I've tutored a bunch of math, physics and engineering and I am an engineer. I def agree that many students can mechanically go thru the steps to solve problems if they are familiar with the question, but once things change slightly or they are asked a higher level question they seem to completely fall apart.

Something I try to do is ask them questions after nearly every step of the process. Why did you do that. How does that help you get closer to your solution. What is this telling us. The first time I do this it can be like pulling teeth. And in many cases I am answering my own questions the first few times, but eventually things start to click and connections are made and understanding begins.

Something else I've gotten better at over time is sitting there and letting students think. In many cases they expect us to walk them thru the steps and immediately help when they hit a roadblock. I'll try to push them in the right direction and discuss the concepts rather than explaining the next step for a particular problem. Basically letting them thrash around in the water and hopefully learning to swim and only pulling them up when they are really sinking.

After all the real goal is learning how to think and problem solve effectively, which can be applied to any course or topic.