I think it is impossible to get to the "higher orders of thinking" without slogging through some of the grunt work to see what is really going on. Unfortunatel

I think it is impossible to get to the "higher orders of thinking" without slogging through some of the grunt work to see what is really going on. Unfortunately there is no magic number of cases or fixed level of calculation work that guarantees understanding. It varies by the nature of the maths being taught and the context, interest and intelligence of the student. Moreover, with today's dependence on calculators and computers many practicing engineers and scientists are often missing the meaning behind the numbers despite knowing the math. I had this last point revealed to me when I gave my first engineering review lecture to an audience of about ~80 engineers. During the Q&A one of the best (and older) engineers asked a question that indicated I had made a mistake. However, the mistake could only be seen by connecting information from PPT slide 5 and slide 40. After the lecture I asked the guy how he ever saw my mistake, I was really curious. He said he tends to think in ratios and showed me that some (easy to calculate) ratios from early in my lecture did not agree with ratios calculated later in the lecture, which revealed the mistake. He explained that he acquired the habit from using a slide rule when he was in school. He noted that young engineers (me), with their over-dependence on calculators and computers, generally did not have this skill. He was not advocating using slide rules and banning computers but thought that using a slide rule had an epistemological value in understanding the meaning behind the numbers which was becoming a lost art.