Charles is absolutely correct. This business of doing a zillion examples,
making certain that there's an example done in class that covers every
possible variation on a problem type, actually complicates things rather
than make them more transparent to the student. It's not (IMHO) simply the
point of diminishing returns that kicks in; rather, the student is
encouraged to see each variation as a new type of problem, and makes it
vastly more complicated to compress it and make it part of his/her
conceptual structure.
And we compound this phenomenon by repeating the process every time there's
a new topic. Each topic acquires lots and lots of sub-topics or
variations, each of which the student is tempted to treat as an entirely
new problem to be learned.
Lillie
p.s. this is what I'm working on for my dissertation research.
At 09:32 PM 10/27/99 -0400, Jane Sieberth wrote:
>This one fascinates me, too. I'd love to see a research study on this.
>Jane
>
>Nancy Sattler wrote:
>
>> Maybe we should collect information and do a study?
>> Nancy
>>
>> -----Original Message-----
>> From: owner-mathedcc@archives.math.utk.edu
>> [mailto:owner-mathedcc@archives.math.utk.edu]On Behalf Of Lindsey, Dr.
>> Charles
>> Sent: Tuesday, October 26, 1999 9:12 AM
>> To: mathedcc@archives.math.utk.edu
>> Subject: RE: [MATHEDCC] Why Johnny can't read.
>>
>> Another anecdote: several years ago, I was teaching two sections of
Calculus
>> I one semester, and I decided to try an experiment. In one class i spent
the
>> usual one or two full days going over these problems; in the other I did
one
>> example and assigned everything else for homework. Result? no significant
>> difference in test performance.
>>
>> Now, this was not a carefully controlled experiment, but it does suggest an
>> interesting hypothesis: that the "point of diminishing returns" for class
>> time spent on these type of problems is reached very quickly. Those that
>> will get it, will get it after one or two examples; those that don't, will
>> most likely not become proficient even if you spend a month on it. If true,
>> this has significant implications for the way we conduct our classes.
>>
>> I have a second hypothesis that the difference between the groups
correlates
>> closely with *general* reading comprehension skills (as opposed to the
>> specific tactics that we teach for word problems), but have not tested this
>> yet. I think these hypotheses would make a good dissertation topic for some
>> erstwhile grad student in education...
>>
>> Chuck Lindsey, Ph.D. clindsey@fgcu.edu
>> Director of General Education
>> Associate Professor and Program Leader, Mathematics
>> Florida Gulf Coast University
>> 10501 FGCU Blvd South
>> Fort Myers, FL 33965-6565
>> Phone: (941) 590-7168 FAX: (941) 590-7200
>> http://itech.fgcu.edu/faculty/clindsey
>>
>> > -----Original Message-----
>> > From: Bret Taylor [SMTP:bret@IAG.NET]
>> > Sent: Monday, October 25, 1999 9:42 PM
>> > To: mathedcc@archives.math.utk.edu
>> > Subject: [MATHEDCC] Why Johnny can't read.
>> >
>> > Anybody remember that book? Popular about 25 years ago.
>> >
>> > Well, I'm gonna write a sequel: Why Johnny can't do math.
>> >
>> > And the reason will be, Johnny can't read.
>> >
>> > I've been reading with interest some of the wonderful posts here that
were
>> > spawned with the thread started a few days ago. Vern, Martha, and
several
>> > others (sorry, I don't remember all the names) have made very good points
>> > about students seeming lack of ability and desire to do any critical
>> > thinking. And Martha mentioned that the problem may even get worse as
the
>> > level of math increases.
>> >
>> > Permit me to give another anecdotal stroy to support this. I'm
teaching a
>> > Calc I class. We are presently doing optimization problems. If the
>> > students are given an equation they can (fairly well) find maxes, mins
and
>> > points of inflection and distinguish between them (algebraically and
>> > graphically). But understanding what they mean with respect to a "real
>> > world problem" is a completely different story. This class is having
more
>> > trouble than any other I have ever taught in finding the proper equation
>> > to
>> > model the problem.
>> >
>> > And today, one problem completely befuddled them. A standard max/min
>> > problem: How long a pipe can be carroed down a hall that has a right
>> > angle
>> > in it. I told them to do it for homework. (We had spent two days on
>> > applications problems in class.) One student asked me to work it today.
>> > I
>> > asked for help and no one had a clue. They saw the diagram in the book,
>> > showing the pipe touching both outside walls of the hall and the inside
>> > corner. They knew the length of pipe was a funciton of theta. But, the
>> > problem asked to find the maximum lentgth of pipe. So, they wanted to
>> > find
>> > the length of pipe as a function of theta and then find the value of
theta
>> > that maximized the length. When I showed them that theta equalling 90
>> > degrees or zero degrees the maximum length for the pipe was infinite,
they
>> > thought we had solved the problem and that it was a pretty stupid
problem.
>> > When I asked them to read the problem again and explain what was wrong
>> > with
>> > our thinking, they couldn't.
>> >
>> > I honestly believe the their problem was more a reading comprehension
>> > problem than a mathematical comprehension problem. Not a single
person in
>> > the class could explain to me what the problem was actaully asking. Even
>> > when I asked them if they had ever tried to move a piece of furniture (or
>> > a
>> > ladder or a bed frame) out of a room into a hall could they see this was
>> > that type of problem, they had difficulty seeing it.
>> >
>> > We worked the problem, and they never really understood that the minimum
>> > vlaue of the length of the pipe was the maximum length of pipe that could
>> > be
>> > carried down the hall and turn the corner.
>> >
>> > I've been using this type of problem for 15 years and don't ever remember
>> > this much of a struggle trying to explain it before. I'm convinced the
>> > problem is primarily a reading comprehension problem and also a lack of
>> > desire to try and understand the problem. The problem said find the
>> > maximum
>> > length, so we had to find a relatvie maximum. Simple.
>> >
>> >
>> >
>> > Bret Taylor "It matters not the subject taught,
>> > Lake-Sumter Community College nor all the books on all the shelves.
>> > Leesburg, FL What matters more, yes most of all,
>> > John 3:3^3+3 is what the teachers are themselves."
>> > John Wooden
>> >
>> >
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>
>
>
>--
>Jane Sieberth
>Franklin University
>201 South Grant Avenue
>Columbus, OH 43215
>
>email: sieberth@franklin.edu
>phone: (614)341-6269
>fax: (614)224-4025
>
>
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*************************************************************
Lillie R.F. Crowley
Professor, Mathematics
138 Moloney Building
Lexington Community College
Cooper Drive
Lexington, KY 40506-0235
(606) 257-4872 x 4115 Phone
(606) 257-4988 Fax
lillie@pop.uky.edu e-mail
http://www.uky.edu/LCC/MATH/Crowley
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