2000, Vol. 12, No.3, 254-268
Mathematics Edl/cation Research Journal
Students' Difficulties in Operating a Graphics
Calculator
Michael Mitchelmore and Michael Cavanagh
Macquarie University
We investigated how students interpret linear and quadratic graphs on a graphics
calculator screen. Clinical interviews were conducted with 25 Grade 10-11 students
as they used graphics calculators to study graphs of straight lines and parabolas.
Student errors were attributable to four main causes: a tendency to accept the
graphic unage lU1critically, without attemptulg to relate it to other symbolic or
numerical ulformation; a poor lU1derstandulg of the concept of scale; an uladequate
grasp of accuracy and approxnnation; and a limited grasp of the processes used by
the calculator to display graphs. Implications for teaching are discussed.
Graphics calculators (GCs) were first developed in the mid 1980s and since that
time have become steadily cheaper, user-friendlier, and more powerful. As a result,
they are being increasingly used in mathematics teaching (Waits & Demana, 2000).
Although there has been considerable research on GCs in the classroom
(Dunham & Dick, 1994), the research has been somewhat inconclusive. As Penglase
and Arnold (1996, p. 82) conclude,
Research results have consistently indicated that the use of graphic calculators can
be associated with significant gains [in some areas]. ... It is stilltmclear, however,
whether the graphic calculator is an effective tool [in other areas]. ... [Some studies]
suggest that the use of the graphic calculator may create difficulties for some
students.
Furthermore, they add few studies distinguish carefully between the use of
the tool and the context of that use" (p. 82), that is, between the use of graphics
calculators and the alternative content and teaching methodology which they make
possible. Consequently, "there is an urgent need to understand which aspects of
graphic calculator usage best facilitate learning" (p. 82).
The majority of research into the use of GCs consists of teaching experiments
which assess the effectiveness of using a GCinstead of traditional pen and paper
methods. Researchers have rarely investigated how individual students actually
use a GC. In particular, although there is anecdotal evidence that students
occasionally misinterpret the graphic image, we have found no systematic research
on the types of misconceptions which arise or their causes. The research reported
below is an attempt to rectify this shortcoming. We focus on the use of GCs for
graphing functions prior to calculus.
/I
Background
Graphs are commonly regarded as a crucial weapon in the mathematician's
arsenal. In precalculus mathematics, major attention is paid to graphs of linear and
quadratic functions. Students learn to recognise the important features of such
StudC1lts' Difficulties in Operating a Graphics Calculator
255
graphs: intercepts and gradients of lines, and axis, vertex, and concavity of a
parabola. They do this by drawing and studying a number of linear and quadratic
graphs.
When GCs are not available, the number of graphs that students can draw is
rather limited. Examples are therefore carefully chosen to minimise distractions.
Scales are often given in advance; they are always "nice" (e.g., 1 unit to 20 mm)
and usually the same on both axes. When scales are not given, students are
encouraged to choose nice scales in order to simplify plotting and reading off
values from the graph. The functions are chosen so that the interesting parts of
their graphs occur fairly near the origin and the coordinates of critical features
(e.g., intersections with the axes) are almost always small integers.
Proponents of the use of GCs in precalculus mathematics (e.g., Asp, Dowsey, &
Stacey, 1993) argue that one of their main advantages is that they allow students
greater freedom to explore. Students can use many more examples than they can
when graphs are drawn by hand, and they can easily change the scales once the
graph is drawn (Demana, Schoen, & Waits, 1993). However, such explorations
inevitably lead to graphs with "nasty" and unsymmetrical scales, non-integer
coordinates, and blank screens and partial views. Because graphs are produced at
the push of a button, students are not plotting individual points or using gridlines,
and the coordinate axes are not labelled, there is a real danger that the whole
process might appear somewhat arbitrary or magical (Dion & Fetta, 1993) and that
fundamental misconceptions might arise (Mueller & Forster, 1999). Students using
GCs also have to deal with the non-trivial task of learning how to operate the
machine itself.
A small number of studies have noted student difficulties in using a Gc.
Several of these point to difficulties with scale. Goldenberg (1988) and Goldenberg
and Kliman (1988) identified problems interpreting changes in scale when
zooming. Student difficulties may be exacerbated when the scales are unequal; for
example, Vonder Embse and Engebretsen (1996) note that pairs of perpendicular
linear graphs do not necessarily appear at right angles on the GC screen. Students
may also fail to zoom out to obtain a true picture of a graph; Steele (1994) noted the
ready acceptance by students of the initial graph shown in the default window.
The use of pixels as representations of points can cause several difficulties, as
Goldenberg (1988) noted, and Williams (1993) remarked on a number of difficulties
students have when points of discontinuity are not visible due to the way the pixels
are connected. At a more basic level, Vonder Embse and Engebretsen (1996) noted
the confusion caused by the awkward-looking coordinates which often appear
when tracing graphs on a Gc. They demonstrated the advantages of being able to
create so-called "friendly windows", in which each pixel has an x-coordinate with a
small number of decimal places. (More modern GCs have a limited facility to do
this at the press of a button.)
Unfamiliarity with the technology was identified by Giamati (1991) as a further
source of difficulty. She found that the students' limited experience in using a GC
might have accounted for their lack of progress in a unit on transformations of
graphs.
Tuska (1993) analysed the responses of first year college students on multiplechoice examinations to determine which errors might be related to GC use. She
Mitchell110rc [...- Cavanagh
256
identified misconceptions in students' understanding of the domain of a function,
asymptotes, the solution of inequalities, and the belief that every number is
rational. It is not clear, however, whether these misconceptions were caused by GC
use, or whether they just became more obvious when using a Gc. The same could
be said of several of the student difficulties identified in other studies.
The present study was designed to investigate student difficulties in using a
GC in much more detail than has previously been done. In particular, we
attempted to distinguish difficulties which are caused by lack of understanding of
the technology and those which are caused by more fundamental, mathematical
misconceptions. Dunham and Dick (1994) suggest that "by probing students'
conceptual understanding through interviews, researchers can paint a more
detailed picture of the effects of graphing calculator-based instruction on students'
learning" (p. 441), and that was the method we used.
Method
Participants
Extensive clinical interviews were conducted with 25 students,S students from
each of 5 Sydney metropolitan high schools (15 Year 10 students: 8 girls and 7 boys;
and 10 Year 11 students: 5 girls and 5 boys). The students were all studying
mathematics at the highest level available to them (15 students in the Year 10
Advanced Course,S students in the Year 11 3-Unit Course, and 5 students in the
International Baccalaureate Mathematics HL Course). It was felt that high~r­
achieving students would be better able to respond to the challenge of the
interview tasks and to articulate their ideas cleady. It was assumed that the
difficulties which these students demonstrated would also be found, to a greater or
less degree, in lower-achieving students.
All the students had studied graphs of straight lines given by equations of the
form y = mx + c and parabolas of the form y = ax 2 + bx + c. They were also familiar
with the quadratic formula, and had used it to solve quadratic equations and to
locate the vertex and x-intercepts of a parabolic graph.
The students had used a Casio fx-7400G graphics calculator in their
mathematics lessons for between 6 and 12 months prior to the first interview. In
one school, students owned their own !t"-7400G and could use it in all lessons and
examinations. In all other schools, the students had only limited access to a class set
of GCs owned by the school. Most of the students (even those who owned their
own GCs) were inexperienced users of the technology, having only used them to
display graphs of linear and quadratic functions.
Interview Tasks
The interview tasks in this study were designed to create for students
problematic situations which directly confront some of the limitations of the GC.
Some of the tasks were similar to those used in previous studies (Steele, 1994;
Vonder Embse & Engebretsen, 1996). Other tasks were designed by the authors to
investigate the misconceptions reported in the literature, summarised above, or
Studellts' Difficulties ill Operating a Graphics Calculator
257
other misconceptions discovered in the course of pilot testing.
The instructions and the initial GC screens students were expected to obtain
(or, in Tasks 2, 5 and 7, those given to the students) are shown in Figure 1. Note
that, on the fx-7400G, the initial window is friendly with pixels at intervals orO.1
along the x-axis; tick marks are shown at intervals of 1 on both axes, their values
ranging from -3 to 3 on the x-axis and from -2 to 2 on the y-axis.
Task 1. The quadratic formula was chosen in such a way that the portion of the
graph in the initial window appeared straight. The task tested whether students
would refer to the algebraic representation of the function and recognise that the
GC screen did not display a complete graph of the parabola. Task 1 also examined
what students might do to obtain a more representative graph.
Task 2. For this task, students were shown a GC with the two graphs'displayed.
The window setting had been changed, so that both axes were displayed from -10
to 10. Task 2 was designed to investigate how well students understood the effect
of scaling the two axes differently.
Task 3. This task was designed to see how the students would move beyond the
initial window to locate the intersection of the two lines. Because the two lines have
a similar slope, they appear to meld together over a number of pixels. The task also
investigated how students would resolve this apparent contradiction.
Task 4. This task was designed to see how students would approximate the xintercepts of a quadratic function with irrational roots. Further, to investigate how
the students would deal with a situation where the vertex was not given by a single
lowest pixel, the coefficients were chosen in such a way that a flat line of six pixels
occurred near the vertex in the initial window.
Task 5. Students were shown two GCs. On one GC, the line y = x was displayed
in the initial window. On the other GC, the line y = x was displayed in a window in
which both the scale and the interval between the tick marks on the y-axis had been
doubled. The values of the. tick marks on the second GC therefore ranged from -6
to 6 on the x-axis and from -4 to 4 on the y-axis, but this was not apparent to the
casual observer. The purpose of this task was to investigate any assumptions which
the students held about the format of the GC window. Task 5 also provided a
further opportunity to discuss the students' notions of scale.
Task 6. Students were given two GCs-one set to the usual initial window and
one set to a window in which both the x- and y-axes were displayed from -10 to
10-and their attention drawn to these settings before they drew the line y = 2x - 1.
On both calculators, it was possible to move the cursor to the point (0, -1).
However, on the second calculator, it was not possible to move the cursor to obtain
the coordinates (I, 1); because the window was not friendly, almost all coordinates
were displayed with 4 decimal places. This task was designed to investigate
whether the students were aware of the procedure used by the GC to assign xcoordinates to pixels. Knowledge of this procedure is essential if students are to
understand how "friendly windows" might be created.
258
Mitchel71lorc & Cavanagh
"
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1. Draw a sketch of y = 0.lx 2 + 2x - 4.
You may use the graphics calculator
to help you.
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2. Explain why the graphs of the lines
y = 2x + 3 and y = -O.Sx - 2.5 do not
appear at right angles on the screen.
What could you do to make the
lines look more perpendicular?
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3. Use the graphics calculator to find
the intersection of y = 2x - 1.5 and
y = 3x + 0.8.
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4. Display the graph of the fllilction
y = 0.7Sx 2 - 1.4SSx - 1 on the
graphics calculator. Find the intercept with the positive x-axis and the
coordinates of the vertex.
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5. Which, if any, of the calculator
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7. Look at the graph of y = x 2 . What
do you notice about the way that
the groups of pixels are arranged?
What might this suggest about the
gradient of the parabola as you
move along the curve?
8. Display the graph of y = x 2 - 2x + 3
on the graphics calculator. Can you
change the window settings of the
calculator so that the graph appears
as a horizontal line?
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6. Display the graph of y = 2x - 1 on
each graphics calculator. Move the
cursor to the points (0, -1) and
(I, 1). What do you notice? Can you
explain what has happened?
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Figure 1. Interview tasks and initial or given GC screens.
V=I.D512
Students' Difficulties in Operating a Graphics Calculator
259
,.,
Task 7. In this task, students were shown the graph of the parabola y = x- .
They were asked to describe the arrangement of the highlighted pixels and to
compare this with the pixel groupings in some of the straight-line graphs they had
seen previously. If needed, the students were shown some linear graphs on another
GC so that a more direct comparison could be made. The purpose of the task was
to investigate whether the students might notice that the pattern of highlighted
pixel groupings on straight-line graphs remained the same, while the number of
pixels grouped together on the parabola did not. The students were then asked to
make inferences about the gradients of straight lines and parabolas based on what
they had noted about the arrangement of the pixels.
Task 8. This task was included as a challenge for those students who finished
the first seven tasks relatively quickly. Its successful completion requires a good
understanding of scale and the role of the various window settings.
Interview Procedures
Each student was interviewed individually by the second author for fifty
minutes on three separate occasions, each approximately two weeks apart. The
eight tasks were administered in a fixed order, spread over the three sessions. For
each task, a semi-structured interview schedule was prepared highlighting the
critical points that we wished to investigate and allowing for various follow-ups
depending on students' responses. If students did not reach one of the critical
points, they were prompted to perform further calculator operations which led to
that point. The students were asked to think aloud and explain the processes they
used to interpret the graphs they saw, but errors were not corrected. Neutral
prompts were used whenever a students' reasoning was not clear to the
interviewer.
For example, if students zoomed out and sketched a parabola in Task I, they
were asked why they had done so. If they sketched a straight line, no comment was
made at the time. However, at the end of that day's session, they were again shown
the function and asked if their sketch was a reasonable representation of it. If
students recognised that the graph of the function was a parabola, they were asked
to explain why they had earlier drawn a line instead of a parabola.
All of the interviews were videotaped, and selected segments were transcribed
to allow a detailed analysis of students' responses.
Results
Task 1. Only seven students (28%) sketched a parabola; all of these students
commented that the x 2 term in the function indicated that the graph must be a
parabola and then zoomed out until they saw the V-shaped curve. The remaining
18 students (72%) drew a straight line as their sketch of the quadratic function. Of
these,S students (20%) simply copied the straight line directly from the calculator
screen; the other 13 students also either zoomed out (n = 6) or referred to the
constant term in the function (n = 7) to find the y-intercept and then marked it on
their sketch.
'
260
Mitchell110re & Cavanagh
Task 2. Only 4 students (16%) explained
~ithout
prompting that the reason
why the lines were not perpendicular was the unequal scaling of the coordinate
axes. The remaining 21 students (84%) recognised after prompting that the axes
were not scaled equally; they were then all able to explain why the lines did not
appear perpendicular. All 25 students were eventually able to change the window
settings in some way so that the angle between the two lines appeared closer to a
right angle.
The students were also asked to discuss how they could calculate settings for
the coordinate axes so that the viewing window displayed the pair of lines at
exactly ninety degrees. Thirteen students (52%) were unable to respond to the
question. Twelve students (48%) recognised that they would need to establish the
relative proportions of the GC screen, but they found it difficult to say how this
might be done: 1 student could not give any method of doing so; 7 students could
think only of measuring the screen with a ruler; and 4 students suggested using the
number of pixels which comprise the screen to establish appropriate window
settings. However, the only method proposed by these last 4 students to find such
settings was to count the pixels by eye. They were unable to relate the default
settings of the initial window to the number of pixels even when prompted to do
so.
Task 3. All 25 students were able to zoom out or scroll the window down until
the region near the intersection of the two lines was displayed on the screen.
However, only 7 students (28%) recognised that the low resolution of the screen
and the fact that the two lines had very similar slopes was the reason why their
images melded together over a number of pixels. The remaining 18 students (72%)
expressed surprise at the image they saw and explained that they were expecting
to see a distinct, easily identifiable pixel at the point of intersection. When this
single pixel did not appear even after they had zoomed in once, 5 students (20%)
explained that the pair of lines briefly ran alongside each other and had more than
a single intersection point. Of the others, 6 students (24%) continued zooming and
tracing until they obtained coordinates which they felt were "nice" (integer or nearinteger values), while 7 (28%) simply chose a point close to the centre of the group
of pixels where the lines appeared to merge.
Task 4. Nineteen students (76%) averaged the x-coordinates of the pixels
immediately above and below the x-axis to obtain an estimate for the intercept,
despite the fact that the y-coordinate displayed for one pixel was considerably
closer to zero than the other. In a similar way, 21 students (84%) explained that the
x-coordinate at the vertex of the parabola could be found by averaging the x-values
of the two centre pixels in the row at the base of the graph. The students did not
seem to realise that the y-coordinate of this point was greater than the y-coordinate
of the lower of the two centre pixels.
All the students then zoomed in to find the intercept and vertex more
accurately. Most (88%) continued zooming until the interviewer stopped them; all
these students expressed the belief that the horizontal line at the base of the
parabola would become shorter each time they zoomed in until eventually one
pixel would be seen at the vertex. Only three students (12%) correctly stated that
the ropts of the parabola were irrational and that the x-intercept was therefore a
Students'
D~fficulties in
Operating a Graphics Calculator
261
non-terminating decimal which could not be found exactly. Only one of these
students could also explain that the x-coordinate of the vertex of the parabola was a
rational number which could be expressed by a terminating decimal.
Only 3 students (12%) correctly stated that the scale of the graph they saw
displayed on the GC screen had changed after they had zoomed. The other
students spoke of zooming as akin to using a more and more powerful microscope.
Task 5. Almost all the students (92%) thought that the first screen showed the
line y = x and that the second must represent the line y = O.5x. Each of these
students explained that the tick marks calibrated on the coordinate axes must
always represent unit values. Only 2 students (8%) recognised that the apparent
slope of the graphs depended on the scaling of the axes.
Task 6. Explaining why it was not possible to locate (I, 1) on the second
calculator proved to be an extremely difficult task. In fact only 2 students (8%)
gave a satisfactory explanation which included a reference to the relationship
between the number of pixels across the screen and the window settings. Nine
students (36%) were unable to answer at alL and the remaining students often gave
the impression that they made a particular response because they could think of
little else. For instance, 6 students (24%) regarded the unequal scaling of the axes as
the main source of the problem; 3 students (12%) thought that the problem was due
to the linear function itself and that if another line which contained the point (1)
was graphed then it would be found; 3 students (12%) said that there was a pixel
assigned to (1,1) but the low resolution of the screen meant that it could not be seen
until they had zoomed in a number of times; and 2 students (8%) attributed the
problem to the settings of both coordinate axes rather than relating it to the x-axis
alone.
Task 7. All of the students were able to explain the arrangement of the pixels in
2
the graph of the parabola y = x • They noted the regular patterns of the pixel
groupings in the linear graphs which they had seen previously, and contrasted
these with the way that the number of pixels grouped together on the curve
continually changed. All of the students concluded that, whereas the slope of a line
remains constant, the slope of a parabola changes as you move along the curve and
increases in absolute value as you get further away from the vertex.
Task 8. This task was only attempted by 12 students. Only one student was able
to change the window settings so that the parabola appeared as a horizontal line
across the screen. The remaining 11 students could not solve the problem because
of the restrictions they imposed on their choices for the window settings. These
students claimed that the minimum values for the coordinate axes must always be
a negative quantity and the maximum values m,ust always be positive, but they
could not explain why they felt that this should be the case. Moreover, even though
the horizontal line of pixels on the parabola lay wholly within the first quadrant of
the number plane, these students continued to choose equal and opposite values
for the minimum and maximum. They stated that they did so in order to maintain
the origin at the centre of the screen.
262
Mitchell1lOre & Cavanagh
Discussion
The student responses obtained in this study may be influenced by any
number of factors, including inattention, inexperience, and nervousness. Many
responses showed lack of reflection, perhaps due to the unfamiliarity of the tasks
and the interviewer. So our findings are indicative of what students have to learn
rather than definitive statements of levels of their ability to think and reason using
the Gc.
With these reservations in mind, students' responses to the tasks described
above suggest that their difficulties could be traced to four key areas of
understanding.
Scale
Students' responses to Tasks 2, 5, and 8 show that they had a very limited
conception of scale as applied to graphs. The reason seemed to be that they had
had little or no experience in dealing with graphs where the axes were not scaled
equally. Whenever students were asked to draw a graph by hand during the
interviews, they always did so using equally scaled axes which were invariably
marked at unit intervals. A number of students echoed this comment: "The axes
must be equally scaled because that's the way I always do it, and the graph won't
look right if I don't". A quick inspection of the mathematics textbooks commonly
, used in Years 10 and 11 confirmed that most diagrams of graphs are drawn in this
way. Leinhardt, Zaslavsky, & Stein (1990) report a similar phenomenon.
An important distinction can be made between what might be called relative
and absolute understanding of scale. The former correctly regards scale as a ratio of
distance to value, while the latter interprets scale solely as either the distance
between adjacent marks or the value represented by this distance. The vast
majority of the students in this study had only developed an absolute
understanding of scale. They found it difficult to give appropriate or meaningful
descriptions of the scale of the coordinate axes displayed in the GC window, and
they explained their ideas almost exclusively in the language normally associated
with simple scale drawings. Unfortunately, scale drawings preserve shape and can
be quantified by a single ratio-a much too limited context for developing the skills
needed to deal with the graphs which inevitably arise within the GC environment.
Students' limited conceptions of scale were also manifested in poor
understanding of the zoom operation. As Goldenberg and KIiman (1988) also
found, there was a marked tendency among the students to disassociate the zoom
operation from any change in the scale of the graphs they saw. Instead, the
students most often described zooming in terms similar to the use of a magnifying
glass on a physical object when, as one continues to zoom in, previously obscured
details are gradually revealed. As one student remarked, "If you zoom in on a
certain part of a graph it seems bigger. However, you're only enlarging it to seeit
more closely. You're not actually changing any of the values, so it would still be the
same. You're not changing the scale, just making it [the graph] seem larger". So, for
example, many students expected that zooming in on the vertex in Task 4 would
eventually produce a curve with a single minimum point. They were, naturally,
Stude11ts' Difficulties ill Operatillg a Graphics Calculator
263
surprised to find that the curve only showed more and more points in a line at the
vertex.
On several tasks, students tended to try to solve any problem they encountered
by zooming and to persist with this approach long after it should have been clear
to them that it was unproductive. For example, it is not possible to find an exact
value for an irrational root by zooming. Another indicator of students' poor
understanding of the zoom operation was demonstrated on several tasks: After
zooming in several times, many students expressed surprise that the same
coordinate values should be repeated over and over again. In other words, they did
not readily recognise the effect of the zoom operation on the values assigned to the
pixels.
Students also did not know how to change the spacing of the tick marks
displayed on the axes of a graph using the scl parameter in the window setting. All
the students we interviewed assumed that the scl value was connected with the
scale of the axes, causing considerable confusion and reinforcing their absolute
concept image for scale.
Accuracy and Approxinzation
The students often had great difficulty in making appropriate numerical
estimates for the values they were looking for. This was especially marked in Task
4. The most common approach used by the students was to average the coordinate
values on either side of a point of interest regardless of whether the actual value
was closer to one side or the other.
Students also regularly made a direct correlation between the greater number
of significant figures given in a decimal value and its accuracy, without any real
attempt to consider the specific context of the problem they were being asked to
solve. Somewhat paradoxically, however, the students showed a marked
preference for integer or other "nice" values when locatihg points of intersection.
As one student remarked when tracing to find the intersection point in Task 3, "It's
that point because there weren't any decimal places and it didn't do that for any
other point ... If it gives you a whole number, it means it's more accurate".
Few students were aware of the differences in the decimal representations of
rational and irrational numbers. In general, the students thought that all points on
a graph must always be expressed by finite decimal values because they
represented exact distances from the origin on the number plane. In Task 4, a
student claimed that the x-intercept of the parabola must be a terminating decimal
because "there must be a point where they [the parabola and the x-axis] both
intersect and whether it's like five decimal places or not it will stop ... because
that's where it touches each other and it has a point where they both meet and that
point is a decimal that has to stop".
Linking Representations
Research on student understanding of graphs in general (Leinhardt, Zaslavsky,
& Stein, 1990) has led to the conclusion that a major problem is learning the links
between graphs and other representations of functions (Dugdale, 1993;
264
Mitchcll7Iorc & Cavanagh
Moschkovich, Schoenfeld, & Arcavi, 1993). This,problem was also evident in the
present study as students frequently failed to relate graphs drawn on a GC to their
symbolic representation. Utilising this link would have prevented the
representation of a quadratic function by a straight line in Task 1 and the belief that
two lines might meet in more than one point in Task 3.
Throughout the interviews, students tended to accept whatever was displayed
on the GC screen without reference to their general knowledge of functions. The
power of the visual image was also evident in Tasks 2 and 5. The only time when
the visual image was not misleading was in Task 3, when the visual image
correctly indicated where an intersection would be found. In this case, an
appropriate solution could be found without reference to the symbolic
representation of the functions.
Representation by Pixels
Throughout the interviews, the students regularly commented on the jagged
appearance of the graphs they saw on the GC and demonstrated that they were
aware of the pixel approximations associated with the low resolution of the screen.
For example, most students were able to explain why the calculator displayed the
parabola shown in Task 4 with a horizontal line of pixels near its vertex and why
the patterns of pixels in the graphs of linear and quadratic functions are different.
However, even though they recognised the inconsistencies which arose from
time to time between the position of the cursor and the coordinates, the students
were still likely to base their answers exclusively on the visual image formed by the
highlighted pixels. Furthermore, despite the fact that the students claimed to have
more confidence in the coordinates displayed on the screen, they needed constant
reminders to consider these values before deciding on an answer. For example, the
widespread tendency to average the coordinate values on either side of a point of
interest seemed to arise because the students were so focused on watching the
movement of the cursor that they did not sufficiently attend to the y-values
displayed at the ,bottom of the screen. As one student said, "I've taken the number
in between 2.4 and 2.5 because those are the numbers which are above and below,
on the points which are above and below the x-axis, because that's where the
cursor was when it traced". Here was another example of failure to link graphs to
other representations of functions (in this case, numerical).
Students seemed to have little understanding of how a GC produces a graph.
Many difficulties would have been overcome (especially in Tasks 3, 4, and 6) if
students had realised that (a) the x-coordinates a GC assigns to the pixels depend
on the specified range of x-values to be displayed; (b) the GC displays the
calculated y-coordinate (i.e., the value calculated from the given function and the xcoordinate of the pixel) and not the y-coordinate of the pixel where the cursor is
positioned; and (c) the coordinates displayed on the screen are often
approximations rather than exact values. It was also clear in the interviews that
many students did not know that (in the default setting for Cartesian graphs) the
GC calculates and displays one pixel in each column of pixels and then "joins up"
these pixels to display the graph.
Students' Difficulties in Operating a Graphics Calculator
265
Implications and Conclusions
The results of this studysuggest that many difficulties in using a GC-even
among higher-achieving students-may be due to inadequate understanding of
some fundamental mathematical ideas: scale, accuracy and approximation, and the
link between different representations of functions. These weaknesses point to
shortcomings in the present curriculum which may have adverse effects whether or
not GCs are used.
There is probably a need for a stronger curricular emphasis on scale. Students
need more experience in controlling scale and watching for its effects on graphs.
There is also no reason why graphs should be limited to equal scales on both axes;
in fact, as soon as one plots any empirical data, equal scales are rarely appropriate.
As a companion to this change, geometric work on enlargements and similarity
could also be extended to include stretching. Number theory, particularly relating
irrational numbers to their decimal approximations, may also need more attention
in the syllabus. Now that scientific calculators are common from Year 7, there is no
reason why students should give integer values in their answers simply because
they "look nice"; nor should they be permitted to think that more significant
figures in a number necessarily means that it is more accurate.
That said, it is clear that (as argued by Dick, 1992) students will need to apply
their mathematical understanding in new ways if they are to use GCs effectively as
problem solving tools. Our results suggest three areas that need particular
attention. Firstly, students will need to learn the link between zooming and scale.
Our work suggests that students regard these operations as distinctly different and
do not understand the effects of zooming on the range and scale of the axes.
Secondly, they will need to develop better approximation techniques-especially
the efficient use of interpolation. Thirdly, students will have to learn to integrate
symbolic and numerical information more closely with graphical information.
It is often claimed that the visual images provided by a GC can strengthen
students' conceptual understanding, but our results suggest that these visual
images can also be misleading unless they are interpreted with care. On the other
hand, analysis of visual problems such as those students encountered in our tasks
could lead to greater understanding of both numerical and symbolic
representations. The relatively small number of pixels on the GC screen might even
prove to be an advantage. Given that the majority of the students in this study had
only just begun their study of parabolas and none of them had learnt about the
slope of a curve, the fact that they could all use the pixel groupings to deduce that
2
the gradient of y = x varied at different points along the graph is encouraging.
The GC screen may provide a useful visual support in the early development of
differential calculus concepts.
One should beware that what may seem obvious to an experienced user may
not be at all apparent to novice learners. For instance, more than one student in the
present study struggled to explain how the coordinate axes in Task 2 could both
range from -10 to 10 and yet look so different-they had simply failed to notice the
rectangular shape of the window.
Kissane, Bradley, and Kemp (1994) argue that students also need to develop
the technical skills required to operate a GC effectively. Our work suggests that
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Mitchel17lore & Cavanagh
they also need to acquire some technical understanding of how the GC works. It is
particularly crucial that they learn how each pixel is assigned coordinates and how
these coordinates are related to the coordinates displayed on the screen. This
knowledge would help them recognise why a particular coordinate value is not
represented by a pixel on the screen, and might assist them in creating their own
friendly windows rather than being constrained by the small number of default
window settings that are available on the GC (Dowsey & Tynan, 1997). The
discussions we had with students during the interviews indicate that they can gain
insight into the technical operation of the GC/ but that this does not come naturally
to the majority of students. It must be taught explicitly.
Given the difficulties we have identified, the question arises: Should
mathematics teachers attempt to structure their lessons and choose examples so
that students avoid such difficulties? Or should they deliberately plan activities
which force students to face them'? We agree with Dick (1992) that students would
benefit greatly from confronting the limitations of the technology and attempting to
explain them. Doing so would not only strengthen students' basic mathematical
understanding but could lead to much interesting mathematics. But dearly, we
need to take care in how we challenge students' misconceptions. There might be a
case for avoiding difficulties in the early stages and then structuring challenges to
draw explicit attention to old misconceptions or new developments.
We might also ask whether mathematics teachers understand GCs sufficiently
well to be able to structure their lessons effectively in the way we have suggested.
A follow-up to the present study (Cavanagh & Mitchelmore, 2000c) suggests that,
provided in-service instruction in the use of a GC addresses the problems we have
described, teachers do indeed gain sufficient confidence and proficiency to do so.
. Finally, we note that the students in the present study who owned their own
GCs tended to exhibit a critical awareness of the calculator's output more
frequently than those who had merely borrowed school calculators. There is
therefore to be hoped that, just as regular access to the technology can have a
positive influence on linking different representations of functions (Ruthven, 1990)/
so other difficulties and misconceptions may also lessen with greater exposure to
GCs.
Acknowledgments
The research reported in this paper was supported by a Strategic Partnerships
with Industry-Research and Training (SPIRT) grant from the Australian Research
Council. The industry partner was Shriro Australia Pty. Ltd. (distributors of Casio
calculators). Parts of this research have previously been reported at MERGA and
PME annual conferences (Cavanagh & Mitchelmore, 2000a, 2000b).
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Authors
Michael Mitchelmore, Australian Centre for Educational Studies, Macquarie University
NSW 2109. E-mail: <[email protected]>.
,Michael Cavanagh, Australian Centre for Educational Studies, Macquarie University NSW
2109. E-mail: <[email protected]>.