Using combinations of software to develop mathematical ...



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 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).

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.

[pic]

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.

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 y-intercept 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.

[pic]

Figure 2: y = x2 + bx + 1 for various values of b.

[pic]

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 built-in 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.

[pic]

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 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 Mid-Career 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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