EMAT 8990 Discussion
Nicholas Oppong
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USING COMBINATIONS OF SOFTWARE TO ENHANCE PRESERVICE
TEACHERS’ CRITICAL THINKING SKILLS*
Nicholas Oppong
Educational software has been used in college classrooms to enhance the study of
mathematics [1] and enhance critical thinking [2]. We have found that using
combinations of software in mathematics classrooms produces inquiring minds thereby
enhancing our pre-service teachers critical thinking skills. During explorations or
demonstrations with a single software, our students have always raised questions that
have called for the use of a second or third software. They not only defend their choice of
second and third software, they also argue against the use of other software. Their
choices of software have been based on the objectives for the day and the ability to
explore, conjecture and discover. The combination of software helps them to explore
mathematical concepts with greater ease and discuss the results of the exploration with
more instructional power. In this paper, we will use the study of quadratic concepts to
illustrate the use of combinations of software to enhance critical thinking in pre-service
mathematics teachers.
Computers in Mathematics Education is a required course for our pre-service secondary
mathematics teachers. The course prepares students to become teachers in the 21st
century. It concentrates on using various software applications to pose, extend, and solve
mathematics problems, to organize pedagogical demonstrations, to set up problem
explorations, to communicate mathematical demonstrations, and to help our students
think critically all in a technologically enhanced classroom. The applications used in the
course included Geometer's Sketchpad (GSP) (Key Curriculum, 1996); Algebra Xpresser
(Alan Hoffer, 1990); Calculus (Broderbund Software, 1989); Microsoft Excel (Microsoft,
1996); and Microsoft Word (Microsoft, 1996). The students utilized the Internet for
classroom assignments and explorations. We felt strongly after teaching the course that
using multiple applications benefits students. Only with a broad knowledge of each
package’s strengths and weaknesses together with the ability to use multiple applications
USING COMBINATIONS OF SOFTWARE - REVISED
to achieve certain goals could our students succeed in a technological environment, and
become critical thinkers.
USING ANIMATION FEATURES IN SOFTWARE
Our objective was to reacquaint our students with the behavior of the graph y = a x2 + b x
+ c as the parameters a, b, and c were changed. We also wanted them to think about how
they will teach this concept to high school students. Several avenues were open for the
initial exploration of y = a x2 + b x + c. We wanted to begin with a graph that moved
instantaneously with changes in a, b, and c. We could find this feature in GSP, Theorist,
or the graphing calculator package that ships with the PowerPC. The animation features
in Theorist, or the PowerPC graphing calculator can be used to slide graphs on the screen.
Of the three, we felt that GSP was the most widely available software in the high schools.
The graphing calculator works only on a PowerPC and few are likely to be found in
today’s schools. Theorist is available in select high school classrooms. These are all
limitations of the high school classroom, but are important to our students and their early
teaching experiences. So, we began exploring quadratics with a GSP sketch. We
encourage you to construct your own sketch using the description below (Your sketch
should be similar to Figure 1.).
Given:
1. Point O (origin).
2. Point Endpoint of segment c.
3. Point Endpoint of segment b.
4. Point Endpoint of segment a.
Steps:
1. Let 1= Unit Point of coordinate system with Origin 0.
2. Let x= the horizontal axis.
3. Let y= the vertical axis.
4. Let [j] = Parallel to Axis x through Point Endpoint of segment c. (hidden).
5. Let [k] = Parallel to Axis x through Point Endpoint of segment b. (hidden).
6. Let [l] = Parallel to Axis x through Point Endpoint of segment a. (hidden).
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USING COMBINATIONS OF SOFTWARE - REVISED
7. Let [A] = Random point on Line [l].
8. Let [B] = Random point on Line [k].
9. Let [C] = Random point on Line [j].
10. Let a = Segment between Point[A] and Point Endpoint of segment a.
11. Let b = Segment between Point[B] and Point Endpoint of segment b.
12. Let c = Segment between Point[C] and Point Endpoint of segment c.
13. Let x = Random point on Axis x.
14. Let x^2 = Image of Point x dilated by ratio |0x|/|01| about center Point 0 (hidden).
15. Let a = Image of Point [A] translated by vector Endpoint of segment a.->0 (hidden).
16. Let b = Image of Point [B] translated by vector Endpoint of segment b.->0 (hidden).
17. Let c = Image of Point [C] translated by vector Endpoint of segment c.->0 (hidden).
18. Let bx = Image of Point b dilated by ratio |0x|/|01| about center Point 0 (hidden).
19. Let ax^2 = Image of Point a dilated by ratio |0x^2|/|01| about center Point 0 (hidden).
20. Let [D]= Image of Point ax^2 rotated by 90.00 degrees about center Point 0 (hidden).
21. Let [E]= Image of Point bx rotated by 90.00 degrees about center Point 0 (hidden).
22. Let [F]= Image of Point c rotated by 90.00 degrees about center Point 0 (hidden).
23. Let [G]= Image of Point [D] translated by vector 0->x (hidden).
24. Let [H]= Image of Point [E] translated by vector 0->x (hidden).
25. Let [I]= Image of Point [F] translated by vector 0->x (hidden).
26. Let bx+c = Image of Point [H] translated by vector x->[I] (hidden).
27. Let ax^2+bx+c = Image of Point [G] translated by vector x->bx+c.
28. Let Locus ax^2+bx+c = Locus of Point ax^2+bx+c while Point x moves along Axis x.
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USING COMBINATIONS OF SOFTWARE - REVISED
Figure 1: Quadratic Sketch
We constructed the quadratic sketch and provided it to students in class. In pairs, they
opened the sketch and manipulated a, b, and c to refine their understanding of quadratics.
Throughout subsequent discussions the word “slide” was used for movements of the
graph. Student’s talked about continuous, dynamic movements as they changed a, b, or c.
The description of the slide, its direction and behavior were discussed in more depth.
The dynamic nature of GSP allowed the students to see the effects of change in a, b, and
c. The speed of the software showed continuous deformation of the graph. The students
were able to see a vertical shift with change in c. They noted that the quadratic deformed
from concave upward to a line to concave downward as a went from positive through 0 to
negative values. The positive and negative orientations are possible in GSP through
directed ratios. As they changed only the b value, students began conjecturing that the
vertex of the parabola traveled along a circle, an ellipse, or another parabola. This raised
a critical question [3] in the class: Is GSP the appropriate software for this investigation?
An issue must be considered real by those involved for critical thinking to exist. As
future teachers, the students saw the need to discern appropriate software for themselves
as a real issue. They stated their positions and gave sufficient evidence to make their
perspectives clear. There was a desire for dialogue in which students could clearly state
opposing viewpoints. The preservice teachers felt that high school students might not be
able to describe or give the equation of the locus of the vertex of the parabola because
there was only one graph, and it was moving. The students felt that this behavior could
best be observed in a more static environment. To them, a static environment was one
where a graph could not be moved, instead multiple graphs could be viewed. Critically
thinking about their options, they began a discussion of other software packages such as
Algebra Xpresser, and hand-held graphing calculators. Ultimately, after discussing
several factors including accessibility and performance, they chose Algebra Xpresser to
continue their exploration . A large consideration for our students was the availability of
color in Algebra Xpresser to distinguish the different graphs.
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USING COMBINATIONS OF SOFTWARE - REVISED
USING STATIC FEATURES IN SOFTWARE
Algebra Xpresser is a relation grapher that allows multiple graphs in color on a single set
of axes. Graphs were drawn to compare several integer values for b. We began with the
general case of a=1, and c=1. We graphed 11 values of b from -5 to 5 (Figure 2). The
students began to agree that the locus of vertices formed a new parabola. They found that
the equation of this particular parabola was y = - x2 +1. Hence, the parabola had a yintercept of 1 and that it was concave down (Figure 3). They went on to prove that it was
always a parabola and to find the general form based on a and c. Again, the students
refined their understanding of quadratic graphs, and were confident that they would be
able to engage their high school students in similar explorations.
Figure 2: y = x2 + bx + 1 for various values of b.
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USING COMBINATIONS OF SOFTWARE - REVISED
Figure 3: y = - x2 +1 superimposed.
Then, we posed another critical question: Have GSP and Algebra Xpresser satisfactorily
covered the subject of graphs of quadratic equations? This opened up a discussion of
other necessary dimensions of quadratic graphs. Several students felt the material had
been covered. Others, relating to high school students, felt there were deficiencies in both
applications. They felt strongly that whenever they were unsure of a definition, such as of
a function, or a manipulation, such as evaluating a function at a specific value, GSP and
Algebra Xpresser offered no assistance. The follow-up discussion, taking into
consideration the differing viewpoints, called for the use of software packages with builtin tutorials. Their choice was Calculus, which offered a clear demonstration of
definitions and manipulations.
USING TUTORIAL BASED SOFTWARE
Calculus is a tutorial package with definitions, dynamic illustrations and quizzes.
Students explored the program by following the closed instructions and reviewing the
accompanying text, see Figure 4 for example. During the exploration of curve shifts in
the chapter on functions, students noticed the general attributes that caused shifts. For
instance, f(x)+k shifts the curve up for k > 0 and down for k < 0 in the same fashion as the
c encountered in the other packages. The students used this package to generalize the
information they had collected in other software packages.
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USING COMBINATIONS OF SOFTWARE - REVISED
Figure 4
As we moved through the different packages, the reader may note that each exploration is
shorter than the previous exploration. The GSP exploration, which could have been done
with Theorist or PowerPC graphing calculators allowed each student to get a good feel
for changes in a, b, and c. We covered most of the material with few “holes” or
limitations. To explore the behavior of b, we found it necessary to compare several static
graphs. The construction of a graph on GSP amounted to constructing a locus of points
as x moved along the x-axis. To construct multiple graphs on GSP is to tie up the
package’s memory with locus constructions. The speed drops greatly with each new
graph. To overcome this limitation, we chose Algebra Xpresser. Multiple graphs were
no problem for this package. Values for b were controlled precisely and it answered the
question at hand. Together these two packages had only one noticeable limitation: they
offered no help with definitions or manipulations. Calculus was used to fill in these
deficiencies. The preservice teachers’ critical thinking was enhanced by analyzing their
own points of view together with the viewpoints of their classmates as they decided the
combination of software to be used.
There are strengths in all the packages used in Computers in Mathematics Education. We
see the true strength of technology in mathematics when the applications are used in
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USING COMBINATIONS OF SOFTWARE - REVISED
combination. Our students were able to see dynamic, continuous deformations as the
values of a, b, and c were continuously changed. They began to conjecture based on this
motion. The students perceived limitations when it became necessary and beneficial to
look at several different static graphs. Finally, to tie up loose ends, we used a package
with built-in definitions and abstractions of our previous work. From our students’
reactions to classroom activities and assessment, we feel that multiple software use serve
to enrich their learning experience and made them critical thinkers. They were confident
that they will be able to model their teaching to help high school students become critical
thinkers.
REFERENCES
1. Blubaugh, W. L., “Use of software to improve the teaching of Geometry”,
Mathematics and Computer Education, Vol. 29, pp. 288-293 (1995).
2. Akbari-Zarin, M. & Gary, M. W., Computer assisted instruction and critical thinking.
Journal of Computers in Mathematics and Science Teaching, Vol. 9, pp. 71-78
(1990).
3. Ashby, W. A., “Questioning critical thinking: Funny faces in a familiar mirror”,
Issues of Education at Community Colleges: Essays by Fellows in the MidCareer Fellowship Program at Princeton University (1996).
*Oppong, N. K., & Russell, A. (1998). Using combinations of software to enhance
preservice teachers’ critical thinking skills. Mathematics and Computer Education. 32(1),
37-43.
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