I have been following the back-and-forth on this mathematics vs. concept
stuff with some interest. Some will recall my earlier postings on my
diagnostic testing indicating that the general level of mathematical
cognition among our physics (and astronomy and physical science) is
generally far worse than most of our faculty have any idea of.
The antithesis between physics concepts and mathematics is indeed a
straw man as others have pointed out. The real questions are these:
1. Can we teach any sort of "College Level" (whatever that means)
physics or science course with any sort of standards to the real
students who are in our classrooms?
2. Is "conceptual physics" merely a way of avoiding the issue of the
real skill and cognition level of our students?
3. What is the MINIMUM level of mathematical skill and cognition that we
must have before any attempt at a serious physics course can be done
honestly?
4. Have our attempts at reducing the mathematical demands in favor of
"conceptual understanding" left us participating in the fraud (I can't
find a more polite word) that permeates the educational system, where we
pass people in courses where we know they have attained no serious
understanding or mental growth whatsoever?
I think everyone would agree that all of physical science is
quantitative. It has to involve numbers, measurements, relations between
variables (the heart of all science), proportion, ratio, rate of change,
area, volume, very large and very small numbers, and on and on.
One can fairly ask the questions: Do I need calculus?, Do I need
trigonometry?, Do I need to be able to solve quadratic equations? to be
able to do something of real meaning in a physics or physical science
course.
My answer to all the above is "No." But I cannot do anything serious in
these courses when I have students who cannot put fractions and decimals
in order from smallest to largest, who don't know how many times 1/2
goes into 3, who can't solve a linear equation in one variable (if the
coefficients are not integers), who have no operational sense of the
meaning of area and volume, who cannot judge whether the answer that
appears off the calculator is even plausible or wildly off. There are
far more such students in every institution where I have tested than the
faculty and administration in those institutions dreamed of.
The students who are stuck at the skill and cognitive level of a 10 year
old (yes, you have them in your classes) have passed many previous math
and science courses in highschool and even in college.
The level of what Piaget called "formal reasoning" ability, is
absolutely prerequisite to any serious college (or even high school)
level physics course if the course is honestly intended to communicate
any sort of real comprehension of basic science. The majority of
students leaving high school do not have such level (large majority I
think). They can thus only memorize meaningless rubricks to attempt to
pass. This is usually sufficient.
One example will suffice: Perhaps the origin of modern physics as a real
science comes with understanding that in mechanics acceleration is more
fundamental than velocity. The distinction between these is far more
difficult than most teaching gives credit for, as the work of Hestenes,
Hake and others makes clear (and this was stated by Arnold Arons nearly
30 years ago). While one doesn't need calculus to understand this
distinction (calculus often masks real lack of understanding by getting
answers too easily), the distinction is inherently mathematical -- a
level of mathematical thinking that is quite abstract and a level that
is not achieved by a large fraction of elementary physics students.
I know I sound like a broken record. But we are perpetually asking about
the side issues. Mostly we have been dumbing down courses to avois
unacceptable failure rates. The debate should not be whether we need
mathematics, but how best we can bring students to the cognitive level
where the whole subject has a prayer of meaning anything. The research
clearly shows that "Interactive Engagement" (Hake's term I think) is a
critical component of this (necessary but perhaps not sufficient).
Mazur's work clearly shows that emphasis on conceptual mastery does NOT
decrease problem solving skill (at least for Harvard students!).
I would dearly love to see a debate on this forum on the reality of our
students and what indeed is possible. Quite a few have now received my
diagnostic test. I'm sure others can find tests, or design their own,
but they must go back to test for understanding of very basic things,
much more basic than many will realize. I hope people will post here
results of the use of my (or other) test. Perhaps at some point soon, I
will start doing so.