One of the aspects of education that is most difficult to consciously think about is how do we teach (and learn) critical thinking skills. Without a doubt, critical thinking skills are key to good education. But what are they?
Critical thinking cannot be taught the same way mechanical rote learning can be "taught". It requires a certain amount of comprehension and active engagement and, ultimately, imagination is demanded of the student. For mathmatical (or analytical) thinking, memorization of multiplication tables will not do. Analytical thinking requires freedom to explore and experiment to test the boundaries of accepted forms.
Analytical thinking is an ability to use the constraints and restrictions of a discourse/study and its first principles to arrive at a tenable position and/or original insights. This is true for constitutional politics, policy development, science and maths. Analytical thought does not resort to "dirty tricks" or "sophistry"; the ends do not justify the means, rather, the means are demanded to justify their efficacy and intellectual "purity" for arriving at a certain end.
Having said that, mistakes and errors in judgement are unavoidable in analytical thinking and a certain amount of self-confidence is required to make and defend arguments, and one must be willing to admit error so one may advance (to learn from one's mistakes). It takes character. This character is a cultivation of critical and analytical thinking.
When I started thinking about linguistics and, later, number analysis I was drawn in by the fact that one can make general statements about things and processes using the principle of parsimony, and the veracity of these statements may be judged according to the particular logical consequences of them. This is critical thinking par excellence.
Doing arithmetic is boring - nay, mind-numbing. There is a better way to teach maths after the student has acquired and mastered the basic arithmetic operations of adding, subtracting, multiplying and dividing. This better way is to examine and explore the structure of numbers.
For instance, I modified some equations, such as: 4n - 1; 4n + 1 into
4n - 2n +/- 1 (for odd numbers)
4n - 2n + 0 or 2 (for even numbers)
and started generating numbers starting from 1 to infinity, thus:
4 - 4 + 1 = 1
4 - 4 + 2 = 2
8 - 4 - 1 = 3
8 - 4 + 0 = 4
8 - 4 + 1 = 5
8 - 4 + 2 = 6
12 - 4 - 1 = 7
12 - 4 + 0 = 8
12 - 4 + 1 = 9
12 - 4 + 2 = 10...
This is a bit more interesting than just teaching simple arithmetic. Using variations of the equations and colour-coding by whether the component asks to subtract or add a number also brings out some very interesting patterns which have regularity but some magical disproportionality about them as well, especially the prime numbers whose waves upon waves one may see and appreciate but never quite predict (or, at least, I couldn't). These types of exercises can make maths fun.
In my career as an analyst, I've sought to perceive underlying structures of things and processes, and how rearranging them affects the truth of them (whether it perverts or brings out its beauty). I have, in fact, spent many hours mesmerised by these thoughts. When I perceive a pattern, I try and figure out what the motivating principles are for what I see.
For example, I tried to capture the patterns in the periodic table of elements on my own. I didn't succeed obviously but the mere exercise of contemplation made me appreciate its inner order all the more. It's the same with music... with physics.