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Volume 1, Issue 3
ISSN 1528-5804
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Harper, S.R., Schirack, S.O., Stohl, H.D., &
Garofalo, J. (2001). Learning mathematics and developing pedagogy
with technology: A reply to Browning and Klepsis. Contemporary
Issues in Technology and Teacher Education, [Online serial]
, 1 (3) . Available:
http://www.citejournal.org/vol1/iss3/currentissues/mathematics/article1.htm
Learning Mathematics and Developing Pedagogy with
Technology: A Reply to Browning and Klespis
SUZANNE R. HARPER and SHANNON O. SCHIRACK
Center for Technology and Teacher Education, University
of Virginia
HOLLYLYNNE DRIER
STOHL
North Carolina State University
JOE
GAROFALO
Center for Technology and Teacher Education, University of
Virginia
Browning and Klespis (2001) raised a number of
thoughtful and important issues in their reaction to our article
concerning guidelines for the preparation of preservice secondary
mathematics teachers (PSTs) to use technology appropriately in
their teaching (Garofalo, Drier, Harper, Timmerman, & Shockey,
2000). In this reply, we provide examples of our work with PSTs and
illustrations of actual work done by PSTs to address their two main
points, namely that (a) preservice teachers need to experience the
learning of 'new' mathematics with technology, and (b) preservice
teachers usually struggle with appropriate use of technology as
beginning teachers.
Experiencing Mathematics Topics Again, for the
First Time
Most secondary mathematics teacher education
programs require either a major or a strong concentration (e.g., 30
credits or more) in mathematics. At the University of Virginia, for
example, PSTs are mathematics and education majors and are awarded
both a BA in mathematics and an MT in education concurrently at the
end of five years of study. Hence, they come into our mathematics
pedagogy course in their fourth year having relatively strong
mathematics backgrounds, including exposure to many advanced
mathematics topics in a variety of areas. Despite this background,
a number of PSTs come to us without knowledge of some topics that
they will teach in secondary classrooms, such as recursion,
regression, and fractals. Their learning of these topics, enhanced
by technology, is very different from their learning of topics in
both their high school and college mathematics courses. This allows
them to reflect on various aspects of the experiences they have had
learning these topics and to compare them to learning experiences
they have had with other mathematics topics.
Even though PSTs come to us having completed eight
or nine college mathematics courses, they often lack conceptual
knowledge of some topics they have studied, including some at the
secondary level. Too often, their knowledge of these topics can be
characterized as procedural (Hiebert & Lefevre, 1986) or
instrumental (Skemp, 1978). Skemp went so far as to say that
instrumental mathematics (rules without reason) is different
mathematics from relational mathematics (knowing what to do and
why)!
When PSTs are learning these topics conceptually
for the first time with technology, they often experience 'Ah ha!'
or 'Eureka!' moments. These moments provide opportunities to
discuss and compare methods of teaching and quality of learning.
One example of such a topic is Taylor's theorem. Most PSTs recall
the theorem and are able to tell us that it is about approximating
functions with series, and many remember the expansion formula and
can generate terms. But rarely are PSTs able to conceptually or
graphically explain the theorem. They are usually confused about
points of expansion and the remainder terms. However, once they
algebraically expand a function about a point, generate graphs and
tables for several approximations, and analyze their results, they
develop understandings that they never had. (Figure 1.)
 
Figure 1 . Graphs and table of sin( X
) (blue), X - X 3 /3! (green), and
X - X 3 /3!+ X 5 /5!
(orange)
Another such topic is linear programming. All our
PSTs remember that to find the extreme values of an objective
function all one has to do is evaluate it at the intersection
points of the graphs of the constraining functions defining the
feasibility region. No PST has ever been able to explain to us why
this works. However, once they explore the interactive linear
programming applet at ExploreMath.com by dynamically observing
values of the objective function as the cursor is moved over points
in the feasibility region and interpreting and explaining what is
happening, they begin to understand that they are evaluating a
linear surface defined over a two-dimensional region. (Figure 2.)
The Linear Programming applet, requiring a Shockwave plug-in, can
be found at
http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=31
.
Figure 2. The ExploreMath.com Linear
Programming applet
We believe, following Skemp, that in a very real
sense, PSTs are learning new mathematics in each of these
instances. Furthermore, they are in a position to reflect on
differences between their 'old' learning and their 'new' learning
and connect them to differences in teaching approaches.
Learning to Teach with Technology
The other point Browning and Klespis made is that
it takes time to learn how to incorporate technology appropriately
into one's teaching. We wholeheartedly agree. We believe that the
approach we take, characterized by the guidelines discussed in our
previous article, helps PSTs begin to learn how to use
technology in their subsequent teaching. Not only are they learning
features of various technologies, but they are experiencing some of
the potential benefits afforded by appropriate use of technology in
a classroom where using technology as a teaching and learning tool
is explicitly modeled and discussed. We have not yet conducted
formal research on the impact of these experiences on our PSTs, but
we do have examples of PSTs' use of technology to enhance teaching.
The following six examples illustrate a variety of ways that our
PSTs have used their learning experiences with technology as
building blocks to develop lessons and tools they believe embody
appropriate use of technology in a mathematics classroom.
Mitch's Linear Programming Lesson .
When Mitch was a student teacher introducing the concept of linear
programming to his students in an Algebra II class, he proposed the
following situation:
You are organizing the items to be sold at
the concession stand for the next home football game. You notice
that you have 2,500 cups that are used for both hot chocolate and
coffee. Hot chocolate sells for 50 cents while coffee sells for 30
cents. If you must sell at least 300 cups of hot chocolate and 500
cups of coffee, how many cups of each must you sell in order to
maximize your profit?
After his students generated the constraint
functions from the given information and graphed them on their
calculators, they interpreted the area bounded by these three
constraints as a planar region that represents the possible
solution set. The students knew that they needed to evaluate the
profit function at the intersections of y = 2,500 ' x
and y = 500 and of y = 2,500 ' x and x
= 300, and were able to find them with their calculators (Figure
3).
Figure 3 . The intersection of y =
2,500 ' x at y = 500
Mitch was not satisfied with his students merely
knowing how to determine the maximum values of the profit function
procedurally, as he did prior to working with the
ExploreMath applet (Figure 2). He wanted them to understand
why the method they used was mathematically justified. He did not
have access to ExploreMath in his classroom, so he had to
improvise. To assist his students in seeing that the maximum value
of the profit function, a three-dimensional surface, occurred at a
vertex of the feasibility region, Mitch asked three students to
hold a plank of wood up above their heads. With each student
representing one of the three points of intersection, Mitch
challenged them to make the highest point on the board not at one
of the corners. This hands-on experiment demonstrated to the
students that the highest point always occurs at one of the points
of intersection. Mitch was influenced by his use of technology in
his pedagogy course to assist his students to move beyond rote
understanding and was able to do so even without access to that
same technology.
Ann's Probability Simulation .
Ann, a preservice teacher in our mathematics methods course, wanted
to design a lesson for students to use experimental probabilities
to closely approximate theoretical ones and to discuss the law of
large numbers. She created an interactive spreadsheet that would
quickly generate a large amount of random data. The experiment in
Ann's lesson was to draw a random card from a standard deck 2,000
times. The user is able to specify a card (e.g., king of spades),
and the spreadsheet tallies the number of times that card was
picked out of the 2,000 trials. In addition, the tallies shown in
Figure 4 allowed the students to see the similarity in results for
picking each of the 52 cards, as well as the 4 suits and the 13
different face values. She used this spreadsheet in a peer teaching
episode to help other mathematics education students calculate
experimental probabilities and to lead them to thinking about the
theoretical probability of picking a single card (1/52), a specific
suit (1/4), and a specific face value (1/13).
Figure 4. Frequency chart for all cards
chosen from 2,000 random draws
Matt's Geometry Web Resource .
After completing his student teaching, Matt created the Geometry
Gala, an interactive web-based geometry resource for students and
teachers. Geometry Gala contains statements of many of the more
commonly studied theorems concerning angles, circles, triangles and
quadrilaterals, a collection of downloadable Geometer's
Sketchpad interactive sketches to illustrate each of these
theorems, and movies of these sketches with sound descriptions.
(Figure 5.) Matt's Geometry Gala can be accessed at
http://curry.edschool.virginia.edu/teacherlink/math/geometrygala/ggg_contents.html
.
Figure 5. A screen shot from the Geometry
Gala
Mike's Calculator-Based Ranger Lesson
. Mike made regular use of graphing calculators and CBLs during his
student teaching. On one occasion he asked students to work in
small groups to 'create' graphs by 'walking them' in front of a
TI-Ranger. He began by asking students to walk the 'usual' graphs
(e.g., increasing with constant slope, horizontal) and asked them
to discuss their methods and observations and to interpret and
analyze their findings with respect to slope and distance-rate-time
relationships. In order to assess whether or not his students
really understood these relationships, he asked them to create more
difficult and even impossible graphs. For example, he asked them to
create parallel lines, spikes, and loops. In each case he allowed
students to experiment with walking the graphs. He asked them to
explain why they walked what they walked, why they thought the
graphs were difficult or impossible, and, most importantly, how
their methods and findings related to the concept of slope and
distance-rate-time relationships.
Kris's Geometry Lesson . One
day while student teaching, Kris was teaching a lesson on the
angles formed when two parallel lines are cut by a transversal.
During the first half of class, the students were in the computer
lab using The Geometer's Sketchpad . Students worked
individually at the computer and referred to a written exploration
guide Kris had created. Not only did the guide provide the students
with instructions of the mathematical exploration but also of the
technology features of Sketchpad necessary to complete the
exploration. As the students were following the written guide, Kris
facilitated the students to make conjectures about pairs of
congruent angles in their sketches. Students recorded their
conjectures as they followed through a series of explorations, as
seen in Figure 6. After approximately 45 minutes in the computer
lab, the students went back to their classroom with their completed
exploration guide in hand. The class summarized the conjectures
they had made in the computer lab before Kris gave each student a
printed sheet with the Corresponding Angle Postulate , the
Alternate Interior and Exterior Angles Theorems , and the
Consecutive Interior Angles Theorem formally stated. She
purposefully left space under each of the theorems for the students
to provide a formal proof. She carefully explained how their
explorations in the computer lab did not provide mathematical proof
of the theorems; their explorations were used to conjecture the
theorems. Kris used the remaining class time to have students
formally proof the conjectures they had made in the lab. Although
Kris had seen many activities in her methods class where The
Geometer's Sketchpad was used, the particular activity she used
in her class was not one of them. She transferred what she had
learned in her methods class to create a short exploration that was
useful for the objectives of her particular lesson.
Figure 6. Kris' Exploration Guide
Jerry's Model of Projectile Motion
. For a class project in our course, Jerry, then a
preservice teacher, wanted to use the interactive capabilities of a
spreadsheet to help students explore projectile motion and make
connections with parametric equations, trigonometry, and quadratic
functions. Jerry created the projectile motion environment (Figure
7) to allow students to manipulate various parameters in a
parametric equation (initial altitude, initial velocity, and launch
angle). With the time scroll bar, students can animate the motion
of the object and explore how long it takes for the object to reach
its maximum altitude and when the object will hit the ground. By
varying the other parameters, the students can explore how each
will affect the path of the object and which values will maximize
or minimize altitude and horizontal distance. Jerry believed that
manipulating different parameters and visualizing the path of the
object could make the mathematical equations used to describe
projectile motion relevant and meaningful to students. During his
peer lesson, he engaged the class in a rich discussion comparing
the parameters in the parametric equation to the parameters in
trigonometric and quadratic functions.
Figure 7. Interactive spreadsheet to explore
projectile motion
Final Thoughts
We realize that these PSTs are just beginning to
develop into teachers who can incorporate technology into their
teaching. We are encouraged by their initial efforts to help
students learn mathematics conceptually and apply mathematics
meaningfully. Other researchers (Olive & Leatham, 2000) have
documented that using technology as a tool for learning mathematics
is not enough to ensure PSTs will use technology as a teaching and
learning tool in their own classrooms. Many factors affect the
development of PSTs' use of appropriate technology tools. Many PSTs
need sustained interactions with technology throughout their
teacher education programs, especially in the context of pedagogy
courses, combined with meaningful field experiences and support to
learn to value the affordances of a technology-enabled classroom
and embrace teaching and learning opportunities with technology. In
addition, PSTs' beliefs about teaching and learning are central to
their development. However, PSTs' past learning experiences with
technology, as well as their engagement and interactions in
technology-enabled learning activities like the ones we advocate,
have strong influences on their belief systems. For a discussion of
how PSTs' beliefs about teaching and learning mathematics are
affected by experiences with technology, see Drier, 2001. We hope
that our guidelines, learning activities, and examples motivate
mathematics teachers and educators to reflect on and discuss the
use of technology in their teaching.
References
Browning, C., & Klespis, M. (2001). A
reaction to Garofalo, Drier, Harper, Timmerman, & Shockey.
Contemporary Issues in Technology and Teacher
Education , [Online serial], 1 (2). Available:
http://www.citejournal.org/vol1/iss2/currentissues/mathematics/article1.htm
.
Drier, H. (2001, March). Beliefs,
experiences, and reflections that affect the development of
techno-mathematical knowledge . Proceedings of the Twelfth
International Meeting of the Society for Information Technology and
Teacher Education, Orlando, FL.
Garofalo, J., Drier, H.S., Harper, S.,
Timmerman, M.A., & Shockey T. (2000). Promoting appropriate
uses of technology in mathematics teacher preparation .
Contemporary Issues in Technology and Teacher Education
, [Online serial], 1 (1). Available:
http://www.citejournal.org/vol1/iss1/currentissues/mathematics/article1.htm
.
Hiebert, J., & Lefevre, P. (1986).
Conceptual and procedural knowledge in mathematics: An introductory
analysis. In J. Hiebert (Ed.), Conceptual and procedural
knowledge: The case of mathematics (pp. 1-27). Hillsdale,
NJ: Lawrence Erlbaum.
Olive, J., & Leatham, K. (2000). Using
technology as a learning tool is not enough . Paper presented
at the International Conference of Technology in Mathematics
Education, Auckland, New Zealand.
Skemp, R. (1978). The psychology of
learning mathematics . Hillsdale, NJ: Lawrence Erlbaum.
Contact
Information
Suzanne Harper
Center for Technology and Teacher Education
University of Virginia
1912 Thomson Road
PO Box 400279
Charlottesville, VA 22904-4279
spr4z@virginia.edu
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