The Tortoise and the Hare
I see this phenomenon with enough frequency that I think it deserves to be written about. I don't see the ridicule from the hare directed at the tortoise as in the fable but I do see similarities. Here is my description of some of the characteristics the tortoise and the hare have with respect to their study of mathematics:
Tortoise - isn't quick to understand concepts and has a difficult time with the material; is dedicated and does homework practicing a variety of computations over and over again; knows he needs to work hard and is willing to do what it takes to gain a descent understanding and get good grades.
Hare - quick thinker and possesses a better than average command of 'some' of the fundamental concepts studied; is able to solve many problems without doing computations; is not willing to do computation - if he's unable to simply think through the problem and get the solution, he resists doing the problem altogether; tends to rest on his laurels.
Eventually and inevitably material is introduced to the hare for which he can not figure out the solution in his head. Now the hare has a problem; he is not practiced with the computational techniques required to do the problem. The result is the hare can no longer keep up with the class and is repeatedly confronted with problems he can not do in his head. Luckily for the hare he is quick witted and is capable of learning the computations. He only needs to put in the effort required to catch up. But the hare does have an obstacle to overcome - that obstacle is himself. The hare may tenaciously resist doing/learning computation. He may believe he should be able to do the problems without learning the computations because he has always been able to do it. Math has always come easy to him and he wants it to stay that way. The hare either acquiesces and learns the computations or is left behind.
During this time while the hare resists figuring out how to do the computations the tortoise has been quietly going about his business of figuring out how to accomplish the tasks at hand. The tortoise has become proficient at doing computations and is ready to move on to (learn) the next topic on his way to becoming adept at mathematics. This is where the tortoise pulls ahead of the hare.
The point at which the math becomes too involved for the hare to find the solution in his head is different for all people who possess the qualities attributed to the hare. But that point inevitably comes. The tortoise is use to working hard and he continues on his way throughout his mathematical education.
Letter Written to Friends Daughter
A personal note to Students thinking about College