I like teaching the occasional gifted student, but who I really value in class are students who are willing to work. Those who are willing to put in the time and effort, and who go over the subject matter is a wonderful thing to have in any class. There is a big difference between hard work and smart work however.
With regard to learning mathematics, rote works. Repeating basic facts and methods is one of the things that cannot be replaced in teaching. Familiarity helps students get comfortable with specific processes in math, but it has a limited utility. After a certain point repeated practice is tedious, doesn’t help to convey greater understanding of underlying concepts or help in refining technique. At times I see students who try to learn every single possible variation of an equation. Grinding through question after question trying to memorize patterns isn’t the goal. Students need to apply some critical thinking, and learn broader scopes of methods.
There are some crucial differences between simply working hard, and working smart. The easiest difference to spot is knowing what steps to skip… and this is where there can be a hand-off between rote work and critical work. Getting familiar with the basics is one thing, but showing every little detail is just busy-work. Memorization is a good foundation. After a certain point critical thinking needs to take over.
The bigger difference between memorization and critical work is play. When students start noticing differences on their own without prompting, then they start to play with variations. Curiosity more than need drives students to go off script and expands (or deepens) their understanding of the topic.
Minds at work only examine what is in distinct categories, but minds at play will color outside the lines.
No student is going to perfectly follow a teacher’s script to learning. That is as it should be! If a student finds a path by themselves, they will remember it better. One of the hard things for a teacher to do is to stress the need for discipline to get work done, but also leave enough flexibility so students can grow on their own. Even in classes with prescribed online work, this is possible. I like to emphasize looking at off site resources and time spent over working on specific objectives.
Getting into learning mode is one thing – this is one of the hidden uses of rote work. Starting work with some simple practice, then move on to some simple process questions. As strange as it sounds, boredom can be helpful. It is really satisfying to see students start looking for more interesting or more challenging questions.
Discipline and curiosity are not at odds, they are the hallmarks of the best students. Hard work and play really need to go together.