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- A Nation of Fast Answers
- (reprinted from Today's Catholic Teacher)
By Gary R. Gruber, Ph.D.
The very mechanism that tries to
assess the performance and progress of our nation’s students, tests, may
well be the cause for actually creating uninspired shoot-from-the-hip
students and teachers. Now even high performers, as well as low achievers,
rush into answers without really doing any in-depth thinking or paying
attention to the process used. What differentiates the true
creative student from the run-of-the-mill student is that the former
enjoys the challenge and excitement of solving a problem and is only
secondarily concerned with the actual answer, whereas the latter is
wholly interested in the answer. Testing, the way it is presently
conducted reinforces this deplorable need for the student to rush into an
answer without providing the incentive for him or her to enjoy the
thinking process used to solve the question. What’s worse, current
standardized tests do not attempt to find out how the student
arrives at the answer—whether a rote approach or an inventive one.
Having spent more than 20 years critically analyzing
exams such as the SAT (Scholastic Aptitude Test), I have seen the
uselessness of some things schools do with these exams. For example, in
a vocabulary test a student may get six of the ten antonym questions
correct. Simply knowing that a student gets a certain percentage right
does not really tell us much. What I’d be more interested in is to find
out what method the student used to get a right answer and what method was
used to get a wrong answer. Did he or she guess? Use knowledge of prefixes
and roots to get the meanings of words? Knowing the student’s pattern of
thinking is essential to helping him or her correct any weaknesses and
reinforce areas of strength. Such methods of assessment impress the
student with the need for paying attention to process rather than just
getting the right answer. Gradually the student will see that this process
method will provide correct answers, along with greater interest,
confidence, and self-esteem.
The situation is more serious than we can imagine.
Consider this: In 1958, at Brooklyn Technical High School, 20 gifted
students in the math area (scores on the Math SAT—over 700 out of 800)
were given the following problem:
You
are given a triangle with sides 3, 4, and 5 (a right
triangle). Suppose you drew a square such that one corner of the square
touched the side 3 of the triangle, another corner touched the side
4 of the triangle, and the base of the square rested on the longest side
of the triangle. The question is, What is the side of the square?
Here’s what I found. Ten
out of the 20 students were satisfied in just trying to figure out the
side of the square from the 3,4,5 triangle information. But the other 10 wanted to
discover for any sides a, b, and c, what the length of the side of the
inscribed square was in terms of a, b, and c. That is, those students were
curious enough to want to see a pattern of how the sides of the triangle
affected the length of the side of the square inscribed.
The curious students were more interested in the
problem-solving process and discovering a mechanism which would describe
a general pattern in the math problem—as opposed to just getting an
answer to the problem. And I knew why—because a truly creative person
looks for patterns. Just getting an answer of something like 7 does not
give the inventive person true satisfaction. However, to discover how the
side of the square varies as I vary the sides of the triangle is rewarding.
Now in 1988, I gave that same problem to the same
number of gifted students (again in the range of over 700 out of 800 on
the SAT). No one of the 20 was interested in finding the pattern
that the latter ten in the earlier 1958 study were interested in. No one.
All were just out to get a fast answer to the question. When I asked why
that was the case, they told me that “that’s all I asked them to
do!” So I’m convinced that we are creating a group of “robots”
or “rote mechanics.” And of the gifted and high achievers, we are
creating a group of very bright robots and rote mechanics. No wonder we
can’t find another “Einstein.” This little experiment should make
us realize that we must start getting students interested in the
creative aspects of problem solving, rather than maintaining their need
for just seeking fast answers to questions.
Evidence shows that this educational sickness of
rushing into answers without “process” thinking is mainly due to the
tremendous emphasis on testing. Because of that emphasis, and because
students do not want to show their “stupidity” (even on tests where no
one is directly watching them), they will not take “thinking” chances.
Here’s a striking example. I recently gave this question to 300
students. Complete the analogy:
SCISSORS is to SEVER as (A) stapler is to strike (B) arrow is to direct
(C) tweezers is to point (D) pencil is to write or (E) shears is to mend.
Of those that knew what the word SEVER meant,
a very high percentage got the answer to the analogy (Choice D) However,
of the 160 students that did not know what SEVER meant, only one
got the right answer by a clever thinking method, and three guessed and
got the right answer. I confronted the 160 and asked why they didn’t
think that SEVER would mean “to cut,” since the only thing that you
really associate SCISSORS with is cutting. All but the one who
used that approach said that relating SEVER to cut was too easy. They’d
be made a fool of by answering the question by such an “easy”
method. However, they didn’t realize that they had nothing to lose since
they would get the wrong answer anyway or have to guess blindly if they
didn’t take a chance. Had those students been more aware of
“process,” they would have learned through analysis that perhaps the
only approach would be to assume that SEVER means CUT, since
SCISSORS are used for cutting.
Having always been interested in process, I decided
to try to, in some measure, reverse the general problem, using the test
maker’s own ammunition. I created a test which measures how a
student is approaching a question: that is, what process, if any, the
student is using. I then have the student analyze his or her answer to the
particular question and then lead the student to the correct methods
and thinking processes the student should use. Subsequently, I give the
student practice with those same thinking processes through further
examples involving the same thinking process. Here’s an example of a
question:
Is Column A greater, less than, or equal to Column B?
Column A Column
B
33X22
34X21
The “answer-only prone” probably will not take
time to observe the closeness of the numbers in both columns, the 33 in
Column A, the 34 in Column B, the 22 in Column A, and the 2]
in Column B. Those students will probably just multiply 33 X 22 and
34 X 2] and compare results. Not only does that take some time, but
there’s a chance of error. It also is not a gratifying way to solve
the problem. Here’s how the “process person” would solve it: First
he or she would say, since I have to compare the two columns, I don’t
necessarily have to calculate the quantity in each column, which is
tedious. Let me divide both columns by 21 and by 33:
Column A ColumnB
33X22 34X21
33X21
33X21
This gives:
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Column A Column
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which is just:
Column A Column B
1 1/21
1 1/33
Column A is greater than Column B. To find out in
the above example, whether a student used the right process for solving
the question, ask a follow-up question:
How did you answer the above question:
1) I multiplied the 33 X 22 in
Column A and the 34 X 21 in
Column B, then compared results.
2) I divided both columns by 22 and by
34.
3) I divided both columns by 21 and by
33.
4) I guessed.
5) By none of the above
methods.
If the student gets the right answer to the original
problem, but answers 1) in the follow-up problem, we realize that the
student had used the “rote” multiplication method. If the student
gets the right answer and answers 2), the student used a good process for
solution, coupled with a clever rationalization that 33/34 is
greater than 21/22. If the student answers 3), the student used a
different, but good process in solution. If the student answers 4), we
know that the student guessed at the right answer. If the student answers 5),
the student used a method of his or her own that was not a normal or
standard method. In any case, whatever the follow-up answer the student
chooses, we are in a very good position to direct the student to the best
method to use for solution, knowing “how” the student solved the
question. A test like this reinforces the problem-solving process and
interest in solving a problem by the best method. It also actually does
something more subtle: It gives the student the mechanisms of
“observation” and “connection,” which are the two key elements of
solving any problem.
Gary R. Gruber has worked with many school districts
to improve student thinking skills and test scores.