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Why do calculus students have difficulty with understanding a problem that
asks them to find the maximum length of a ladder that can fit around the
corner of a hallway? Why do arithmetic and algebra students have difficulty
answering a question like "how many buses are needed to transport 1128 Army
soldiers if each bus can carry 36 soldiers"? Reading comprehension may be
(probably is) part of the problem, but let's look at our own house before we
start blaming the reading teachers for this - have you looked lately at
arithmetic and algebra texts that are used in elementary, middle, and high
schools, or community colleges that teach remedial math? Look at the
chapters - 90% of the material in each chapter consists of rote exercises on
skills - manipulation of numbers or algebraic symbols with no context given.
At best, 10% of the material contains what we call "word problems", which
are almost always separated into a section of their own (as though they are
separate somehow from the calculations). Also, have you looked lately at
the majority of these word problems - they are contrived, dull problems that
are made up strictly in order to have students "apply" the skills they have
been practicing in the other 90% of the chapter. Tell me, when was the last
time you knew that the value of your after the first year you owned it was
$15,500, that it depreciated by 30% that year, and needed to figure the
original price of your car? No one does this kind of thing in real life (at
least not with cars!), so when we present absurd sorts of problems for
students to "critically think" about, they do think about them ("this
problem is stupid" is indeed critically thinking about the problem, because
it is stupid!! If we can't come up with more realistic problems to justify
some of the skills we're teaching, we have to begin to ask ourselves if
these skills are necessary). After being fed a steady diet of this kind of
nonsense as the only real "application" problems for math (algebra in
particular), is it any wonder that our students don't see math as applying
to anything much in the real world, and think nothing of giving an answer of
"31 remainder 12" to the bus problem mentioned above - we have conditioned
them to think that word problems that involve calculation are frequently
stupid, so they don't recognize a real-life sort of reasonable problem when
they finally see one. The fact of the matter is this: our students only
have the chance to get good at what they practice - if we have them practice
90% of the time on manipulation of symbols, 10% of the time on contrived
word problems, why do we then expect that they will be whizzes at solving
realistic problems that involve calculation?
If we want students to know and enjoy and use the power that mathematics
offers them, we must present them with more realistic problems, use
technologies (such as spreadsheets) that allow them to confront realistic
problems like those they will see in their lives after school, and we are
going to have to make some hard choices about the topics we teach them -
both the number (too many to promote good understanding in most courses) and
the topics themselves (not all of the topics we teach are really as
important as others - one quick example - when I started teaching 10 years
ago, teaching interpolation for trig and logarithmic functions was thought
to be sacred - fortunately, $5 calculators finally made folks realize that
interpolation is no longer a necessary skill to teach).
Deborah Cohen
Coordinator of Assessment, Research, and Planning
Associate Professor of Mathematics
John Tyler Community College
Chester, VA
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