A long while ago, a professor enlightened my class with his theory on the 3 stages of learning. At that point and since then, it has held true time and time again. Without further ado, here are the stages:

1. Duplication - Ability to take content and repeat it back
2. Comprehension - Ability to take content, understand it, and state it in another way
3. Improvisation - Ability to take content and create new conclusions from it

Examples

Learning Chemistry

For the example where this originated, here are some examples of each stage:

1. Duplication
   • Reading content from a textbook and reciting it back
   • Taking notes in class and repeating them back
2. Comprehension
   • Taking notes in class and rephrasing them in your own words
   • Asking deeper questions about content
3. Improvisation
   • Deriving new formulas
   • Creating new conclusions

Learning Math

Learning Math is very similar to learning Chemistry. Some more concrete examples of improvisation would be:

• Breaking down proofs into smaller sections for easier reuse
• Creating a new proof based off the combination of other proofs

Learning Programming

For learning Programming, some examples of each stage would be:

1. Duplication
   • Taking an existing program and rewriting it line for line
   • Writing a program that follows a specification
2. Comprehension
   • Taking an existing program and swapping variable values
   • Explaining why choices were made in a specification
3. Improvisation
   • Rewriting a program but maintaining the same results
     • For example, moving from iterative to recursive
   • Create a new specification for a problem