This debate has been around before, and has never been resolved. Why is
this?
I think the answer lies in the way in which the debate has proceeded. Most
arguments have been based on particular theoretical considerations about how
energy should be taught. Often a particular model for energy is invoked,
and in that model work may be considered to be irrelevant. Absolutely no
empirical evidence has been presented to show that students either
understand energy or work better by taking a particular approach. As a
result none of the answers can be shown to result in better student
learning, so the question can not be decided.
Let me cite an example. Various texts consider the topics of momentum and
energy in different orders. Sometimes the author introduces energy first,
and sometimes momentum first, and the choice until now has been a purely
personal preference. Priscilla Laws tried both approaches in her Workshop
Physics course and has used test scores to see which one results in better
student understanding. She found better student learning by introducing
momentum first. Admittedly, whether or not this result applies to other
pedagogical approaches is unknown. However this is, as far as I know, the
only firm reported evidence in favor of a particular sequence. As a result
a number of texts have been reordering topics to adhere to what has now been
dubbed "the new mechanics sequence".
The only way one can firmly decide this question is by teaching energy both
ways, with and without work, and then using testing to see if one approach
is superior. I have not seen any papers on this experiment. However, most
of the reformed curricula do use work as an entry into energy, just the same
as they use impulse as an entry into momentum. The reformers seem to think
that it is helpful to teach it, but I do not know if they have any evidence
for this sequence.
On the basis that it seems to be an approved method of teaching, I would
definitely include it. However, the amount of time that you spend on it
could be determined by how important you feel it is.
In my class the topic of work has actually been very beneficial. Since my
class is not calculus based the students are presented with some very
interesting calculations which challenge them considerably. One problem
involves calculating the work a father does as he pushes a playground
merry-go-round twice around. The students are given the force in the
problem statement, and a diagram shows the diameter of the merry-go-round.
Some students come up with the answer zero, some calculate 4 F d, some 8 F
d and a few 2 Pi d F. The students who realize that it is not 4 F d for
once around often do not know that you calculate the distance around as Pi
d. I make them measure the drawing using the edge of a sheet of paper.
They are also puzzled by that, as they say you can not wrap the edge around
the circle. Eventually the roll it and figure out that the distance is 3 d
plus a bit more. With some questions such as is there anything you learned
in math that might be helpful they come up with Pi d. Notice that I am
really teaching them how to think and not just work.