If one wishes a subject to be taught with maximal effectiveness, he should:
Form patterns of computation in the individual with regard only to their usefulness.
That seems like a no-brainer, doesn’t it? Why would you teach someone something that wasn’t useful? Or give them a tool or show them a short cut to an answer that isn’t as effective as the “long” way?
The bad news is we teach students “helpful” nonsensical short cuts and study gimmicks a lot of the time. An example would be showing students how to use context clues to determine the meaning of a reading passage. How is guessing and making up definitions (instead of learning to use a dictionary) useful? Then we sit around and wonder how it is that overall student comprehension is low. Relying on context clues is not a useful pattern of computation and actually creates bad study habits, not to mention creates a lot of unnecessary confusion for students.
The good news is we also get it right a lot of the time. We drill drill drill students on math facts and multiplication tables as automatic recall so that calculating more complex math problems is easier and faster. We provide them with manipulatives to show them how these concepts work and give them tools beyond just the use of their fingers for counting. We know that a solid foundation and practice in basic arithmetic skills is fundamental to all other math concepts.
Tools are only useful to a person if they are reliable and they get the job done. Otherwise they aren’t real tools and instead serve to inhibit real learning and the acquisition of valuable skills.
Teaching useful patterns of computation to students leads to independent thinking and their ability to figure out more complex problems. These are the tools that will serve them for life.