || NOT-THAT 1.3 Identity as absence of consequential distinction 1.4 Multiplicity from repeated distinction 1.5 Possibility domain U U as accumulated distinctions, not a primitive container 1.6 Inclusion, exclusion and complement 1.7 Symmetry versus oriented distinction II. Distinctions Acting on Distinctions 2.1 ḓ 1 → ḓ ( ḓ 1 ) ḓ_1\rightarrowḓ(ḓ_1) 2.2 Relations as distinc...
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Semantic Cloud

  Semantic Cloud Research TOC 0. Primitive Alphabet 0.1 Possibility domain U \mathcal U 0.2 Primitive distinction ḓ ( x , y ) : = x ∣ ¬ y ḓ(x,y):=x|\neg y 0.3 Context/perturbation C t C_t 0.4 Active distinctions D t D_t 0.5 Frozen invariants I t I_t 0.6 Refinement ⪯ \preceq , incompatibility ⊥ \perp , conditional independence ∥ \parallel 0.7 Directed transition a → b a\to b 0.8 Directional cost ω t ( a → b ) \omega_t(a\to b) 0.9 Propagation limiter m m 0.10 Surviving influence π t ( a → b ) \pi_t(a\to b) 0.11 Active relation A t = { ( a , b ) : π t ( a → b ) > 0 } A_t=\{(a,b):\pi_t(a\to b)>0\} 0.12 Epistemic type ϵ \epsilon 0.13 External discriminator Γ \Gamma 0.14 Witness W W , debt δ \delta , residue ρ \rho , counterkernel ϰ \varkappa 0.15 Persistent state Σ \Sigma 0.16 Terminal algebra Θ \Theta   I. Fundamental Laws 1.1 ¬ ḓ ⇒ \negḓ\Rightarrow no semantics 1.2 Meaning is distinction-relative, not token-intrinsic 1.3 Directionality: ω ( a → b ) ≠ ω ( b → a ) \omega(a\t...
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Statistical Estimation and Prediction

  Statistical Estimation and Prediction Table of Contents Part I — The pre-statistical situation: what must exist before statistics is possible 1. Why statistical reasoning begins with multiple possible states of the world 1.1 How uncertainty requires more than one possible state 1.2 How a realized state differs from the alternatives that could have occurred 1.3 Why the set of imagined possibilities can itself be incomplete 1.4 How impossible, possible, plausible, and observed states differ 1.5 Why probability cannot be introduced before possibilities have been distinguished 1.6 What happens when the assumed possibility space excludes the actual state 2. How one realized world becomes only partially accessible to an observer 2.1 The difference between what exists and what becomes observable 2.2 How observation exposes some properties while hiding others 2.3 Why every observation system performs selection and transformation 2.4 How measurement resolution determines which distinction...
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