page 00015
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Two days after each opener, I presented a short mini-lesson about that
particular problem, discussing both common mistakes and one or more correct
ways to do it. I emphasized deciding whether a calculator would be useful.
At the end of the project, I gave the quiz depicted on the previous page,
to determine if the students' abilities had improved as a result.
The problems on the quiz were similar, but not identical, to the
problems on the openers. All 23 of my students took the quiz.
I graded each problem on the quiz using the same scale I used on each
openers, ranging from 1 (completely wrong) to 4 (completely correct), with
a zero designating no attempt whatsoever. Using the same scale allowed me to
compare the scores for corresponding problems to determine if students improved.
I noticed, on average, a minor improvement on three of the problems and a minor drop
on two of the problems. There was a major drop on the factorial problem, but this is
likely an error (see below). There did not seem to be any correlation between
improvement and usefulness or uselessness of calculators.
Solving an algebraically unsolvable equation
On average, students' scores on this problem dropped 0.3 points from the already low
scores on the equivalent opener. It was essentially the same problem, but with two
differences that may partially account for the lack of improvement: first, there was
an x already by itself on one side of the equation; second, the other side of the
equation involved a trigonometric function. Since the students had just been working
with trigonometric functions in other contexts, they may have misunderstood what the
problem was asking, or perhaps tried to treat it like some other sort of problem.
Graphing a polynomial
Average score change on this problem was a minor drop of 0.15 points, which is the result
of a balance of increases in 6 students, decreases in 10, and 5 staying the same.
Many students were still trying to use graphing calculators in spite of the explanation.
This problem was almost identical to the opener problem, except with a triple root added in.
Solving a system of three equations
Overall scores improved noticeably on this problem, with 4 students raising their performance
significantly and only one score decreasing. Average score change was an increase of 0.4 points.
The numbers were changed, but this problem was not significantly different from the opener.
Simplifying fractions with factorials
This problem showed the worst drop in scores, with an average drop of 1.5 points despite being
an equivalent problem which could be solved in exactly the same way as the opener. As mentioned
earlier, though, data on said opener was likely corrupted by students sharing answers. It is
evident from the quiz answers that perhaps half of the class did not in fact understand it.
Solving an exponential equation
Average score change here was an insignifcant increase of 0.06 points; again, the balance of a
few minor increases and a few minor decreases. Apart from a change in the numberes used, the
problem was identical to the one used on the corresponding opener.
Getting creative with place value
This quiz problem was most different from the corresponding opener, although superficially it
may appear similar. It asks students for the ones digit of 3^500, a number clearly too big for
a typical graphing calculator to handle; thus it can only be solved by seeking patterns in
powers of 3. 2 students got the answer exactly right, 1 came close but made a minor mistake,
and 2 others at least showed evidence of looking for a pattern. If this problem can legitimately
be compared to the corresponding opener, the average score change was an increase of 0.2 points.