Graphing Calculator-Associated Strategies Used by and Misconceptions of High School Students

ABSTRACT

Many researchers and math educators have identified graphing
calculators as catalysts in improving student understanding and
achievement; however, some question whether or not their use may impede
student understanding by promoting errors or misconceptions. In an effort
to determine whether or not graphing calculators really are a "good
thing" for mathematics education, several researchers have
administered tests to measure achievement gains. Although their findings
indicated mixed, but encouraging results, a large portion of these studies
were *quantitative* in nature. Consequently, we only know that
achievement gains were or were not made but not necessarily *why*.
For research to effectively guide curriculum development and instruction,
we need to find out *why* graphing calculators are or are not
enhancing student understanding and achievement.

In the fall and winter of 1996-97, I carried out a
*qualitative* study of eighteen high school students so that I could
discover how and why using graphing calculators may or may not contribute
to student achievement. As the students completed graphing problems, I
identified the various strategies students used to interpret and generate
graphs and the scales of graphs when using graphing calculators, and how
they reconciled conflicting information due to perceptual illusions. I
also documented the misconceptions that students possessed when
interpreting graphs displayed on graphing calculators.

Some of the major findings of this study included: the lack of understanding of the effects of scale and range changes; the belief that a finer scale will result in a more accurate trace; heavy use of the "press and pray" strategy; "setting the window using the equation's coefficients" strategy; mis-identification of points of discontinuity; explanations for the non-appearance of points of discontinuity; and the graph interpretation bias due to the left-to-right generation of graphs.

Based on these findings, recommendations were given for teachers who use, or who are considering using graphing calculators in instruction.

Despite wide recommendations given by various organizations (NCTM, 1980, 1987, 1989; NRC, 1989, 1991; MSEB & NRC, 1990) to use graphing technology, many researchers have cited several areas deserving of research because "certain problems or pitfalls must be accounted for if students are to gain maximum benefit from the technology" (Wilson & Krapfl, 1994, p. 260). For example, when using graphing technology, many researchers have emphasized the importance of student awareness of and exposure to scaling (Goldenberg, 1987, 1988, 1991; Harvey, Lewis, Umiker, West, & Zodhiates, 1988; Demana & Waits, 1990; Dion, 1990; Dunham, 1991a; Dunham & Osborne, 1991; Dick, 1992a; Hector, 1992; Williams, 1993). Still, others have cited potential problems when graphing calculators were used to view graphs of discontinuous functions and asymptotes (Demana & Waits, 1988; Dion, 1990; Dick, 1992a; Hector, 1992; Tuska, 1993; Williams, 1993). Additionally, concern has been expressed that students may become over-reliant on technology (Fey, 1990; Dunham, 1991b; Quesada & Maxwell, 1993, 1994), resulting in a possible displacement of skills.

Is the use of graphing calculators really a good thing for
mathematics education? Several researchers have sought to answer this
question by administering tests to measure achievement gains, and their
findings indicated mixed, but encouraging results. Because a large portion
of these studies were *quantitative* in nature, they reported only
that gains were or were not made in achievement, but not necessarily
*why*. "For research to effectively guide curriculum development and
instruction, we need to find out why" (Dunham & Dick, 1994, p. 443).

In an attempt to "find out why" graphing calculators may or may not
contribute to student achievement, I carried out a *qualitative* study
in the fall and winter of 1996-97. The students who participated in this
study were eighteen high school students from Central Virginia enrolled in
either an algebra 2, a precalculus, or a calculus class. The students were
videotaped individually for approximately thirty minutes as they completed
various graphing exercises using a graphing calculator of their choice. As
the students completed each problem, each keystroke and window setting were
carefully recorded. Also, with each and every keystroke, students were
asked to justify their reasoning.

The focus of this study was how high school students, when using graphing calculators, deal with issues of scaling, obtain appropriate viewing windows of graphs, and interpret and resolve perceptual illusions. It was hoped that by identifying the strategies students used to generate and interpret graphs and the scales of graphs, and how they reconciled conflicting information due to perceptual illusions, teachers would be given insight into how students think and reason, which may explain and account for gains, or the lack thereof, in achievement scores. Also, by discovering what misconceptions graphing calculators give rise to on the part of students, teachers could prevent these misconceptions from developing (or dispel them if they exist already!) by better informing students of the limitations of graphing calculators.

Described below are some of the major findings of this research study, followed by recommendations for teachers.

Scale and Range Confusion

When using graphing calculators, several students in this study demonstrated their lack of understanding of the relationship between the range (Xmin, Xmax, Ymin, and Ymax) and scale (Xscl and Yscl) of a viewing window and the effects of changing these window values. Similar results have been reported by others (Kerslake, 1981; Goldenberg, 1987, 1988, 1991; Leinhardt, Zaslavsky, & Stein, 1990; Yerushalmy, 1991; Williams, 1993). In six of nine instances, students demonstrated the misconception that by changing the scale (Xscl or Yscl), the inclination of a line displayed on their calculator's screen would appear to be more or less steep. Their predictions included: "It's gonna be more diagonal" and "It would tilt down more."

The students in this study may have conjectured that the inclination
of a line would change if the scale (Xscl or Yscl) were altered because, if
this were a paper and pencil graph, adjusting the scale (placement and
value of tick marks) *could* result in a graph of a different
appearance. However, in a graphing calculator environment, changing the
scale (Xscl or Yscl) will *not* affect the shape of a graph; instead,
what changes is the number of tick marks appearing on the calculator’s
screen. These students failed to see that when using graphing calculators,
the graph of an equation is *independent* of one’s choice for
scale (Xscl or Yscl) and *entirely* dependent on the ratio of the
bounds on the axes.

Williams (1993) cited that the students in her study demonstrated a
possible
semantic confusion with the word "scale." What do we mean when we ask
students
what the "scale" of the axes is? Do we mean the *value* of the tick
marks,
or do we mean the *bounds* of the axes? One confounding factor is
the fact
that the variables in the viewing window on graphing calculators that result
in the placement of tick marks on the screen are labeled "Xscl" and "Yscl."
When the "scale" of a map is made larger, the map increases in size; yet, if
Xscl and Yscl are increased, the graph does *not* increase in size.
Careful
consideration should be given, therefore, when using the word "scale" in
a graphing
calculator environment because of its ambiguity. Perhaps the
manufacturers of
graphing calculators should consider renaming "Xscl" and "Yscl" to "X-tick"
and "Y-tick", respectively, to prevent this confusion!

A Finer Scale Will Result in a More Accurate Trace

Two students demonstrated the misconception that by making the
Xscl finer, the trace cursor would be more precise when locating a point of
interest. For example, when trying to find the point of discontinuity in
the graph of the equation y = (x - 3) / (x - 3), one student justified his
reasoning for choosing a finer Xscl setting stating, "I guess the smaller
you make your intervals, your um scale, the more precise the graph will be.
The more precise your little cursor here will be as far as finding a point
on the graph when tracing. And the wider they are, the less accurate they
will be." Although both students were correct in believing that a finer
scale would enable them to better *visually* approximate the location
of a particular point on a graphing calculator or on a paper and pencil
graph, they possessed the misconception that creating a finer scale would
increase the accuracy of the trace feature.

Strategies Used to Obtain Graphs

When asked to find an appropriate viewing window for the graph of an equation, in 51 of 73 instances (70%) students used the "press and pray" strategy in which they immediately pressed GRAPH or DRAW to display the graph. Students responded by immediately displaying the graph in a default window (standard or initial) in 12 of 73 instances (16%). Only in 10 instances (14%) did students begin by first hand-setting a viewing window, after first considering the critical points and features of the equation.

Another recurring strategy used by students, which has been documented by others (Dick, 1992a; Donley & George, 1993; Tuska, 1993), was that "If I make the window large enough, I will find the graph." That is, several students expressed the belief that if they set the bounds on the viewing window to be very large, they would eventually capture the graph's end behavior and, similarly, if the graph were made small enough, local behavior could be observed. Instead of reflecting on the mathematics of the equation to determine an appropriate window setting, students continually used the zoom features to zoom in or out on the graph until an appropriate view of the graph was obtained. Thus, very little critical thinking and much button pushing was occurring.

Although one salient feature of a graphing calculator is the window feature, which allows students to quickly and easily adjust the bounds and scale of graphs, to what extent does access to this feature discourage or squelch mathematical thinking? It appeared as though students in this study were doing less mathematical thinking when using graphing calculators than when using paper and pencil. When asked to graph an equation using paper and pencil, students had to first analyze the equation for critical points in order to determine an appropriate scale for their axes. For example, when asked to manually graph such linear equations as y = 3x + 400 and y = 20x + 1000, 11 out of 13 struggled with how to design a scale on the axes that accommodated both a small slope value and a rather value for the large y-intercept. They asked, "Does there have to be a scale at all?" and commented, "The slope is going to look bad." When using their graphing calculator, most students fell victim to initially using the "press and pray" strategy and thus did not analyze the equation for critical points, or consider what might be an appropriate window setting. In fact, one student stated, "I usually don't bother fiddling with the window. The less thinking the better because it's needless keystrokes." Furthermore, when students were asked to identify the y-intercept in the above linear equations, several students relied on "CALC" and "GSOLV" features of their calculator to provide the answer, even though the y-intercept was clearly identifiable in the algebraic representation of the equation.

"Setting the Window Using the Equation's Coefficients" Strategy

Several students over-generalized that any or all of the coefficients of an equation could be used to set the viewing window. For example, when asked to find an appropriate viewing window for the graph of y = -.0001x2 + .002x + 250, an algebra 2 student correctly identified the y-intercept and, consequently, set the y-axis to range between 0 and 300. However, she incorrectly used the x-coefficients in the equation to set the x-values in the window. She set the Xmin equal to -.0001 and the Xmax equal to .002 because, "These [pointing the coefficients in the equation] are really tiny numbers and -.0001 is the coefficient in front of the x2 and it's the smallest number."

The Calculator Displays the Asymptotes

Several researchers have warned of the potential problem with graphing calculators, where points to the left or right of a vertical asymptote of a rational function may be connected, giving an impression of continuity where there is discontinuity (Demana & Waits, 1988; Hector, 1992; Tuska, 1993). Ten students were given tasks in which they were asked to graph an equation that contained an asymptote and, when the graph appeared on the screen, all but one pre-calculus student incorrectly identified the "line" connecting the two pieces of the graph as the asymptote.

I believe that because some of the students in this study predicted
from their visual inspection of the equation that it contained a point of
discontinuity in the form of an asymptote, they therefore anticipated
*seeing* an asymptote on their calculator's screen. When what looked
like an asymptote appeared on the screen, this confirmed the students'
predictions. Thus, their preconceptions influenced their perception of the
graph (Goldenberg, 1987). When I explained to one particular precalculus
student that the "line" appearing on the screen was *not* an
asymptote, she replied sarcastically, "Could have had me fooled!"

Another reason why students believed that the line appearing on the screen was an asymptote may be due to the fact that, when graphing an asymptotic equation using paper and pencil, students are normally requested to draw the asymptote (usually represented by a dashed line). Consequently, when a "line" appeared on their calculator's screen, I believe students recognized its familiar shape and assumed it was an asymptote generated by their calculator.

Non-Appearance of Points of Discontinuity

Several students gave very interesting reasons for why a point
of discontinuity did not appear on their calculator’s screen. One
pre-calculus student explained the non-appearance of the hole in the
equation y = (x^{2} - 4) / (x - 2), claiming, "It’s just a
single point. If it showed a hole on the graph, it would show an area
larger than the point. It would include things above and below 2." An
algebra 2 student attributed the fact that he could not see the hole to the
manufacturer of his calculator claiming, "I've seen it before....like a
graph of this form, you know, with a hole, that it usually puts a mark.
But that was on an [TI-] 82, so I don't know if whether that made the
difference or not."

Several researchers have expressed concern over the fact that a point of discontinuity may not show up as a "hole" on the calculator's screen, depending upon the window setting (Dion, 1990; Dick, 1992a; Hector, 1992; Tuska, 1993; Williams, 1993). Goldenberg (1988) stated that because the "identifiable size of a pixel makes the hole in a graph....seem to have size" (p. 165), this may reinforces students' apparent expectation that they can see a hole if they magnify it sufficiently. Dick (1992a) has recommended that teachers encourage students to change viewing windows to make certain that they experience "these phenomena and explain why it happens" (p. 154).

Left-to-Right Generation of Graphs

Several students seemed to be "visually enticed" as they watched graphs
being generated from left-to-right on their calculator's screen. For
example,
after watching the graph of the cubic equation y = 2x^{3} -
16x^{2}
+ 12x + 6 appear on her calculator's screen, a precalculus student
conjectured
that the non-displaying piece of the graph would appear off to the
*right*
of the screen, "because that's the way the calculator drew it. It started
from
the left and went to the right." Despite the fact that the "missing
piece" was
indeed to the right of her viewing screen, the mathematics of the
equation (namely,
the coefficient of the x3 term) did not suggest this to her but, instead,
the
left-to-right generation of the graph. Similarly, when asked to find an
appropriate
viewing window for y = 2px^{2} + 2000/x,
another precalculus student, who was also visually enticed by the
left-to-right
generation of the graph continued to adjust the Xmax in his viewing
window in
hopes to find "the other edge" of the graph.

Conclusions and Recommendations

The goal of integrating graphing calculators, or any technology
or manipulatives, into the mathematics curriculum is to enhance student
learning and achievement. Based on the results of this study, it may
appear as though graphing calculators may do more harm than good, given the
students' underdeveloped understanding of scale and range changes, their
over-reliance on the window feature, and their explanations for the
appearance of points of discontinuity. On the contrary, it was *not*
this technology that was a detriment, but the students' lack of experiences
with working with graphing calculators. In particular, I attribute a large
portion of the students’ underdeveloped understandings, strategies,
and misconceptions to the use of restricted examples.

Several students in this study mentioned that the equations they were assigned to graph in class and for homework "usually come up in the standard window." This use of "restricted examples" (Tuska, 1993, p. 114) denies students the chance to visually see and experience the difference a window can make and thus, most probably accounted for the lack of students' understanding of scale and range changes. For example, when asked to graph the equation y = 20x + 1000, several students did not think what appeared on their calculator’s screen was correct because the line was "not steep enough." Because the students had graphed lines primarily in default viewing windows, students developed a fixed, mental notion of what a line with a slope of 20 should look like: It should be very steep. When what appeared on their calculator's screen did not match this belief, students began doubting the graph generated by their calculator and fumbled with TRACE and other features to confirm the slope and y-intercept. Similarly, had students been encouraged to view graphs of discontinuous functions in various windows, students would have seen firsthand the "difference a window can make;" that is, a hole may or may not appear due to the window's setting.

The restricted use of examples is perhaps a primary reason why students heavily used the "press and pray" strategy to obtain appropriate viewing windows of graphs and why they also relied on using default window settings. One benefit of graphing calculators is that they can relieve students of the burden of cumbersome computations and algebraic manipulations. Certainly, if students want to quickly obtain a graph of an equation and the calculator offers features that accomplish this, why shouldn't students forge ahead and take advantage of these features? The danger arises when students become overly reliant on these features to the point of complete dependence, something I witnessed in this study. It appeared as though the beneficial features of graphing calculators that allowed for the automatic generation of graphs consequently removed the need for students to think critically about the equation. Thus, mathematical thinking was being replaced by button pushing.

The implication here is that teachers need to expand their
repertoire of examples to include those equations that, when graphed, do
*not* necessarily display in default windows. This would, in turn,
obligate and promote students to perform more exploration and analysis when
challenged to find the domain and range of graphs. Students need to learn
the navigational skills to position the window for the best scenic views of
the graph (Dick, 1992b) and they must have experience with controlling
range and scale, shifting figures around on the screen, and observing and
describing the effects of those changes and shifts. Because graphing
calculators have a finite viewing screen, this inherent limitation gives
rise to the need for graphing skills which have long gone under-emphasized.
Had the students in this study been provided with more opportunities to
graph equations for which default viewing windows were *not* the most
appropriate, perhaps more students would have been able to correctly
explain the effects of scale and range changes, the appearance of so-called
asymptotes, and the non-appearance of points of discontinuity.

In conclusion, I strongly believe that graphing calculators are
excellent tools which can foster exploration and investigation and can
enhance student understanding and therefore achievement; however, teachers
need to carefully consider how to implement graphing calculators in
instruction so that students' mathematical understanding of graphing is
enhanced and reinforced, and not replaced by the memorization of a sequence
of button pushing. We as teachers need to insure that students are using
the calculators in the way in which we intend. More importantly, teachers
should encourage students to *first* conjecture and predict the
behavior of graphs by using their mathematical knowledge, and then use the
calculator as a *secondary* means of verification.

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