Here are the best practices for explaining algorithms and data structures effectively:

Core Principles

1. Build from Intuition to Formalism Start with the "why" before the "how." Explain the problem the algorithm solves and the key insight that makes it work. For example, before diving into binary search code, explain why splitting the search space in half is powerful.

2. Use Multiple Representations

• Visual diagrams showing state changes step-by-step
• Concrete examples with small inputs worked through manually
• Pseudocode to show logic without syntax clutter
• Actual code with clear variable names and comments
• Analogies to real-world scenarios

3. Progressive Complexity Present the simplest version first, then add optimizations or handle edge cases. For instance, explain naive recursive Fibonacci before introducing memoization.

What Else Beyond Examples and Diagrams?

Interactive Elements

• Walk through specific test cases line-by-line
• Show variable states at each step
• Trace recursive call stacks visually
• Demonstrate what happens with edge cases (empty input, single element, duplicates)

Complexity Analysis

• Explain time and space complexity with reasoning, not just Big-O notation
• Show why certain operations dominate
• Compare with alternative approaches

Common Pitfalls

• Point out typical bugs or misconceptions
• Explain boundary conditions carefully
• Show incorrect approaches and why they fail

Pattern Recognition

• Connect to related algorithms and data structures
• Highlight when to use this technique
• Provide decision criteria for choosing between alternatives

Incremental Building

• Break complex algorithms into digestible components
• Explain helper functions separately
• Show how pieces fit together

Proof Sketches (for advanced topics)

• Explain correctness through invariants
• Show why the algorithm terminates
• Demonstrate optimality when relevant

The key is addressing different learning styles: some people need to "see it," others need to "trace it," and still others need to understand "why it works mathematically."