Matt <matt42287@hotmail.com> wrote, in his Math-Teach post of 10 Mar
04 18:51:15-0500 titled "Rule to Apply for Rates and Ratios":
"I got a problem: If you have a dozen eggs that costs $2.40 - what
rule can we apply to KNOW that we're looking for an answer of cost
per egg; not eggs per $1.00?"
In his response of 10 Mar 2004 21:41:49-0500, Ralph Raimi wrote
[bracketed by lines "RRRRR. . . .; two comma's have been inserted in
the interest of "good English prose style"]:
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
There is no rule until there is a question. Your opening clause,
"If you have a dozen eggs that costs $2.40" is not a question, certainly
not a mathematical question. Ask a question, and I believe you will find
that the necessities of good English prose style, in framing both the
information given and the subsequent question, will cause the answer to
that question to become obvious.
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
Ralph Raimi is in good company. Werner Heisenberg (1999) put it this way:
"In the course of coming into contact with empirical material,
physicists have gradually learned how to pose a question properly.
Now proper questioning often means that one is more than half the way
towards solving the problem."
But in response to the question that Matt may have intended to pose,
correctives for students' deficient reasoning regarding ratios have
been masterfully elucidated by the late Arnold Arons (1990, 1997) in
Sections 1.6 - 1.9. For a review of Arons' insights in education see
Hake (2003).
REFERENCES
Arons, A.B. 1990. "A Guide to Introductory Physics Teaching." Wiley;
reprinted with minor updates in Arons (1997).
Arons, A.B. 1997. "Teaching Introductory Physics." Wiley. Contains a
slightly updated version of Arons (1990), plus "Homework and Test
Questions for Introductory Physics Teaching" (Arons
1994), plus a new monograph "Introduction to Classical Conservation Laws."
Hake, R.R. 2003. "The Arons-Advocated Method"; to be submitted to the
"American Journal of Physics," online as ref.31 at
<http://www.physics.indiana.edu/~hake>.
Heisenberg, W. 1999. "Physics & Philosophy: The Revolution in Modern
Sciences." Prometheus Boooks (reprint edition - originally published
as "Physik und Philosophie").