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On Monday, May 29, 2006 9:33 AM, Richard Tarara wrote:
My main discomfort is in the attempt to turn all student
failure into some kind of learning disability, stage of
development excuse.
...And I suspect there are people who have discomfort when someone
attempts to turn all student failure into some kind of student laziness
excuse. Can we at least agree that there are students of each type?
We have a problem: an inability to do unit conversions. Some students
seem to pick it up right away. Others do not. I remember learning
ratios in 6th grade and wondering why everyone was having difficulty
with them. I suspect that most people on the list were like me. Most
of my bio, pre-pharm, physical therapy, etc., students are more like my
6th-grade classmates. So what JC says seems to resonate with my own
experiences.
On the other hand, I don't think they are *incapable* of doing it (call
me an idealist). Rather, they appear to have an over-reliance on
following rote methods of problem solving. In other words, situation
"A" calls for procedure "B", and if you "do it right" you will get the
"right answer". If you ask WHY procedure "B" happens to work for
situation "A" they will tell you that it is just the way it works.
I have seen lots of examples of this but the most striking one, perhaps,
was the student who could divide any number by two in their head but had
to use paper and pencil to "cross multiply" in order to divide by three.
This student would take one-third of three and get nine, and then be
unable to tell you the answer (nine) was incorrect.
This is not a problem of laziness, as far as I can tell. It may not be
a problem with proportional reasoning, either, but then what is it?
Getting back on-topic, I have found that "unit replacements" tends to
work better with some students (I haven't done this enough to determine
which students that may be). For example, suppose one wants to convert
(picofarad-millimeter per square-centimeter) to (farad-per-meter).
First, replace "picofarad" by an equivalent number of "farads". A
picofarad is small, so 1 picofarad must be 1 farad divided by some big
number. In this case, pF = F/10^12 (for some reason, my students prefer
dividing by a big number to multiplying by 10 to a negative exponent;
I'm not sure why). Replacing "pF" by "F/10^12", we have that
(pF)mm/cm^2 = (F/10^12)-mm/cm^2
Next, replace "millimeter" by an equivalent number of "meters". A
millimeter is one-thousandth of a meter: mm = m/1000. Replacing "mm" by
"m/1000" we get
(F/10^12)(mm)/cm^2 = (F/10^12)(m/1000)/cm^2
Finally, replace "centimeter" by an equivalent number of "meters". A
centimeter is one-hundredth of a meter: cm = m/100. Replacing "cm" by
"m/100" we get
(F/10^12)(mm)/(cm)^2 = (F/10^12)(m/1000)/(m/100)^2
And then simplify.
I'm not saying this will work for everyone. In fact, I am saying it
won't. However, when I ask students to explicitly write down the steps
they do for a simple conversion, I find many do it this way.
Consequently, the above method makees "more sense" to them somehow.
----------------------------------------------------------
Robert A. Cohen, Department of Physics, East Stroudsburg University
570.422.3428 rcohen@po-box.esu.edu http://www.esu.edu/~bbq
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