[romwell]

When I taught Calculus, I taught understanding over memorizing steps to arrive at a solution for a particular type of problem.

The very best students loved it, but most of the people didn't like it at all.

With mathematics, like with gym, you gain when you put in effort. Most people don't enjoy either.

[thr0w__4w4y]

Yes indeed. Outside of work, I'm an endurance sports person, so basically performance is correlated strongly with training hard and suffering. There is a saying, "Pain is weakness leaving the body", I first heard it in high school (team went on to win a state championship in a highly competitive state). When I was suffering on workouts I just pictured myself getting stronger.

OK hopefully I didn't get too far afield. To me, the analogous concept in learning, particularly in technical fields, is that "learning is ignorance leaving the mind".

In college, particularly math and physics, I /always/ focused on understanding the underlying principles. Initially it was out of fear that if I forgot the formulas, I could re-derive them. But a strange thing happened... through that process, I developed an intuition and an ability to "see" what formulas and concepts to apply when. Once I got to that point in a problem, "seeing it for what it was", finishing to the solution became busywork.

[heymijo]

You're between a rock and a hard place.

The rock are the incentives, how your performance is measured, and the short duration you will have teaching these students.

The hard place is students who have likely spent 13 years in K-12 learning without understanding and are now being asked to do engage in practices they have little to no experience with.* They also have incentives to get good grades and a good GPA, which can be at odds with actual learning.

*To get more concrete, the practices have a name--Standards for Mathematical Practice (SMPs). The National Council of Teachers of Mathematics developed them and considers them the "heart and soul" of the Common Core Mathematics Standards. Not only are these practices absent from most classrooms, all too many teachers are not even aware of them! (see my Notch Generation reply to Sriram to understand why)

https://www.teacherstep.com/breaking-down-the-common-cores-8...

[jkmcf]

Did you happen to explain why you were teaching this way?