You may wish to note in my post (under the subject Equations) when I
mentioned that I wished textbooks would use delta more often, I used the
example of the energy in a compressed/stretched spring...
delta-U(spring) = 1/2 k delta-(x^2)
The first problem some students run into if the book and professor are
not in the habit of using delta when they should, is that students will
miss a problem when the initial value of the variable in question is not
zero. For example, time-initial is 3 s , time-final is 7 s, and they
total miss the 3 and just plug the 7 into their equation.
The second problem, which is more insidious, is that students are not
used to the fact that (delta-something)^2 is not the same thing as
delta-(something^2).
Of course I am saddened by how many students still goof even after I
have told them about it. Consider the problem... A spring is further
compressed from 0.02 m of compression to 0.07 m of compression. What
work is required to do this (or what is the change in spring potential
energy)? Many students will do (0.05)^2 rather then [(0.07^2 -
(0.02)^2] in spite of the fact that I do use delta all the time, and
when teaching this particular case I not only warn them not to forget
the initial compression, I also warn them that delta-(x^2) is what the
equation is, not (delta-x)^2. I even say, "I am sad to predict that
several of you will get this wrong on the test, and if you prove me
wrong by everyone getting this correct on the test, I will bring
home-made cookies to class as a celebration."
Of course I make my cookie promise a week or so before the test, and I
don't go out of my way to remind them about the promise. I've been
teaching this for 28 years and have probably made the promise in 20 of
those 28 years. I have never had to make cookies.
It was precisely for the reason of confusing (delta-x)^2 and delta-(x^2)
that I chose the spring-energy equation for my example in my post.
And then there was the student who, on the day of the promise, asked
what kind of cookies would I bring. I said my favorite cookie is
oatmeal and my second favorite is peanut butter. That student made the
mistake on the exam. When she was leaving class on the day I handed
back the exam, I said, "Melanie, you even asked what kind of cookies I
would bring, and you still missed that spring problem." She replied,
"Well I wouldn't have missed it if you had promised chocolate-chip
cookies."
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu