Powers, R., & Blubaugh, W. (2005). Technology in mathematics education: Preparing
teachers for the future. Contemporary Issues in Technology and Teacher Education [Online serial], 5(3/4). Available: http://www.citejournal.org/vol5/iss3/mathematics/article1.cfm
Technology in Mathematics Education: Preparing
Teachers for the Future
Robert Powers and William Blubaugh
University of Northern Colorado
The preparation of preservice teachers to use technology is one of the most
critical issues facing teacher education programs. In response to the growing
need for technological literacy, the University of Northern Colorado created
a second methods course, Tools and Technology of Secondary Mathematics. The
goals of the course include (a) providing students with the opportunity to
learn specific technological resources in mathematical contexts, (b) focusing
student attention on how and when to use technology appropriately in mathematics
classrooms, and (c) giving opportunities for students to apply their knowledge
of technology and its uses in the teaching and learning of mathematics. Three
example activities are presented to illustrate these instructional goals of
The preparation of tomorrow's
teachers to use technology is one of the most important issues facing today's
teacher education programs (Kaput, 1992; Waits & Demana, 2000). Appropriate
and integrated use of technology impacts every aspect of mathematics education:
what mathematics is taught, how mathematics is taught and learned, and
how mathematics is assessed (National Council of Teachers of Mathematics
[NCTM], 2000). Changes in the mathematics curriculum, including the use
of technology, have been advocated for several years. The Mathematical
Sciences Education Board (MSEB) and the National Research Council maintain
that “the changes in mathematics brought about by computers and calculators
are so profound as to require readjustment in the balance and approach
to virtually every topic in school mathematics” (MSEB, 1990, p. 2). Future
mathematics teachers need to be well versed in the issues and applications
Technology is a prominent feature of many mathematics classrooms. According
to the National Center for Education Statistics (NCES, 1999), the percentage
of public high school classrooms having access to the Internet jumped from
49% in 1994 to 94% in 1998. However, the use of computers for instructional
purposes still lags behind the integration of technology in the corporate
world and is not used as frequently or effectively as is needed. One way
to close the gap and bring mathematics education into the 21st century
is by preparing preservice teachers to utilize instructional tools such
as graphing calculators and computers for their future practice.
In the past at our campus, technology issues and "training" in mathematics
education were addressed within the confines of a regular three-semester-hour
mathematics methods course, taught by a professor of mathematics education within
the College of Arts and Sciences. With increasing demands placed on the teacher
preparation program by state legislation, which has become common throughout
all of education, the amount of content in the methods course was becoming overwhelming.
As a result, little time was available to address the issue of the technology
required for effective mathematics instruction.
Even before the additional state requirements, relatively
little time was spent providing preservice teachers with hands-on experience
using graphics calculators and mathematics software. Secondary mathematics
majors occasionally used a computer algebra system (CAS) for different
projects within their calculus courses, as well as spreadsheets and software
applications in their statistics course. Additionally, most teacher candidates
have had experience using graphics calculators at different points within
various mathematics courses. However, little time was spent preparing
preservice mathematics teachers to use technology in their future classrooms. Our
program has required all secondary education majors to take two one-credit
general education technology courses that address spreadsheet, word processing,
and Web-page development, but none of these college technology experiences
provided them with content specific or classroom specific experiences they
will need as future mathematics teachers.
Our response to the growing need for technological literacy
was to create a second methods course entitled, Tools and Technology of
Secondary Mathematics. This course supplements the content and methods
of our existing methods course, but focuses on the utilization of technology
in secondary mathematics classrooms. In keeping with the philosophy of
our Secondary Professional Teacher Education Program, the course has three
broad aims. First, teacher candidates receive hands-on training in using
software tools, graphing calculators, and the Internet for mathematics
instruction focused at the secondary school level. Second, they learn
how and when to use appropriate technology to enhance their mathematics
instruction of topics that are taught at the middle and high school grades. Third,
they develop and teach lessons to their peers with equipment available
to a typical public school mathematics classroom, using the technology
learned in this course.
One purpose of the technology methods course is to provide the opportunity
for preservice teachers to use specific technological resources in mathematical
contexts. That is, teacher candidates are presented with a task involving some
mathematical problem or situation and are required to learn to use and apply
an appropriate piece of technology in completing the task. For example, one
activity used in the methods course is found on the NCTM (2004) Illuminations
Web site (available at http://illuminations.nctm.org/lessonplans/9-12/webster/index.html).
The activity, titled “The Devil and Daniel Webster” and adapted
from Burke, Erickson, Lott, and Obert (2001), has teacher candidates explore
recursive functions using technology. The undergraduates are presented a scenario
in which each person earns an initial salary of $1,000 on the first day, but
pays a commission of $100 at the end of the day. On subsequent days, both amount
earned and commission are doubled. Preservice teachers complete a chart using
either handheld or computer technology to determine if it is profitable to work
for one month under these conditions. Additional questions require the undergraduates
to graph the data from the chart. In this way, teacher candidates not only
learn to use the kinds of technological tools that are available for use in
instruction, but also learn them in the context of examining mathematics, which
helps increase their content knowledge.
In addition to learning to use the technology, pedagogical
issues associated with the instructional tools are emphasized. Specifically,
the course focuses attention on how and when to use technology appropriately
in mathematics classrooms. Misuses of technology are discussed and discouraged,
such as using calculators as a way to avoid learning multiplication skills
and using computers to practice procedural drills rather than to address
conceptual understanding. Rather, preservice teachers discuss the uses
and benefits of commercial software and handheld devices to explore different
content topics that have become possible with technology and consider pedagogical
issues. Some time is also spent previewing national curriculum projects
that have a high involvement with technology (e.g., Key Curriculum Press,
2002). As a result, preservice teachers address and discuss issues of
teaching prior to their clinical experience, which helps these students
focus attention on these matters when participating in their practicum.
Teacher candidates in the technology methods course apply
their knowledge of technology and its uses in the teaching and learning
of mathematics. These future mathematics teachers create several lesson
plans using technology as an instructional tool. Lesson plans center around
concepts and skills found in pre-algebra, algebra, geometry, precalculus
and calculus that are enhanced using technology. Once a topic is selected
for the lesson plan, preservice teachers determine an appropriate piece
of technology that facilitates instruction. They develop and write
instructional lessons using graphing calculators, an interactive mathematics
computer environment, an interactive geometry application, computer spreadsheets,
and the Internet. However, based on a selection of specific mathematics
topics, each teacher candidate creates lessons using additional forms of
technology examined in the course, including dynamic statistical software
and a CAS. As a result, each teacher candidate has a unique experience
of using technology to enhance mathematics instruction at the secondary
Depending on time constraints, preservice teachers teach at
least one of their lessons with their peers as students. Our course ensures
that these future mathematics teachers are able to write and deliver lesson
plans that incorporate appropriate technology for mathematics courses at
the level for which they are seeking licensure.
It is important that teachers
are able to develop well-conceived lesson plans that are structured and
detailed, focusing on specific mathematics topics and using multiple representations,
such as the examples in the appendices. Open-ended exploration
and inquiry-driven mathematics lessons using such software as interactive,
dynamic geometry or algebra software are also developed after the teacher
candidates are able to develop a detailed lesson that explores the topic
with some depth. For students to experience a mathematics topic in depth,
specific “guided” discovery lesson planning is required. Part of the objective
is to counter a pervasive disposition of the mathematics curriculum in
this country as being a mile wide and an inch deep.
Since the creation of the technology methods course, we believe
that our program adequately addresses the needs of many preservice teachers
to be competent at integrating these instructional
tools for teaching and learning mathematics. The growth of future teachers’
ability to use technology appropriately in the mathematics classroom during
the course becomes evident in observations. The following illustrations
provide detailed descriptions of the process in which preservice teachers
engage as they learn, analyze, and apply a particular piece of technology
in the course.
Interactive Computer Environment
One important feature
of the course is to introduce future teachers to the world of possibilities
open to instruction when computers are used effectively. The vast majority
of our preservice teachers have had some experience using computers within
and outside their high school mathematics courses, but few have had the
opportunity to learn mathematics in an interactive computer environment. Providing
this experience for our teacher candidates has created a template on which
they can draw as future teachers.
|Figure 1. Changing the value of v0
in the function v(t) is apparent in the graphs and tables. (Click anywhere
on figure to view the enlarged image.)
For one activity, the preservice teachers use an interactive mathematics computer
environment as an electronic textbook. Embedded in the text is the derivation,
using calculus, of the velocity of an object under the influence of earth’s
gravity as a function of time (i.e., v(t) = gt + v0).
Through this interactive environment preservice teachers manipulate parameters
and see, in real time, the effects of those changes on the graphs and data tables
of the function. For example, after explaining that the value of the gravitational
constant, g, is 9.8 meters per second per second, teacher candidates
integrate the gravitational constant with respect to time, t, to obtain
the velocity function: v(t) = -gt + v0. This function
illustrates the physics principle that the velocity of an object is the integral
of its acceleration. In Figure 1, the result of changing
the initial velocity from 49 meters per second to 4.9 meters per second is apparent
by the graphs and tables. After completing this assignment, preservice teachers
learn how to create an activity using the interactive computer environment.
The potential of such
an instructional tool is readily apparent to teacher candidates. Instead
of using a static textbook in which authors determine examples and illustrations,
using an interactive computer environment in instruction allows the preservice
teachers to choose their own examples and participate in dynamic illustrations. Additionally,
the undergraduates can type and check spelling, as in any common word processor,
respond to problems and questions embedded in the computer application,
and print copies for classroom use or assessment purposes by the teacher.
Teacher candidates then
develop lessons or activities using this technology that are appropriate
for their future middle school or high school students. One possible activity
applies the knowledge gained in the initial experience with the interactive
computer environment. Appendix A contains an example of one such
activity used in our program as a guide for preservice teacher generated
work that uses the height of an object acted upon only by the force of
gravity as an application of quadratic equations. The scenario involves
the launch of a model rocket into the air and requires high school students
to model the height of the object as a function of time in tabular and
graphical form. Such an activity demonstrates the multiple uses of important
components of the interactive computer environment within an appropriate
context of secondary mathematics.
Interactive Geometry Application
One way to introduce
teacher candidates to a particular piece of technology is through classroom-ready,
published materials. This is particularly useful when the software is
well established and used regularly in classrooms, because teachers could
adopt the activity for future classroom use. In one case, we use Bennett
(2002) to introduce undergraduates to interactive geometry software on
the computer. For example, the following problem could be posed at the
beginning of a class session: How could you determine the height of a
tree without measuring it directly? At the time they take the technology
methods course, teacher candidates typically have an extensive cache of
techniques to solve such a problem from prior geometry and trigonometry
courses. Bennett (2002) utilizes interactive geometry software to find
such indirect measures using lengths that are easy to measure and proportions
in similar triangles. Specifically, the worksheet directs the learner
to create line segments to represent the tree’s height and the learner’s
height in the application; then learners construct parallel lines to simulate
the rays of the sun. Finding the tree’s height is a matter of calculating
the unknown length (tree’s height) in the proportion of ratios of object
height to shadow length. Although preservice teachers often know this
technique, constructing the solution in the interactive environment helps
clarify concepts and procedures learned in prior courses.
After becoming familiar
with the software from the activity, discussions take place on the appropriate
uses of the technology. In the case of the interactive geometry software,
teacher candidates should recognize several potential uses of the software
in a high school geometry course. For example, appropriate use of the
software can reinforce properties of similar triangles in students’ minds. The
preservice teachers should also recognize
that the interactive component of the software allows their students to
see that corresponding angle measures remain equal and that corresponding
ratios of sides remain equal during actions that change the dimensions
of the similar triangles. Preservice teachers
reflect on the ability of the software to have students discover these
properties, rather than simply telling their students, thus creating a
more student-centered classroom environment. These future teachers should
also recognize the need to transfer the knowledge gained from the interactive
domain to problem situations away from the technology,
which leads to discussions of how this might be accomplished.
As a culminating experience
with the technology, preservice teachers create
lessons using the software that are applicable to a secondary mathematics course. Often, ideas for these activities
are generated by recognizing alternative solution methods for problems
already considered. After exploring the interactive geometry software
while solving the tree problem, teacher candidates are encouraged to develop
alternative solution methods for solving indirect heights. Appendix
B presents a follow-up activity for finding indirectly unknown heights
of objects. The problem involves finding the height of a flagpole when
a mirror is placed on the ground between an observer and the flagpole. The
activity leads learners to find an indirect height using similar triangles
formed by the reflection in the mirror because the angle of incidence equals
the angle of reflection for light. Additionally, the solution plan requiring
learners to reflect a ray across a line demonstrates the principles involved,
as well as a more sophisticated feature of the interactive geometry environment.
Handheld Data Analysis
One of the easiest technologies
for preservice teachers to learn, and yet one of the most adaptable for
classroom instruction, is graphing calculator technology. Still, too few
secondary school mathematics teachers are comfortable using graphing calculators
or know how to use them effectively for classroom instruction. A primary
goal of the technology methods course is to provide instruction and experience with handheld
technology. Utilizing graphing calculators in a statistical application
is one way to meet this goal.
Recording, graphing, and analyzing data are important skills in mathematics,
as well as in everyday life. The notion that data exist everywhere in the world
is important for students to realize. Additionally, the ability to organize
data provides a person with quick numerical and visual representations of the
data and the power to predict, to within a predetermined degree of accuracy,
future related events based on the data. An introductory lesson for managing
data using handheld technology is to enter and graph party affiliations of the
presidents of the United States. Two common representations of the data are
bar graphs and circle graphs (see Figure 2).
Figure 2. Circle graph and bar graph of presidential party
affiliations in TRACE mode.
One of the issues that should be raised by preservice teachers involves the
best visual representation of the political parties of the presidents. They
should discuss the advantages and disadvantages of their bar graphs and circle
graphs, as well as other common graphical representations. Although the graphs
can be obtained from computer spreadsheet technology, students must recognize
the importance of being familiar with handheld technology as well. We want
our teacher candidates to be capable and experienced with various technological
tools so that they are comfortable using the technology available to them in
the schools in which they will be teaching.
One required activity
of the course is to develop a problem involving the collection, graphing,
and analysis of data for middle school or high school mathematics students
to complete. Appendix C contains an authors’ example of one such activity
used in the technology methods course but applicable for a high school
class. This activity uses presidential data, similar to the introductory
activity, but involves the ages of the presidents at the time of inauguration. The
activity extends the relatively simple task of representing data using
handheld technology and includes more statistically rigorous analysis of
the presidential ages. The activity highlights the mathematical power
available to most students to make sense of the world around them using
Teachers will use technology
appropriately and effectively in their mathematics classrooms if they are
familiar and comfortable with the technology and, especially, if they have
had successful experiences with the technology in an instructional environment. Additionally,
teachers who are able to use today’s technology in the classroom will be
prepared to learn and utilize tomorrow’s technology. This core course
for the secondary teacher education program provides that experience. After
this course, the teacher candidates integrate the technology in their field
experiences conducted in one of the university’s partner schools. In one
instance, preservice teachers use technology during their first clinical
teaching experience. At another time, during their semester-long student-teaching
experience, host teachers and university faculty members evaluate student
teachers on their ability to integrate technology in the classroom. Upon
graduation, these future teachers should not only be knowledgeable as to
which mathematics concepts are best learned through technology, but also
will have had many successful experiences in developing and carrying out
lesson plans that involve a variety of different technologies.
Since the creation of
our technology-based methods course, its need is apparent. Although technology
in typical secondary schools is sparse, several of our partnership schools
are dedicated to utilizing technology in mathematics education. From interactive
chalkboards to data-sharing hubs for handheld devices, our preservice teachers
are beginning to experience these instructional tools during their field
experiences. Consequently, we think it is important to prepare them for
these eventualities. Our preservice teachers’ experience with technology
in our program makes them attractive to secondary school selection committees.
The quality of our preservice teachers since our program emphasized technology
in the mathematics classroom is apparent. As university supervisors, we often
hear from the host teachers that our graduates are highly knowledgeable in dealing
with technological instructional tools. Many host teachers admit to learning
valuable teaching strategies using technology from individuals in our program.
Although most of our preservice teachers receive favorable technology evaluations,
we think we can do better. Our preservice teachers continue to think pedagogically
in ways that they were taught rather than to think of the potential learning
gains using technology. This course does lay the foundation for these teachers
as they become more comfortable with their teaching practices and different
ways to educate their students.
Today’s middle school
and high school students were born into a world with technology. Using
technology during mathematics instruction is natural for them, and to exclude
these devices is to separate their classroom experiences from their life
experiences. One objective in preparing teachers for the future is to
ensure that their classrooms will include the technology that will be commonplace
for a future generation of mathematics learners, thus ensuring that the
mathematicians, mathematics educators, and citizens of tomorrow experience
harmony between their world of mathematics and
the world in which they live.
Bennett, D. (2002). Exploring geometry
with Geometer’s Sketchpad. Emeryville, CA: Key Curriculum Press.
Burke, M, Erickson, D., Lott, J. W., & Obert,
M. (2001). Navigating through algebra in grades 9 – 12. Reston,
VA: National Council of Teachers of Mathematics.
Kaput, J. J. (1992). Technology and mathematics
education. In D. A. Grouws (Ed.), Handbook of research on mathematics
teaching and learning, (pp. 515–556). New York: MacMillan Publishing
Key Curriculum Press. (2002). IMP sample
activities. Retrieved November 15, 2004, from http://www.mathimp.org/curriculum/samples.html
Mathematical Sciences Education Board.
(1990). Reshaping school mathematics: A philosophy and framework for
curriculum. Washington, DC: National Academy Press.
National Center for
Education Statistics. (1999). Digest of education statistics 1998. Washington,
DC: U.S. Department of Education.
National Council of Teachers of Mathematics.
(2000). Principles and standards for school mathematics. Reston,
National Council of Teachers of Mathematics. (2004). Illuminations.
Retrieved November 8, 2005, from http://illuminations.nctm.org/
Waits, B. K., & Demana F. (2000). Calculators
in mathematics teaching and learning: Past, present, and future. In M.
J. Burke & F. R. Curcio (Eds.), Learning mathematics for a new century (pp.
51–66). Reston, VA: National Council of Teachers of Mathematics.
University of Northern Colorado
University of Northern Colorado
Figure 1. Changing the value of v0 in
the function v(t) is apparent in the graphs and tables.
Computer Algebra System: Rocket’s Flight
Statement of the Mathematical Situation
A model rocket blasts off from
a position 2.5 meters above the ground. Its starting velocity is 49 meters
per second. Assume that it travels straight up and the only force acting
on it is the downward pull of gravity. Describe the rocket’s flight path
and any key aspect that may be significant or important to the problem. Also,
note that the acceleration due to gravity is 9.8 m/sec2.
Directions for Students
Analyze the above situation and describe it using tabular,
graphical, and symbolic representations. After you have completely analyzed
the situation in two different ways, describe in your own words the biggest
challenge you had in analyzing and completing the task. You may choose
(but are not required) to follow the steps below.
Step 1: What is the value
of h(0)? What is the real-world meaning of h(0)?
Step 2: What is the initial value of the velocity,
given by v(0)?
Step 3: What is the acceleration due to gravity,
or g? How would any equation describing the rocket’s flight show
that this force is downward?
Step 4: Write the unique quadratic function
that represents the height, h(t), of the rocket identified in the
problem statement t seconds after liftoff. Hint below.
Step 5: In the space below: (a) Graph your
function h(t) using the best viewing window that shows all important
parts of the parabola. (b) Make a table of heights above the ground, for
the first 10 seconds of flight, increment by 1 second.
Step 6: How high does the rocket fly before
falling back to Earth? When does it reach this highest point?
Step 7: How much time passes while the rocket
is in flight?
Step 8: Write the equation you must solve to
find when h(t) = 50.
Step 9: When is the rocket 50 meters above the
ground? Approximate your answer to the nearest tenth of a second.
Step 10: Describe in words and show graphically your answer to Step
Related Online Resource:
Online Graphics Calculators:
Geometry Application: Flagpole Problem
In this activity, you
will solve the flagpole problem using an interactive geometry application.
To find the height of a flagpole,
a student placed a mirror on the ground and stood so that she could look
in the mirror and see the reflection of the top of the flagpole. See figure
on the right.
Sketch and Investigation
1. Construct line segment to represent
the line j perpendicular to segment at point B to represent the ground.
point C on line j to represent the location of the observer.
line l perpendicular to line j at point C.
point D on line l to represent the eye level of the observer.
6. Construct point E between points C and B on line
j to represent the location of the mirror.
the line m perpendicular to line j through E.
line m as a mirror.
ray about line m.
the intersection of the reflected ray and the line l. Label it point F.
the lengths of , , and .
angles and .
1. Drag point E between C and B. What
do you notice about angles and ? What does this imply about triangles and ?
2. Drag point E until point D and F coincide. What
are the measures of , , and ? Describe these measures in the context
of the flagpole problem.
3. Using the relation between triangles and , determine a
formula for calculating segment . What is the measure of ?
4. The student measured the distance
from herself to the mirror to be 1.19 meters, from the mirror to the base
of the flagpole to be 6.65 meters, and her eye level height to be 1.70
meters. How tall is the flagpole?
Related Online Resources:
Dynamic Geometry for the Internet: http://www.keypress.com/sketchpad/javasketchpad/about.php
Data Analysis: The Presidency of the United States
The Presidency of the United States
1st: Using the Internet, school library,
or a history book obtain the ages, at inauguration, of the presidents of
Question: What do you initially observe about
the ages of our presidents?
2nd: Using the statistical lists of your
graphics calculators, enter the presidential order (1st, 2nd,
3rd, etc.) and the inauguration age for each president in "Order" and "Age" lists.
Question: To enter the data in the "Order" list,
do you remember the quick way of entering an arithmetic sequence using
the seq( command?
3rd: Construct and display a histogram of
ages at inauguration on the screen of your calculator.
Question: From this graph, what do you notice about the inauguration
ages of the presidents?
4th: Construct a box-and-whiskers plot of
their ages at inauguration on the screen of your calculator.
(1) What five-number summary is obtained from the box plot?
(2) What does the box plot reveal
regarding the spread of the data?
(3) By looking at its shape
and length, what else does the box plot reveal?
5th: Display your histogram and a box-and-whiskers
plot of the above ages as two different plots, and display them on the
(1) By looking at two different graphical representations at the same time,
what additional information or reinforcing comments can you make?
(2) Which of the two graphs
provide the more important information and why?
(3) Support your conclusions from Question 2 above by calculating 1-Var
Stats of the Presidents’ ages.
6th: Now graph the ages at inauguration by
the order of their presidency.
(1) Describe any patterns that you see in the points.
(2) If we divide the presidency
into three parts, say the first 14, the middle 15, and the last 14 presidents,
how do the 3 parts compare with each other?
(3) Verify your observations,
using the statistics available in your calculator.
(4) Based on the ages of the
last 14 presidents, what "predictions" can you make regarding
the next president?
7th: Knowing the inauguration ages and political
party affiliation of each of our presidents, what questions were not asked
that you would like answered?
Extension: If time permits, perform the additional analysis involving
confidence intervals and tests of significance.
8th: Determine the Confidence Interval, at
the 95% level for the mean, for the age at inauguration of our next president.
(1) What does a 95% Confidence Interval mean?
(2) Would the 95% Confidence Interval be the same for the mean and the median
of the ages? Elaborate.
(3) Based on the ages of all U.S. presidents at inauguration, what is the
interval that will likely contain the age of the next president at the 95% confidence
(4) Using only the ages of the last 14 presidents, what is the 95% confidence
interval that will likely contain the age of the next president?
(5) What is the difference between
3 and 4 above, and why?
9th: Suppose we were to randomly select the age of one of our past
presidents to help determine the likely age of the next president.
Question: What is the probability that the next
president would be between 40 and 46 years old (a) using the ages of all
43 presidents, and (b) using the ages of the last 14 presidents?
10th: Use a t-test of independent sample means available on
your calculator to determine if the ages of the first 14 presidents at inauguration
are significantly different from the ages of the last 14 presidents at inauguration.
(1) What is the meaning of this calculator result?
(2) Why was this test used?
Related Online Resources:
Political Party Data: http://www.presidentsusa.net/partyofpresidents.html
at Inauguration: http://www.campvishus.org/PresAgeDadLeft.htm
Data Plots/Graphs: http://matti.usu.edu/nlvm/nav/category_g_4_t_5.html