A former semiconductor researcher, and later physics professor, ponders our human predicament, recalling the depth of understanding of his former students... 
F = ma
Not
really what Newton discovered: the REAL second law is F = dp/dt.
Gateway to revolution. 
Affects
billions.


It's
mathematical.
Simple
math!
Subtle
math!!
Very
simple...very subtle.

click
on the door
to further explore 

Some tax increases are on the way.
Your state of three million people is going to fund a new freeway costing
six billion dollars. Your city of one hundred thousand people is
going to fund a new soccer stadium costing four million dollars.
Your country of 250 million is going to fund a new Coast Guard rescue vessel
costing six hundred million dollars.
If
the costs are proportioned equally among the populations, which one costs
you the most? Which one the least?

¬ Those who understand ratio and proportion find this problem very easy, the route to the answer obvious. Those who don't understand, often add numbers where they should be multiplied and subtract where they should be divided. F = ma is a ratio. 

A radio reporter at the beginning of the noon news reported, "The stock market is down sharply today: The Dow is currently down one hundred and eightythree points." An hour later the same reporter finished the hour's news with, "The stock market is still falling, but not as fast. The Dow is now down ninetyseven points." How
would you explain to the reporter what's wrong with the second report?

¬
"Seeing" rates of change makes seeing the error here easy, the explanation obvious. F = ma tells something about acceleration, which is a rate of change. 

You need to know with some precision how fast the market is changing. How many points per second is it falling or rising—at some specific time of day. How
would you go about calculaing the "instantanenous" rate of change?

At the end of the 17th century, Isaac Newton saw how to define and calculate
instantaneous rates of change. Those who understand Newton's insight
see the answer to this problem as "obvious."
¬ F = ma is about the instantaneous. 

The coach asks the athletes to line up by size. Five of the players are on both the basketball team and the football team. But when they lined up ("by size") for the football coach they are in a different order than when they lined up for the basketball coach. The wrestling coach didn't find either order very useful when those five reported for the wrestling team. How
might you help the many coaches rank their athletes?

"It
depends on..." is an insight that dips into many dimensions. Those who
have that kind of insight see ordering by most measures as being in a manydimensional
space.
Those who don't, pervasively and persistently see every measure as scalar, all orderings in lines. F = ma relates two vectors (one of the multicomponent measures). 

What if the temp is 4 degrees F outside,
and then the temperature drops 7 degrees. What is the new temperature?
What is the direction, up or down, of the acceleration of a freely bouncing
ball at the bottommost point of its bounce, that is, at the instant its
velocity changes from down to up?
How
many images can you associate with simple math concepts?

Those who don't in some way "see" the abstract concepts are doomed to rote memorize vast quantities of specific cases. F = ma is about the vector, acceleration (an abstraction that is visualizable in many ways). 


F = ma is not Newton's second law as Newton showed it to us. Newton's Law is hidden under the "F = ma" at the top of this page. 
If you are not sure of your answers to the above math problems, try working them out with friends.
The answers are unimportant; becoming able to see the routes to the answers is very important.
Teachers and learners alike need to explore ways to improve such ability.
How should answers be presented?