Editor’s Note: Marshall McLuhan’s concept of tools as an extension of man’s capabilities is illustrated by this research, where technology short-circuits a time consuming manual procedure and improves conceptual learning and problem solving.
The Effect of Technology on the Classroom Discourse
The Teacher’s Initiation (I)
The Students’ Responses (R)
The Teacher’s Reaction (E)
What happens to Y1 when we add three to ex?
(Pause)The graph is shifted to the left three units
What happens to Y1 when we subtract three units from ex?
(Pause)The graph is shifted to the right three units
What do we call this type of shifting?
(Pause) We call it horizontal shifting because it shifts Y1 three units in the horizontal direction
The third sequence of the classroom discourse is the teacher’s reaction to students’ answers. For most of the class period, the teacher reacted to the students’ answers by posing other questions. These questions were used effectively by the teacher to guide the students toward understanding the mathematical concepts or resolving their mathematical misconceptions.
In one case, the teacher discovered that the students were misled by using the visual representation of functions. The teacher and students solved the problem analytically to verify whether the function Y = X4 + X3 + 3 is odd, even, or neither, and said “The calculator here is a tool to show, not to verify”.
From the data analysis of classroom discourse, one came to realize that the lecture’s pattern did not dominate the class that used the graphing calculator and the teacher’s talk did not dominate the students’ talk.
In order to investigate the source of classroom discussions between the teacher and students in this research site, the researchers visited another section which studies the same course without using the graphing calculator. In that classroom, the teacher was talking most of class period while the students were taking notes. The following paragraph represents segments of classroom talk in this course. The teacher introduced the concept of graphing functions by talking to the students while writing on the board:
“We are going to approach graphing by learning various basic shapes ..... Some of the basic shapes are:
Lines / X and Y both have power 1.
Parabolas / One variable has power 2, and the other has power 1.
Cubic/ One variable has power 3, and the other has power 1.
Circles / both variables have power 2.
Once you recognize the basic shape of the graph of a function, you can obtain detailed information about it. For instance, you can find X- and Y-intercepts.”
Despite the fact that the research interest lies only in classroom discourse in a graphing calculator college introductory mathematics course, the information that was collected from the research site could help to reach the conclusion that the graphing calculator was an important agent in creating a departure from teacher-centered instruction to student-centered instruction.
Regarding the second analytical research question “How does the class of the college introductory mathematics course use the graphing calculator to study mathematical concepts?”, the analysis of observations data indicated that the graphing calculators were used to display graphs by entering the scales of the axes and the rule of a function. Students may change the scales until they are able to see the graph on the screen. Then, by using the “zoom in” facility, the students were able to view smaller parts of the graph. These features of the graphing calculator were used effectively to study different functional concepts, such as even and odd functions, symmetric functions, and vertical and horizontal shifts of graphs of given functions.
In one situation, the teacher gave different examples to introduce the concepts of vertical and horizontal shifts. For example, in teaching horizontal shifting, the teacher and the students used their graphing calculators to graph the following three graphs on the same coordinates: Y1 = X2, Y2 = (X + 3)2, and Y3 = (X - 3)2.
The teacher moved around in the classroom to check students’ graphs and help anyone who needed it to set up his or her calculator to graph these functions. In addition, the teacher’s graphs of these functions could be seen clearly on the overhead projector.
T: What happens to Y1 when we add three to X?
S1: (Pause) The graph is shifted to the left three units.
T: What happens to Y1 when we subtract three from X?
S2: (Pause) The graph is shifted to the right three units.
T: What we call this type of shifting?
S3: (Pause) We call it horizontal shifting because it shifts Y1
three units in the horizontal direction.
An interesting issue raised by the teacher about the reasoning behind the horizontal shifts was “Why does the function shift to the left when we add a number to X, and to the right when we subtract a number from X?” This part of the classroom discourse could help to see the benefit of using graphing calculators to resolve students’ misconceptions about the horizontal shifting. This misconception is embedded in students’ beliefs that the function would be shifted to the right when one adds a number to X and to the left when one subtracts a number from X.
The results of this study showed that the effective use of graphing calculators requires a solid understanding of the mathematical concepts involved. Otherwise, visual representation of the graph of a given function generated by the graphing calculators becomes misleading. Therefore, the instructors who teach functions by the use of graphing calculators should be aware of this issue. In this situation, the instructor introduced the concepts of even and odd functions as follows: First, the teacher and the students graphed f(X) = X2 as an example of an even function. This function is called an even function because its graph is symmetric with respect to the y-axis, and satisfies the condition: f(-X) = f(X). For example, f(-2) = f(2) = 4.
Second, the teacher and the students graphed f(X) = X3 as an example of an odd function. This function is called an odd function because its graph is symmetric with respect to the origin, and satisfies the condition: f(-X) = - f(X). For example, f(-2) = -8, but f(2) = 8. Therefore f(-2) = -f(2).
Also, among the questions that the teacher asked was whether the function f(X)= X4 + X3 -2 is even, odd, or neither. The teacher and the students used their graphing calculators to graph this function. The teacher said: Now, is this function even, odd, or neither? Some students said it was even, others said it was odd, others said it was neither. The teacher, therefore, found that it was time to tell the students the importance of solving this problem analytically and not to rely on the visual representation of the function only. The teacher, thus, told the class “the calculator in this case is a tool to show, not to verify” and then he discussed the analytical solution of this problem with the students.
Finally, regarding the third research question, “How does the class of the college introductory mathematics course use the graphing calculator to solve real-life problems?”. The analysis of observations data indicated that the features of the graphing calculator were used effectively by the class in solving real-life problems; the teacher provided the students with real-life problems and encouraged them to solve such problems using their calculators.
In one situation, one could enjoy listening to the teacher and the students discussion while solving the following real-life problem, “Book Value Problem”:
A photocopying machine was sold for $3,000 dollars in 1988 when it was purchased. Its value in 1996 had decreased to $600.
After posing the problem, the teacher asked how many of the students were majoring in business. Three of the twenty students raised their hands, which pleased the teacher because this type of problem is important to them; it introduces what is called book value in the field of business. The teacher states that BV (bookvalue) = mX + b, and asked “what is m and what is b? One student indicated that m is the slope, and b is the y-intercept.”
T: What is the book value when X equals zero years?
S1: (Pause) Three thousand dollars.
T: What is the book value when X equals eight years?
S2: (Pause) Six hundred dollars.
T: I want you to find the slope.
One student talked and the teacher wrote the following on the board:
m = (3000-600)/(-8) = -300.
T: What does it mean to have a negative slope?
S3: (Pause) The value of the machine decreases its value three hundred dollars each year.
T: How can we find the y intercept?
S4: (Pause): If we plug in X equals zero in the book value equation, we get b equals three thousand, which is the y intercept.
T: What does that mean?
T: This is the original price of the machine.
T: Now, what is the book value function?
S5: (Pause) Y = minus three hundred X plus three thousand.
The teacher wrote “-300X + 3000” on the board:
Now, the teacher and the students were ready to graph the book value function. Therefore, they used the information in the problem to set up their calculators as follows:
Y1 = - 300X + 3000
X min. = 0. (The minimum value on the X-axis).
X max. = 10. (The maximum value on the X-axis).
X scl. = 1. (The distance between 0 and 10 is divided into tick points with
a length of 1 unit each).
Y min. = 10. (The minimum value on Y-axis is 10).
Y max. = 4000 (The maximum value on Y-axis is 4000).
Y scl. = 100. (The distance between 10 and 4,000 is divided into tick points
with a length of 100 units each).
T: How would you interpret the y intercept?
S6: (Pause) The original cost, which is three thousands.
T: What does the X intercept mean?
S7: (Pause) X intercept means how old the machine is when it is worth
T: How can we determine the book value of the machine in nineteen ninety-two?
All Students and the teacher used their graphing calculators to find the value of y when X = 4, and they got the result y = $1,800.
T: What does this mean?
S8: (Pause) It is four years old when its worth is eighteen hundred dollars.
T: How can we verify this analytically?
S9: (Pause) We can use the book value formula and plug in X equals four to get
y “equals minus three hundred times 4 plus three thousand”.
The teacher wrote the following on the board:
y = - 300 (4) + 3000 = 1800.
T: Now, we get the same result graphically and analytically.
From the above mentioned real-life situation, one could understand the importance of solving the problems analytically and supporting the solution graphically.
This study was conducted to gather information regarding classroom discourse associated with the use of graphing calculators. Overall findings of data analysis of classroom observation revealed a different classroom discourse. The graphing calculator could be considered an important agent in creating a departure from teacher-centered instruction to student-centered. Three possible reasons could be given as evidences to support this conclusion: First, the teacher’s role shifted from producing a graph of a given function to questions of what the graph is saying about. Also, the students’ role shifted from plotting of points and drawing a graph of a given function to gleaning information from the graph. As a result, neither the lecture’s pattern dominated the climate of the class that used the graphing calculator nor the teacher’s talk dominated the students’ talk. This result is consistent with findings of other research conducted by Wheatley (1994), Yates (1995), Wood (1999), Herbst (2002), Ackles, Fuson, and Sherin, 2004, Marrongelle and Larsen (2006), and Mesa (2008) in which they indicated that the constructivist learning environment could help students to become more engaged in classroom discussions.
Second, the graphing calculator provided students with graphical representations of functions freeing them from burden of plotting and drawing functions; thus, providing them with more time to focus on conceptual understanding of function. As a result, the focus was shifted from procedural works associated with graphing functions by hand toward conceptual understanding of functions and study important features of functions such as vertical and horizontal shifting of functions and whether the function is odd, even, or neither. This finding is consistent with findings of other research such as Shore (1999), Knott, Srirman, & Jacob (2008), and Mesa (2008) which indicted that teaching functions with graphing calculators offers teachers and students opportunities to shift the emphasis away from procedural-oriented toward conceptual-oriented.
Third, accurate graphs obtained quickly with the graphing calculator help the teacher and students to focus on problem-solving processes. As a result, students become more engaged in the problem-solving situations until the solution of the problem is reached after the calculator carries out the tedious and time-consuming manipulation of plotting points and drawing functions associated with real-life situations. This finding is consistent with findings of other research such as Dunham and Dick (1994), Wheately (1994), Yates (1995), Shore (1999), and Mesa (2008) which raveled that teaching functions with graphing calculators offers teachers and students more time to focus on problem-solving processes.
Based on the results of this study and the discussions made, many classroom implications and suggestions for further research could be made. Some of these are:
Ackles, K., Fuson, K., & Sherin, M. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education. 35(2), 81-116.
Bellon, J. J., Bellon, E. C. & Blank, M. A. (1996). Teaching from a research knowledge base: A development renewal process. New York: Macillan Publishing Company.
Cazden, C. B. (2001). Classroom discourse: The language of teaching and learning. Northmoth: Heinemann.
Dunham, P. H. & Dick, T. (1994). Research on graphing calculators. Mathematics Teachers, 87(6), 440-445.
Good, T. & Brophy, J. (2003). Looking in classrooms (9th ed.). Boston: Allyn & Bacon.
Herbst, P. G. (2002). Engaging students in proving: A Double bind on the teacher. Journal for Research in Mathematics Education, 33(3), 176-203.
Knott, L., Sriraman, B., Jacob, I. (2008). A Morphology of teacher discourse in the mathematics classroom. The Mathematics Educator, 11(1,2), 89-110.
Marrongelle, K. & Larsen, S. (2006). Generating mathematical discourse among teachers: The role of professional development resourse. Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, TBA, Mérida, Yucatán, Mexico. 2009-05-24 Available at http://www.allacademic.com/meta/p115453_index.html
Mesa, V. (2008, July). Classroom participation in pre-college mathematics courses in a community college. Paper presented at the International Congress of Mathematics Education-DG 23, Monterrey, Mexico.
National Council of Teachers of Mathematics. (1996, Second Edition). Professional standards for teaching mathematics, Reston,
National Council of Teachers of Mathematics. (1991). Principles and standards for school mathematics. Reston, VA.
Rubin, H., & Rubin, S. (2004). Qualitative interviewing: The art of hearing data. New Delhi, SAGE Publications.
Shore, M, (1999). The Effect of graphing calculators on college students’ ability to solve procedural and conceptual problems in developmental algebra. (Unpublished doctoral dissertation, West Virginia University).
Wheatley. G. (1994). Calculators and school mathematics. In R. Rey, N. Noda, K. , & D. Smith (Eds.), Computational alternatives for the twenty-first century: Cross-cultural perspectives from Japan and the United States (Pp. 115-124)., Reston: VA. National Council of Teachers of Mathematics
Wood, T. (1999). Creating a context for argument in mathematics class. Journal for Research in Mathematics Education, 30 (2), 171-191.
Yates, B. (1995). The individual and group technology-based problem-solving. In H. Folye (Ed.), Interactive learning in the higher education classroom. (Pp. 41-60).Washington, DC: National Education Association.
Ahmad Moh’d Al- Migday received his Ph. D from University of New Orleans in Math Education. He is now an associate professor in the Dept. of Curriculum and Instruction at the College of Educational Sciences, the University of Jordan, Amman, Jordan. Al- Migdady’s research interests are technology application in education, Constructivism and methods of teaching mathematics. firstname.lastname@example.org
Abdelmuhdi Ali Aljarrah received his Ph.D. from Colorado State University Fort Collins CO with a major in Education and Human Resource Studies and an emphasis in educational technology and distance education. He has taught from 2002 to the present as an assistant professor in the Faculty of Educational Sciences, University of Jordan, Amman, Jordan. email@example.com
Faisal M. Khwaileh received his Ph.D. from Michigan State University in Curriculum, Instruction and Educational Policy. He is now an assistant professor in the Dept of Curriculum and Instruction at the College of Educational Sciences, The University of Jordan, Amman, Jordan. firstname.lastname@example.org