page 00013
previous . . . . .
contents . . . . .
next
Overall, students performed best on problems of types that they had seen in class
recently or that had been reviewed frequently. Conversely, the class as a whole performed
very poorly on problems of types that they hadn't seen recently or at all. Many students
were able to adequately determine whether a calculator would be useful or not, but few
understood how to use a calculator to solve the problems that needed it. The one exception
is solving a system of equations with matrices, which about half of the students did
perfectly using a graphing calculator. It is possible, though, that they have just
memorized the procedure, and are not yet comfortable with how it actually works.
Solving an algebraically unsolvable equation
Solving this problem required graphing both sides of the equation and
finding both intersections of the curve graphically, or moving all terms to
one side of the equation, graphing it, and finding both roots graphically.
Both methods were taught to the class on the graphing calculator several
months earlier. However, the concept does not seem to have been retained
well. None of the 23 students found both solutions. One student used graphs
to find one solution, and three found one approximate solution by graphing or
guessing and checking; the rest had no progress towards a correct solution.
9 of the students overrated their own accuracy, and 2 underrated it.
Graphing a polynomial
Since the polynomial was given in fully factored form, the important features
of its graph should be evident from examining the equation. The necessary ideas
were taught several months earlier without the use of a graphing calculator.
However, many students still tried typing the equation into the calculator.
I know this because I specifically designed this polynomial to lead to
problems if graphed on a graphing calculator using the standard window
settings; it ends up being too steep to notice certain critical features.
Of the 20 students that participated in this opener, 10 sketched an accurate
graph showing all roots and correctly depicting the graph's behavior in-between,
3 got the general idea with one or two minor errors, and the remaining 7 either
were completely off or else obviously just copied an erroneous graph from their
calculator screen. I discovered later, however, that one of my students has been
using his Casio graphing calculator in a refreshingly resourceful manner for this
sort of problem: specifically, he entered the equation into the calculator, but
upon viewing the graph, immediately realized that it was too steep to make any
definite conclusions. Instead of experimenting with the window settings, he
multiplied the entire equation by a very small stretch factor, knowing that
this would flatten the graph to a more reasonable steepness while leaving the
x-intercepts and other important features exactly where they were! This is one
example, at least, of a graphing calculator being used as a true learning
instrument rather than just a crutch.
6 of the students overrated their own accuracy, and 8 underrated it.
Solving a system of three equations
Much earlier in the year, the class was reminded how to solve systems of three
equations with three unknowns; a short while later, they were introduced to
matrices and how these may be used to solve such a system. This latter method
superficially requires the use of graphing calculators, as the curriculum in
use makes no attempt to teach the students how to invert a matrix without one.
Of the 15 students that participated, 8 were exactly right and the remaining 7
at least had some steps in the right direction. There was a rough balance between
the two different methods, but those using matrices were more likely to be correct.
3 of the students overrated their own accuracy, and 2 underrated it.
Simplifying fractions with factorials
This opener could not in fact be evaluated by a graphing calculator, because the
numbers involved (99!, 3^98, etc.) were far too big for the calculator to handle.
Instead, students were expected to use their knowledge of factorials from the
previous chapter on combinatorics to simplify the expression algebraically.
Of the 19 students that participated in this opener, 18 got it exactly right.
However, this figure may not be accurate; I noticed much conversing amongst
the students during this opener, and I suspect that they may have been sharing
answers. Their work was fine, but many seemed unconfident about their answers.
None of the students overrated their own accuracy, and 9 underrated it.
Solving an exponential equation
This was an equation to be solved by taking the logarithm of both sides, a
technique covered five months earlier. The first steps are algebraic, but
finding a numerical answer requires use of a calculator. Although this topic
was introduced nearly half a year earlier, it was frequently reviewed afterwards,
and of the 17 students, 12 got it exactly right, 1 came close but made a minor
mistake, and only 4 were completely off (3 of whom didn't use a calculator).
3 of the students overrated their own accuracy, and 2 underrated it.
Getting creative with place value
This problem challenged students to determine how many digits are in 7^500,
clearly a number far too big to be represented on any pocket calculator.
Solving this problem requires a clever and innovative use of logarithms that
none of my students figured out, but several of them were looking for patterns.
Some students just made wild guesses; one just wrote "Error: overflow."