Graphing Calculator-Associated Strategies Used by and Misconceptions of High School Students
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 calculators screen. These students failed to see that when using graphing calculators, the graph of an equation is independent of ones 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 calculators screen. One pre-calculus student explained the non-appearance of the hole in the equation y = (x2 - 4) / (x - 2), claiming, "Its 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 = 2x3 - 16x2 + 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 = 2px2 + 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 calculators 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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