Forum 10 - Lesson
10
L10-Q1 Problem Solving
Instructor: Tom Anderson
Submitted By Kim Fitzer
For the following
problem-solving exercises, I
chose the two problems listed below:
1. You
wake up in a pitch-black room in a hunting lodge, and there's no light handy.
In your duffel bag there are six black socks and six
white
ones,
all mixed together. You want to pick out a matching pair. What is the smallest
number of socks you can take out of the bag and be sure of
getting
a pair of the same color? How many socks _3____?
This problem was fairly easy to solve.
Fortunately, there are only two colors, if there were more, this problem
would be considerably more difficult. We
know from the question that there are three pairs of black and three pairs of
white socks. In order to solve this
problem, we must generate a set schema. The set schema has four components: the object
slot (socks), the quantity slot
(6 of each color), the specification
(6 black, 6 white), and the role slot
(smallest number of socks to make a pair).
This also requires knowing how many socks are in a pair, so we are
scaffolding information: old on top of
new. Our action strategy is to determine how many socks we must remove in
order to make a pair (Bruning, Ronning, & Schraw,
1999).
If we remove just two
socks, the chances are fairly good that we may be
removing one of each color. So good are
the odds in fact, that we cannot be sure that
we will be getting a pair, so based on this uncertainty, two cannot be the
answer. However, by removing one more,
we will be adding one more of either color sock we originally selected,
therefore creating a matching pair. Of
course, the answer of the problem is also text dependent. The language of the problem states that we
must create a pair of either color, not specifically black or white. This does make it easier to solve, and the
answer would be would be considerably different if we needed to be choosy. So, being able to
decode the text is an important step in being able to solve mathematical word problems.
Of course, there are
other questions, such as: If you wake up
in a pitch black room, how do you know you are in a
hunting lodge? Or
how do yuo find your duffle bag. But these questions
cannot be answered here.
2. A camp cook wanted to measure
four ounces of syrup out of a jug but he had only a five-oz and a three-oz
bottle. How did he manage it?
Write out the steps, such as pour
3oz into 5oz bottle, and such.
The Answer:
Solving this problem
was considerably more difficult, because of inconsistencies in schema
selection. At first, we knew that at
some point, there needed to be a third bottle.
However, the third bottle is not mentioned in
the text. Also,
text comprehension also played a part.
We knew that three oz. was less than five oz., and that four oz. was in
between. We also knew that the
difference between five and three were two, and the difference between three
and four and four and five were one. But how to isolate the one oz., on which the whole problem
would depend?
The schema used to
solve the problem proceeded from the problem
schemata, by drawing a mental image of a three and a five oz. bottle. We began by mentally pouring three oz. into
the five oz. bottle. But
then, how to find the one oz.? This
presents the action schemata, or the essential question in the
problem. At first, we tried dividing the
two oz. left in the five oz. bottle visually in half, and filling up the
remaining two oz. halfway. However, this
was inaccurate. The strategic schemata used in the first attempt, to divide the
remaining two oz. in half, proved to be inconclusive. A different strategy would have to be formed
(Bruning, et. al., 1999). Suddenly, it became clear that if the three
oz. bottle were to be filled again, and two of the three oz. were poured into
the five oz. bottle, the elusive one oz. of syrup would be left in the three
oz. bottle. At this point, it was simply
a matter of figuring out how to get the one oz. and an additional three oz.
into the five oz. bottle. The third
bottle, into which the five oz. bottle is poured in
step 4, is the original jug of syrup. And of course, by refilling the three oz. bottle and adding
it to the one oz. already in the five oz. bottle, the four oz. measure is
created.
Students often use
faulty schemata to solve problems. They often
do not utilize estimation schema to
evaluate their answers. S.K. Reed (1984)
conducted several studies to determine students’ abilities to isolate and
select the correct strategic schema for solving word problems. Students often select the simplest solution,
or schemata, without regard for appropriateness of their answers (this does not
apply just to mathematics and word problem solving). Students also often have difficulties in
applying previously learned information to solve new problems. In a study that Gick
and Holyoak (1980, 1983) conducted, students were given practice problems to solve prior to being given a
new set of problems. While students were
able to solve the practice problems, they were not successful in solving the
new problems because they failed to connect the old informatioin
with the new. The new problems differed
from the old just enough to prevent the students from drawing an analogy between
the two. It became apparent to the
researchers that while the selection and implementation of schema may occur easily
at lower-level problem-solving situations, higher-order problem-solving
such as algebraic word problems often failed to generate the correct schema. Students continued to apply the same schema
that worked well for the lower-level problems to the more difficult problems,
and found these techniques and strategies inadequate (Bruning,
et. al. 1999). If this information is
true, and classroom observations appear to support these findings, then new
strategies and problem-solving methodologies must be taught.
References
Bruning, Roger H., Schraw, Gregory J. and Ronning, Royce R. 1999. Cognitive Psychology and Instruction: Third Edition.
Upper
Saddle