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Technology Used to Teach. . .

M.N Guntharp 1

Technology Used to Teach Preservice Mathematics Teachers

Marsha Nicol Guntharp, Ph.D.

Abstract

Preservice mathematics teachers often do not have the kind of deep understanding of mathematics

concepts desired for teaching the subject to precollege students. To facilitate this understanding, I have

included worthwhile mathematics tasks in the mathematics pedagogy course that I teach. Since

technology is important in the teaching and learning of mathematics in the 21st century, I often use

technology as a tool to facilitate making conjectures, exploring, making connections, using

representation, and problem solving; and to model instructional strategies necessary for our students to

learn. The technology explorations mentioned in this paper include graphing calculators, computer

spreadsheets, Calculator-Based Laboratories (CBLs) with probes, and dynamic geometry software on a

calculator.

Many of the students I teach are middle-childhood and adolescent-to-young-adult mathematics

education candidates, and often they are confident of their understanding of mathematics concepts.

However, when confronted with their understanding of many mathematics concepts, their understanding

would be classified as instrumental (i.e., rules without reasons), rather than relational (i.e., knowing

both what to do and why (Skemp, 1978, p. 9).

As an example, we talk about driving in the mountains and seeing the signs that indicate the

steepness of the road ahead. When asked what the 5% represents, students can often indicate that the 5%

represents the slope of the road, but most students cannot say

what that means physically. Their memorized definition of rise

over run doesnt transfer into real-life situations. When prompted

and guided, some students can eventually see that for every 100

units (feet, yards, miles, etc.) the road drops five of the same units,

but even the units seem to stump some students. Questions such as Should we use feet or yards?

indicate a lack of relational understanding.

[As a side note, there is a rather humorous story associated with the 5% grade sign and the

definition of gradea story that could save you time and effort if you decide to use the previous


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example. I used the example in a presentation at a national conference, and at the end of the presentation,

I was confronted by a gentleman who said that I used the incorrect definition of grade. He indicated

that, by my definition, 100% grade would be a 45o

angle, but, in fact, it should be 90o. I really did think

that the definition I used was correctbut, he sounded so convincing. So, of course, I immediately went

into research mode. After doing much research and finding much conflicting evidence, I e-mailed the

Ohio Department of Transportation and the U.S. Department of Transportation. I received replies from

both showing that they had sent my e-mail around to their respective engineers. My e-mail, it seems, had

made many trips through various people at the state and national levels! The end result was gratifying and

vindicating, as each e-mail agreed with my definition of gradethat it does, indeed, mean rise over

run.]

Rusch and Nicol (2004) have found in their research that most of their mathematics preservice

teachers who have bachelors degrees in mathematics struggle with relational understanding when

confronted with absolute-value inequalities and with simple permutations. Most of the students in my

classroom admit that their ideas of mathematics are rule-based and algorithmic, and that they have gotten

those ideas from the way they were taught mathematics throughout their lives.

Just how relevant are mathematics instructors' beliefs to their instructional practices? Hersh

(1986) maintains that their beliefs are quite relevant: One's conceptions of what mathematics is [sic]

affects one's conception of how it should be presented. One's manner of presenting it is an indication of

what one believes to be most essential in it (p. 13).

Cooney and Wilson (1993) give an example of how teachers' beliefs affect their use of calculators

in their instructional practices:

If teachers believe that graphical representations are fundamental to the teaching of functions,then technology may be viewed as an indispensable tool. On the other hand, if teachers consider

graphical representations as interesting but not necessarily a central consideration, then the use of

technology may be viewed as a secondary consideration. (p. 146)

Thompson (1992) says that many studies in mathematics education have indicated that teachers' beliefs

about mathematics and its teaching play a significant role in shaping the teachers' characteristic patterns


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of instructional behavior(p. 131). So, the goal of teaching concepts to preservice teachers is to model

appropriate instructional strategies and, thus, help to shape their beliefs.

One of the beliefs I am interested in fostering is that technology is, indeed, an indispensable tool

in the teaching of mathematics in the 21st century. Steen (1990) discusses the rapid growth of

mathematics in the 20th century and its changing focus from number and shape to pattern and order.

Because of computer graphics, one cannot only understand with the mind, but also perceive with the eye;

and understanding of mathematics is enhanced.

Change in the practice of mathematics forces re-examination of mathematics education. . . .

Students who will live and work using computers as a routine tool need to learn a different

mathematics than their forefathers. Standard school practice, rooted in traditions that are several

centuries old, simply cannot prepare students adequately for the mathematical needs of the

twenty-first century. (Steen, 1990, p. 2)

The National Council of Teachers of Mathematics (NCTM; 1989) discusses the effect that

technology has had on the discipline of mathematics: The new technology not only has made

calculations and graphing easier, it has changed the very nature of the problems important to mathematics

and the methods mathematicians use to investigate them (p. 8). The Technology Principle, included in

NCTMs Principles and Standards for School Mathematics (2000), says that Technology is essential in

teaching and learning mathematics; it influences the mathematics that is taught and enhances students

learning (p. 24). Because of NCTM's stand, they have advised that technology should be used widely

and responsibly, with the goal of enriching students learning of mathematics, and that technology

enriches the range and quality of investigations by providing a means of viewing mathematical ideas from

multiple perspectives (p. 25).

Cornu (1992) claims that, if the role of the computer is that of an electronic blackboard, in

which the teacher uses it as a tool, much as any blackboard would be used, then it does not upset the

traditional balance in the classroom . . . [and] it will not revolutionize the classroom. If, however,

students are encouraged to interact with the computers, then the focus changes and the teacher is no

longer the imparter of knowledge, but, rather, a facilitator of learning. Such a change would produce a

revolution in most class- and lecture-rooms. Teachers would need to acquire new knowledge, radically


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change their present aims and emphases, and would be required to teach in ways that are foreign to them.

This last demand means a sacrifice of traditional security. . . . It would be foolish to underestimate the

challenge this presents (p. 27).

A book edited by Sutton and Krueger (2002) discusses how using instructional technology can

affect mathematics reasoning and problem solving. Technology allows us to teach some traditional

topics in a new way as well as teach new topics that are not accessible to our students without the

technology (p. 62). In addition, technology can make mathematics teaching more student-centered by

freeing students to conjecture, solve problems, analyze, synthesize, and evaluate. . . . technology creates

an active environment in which students can communicate with working mathematicians and gather data

in various environments (p. 66).

Carpenter and Romberg (2004) note that key practices in mathematics and science study are

modeling, generalizing, and justifying. To fully understand mathematics and science, students must also

participate in the practices used by mathematicians and scientists. Typically, these practices go unnoticed

and unlearned by most students because they are not made an explicit focus of attention (p. 3). Although

the modeling discussed by Carpenter and Romberg does not necessarily include the use of technology,

there is no doubt that mathematicians and scientists do, indeed, use technology for modeling. Fisher

(2001) devotes an entire book of lessons in high-school mathematics to modeling with STELLA

software, which is used to model change. In her words, By looking at functions from the standpoint of

their behavior over time, one gains an understanding of the function itself, how it is different from other

functions, and how it can be applied to understanding real world phenomena. I think this is the primary

purpose of the study of algebra (pp. 01).

Von Glasersfeld (1990) discusses Piaget's theory: Knowledge is not passively received either

through the senses or by way of communication. Knowledge is actively built up by the cognizing subject

(p. 22). Piaget believed that people developed schemas, or conceptual structures, about their world. He

believed that underlying all of human existence was the desire for equilibration: the drive for

equilibrium, a state of being able to explain new experiences by using existing schemes. (Eggen &


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Kauchak, 2007, p. 34). Humans are motivated by a desire to understand their world and to order and

categorize their experiences so that they fit or relate to what they already know.

Mathematics preservice teachers must construct their own understanding of mathematics and

their own understanding of mathematics education. Their beliefs, which are entrenched, become

perturbed when confronted with beliefs inconsistent with their ownthey experience cognitive conflict.

Their equilibrium becomes disturbed, and they seek to restore that equilibrium. The process is ongoing

and difficult, as they strive for equilibrium in their world.

Nicol (1997, 1999, 2000) found that use of a CBL and a graphing calculator dramatically affected

a high-school physics teachers understanding of mathematicswhich, in turn, affected his instructional

strategies. Dunham (2000) reviewed research on graphing calculators and says the following:

Research reviews by Dunham(1993, 1995), Dunham and Dick (1994), Heid (1997), Marshall

(1996), and Penglase and Arnold (1996) indicate mostly positive benefits for achievement in

algebra, trigonometry, calculus, and statistics. The consensus of the reviews is that students who

use graphing calculators display better understanding of function and graph concepts, improved

problem solving, and higher scores on achievement tests for algebra and calculus skills. (p. 40)

Vonder Embse (1997) talks about the algebraic, graphical, and numerical connections that students can

make among mathematics concepts, facilitated by the use of graphing calculators. Griffith (1998)

indicates that technology impacts mathematics teaching by rendering some content obsolete, increasing

some content in importance, and allowing some new content to become possible.

A Graphing Calculator Approach to Slopes of Linear Equations

Developing a deep understanding of slope or rate of change is crucial for teachers of algebra.

We first look at an actual staircase and discuss what slope means physically. It should become obvious

that the railings and the stairs should have the same slope. Talking about

each individual step, with its rise and run, helps to solidify the idea of

slope for the staircase, and helps the kinesthetic learner to better develop

the concept. We can then look at a picture of stairs and determine the


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slope of the stairs in that picture by finding the slope of the railing or of the border running along the

stairway. [That the railing, border, and stairs might not have exactly the same slope in the picture because

of the two-dimensional perception can be a topic of discussion.]

Next, we find the actual slope of the pictured road in the road sign shown previously, and again

here, and compare it with the 5% grade mentioned in the sign. Putting these slopes into linear equations to

model the steepness of the measured items can help visual learners see a comparison. In the screen dump

shown here, the bold graph of the linear function with the smallest slope models the road with a 5%

grade. The solid graph of the linear function models the road depicted in the road sign. The dotted graph

of the linear function with the steepest slope models the steepness of the stairs.

Next, to reinforce the concept, students measure objects in the room, using both centimeters and

inches. Together, we input the measurements in lists in a graphing calculator. Each object measured

corresponds to an (x,y) coordinate derived from the inch and centimeter lists. We then make a stat plot

and graph the results. The screen dumps below show the results of actual data collected in one of the

classes.

Data Collected Setting up the Graph Graph of Data

Before showing the graph, I ask the question, Do you think the plotted points will be in some

sort of pattern, or will they be random? Surprisingly (or perhaps not so surprisingly), several of these


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preservice mathematics teachers believe the points will be random. A discussion then ensues as to why

the points appear to be linear and what function will be the appropriate linear regression for these data.

We then discuss how the slope of the function would change if the list ofx coordinates were the

centimeter measures, and the list ofy coordinates were the inch measures. This leads to a brief discussion

about inverse functions, which we build on later.

Another depiction of rates of change or slopes occurs when we use the Calculator-Based

Ranger (CBR), which uses a motion sensor to determine distance of an object. Students walk in front of

a motion detector, trying to reproduce a graph shown on a screen. In the following screen dump, such a

graph is shown in which distance is represented on they-axis, and time (in seconds) is represented on the

x-axis. We do many activities with the CBR, including trying to reproduce letters of the alphabet.

To solidify the concept, students draw pictures on graph

paper, and then duplicate the pictures using piece-wise defined

functions (Davidson, McGill, & Stephens, 2004), as in the following depiction of a whale tail. By doing

this activity, students need to figure out domain and range, as well as functions.

Integrating Mathematics and Science

Mathematics and science have been intertwined for centuries, with mathematics being the means to

explain nature. For several centuries, mathematics was definitely the language of science, but in the 1800s,


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it began to become more abstract and analytical. This drift away from the constraints of the real world

may have had as its declaration of independence the emergence of non-Euclidean geometry in the first third

of the nineteenth century(Dunham, 1990, pp. 245246). Mathematics became more abstract and, to some,

more pure. However, in spite of all its purity, mathematics often remains linked to the real world. Even

non-Euclidean geometry was used by Einstein in advancing his theories of relativity (Dunham).

Today, there seems to be an increasing effort to, again, link science and mathematics. Not only are

mathematicians becoming more concerned about grounding their discipline in reality, but mathematics

educators are becoming more concerned about mathematics curriculum that is relevant to students and

teachers alike. House (1994) suggests several advantages in using the science context in the teacher

education curriculum: (a) Using such a context gives prospective teachers a new perspective from which to

reconstruct important mathematics topics; (b) it allows demonstrations that can, in turn, be used with high

school students; (c) the laboratory setting stimulates discourse that models the communication that we

expect teachers to engender in their own classrooms (p. 290); and (d) using the science setting promotes

the idea that mathematics can be studied concretely before being mastered abstractly.

Continuing with the theme of exploring rates of change, our class next looks at finite differences

using a spreadsheet and incorporating science topics. We begin by filling in our spreadsheet in the time

column. Our change in time must be a constant, so for the sake of simplicity, we make the time increments

1 second. Next, we look at the acceleration column and discuss the acceleration of a falling object due to

gravity. On Earth objects accelerate at a constant rate of 9.8 meters per second squared (m/s2), so we fill the

column with -9.8 in each cell. The next column will show velocity, and students must figure out a formula

for finding velocity based on the previous velocity. Since the object is accelerating 9.8 m/s every second,

the finite differences between each of the cells of the velocity column is -9.8 m/s. Since our time increment

is one second, we take the previous velocity and add -9.8 m/s to it. So, the initial velocity is 0 m/s, and at

the end of the first second, the velocity of the object is 9.8 m/s. At the end of the second second, the

velocity of the object is 9.8 m/s + 9.8 m/s, or 19.6 m/s.


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Time Acceleration Velocity

1 -9.8 -9.8

2 -9.8 -19.6

3 -9.8 -29.4

4 -9.8 -39.2

5 -9.8 -49

6 -9.8 -58.8

7 -9.8 -68.6

8 -9.8 -78.4

9 -9.8 -88.2

Discussion then ensues about the way in which finite differences relate to slopes or rates of change.

Graphs can be made of velocity vs. time, and regression equations can be calculated.

Next, we look at exponential decay, relating it to, for instance, radioactive decay, by using some

sort of multiple-sided objects. The easiest objects to use are candies, such as M&Ms

or Skittles, or

coins, such as pennies or nickelsall of which are two-sided. Using M&Ms, for instance, we shake

them and remove the ones with the M up, and count the remaining candies. Counting that as Trial 1, we

repeat trials until all candies have been removed (not counting, of course, the zero remaining, since this is

an exponential function). The more candies used, the better the experiment will fit the theoretical

Velocity vs. Time

y = -9.8x + 2E-13-800.00

-700.00

-600.00

-500.00

-400.00

-300.00

-200.00

-100.00

0.00

0.00 20.00 40.00 60.00 80.00

Time in Seconds

Ve

locityinFt/Sec

Velocity

Linear (Velocity)


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Number of Candies Left After Each Trial

0

20

40

60

80

100

120

0 2 4 6

Trial Number

NumberofC

andiesLeft

Number of Candies Left

statistics, so all students report their data to me, and then we all enter the total data into columns in the

spreadsheet. Next, we look at finite quotients of the number of remaining objects divided by the number

of objects in the previous cell, and note that the quotients (in an ideal world) are the same. (See the

example.)

Trial Candies

Finite

quotients

0 100

1 50 0.5

2 25 0.5

3 12 0.48

4 6 0.5

5 3 0.5

Discussion ensues about what a regression equation would be and why. The graph of the above data

follows.

Exponential equations are also explored by using the CBL and a temperature probe. The probe is inserted

in a cup of hot water and then pulled out, and temperature data is collected as the thermometer cools. The

result is graphed in a graphing calculator, and is then analyzed and discussed.

Using the CBR and a mass on a spring, we look at data of distance vs. time, velocity vs. time, and

acceleration vs. time. Comparing the sinusoidal graphs, we see that velocity is zero when acceleration is


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greatest. To most of us in the class, this makes little sense intuitively; however it is an excellent

demonstration of how the mathematics describes the physics.

Another demonstration of graphical analysis describing physical phenomena occurs when we

analyze the quadratic equations that develop from data collected with a CBR (using the ball bounce

application) and a bouncing ball. Analysis of each quadratic equation gives the coefficient of the

quadratic term to be approximately -4.9, or half gravity. Comparing that with the previous finite

difference formula we developed on the spreadsheet, we find that they confirm each other. Students who

have had calculus can then validate the answers and make connections using algebraic computations. By

doing activities like these, students can conclude that an important part of learning mathematics is to use

multiple representations by approaching problems numerically, visually supporting graphically using

technology, and confirming algebraically (using paper and pencil techniques) problems solved using

technology (Waits & Demana, 2000).

The connection between mathematics and science can also be made at a more elementary level.

On the NCTM Web site, one can find theIlluminations area(http://illuminations.nctm.org/), which is

designed to illuminate the new vision for school mathematics as presented in the Principles and

Standards. Lessons are available in a listing searchable by grade level or content standard. Interactive

tools demonstrate and explore concepts. Selected Web resources support teachers in the teaching and

learning process (http://www.nctm.org/profdev/resources.asp#electronic).

One of theIlluminations lessons involves connecting mathematics, literature, and science by

using the bookCounting on Frank, in which a child wonders how many humpback whales can fit inside

his room (Clement, 1991; http://illuminations.nctm.org/LessonDetail.aspx?ID=L203). Preservice students


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can then learn ways to adapt lesson plans. Using the Web as a resource, students can find information

about humpback whales. Many resources appear to include length of whales, but not their width and

depth. The lesson can be adapted by using humpback whale models (plastic whales), so that the width and

depth of the whales can be determined using proportions, and then calculations can be made to determine

the volume of the room and the capacity of whales that the room will hold. Connections can be made with

the movie Star Trek IV: The Journey Home, in which Captain Kirk, Spock, Scotty, and the rest of the

crew go back in time from the 23rd century to 1986 to get humpback whales. To take the whales back to

the 23rd century, they must build a tank to hold the whales and the water (Bennett, 1986). Showing clips

from the movie introduces another media into the classroom, to capture students interest, while making

connections between mathematics and science.

Inverse Functions and Inversely Proportional Relationships

Building on the previous centimeter/inch discussion of inverse functions, we also use parametric

equations to demonstrate inverse functions. The following screen dumps show the way we can use

parametric equations to demonstrate inverse functions and then graph the results with a square window.

We discuss the difference between inverse functions and inversely proportional relationships, and

then use the CBL and a pressure probe to collect data demonstrating Boyles law (which shows that the

volume of a fixed quantity of an ideal gas is inversely proportional to pressure, given constant

temperature).


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Geometry

Although our preservice mathematics teachers take a course in geometry, we still look at some

simple dynamic geometry using Cabri

Jr. App interactive geometry on the Texas Instruments graphing

calculator. Dynamic geometry software allows students to make conjectures and explore properties and

relationships. To help the mathematics preservice teachers explore firsthand how dynamic geometry

software can enhance understanding, we use construction tools to create a quadrilateral, then a

parallelogram, a rhombus, and a square. Students need to think about the properties in order to make

the constructions so that, for example, a square remains a square under animation. Following is a square

that has been constructed and then animated by being made larger and then rotateddemonstrating that

the construction holds up under animation. Using the constructions and looking at the properties, we use

Venn diagrams to show the relationships among the quadrilaterals. [Note that some of these preservice

mathematics teachers struggle with these Venn diagrams!]

Another construction we perform is a demonstration of a Third International Mathematics

and Science Study (TIMSS) video in which a Japanese teacher is using dynamic geometry software (U.S.

Department of Education, 1997). We try to reproduce the constructions in which triangles situated

between two parallel lines can be animated to show that, if the bases are the same, then the areas are the

same measure. This exploration demonstrates that the determining factor in finding the areas of the

triangles is the height.


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Making Connections to the Culture of the Day

Texas Instruments, in partnership with CBS and working in association with NCTM, has created

an educational outreach program for grades 912 ("We All Use Math Every Day") that promotes the

many uses of mathematics and supports math teaching (http://www.cbs.com/primetime/numb3rs/ti/). The

math education program is based on episodes of theNUMB3RS television show, in which the FBI teams

with a mathematics professor to solve crimes. Teachers can be placed on an e-mail alert to receive

information about the third-season weekly programs, or they can simply go to the Web site and download

the lessons (for use during the 20062007 school year only). Our class watches an episode and then goes

through an activity or two to make similar connections. Activities involve parabolas and curve fitting,

histograms, relative frequency, probability, vector fields, implicit and explicit functions, and more; and

many of the activities require the use of a graphing calculator. DVD sets of the complete first and second

seasons ofNUMB3RS are available, so that teachers dont need to depend on the weekly show schedule,

and instead can pick and choose. Preservice teachers find the activities challenging and engaging; and the

fact that they are connected to real-life FBI-type of situations is appealing to students of all ages.

Conclusion

Preservice mathematics teachers often have little relational understanding of concepts they will be

required to teach (Rusch and Nicol, 2004). To facilitate understanding and to model appropriate, inquiry-

based, instructional strategies, I use worthwhile mathematics tasks with our students. Those who aspire to

be teachers then perform the dual role of teacher and student as they go through the tasks, acquiring

deeper understanding of concepts while learning how to teach those same concepts to their future


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students. Technology is used as a crucial tool in these explorations because such use is strongly

encouraged, if not mandated, by the NCTM (2000) and by Ohios Academic Content Standards (Ohio

Department of Education, 2003). The use of technology impacts not only what we teach, but how we

teach it.


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