Examples of Using Computer Technology in Mathematics Education

Yixun Shi Department of Mathematics, Computer Science and Statistics

Bloomsburg University of Pennsylvania Bloomsburg, PA 17815, USA

yshi@bloomu.edu

Abstract—This paper presents several examples of using computer technology to make teaching and learning mathematics more effectively and efficiently. These examples show that technology will allow teachers to lead much more in-depth discussions in class. The use of computer technology in classrooms may in general raise students’ interests in mathematics, and enhance their problem solving skills.

Keywords-Mathematics education, computer technology, algebra, calculus, probability

I. INTRODUCTION Several years ago, the National Council of Teachers of

the United States included in its Standards that the computer technology should be used as one of the essential tools in teaching and learning of mathematics. When time goes by, more and more mathematics teachers have involved the use of computer technology in their teaching, and many of them have developed various innovative ways of using technology to make their classroom instructions more effective and efficient. The use of computers in a classroom would make it possible for students to see much more vivid graphical demonstrations of mathematical concepts and facts, and would allow students more opportunities to perform hands-on exercises. Technology will also allow teachers to lead much more in-depth discussions in class, which would be impossible without the aid of computers. The use of computer technology in classrooms may in general raise students’ interests in mathematics, and enhance their problem solving skills.

At Bloomsburg University, our department offers a course for the mathematics education majors to introduce them to the use of accessible technology in the classroom. Through the years of teaching this class, I have established, and illustrated to my students, various cases where computer technology may be used to teach mathematics subjects more effectively and to make the learning process more interesting. In the following sections, I would like to present a few such examples.

II. USING MATHEMATICA IN TEACHING ALGEBRA MATHEMATICA is a popular computer software

developed for mathematical computations and mathematics education. As an example of its applications in an algebra

class, let us see how it may be used in teaching the subject of quadratic functions.

In general, a quadratic function has the form cbxaxxp ++= 2)( where a, b, c are real numbers with .0≠a The graph of p(x) is a parabola, which opens up when a > 0 and down when a < 0. At the times when computers were not accessible in the classroom, the teacher would normally draw two parabolas on the blackboard, one for a quadratic function with a > 0 and the other with a < 0, so that students could see that the sign of a indeed determines the direction of the parabola. In order to draw these parabolas, which are the graphs of corresponding functions, the classical two column X-Y value tables would have to be computed. Due to the class time limitation, usually the graphic display would stop there. When computer technology becomes available in the classroom, the graphs can be made so rapidly that the teacher now would be able to apply a lot more graphic demonstrations to lead students into much more in-depth discussions. For instance, the teacher may want to illustrate to the class how each of the three coefficients a, b, and c would make its impacts on the graph of the function cbxaxxp ++= 2)( . We may start with the coefficient a. Suppose we fix the values of b and c, and let the value of a change. Ask students to observe the corresponding change in the parabola. Note that a is required to be non-zero, because otherwise the function would become a linear function. So what happens if the value of a is very close to zero? What happens if a is getting further and further away from zero? Using computer software such as MATHEMATICA, the teacher can easily show students the corresponding changes in the parabola. Figure 1 is a MATHEMATICA screen shot that illustrates part of this discussion. Here we use an example 4)( 2 −+= xaxxp with a = 1, 0.5, and 0.1. From the display students see that for all these three positive values of a, the parabola is opening up. When the value of a is close to zero, the parabola is rising up slowly and it looks somewhat like a straight line. When the value of a becomes larger, the parabola rises up more rapidly.

2009 First International Workshop on Education Technology and Computer Science

978-0-7695-3557-9/09 $25.00 © 2009 IEEE

DOI 10.1109/ETCS.2009.541

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2009 First International Workshop on Education Technology and Computer Science

978-0-7695-3557-9/09 $25.00 © 2009 IEEE

DOI 10.1109/ETCS.2009.541

79

2009 First International Workshop on Education Technology and Computer Science

978-0-7695-3557-9/09 $25.00 © 2009 IEEE

DOI 10.1109/ETCS.2009.541

79

2009 First International Workshop on Education Technology and Computer Science

978-0-7695-3557-9/09 $25.00 © 2009 IEEE

DOI 10.1109/ETCS.2009.541

79

What if we fix the values of a and c, and let b change? The graphic displays would allow students to observe these two facts: First, the change in b will result in a horizontal shift of the symmetric line of the parabola. Second, the absolute value of b determines the distance between the x-intercepts of the parabola. A bigger |b| pushes the two x-intercepts further apart, while a smaller |b| drags them towards each other. When |b| gets down to a certain point, the two x-intercepts will meet each other and become one point. Beyond this point, there will be no x-intercept at all. Figure 2 is another MATHEMATICA screen shot that demonstrates this phenomenon. The function used here is

14)( 2 ++= bxxxp with b = 8, 6, 4, and 2. These graphs should also well remind students of the discriminant theorem on the solutions of quadratic equations. Perhaps it is time now for students to conduct a hands-on exercise by fixing the values of a and b, and letting c change. Ask students to make examples similar to the ones above and observe the change in graph when the value of c changes. Does the change of c cause a shift of the graph? If so, what kind of shift (horizontal or vertical)? What is the change in the distance between x-intercepts? How is this related to the discriminant theorem?

Figure 1. Changes in the parabola when a changes

Figure 2. Changes in the parabola when b changes

III. USING MATHEMATICA IN TEACHING CALCULUS A typical example here is to use MATHEMATICA to

illustrate the relation between a difference quotient

xxfxxf

Δ−Δ+ )()( 00 and the derivative )(' 0xf , or

equivalently the relation between the slopes of a secant line and the tangent line. More precisely, using MATHEMATICA the teacher can easily show students that when xΔ approaches zero, the corresponding secant line will get closer and closer to the tangent line at the point

))(,( 00 xfx . The following MATHEMATICA screen shot, Figure 3, demonstrates such an example where

23)( 2 += xxf and 00 =x . Displayed there are the

tangent line and three secant lines corresponding to xΔ =1, 0.5, and 0.1. In a real classroom setting, these secant lines may be displayed one after another, with the later one replacing the former one, so that students can view a vivid “movement” of the secant line. MATHEMATICA may be used in many cases when teaching a calculus class. For instance, it can effectively demonstrate the fact that a differentiable function f(x) is strictly increasing (or decreasing) if and only if its derivative f ’(x) is positive (or negative). By displaying graphs of f(x) and f ’(x) simultaneously in a MATHEMATICA screen, this fact gets visualized and efficiently described. In all such cases, many hands-on exercises can be assigned to students. With the help of MATHEMATICA, they may explore a lot more in this course, and get more in-depth understanding of the subjects.

IV. USING EXCEL IN TEACHING BASIC PROBABILITY THEORY

When teaching a basic probability theory class, in many occasions the MS EXCEL can be used to make the classroom instruction more effective. A typical example is to use EXCEL to simulate random experiments, and then

Figure 3. Secant line approaches tangent line when xΔ →0

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have students to compare the sample relative frequencies against the theoretical probability values. Ask students questions like these:

• Randomly toss two fair coins, what is the probability that exactly one coin shows the head? (1/2, or 0.5)

• Randomly throw two fair dice, what is the probability that the second die shows a larger number than the first one does? (15/36, or 0.417)

• Randomly pick two books without replacement from a set of five books, two of them mathematics books and the other three history books, what is the probability that both books picked are history books? (3/10, or 0.3)

After students find out the theoretical values of these probabilities, ask them to use EXCEL to make simulations of corresponding random experiments. For example, simulate the toss of two fair coins for 200 times and see how many times exactly one head is obtained. They may then compare the sample relative frequency with the probability value. What happens if we simulate the toss for 500 times? 1000 times? These in-class hands-on simulation exercises would never be made possible without the use of computer technology. These exercises would not only make students more enthusiastic in class, but also lead them to a better understanding of the conceptual as well as practical meanings of the probability. Figure 4 is a sample EXCEL screen shot showing the results of one simulation of tossing two fair coins for 200 times, where 1 stands for “a head” and 0 for “a tail”. The first two columns are generated using random numbers, simulating the random tosses of the coins. In this simulation, exactly one head is obtained for 104 times, and hence the relative frequency from this sample equals 0.52, which is quite close to the probability value 0.5.

Figure 4. Simulation of tossing two fair coins for 200 times

V. CONCLUSION COMMENTS Many cases may be presented where computer

technology can help making mathematics teaching and learning more effective as well as more entertaining. It is extremely important for mathematics teachers to keep in mind that the technology is much more than just an effective demonstration tool or a fast computing hand. The computer technology allows teachers and students to do tons of things that would not be possible without the use of technology. It is a responsibility as well as a privilege of the teachers (including the future mathematics teachers in my class) to explore these new possibilities. One of these possible areas, for instance, is to include case studies of certain sports and mathematical games in the classes. These studies normally involve large data sets or lengthy computations. The availability of technology would make these studies possible in a classroom setting. See [2], [4], [5], [6] for a few examples. Since most students tend to enjoy sports and games, such studies would further raise students’ interests and enhance their problem solving skills.

In conclusion, computer technology has opened a new door for mathematics educators and students. Time will witness the significant and positive impacts that the use of technology will have on the way mathematics is taught and learned.

REFERENCES [1] M. Bittinger and J. Beecher, College Algebra, 2nd ed.,Pearson

Addison Wesley, New York, 1993. [2] E. Mauch and Y. Shi, “Using a Sequence of Number Pairs as an

Example in Teaching Mathematics”, Mathematics and Computer Education, 39 (3), 2005, pp 198 – 205.

[3] J. McClave and T. Sincich, Statistics, 9th ed., Prentice Hall, New Jersey, 2003.

[4] Y.Shi, “Using volleyball games as examples in teaching mathematics”, Teaching Mathematics and its Applications, 22(2),2003, pp 53 -- 62.

[5] Y.Shi, “The game PIG: making decisions based on Mathematical thinking”, Teaching Mathematics and its Applications, 19(1), 2000, pp 30 – 34.

[6] Y. Shi, “A mathematical study of the game Twenty Four Points”, Mathematics Teacher, 92(9), 1999, pp 828-832.

[7] Y. Shi, C. H. Brase, and C. P. Brase, Technology Guide to accompany Understandable Statistics, Houghton Mifflin, Boston, 2006.

[8] J. Stewart, Calculus, 5th ed., Thomson Brooks/Cole, California, 2003.

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