I agree with most of what you state here. I am jaded however to any new mathematics. I saw what happened to a whole generation of school kids when some "educrat" thought it best to help kids understand "what numbers mean" That went over like a lead balloon and did not help kids worth a wit. Sort of what happened to this thread. The math,numbers, what "words" mean such as torque and power really do not add to an understanding beyond an engineers or mathematicians mind set of what actually happens in the field. Why is 6 more hp helpful or is it? When is it helpful when working a tractor? Simple enough of a question. People have all kinds of varied interests and brain wiring. Of course people are going to gravitate to their wiring and love of certain mind pondering. They want to share this with the world as everyone else would like to share. A practical sense (or I should say a practical sense of the majority) of things however can sometimes get lost in these myriads of thought processes so I do hope any new math improves upon 2+2 is 4 and the like.
As someone who puts his nose into pure mathematics for a while, I can say that pure mathematicans live in another world. You can call that they belittle pure physicans and even applied mathematicans. Pure mathematicans are really in another worlds. just to give an example. For physicans, there is only one infinity. For a pure mathematican, there are infitine number of infinities. But, they do somethings real even though they themselves are not aware of what they do really and they communicate with physicans and applied mathematicans about what they are doing really, their means in the real life, etc. Sometimes, especially in last decades, big inventions (such as Einstein's general relativity, black holes, etc which are not proved physically yet) were made in pure mathematics first. Anyway, pure mathematicans are in another worlds and the gap between them and common people has increased a lot. But, this is a reason for a pure mathematican to be proud. Common should actually thank a little to physic and applied mathematicans who try to make common understand things in science and mathematics more. That's why we see different expressions of power, force, etc etc. due to their efforts of physic and applied math people, not pure mathematicans at all who live in a world they have built for themselves.
Anyway, maybe, a new math teaching way is really necessary, maybe, should be started at the beginning. Till 2-3 decades ago, there were philosophy courses at high school, now, which are omitted. For example, without understanding what the force is, we are memorizing it and errors in our understanding are getting bigger in later stages like snowballs. For example, we often said here that force is a static thing. What an average Joe understands from this when it is called static? He'll probably understand that there is no motion. Right, engineers too understand same. But, no motion ideally means timeless, infinitely same state in time. Again, ok. But, then, we should not see "time" in its unit of force then if it is same in "all time". Now, lets look at the unit of force (in si unit system.) It is N, that's, Newton. Now, we don't see time here. See how we are forced to forget its background, hidden time in this unit.. Lets rewrite it explicitly. N = Kg x m / s / s. (here, m/s/s or m/s^2 as you know is unit of change of velocity, or, acceleration.) So, we use time even for a force on a body which always sits on the ground. We say this force is constant, and we call it weight. This is so because of our one assumption we make in our early schools, that's, gravitational acceleration is constant, which is not actually, changes a little depending on its coordinates in the world vertically and horizontally. This is another story, but, I am trying to come to a point.
If you looked into our talks here, what thing an average Joe noticed? He probably noticed that whenever we divide a quantity by time, we called it power. So, in my words above about the force, now, he noticed that another quantity too is divided by time and he probably thought that force too is power. Technically and mathematically, correct according to grouping math method. Actually, now, saying force is power isn't a strange expression for his logic of any Joe who hasn't seen any school in his life. Now, let me give another example to tell him what we technicans call a power rather than telling him what the power really is.. When you see a man keeping a big rock in his hands above his shoulders and stays so, what do we say? What a powerful man! Well, this was in very old days, centuries ago, in pre-engineering life. Today, correct expression is "strong man", not powerful man. To call him hercule or powerful man, we technics look at "how fast" he lifted the rock up above his shoulders, which means there is "time" in ranking. This is why today we divide quantities by time to call the output as power. So, power in old days was like a static concept while today it requires a dynamic to have a power. But, even at Newton's times, 5 century ago, we forget there is a hidden "time" even in static force like gravitational force, weight. People at Newton's time were understandable as they didn't know gravitational acceleration isn't constant actually. But, although we learn it in early school days, we forget our early assumptions, neglecting things like change in gravitational accelerations which is indeed small. But, forgetting this assumption make us lose our base, and errors in technique accumulate in further stages of educations and communication language gap between average Joe and engineers gets bigger and miscommunications happen a lot. But, engineers should be respected more, even more than pure mathematicans who try to pull engineers toward themselves and they open the gap more. Engineers are like middle-men, who are blamed or critisized more than others, so, because they are close to average Joe. Average Joe critisizes engineers not by telling things to him clearly. On the other end side, pure mathematican critisizes engineer for his low level of science, and them pure mathematicans even belittle them engineers, even physicans and applied mathematicans. Average Joe living in a "very real" life and pure mathematican Joe living in totally another world out of totally real life should have a meeting and lets see what these two lazies can do about what engineers couldn't.
