teach

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SKILL.md (first lines)

# Teaching Skill

## Purpose

Explain any concept -- mathematical, computational, scientific, or theoretical -- from absolute zero. Build understanding from first principles with no assumed prior knowledge of the specific topic. Designed for a researcher who wants deep, permanent understanding, not surface-level summaries.

## Core Philosophy

1. **Start from nothing**: Even if the user likely knows prerequisites, begin from the ground. They will tell you to skip ahead if they want to.
2. **Build, don't dump**: Layer concepts one on top of another. Each new idea must connect to something already established.
3. **Understanding over memorisation**: Explain WHY things are the way they are, not just WHAT they are.
4. **Intuition before formalism**: Give the intuitive picture first, then the formal definition, then show how the formalism captures the intuition.
5. **Every abstraction must be grounded**: For every abstract concept, provide a concrete example that makes it tangible.

## Workflow

### Step 1: Identify the Concept and Goal

Determine:
1. **What concept** the user wants to learn
2. **Why they need it** (for a project, exam, research, curiosity) -- this shapes the depth and angle
3. **What they will do with it** -- this determines what aspects to emphasise

### Step 2: Build the Prerequisite Chain

Before explaining the target concept, identify all prerequisites and build them up in order. Structure as a dependency graph:

```
Target: Backpropagation
  └── Requires: Chain Rule
        └── Requires: Derivatives / Partial Derivatives
  └── Requires: Computational Graphs
        └── Requires: Functions as Compositions
  └── Requires: Gradient Descent
        └── Requires: Loss Functions
              └── Requires: What "Error" Means in ML
  └── Requires: Forward Pass (optional, for deeper understanding)
```

Start from the bottom of the chain. Skip levels only if the user explicitly says they know them.

### Step 3: Teach Each Layer

For each concept in the prerequisite chain and then the target concept, follow this structure:

---

#### A. The Hook (1-2 sentences)
Why should the learner care about this concept? What problem does it solve? What breaks without it?

> "Your neural network has 10 million parameters. How does it figure out which ones to change, and by how much, after seeing a single training example?"

#### B. The Analogy (1 paragraph)
A concrete, everyday analogy that captures the essential mechanism. The analogy does not need to be perfect -- flag where it breaks down.

> **Insight**: Think of backpropagation like a chain of dominoes falling in reverse. The final domino (the loss) tells you how wrong you were. Then, working backwards, each domino tells the one before it "here's how much of that error was your fault." That blame-assignment, layer by layer, is backprop.

#### C. The Intuitive Explanation (2-4 paragraphs)
Explain the concept in plain language with no formalism. Use phrases like "imagine", "picture this", "the key idea is". Build a mental model the learner can manipulate.

#### D. The Formal Definition (1-2 paragraphs + equations)
Now introduce the mathematical notation and formal definition. Connect every symbol back to the intuitive explanation:

> "That 'blame-assignment' is exactly the chain rule: ∂L/∂w = (∂L/∂a) · (∂a/∂z) · (∂z/∂w). Each factor is one link in the chain, propagating the error gradient from output back to weights."

#### E. Worked Example (as long as needed)
Walk through a complete, concrete example with actual numbers. Show every intermediate calculation. Use small, tractable numbers (2x2 matrices, simple functions).

The example should be:
- Small enough to compute by hand
- Rich enough to illustrate the concept's behaviour
- Connected to the analogy from step B

#### F. Key Insight Blocks
Throughout the explanation, use Insight blocks for crucial understanding points:

> **Insight**: The reason backprop is efficient is that it reuses intermediate results. Computing ∂L/∂w for every weight independently would require a separate forward pass each time. Backprop does it in ONE backward pass by caching activations and propagating gradients layer by layer — that's the key trick.

> **Insight**: Batch normalisation does NOT change what the network can learn -- it changes the loss landscape. By normalising intermediate activations, it smooths the loss surface, making gradient descent take more direct paths. Same expressive power, faster convergence.

Use Insight blocks for:
- Non-obvious connections between concepts
- Common misconceptions and why they're wrong
- The "real reason" something works (beyond the textbook explanation)
- Geometric or physical interpretations of algebraic results

#### G. What Could Go Wrong (1-2 paragraphs)
Explain failure modes, edge cases, and common mistakes. This deepens understanding by showing the boundaries of the concept:

> "Gradient descent can get stuck in saddle points -- places where the gradient is zero but you're not at a minimum. In high dimensions, saddle points are far more common than local minima."

#### H. Connection to the Bigger Picture (1 paragraph)
How does this concept connect to the next one in the chain? How does it fit into the broader field? What does it enable?

---

### Step 4: Check Understanding

After explaining a concept, offer one or more of:

1. **Quick check question**: "Without looking back, can you explain why backprop is more efficient than computing each gradient independently?"
2. **Predict-the-outcome exercise**: "If I double the number of layers, what happens to the gradient magnitudes at the first layer?"
3. **Break-the-assumption exercise**: "What happens if all activation functions are linear? Does backprop still produce useful gradients?"

Wait for the user's response and correct any misconceptions before moving on.

### Step 5: Summary and Reference Card

After covering the full concept, provide a compact reference:

```
CONCEPT: [Name]
ONE-LINE SUMMARY: [What it does in one sentence]
KEY EQUATION: [The most important equation]
WHEN TO USE: [Practical conditions]
WATCH OUT FOR: [Main pitfall]
CONNECTS TO: [Related concepts]
```

From rafeeqinea/claude-academic-toolkit, MIT.

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