Formal discipline theory led to views that learning Latin and geometry were important, even if few students would use these skills in their lives, because by their formal character they acted as the ideal dumbbells for mental ...

Provided a person has the ability to see the correspondence, the problems “three times three” and “3 x 3” are the same, even if they look different. In short, an idea, as a more abstract, general notion, can influence your thinking on a wider range of problems than a procedure you memorize.

• We need to be able to recognize the idea in different contexts
• We need to be able to modify the idea to suit our current purposes.
• We need all the specific skills of implementation.

The easy-to-spot ideas in a field may really just be the tip of an iceberg of invi...

Few keys to acquiring more broadly useful skills:

• Breadth comes from specificity. All general skills are built from specific knowledge and procedures. There’s no shortcut.
• Deeply understanding ideas helps. While not a panacea, deeply understanding more abstract ideas extends t...