This is a precursor to 'how to make math class interesting'.
There are at least 4 levels at which math is typically taught.
1. Math process
2. Math understanding
3. Math application
4. Math abstraction
To solve an equation 2x - 70 = 30 there is a process. If we use that process blindly, we can find the value of x. This is math process. This is very important in math, if students don't learn this well, they will fail in math.
If we put different values for x in the equation and the one that fits would be the solution. This is the math understanding. Once the process is practiced and learnt, there should be no compulsion to use the process for every simple problem. Students should be encouraged to use understanding to quickly arrive at the solution.
If we buy something worth Rs. 70 and give 2 notes of Rs. 50, the shop keeper returns Rs. 30. This transaction can be modelled using the above equation. This is math application. Teaching children how to model real life scenarios into equations of math is an important step in learning math and also in appreciating its uses for the real world.
Math abstraction is about disconnecting the real world entities from math and appreciating the connections between patterns of various things in math. Things like 'we need to have at least n distinct equations to solve for n variables' is an abstract concept and it doesn't care about what the variables represent in any application. When a student gets the taste of this kind of abstraction, she gets hooked to math.
We as traditional school teachers of math tend to spend most time in teaching math process. There are reasons for this; this is the most boring aspect of math almost like running on a treadmill! This has to be perfected to pass exams.
The other 3 steps are naturally interesting if a student can move up to those steps after becoming good in the process.
Making the math process interesting without resorting to rewards, punishments and morals is hard.
Any ideas in this regard without slowing down the teaching process is highly appreciated.