Concept Explainer -- Domain Knowledge

Concept Explainer -- Domain Knowledge

Reference knowledge for generating rich, principle-first concept explanations within TTT sessions. Replaces simple teaching blocks with structured concept breakdowns when appropriate, while preserving the gap-driven, one-at-a-time delivery model of TTT.

Standalone Invocation Rule

When this skill is invoked directly (e.g., via /training-plan:concept-explainer or when a learner declares "not familiar" with a topic), ALWAYS deliver the Full 7-Section Concept Explanation at intermediate depth (default). NEVER use the abbreviated form for standalone invocations -- the abbreviated form is reserved exclusively for simple gaps within TTT session Teach phases. A standalone concept explanation must include all 7 sections: The Problem, Core Principles, Key Innovations, Worked Example, Intuitive Formalization, CTQ Mapping, and Conceptual Map.

The 7-Section Concept Explanation Structure

Every concept explanation follows this structure (7 sections). Sections may be abbreviated or omitted based on the adaptive depth logic below, but the Worked Example (3.5) is ALWAYS included -- it is the core teaching artifact.

Section 1: The Problem

What observation, need, or contradiction made this concept necessary?

• What was broken, paradoxical, or limiting?
• What did people try before, and why did it fail?
• What was at stake -- practical, theoretical, or both?

Write 1-3 paragraphs in plain language. The learner should feel the tension the concept resolves.

Within TTT: Frame the problem in terms of the learner's gap. "You got this wrong because [gap]. Here's why that matters..."

Section 2: Core Principles (3-5 principles)

For each principle:

### Principle N: [Name]
**Statement**: [One clear sentence]
**Formally**: $$[LaTeX equation]$$
**Example**: [Concrete, vivid illustration]

Order from most fundamental to most derived. At introductory level, omit the Formally line.

Section 3: Key Innovations (2-4 innovations)

### [Innovation Name] ([Person/Group], [Year/Era])
**What**: [What was introduced or discovered]
**Why it mattered**: [What problem it solved, what it unlocked]
**Example**: [Concrete worked illustration]

Order chronologically or by logical dependency.

Section 3.5: Worked Example

A FULLY DEVELOPED example that walks through the concept from start to finish. Every intermediate step is shown and annotated. Nothing is skipped. Nothing is left as "exercise for the reader."

The worked example is the heart of the teaching block. It must be:

• Complete: every algebraic step, every matrix multiplication, every word transformation -- shown explicitly
• Annotated: each step has a note explaining WHY that step is taken, not just WHAT is done
• Domain-appropriate: the format adapts to the domain (see domain templates below)

### Worked Example: [Title describing the specific scenario]

**Given**: [All input data, clearly stated with units]

**Step 1**: [Name of this step]
$$[LaTeX showing the operation]$$
> **Why**: [1-2 sentences explaining why this step is needed and what it accomplishes]
> **Note**: [Optional: gotcha, unit warning, common mistake at this step]

**Step 2**: [Name of this step]
$$[LaTeX showing the operation, with intermediate result]$$
> **Why**: [Explanation]

**Step 3**: [Continue until the final result]
$$[Final result with units]$$
> **Check**: [How to verify this result makes sense -- sanity check, order of magnitude, dimensional analysis]

**Result**: [Final answer, boxed or bold, with units and interpretation]

Rules for worked examples:

• NEVER skip intermediate steps. If a step involves expanding $(a+b)^2$, write $a^2 + 2ab + b^2$ explicitly.
• NEVER write "it can be shown that" or "after simplification" -- show the simplification.
• Every variable must be defined on first use with its units.
• The "Why" annotation explains the REASONING, not just restates the math in words.
• The "Note" annotation flags traps: unit mismatches, sign conventions, boundary conditions.
• At introductory level: use concrete numbers throughout, minimize symbols.
• At intermediate level: start with symbols, substitute numbers, show both.
• At advanced level: prove each step, show alternative approaches, discuss edge cases.

Domain Templates

Mathematics / Physics / Engineering: Show every algebraic manipulation, every matrix operation, every integral evaluation. Use LaTeX for all expressions. Annotate each transformation.

**Step 2**: Multiply the matrices
$$\mathbf{A}\mathbf{B} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \\ 6 \end{pmatrix}$$

Row 1: $1 \times 5 + 2 \times 6 = 5 + 12 = 17$
Row 2: $3 \times 5 + 4 \times 6 = 15 + 24 = 39$

$$\mathbf{A}\mathbf{B} = \begin{pmatrix} 17 \\ 39 \end{pmatrix}$$

> **Why**: Matrix-vector multiplication gives us the transformed coordinates. Each row of $\mathbf{A}$ acts as a weight vector that linearly combines the input components.
> **Note**: Row-by-column, NOT element-wise. A common error is multiplying corresponding elements (Hadamard product), which is a different operation.

Language Learning / Translation: Show word-by-word breakdown, morphological analysis, grammatical annotations, then natural translation.

**Step 1**: Word-by-word analysis
| Original | Root | Grammar | Literal meaning |
|----------|------|---------|-----------------|
| Il | -- | definite article, masc. sing. | the |
| ragazzo | ragazzo | noun, masc. sing. | boy |
| mangia | mangiare | verb, 3rd pers. sing., presente indicativo | eats |
| la | -- | definite article, fem. sing. | the |
| mela | mela | noun, fem. sing. | apple |

> **Why**: Italian word order here is SVO (Subject-Verb-Object), same as English. The article agrees in gender with its noun: "il" (masc.) for "ragazzo", "la" (fem.) for "mela".
> **Note**: "Mangia" alone tells you the subject is he/she/it -- Italian often drops the pronoun entirely ("Mangia la mela" = "[He] eats the apple").

**Step 2**: Structural translation
Il ragazzo | mangia | la mela
The boy    | eats   | the apple

**Result**: "The boy eats the apple."

Cost Engineering / Business: Show every line item, every subtotal, every percentage application with the base it applies to. Use SI/ISO units throughout (kg, m, m2, m3, t for tonnes).

**Step 1**: Calculate direct costs
| Item | Quantity | Unit | Rate | Cost |
|------|----------|------|------|------|
| Concrete | 450 | m3 | $180/m3 | $81,000 |
| Rebar | 38.0 | t | $2,100/t | $79,800 |
| Formwork | 1,200 | m2 | $45/m2 | $54,000 |
| **Direct subtotal** | | | | **$214,800** |

> **Why**: Each line is Quantity x Unit Rate. Units must cancel: m3 x $/m3 = $.
> **Note**: Verify that your rate card units match your quantity takeoff units. Rebar may come in kg or tonnes (1 t = 1,000 kg) -- always confirm which unit the rate card uses and convert if necessary.

**Step 2**: Add indirect costs (itemized, NOT a percentage)
| Item | Basis | Cost |
|------|-------|------|
| Site security | 6 months x $4,500/month | $27,000 |
| Builder's risk insurance | 0.8% of direct costs | $1,718 |
| Mob/demob | lump sum | $35,000 |
| **Indirect subtotal** | | **$63,718** |

> **Why**: Indirects are project-wide costs not traceable to a single scope item. Each must be estimated individually -- never use a blanket percentage.
> **Note**: Insurance is calculated on direct costs ($214,800 x 0.008 = $1,718), not on the total. Applying it to the total including itself creates a circular reference.

Computer Science / Programming: Show every line of logic, every variable state change, every iteration of a loop.

**Step 1**: Initialize variables

array = [3, 1, 4, 1, 5] n = 5 swapped = True

> **Why**: Bubble sort needs a flag to know when to stop. If no swaps occur in a full pass, the array is sorted.

**Step 2**: First pass (i = 0)

Compare array[0]=3 and array[1]=1 --> 3 > 1, SWAP --> [1, 3, 4, 1, 5] Compare array[1]=3 and array[2]=4 --> 3 < 4, no swap Compare array[2]=4 and array[3]=1 --> 4 > 1, SWAP --> [1, 3, 1, 4, 5] Compare array[3]=4 and array[4]=5 --> 4 < 5, no swap

> **Why**: Each pass "bubbles" the largest unsorted element to its correct position. After pass 1, the last element (5) is guaranteed in place.
> **Note**: We made 2 swaps, so swapped=True and we need another pass.

Section 4: Intuitive Formalization

NOT the most general form -- minimum math needed to make predictions.

### Starting Point
[The simplest mathematical statement, with all terms defined]

### Building Up
[Step-by-step development, each step grounded in earlier examples]

### Key Equations
| Name | Equation | What it tells you |
|------|----------|-------------------|
| [name] | $[LaTeX]$ | [plain-language meaning] |

At introductory level, this section is a brief paragraph with minimal notation.

Section 5: CTQ Mapping

| CTQ | Source | Mastery Test | Common Failure Mode |
|-----|--------|-------------|---------------------|
| [specific understanding] | Principle N / Innovation N | [question or task] | [failure pattern] |

See the CTQ Derivation Method and Failure Mode Taxonomy below for how to populate this table.

Section 6: Conceptual Map

Markdown nested-list visualization of the concept's structure. See the Concept Map Format section below.

---

Depth Levels

Level	Math	Principles	Worked Example	Innovations	CTQ
introductory	Minimal notation, intuition-first	3, plain-language only	Concrete numbers throughout, all steps shown, plain-language annotations	2, brief	Short table (2-3 rows)
intermediate	Full LaTeX throughout	3-5 with LaTeX	Symbols + numbers, all steps with LaTeX, "Why" and "Note" annotations	2-4 with worked examples	Full table with mastery tests
advanced	Proofs, edge cases, competing forms	5 with derivations	Multiple approaches shown, edge cases explored, proof annotations	4 with detailed proofs	Advanced failure modes + edge cases

Default within TTT sessions: intermediate.

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Adaptive Teaching: When to Use What

Abbreviated Concept Explanation (for simple gaps)

When a gap is narrow, concrete, and does not involve deep conceptual misunderstanding:

---

**The Problem**
<1-2 sentences: why this gap matters, framed in the learner's context>

**Core Principles**
<1-2 principles, plain language, no LaTeX>

**Worked Example**
<Short but COMPLETE worked example: 2-4 steps, every step shown, annotated with "Why" notes. No steps skipped. Domain-appropriate format.>

**Quick Check** (CTQ: [source principle])
<CTQ-derived verification question>
(Expected answer: <answer>)
(Watch for: <failure mode>)

---

Even in abbreviated form, the worked example must be COMPLETE -- every intermediate step shown and annotated. It is just shorter (fewer steps, simpler scenario) than in the full form.

Full Concept Explanation (for complex gaps)

When a gap involves conceptual confusion, wrong mental models, or multi-layered misunderstanding, deliver all 6 sections at the appropriate depth level.

Concept Comparison (for conflation gaps)

When the gap involves confusing two related concepts, use the comparison format (see Concept Comparison Protocol below) instead of a standard explanation.

Decision Criteria

A gap is complex if ANY of these apply:

1. The gap matches a CTQ failure mode pattern (conflation, overgeneralization, missing prerequisite, procedural-without-conceptual)
2. The gap is flagged as a persistent pattern in the learner profile (2+ occurrences)
3. The gap involves a prerequisite the learner is missing
4. The gap was a "verbal-without-formal" or "formal-without-intuitive" failure

A gap is a conflation specifically if the learner confused two related but distinct concepts.

Otherwise, the gap is simple -- use the abbreviated form.

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Units Convention

All examples and worked examples use SI / ISO units by default:

• Mass: kg, t (metric tonnes, 1 t = 1,000 kg)
• Length: m, mm, km
• Area: m2
• Volume: m3, L
• Force: N, kN
• Pressure: Pa, kPa, MPa
• Temperature: C (or K for thermodynamics)
• Currency: $ (or the learner's currency)

Never use imperial units (lb, ft, in, gal, F) unless the learner explicitly requests them or the domain requires them (e.g., US aviation uses feet).

LaTeX Conventions

All mathematical expressions use LaTeX notation:

• Inline: $...$
• Display (centered, separate line): $$...$$
• Variables in prose: $x$, $\theta$, $\mathbf{v}$

Standard notation reference:

Category	Notation
Vectors	$\mathbf{v}$, $\vec{x}$
Matrices	$\mathbf{A}$, \begin{pmatrix}...\end{pmatrix}
Partial derivatives	$\frac{\partial f}{\partial x}$
Integrals	$\int_a^b f(x)\,dx$
Summations	$\sum_{i=1}^{n}$
Greek letters	$\alpha$, $\beta$, $\theta$, $\lambda$
Set notation	$\in$, $\subset$, $\cup$, $\cap$
Probability	`$P(A
Norms	$|x|$, $|A|_F$

At introductory level, use minimal notation and explain every symbol on first use.

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CTQ Derivation Method

CTQ (Critical-To-Quality) criteria define what the learner MUST understand to claim mastery. They are verifiable -- not topics to study, but criteria to meet.

Good CTQ: "Can derive the relationship between entropy and the number of microstates" Bad CTQ: "Understand entropy" (too vague, not verifiable)

Step 1: Trace from Principles

Principle	->	CTQ
Energy is conserved	->	Can track energy transformations across heat, work, and internal energy in a closed system
Direct vs indirect costs	->	Can classify any project cost line item as direct or indirect with justification

Step 2: Trace from Innovations

Innovation	->	CTQ
Boltzmann's $S = k_B \ln W$	->	Can calculate entropy from microstate counting for a simple system
Earned Value Method	->	Can compute CPI and SPI from BCWP, BCWS, and ACWP

Step 3: Identify Failure Modes

For each CTQ, identify the most likely way a learner will fail it. See the Failure Mode Taxonomy below.

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CTQ Failure Mode Taxonomy

Failure Pattern	Description	Example
Conflation	Mixing up two related but distinct concepts	Confusing indirect costs with contingency
Overgeneralization	Applying a principle beyond its valid domain	Assuming all nested loops are $O(n^k)$
Missing prerequisite	Lacking foundational understanding needed for this concept	Cannot classify costs because unfamiliar with WBS
Procedural without conceptual	Can follow steps but cannot explain why they work	Applies the formula correctly but cannot explain when it breaks
Verbal without formal	Can describe in words but cannot write the equation	Explains entropy conceptually but cannot write $S = k_B \ln W$
Formal without intuitive	Can manipulate symbols but cannot explain what results mean	Solves the integral but cannot interpret the physical meaning

When identifying gaps in the EVALUATE phase, classify each gap using this taxonomy:

• "Gap [conflation]: confused indirect costs with contingency"
• "Gap [procedural-without-conceptual]: applied formula correctly but cannot explain why it works"

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CTQ-Enhanced Quick Check Design

Quick Checks are now derived from CTQ mastery tests:

**Quick Check** (CTQ: [source principle or innovation])
[Question from the CTQ's Mastery Test column]
(Expected answer: [answer])
(Watch for: [failure mode pattern from CTQ table])

The Quick Check question comes directly from the CTQ table's "Mastery Test" column. The "Watch for" line alerts the evaluator to the most likely failure mode for this specific check.

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Concept Comparison Protocol

Triggered when a gap is classified as conflation -- the learner confused two related but distinct concepts.

Format

---

## [Concept A] vs [Concept B]

### What They Share
| Shared Principle | In [Concept A] | In [Concept B] |
|-----------------|----------------|----------------|
| [principle] | [how it manifests] | [how it manifests] |

### Where They Diverge
| Dimension | [Concept A] | [Concept B] |
|-----------|-------------|-------------|
| Core assumption | [what A assumes] | [what B assumes] |
| Primary strength | [where A excels] | [where B excels] |
| Typical use case | [when to use A] | [when to use B] |

### CTQ: The Critical Distinction
| CTQ | Mastery Test | Failure Mode |
|-----|-------------|-------------|
| Can distinguish [A] from [B] in [context] | [specific test] | Conflation: [specific confusion] |

**Quick Check** (CTQ: distinction)
[Question that specifically tests the A vs B distinction]
(Expected answer: [answer])
(Watch for: conflation -- [specific confusion to watch for])

---

When to Use

Use instead of a standard teaching block when:

• The gap is classified as [conflation]
• The learner explicitly confused two specific concepts
• The evaluation identified "confused X with Y" as a gap

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Deep-Dive Protocol

During a TTT session, the learner may request deeper explanation of a principle just taught.

How It Works

1. Identify the principle or innovation to expand (from the most recent teaching block or concept map)
2. Generate a full 6-section explanation for that specific principle, at the session's depth level
3. Frame Section 1 (The Problem) in context of the parent concept: "Within [parent concept], [this principle] matters because..."
4. Deliver the deep-dive inline -- it does NOT interrupt the session
5. After the deep-dive, return to the TTT flow (next gap, or Phase 3.5 if all gaps done)

Rules

• Deep-dives are offered, never forced: mention the option once after the first teaching block
• Only one deep-dive per teaching block (prevent rabbit holes)
• The deep-dive is included in the session transcript and concept file
• Deep-dives can chain: a deep-dive may reveal new principles the learner can further explore

---

Concept Map Format

Node Types

Type	Meaning	Usage
(core)	Central concept being mapped	Exactly one per map
(principle)	Foundational truth or constraint	The 3-5 load-bearing ideas
(innovation)	Conceptual breakthrough or key contribution	Creative leaps that built the concept
(application)	Practical use or derived concept	Where theory meets reality
(prerequisite)	Required prior knowledge	What you need before this concept
(shared)	Common to both concepts in a comparison	Only used in comparison maps

Relationship Labels

Label	Meaning	Direction
rests on	Built on these principles	core -> principle
requires	Needs these prerequisites	core -> prerequisite
led to	Produced these innovations	principle -> innovation
enables	Enables these applications	innovation -> application
generalizes	Generalization of the child	parent -> child
is a special case of	Specialization	child -> parent
contrasts with	In tension or opposition	sibling <-> sibling
equivalent to	Different formulations of same idea	sibling <-> sibling

Structure

## Concept Map: [Topic]

- **[Topic]** (core): [one-line description]
  - requires:
    - **[Prerequisite]** (prerequisite): [description]
  - rests on:
    - **[Principle 1]** (principle): [description]
      - led to:
        - **[Innovation A]** (innovation): [description]
          - enables:
            - **[Application X]** (application): [description]
    - **[Principle 2]** (principle): [description]
  - enables:
    - **[Application Y]** (application): [description]

Design Rules

1. Start from the core concept at the top level
2. Group children by relationship type: requires first, then rests on, then enables
3. Every principle and innovation from the teaching blocks MUST appear in the map
4. Nest innovations under their parent principle using led to
5. Add 2-4 applications showing practical impact
6. Add 1-3 prerequisites for context
7. Keep descriptions to one line

Depth Scaling

Depth	Total Nodes	Levels	Include
shallow	5-8	2	Core + immediate principles/innovations
standard	10-15	3	+ applications + some prerequisites
deep	15-25	4	+ cross-connections + secondary applications + all prerequisites

Default for TTT sessions: standard (10-15 nodes).

Comparison Maps

For conflation gaps, use a three-section map:

- **Shared Foundation**
  - **[Shared Principle]** (shared): [description]
- **[Concept A]** (core): [description]
  - rests on:
    - **[Shared Principle]** (shared): see above
    - **[A-only Principle]** (principle): [description]
- **[Concept B]** (core): [description]
  - rests on:
    - **[Shared Principle]** (shared): see above
    - **[B-only Principle]** (principle): [description]

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Concept File Export Format

Every Teach phase that delivers content generates a concept file during the RECORD phase.

Filename

concept-<domain>-<topic>.md in kebab-case.

• Example: concept-cost-engineering-direct-vs-indirect-costs.md
• Example: concept-physics-entropy.md

YAML Frontmatter

---
title: "Concept Report: [Topic]"
domain: [domain in kebab-case]
date: [YYYY-MM-DD]
type: ttt-session
session: SG-[N]
plan: [training plan filename]
level: [introductory/intermediate/advanced]
tags: [domain, topic, sub-goal-title]
---

The type: ttt-session field distinguishes files generated during TTT from standalone /explain files.

Content

The file contains:

1. All teaching blocks delivered during Phase 3 (abbreviated or full)
2. The concept map from Phase 3.5 (if generated)
3. The CTQ table aggregated from all teaching blocks
4. A "Session Context" section noting the plan, sub-goal, initial test result, and gaps found

Compatibility

These files are structurally compatible with concept-builder's /library command. If concept-builder is also installed, its /library list, /library search, and /library path commands will find and index these files.