____ THE ____________

_____ MATHEMATICS ___

________ EDUCATOR _____ Volume 21 Number 1

Summer 2011 MATHEMATICS EDUCATION STUDENT ASSOCIATION

THE UNIVERSITY OF GEORGIA

Editorial Staff

Editors

Allyson Thrasher

Catherine Ulrich

Associate Editors

Zandra de Araujo

Amber G. Candela

Tonya DeGeorge

Erik D. Jacobson

Kevin LaForest David R. Liss, III

Laura Lowe

Patty Anne Wagner

Advisor

Dorothy Y. White

MESA Officers

2011-2012

President

Tonya DeGeorge

Vice-President

Shawn Broderick

Secretary

Jenny Johnson

Treasurer

Patty Anne Wagner

NCTM

Representative

Clayton N. Kitchings

Colloquium Chair

Ronnachai Panapoi

A Note from the Editors

Dear TME readers,

On behalf of the editorial staff and the Mathematics Education Student Association

at The University of Georgia, I am happy to share with you the first issue of the 21st

volume of The Mathematics Educator. As we embark on the second decade of TME,

this issue gives our readers both a view into some up-and-coming trends in

mathematics education and harkens back to the roots of our field. In lieu of a

traditional editorial, as our opening article, we present the first English-language

publication of an interview of George Pólya, captured by his former student Jeremy

Kilpatrick. In the interview, Kilpatrick delves into the ideas of one of our field’s early

prominent leaders, introducing us to Pólya's ideas about the nature of mathematical

thinking and ability. The remaining articles in this issue highlight current trends in

preservice mathematics teacher education: using technology to enrich preservice

teachers’ mathematical learning, developing curricula for building preservice teacher

understanding of statistics, and exploring what preservice secondary teachers value in

their undergraduate mathematics courses.

More specifically, José N. Contreras offers a description of how he used

Geometer’s Sketchpad (GSP) to help preservice teachers discover geometric theorems,

develop proofs for those theorems, and deepen conceptual understanding by exploring

connections between theorems. He explains the different functions GSP served in

facilitating his students’ understanding. Hollylynne and Todd Lee provide an inside

view of how they used research to inform curricular revisions in their article,

“Enhancing Prospective Teachers’ Coordination of Center and Spread.” They provide

an excellent model of how to analyze and refine the development of mathematical

themes in curricular materials. Finally, Lee Fothergill adds to the on-going debates

about what mathematics teachers need to know. He examines perceptions about the

content of calculus courses for preservice teachers among both student teachers and

mathematics department faculty, and he finds some interesting areas of agreement.

Publishing TME requires the help of many people: authors, editors, and faculty

advisors. But the backbone of our journal is no doubt our reviewers who provide the

first critical feedback on submitted manuscripts and often receive far more requests for

reviews than acknowledgement of their work. At the conclusion of this issue, Katy and

I offer the tireless reviewers for this issue a long-overdue thanks. We hope that you

enjoy this issue and share it with your colleagues.

Allyson Hallman Thrasher

Catherine Ulrich

105 Aderhold Hall tme@coe.uga.edu The University of Georgia www.ugamesa.org Athens, GA 30602-7124

About the cover: Graph by Kylie Wagner, rendered in Illustrator by Jeff Sawhill A predictive model can be fitted to the random variable y by minimizing the vertical distance between the fitted line and observed

y-values. We can calculate the probability of y occurring within a certain distance of the predicted y-values by using a series of

normal curves; where the mean of the curve is equal to the predicted y-values. This three-dimensional graph of the error

distribution of a regression line more accurately captures this probability function than the two-dimensional diagram (shown in the

upper left corner of cover).

This publication is supported by the College of Education at The University of Georgia

____________THE ________________

___________MATHEMATICS________

______________EDUCATOR ____________

An Official Publication of

The Mathematics Education Student Association

The University of Georgia

Summer 2011 Volume 21 Number 1

Table of Contents

3 A Look Back…. Pólya on Mathematical Abilities JEREMY KILPATRICK 11 Using Technology to Unify Geometric Theorems About the Power of

a Point JOSÉ N. CONTRERAS 23 Aspects of Calculus for Preservice Teachers LEE FOTHERGILL 33 Enhancing Prospective Teachers’ Coordination of Center and Spread:

A Window Into Teacher Education Material Development HOLLYLYNNE S. LEE & J. TODD LEE

48 A Note to Reviewers 49 Submission Guidelines 51 Subscription form

© 2011 Mathematics Education Student Association

All Rights Reserved

The Mathematics Educator

2011, Vol. 21, No. 1, 3-8

3

Dr. Jeremy Kilpatrick is Regents Professor of Mathematics

Education at The University of Georgia. His research interests

include mathematics curricula, research in mathematics education,

and the history of both.

A Look Back…

Pólya on Mathematical Abilities1

Jeremy Kilpatrick

In April 1978, I interviewed George Pólya about his views on mathematical abilities. I was in California for

the annual meeting of the National Council of Teachers of Mathematics in San Diego and arranged to stop by

Pólya’s house in Palo Alto after the meeting to discuss his views on mathematical abilities as well as the articles

on mathematics education to be included in his collected papers (Rota, Reynolds, & Shortt, 1984). The

following article is abridged from that interview and focuses on mathematical abilities.

For me, the most unexpected feature of the interview was that although Pólya had obviously reflected

throughout his long life on the question of how he and others do mathematics, he had apparently not given

much thought previously to the abilities they were drawing on when they did it. Nonetheless, Pólya’s wit and

charm come through clearly as he patiently struggles with his former student’s awkward questions.

JK: What are the qualities that you think make

someone capable in mathematics? In other words,

what are the mental abilities that distinguish

someone who is capable in mathematics from

someone who is not so capable?

GP: I couldn’t give you a good description, you see. I

never made any clear ideas about that. Moreover,

there are so many different kinds of

mathematicians.

JK: What different kinds?

GP: Well, I wrote a little article about it once where I

mentioned Emmy Noether.i I made a joke about it.

She was for generalization; I was for

specialization.

….

JK: Do you think it’s important to have good spatial

ability to be a mathematician?

GP: To a certain extent, yes, but that’s also so

different. Hadamard tells about—. Do you know

the book of [Jacques] Hadamard?ii

JK: Yes, I know the book

GP: If he were here, he would give you much better

answers—anyway, more answers. He thinks

sometimes you are the “auditive” type, or you are

the “visual” type. And he himself is more an

auditive type. I don’t know. It certainly helps,

especially—. There is Jean Pedersen;iii she

certainly has spatial ability.

JK: What about memory? Do you think

mathematicians have a special memory? For

mathematical things?

GP: Yes, sure.

JK: Do you have to have a very good memory?

GP: Well, sure, for everything. Horace says in the Ars

Poetica, “Mendacem oportet esse memorem”iv—

my Latin still works a little. He says, “A liar must

have a good memory.” A poet is a liar. He invents

everything. He must very well remember what he

did before. So a good memory, that is necessary

for everything.

JK: A specially organized memory? Do you think

mathematicians have a memory that is organized

in a different way?

GP: Yes, exactly. What is organized? I find, you see,

the general terms in which you could describe it,

they are either lacking or they are vague.

JK: I can see that. But people have tried to—. Well,

one question is whether mathematicians have

certain special kinds of abilities, or they just have

ordinary abilities, but they apply them to

mathematics.

1 This interview is abridged from the original transcript, which is available in Portuguese from Guimarães, H. (2010). Jeremy

Kilpatrick: entrevista a George Pólya [Jeremy Kilpatrick: interview with George Pólya]. Quadrante, 19(2), 103–119.

Mathematical Abilities

4

GP: The second is probably a little better. No one is

completely true, but the second is better. For

instance, I can tell you, I have a pretty good

memory—. Anyhow, for the mathematics I did, I

have a pretty good memory. Well, now it goes

downhill like the rest of it, but I could remember

pretty much everything what I did. Not what other

people did. …But I have also a good memory for

poetry and a good memory for jokes. So it is not

specialized for numbers. I have a good memory

for poetry, but I recall it so: It comes often; I

recall it, in between, for any reason or without

reason. I just ask you whether you know German.

Because I recall something very pretty what

Schiller said about it.

JK: And you recall the whole thing?

GP: There are just two lines. He describes very well

what he—. I will tell it to you in German. It is

very good German. He means it probably for

poetry, or possibly, he was also a historian—he

wrote history. But it is good for mathematics. I

say it in German:

Nur Beharrung führt zum Ziel,

Nur die Fülle führt zur Klarheit,

Und im Abgrund wohnt die Wahrheit.v

He said, “Only—.” Ah, “Beharrung”—how do

you say it? “Who always—.”

Well, now, I have four languages; it’s very

difficult to find the right—. “Beharrung.” So, if

you are working all the time in the same direction,

you must go ahead all the time. “Nur die Fülle”—

if you know many things, keep together—”führt

zur Klarheit”—then you may be clear. If your

knowledge is based on many things. “Und im

Abgrund wohnt”—and the truth is in the deep.

You can say the same thing about mathematics,

but Schiller certainly meant it for poetry or for

history, and not for mathematics. …

….

JK: But different mathematicians have different

strengths and weaknesses.

GP: Different people have different strengths and

weaknesses.

JK: What are your strengths and weaknesses as a

mathematician?

GP: ….. I like to go down to something tangible. And

I start from something tangible. From some

physics, or even from some everyday things. …. I

say the same thing about—have you read it?—

about Emmy Noether.vi

JK: Yes, I’ve read the paper.

GP: So there are two kinds of monkeys: up monkeys

and down monkeys.

JK: And you’re a down monkey.

GP: I’m a down monkey, and she was an up monkey.

They are different; so are people.

JK: What were the parts of mathematics that you had

the most difficulty understanding?

GP: I don’t know. Perhaps, well, oh, I appreciate—.

It’s not the difficulty of understanding. For

instance, I appreciate foundations, but I couldn’t

work on it.

JK: Why not?

GP: Not my line, you see.

JK: Because it deals with generalization? Because it’s

too general?

GP: Well—.

JK: Too abstract?

GP: It cannot be expressed in words, you see. It is

simply not my line. Oh, I admit it is important,

but I just couldn’t work on it. It was very, very

fortunate, you see. ….[David] Hilbert came to

visit Hurwitz in Zurich. He was very old, you see.

He felt …he needs a good assistant. And there

were proposed two: [Paul] Bernays and myself.

It’s a great luck that they have chosen Bernays

and not me. Because I was not good for

foundations, and Bernays was excellent, you see.

They wrote the book: Bernays, Hilbert, and

[Wilhelm] Ackermann.vii It is hundred percent

written by Bernays. Of whose thought, I don’t

know. By Hilbert, you see, maybe it was

organized, probably. And it is enormous luck for

science and for myself that I was not chosen, you

see. It would have been, of course, in a way, it

would have been very flattering to be an assistant,

but it was much better not to be.

JK: Let’s talk about problem solving. Where did the

rules and heuristic methods that are in How to

Solve It,viii where did those come from? What’s

the source?

GP: This I gave in print. ….This is, I think, my first

paper about problem solving.ix And this is told in

detail here in the first lines. I had a kid, a stupid

kid to prepare for a high school examination. And

Jeremy Kilpatrick

5

I wished to explain him some—. Almost this

problem.x And I couldn’t do it. And the evening I

sat down, and I invented that [representation]. So

that was the starting of my explicit interest in

problem solving.

JK: So, trying to teach him, you came up with these

questions.

GP: No, no, that came afterwards, you see. But just the

main thing, the representation by a graph. I didn’t

know the word graph, and so on, but I invented

this representation. Then I made it better. I made a

geometric figure. ….And that was the beginning

of my explicit interest.

Implicitly, I was probably interested before. I

was also interested: How did people discover it?

And then Mach, Ernst Mach, he said, “To

understand a theory, you must know—. It is really

understood if you know how people discovered

it.” I read his book,xi and this influenced me

enormously. This brought me from philosophy to

physics. …..

JK: The graph came before your questions or your

suggestions like, “What is the unknown?” “Can

you draw a figure?”

GP: Oh, yes. The graph came first. Then I was also

very much interested by Descartes. By the

Regulae.xii

JK: The Rules, yes.

...

GP: ….. Oh, have you seen the number of the Journal

of Graph Theory? …..

JK: No, I haven’t seen that.

GP: There are two articles in it.xiii The first, by

Harary—I don’t have a reprint. And the other, by

Albert Pfluger. I don’t know whether you know

who he is.

JK: No.

...

GP: …He was a student. He made his Ph.D. with me. I

knew him, his daughter, and so on, and so on.

JK: And he tells the story.

GP: And he pretty much describes the story.

….

JK: When you solve problems, do you use your

advice from How to Solve It? Consciously?

GP: Yes. Well, even more than that. ….I had the rules,

and I tried it out on myself. So, for instance, I

edited the works of Hurwitz. ….He had a

mathematical diary, and it is beautifully written,

you see. It is written very comp1ete1y—not just

scribb1ed, but clearly written, well-formulated,

you see—where he describes what he thought of:

sometimes his conversations; sometimes what he

read. And then I thought about editing it, you see.

And so, I found among others, this problem which

falls me to … this [Pólya] Counting Method, you

see. And I chose this counting method just to

check my own rules. Whether my own rules

would work. …

…

GP: ….. And this problem of Hurwitz, it was just good

for that. Obviously an interesting problem

because Hurwitz and Cayley had worked on it,

and [it is] connected essentially with chemistry.

That I like, you see: connected with something

important and with the practice. But, on the other

hand, very little preliminary knowledge is

needed.….

JK: Yes.

…

JK: Some people say that they cannot use the rules.

Or that—.

GP: Well, that’s okay. People are different. People are

different.

JK: Do you think it’s possible to develop somebody’s

ability to solve problems?

GP: I think so.

...

GP: Well, I think it is not so much “develop” as it is

“awaken,” I would say.

JK: It’s there.

GP: It is somewhere there. If there is nothing there,

you cannot—. But you can awaken it, you see. A

good teacher, and so on, a good opportunity to

awaken it, you see. Well, my own case—. I had

obviously some probability for it, but it was

awakened very lately. I would have been probably

a much better mathematician if I had had in the

gymnasium a good teacher. It can be awakened—

this I think so. This may be too optimistic—. I

think even [with] my rules can a teacher, a good

teacher emphasizing a little my questions can help

awaken it. Alan Schoenfeld has some ideas how

Mathematical Abilities

6

to do it. I don’t quite agree with what he says, but

anyhow, I think so. This I believe. That is no

proof, of course. But it would be very difficult to

prove or disprove it.

JK: Do you think it is important for the teacher to

demonstrate in front of the class how to, to show

the class—. Is it important, for the teacher to

show in front of the class how to solve the

problem? The teacher should be an actor?

GP: The most important for the teacher that he should

himself have the experience of solving. In …

Belmont [CA], there is a Catholic college, the

College of Notre Dame. There we had a meeting.

…And there we had Ed Teller, the father of the

atomic bomb. He gave a talk, and even a very

interesting talk.xiv I don’t agree with everything

what he said, but it was good. He said the most

important is the teacher; the teacher should amuse

the kids. Mathematics should amuse the kids.

JK: Do you agree?

GP: Yes, sure. To awaken them, the problems should

be amusing; the problems should be challenging.

They should be amusing—not faraway problems,

not “practical” problems: how to pay your income

tax.

JK: That’s not amusing.

GP: (Laughs.) Definitely not. The Infernal Revenue

Service: It’s not amusing.

JK: How did you identify the students you had who

were best in mathematics? You taught some

students who were good in mathematics. How

could you tell who were the best ones?

GP: Who was the best one, I can’t tell you.

JK: Well, among the best, how could you identify

their talent? They were quicker?

GP: Anyhow, they asked good questions. So they

found out something by themselves. And so on.

There is no simple way—. You see, people are

too different. Mathematicians are too different.

There is no simple way of describing it. I don’t

think so.

JK: What about people who are creative in

mathematics as opposed to just being able to learn

it? What does that take? What does that require?

Just great interest?

GP: I don’t know.

JK: Not everyone could be creative in mathematics.

GP: I said somewhere, “What is the difference

between productive and creative?” If you think

about a problem, if you produce a result, then you

are productive. If in working you get into a

method with which you can solve also other

problems, then you are creative. That’s the

difference. And that is difficult to say. I don’t

think there are obvious signs to recognize this. I

don’t think so.

JK: Are these things that kids are born with?

GP: That I am pretty sure: You must have a genetic—.

That must be somehow born to it, that is clear.

JK: And it helps if you have a teacher—.

GP: Oh, that helps, to awaken it.

JK: But even if you don’t have a teacher to awaken it,

you could be—.

GP: Oh, you could.

JK: As your own case.

GP: …. Well, I had Mach as a teacher. A little late, but

…Mach said it, and he illustrated it very strongly:

“If you wish to understand the theory, you should

know how it was discovered.” And this I

understood.

JK: Do you think that’s one of the problems with

teaching mathematics in school, that we present it

to the kids—? We present mathematics to the

kids, but we don’t show them how it has been

discovered? In other words, teaching should be

more genetic?

GP: You should illustrate it, you see. You make a little

theatre, and you pretend to discover it. This I

printed it even somewhere. You pretend to

discover it.

JK: And you think that’s important for—.

GP: If you do that well, then they learn much more

than just this problem.

JK: You have collaborated with other mathematicians.

...

GP: ….I collaborated with very good mathematicians,

better than myself. With Hurwitz, with [Godfrey

Harold] Hardy, with [Gábor] Szegö. They are

here around me (points to pictures on the wall of

his study). Of course, I collaborated most with

Szegö.

JK: Does Szegö approach mathematics as you do?

Jeremy Kilpatrick

7

GP: Well, on the contrary—we were to some extent

complementary.

JK: How?

GP: For instance, he is an excellent calculator; he is

excellent at calculating.

JK: And you’re not so good?

GP: Oh, I am not so bad, but he—. Anyhow, we

somehow complemented each other. He knew

some subjects, for instance, he knew polynomials

better than me. About Legendre, and so on. We

somehow—. Our interests were sufficiently

similar, but also sufficiently different, and I

couldn’t enumerate all the points, but it was more

complementing. We had, of course, some very

similar interests, but also different. Also, similar

backgrounds. We were both students of [Leopold]

Fejér, and so on, but—.

JK: What kind of a teacher was Fejér?

GP: Oh, he was very good, very good. I scarcely had a

class by him, but I talked with him a lot. He was

excellent. Oh, this is printed somewhere; I have

an obituary of Fejér, where I tell about this.xv He

could tell so good stories.

….

JK: When you work on mathematics, when you try to

do mathematics or solve a problem, do you find

the advice to let the problem go for awhile and—

is that good advice?

GP: Not before I did something.

JK: [You need to] try a little. Have you ever had the

experience of having a solution come to you in

the unconscious?

GP: Oh, yes, sure. There is even—. “Waiting for the

good wind”—this is a usual expression.

JK: Have you had the experience?

GP: I don’t know by whom I heard it, but I didn’t

invent it, I am sure. So, if you are a sailor—not if

you have a boat with a machine, but if you have a

sailing boat—then you have to wait for the good

wind. So, “waiting for the good wind”—I didn’t

invent this expression; that must be somehow

traditional in English.

JK: People like Poincare and others tell—.

GP: And that is waiting. Sleep on your problem. That

is international. It is said in all languages.

JK: Have you had the experience of waking up with a

solution?

GP: Oh, yes, now and then. Even this I describe

somewhere in one of my papers.

JK: It came that way to you.

GP: But very seldom. And I heard it from Hurwitz the

same. You wake up with a solution, but it is just

phantasmagoria.

JK: It’s not really a solution?

GP: It doesn’t; it is not so. It happened very seldom.

That really I wake up with a solution that was so.

A simple thing is in the Inequalities, one solution

for the—. It is mentioned, I think, in one of my

late papers.xvi (Gets paper.)

...

GP: ….. But once or twice—once I remember it

definitely happened; I really dreamt it correctly. I

just had to write it out, the details, in the morning.

And Hurwitz had the same, I heard. I’m pretty

sure it is described there.

JK: Do you draw a lot of figures when you work on

problems?

GP: Sometimes, yes. Oh, I draw a lot of figures.

Sometimes very carefully.

JK: Even when the problem doesn’t require a figure?

GP: Sure. It may be a beginning of the idea. That you

come to a figure which is connected with the

problem.

…

GP: [The conversation turns back to the talk by Teller]

But it was good that somebody told it to the

teachers. Especially that the main thing of the

teacher should be the interest; he should amuse.

He should convince the kids that mathematics is

amusing.

JK: How can the kids ever learn mathematical skills,

then?

GP: They will learn it. If he plays Nim, he will learn to

make additions very quickly. And learn to

combine things, and so on. Teller is surely a much

greater scientist, and by the way, Teller is not

only that. You know there was a mathematical

competition in Hungary.xvii

JK: Yes.

GP: Teller won this competition as a kid. So he knows

it, when he talks about learning mathematics,

Mathematical Abilities

8

about the mathematics at high school age, he has

real experience, first-rate experience. But Jean

Pedersen, who is a very successful teacher, goes

to high schools, or they come to the University of

Santa Clara. And she shows the kids how to make

models. Then they are anxious to make models.

And once she photographed each kid with the

model he made. So that is also something. That is

also a mathematical occupation. They learn

geometric figures, and so on. “Learning starts by

seeing and doing”—this I also quote

somewhere.xviii

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direction of the mind). In Des-Cartes Opuscula posthuma,

physica & mathematica. Amsterdam, The Netherlands: P. & J.

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Hadamard, J. (1945). The psychology of invention in the

mathematical field. Princeton, NJ: Princeton University Press.

Harary, F. (1977). Homage to George Pólya. Journal of Graph

Theory, 1, 289-290.

Hardy, G. H., Littlewood, J. E., & Pólya, G. (1934). Inequalities.

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Hilbert, D., & Ackermann, H. (1928). Grundzüge der theoretischen

Logik [Principles of mathematical logic]. Berlin, Germany:

Springer.

Hilbert, D., & Bernays, P. (1934. Grundlagen der Mathematik

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Hilbert, D., & Bernays, P. (1939). Grundlagen der Mathematik

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Mach, E. (1883). Die Mechanik in ihrer Entwicklung [The science

of mechanics]. Leipzig, Germany: Brockhaus.

Pfluger, A. (1977). George Pólya. Journal of Graph Theory, 1,

291–294.

Pólya, G. (1919). Geometrische Darstellung einer Gedankenkette

[Geometrical representation of a chain of thought].

Schweizerische Pädagogische Zeitschrift, 2, 53–63.

Pólya, G. (1957). How to solve it. Princeton, NJ: Princeton

University Press.

Pólya, G. (1961), Leopold Fejér. Journal of the London

Mathematical Society, 36, 501–506.

Pólya, G. (1969). Some mathematicians I have known. American

Mathematical Monthly, 76, 746–753.

Pólya. G. (1970). Two incidents. In T. Dalenius, G. Karlsson, & S.

Malmquist (Eds.), Scientists at work: Festschrift in honour of

Herman Wold (pp. 165–168). Stockholm: Almqvist &

Wiksell.

Pólya, G. (1981). Mathematical discovery: On understanding,

learning and teaching problem solving (Combined ed.). New

York, NY: Wiley.

Pólya, G. (1984). A story with a moral. In G.-C. Rota, M. C.

Reynolds, & R. M. Shortt (Eds.), George Pólya: Collected

papers (Vol. 4: Probability; combinatorics; teaching and

learning in mathematics, p. 595). Cambridge, MA: MIT Press.

(Reprinted from Mathematical Gazette, 57, 86–87, 1973)

Rota, G.-C., Reynolds, M. C., & Shortt, R. M. (Eds.). (1984).

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Cambridge, MA: MIT Press.

Schiller, F. von. (1796). Sprüche des Konfucius. In F. von Schiller

(Ed.), Musen-Almanach für das Jahr 1796 [Muses Almanac

for 1976] (pp. 39–47). Neustrelitz, Germany: Michaelis.

i Pólya, 1973/1984. ii Hadamard, 1945.

iii Professor of mathematics at Santa Clara University.

iv The quotation actually comes from Quintilian (De

Institutione Oratoria, IV. ii). v “Naught but firmness gains the prize, Naught but fullness

makes us wise, Buried deep, truth ever lies!” (Schiller,

1796). vi Polya, 1973/1984.

vii Hilbert & Ackermann, 1928; Hilbert & Bernays, 1934,

1939. viii Pólya, 1957.

ix Pólya, 1919. The improved representation can be found in

Mathematical Discovery (Pólya, 1981, Vol. 2, p. 9) and

inside the front cover of Vol. 2 of the original edition. x The problem is to find the volume of a right pyramid with

square base given the altitude and the lengths of the sides of

the upper and lower bases (see Polya, 1981, Vol. 2, p. 2). xi Mach, 1883.

xii Descartes, 1701.

xiii Harary, 1977; Pfluger, 1977.

xivxiv At the February 1978 meeting of the Northern

California Section of the Mathematical Association of

America, held at the College of Notre Dame, Edward

Teller’s talk was entitled “The New (?) Math.” xv Pólya, 1961. See also Pólya, 1969.

xvi It was the proof of the inequality between the arithmetic

and geometric means given in Hardy, Littlewood, and Polya,

1934, p. 103. See Pólya, 1970. xvii

The Eötvös Competition. xviii

Pólya, 1981, Vol. 2, p. 103. Pólya’s paraphrase of Kant:

“Learning begins with action and perception.”

The Mathematics Educator

2011, Vol. 21, No. 1

9

The Mathematics Educator

2011, Vol. 21, No. 1

10

The Mathematics Educator

2011, Vol. 21, No. 1, 11–21

11

Using Technology to Unify Geometric Theorems About the Power of a Point

José N. Contreras

In this article, I describe a classroom investigation in which a group of prospective secondary mathematics

teachers discovered theorems related to the power of a point using The Geometer’s Sketchpad (GSP). The

power of a point is defines as follows: Let P be a fixed point coplanar with a circle. If line PA is a secant line

that intersects the circle at points A and B, then PA·PB is a constant called the power of P with respect to the

circle. In the investigation, the students discovered and unified the four theorems associated with the power of a

point: the secant-secant theorem, the secant-tangent theorem, the tangent-tangent theorem, and the chord-chord

theorem. In our journey the students and I also discovered two kinds of proofs that can be adapted to prove each

of the four theorems. As teacher educators, we need to design learning tasks for future teachers that deepen their

understanding of the content they are likely to teach. Having a profound understanding of a mathematical idea

involves seeing the connectedness of mathematical ideas. By discovering and unifying the power-of-a-point

theorems and proofs, these future teachers experienced what it means to understand a mathematical theorem

deeply. GSP was an instrumental pedagogical tool that facilitated and supported the investigation in three main

ways: as a management tool, motivational tool, and cognitive tool.

The judicious use of technology enhances the

teaching and learning of mathematics. Technology

frees the user from performing repetitive and

computational tasks, and thus, it allows more time for

action and reflection. As a consequence, when students

use technology as a cognitive tool, they develop a

deeper understanding of mathematical concepts,

patterns, and relationships (Battista, 2007; Clements,

Sarama, Yelland, & Glass, 2008; Hollebrands, 2007;

Hollebrands, Conner, & Smith, 2010; Hollebrands,

Laborde, & Sträβer, 2008; Hoyles & Healy, 1999;

Hoyles & Jones, 1998; Koedinger, 1998; Laborde,

1998; Laborde, Kynigos, Hollebrands, & Sträβer,

2006).

For example, Battista (2007) describes how two

fifth graders constructed meaning for a spatial property

of rectangles--each of the four angles of a rectangle

measures 90°--within the Shape Makers environment

(Battista, 1998), a GSP microworld for investigating

geometric shapes. In their review of research on

learning and teaching geometry within interactive

geometry software (IGS) environments, Clements,

Sarama, Yelland, and Glass (2008) concluded that IGS

“can be beneficial to students in their development of

understandings of geometric shapes and figures” (p.

131). Similarly, research reviewed by Hollebrands,

Conner, and Smith (2010) suggests that IGS

environments “enable students to abstract general

properties and relationships among geometric figures”

(p. 325).

IGS such as The Geometer’s Sketchpad (GSP)

(Jackiw, 2001) and Cabri Geometry II (Laborde &

Bellemain, 1994) are powerful instructional technology

tools. IGS allows the user to construct dynamic figures

that can be manipulated or moved without altering the

mathematical nature of the geometric figure. This

feature allows the user to quickly generate many

examples of a geometric diagram. This feature is in

marked contrast to the static nature of textbook and

paper-and-pencil illustrations. A diagram that can be

resized by dragging flexible points also motivates the

user to investigate invariant geometric relationships.

As a result of motivation, action, and reflection,

students construct a more powerful abstraction of

mathematical concepts (Battista, 1999).

This article describes a classroom activity in which

a group of 13 prospective secondary mathematics

teachers (hereafter referred to as students) investigated

the power of a point with GSP. My objective was to

guide my students to discover and unify several

geometric theorems related to the power of a point.

The power of a point is defined as follows: Let P be a

fixed point coplanar with a circle. If PA is a secant

line that intersects the circle at points A and B, then

Dr. José N. Contreras, jncontrerasf@bsu.edu, teaches mathematics

and mathematics education courses at Ball State University. He is

particularly interested in integrating problem posing, problem

solving, technology, history, and realistic mathematics education in

teaching and teacher education.

Technology to Unify Power of Point Theorems

12

PA·PB is a constant called the power of with respect to

the circle.

The Classroom Setting

The students were enrolled in my college geometry

class for secondary mathematics teachers. The

textbook I used was Geometry: A Problem-Solving

Approach with Applications (Musser & Trimpe, 1994).

All of my students had completed the calculus

sequence, discrete mathematics, and linear algebra. In

addition, by this point in the course, my students were

proficient using GSP, as they had employed it to

complete several tasks involving constructing

geometric figures (e.g., centroid of a triangle, squares,

etc.), detecting patterns, and making conjectures. We

conducted our power of a point investigation in the

computer lab where each student had access to a

computer with GSP. To facilitate and manage the

investigation more efficiently and accurately, I

provided students with geometric files relevant to the

investigation. I had my laptop computer connected to

an LCD projector.

Starting the Investigation: Discovering the Power of

a Point

We began our investigation with the problem

shown in Figure 1.

Find the value of PD in the configuration below

where 60.1=PA cm, 50.1=PC cm, 30.3=PB cm.

Justify your method.

C

A

P

B

D

Figure 1. The initial problem.

Some students used the proportion 50.160.1

30.3 PD= or one

of its equivalent forms, others said that they did not

remember how to do this type of problem, while a third

group claimed that they had never seen a problem like

that before. I then asked students to open the “power of

a point” file to investigate this problem using GSP. I

had hoped for students to attempt to discover the

general relationship. A few students quickly used the

measurement capabilities of GSP to find or verify their

solution. When they realized that their solution was

incorrect, they concluded that their proposed

relationship 50.160.1

30.3 PD= did not hold. Another student

reached this conclusion by noticing that dragging point

B changed PA, PB, and PA

PB, but did not influence PC

and PD. Therefore, the proportion PD

PC

PA

PB= did not

hold. The measurement and dragging capabilities of

GSP allowed students to disconfirm their initial

conjectures.

After confirming that dragging point B changed PA

and PB, I told them that a hidden quantity involving

only PA and PB remained constant and challenged

them to find it. Some students tried PA+PB and

PB–PA. One of the first students who discovered that

PA·PB remains constant said, “I can’t believe it. PA·PB

remains the same no matter where points A and B are.”

Other students verified this hypothesis by dragging

point B and calculating PA·PB (see Figure 2). One

student was puzzled because she noticed that PB

increases in some instances but the product remained

the same. Another student said, “Yes, but PA

decreases. When one number increases the other

decreases. So they balance each other.” At this time I

mentioned that the constant PA·PB is called the power

of point P, P(P), with respect to the circle. In this case,

the computational and dynamic capacities of GSP

allowed some students to discover that PA·PB remains

invariant regardless of where points A and B are

located in the circle.

Continuing the Investigation: An Unanticipated

Discovery

As we did with other investigations involving GSP,

we systematically tested our conjecture for different

circles and points. To test our power-of-a point

conjecture for a given circle, we dragged point P and

then point B to verify that PA·PB is constant. Students

also noticed that for a given circle, the farther point P

was from it, the greater its power. A couple of students

also dragged the point controlling the radius of the

circle and noticed that the radius influenced the power

of a point as well. I had originally planned to just test

our conjecture for different points and different circles,

but our systematic testing led us to investigate an

unexpected conjecture related to how both the length

of the radius (r) of the circle, and the distance from P

to the center (O) of the circle impacted its power.

José N. Contreras

13

60.1=PA cm 30.3=PB cm 50.1=PC cm

30.5=⋅PBCPA cm2

C

A

P

B

D

44.1=PA cm 68.3=PB cm 50.1=PC cm

30.5=⋅PBCPA cm2

C

A

P

B

D

Figure 2. PA·PB seems to be constant for a given point P and circle.

I hid the product PA·PB on my GSP sketch and

asked students to predict the behavior of the power of

point P as I increased the radius of the circle from 0

with both its center O and point P fixed. A student

claimed that the power of the point would remain

constant because PB increases and PA decreases.

Another student refuted this explanation saying that the

power would decrease because PB increases but PA

approaches zero and becomes zero when the circle

goes through P. The second student added that the

power would increase as the radius of the circle

increased “beyond P”. Students confirmed this

conjecture on their GSP sketches. At this time, it

occurred to me to ask students for the maximum value

of the power of the point when the point is still in the

exterior of the circle (i.e., the radius of the circle is less

than PO). Some students provided a numerical value

while others argued that the maximum value did not

exist because PA, PB, and PB·PA disappear when the

circle becomes a point. One student said that we could

still consider a point as a circle of radius zero, and

another student mentioned that a point could be

considered as the limiting case of a circle when the

radius approaches zero. However, most students in the

class agreed that a point is not a circle because the

radius has to be greater than zero. I then asked students

to consider what conception would be more helpful or

convenient to describe the behavior of PA·PB. We then

formulated the following conjecture:

Let P be a fixed point and C a circle with fixed

center O but variable radius r. As the radius of the

circle increases from zero, the power of the point

with respect to C

a) decreases from a maximum, the square of the

distance from the point to the center of the

circle (when the radius of the circle is zero), to

zero (when the circle contains P) as the radius

increases from 0 to OP.

b) increases from zero without limit as the radius

increases without limit from OP (P is an

interior point).

At this point, I wanted to investigate the

relationship between the power of a point and the

radius of a circle. Since I knew my students were not

familiar with the graphing capabilities of GSP, I asked

them to use pencil and paper to sketch a graph of the

power of a point as a function of the radius. While they

did this, I constructed the graph in GSP using the trace

feature. I asked students how we could conveniently

position a circle in the coordinate plane to simplify the

computations. One student suggested putting the center

of the circle at the origin and points P, A, and B on the

x-axis. This student provided the table shown (see

Figure 3) for the point P whose coordinates were (2,

0). Other students constructed similar tables using the

same or different coordinates for point P.

P(P)

0 2(2) = 4

1 3(1) = 3

2 4(0) = 0

3 5(1) = 5

4 6(2) = 12

5 7(3) = 21

Figure 3. Student-constructed table examining the

relationship between radius of a circle and power of

Technology to Unify Power of Point Theorems

14

point.

All students agreed with the GSP graph (see Figure

4) since it looked like their sketches, and that the first

piece of the graph seemed to be a parabolic arc. To

better visualize the nature of the second piece of the

graph, I changed the scale of the y-axis. Notice that the

circle is not shown on the second graph. We

conjectured that the graph appeared to be two pieces of

parabolic arcs.

As we tried to make sense of the table in Figure 3

and the graphs in Figure 4, we generalized the pattern

depending on whether P is an exterior or an interior

point as:

P(P) = (2 + r)(2 – r) = 4 – 2r

or

(r + 2)(r – 2) = 2r – 4.

I then asked students for the geometric interpretation of

the number 2 in this formula. After some reflection and

discussion, students realized that 2 was the distance

from the point P to the origin, which is the center of the

circle O. Therefore we could rewrite our equations as:

P(P) = PA·PB = (OP – r)(OP + r) = OP2 – r

2

and

P(P) = r2 – OP

2 .

when P is exterior to the circle and when P is interior

to the circle, respectively. Since my objective for this

activity was to unify theorems related to the power of a

point, I asked the students, “How can these two graphs

be unified? How we can have one parabolic arc instead

of two pieces?” In a previous activity we had unified

the theorems related to the measures of angles formed

by secant lines when the vertex of an angle is an is an

exterior point and when the vertex is an interior point

by considering directed arcs, so it was natural for a

student to suggest using directed distances. Another

student said that using directed distances could “flip”

the second piece across the x-axis. The first student

inferred from the graph that we could unify the two

formulas by considering the power of an exterior point

to be positive and the power of an interior point to be

negative. In order to do this, we needed to consider PA.

and PB as directed distances, similar to directed arcs.

As a result, we obtained the graph displayed in Figure

5. The equation of this graph is P(P) = 22 rOP − .

I was particularly delighted that we had also

discovered a formula for the power of a point in terms

of its distance to the center of the circle and the radius.

The interactive, graphing, and dynamic capabilities of

GSP motivated us to follow our intuitions and test the

resulting conjectures. It minimized the managerial and

logistic difficulties of performing this part of the

investigation with paper and pencil.

I was particularly delighted that we had also

discovered a formula for the power of a point in terms

of its distance to the center of the circle and the radius.

The interactive, graphing, and dynamic capabilities of

GSP motivated us to follow our intuitions and test the

resulting conjectures. It minimized the managerial and

logistic difficulties of performing this part of the

investigation with paper and pencil.

4

3

2

1

1

2

2 2 4

PA·PB = 2.18 cm2

OA = 1.33 cm

PB = 3.32 cm

PA = 0.66 cm

B

AO

E

P

40

30

20

10

10

2 4 6O

P

Figure 4. The power of a point as a function of the radius of the circle.

66.0=PA cm

32.3=PB cm

18.2=⋅PBPA cm2

33.1=OA cm

José N. Contreras

15

4

2

-2

OP2-OA2 = 3.04 cm2

OP = 2.01 cm

OA = 1.00 cm

B

O

P

A

Figure 5. The unified graph of the power of a point

as a function of the radius of the circle.

Continuing the Investigation: Establishing the

Secant-Secant Theorem

After these unexpected but productive digressions,

we came back to our original problem. Two students

admitted that they did not know how to use PA·PB to

find PD. After I dragged point B around the circle

hoping that these students could see the connection that

PA·PB = PC·PD because PA·PB is a constant, only one

student still failed to see the connection. A classmate

provided the following explanation: “PA times PB is a

constant no matter where points A and B are. So if A =

C and B = D we have that PA·PB = PC·PD.” The

student computed the product PC·PD to see the

pattern. After we established the relationship PA·PB =

PC·PD, I asked the class how we could prove it. Since

nobody provided any hint or suggestion about how to

prove the relationship, I suggested rewriting PA·PB =

PC·PD in another way. Some students suggested

rewriting PA·PB = PC·PD asPB

PD

PC

PA= . This

prompted one student to suggest using similar

triangles. Several students immediately proved the

equality by using the AA similarity theorem to prove

∆APD ~ ∆CPB (see Figure 6), and one student shared

his proof with the rest of the class.

By proving that PA·PB = PC·PD for arbitrary B

and D on the circle, we established that PA·PB is a

constant for a particular exterior point of a given circle.

We then formulated the corresponding theorems in the

following terms:

C

A

P

B

D

Figure 6. ∆APD ~ ∆CPB.

(i) The secant-secant theorem: Let P be an

exterior point of a circle. If two secants PA

and PC intersect the circle at points A, B, C,

and D, respectively (see Figure 6), then

PA·PB = PC·PD.

(ii) P is an exterior point and PA is a secant of a

circle. If the secant PA to the circle intersects

the circle at points A and B, then PA·PB is a

constant. This constant is called the power of

P with respect to the circle.

GSP allowed students to dynamically manipulate

and interact with the power of a point, an abstract

object, in a “hands-on” manner. By moving points

along the circle, they gained experience with one of the

representations of the power of a point.

Modifying the Secant-Secant Theorem: The

Tangent-Secant Theorem

Since my goal was to formulate theorems related to

the secant-secant theorem, I asked students what other

theorems could be generated from this theorem. The

class listed the following possible cases to consider:

1. P is on the exterior

2. One secant and one tangent

3. Two tangents

4. P is on the circle

5. P is in the interior of the circle

We then proceeded to investigate the case when P

is an exterior point of a circle, one line is a secant, and

the other is a tangent. With my computer, I illustrated

the situation as D approaches C (see Figure 7a) and

01.2=OP cm

00.31=OA cm

04.322 =⋅OAOP cm2

Technology to Unify Power of Point Theorems

16

asked students to predict the relationship PA·PB =

PC·PD when line PC (or PD ) is a tangent line to the

circle. Most students predicted that PA·PB = 2PC (or

2PD ). To further test their conjecture, I had my

students open a file containing a pre-constructed

configuration to illustrate the “secant-tangent” situation

(see Figure 7b). After testing our conjecture for several

cases by dragging point P and varying the size of the

circle (see Figure 7c), students were confident that the

conjecture was true and, therefore, that we could prove

it.

Since ∆APD approaches ∆APC (see Figures 7a and

7b), I was expecting students would use the similarity

of ∆APC and ∆CPB to prove the tangent-secant

conjecture. However, only two students thought of

using the fact that ∆APC ~ ∆CPB (see Figure 8a) to

prove our conjecture. Since I wanted to unify the two

theorems (the secant-secant theorem and the tangent-

secant theorem), I illustrated on my computer how, as

line PC approaches a tangent line, ∆APD approaches

∆APC (see Figure 8b). All students were able to justify

that ∆APC ~ ∆CPB by the AA similarity theorem and

derived the tangent-secant relationship. Initially two

students measured angles ACP∠ and CBP∠ to

convince themselves that those angles are congruent.

Eventually both of them “saw” why they are

congruent: By the inscribed angle theorem

( )⌢

ACCBPm21=∠ and, by the semi-inscribed angle

theorem, ( )⌢

ACACPm21=∠ . We formulated our

theorems as follows:

(iii) The tangent-secant theorem: Let P be an

exterior point of a circle. If a secant PA and

a tangent PC intersect the circle at points A,

B, and C, respectively, then PA·PB = 2PC .

(iv) If P is an exterior point and PA is a tangent

line of a circle with point of tangency A, then

the power of the point is = 2PA .

C

A

P

B

D

A

C

P

B

A

C

P

B

(a) (b) (c)

Figure. 7: Discovering the tangent-secant theorem.

A

C

P

B

C

A

P

B

D

(a) (b)

Figure 8. ∆APD approaches to ∆APC as C and D get closer.

96.0=PA cm

47.2=PB cm

36.2=⋅PBPA cm2

09.1=PA cm 42.2=⋅PBPA cm2

22.2=PB cm 42.22 =PC cm2

56.1=PC cm

José N. Contreras

17

The dynamic geometry environment facilitated our

examination of what varied and what remained

invariant as one secant line approached and eventually

became a tangent line. Students gained experience with

a second representation of the power of a point. They

were also able to see similarities and differences

between the new proof and the proof for the secant-

secant theorem.

Modifying the Secant-Secant Theorem: The

Tangent-Tangent Theorem

Our next task was to investigate the case when

both lines are tangent (see Figure 9a). I asked students

to conjecture a new relationship by applying our

knowledge of the power of a point to Figure 9a. One

student said that PA = PC but he was unable to explain

the connection between this relationship and the

tangent-secant theorem. He could only say that the

figure suggests such a relationship. As a hint, I used

the tangent-secant configuration, dragging point B until

it got close to point A (see Figure 9b), and asked

students what would happen when PA becomes a

tangent. After some reflection, two students were able

to deduce that PA = PC. One of the arguments was as

follows: By the secant-tangent theorem, P(P) = 2PA

and P(P) = 2PA , so 2PA = 2PC . After taking the

square root of both expressions, we got PA = PC. We

formulated our theorem as follows:

(v) Let P be an exterior point of a circle. If PA

and PC are tangent lines to the circle, with

tangency points A and C, then PA = PC (see

Figure 9a).

To illustrate the interconnectedness of these

mathematical theorems, I challenged my students to

find as many additional proofs as they could that

PCPA = . As a group, students provided two more

proofs, which refer to the diagram in Figure 10.

C

A

P

A

C

P

B

(a) (b)

Figure. 9: Discovering the tangent-tangent theorem.

C

A

OP

Figure 10. Diagram students used to prove PCPA =

Sketch of proof 1. Since lines PA and PBC are

tangent lines, they are perpendicular to the radii that go

through their points of tangency. Therefore, triangles

∆AOP and ∆COP are right triangles. Since AO = CO

(by definition of a circle), ∆AOP ≅ ∆COP by the

Hypotenuse-Leg congruence criterion. As a

consequence, AP = CP.

Sketch of proof 2: As in proof 1, ∠OAP and

∠OCP are right angles. In addition AO = CO. Since O

is equidistant from the sides of ∠APC, it belongs to its

angle bisector. Therefore, PCO is the angle bisector of

∠APC, which means that CPOAPO ∠≅∠ . We

conclude that ∆AOP ≅ ∆COP by the AAS congruence

criterion. By definition of congruent triangles,

CPAP = .

O

A

P

A

P

C

C

Technology to Unify Power of Point Theorems

18

Since one of my objectives was to unify the

theorems related to the power of a point, I asked

students to prove that PA = PC by modifying the proof

for the tangent-secant theorem. Since ∆APC ~ ∆CPB

and points A and B collapse into one point, all of the

students were able to see that ∆APC ~ ∆CPA. Some

students established that 22 PCPA = using the

proportionPA

PC

CP

AP= , another established directly that

PCPA = using the proportion 1==CA

AC

CP

AP, and

others used the fact that ∆APC ≅ ∆CPA by the ASA

congruence criterion. Finally, following my

suggestion, the class proved that PCPA = using the

converse of the isosceles triangle theorem since

∠PAC ≅∠PCA.

C

A

P

Figure 11. The tangent diagram.

GSP was a powerful pedagogical tool because it

allowed students to adapt the proof of the tangent-

secant theorem to develop another proof of the tangent-

tangent theorem. They were able to dynamically see

how the two original triangles were continuously

transformed into a single triangle.

GSP was a powerful pedagogical tool because it

allowed students to adapt the proof of the tangent-

secant theorem to develop another proof of the tangent-

tangent theorem. They were able to dynamically see

how the two original triangles were continuously

transformed into a single triangle.

The Secant-Secant Theorem Again: The Chord

Theorem

As we continued working towards the unification

of all the theorems related to the power of a point, I

had my students consider the case when P is an interior

point of the circle and both lines are secant to the given

circle (see Figure 12a). The theorem states:

(vi) If AB and CD are two chords of the same

circle that intersect at P, then PA·PB =

PC·PD.

By now, all of my students were able to predict

that PA·PB = PC·PD. As I expected, all but two

students proved this relationship by using the fact that

∆APD ~ ∆CPB (see Figure 12b).

C

AP

B

D

C

AP

B

D

(a) (b)

Figure 12. Proving that PA·PB = PC·PD using ∆APD ~ ∆CPB.

A

P

C

José N. Contreras

19

The Investigation Concludes: The Unification and

Another Discovery

At this point, the investigation took another

unexpected turn: Two students proved the power-of-a-

point relationship using triangles ∆ACP and ∆DBP

(see Figure 13a). At that time, it occurred to me that

this proof could be extended to the other cases, so I

challenged the class to adapt the proof to the other

situations. While there were no changes for the

tangent-secant theorem and the tangent-tangent

theorem, all of my students were challenged by the

secant-secant theorem (see Figure 13b).

Some students argued that the proof could not be

adapted to the secant-secant theorem because triangles

∆ACP and ∆DBP did not look similar. I myself was

not sure whether triangles ∆ACP and ∆DBP were

similar. Based on visual clues, one student thought that

BDPACP ∆∆ ~ , but another student refuted her

proposition because lines AC and BD are not

necessarily parallel. To investigate whether triangles

∆ACP and ∆DBP were similar, we measured their

angles and, to our surprise, we found that

DBPACP ∠≅∠ and ∠CAP ≅ ∠BDP. Our next task

was to explain these congruencies. After some

reflection and discussion, and without my guidance, a

student concluded that ∠CAP ≅ ∠BDP if and only if

°=∠+∠ 180)()( CABmBDPm . Since we had not

proved that angles ∠BDP and ∠CAB are

supplementary, I challenged the class to prove their

claim. Some students were able to prove the claim

using the inscribed angle theorem as follows:

°==

+=∠+∠

° 180

)()()()(

2360

21

21

⌢⌢

BDCmCABmCABmBDPm

We stated our theorem as follows:

(vii) The opposite angles of a cyclic quadrilateral

are supplementary.

C

A

P

B

D

C

A

P

B

D

(a) (b)

Figure 13. Triangles ∆ACP and ∆DBP support our theorems.

We concluded our investigation of the power of the

point by combining our theorems into one theorem that

we called the power-of-the-point theorem:

(viii) Let C be a circle and P be any point not on

the circle. If two different lines PA and PC

intersect the circle at points A and B, and C

and D, respectively, then PA·PB = PC·PD.

In addition, we came back to our formula for the

power of a point in terms of its distance to the center of

the circle and the radius of the circle:

(ix) The power of a point with respect to a circle

with center O and radius r is 22 rOP − .

GSP was instrumental in investigating the

possibility of developing a second proof for the secant-

secant theorem based on two triangles that did not look

similar to us at first sight. GSP motivated us to

question our initial impression that the triangles are

non-similar and to go beyond empirical evidence to

justify mathematically why those two triangles are

similar.

We then discussed why textbooks presented the

four theorems (secant-secant, secant-tangent, tangent-

tangent, and chord-chord) separately if they could be

stated as a single theorem. My goal was to help my

students recognize that our knowledge of a

mathematical theorem deepens as we discover or come

to know the new relationships or patterns that emerge

Technology to Unify Power of Point Theorems

20

in special cases of a theorem. If we do not make

explicit that the four theorems can be unified, we tend

to learn each one as a separate, compartmentalized

theorem. As a consequence, we may fail to remember

one case (e.g., the tangent-secant case) even when we

know another case (e.g., the secant-secant case).

Discussion and Conclusion

In the power of the point investigation, we used the

power of the dynamic, dragging, computational,

graphing, and measurement features of GSP to

discover and unify several theorems related to the

power of a point. We all discovered some theorems.

my students, under my guidance, discovered the main

theorems related to the power of a point and the

supplementary property of the opposite angles of a

cyclic quadrilateral, and I discovered alongside my

students the formula of the power of a point in terms of

both the distance from the point to the center of the

circle and the length of the radius of the circle. In

addition, we unified the five main power-of-a point

theorems. As I have shown, GSP was an essential

pedagogical tool that was instrumental in our

investigation.

I used GSP as a pedagogical tool in three main

ways: as a management tool, a motivational tool, and a

cognitive tool (Peressini & Knuth, 2005). As a

management tool, GSP allowed us to perform the

investigation more efficiently and accurately avoiding

computational errors and imprecise drawings and

measurements associated with lengthy paper and pencil

constructions needed to examine multiple examples.

As a motivational tool, GSP enhanced our dispositions

to perform the investigation. The dynamic and

interactive capabilities of GSP allowed us to follow our

intuitions, question our predispositions, and test the

resulting conjectures easily and accurately. As a

cognitive tool, GSP provided an environment in which

all of us were active in the process of learning the

concepts and procedures at hand. We were able to

actively represent and manipulate this abstract

geometric object in a hands-on mode. As we

experienced first hand the meaning of the power of a

point, we reflected on the factors that influenced its

behavior. As a result of our actions and reflections, we

constructed a more powerful abstraction of this

concept, and, thus, we developed a deeper

understanding of it.

Understanding the unification of the four theorems

is important from both pedagogical and mathematical

perspectives. From a pedagogical point of view,

understanding the relationships among different

representations of mathematical theorems and concepts

helps us to generate the special cases, to remember the

different forms that a theorem can take, to reduce the

amount of information that must be remembered, to

facilitate transfer to new problem situations, and to

believe that mathematics is a cohesive body of

knowledge (Hiebert & Carpenter, 1992). From a

mathematical point of view, doing mathematics

involves discovering special relationships as well as

unifying known theorems. Even concepts that are

apparently different can be unified when examined

from another viewpoint. For example, from the

perspective of inversion theory, lines and circles are

the same type of geometric objects. Yet, from a

Euclidean point of view, the circles and lines are

absolutely different geometric entities. In our case, the

power of a point P with respect to a circle with center

O and radius r is the product of two directed distances

from P to any two points A and B of the circle with

which it is collinear. By allowing A = B, the theorem is

transformed into useful instances from which we

derive special and useful corollaries. By considering

the case when points P, A, B and O are collinear, we

obtain another useful instance of the theorem (i.e.,

P(P) = 22 rOP − ).

In this mathematical investigation, students

experienced learning mathematical concepts with a

specific piece of technology. They were engaged in the

process of constructing mathematical knowledge by

discovering and justifying their conjectures and

making sense of classmates’ explanations. They

justified their conjectures not only with the

technological tool (i.e., testing a conjecture for several

instances), but also with mathematical theory (i.e.,

justifying why a conjecture is plausible and proving

that a theorem is true). By learning mathematical

concepts within technology environments, these future

teachers further developed not only specific content

knowledge but also their conceptions about the nature

of mathematical activity and their pedagogical ideas

about learning mathematics with technology. They

deepened their knowledge of the connections among

the various special cases of the secant-secant theorem.

They experienced that doing mathematics involves

formulating and testing conjectures and

generalizations, as well as discovering and proving

theorems. From a pedagogical point of view, these

future teachers experienced what it means to teach and

learn mathematics within IGS environments. The

students take a more active role in their own learning

under the guidance of the teacher whose main

responsibility becomes facilitating. Making

connections among mathematical ideas is a powerful

José N. Contreras

21

tool for prospective teachers’ learning that they can

transfer to their own teaching practice.

REFERENCES

Battista, M. (1998). Shape makers: Developing geometric

reasoning with the Geometer’s Sketchpad. Berkely, CA: Key

Curriculum Press.

Battista, M. (1999). The mathematical miseducation of Americaʼs

youth. Phi Delta Kappan, 80, 424–433.

Battista, M. (2007). Learning with understanding: Principles and

processes in the construction of meaning for geometric ideas.

In W. G. Martin & M. E. Strutchens (Eds.), The Learning of

Mathematics, 69th Yearbook of the National Council of

Teachers of Mathematics (pp. 65–79). Reston, VA: National

Council of Teachers of Mathematics.

Clements, D. H., Sarama, J., Yelland, N. J., & Glass, B. (2008).

Learning and teaching geometry with computers in the

elementary and middle school. In M. K. Heid & G. H. Blume

(Eds.), Research on technology and the teaching and learning

of mathematics: Vol. 1. Research syntheses (pp. 109–154).

Charlotte, NC: Information Age.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with

understanding. In D. A. Grouws (Ed.), Handbook of research

on mathematics teaching and learning (pp. 65–97). Reston,

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Hollebrands, K. (2007). The role of a dynamic software program

for geometry in the strategies high school mathematics

students employ. Journal for Research in Mathematics

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Hollebrands, K., Conner, A., & Smith, R. C. (2010). The nature of

arguments provided by college geometry students with access

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and the learning of geometry at the secondary level. In M. K.

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teaching and learning of mathematics: Vol. 1. Research

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with formal proof through computer-integrated teaching

experiments. In O. Zaslavsky (Ed.), Proceedings of the 23rd

conference of the International Group for the Psychology of

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Technion.

Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry

contexts. In C. Mammana & V. Villani (Eds.), Perspectives on

the teaching of geometry for the 21st century: An ICMI study

(pp. 121–128). Dordrecht, The Netherlands: Kluwer.

Jackiw, N. (2001). The Geometer’s Sketchpad. Software. (4.0).

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Koedinger, K. (1998). Conjecturing and argumentation in high

school geometry students. In R. Lehrer & D. Chazan (Eds.),

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Mahwah, NJ: LEA.

Laborde, C. (1998). Visual phenomena in the teaching/learning of

geometry in a computer-based environment. In C. Mammana

& V. Villani (Eds.), Perspectives on the teaching of geometry

for the 21st century: An ICMI study (pp. 113–121). Dordrecht,

The Netherlands: Kluwer.

Laborde, C., Kynigos, C., Hollebrands, K., & Sträβer, R. (2006).

Teaching and learning geometry with technology. In A.

Gutiérrez & P. Boero (Eds.), Handbook of research on the

psychology of mathematics education: Past, present, and

future (pp. 275–304). Rotterdam, The Netherlands: Sense.

Laborde, J., & Bellemain, F. (1994). Cabri Geometry II. Dallas,

TX: Texas Instruments.

Peressini, D. D., & Kuth, E. J. (2005). The role of technology in

representing mathematical problem situations and concepts. In

W. J. Masalski (Ed.), Technology-supported mathematics

learning environments, 67th Yearbook of the National Council

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National Council of Teachers of Mathematics.

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Macmillan.

The Mathematics Educator

2011, Vol. 21, No. 1

22

The Mathematics Educator

2011, Vol. 21, No. 1, 23–31

23

Aspects of Calculus for Preservice Teachers

Lee Fothergill

The purpose of this study was to compare the perspectives of faculty members who had experience teaching

undergraduate calculus and preservice teachers who had recently completed student teaching in regards to a first

semester undergraduate calculus course. An online survey was created and sent to recent student teachers and

college mathematics faculty members who had experience teaching a first semester calculus course to help

determine the aspects of calculus that they deemed most important in the teaching of calculus to pre-service

mathematics teachers. Faculty members with experience teaching at the secondary level, faculty members

without experience teaching at the secondary level, and recent student teachers’ survey results were compared

and there were some notable differences between the groups. The aspect that was ranked the highest among all

groups was problem solving which is consistent with the views of major mathematical organizations, such as

the Mathematical Association of America (MAA) and National Council of Teachers of Mathematics (NCTM).

While all groups’ views were similar and consistent with research, recent student teachers’ responses suggest

that when preparing future teachers in undergraduate calculus, more emphasis should be placed on connections

to the secondary curriculum and applications in technology.

Since Calculus is an undergraduate entry-level

course for many fields of study, instruction is

generally not geared toward preservice

mathematics teachers. This raises the question

whether this type of learning environment is

conducive to the preparation of a secondary

mathematics teacher. Originally a doctoral

dissertation (Fothergill, 2006), this study examines

mathematics faculty and student teacher responses

to a survey designed to obtain their perceptions of

a theoretical first-semester undergraduate calculus

course specifically designed for preservice

secondary mathematics teachers. While many

aspects of student understanding of calculus have

been researched, this study examines the aspects

to be emphasized in an undergraduate calculus

course designed for preparing preservice

mathematics teachers.

Background

According to the United States Department of

Education (2000), the demand for certified

mathematics teachers is growing at a quicker rate

than the supply. Moreover, Brakke (2000) argued

that to increase the interest in the mathematics

field, higher education must help improve the

quality of K-12 mathematics education programs.

The National Research Council (NRC, 1989)

stated, “No reform of mathematics education is

possible unless it begins with the revitalization of

undergraduate mathematics in both curriculum

and teaching style” (p. 39). While reform in

undergraduate mathematics has started, it has not

gone far enough to incorporate the needs of

preservice mathematics teachers.

As stated by Ferrini-Mundy and Findell

(2001) and Clemens (2001) mathematics faculty

ignored the needs of the preservice mathematics

teachers who were becoming an increasing part of

their department. Though mathematics faculty

focus on mathematics content, Wu (2011) claimed

that they should also focus on the professional

development of future teachers. According to a

RAND Corporation funded Mathematical Study

Panel (Ball, 2003), preservice mathematics

teachers should be prepared for teaching which is

completely different from preparing students to

conduct mathematical research. The report did not

advocate less rigor; instead, it suggested that

preservice teachers needed preparation for the

specific mathematical demands they will face in

the K-12 classroom.

Dr. Lee Fothergill joined the division of Mathematics and

Computer Science at Mount Saint Mary College in Newburgh, NY

following ten years of classroom teaching at the secondary level.

His research interests include the role that mathematics faculty

have in teacher preparation and the connection between

undergraduate and secondary mathematics curricula.

Calculus for Preservice Teachers

24

The Conference Board of Mathematical Sciences

Report (2001) stated that the mathematics department

is partially responsible for the education of

mathematics teachers. Similarly, the NRC (2001)

recommended that mathematics departments assume

greater responsibility for offering courses that provide

preservice mathematics teachers with appropriate

content that is taught using the kinds of pedagogical

approaches that preservice mathematics teachers

should model in their own classrooms. Papick (2011)

suggested the need for specialized courses for future

teachers that address the connection of mathematical

ideas to the topics that are taught in K-12 mathematics

classrooms. According to Bell, Wilson, Higgins and

McCoach (2011), professional development for

inservice teachers has been shown to include

illustrations of pedagogy and connections across

mathematics concepts which lead to growth in

mathematical knowledge for teaching; therefore

undergraduate courses that reflect these qualities

should be available for preservice teachers.

Rationale

With calculus being the capstone course for

mathematics studied at the secondary level, it is

important that preservice teachers have a strong

mathematical teaching knowledge of calculus.

Although not all preservice teachers will teach calculus

at the high school level, it is still imperative that they

understand how the content they are responsible for

teaching relates to their students’ further study in

mathematics. The U.S. Department of Education

(2000, 2002) stated that highly qualified teachers need

to have a deep understanding of subject matter to be

successful in the classroom. This requires developing

teachers who are independent learners who can read,

write, and communicate mathematics. It can be argued

that a teacher with these qualities will be more

confident in making curriculum decisions. Since

calculus is often a preservice teacher’s first college

mathematics course, it is reasonable to study how we

can improve the teaching of calculus to influence the

preparation of preservice mathematics teachers.

The purpose of this study is to explore the

perspectives regarding aspects of calculus that

mathematics faculty and student teachers deem

important and, therefore, that should be

emphasized in an undergraduate calculus course

for preservice teachers. The study was based on

the following questions:

1. What aspects of a calculus course do

undergraduate professors deem most important

when preparing preservice mathematics

teachers?

2. What aspects of calculus do student teachers

deem most important in preparing them to

teach at the secondary level?

Methods

Both faculty and recent student teachers responded

to a survey to rank aspects of calculus they deemed

most important to the undergraduate mathematics

preparation of preservice teachers. Faculty members

who had experience teaching undergraduate calculus

were chosen for the study. In addition, some had

experience teaching secondary mathematics, but this

was not one of the study selection criteria. Recent

student teachers’ perspectives are of interest because,

with their fresh experience in the classroom, and not so

distant experience in a calculus course, they can

discern how their calculus course helped or did not

help them in becoming a secondary mathematics

teacher. Therefore, they can give insightful

recommendations for a calculus course designed

specifically for secondary education mathematics

students.

Survey Development

Recommendations from major mathematical

organizations were used to determine aspects of

calculus that should be emphasized and included in the

survey. The Mathematics Education of Teachers, a

Conference Board of Mathematical Sciences (CBMS)

report (2001), gives specific recommendations for the

mathematical content and pedagogy for the preparation

of secondary school mathematics teachers. It gives the

most detailed outline of the college-level mathematics

that secondary school teachers should be studying and

recommends that preservice teachers’ undergraduate

study should develop:

1. Deep understanding of the fundamental

mathematical ideas in grades 9-12 curricula

and strong technical skills for application of

those ideas.

2. Knowledge of the mathematical

understandings and skills that students acquire

in their elementary and middle school

experiences, and how they affect learning in

high school.

3. Knowledge of mathematics that students are

likely to encounter when they leave high

Lee Fothergill

25

school for collegiate study, vocational training,

or employment.

4. Mathematical maturity and attitudes that will

enable and encourage continued growth of

knowledge in the subject and its teaching. (p.

122)

The report summarizes the benefits of the study of

calculus for preservice secondary level mathematics

teachers, recommending that first year mathematics

education majors take calculus because:

Calculus instructors can provide a useful

perspective for future high school teachers by

giving more explicit attention to the way that

general formulations about functions are used to

express and reason about key ideas throughout

calculus. Its central concepts, the derivative and

the integral, are conceptually rich functions. (p.

133)

More generally, the report suggests the following

goals for the study of mathematics: developing

mathematical maturity, understanding functions, and

having a deep understanding of mathematical ideas and

the skills needed to apply those ideas.

This CBMS report (2001) is aligned with the

National Council of Teachers of Mathematics (NCTM)

standards (2000) and the Undergraduate Programs and

Courses in the Mathematical Sciences 2004 CUPM

Curriculum Guide (Barker, Bressoud, Epp, Ganter,

Haver, & Pollatsek). The NCTM process standards

(2000) include problem solving, reasoning and proof,

connections within and outside mathematics, and

representations of functions. The CUPM curriculum

guide, which helps mathematics departments in

designing undergraduate curricula, recommends

making connections, developing mathematical

thinking, and using a variety of technological tools as

goals for undergraduate calculus. These

recommendations together with trends in calculus

textbooks (Stewart, 2003; Strauss, 2002), informed the

list of aspects that should be used when teaching

calculus to preservice teachers. The survey included

the following aspects:

• proof writing skills using formal definitions

and theorems;

• mathematical reasoning and problem solving

skills;

• strengthen the students’ algebraic skills;

visualization of functions and multiple

representations of functions;

• mathematical maturity and prepares students

for upper-level mathematics;

• mathematical-based technology skills (i.e.

graphing calculator and calculus based

software programs);

• connection between undergraduate

mathematics and high school mathematics

curriculum; and

• application to fields outside of mathematics

Both the faculty and pre-service teacher survey obtained demographics such as professional

backgrounds, gender, years of experience, and highest

degree obtained, as well as opinions about what aspects

of calculus they considered important when teaching

calculus to preservice teachers. The survey student

teachers asked them to rank the top three aspects of an

undergraduate calculus course that would be most

beneficial to pre-service mathematics teachers. In

addition, the student teachers were asked open-ended

questions about their experience in calculus and how it

related to their first teaching experience. Faculty

participants’ survey asked them to rank in order of

importance what they thought were the top three

aspects of calculus that help preservice teachers

become effective educators of secondary school

mathematics. Both faculty and student teacher

participants were asked to give any suggestions for the

creation of a calculus course for preservice teachers.

Participants and Data Collection

The online survey was sent via e-mail to

mathematics departments’ faculty members from four-

year colleges and universities in the United States that

were randomly selected from a list maintained by

University of Texas at Austin (2005). Colleges were

chosen at random and then all faculty from the

institution was emailed. The e-mail explicitly

requested faculty members that had experience

teaching undergraduate calculus to complete the online

survey. However, since the survey was sent to all

faculty members, it was inevitable that faculty

members without experience teaching calculus were

contacted.

Less than ten percent of the fifteen hundred faculty

members responded, which can be partially attributed

to the likelihood that many of the faculty members that

were e-mailed did not fit the survey criteria. Although

the low response rate could impact the validity and

reliability of the study, the response rate is much

higher if we disregard faculty members who were

Calculus for Preservice Teachers

26

invited to participate but did not meet survey criteria.

Hence, these responses can provide useful information

in regard to aspects of calculus that future and current

educators deem important.

Former student teachers who had completed

student teaching within the last year were sent an

online survey. Using the University of Texas at

Austin’s (2005) website, the researcher chose schools

at random and emailed college representatives from

either mathematics or education departments at over

300 four-year colleges and universities in the United

States. The college representative consisted of one of

the following: a mathematics department chairperson,

mathematics education chairperson, secondary

education chairperson, or student teacher supervisor. In

some instances, more than one representative was e-

mailed from each school. The email asked the college

representative to forward the online survey link to

secondary mathematics education students who

completed their student teaching practicum in the past

year. The response rate cannot be determined because

college representatives did not report how many recent

student teacher received the survey link and each

school has a different number of mathematics

education students each year.

Data Analysis

The aspects of calculus that faculty and student

teachers ranked the highest most often were identified

as the aspects that should be emphasized when

teaching calculus to future mathematics teachers. For

each aspect the percentage of respondents ranking it

first, second, or third most important was calculated.

To investigate potential differences, responses from

faculty with secondary teaching experience were

compared against those without such experience.

Lastly, responses from faculty with and without

secondary teaching experience were compared with

student teacher responses.

Results

The 114 faculty respondents consisted of 88 males

and 26 females, with a mean of 20.1 years teaching

experience. Eighty-five faculty members did not have

experience teaching at the secondary level, while

twenty-nine did have experience. Fifty-seven student teachers responded with 17 being male and 40 female.

Faculty Members

Figures 1 and 2 illustrate the overall results of the

online survey given to faculty members.

Overwhelmingly, problem solving received the highest

number of responses with 68 out of 114, approximately

60%, of the faculty members choosing it as the most

important aspect of calculus that should be emphasized

in a calculus course designed for preservice

mathematics teachers. Visualization of functions and

applications outside of mathematics were also

frequently selected. The aspects that received the least

number of responses were technology skills, proof

writing skills, and connection to the HS curriculum.

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

Proof Writing

Skills

Problem

Solving

Algebraic

Skills

Visualization

of Functions

Mathematical

Maturity

Technology

Skills

Connection to

HS

Curriculum

Applications

Outside of

Mathematics

Figure 1. Faculty Members percentage of (n = 114)1st,

2nd, and 3rd ranked aspects.

0

20

40

60

80

100

120

Proof Writing

Skills

Problem

Solving

Algebraic

Skills

Visualization

of Functions

Mathematical

Maturity

Technology

Skills

Connection to

HS

Curriculum

Applications

Outside of

Mathematics

Figure 2. Faculty members (n = 114) 1st, 2nd, and 3rd ranked aspects.

The examiner combined all first, second, and third

ranked responses selected for each aspect as shown in

chart 2. For clarity, problem solving received 68, 20,

and 15 responses respectively for first, second, and

third ranking; therefore, problem solving received a

combined response of 104 out of 114 faculty members.

Problem solving had the most combined responses

with approximately 91% of the faculty members

choosing this aspect as one of their top three that they

believe should be emphasized in a calculus course for

1st

2nd

3rd

1st

2nd

3rd

Lee Fothergill

27

preservice mathematics teachers. Visualization of

functions and applications outside the mathematics

curriculum were other top combined responses,

approximately 61% and 56% respectively.

Recent Student Teachers

Figures 3 and 4 illustrate their responses were

similar to faculty with problem solving, visualization of

functions, and applications outside of mathematics

being the aspects of calculus they most often deemed

important. A notable difference was that so few of the

recent student teachers considered proof writing skills

important; only four ranked it among their top three.

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

40.0%

Proof Writing

Skills

Problem

Solving

Algebraic

Skills

Visualization

of Functions

Mathematical

Maturity

Technology

Skills

Connection to

HS

Curriculum

Applications

Outside of

Mathematics

Figure 3. Student teachers percentage of (n = 57)1st, 2nd, and 3rd ranked aspects.

0

5

10

15

20

25

30

35

40

45

50

Proof Writing

Skills

Problem

Solving

Algebraic

Skills

Visualization

of Functions

Mathematical

Maturity

Technology

Skills

Connection to

HS

Curriculum

Applications

Outside of

Mathematics

Figure 4. Student teachers (n = 57)1st, 2nd, and 3rd ranked aspects.

Faculty Members With and Without Experience Teaching at the Secondary Level

To investigate possible differences in their

perspectives, the author then divided the faculty

members into two categories: faculty members with

experience teaching at the secondary level and without

experience teaching at the secondary level, hereafter

referred to as faculty with experience and faculty

without experience. The faculty member with no

experience teaching at the secondary level consisted of

68 males and 17 females and had a mean of 20.8 years

experience teaching calculus. The faculty members that

had experience teaching at the secondary level

consisted of 20 males and 9 females, with a mean of

18.1 years experience teaching calculus. Problem solving was chosen by both groups as an

important aspect to emphasize when teaching calculus

to preservice mathematics teachers (see Figure 5). The

chart demonstrates that 92.9% of the faculty without

experience teaching at the secondary level and 86.2%

of the faculty with experience teaching at the

secondary level had selected problem solving as one of

their top three aspects of calculus. Faculty members

with experience had a higher percentage of responses

in visualization of functions, algebra skills, technology

skills, connections to the high school curriculum, and

mathematical maturity as compared to faculty

members without experience. The greatest difference

occurred in the category of visualization of function;

72.4% of faculty with experience had this aspect in

their top three, but only 56.5% faculty without

experience listed it in their top three. It should also be

noted that 10.3% of faculty with experience thought

that connection to high school curriculum was the most

important aspect, whereas not one faculty member

without experience chose that as the most important

aspect.

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

100.0%

Proof Writing

Skills

Problem

Solving

Algebraic

Skills

Visualization

of Functions

Mathematical

Maturity

Technology

Skills

Connection to

HS

Curriculum

Applications

Outside of

Mathematics

Figure 5. Faculty members with (n = 29) and without

(n = 85) teaching experience at the secondary level,

combined 1st, 2nd, and 3rd rankings.

Faculty without experience

Faculty with experience

1st

2nd

3rd

1st

2nd

3rd

Calculus for Preservice Teachers

28

Faculty Members vs. Student Teachers

Figure 6 compares the results of all three groups.

While some aspects seem to have similar results, one

aspect that demonstrated a difference in perceptions

between faculty members with and without experience

and the student teachers was connection to the high

school curriculum. Connection to high school

curriculum was chosen by 8.2% of faculty members

without experience teaching at the secondary level as

one of their top three aspects. Student teachers had

more than doubled the percentage of faculty members

without experience with 17.5% of them choosing the

connection to the high school curriculum as an

important aspect.

Technology skills were chosen by 4.7% of the

faculty members without experience at the secondary

level as one of their top three aspects. In contrast,

10.3% of faculty members with experience put

technology skills as one of their top three aspects more

than doubling that of faculty members without

experience. Moreover, 21.1% of student teachers put

technology skills into their top three aspects making

this percentage four times higher than that of faculty

members without experience.

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

100.0%

Proof Writing

Skills

Problem

Solving

Algebraic

Skills

Visualization

of Functions

Mathematical

Maturity

Technology

Skills

Connection to

HS

Curriculum

Applications

Outside of

Mathematics

Figure 6. Faculty members with (n = 29) and without

(n = 85) teaching experience at the secondary level and

student teachers combined 1st, 2nd, and 3rd rankings.

Discussion

It is interesting to note that faculty and student

teachers agreed with the highest five aspects to be

emphasized in a calculus course designed for

preservice secondary mathematics teachers (see Table

1). With major mathematics affiliations such as the

MAA and NCTM promoting problem solving, it is no

surprise that problem solving was ranked by both the

faculty members and student teachers as the most

important aspect to be emphasized in a first semester

undergraduate calculus course designed for preservice

mathematics teachers.

However, an argument can be made that

undergraduate calculus is not meeting all the needs of

prospective secondary mathematics teachers. While

student teacher perceptions agreed with the faculty’s in

most aspects, student teachers ranked technology skills

and connections to secondary curriculum higher than

did faculty. Since faculty perceptions differ from the

student teachers in these aspects, faculty members may

not be meeting these needs. These results indicate that

preservice teachers value making connections to the

mathematics they will be teaching and that to better

meet their needs college should put greater emphasis

on making connections to the secondary curriculum

and technology in their coursework for preservice

teachers.

Table 1

Comparison of Faculty and Student Teacher Top Three Responses Combined

Faculty

Student

teachers

1. Problem solving 91.2% 77.2%

2. Visualization of functions 60.5% 65.0%

3. Applications outside of

mathematics 56.1% 43.9%

4. Mathematical maturity 37.7% 36.8%

5. Algebraic skills 28.1% 31.6%

6. and 7. Proof writing skills 14.0% 7.0%

7. Connections to HS curriculum 9.6% 17.5%

7 and 6. Technology skills 6.1% 21.1%

Recommendations

A first calculus course can provide an initial

training ground for preservice teachers. It may benefit

colleges with a large secondary mathematics education

population to develop a calculus course designed

specifically for preservice mathematics teachers, so

that vertical connections can be made between high

school and college level mathematics. This can provide

prospective teachers with content knowledge, as well

Faculty without experience

Faculty with experience

Student teachers

Lee Fothergill

29

as pedagogical knowledge that can be used in their

future secondary teaching.

There are many connections that can be made

while teaching calculus to preservice teachers and

these connections need to be explicit. When taught at

the secondary level, logarithmic functions may seem

an abstract concept with limited application. Hence,

when teaching logarithmic differentiation to preservice

teachers, the instructor can make explicit reference to

logarithmic functions and rules of logarithmic

expressions taught at the secondary level. The process

of finding the n-th derivative of the sine function is

similar to finding the value of i to the n-th power, a

common part of algebra in secondary mathematics

curriculum. The instructor can use the derivative to

connect the concept of finding a relative minimum or

maximum value of a function to the concept of finding

the derivation of the formula for the axis of symmetry

of a parabola.

Calculus is the culminating course of high school

mathematics; therefore, preservice teachers should

have a deep understanding of this content. As the

instructors for this course, mathematics faculty

members have a responsibility for preparing future

teachers. Mathematics faculty members teaching

calculus to future teachers should be teaching in a way

that meets the needs of their students and helps them

develop as professional educators.

Limitations & Further Research

While this study suggests that there are

differences in perspectives on calculus between faculty

members and future teachers, further research is still

needed. One might argue that student teachers may not

have enough experience to connect what they learned

in a calculus course to the high school curriculum.

Student teachers have a somewhat limited experience

at the secondary level and their student teaching

experiences can vary greatly. Some may say it is too

early in their teaching career to make judgments about

what is needed in a calculus course for preservice

teachers. On the other hand, the student teachers’

responses mostly matched the faculty responses and

established research, lending credence to their

perceptions of their learning needs. In future studies

one might include more experienced inservice teachers

who are more familiar with what makes teachers

successful and who are better able to reflect on their

learning of calculus. Further research could also

include how other undergraduate courses, required for

preservice teachers, such as linear algebra, abstract

algebra, and geometry could be modified to benefit

them.

REFERENCES

Ball, D. L. (2003). Mathematical proficiency for all students:

Toward a strategic research and development program in

mathematics education. Santa Monica, CA: RAND

Corporation.

Barker, W., Bressoud, D., Epp, S.,Ganter, S., Haver, B.,&

Pollatsek, H. (2004). Undergraduate programs and courses in

the mathematical sciences: CUPM curriculum guide 2004.

Washington, DC: MAA

Bell, C., Wilson, S., Higgins, T., & McCoach, D. (2011).

Measuring the effects of professional development on teacher

knowledge: The case of developing mathematical ideas.

Journal for the Research in Mathematics Education, 41, 497–

512.

Brakke, D. F. (2000). Higher education and its responsibility to K-

12 schools – the essential pipeline for future scientists,

mathematicians, and engineers. AWIS Magazine, 29, 32–33.

Clemens, H. (2001). The mathematics-intensive undergraduate

major. In CUPM discussion papers about mathematics and the

mathematical sciences in 2010: What should students know?

(pp. 21–30). Washington, DC: Mathematical Association of

America.

Conference Board of the Mathematical Sciences. (2001). The

mathematical education of teachers. Providence, RI &

Washington, DC: American Mathematical Society and

Mathematical Association of America.

Ferrini-Mundy, J., & Findell, B. R. (2001). The mathematical

education of prospective teachers of secondary school

mathematics: Old assumptions, new challenges. In CUPM

discussion papers about mathematics and the mathematical

sciences in 2010: What should students know? (pp. 31–41).

Washington, DC: Mathematical Association of America

Fothergill, Lee. (2006). Calculus for preservice teachers: Faculty

members' and student teachers' perceptions. Un published

doctoral Ddissertation), Teachers College Columbia

University, New York.

National Council of Teachers of Mathematics. (2000). Principles

and standards for school mathematics. Reston, VA: Author.

National Research Council. (2001). Educating teachers of science,

mathematics, and technology: New practices for the new

millennium. Washington, DC: National Academy Press.

National Research Council. (1989). Everybody counts: A Report to

the Nation on the Future of Mathematics Education.

Washington, DC: National Academy Press.

Papick, Ira. J. (2011). Strengthening the mathematical content

knowledge of middle and secondary school mathematics

teachers. Notices of the American Mathematical Society, 58,

389–392.

Stewart, J. (2003). Calculus: Early transcendentals (5th ed.).

Brooks/Cole: Pacific Grove, CA.

Strauss, M., Bradley, G., & Smith, K. (2002). Calculus (3rd ed.).

Upper Saddle River, NJ: Prentice Hall.

Triesman, U. (1992). Studying students studying calculus: A look

at the lives of minority mathematics students in college.

College Mathematics Journal, 23, 362–372.

Calculus for Preservice Teachers

30

The University of Texas at Austin, (2005). Universities: by state.

Retrieved from

http://www.utexas.edu/world/univ/state/.

U. S. Department of Education. (2000). Before it's too late: A

report to the nation from The National Commission on

Mathematics and Science Teaching for the 21st Century.

Retrieved from http://www.ed.gov/americacounts/glenn/

U.S. Department of Education, Office of Postsecondary Education

(2002). Meeting the highly qualified teachers challenge: The

secretary's annual report on teacher quality. Washington, DC.

Wu, H. (2011). The mis-education of mathematics teachers.

Notices of the American Mathematical Society, 58, 372–383.

Lee Fothergill

31

APPENDIX

Faculty Member Survey

Gender: M or F Years teaching Calculus: __________

Do you have experience teaching at the secondary level? : _________

Highest Degree Earned: _________

Please rank the following statements about aspects of calculus that you believe helps pre-service teachers become

effective educators of secondary school mathematics. Please put 1 next to the most important, 2 next to the second

most important, and 3 next to the third most important.

____ Calculus helps to develop proof writing skills using formal definitions and theorems.

____ Calculus helps to develop mathematical reasoning and problem solving skills.

____ Calculus strengthens the students’ algebraic skills.

____ Calculus helps develop an understanding and visualization of functions and multiple representations of

functions.

____ Calculus builds mathematical maturity and prepares students for upper-level mathematics.

____ Calculus facilitates the development of mathematical-based technology skills (i.e. graphing calculator and

calculus based software programs).

____ Calculus demonstrates a connection between undergraduate mathematics and high school mathematics

curriculum.

____ Calculus provides insight into its application to fields outside of mathematics.

Please indicate any other aspect that you believe help pre-service teachers.

Do you feel your answers would differ, if asked about non-mathematics education majors?

Student Teachers Survey

Gender: ______

Please rank the following statements about aspects of calculus that you believe helps pre-service teachers become

effective educators of secondary school mathematics. Please put 1 next to the most important, 2 next to the second

most important, and 3 next to the third most important.

____ Calculus helps to develop proof writing skills using formal definitions and theorems.

____ Calculus helps to develop mathematical reasoning and problem solving skills.

____ Calculus strengthens the students’ algebraic skills.

____ Calculus helps develop an understanding and visualization of functions and multiple representations of

functions.

____ Calculus builds mathematical maturity and prepares students for upper-level mathematics.

____ Calculus facilitates the development of mathematical-based technology skills (i.e. graphing calculator and

calculus based software programs).

____ Calculus demonstrates a connection between undergraduate mathematics and high school mathematics

curriculum.

Please indicate any other aspect that you believe help pre-service teachers.

The Mathematics Educator

2011, Vol. 21, No. 1

32

The Mathematics Educator

2011, Vol. 21, No. 1, 33–47

33

1 The work on this curriculum development and research was supported by the National Science Foundation under Grant No.

DUE 0442319 and DUE 0817253 awarded to North Carolina State University. Any opinions, findings, and conclusions or

recommendations expressed herein are those of the authors and do not necessarily reflect the views of the National Science

Foundation. More information about the project and materials can be found at http://ptmt.fi.ncsu.edu.

Enhancing Prospective Teachers’ Coordination of Center and Spread: A Window into Teacher Education Material

Development1

Hollylynne S. Lee & J. Todd Lee

This paper describes a development and evaluation process used to create teacher education materials that help

prepare middle and secondary mathematics teachers to teach data analysis and probability concepts with

technology tools. One aspect of statistical reasoning needed for teaching is the ability to coordinate

understandings of center and spread. The materials attempt to foster such coordination by emphasizing

reasoning about intervals of data rather than a single focus on a point estimate (e.g., measure of center). We take

a close look at several different data sources across multiple implementation semesters to examine prospective

mathematics teachers’ ability to reason with center and spread in a coordinated way. We also look at the

prospective teachers’ ability to apply their understandings in pedagogical tasks. Our analysis illustrates the

difficulty in both achieving this understanding and transferring it to teaching practices. We provide examples of

how results were used to revise the materials and address issues of implementation by mathematics teacher

educators.

Data analysis, statistics, and probability are

becoming more important components in middle and

high school mathematics curricula (National Council

of Teachers of Mathematics, 2000; Franklin et al.,

2005). Therefore, university teacher educators are

challenged with how to best prepare prospective

mathematics teachers to teach these concepts. The

challenge is exacerbated by the fact that many of these

prospective teachers have not had meaningful

opportunities to develop an understanding of pivotal

statistical and probabilistic ideas (e.g., Stohl, 2005).

Although simulation and data analysis tools—graphing

calculators, spreadsheets, Fathom, TinkerPlots,

Probability Explorer—may be available in K-12

classrooms, there is a need for high quality teacher

education curriculum materials. Such curriculum

materials can help teacher educators become

comfortable with and incorporate tools for teaching

probability and data analysis. These teacher education

curricula need to primarily aim for prospective teachers

to develop a specific type of knowledge related to

statistics that includes a deeper understanding of: (a)

data analysis and probability concepts, (b) technology

tools that can be used to study those concepts, and (c)

pedagogical issues that arise when teaching students

these concepts using technology (Lee & Hollebrands,

2008b; Lesser & Groth, 2008).

The authors of this paper are part of a team

engaged in a teacher education materials development

project, funded by the National Science Foundation, to

create units of course materials—modules with about

18-20 hours of class materials with additional

assignments—to integrate technology and pedagogy

instruction in various mathematical contexts. The

project intends to create three modules that could be

distributed separately and used in mathematics

education methods courses, mathematics or statistics

content courses for teachers, or professional

development workshops focused on using technology

to teach mathematics and statistics. The modules are

not designed for teachers to use directly with their

students. Rather, the developers anticipate that after

using the materials teachers will have the knowledge

needed to create their own technology-based activities.

The three modules will focus on the teaching and

Dr. Hollylynne Stohl Lee is an Associate Professor of Mathematics

Education at North Carolina State University. Her research

interests include the teaching and learning of probability and

statistics with technology.

Dr. J. Todd Lee is a Professor of Mathematics at Elon University.

He is interested in undergraduate mathematics education,

including the probability and statistics learning of pre-service

teachers.

Coordination of Center and Spread

34

learning of data analysis and probability, geometry,

and algebra.

The first module focuses on learning to teach data

analysis and probability with technology tools,

including TinkerPlots, Fathom, spreadsheets, and

graphing calculators (Lee, Hollebrands, & Wilson,

2010). This module is designed to support a broad

audience of prospective secondary teachers. For many

prospective teachers, engaging in statistical thinking is

a different process than that which they have been

engaged in teaching and learning mathematics (e.g.,

delMas, 2004). Thus, it is important to engage

prospective teachers as active learners and doers of

statistics. The module incorporates several big ideas

that can support teachers as they learn to teach data

analysis and probability: engaging in exploratory data

analysis; attending to distributions; conceptually

coordinating center and spread in data and probability

contexts; and developing an understanding of, and

disposition towards, statistical thinking as different

from mathematical thinking. For this paper, we focus

solely on one of these big ideas as we discuss the

material development process using the following

guiding question: How can we use technology tools to

enhance prospective mathematics teachers’

coordination of center and spread? We analyzed

several forms of data to revise the teacher education

materials. The results provide insight into ways

prospective mathematics teachers may reason about

center and spread in a coordinated way.

Why Focus on Coordinating Center and Spread?

Coordinating measures of center and spread has

been identified as a central reasoning process for

engaging in statistical reasoning (e.g., Friel, O’Connor,

& Mamer, 2006; Garfield, 2002; Shaughnessy, 2006).

In particular, Garfield (2002) noted that part of

reasoning about statistical measures is “knowing why a

good summary of data includes a measure of center as

well as a measure of spread and why summaries of

center and spread can be useful for comparing data

sets” (Types of Correct and Incorrect Statistical

Reasoning section, para. 11).

Single-point indicators, used as a center of a

distribution of data (e.g., mean or median) or as an

expected value of a probability distribution, have been

over-privileged in both mathematics curricula

(Shaughnessy, 2006) and statistical research methods

(Capraro, 2004). When used with samples, single-point

central indicators may not be accurate signals of what

is likely an underlying noisy process (Konold &

Pollatsek, 2002). Many others argue that attending to

variation is critical to developing an understanding of

samples and sampling distributions (e.g., Franklin et al,

2005; Reading & Shaughnessy, 2004; Saldanha &

Thompson, 2002; Shaughnessy, 2006).

Understanding variability, both within a single

sample and across multiple samples, can be fostered

through attending to intervals: Intervals embody both

central tendency and spread of a data set (Reading &

Shaughnessy, 2004). Attending to intervals aligns well

with the many voices of concern in professional

communities on the limitation of null hypothesis

significance testing, which rely on single-point p-

values. For example, the medical industry has taken

major moves toward examining and reporting data

through alternative tools, confidence intervals being

foremost (Gardner & Altman, 1986; International

Committee of Medical Journal Editors, 1997). Other

areas, such as psychology, ecology, and research in

mathematics education, are also moving in this

direction (Capraro, 2004; Fidler, 2006).

When describing expected outcomes of a random

process, interval thinking can make for a powerful,

informative paradigm shift away from single-point

estimates. Statistics education researchers have

advocated this shift in focus (e.g., Reading &

Shaughnessy, 2000, 2004; Watson, Callingham, &

Kelly, 2007). For example, in a fair coin context,

describing the number of heads that may occur when

tossing a coin 30 times is better described as “typically

about 12 to 18 heads” rather than “we expect 15

heads.” The latter statement does not acknowledge the

variation that could occur. As Reading and

Shaughnessy (2000, 2004) have noted, many students

will initially provide single point values in tasks asking

for expectations from a random process, but this is

likely related to the common use of such questions as

“‘What is the is the probability that …?’ Probability

questions just beg students to provide a point-value

response and thus tend to mask the issue of the

variation that can occur if experiments are repeated”

(p. 208, Reading & Shaughnessy, 2004). Explicitly

asking for an interval estimate may illicit a classroom

conversation that focuses students’ attention on

variation.

Prospective and practicing teachers have

demonstrated difficulties similar to middle and high

school aged students in the following areas:

considering spread of a data set as related to a measure

of center (Makar & Confrey, 2005), appropriately

accounting for variation from an expected value

(Leavy, 2010), and a tendency to have single-point

value expectations in probability contexts (Canada,

2006). Thus, there is evidence to suggest mathematics

Hollylynne S. Lee & J. Todd Lee

35

educators should help prospective teachers develop an

understanding of center and spread that can allow them

and their students to reason appropriately about

intervals in data and chance contexts. The aim of our

materials development and evaluation efforts reported

in this paper is to document one attempt to foster such

reasoning and to reflect upon how the evaluative

results informed improving the materials and

suggestions for future research.

Design Elements in Data Analysis and Probability

Module

From 2005 to 2009, the Data Analysis and

Probability module materials for prospective secondary

and middle mathematics teachers were developed,

piloted, and revised several times. To facilitate

understanding of measures of center and spread in a

coordinated way, Lee et al. (2010) attempted to do the

following:

1. Emphasize the theme of center and spread

throughout each chapter in the material, with the

coordination between the two explicitly discussed

and emphasized through focused questions

covering both content and pedagogical issues.

2. Use dynamic technology tools to explore this

coordination.

3. Place the preference for intervals above that of

single-point values even if the construction of

these intervals is reliant upon measures of center

and spread.

Lee et al., with consultation from the advisory board

and a content expert, attempted to attend to these

elements, along with other design elements aimed at

developing prospective teachers’ understanding of data

analysis and probability, technology issues, and

appropriate pedagogical strategies. A discussion of the

design of the entire module as it focuses on developing

technological pedagogical content knowledge for

statistics is discussed in Lee and Hollebrands (2008a,

2008b).

Methods

The project team followed curricular design and

research method cycles as proposed by Clements

(2007), including many iterations of classroom field-

testing with prospective teachers, analysis of field-

testing data, and subsequent revisions to materials. Our

primary research site, a university in the Southeast

region of the US, has consistently implemented the

module in a course focused on teaching mathematics

with technology serving third- and fourth-year middle

and secondary prospective teachers and beginning

graduate students who need experience using

technology. A typical class has between 13 and 19

students. In Fall 2005, during the five-week data

analysis and probability module, the instructor used the

pre-existing curriculum for the course to serve as a

comparison group to the subsequent semesters. The

students took a pretest and posttest designed to assess

content, pedagogical, and technology knowledge

related to data analysis and probability.

In each of the subsequent semesters from 2006-

2007, the same instructor as in Fall 2005 taught a draft

of the five-week Data Analysis and Probability module

from our textbook (Lee et al., 2010) with a request that

the curriculum be followed as closely as possible. In

addition, the module was implemented in a section of

the course taught by a different instructor, one of the

authors of the textbook, in Spring 2007. During the

first two semesters of implementation, class sessions

were videotaped and several students were

interviewed. In the first three semesters of

implementation, written work was collected from

students and pre- and post-tests were given. Since

2007, many other instructors have used the materials at

institutions across the US and improvements and slight

modifications were made based on instructor and

student feedback, with final publication in 2010 (Lee et

al.).

For this study, we are using several sources of data

for our analysis of how prospective teachers may be

developing a conceptual coordination between center

and spread in data and probability contexts, with a

particular focus on interval reasoning. Our data sources

include: (a) examples of text material from the module,

(b) a video episode from the first semester of

implementation in which prospective teachers are

discussing tasks concerning probability simulations, (c)

prospective teachers’ work on a pedagogical task, and

(d) results from the content questions on the pre- and

post-tests across the comparison and implementation

semesters through Spring 2007.

Analysis and Results

We discuss the analysis and results according to

the four data sources we examined. In each section we

describe the analysis processes used and the associated

results.

Emphasis in Materials: Opportunities to Learn

To begin our analysis, we closely examined the

most recent version of the text materials for

opportunities for prospective teachers to develop a

coordinated conceptualization between center and

Coordination of Center and Spread

36

spread. The materials begin by helping prospective

teachers informally build and understand measures of

center and spread in the context of comparing

distributions of data (Chapter 1) and then explore a

video of how middle grades students compare

distributions (Chapter 2).

In Chapter 3, prospective teachers consider more

deeply how deviations from a mean are

used to compute measures such as variation and

standard deviation. In Chapter 4, the materials build

from this notion in a univariate context to help students

consider measures of variation in a bivariate context

when modeling with a least squares line. The focus on

spread and useful intervals in a distribution continues

in Chapters 5 and 6 where prospective teachers are

asked to describe distributions of data collected from

simulations, particularly attending to variation from

expected values within a sample, and variation of

results across samples. These last two chapters help

prospective teachers realize that smaller sample sizes

are more likely to have results that vary considerably

from expected outcomes, while larger sample sizes

tend to decrease this observed variation.

We only considered places in the text materials

where the authors had made an explicit reference to

these concepts in a coordinated way as opportunities

for prospective teachers to develop a conceptualization

of coordinating center and spread. We closely

examined the text materials to identify instances where

there was an explicit emphasis placed on coordinating

center and spread in (a) the written text and technology

screenshots, (b) content and technology tasks, and (c)

pedagogical tasks. One researcher initially coded each

instance throughout the textbook, the researchers then

conferred about each coded instance to ensure that both

agreed that an instance was legitimate. We tallied the

final agreed-upon instances in each chapter as

displayed in Table 1. We also specifically marked

those instances addressing coordinating center and

spread that placed special emphasis on promoting

interval reasoning as displayed in Table 1. For an

example of instances coded as focused on interval

reasoning, see Table 2. The point of this content

analysis was to identify where and how often the

authors of the materials had actually provided

opportunities for prospective teachers to coordinate

center and spread and engage in reasoning about

intervals. This analysis could also point out apparent

gaps where opportunities may have been missed to the

author team.

As seen in Table 1, every chapter contained

content and technology tasks as well pedagogical tasks

that emphasized the coordination of center and spread.

This coordination was discussed in the text along with

any diagrams and technology screenshots in all but

Chapter 2 (which is a video case with minimal text),

with slightly heavier emphases in Chapters 4 and 5.

Chapters 5 and 6 have the most content and technology

tasks focused on coordinating center and spread. Of

particular importance is that an explicit focus on

interval reasoning only appears in Chapter 1, 5, and 6,

with Chapter 5 containing a particularly strong

emphasis. Although evidence suggests the design of

the materials provides opportunities to build

understanding of center and spread throughout,

attention to this in the early versions of the materials is

uneven, particularly in terms of emphasizing interval

reasoning.

Table 1

Instances in Module of Coordinating Center and Spread

Instances of coordinating center and spread

Text

Content &

technology. task

Pedagogical

task

Percent of

instances with

focus on interval

reasoning

Ch 1: Center, Spread, & Comparing Data Sets 3 5 2 50%

Ch 2: Analyzing Students’ Comparison of Two Distributions using

TinkerPlots

0 2 2 0%

Ch 3: Analyzing Data with Fathom 2 5 3 0%

Ch 4: Analyzing Bivariate Data with Fathom 5 3 3 0%

Ch 5: Designing and Using Probability Simulations 4 13 4 76%

Ch 6: Using Data Analysis and Probability Simulations to

Investigate Male Birth Ratios

1 15 1 59%

Hollylynne S. Lee & J. Todd Lee

37

Table 2

Examples of Instances in Materials Coded as Opportunities to Coordinate Center and Spread and Promote Interval

Reasoning

Written text and screenshots Content and technology tasks Pedagogical tasks

Students may attend to clumps and gaps in

the distribution or may notice elements of

symmetry and peaks. Students often

intuitively think of a “typical” or “average”

observation as one that falls within a modal

clump…Use the divider tool to mark off an

interval on the graph where the data appear

to be clumped.

(Chapter 1, Section 3, p. 11)

Q17: Use the Divider tool and the Reference

tool to highlight a clump of data that is

“typical” and a particular value that seems to

represent a “typical” salary. Justify why your

clump and value are typical. (Chapter 1,

Section 3, p. 13)

Q19: How can the use of the dividers to

partition the data set into separate regions be

useful for students in analyzing the spread,

center and shape of a distribution? (Chapter

1, Section 3, p. 14)

In our context, we are interested in how

much the proportion of freshmen returning to

Chowan College will vary from the expected

50%. To examine variation from an expected

proportion, it is useful to consider an interval

around 50% that contains most of the sample

proportions.

(Chapter 5, Section 3, p.102)

Q11. Given a 50% estimate for the

probability of retention, out of 500 freshmen,

what is a reasonable interval for the

proportion of freshmen you would expect to

return the following year? Defend your

expectation. (Chapter 5, Section 3, p. 100)

Q16. If we reduced the number of trials to

200 freshmen, what do you anticipate would

happen to the interval of proportions from

the empirical data around the theoretical

probability of 50%? Why? Conduct a few

samples with 200 trials and compare your

results with what you anticipated. (Chapter 5,

Section 3, p. 103)

Q19. Discuss why it might be beneficial to

have students simulate the freshman

retention problem for several samples of

sample size 500, as well as sample sizes of

200 and 999. (Chapter 5, Section 3, p. 103)

[Implied emphasis on interval reasoning

because it is one of the follow-up questions

to Q16.]

Classroom Episode from Chapter 5

Because Chapter 5 contained the largest focus on

coordinating center and spread via interval reasoning,

we analyzed a 2.5 hour session of a class engaging in

Chapter 5 material from the first implementation cycle.

The researchers viewed the class video several times

and critical episodes (Powell, Francisco, & Maher,

2003) were identified as those where prospective

teachers or the teacher educator were discussing

something that had been coded as an “instance” in

Chapter 5 as seen in Table 1. Each critical episode was

then more closely viewed to examine how the

reasoning being verbalized by prospective teachers or

the teacher educator indicated an understanding of

coordinating center and spread and the use of interval

reasoning.

It is not possible to present a detailed analysis of

the entire session; however we present classroom

discussions around several of the interval reasoning

tasks shown in Table 2. Consider the following

question posed in the text materials:

Q11: Given a 50% estimate for the probability

of retention, out of 500 freshmen, what is a

reasonable interval for the proportion of

freshmen you would expect to return the following year? Defend your expectation.

This question follows material on the technical

aspects of using technology to run simple simulations

and how to use these simulations as a model for real

world situations. Immediately prior to Question 11

prospective teachers are asked to write (but not run) the

commands needed on a graphing calculator that would

run multiple simulations of this scenario. In answering

Question 11, several prospective teachers propose three

intervals they considered to be reasonable for how

many freshmen out of 500 they expect to return the

following year at a college with a 50% retention rate;

230-270, 225-275, and 175-325. The teacher educator

asked a prospective teacher to explain his reasoning for

Coordination of Center and Spread

38

the interval 230-270. (T denotes teacher educator and

PT denotes a prospective teacher)

T: Can you tell me why you widened the

range?

PT1: I didn’t, I narrowed it

T: Tell me why you narrowed it

PT1: 500 is a big number. So I thought it

might be close to 50%.

T: So you thought because 500 is a big

number it would be closer to

PT1: Half

T: To half, closer to 50%. So, MPT1

[who proposed an interval of 175-

325], why did you widen the range?

This [pointing to 225-275 on board]

was the first one thrown out, why did

you make it bigger?

PT2: Well it’s all according to how long

you’re going to do the simulation.

T: Out of 500 students how many [slight

pause] what range of students will

return? Do you think it will be exactly

50% return?

PT2: Probably not

T: So for any given year, what range of

students might return, if you have 500

for ever year?

PT2: 175 to 325

T: Ok. So can you tell me why?

PT2: Without knowing anything I wouldn’t

go to a tight range.

T: Because you don’t have enough

information.

PT3: It’s like the coin flips; you have some

high and some low, so it might not fall

into the 225 to 275 interval.

PT4: I’d say it will most likely fall into that

first range, but it’s not a bad idea to be

safe and say it can go either way.

First, all intervals were given in frequencies, rather

than proportions. This is likely an artifact of the

wording of Question 11 during that implementation

cycle. In that version of the materials, the question did

not specifically use the word proportion. All intervals

suggested by the prospective teachers are symmetric

around an expected retention of 250 (50%) of 500

freshmen. Two of the intervals have widths less than

10% of the range, or a maximum variation of 5% from

the mean, while the largest proposed interval 175-325

suggests a variation of ±15%. The smaller intervals

have around 93% and 98% chances of containing the

future retention proportion, while the largest interval

will succeed with an almost mathematical certainty.

While one prospective teacher reasoned that 500 is a

large enough sample to expect values “close” to 50%,

another is much more tentative and casted a wider net

due to an uncertainty about the number of times the

simulation would be run. This prospective teacher, and

the two that responded afterward, may be trying to

capture all possible values, rather than consider a

reasonable interval that would capture most values. Or

they may merely be dealing with the difficulties of

estimating the binomial distribution of 500 trials. Only

one prospective teacher justified an interval by

explicitly reasoning from an expected value, and there

were no justifications. The teacher educator did not

question why the intervals were symmetric about the

expected value. The reasoning of the prospective

teacher is similar to that noticed by Canada (2006) in

his research with prospective elementary teachers.

Canada noted, “almost all of my subjects pointed out

that more samples would widen the overall range,

while very few subjects suggested that more samples

would also tighten the subrange capturing most of the

results” (p. 44).

After about 30 minutes of exploration using a

calculator to run simulations, the teacher educator

asked each prospective teacher to run two simulations

of the “50% retention rate of 500 freshmen” and

compute the proportion of freshmen returning. The

teacher educator collected and displayed this data as a

dot plot in Fathom (Figure 1). This is the second time

during this lesson the teacher educator used Fathom to

collect data from individual’s samples and display

them as a distribution. This teacher educator’s move

was not suggested in the curriculum materials;

however its value in indicating a public record and

display of pooled class data is duly noted and used in

revisions to suggest such a way to display class data in

aggregate form.

Hollylynne S. Lee & J. Todd Lee

39

Retetention

0.42 0.46 0.50 0.54

Freshman Classes Dot Plot

Figure 1. Distribution of 34 sample proportions pooled from class and displayed.

The plot in Figure 1 appears quite typical for what

might occur with 34 samples of 500, with a modal

clump between 0.48 and 0.51. The teacher recalls the

predicted intervals and asks:

T: If we take a look at the distribution of

this data in a graph [displays

distribution in Figure 1], is that kind of

what you would assume? We ran the

simulation of 500 freshman 34 times.

So we notice, we assumed 50%. Are we

around 50%? How many times are we

at 50%?

PT: One

T: Here are your predictions from earlier

on the number of students you might

see in a range [three proposed

intervals]. Our proportion range is about

from 0.44 to 0.53. Think any of these

ranges for the students are too wide or

too narrow…?

The teacher educator immediately drew attention

to the expected value of 50% and variation from that

expectation with comments of “around 50%” and “at

50%.” The conversation shifted as the teacher educator

appeared to draw their attention to the entire range of

proportion values, rather than on a modal clump

around the expected value. It appears that both the

teacher educator and the prospective teachers

interpreted the request for a “reasonable interval” in

the textbook question to mean the range of all sample

proportions likely to occur, or that do occur.

The discussion continued as the teacher educator

had the prospective teachers use an algorithm to

convert the proportion range, which was re-estimated

as 0.43-0.55, to frequencies 215-275 so they could

compare the predicted intervals. They noted the

similarity of the sample range to two of the proposed

intervals, and they noted that the range is not

symmetric around 0.5 and therefore is “not like we

thought” [FPT1]. The teacher educator then focused

the class back on the expected value of 50% and asked

why they did not get more samples with a retention of

50%. One prospective teacher offered a reason related

to a low sample size and another suggested the

graphing calculator’s programming may be flawed.

Another prospective teacher countered the idea:

PT: If it [graphing calculator] is

programmed to act randomly, it is not

going to recognize any particular value.

And it will..., point 5 is the theoretical

value. But the actual values don’t have

to be point 5, they should be close to

point 5, which most of them are.

The teacher educator did not pursue the

conversation about the graphing calculator, but instead

asked a question based on Question 16, as seen in

Table 2, and two questions that follow in the text. We

will use this conversation to consider how students

reason about the relationship between sample size and

variation from the expected center.

T: So let’s say instead of doing 500

freshmen, we would decrease this set to

200. How do you think the range might

differ, or if we increased to 999 how

might the range of proportions be

different?

PT: It would be narrower.

T: Narrower for which way, if we reduced

to 200 or increased to 999?

PT: 999

T: Why do you think it would be

narrower?

PT: The more trials there are, the closer it

will be to the true mean.

T: [Asks students if they agree, about half

the class raise their hand.]

… …. [Other prospective teachers make

similar comments.]

T: If we decrease to 200 trials in each

sample from 500 do you expect the

range to be similar or do you expect it

to be wider or narrower or similar??

PT: Wider. With a smaller sample you will

have more variability.

Coordination of Center and Spread

40

T: So you are going with the idea that a

smaller sample will have more

variability. Does everyone agree or

disagree? [many prospective teachers

say agree].

This episode suggests that at least some

prospective teachers were developing an understanding

of the relationship between the freshman class size and

the variation in the distribution of sample proportions

from repeated samples. This suggests that although

they may have not initially approached the task with an

expectation of an appropriate interval for what might

be typical, many came to reason, through the extended

activity and repeated simulations, that the reasonable

interval widths were affected by sample size. This

again aligns with Canada’s (2006) result that his

instructional intervention helped more of the

prospective elementary teachers consider the role of

sample size as an influence on the variation of results

around the expected value.

It seems as though explicitly asking about intervals

provided opportunities for class discussions that went

beyond the discussion of a single expected value, in

this case 50%. Such an opportunity can help develop

the notion that with random processes comes variation,

and that understanding how things vary can be

developed through reasoning about intervals rather

than merely point-estimates of an expected center

value. However, symmetry may well have been

strongly used due to the retention rate being 50%; it

may be beneficial to incorporate an additional question

using retention rates other than 50%.

Pedagogical Task Following Chapter 5

The ultimate goal of these materials is to develop

prospective teachers’ abilities to design and implement

data analysis and probability lessons that take

advantage of technology. Fortunately, there are many

opportunities within the materials to engage in

pedagogical tasks. One such task followed the

previously described prospective teachers’ work in

Chapter 5. As a follow-up to our examination of the

classroom interactions for Chapter 5, we examined

how these same prospective teachers may have applied

their developing understandings in a pedagogical

situation. The task describes a context in which college

students are able to randomly select from three gifts at

a college bookstore and then asks:

Explain how you would help students use

either the graphing calculator, Excel, or

Probability Explorer to simulate this context.

Explicitly describe what the commands

represent and how the students should

interpret the results. Justify your choice of

technology.

Of particular interest to us was whether

prospective teachers would plan to engage their

students in using large sample sizes, using repeated

sampling, and using proportions rather than

frequencies to report data. We also were interested in

whether they would promote or favor interval

reasoning in lieu of point-value estimates.

Each prospective teacher submitted a written

response to this task. Seventeen documents were

available for analysis. Each response was summarized

with respect to several categories: (a) which

technology was chosen and why, (b) how the tool

would generally be used, (c) what use was made of

sampling and sample size, (d) how representations for

empirical data would be used, and (e) what they want

students to focus on in their interpretation. The

summaries were used to identify patterns across cases

as well as interesting cases.

The majority chose to use a graphing calculator (10

of 17), only 5 of the 17 prospective teachers planned

experiences for their students that incorporated

repeated samples, and only 7 used proportions. In

addition, 10 prospective teachers focused explicitly on

a point estimate, one used both a point and interval

estimate for interpreting a probability, while six of the

responses to the task were not explicit enough to tell

what the prospective teacher intended. Thus, the

majority planned for students to simulate one sample

(sample sizes vary across lessons, but many were less

than 50) and to make a point estimate of the probability

from that sample.

The prospective teachers did not provide much

evidence, during the week immediately following their

discussion of the material in Chapter 5, that they were

able to transfer their developing understandings of

interval reasoning in a probability context to a

pedagogical situation. It seems that, for most, any

progress made during the class discussions did not

have a transference effect into their pedagogy.

Pre- and Post- Tests

Pre- and post-tests were used to create a

quantitative measure that might indicate prospective

teachers’ conceptual changes. The 20 questions

comprising the content section of the pre- and post-

assessment were selected from Garfield (2003) and

other items from the ARTIST database

(http://app.gen.umn.edu/artist/index). These items

assess general statistical reasoning concerning concepts

Hollylynne S. Lee & J. Todd Lee

41

included in the text materials (e.g., coordinating center

and spread, interpreting box plots, interpreting

regression results and correlations). These questions

were administered to the prospective teachers both

before and after the Data Analysis and Probability

module, and the scores were combined pair-wise as

normalized gains. By normalized gains, we mean the

percentage increase of a student’s available

advancement from the pre- to post-test (Hake, 1998).

Figure 2. Distribution of normalized gain scores for each group of prospective teachers.

The Comparison group (n=15) plot shows

normalized gains realized in Fall 2005 using the

traditional curricula for the course, prior to

implementation of the new materials. Compared

against this group are the normalized gains from three

different semesters (four total sections) in which the

materials were implemented. There were major

revisions to the text materials between Implementation

I (n = 18) and II (n = 15), but only minor edits before

Implementation III (n = 32, based on two sections).

However, prospective teachers in the Implementation

III group were the first that used the module as a

textbook for reference in and out of class. Other than

exposure to different curricula, it seems reasonable to

assume that the prospective teachers across all sections

came from the same population.

Visual inspection reveals a distinct increase in

gains in the implementation groups with respect to the

comparison group. The gains seem to translate by more

than 0.10, but we see little change in the amount of

variation in the inter-quartile ranges. This assessment

is in agreement with Monte Carlo permutation tests, n

= 50,000, comparing both means, p = .009, and

medians, p = .006, of the comparison group with those

of the pooled implementations. However, comparing

gains across the whole test is not part of our current

focus in this paper.

Looking at the normalized gain scores for the

entire content subsection of the test obscures the

performance on particular questions. Thus, we selected

and closely examined four questions from the test that

address various aspects of our focus on the

coordination of center and spread and the alternative

use of intervals (see Figure 3). In Table 3, we record

the percentage of students who answered the multiple

choice questions correctly on the pre- and post-test.

Coordination of Center and Spread

42

3. The Springfield Meteorological Center wanted to

determine the accuracy of their weather forecasts.

They searched the records for those days when the

forecaster had reported a 70% chance of rain. They

compared these forecasts to records of whether or

not it actually rained on those particular days. The

forecast of 70% chance of rain can be considered

very accurate if it rained on:

a. 95% - 100% of those days.

b. 85% - 94% of those days.

c. 75% - 84% of those days.

d. 65% - 74% of those days.

e. 55% – 64% of those days.

10. Half of all newborns are girls and half are boys. Hospital A records an average

of 50 births a day. Hospital B records an average of 10 births a day. On a

particular day, which hospital is more likely to record 80% or more female

births?

a. Hospital A (with 50 births a day)

b. Hospital B (with 10 births a day)

c. The two hospitals are equally likely to record such an event.

11. Forty college students participated in a study of the effect of sleep on test scores. Twenty of the students volunteered to stay up all

night studying the night before the test (no-sleep group). The other 20 students (the control group) went to bed by 11:00 pm on the

evening before the test. The test scores for each group are shown on the graph below. Each dot on the graph represents a particular

student’s score. For example, the two dots above 80 in the bottom graph indicate that two students in the sleep group scored 80 on

the test.

Examine the two graphs carefully. From the 6 possible conclusions listed below, choose the one with which you most agree.

a. The no-sleep group did better because none of these students scored below 35 and a student in this group achieved the highest score.

b. The no-sleep group did better because its average appears to be a little higher than the average of the sleep group.

c. There is no difference between the two groups because the range in both groups is the same.

d. There is little difference between the two groups because the difference between their averages is small compared to the

amount of variation in the scores.

e. The sleep group did better because more students in this group scored 80 or above.

f. The sleep group did better because its average appears to be a little higher than the average of the no-sleep group.

15. Each student in a class tossed a penny 50 times and counted the number of heads. Suppose four different classes produce graphs for the

results of their experiment. There is a rumor that in some classes, the students just made up the results of tossing a coin 50 times without

actually doing the experiment. Please select each of the following graphs you believe represents data from actual experiments of flipping a

coin 50 times.

a b.

c. d.

Figure 3. Sample pre- and post-test questions on center, spread, intervals, and variability.

Hollylynne S. Lee & J. Todd Lee

43

Table 3 Correct Response Rates on Four Test Questions.

Comparison

n = 15

Implementation I

n = 18

Implementation II

n = 15

Implementation III

n = 32

Question

Correct

Answer Pre Post Pre Post Pre Post Pre Post

3 d 47% 47% 44% 50% 53% 53% 53% 53%

10 b 40% 80% 44% 89% 33% 80% 38% 66%

11 d 53% 20% 11% 22% 33% 20% 25% 25%

15 b & d 47% 40% 56% 67% 40% 33% 41% 56%

Across all implementation semesters and the

comparison group, prospective teachers made little to

no improvement in their ability to interpret the

accuracy of a 70% probability in data as an interval

around 70% (Question 3, answer d), with only about

half of them correctly choosing the interval. Across all

semesters, there was also little change in prospective

teachers’ ability to recognize the two reasonable

distributions for a distribution of outcomes from

repeated samples of 50 coin tosses (Question 15,

answers b and d). As shown in response to Question 10

(answer b), prospective teachers appeared to improve

their ability to recognize sampling variability with

respect to sample size: They typically became more

likely to recognize that Hospital B, with the smaller

sample size, had a higher probability of having a

percent of female births much higher (80%) than an

expected 50%. Because the comparison group made

similar gains on Question 10 as those who had engaged

in using the new materials, it appears that merely

engaging in learning about data analysis and

probability may be helpful in one’s ability to correctly

respond to that question, regardless of curriculum

material.

For Question 11, there was very little change in the

percent of prospective teachers who correctly chose d

to indicate that there was little difference between the

groups with respect to center and the large spread, and

in fact most chose f, a comparison done only on a

measure of center. It is disappointing that more

prospective teachers did not demonstrate a

coordination of center and spread with this task on the

posttest. It is interesting that in the Comparison group,

about half initially reasoned correctly but that after

instruction the majority chose to make a comparison

based only on a measure of center (see Figure 3).

Perhaps the traditional curriculum placed a greater

emphasis on measures of center and decision-making

based on point estimates.

The main lesson we take from examining these

pre- and post-test questions is that our materials, as

implemented in 2006-2007, did not appear to

substantially help prospective teachers improve their

reasoning about center, spread, and intervals. For

although we realized gains in the overall scores on

statistical reasoning, a close look at four questions

demonstrates little change.

Discussion

How do these results help answer our question

about the task of developing prospective teachers’

ability to use a coordinated view of center and spread?

One design element used by Lee et al. (2010) was the

deliberate and consistent focus on the coordination of

center and spread. The module covers a broad range of

material, written by three authors through many

iterations and reviews from external advisors. Though

the theme of coordination was maintained throughout

the material, the emphasis was found to be quite

inconsistent across chapters in an early version of the

materials. Even more sporadic was the preference of

intervals over point values with half the chapters

excluding this theme. Even though the focus on

intervals and modal clumping was consistent in the

probability/simulation chapters, a few of the relevant

test questions did not indicate any gains beyond those

from general exposure to data and probability. To

ascertain if these themes can strengthen the intuitions

of clumping over point-value intuitions, the message

must be reemphasized throughout the material.

Coordination of Center and Spread

44

Prospective Teachers’ Developing Understandings

Developing a coordinated view of center and

spread, or expectation and variation, as others have

called it (e.g., Watson et al., 2007), is difficult. Watson

and her colleagues found that hardly any students from

ages 8 to14 used reasoning that illustrated a

coordinated perspective on expectation and variation in

interview settings. Although Canada’s (2006)

prospective teachers made gains during his course in

reasoning about intervals, it was not uncommon for the

teachers to still give single point estimates as expected

values. If students have difficulty in coordinating

center and spread, then it is important for both

prospective and in-service teachers to work towards

developing their own coordinated views in data and

chance settings.

There are not many studies that follow the

development of prospective teachers’ understandings

of statistical ideas into teaching practices. Batanero,

Godina, and Roa (2004) found that even when gains in

content knowledge were made during instruction on

probability, prospective teachers still prepared lesson

plans that varied greatly in their attention to important

concepts in probability. Lee and Mojica (2008)

reported that practicing middle school teachers, in a

course on teaching probability and statistics, exhibited

inconsistent understandings of probability ideas from

lessons in their classrooms. Thus, it is not surprising

that in such a short time period the prospective teachers

in our study did not develop their own understandings

in ways they could enact in pedagogical situations.

Leavy (2010) noted that a major challenge in statistics

education of prospective teachers is “the

transformation of subject matter content knowledge

into pedagogical content knowledge” (p. 49). Leavy

also noted in her study that prospective teachers who

were able to demonstrate a reasonably strong

understanding of informal inference, including

accounting for variation from expected outcomes, had

difficulties applying this knowledge to create informal

inference tasks to use with their own students.

Informing Revisions to Materials

In accordance with curriculum development and

research recommendations by Clements (2007), the

results discussed in this paper informed the next

iteration of revisions to the materials. Several questions

were revised throughout the text and additional

discussion points were inserted to help emphasize the

coordination of center and spread and to provide

additional opportunities for interval reasoning. For

example, a major change occurred in Chapter 1 with

regard to the focus on interval reasoning. Consider the

original questions on the left side of Table 4 with those

on the right. Fall 2007 Q17 asks prospective teachers

to simultaneously consider spread and center through

use of the divider and reference tools in TinkerPlots.

However, in recent revisions, the series of questions

was recast and developed into a series that first has the

prospective teachers consider intervals of interest in the

upper 50%, middle 50%, and then something they

deem to be a cluster containing many data points, i.e., a

modal clump. After the experience with intervals, they

are asked to use the reference tool to mark a point

estimate they would consider a “typical” value and to

reason how the shaded interval might have assisted

them. This series of questions puts much more explicit

attention on valuing intervals when describing a

distribution. The authors also added Q25, which

explicitly asks prospective teachers to consider how the

use a specific technology feature (dividers) can assist

students’ reasoning.

Other revisions made throughout the chapters

included minor wording changes that could shift the

focus of attention in answering the question. For

example the Fall 2007 version of Chapter 3 posed the

question:

Q9. By only examining the graphs, what would

you characterize as a typical City mpg for

these automobiles?

This question was revised:

Q9. By only examining the graphs, what would

you characterize as a typical range of City

mpg for these automobiles? [bolding added]

Informing Support for Faculty

Making changes in the text material is not

sufficient. Fidelity of implementation is important for

ensuring prospective teachers have opportunities to

attend to and discuss the major ideas in the materials.

The big statistical ideas in the text (e.g., exploratory

data analysis, distributions, variation, and coordinating

center and spread) need to be made explicit to the

course instructor through different avenues, such as a

facilitator’s guide or faculty professional development.

Such a guide has been developed and is available at

http://ptmt.fi.ncsu.edu. This guide includes discussion

points that should be made explicit by the instructor

and includes continual reference to the main ideas

meant to be emphasized in the materials. The guide,

along with faculty professional development, can

hopefully allow teacher educators to better understand

the intended curriculum and implement the materials

Hollylynne S. Lee & J. Todd Lee

45

with high fidelity. Faculty professional development

efforts have been established through free workshops

held at professional conferences and week-long

summer institutes. Evaluations of the week-long

summer institutes in 2009 and 2010 suggest that the

fifteen participants increased their confidence in their

ability to engage prospective teachers in discussions

about center and spread in a distribution, as well as

randomness, sample size and variability.

Future Directions

For this study, we did not examine other sources of

evidence of prospective teachers’ development of

understanding related to coordinating center and

spread. Such data may include prospective teachers’

responses to a variety of content and pedagogical

questions posed throughout the chapters and perhaps

pedagogical pre- and post-tasks such as interpreting

students’ work, designing tasks for students, creating a

lesson plan. In fact, teacher educators at multiple

institutions have collected sample work from

prospective teachers on tasks from each of the

chapters. Analysis of this data with a focus on

coordinating center and spread may yield additional

findings that can help the field better understand of the

development of prospective teachers’ reasoning about

center and spread.

Prospective teachers’ familiarity with expected

ranges of values, their propensity to use these ideas in

conceptual statistical tasks, and their pedagogical

implementation of coordination of center and spread

are three different phenomena. As shown in this work

and in other literature, the transference from the first of

these to the latter two is problematic. Future versions

of these materials may need to engage prospective

teachers’ further into the use of interval thinking about

expectation and variation in a broader range of

statistical tasks. More importantly, prospective teachers

will need to be more consistently challenged to

consider how to create tasks, pose questions, and

facilitate classroom discussions aimed at engaging

their own students in the coordination of center and

spread.

Table 4 Sample Revisions in Chapter 1 to Better Facilitate Interval Reasoning

Text of Questions in Fall 2007 Text of Questions in Fall 2009

Q16. What do you notice about the distribution of average salaries?

Where are the data clumped? What is the general spread of the

data? How would you describe the shape?

Q17. Use the Divider tool and the Reference tool to highlight a clump

of data that is “typical” and a particular value that seems to

represent a “typical” salary. Justify why you highlighted a

clump and identified a particular value as typical.

Q18. Drag the vertical divider lines to shade the upper half of the

data, which contains approximately 50% of the cases. Which

states are in the upper half of the average salary range? What

factors may contribute to the higher salaries in these states?

Q20. Create a fully separated plot of the Average Teacher Salaries.

Either stack the data vertically or horizontally. What do you

notice about the distribution of average salaries? Where are the

data clumped? What is the general spread of the data? How

would you describe the shape?

Q21. Use the Divider tool to shade the upper half of the data, which

contains approximately 50% of the cases. Which states are in

the upper half of the average salary range? What factors may

contribute to the higher salaries in these states?

Q22. Drag the vertical divider lines to shade the middle half of the

data, which contains approximately 50% of the cases. Describe

the spread of the data in the middle 50%. What might

contribute to this spread?

Q23. Drag the vertical divider lines to highlight a modal clump of

data that is representative of a cluster that contains many data

points. Explain why you chose that range as the modal clump.

Q24. Use the Reference tool to highlight a particular value that seems

to represent a “typical” salary. Justify why you identified a

particular value as typical and how you may have used the

range you identified as a modal clump to assist you.

Q25. How can the use of the dividers to partition the data set into

separate regions be useful for students in analyzing the spread,

center, and shape of the distribution?

Coordination of Center and Spread

46

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REVIEWERS FOR THE MATHEMATICS EDUCATOR, VOLUME 21, ISSUE 1

The editorial board of The Mathematics Educator would like to take this opportunity to

recognize the time and expertise our many volunteer reviewers contribute. We have listed below

the reviewers who have helped make the current issue possible through their invaluable advice

for both the editorial board and the contributing authors. Our work would not be possible without

them.

Shawn Broderick

Tonya Brooks

Victor Brunaud-Vega

Amber G. Candela

Zandra DeAraujo

Tonya DeGeorge

Christine Franklin

Brenda King

Ana Kuzle

Kevin LaForest

David R. Liss, III

Laura Lowe

Anne Marie Marshall

Kevin Moore

John Olive

Ronnachai Panapoi

Denise A. Spangler

Patty Anne Wagner

James Wilson

The University of Georgia

Samuel Cartwright

Fort Valley State University

Kelly Edenfield

Kennesaw State University

Ryan Fox

Penn. State, Abington

Brian Gleason

University of New Hampshire

Sibel Kazak

Pamukkale University

Yusuf Koc

Indiana University, Northwest

Terri Kurz

Arizona State University

Mara Martinez

University of Illinois at Chicago

Michael McCallum

Georgia Gwinnett College

Jennifer Mossgrove

Knowles Science Teaching Foundation

Anderson Hassell Norton, III

Virginia Tech

Molade Osibodu

African Leadership Academy

Ginger Rhodes

University of North Carolina, Wilmington

Behnaz Rouhani

Georgia Perimeter College

Kyle Schultz

James Madison University

Susan Sexton Stanton

East Carolina University

Diana Swanagan

Shorter University

If you are interested in becoming a reviewer for The Mathematics Educator, contact the Editors

at tme@uga.edu.

49

Manuscript Submission Guidelines

The Mathematics Educator (ISSN 1062-9017) is a biannual publication of the Mathematics Education

Student Association (MESA) at The University of Georgia and is abstracted in Zentralblatt für Didaktik der

Mathematik (International Reviews on Mathematical Education). The purpose of the journal is to promote the

interchange of ideas among students, faculty, and alumni of The University of Georgia, as well as the broader

mathematics education community.

The Mathematics Educator presents a variety of viewpoints within a broad spectrum of issues related to mathematics

education. Our editorial board strives to provide a forum for a developing collaboration of mathematics educators at

varying levels of professional experience throughout the field. The work presented should be well conceptualized;

should be theoretically grounded; and should promote the interchange of stimulating, exploratory, and innovative ideas

among learners, teachers, and researchers. The Mathematics Educator encourages the submission of a variety of types

of manuscripts from students and other professionals in mathematics education including:

• reports of research (including experiments, case studies, surveys, and historical studies),

• descriptions of curriculum projects, or classroom experiences;

• literature reviews;

• theoretical analyses;

• critiques of general articles, research reports, books, or software;

• commentaries on research methods in mathematics education;

• commentaries on public policies in mathematics education.

The work must not be previously published except in the case of:

• translations of articles previously published in other languages;

• abstracts of or entire articles that have been published in journals or proceedings that may not be easily

available.

Guidelines for Manuscript Specifications

• Manuscripts should be typed and double-spaced, 12-point Times New Roman font, and a maximum of 25

pages (including references and endnotes). An abstract (not exceeding 250 words) should be included and

references should be listed at the end of the manuscript. The manuscript, abstract, references and any pictures,

tables, or figures should conform to the style specified in the Publication Manual of the American

Psychological Association, 6th Edition.

• An electronic copy is required. The electronic copy must be in Word format and should be submitted via an

email attachment to tme@uga.edu. Pictures, tables, and figures should be embedded in the document and must

be compatible with Word 2007 or later.

• The editors of TME use a blind review process. Therefore, to ensure anonymity during the reviewing process,

no author identification should appear on the manuscript.

• A cover age should be submitted as a separate file and should include the author’s name, affiliation, work

address, telephone number, fax number, and email address.

• If the manuscript is based on dissertation research, a funded project, or a paper presented at a professional

meeting, a footnote on the title page should provide the relevant facts.

To Become a Reviewer

Contact the Editors at tme@uga.edu. Please indicate if you have special interests in reviewing articles that address

certain topics such as curriculum change, student learning, teacher education, or technology.

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51

The Mathematics Education Student Association is an official affiliate of the National Council of Teachers of Mathematics. MESA is an integral part of The University of Georgia’s mathematics education community and is dedicated to serving all students. Membership is open to all UGA students, as well as other members of the mathematics education community.

Visit MESA online at http://math.coe.uga.edu/Mesa/MESA.html

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52

In this Issue,

A Look Back…. Pólya on Mathematical Abilities JEREMY KILPATRICK Using Technology to Unify Geometric Theorems About the Power of a Point JOSÉ N. CONTRERAS Aspects of Calculus for Preservice Teachers LEE FOTHERGILL Enhancing Prospective Teachers’ Coordination of Center and Spread: A Window Into Teacher Education Material Development HOLLYLYNNE S. LEE & J. TODD LEE