v21n1_color_FINAL by COETME UGA

____ THE____________ _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 21 Number 1

Summer 2011

MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff

A Note from the Editors

Editors Allyson Thrasher Catherine Ulrich

Dear TME readers,

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

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 The University of Georgia Athens, GA 30602-7124

tme@coe.uga.edu www.ugamesa.org

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

The Mathematics Educator 2011, Vol. 21, No. 1, 3-8

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?

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.

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 about memory? Do you think mathematicians have a special memory? For mathematical things?

JK: What different kinds?

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

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: 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.

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: A specially organized memory? Do you think mathematicians have a memory that is organized in a different way?

JK: Yes, I know the book

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.

….

GP: If he were here, he would give you much better answers—anyway, more answers. He thinks 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.

GP: Yes, sure.

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.

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. 3

Mathematical Abilities

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,

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?

Nur die Fülle führt zur Klarheit, v

Und im Abgrund wohnt die Wahrheit.

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 4

say the same thing about—have you read it?— about Emmy Noether.vi

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

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: 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.

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

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

JK: Do you think it’s possible to develop somebody’s ability to solve problems?

JK: No, I haven’t seen that.

GP: I think so.

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?

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

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 5

Mathematical Abilities

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

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

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

JK: How?

GP: Oh, yes, now and then. Even this I describe somewhere in one of my papers.

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: 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, 7

Mathematical Abilities

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

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). George Pólya: Collected papers (Vol. 4: Probability; combinatorics; teaching and learning in mathematics). 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.

REFERENCES Descartes, R. (1701). Regulae ad directionem ingenii (Rules for the direction of the mind). In Des-Cartes Opuscula posthuma, physica & mathematica. Amsterdam, The Netherlands: P. & J. Blaeu. 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. Cambridge, England: Cambridge University Press. Hilbert, D., & Ackermann, H. (1928). Grundzüge der theoretischen Logik [Principles of mathematical logic]. Berlin, Germany: Springer. Hilbert, D., & Bernays, P. (1934. Grundlagen der Mathematik [Foundations of mathematics] (Vol. 1). Berlin, Germany: Springer. Hilbert, D., & Bernays, P. (1939). Grundlagen der Mathematik [Foundations of mathematics] (Vol. 2). Berlin, Germany: Springer. 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, 1973/1984. 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.” ii

The Mathematics Educator 2011, Vol. 21, No. 1

The Mathematics Educator 2011, Vol. 21, No. 1, 11–21

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. 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.

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

Technology to Unify Power of Point Theorems

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 PA = 1.60 cm, PC = 1.50 cm, PB = 3.30 cm. Justify your method. B A

P C D

Figure 1. The initial problem. 3.30 PD or one = 1.60 1.50 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

Some students used the proportion

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 3.30 PD did not hold. Another student relationship = 1.60 1.50 reached this conclusion by noticing that dragging point PB B changed PA, PB, and , but did not influence PC PA PB PC and PD. Therefore, the proportion did not = PA PD 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

PA = 1.60 cm

PB = 3.30 cm

PC = 1.50 cm

PA = 1.44 cm

PA ⋅ PBC = 5.30 cm2

PB = 3.68 cm

PC = 1.50 cm

PA ⋅ PBC = 5.30 cm2

B A

P C

C D

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

3(1) = 3

4(0) = 0

5(1) = 5

6(2) = 12

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

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 – r 2 or (r + 2)(r – 2) = r 2 – 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 – r2 and P(P) = r2 – OP2 . 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) = OP2 − r 2 . 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.

40 3

PA = 0.66 cm PB = 3.32 0.66 cm PA =cm PA ⋅ PB PB = 2= .3.32 18 cm cm2 PA·PB = 2.18 cm2 OA = 1.33 cm OA = 1.33 cm

30 2

20 1

B 2

P O

P 1

E 2

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

José N. Contreras

B A OPOP = 2=.2.01 01 cmcm = 1.00 cm OAOA = 31 .00 cm 2 2 2 2 OPOP ⋅2OA = 3 .04 cm -OA = 3.04 cm 2

P C D

B O

Figure 6. ∆APD ~ ∆CPB. -2

(i)

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 PA PD rewriting PA·PB = PC·PD as . This = PC PB 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:

(ii)

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. 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 15

Technology to Unify Power of Point Theorems

asked students to predict the relationship PA·PB =

∆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

PC·PD when line PC (or PD ) is a tangent line to the circle. Most students predicted that PA·PB = PC 2 (or PD 2 ). 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 tangentsecant theorem), I illustrated on my computer how, as

⌢

m(∠CBP ) = 12 AC and, by the semi-inscribed angle

⌢

theorem, m(∠ACP ) = 12 AC . 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 = PC 2 . (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 = PA2 .

line PC approaches a tangent line, ∆APD approaches

B A

A A

P D

C PA = 0.96 cm PB = 2.47 cm PA ⋅ PB = 2.36 cm2

(a)

PA = 1.09 cm PB = 2.22 cm PC = 1.56 cm

(b)

PA ⋅ PB = 2.42 cm2 PC 2 = 2.42 cm2

(c)

Figure. 7: Discovering the tangent-secant theorem.

B A

A P

D (a)

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

C (b)

José N. Contreras

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 secantsecant 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) = PA2 and P(P) = PA2 , so PA2 = PC 2 . 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 PA = PC . As a group, students provided two more proofs, which refer to the diagram in Figure 10.

B A

CC (a)

(b)

Figure. 9: Discovering the tangent-tangent theorem.

AA PP

O O

C C

Figure 10. Diagram students used to prove PA = PC

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 ∠APO ≅ ∠CPO . We conclude that ∆AOP ≅ ∆COP by the AAS congruence criterion. By definition of congruent triangles, AP = CP . 17

Technology to Unify Power of Point Theorems

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 PA2 = PC 2 using the AP PC proportion , another established directly that = CP PA AP AC PA = PC using the proportion = = 1 , and CP CA others used the fact that ∆APC ≅ ∆CPA by the ASA congruence criterion. Finally, following my suggestion, the class proved that PA = PC using the converse of the isosceles triangle theorem since ∠ PAC ≅ ∠ PCA.

AA PP

GSP was a powerful pedagogical tool because it allowed students to adapt the proof of the tangentsecant theorem to develop another proof of the tangenttangent 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 tangentsecant theorem to develop another proof of the tangenttangent 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).

CC Figure 11. The tangent diagram.

A D

D (a)

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

(b)

José N. Contreras

The Investigation Concludes: The Unification and Another Discovery At this point, the investigation took another unexpected turn: Two students proved the power-of-apoint 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 ∆ACP ~ ∆BDP , but another student refuted her

necessarily parallel. To investigate whether triangles ∆ACP and ∆DBP were similar, we measured their angles and, to our surprise, we found that ∠ACP ≅ ∠DBP 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 m(∠BDP ) + m(∠CAB ) = 180° . 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:

⌢

m(∠BDP) + m(∠CAB) = 12 m(CAB) + 12 m( BDC ) ° = 360 2 = 180°

We stated our theorem as follows: (vii) The opposite angles of a cyclic quadrilateral are supplementary.

proposition because lines AC and BD are not B

C P

P C

A D

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 OP 2 − r 2 .

GSP was instrumental in investigating the possibility of developing a second proof for the secantsecant 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, tangenttangent, 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 19

Technology to Unify Power of Point Theorems

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 20

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) = OP 2 â&#x2C6;&#x2019; r 2 ). 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 classmatesâ&#x20AC;&#x2122; 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

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, VA: National Council of Teachers of Mathematics. 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 Education, 38,164–192. Hollebrands, K., Conner, A., & Smith, R. C. (2010). The nature of arguments provided by college geometry students with access to technology while solving problems. Journal for Research in Mathematics Education, 41, 324–350. Hollebrands, K., Laborde, C., & Sträβer, R. (2008). Technology and the learning of geometry at the secondary level. In M. K. Heid & G. H. Blume (Eds.), Research on technology and the teaching and learning of mathematics: Vol. 1. Research syntheses (pp. 155–206). Charlotte, NC: Information Age. Hoyles, C., & Healy, L. (1999). Linking informal argumentation 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 Mathematics Education (pp. 105–112.) Haifa, Israel: 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). Emeryville, CA: KCP Technologies. Koedinger, K. (1998). Conjecturing and argumentation in high school geometry students. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 319–347). 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 of Teachers of Mathematics (pp. 277–290). Reston, VA: National Council of Teachers of Mathematics. Musser, G. L., & Trimpe, L. E. (1994). College geometry: A problem-solving approach with applications. New York, NY: Macmillan.

The Mathematics Educator 2011, Vol. 21, No. 1

The Mathematics Educator 2011, Vol. 21, No. 1, 23–31

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 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.

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.

Calculus for Preservice Teachers

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

school for collegiate study, vocational training, or employment.

•

mathematical maturity and prepares students for upper-level mathematics;

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:

•

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

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;

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 fouryear 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 25

Calculus for Preservice Teachers

Data Analysis

70.0% 60.0%

1st

50.0%

2nd 3rd

40.0% 30.0% 20.0% 10.0%

Applications Outside of Mathematics

Connection to HS Curriculum

Technology Skills

Mathematical Maturity

Visualization of Functions

Algebraic Skills

Problem Solving

0.0%

Figure 1. Faculty Members percentage of (n = 114)1st, 2nd, and 3rd ranked aspects. 120

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

1st

100

2nd 3rd

80 60 40 20

Applications Outside of Mathematics

Connection to HS Curriculum

Technology Skills

Mathematical Maturity

Visualization of Functions

Algebraic Skills

Problem Solving

0 Proof Writing Skills

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.

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.

Proof Writing Skills

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 Austinâ&#x20AC;&#x2122;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 emailed 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.

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

Lee Fothergill

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. 40.0% 35.0%

1st

30.0%

2nd 3rd

25.0% 20.0% 15.0% 10.0% 5.0%

Applications Outside of Mathematics

Connection to HS Curriculum

Technology Skills

Mathematical Maturity

Visualization of Functions

Algebraic Skills

Problem Solving

Proof Writing Skills

0.0%

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

50 45

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.

1st

2nd 3rd

35 30

100.0%

Faculty without experience

90.0%

Faculty with experience

80.0%

70.0%

60.0%

50.0%

40.0%

30.0%

Faculty Members With and Without Experience Teaching at the Secondary Level

10.0% Applications Outside of Mathematics

Connection to HS Curriculum

Technology Skills

Mathematical Maturity

Visualization of Functions

Problem Solving

0.0% Algebraic Skills

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

20.0%

Proof Writing Skills

Applications Outside of Mathematics

Connection to HS Curriculum

Technology Skills

Mathematical Maturity

Visualization of Functions

Algebraic Skills

Problem Solving

Proof Writing Skills

Figure 5. Faculty members with (n = 29) and without (n = 85) teaching experience at the secondary level, combined 1st, 2nd, and 3rd rankings.

To investigate possible differences in their perspectives, the author then divided the faculty 27

Calculus for Preservice Teachers

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.

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 facultyâ&#x20AC;&#x2122;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%

100.0%

Faculty without experience Faculty with experience Student teachers

90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 10.0%

Applications Outside of Mathematics

Connection to HS Curriculum

Technology Skills

Mathematical Maturity

Visualization of Functions

Algebraic Skills

Problem Solving

Proof Writing Skills

0.0%

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

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

Lee Fothergill

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 K12 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.

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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, 372â&#x20AC;&#x201C;383.

Lee Fothergill

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 studentsâ&#x20AC;&#x2122; 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 studentsâ&#x20AC;&#x2122; 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

The Mathematics Educator 2011, Vol. 21, No. 1, 33–47

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 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.

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

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. 33

Coordination of Center and Spread

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 34

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 pvalues. 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

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 teachersâ&#x20AC;&#x2122; 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 fieldtesting with prospective teachers, analysis of fieldtesting 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 20062007, 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 teachersâ&#x20AC;&#x2122; 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 35

Coordination of Center and Spread

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

50%

Ch 2: Analyzing Studentsâ&#x20AC;&#x2122; Comparison of Two Distributions using TinkerPlots

Ch 3: Analyzing Data with Fathom

Ch 4: Analyzing Bivariate Data with Fathom

Ch 5: Designing and Using Probability Simulations

76%

Ch 6: Using Data Analysis and Probability Simulations to Investigate Male Birth Ratios

59%

Hollylynne S. Lee & J. Todd Lee

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.

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)

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)

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.]

(Chapter 1, Section 3, p. 11)

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)

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)

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 37

Coordination of Center and Spread

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 175325], 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 information.

don’t

have

enough

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 38

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

Freshman Classes

0.42

0.46

Dot Plot

0.50

0.54

Retetention

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

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:

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

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

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.

Narrower for which way, if we reduced to 200 or increased to 999?

PT:

999

Why do you think it would be narrower?

PT:

The more trials there are, the closer it will be to the true mean.

[Asks students if they agree, about half the class raise their hand.]

…

…. [Other prospective teachers make similar comments.]

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. 39

Coordination of Center and Spread

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 40

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 postassessment 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

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 studentâ&#x20AC;&#x2122;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

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.

85% - 94% of those days.

75% - 84% of those days.

65% - 74% of those days.

55% â&#x20AC;&#x201C; 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)

Hospital B (with 10 births a day)

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 studentâ&#x20AC;&#x2122;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.

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

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

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.

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

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

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

Hollylynne S. Lee & J. Todd Lee

Table 3 Correct Response Rates on Four Test Questions. Comparison

Implementation I

Implementation II

Implementation III

n = 15

n = 18

n = 15

n = 32

Question

Correct Answer

Pre

Post

Pre

Post

Pre

Post

Pre

Post

47%

44%

50%

53%

40%

80%

44%

89%

33%

80%

38%

66%

53%

20%

11%

22%

33%

20%

25%

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 teachersâ&#x20AC;&#x2122; 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 oneâ&#x20AC;&#x2122;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 teachersâ&#x20AC;&#x2122; 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

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 44

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

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?

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?

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?

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

REFERENCES Batanero, C., Godino, J. D., & Roa, R. (2004). Training teachers to teach probability. Journal of Statistics Education 12(1). Retrieved from http://www.amstat.org/publications/jse/v12n1/batanero.html Canada, D. (2006). Elementary preservice teachers’ conception of variation in a probability context. Statistics Education Research Journal, 5(1), 36-63. Retrieved from http://www.stat.auckland.ac.nz/serj Capraro, R. M. (2004). Statistical significance, effect size reporting, and confidence intervals: Best reporting strategies. Journal for Research in Mathematics Education, 35, 57-62. Clements, D. H. (2007). Curriculum research: Toward a framework for “research-based curricula”. Journal for Research in Mathematics Education, 38, 35-70. delMas, R. C. (2004). A comparison of mathematical and statistical reasoning. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 79-95). Dordrecht, The Netherlands: Kluwer. Fidler, F. (2006). Should psychology abandon p-values and teach CI’s instead? Evidence-based reforms in statistics education. Proceedings of International Conference on Teaching Statistics 7 . Retrieved from http://www.stat.auckland.ac.nz/~iase Franklin, C., Kader, G., Mewborn, D. S., Moreno, J., Peck, R., Perry, M., & Schaeffer, R. (2005). A Curriculum Framework for K-12 Statistics Education: GAISE report. Alexandria, VA: American Statistical Association. Retrieved from http://www.amstat.org/education/gaise Friel, S., O’Connor, W., & Mamer, J. (2006). More than “meanmedianmode” and a bar graph: What’s needed to have a statistics conversation? In G. Burrill (Ed.), Thinking and reasoning with data and chance: Sixty-eighth yearbook (pp. 117-137). Reston, VA: National Council of Teachers of Mathematics. Gardner, M. J., & Altman, D. G. (1986). Confidence intervals rather than p values: Estimation rather than hypothesis testing. British Medical Journal (Clinical Research Ed), 292, 746-750. Garfield, J. B. (2002). The challenge of developing statistical reasoning. Journal of Statistics Education, 10(3). Retrieved from http://www.amstat.org/publications/jse/v10n3/garfield.html Garfield, J. B. (2003). Assessing statistical reasoning. Statistics Education Research Journal, 2(1), 22-38. Retrieved from http://www.stat.auckland.ac.nz/~iase /serj/SERJ2(1).pdf Hake, R. R. (1998). Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses. American Journal of Physics, 66, 64-74. International Committee of Medical Journal Editors. (1997). Uniform requirements for manuscripts submitted to biomedical journals. Annals of Internal Medicine 126 (1), 3647, [online] Available at http://www.annals.org/cgi/content/full/126/1/36. Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33, 259-289. Leavy, A. M. (2010). The challenge of preparing preservice teachers to teach informal inferential reasoning. Statistics

Education Research Journal, 9(1), 46-67, Retrieved from http://www.stat.auckland.ac.nz/serj. Lee, H. S., & Hollebrands, K. F. (2008a). Preparing to teach mathematics with technology: An integrated approach to developing technological pedagogical content knowledge. Contemporary Issues in Technology and Teacher Education, 8(4). Retrieved from http://www.citejournal.org/vol8/iss4/mathematics/article1.cfm Lee, H. S., & Hollebrands, K. F. (2008b). Preparing to teach data analysis and probability with technology. In C. Batanero, G. Burrill, C. Reading, & A. Rossman (Eds.), Joint ICMI/IASE Study: Teaching Statistics in School Mathematics. Challenges for Teaching and Teacher Education. Proceedings of the ICMI Study 18 and 2008 IASE Round Table Conference. Retrieved from http://www.ugr.es/~icmi/iase_study/Files/Topic3/T3P4_Lee.p df Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Preparing to teach mathematics with technology: An integrated approach to data analysis and probability. Dubuque, IA: Kendall Hunt. Lee, H. S., & Mojica, G. F. (2008). Examining teachers’ practices: In what ways are probabilistic reasoning and statistical investigations supported? . In C. Batanero, G. Burrill, C. Reading, & A. Rossman (Eds.), Joint ICMI/IASE study: Teaching statistics in school sathematics. Challenges for teaching and teacher education. Proceedings of the ICMI Study 18 and 2008 IASE Round Table Conference. Retrieved from http://www.stat.auckland.ac.nz/~iase/publications/rt08/T2P14 _Lee.pdf. Lesser, L. M., & Groth, R. (2008). Technological pedagogical content knowledge in statistics. In Electronic Proceedings of the Twentieth Annual International Conference on Technology in Collegiate Mathematics (pp. 148-152).Retrieved from http://archives.math.utk.edu/ICTCM/VOL20/S118/paper.pdf Makar, K., & Confrey, J. (2005). “Variation-talk”: Articulating meaning in statistics. Statistics Education Research Journal, 4(1), 27-54. Retrieved from http://www.stat.auckland.ac.nz/serj National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author. Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An analytical model for studying the development of learners’ mathematical ideas and reasoning using videotape data. Journal of Mathematical Behavior, 22, 405-435. Reading, C., & Shaughnessy, J. M. (2000). Student perceptions of variation in a sampling situation. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 89-96). Hiroshima, Japan: Hiroshima University. Reading, C., & Shaughnessy, J. M. (2004). Reasoning about variation. In J. Garfield & D. Ben-Zvi (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 201-226). Dordrecht, The Netherlands: Kluwer. Saldanha, L., & Thompson, P. W. (2002). Conceptions of sample and their relationships to statistical inference. Educational Studies in Mathematics, 51, 257-270. Shaughnessy, J. M. (2006). Research on students’ understanding of some big concepts in statistics. In G. Burrill (Ed.), Thinking

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and reasoning with data and chance: Sixty-eighth yearbook (pp. 77-98). Reston, VA: National Council of Teachers of Mathematics. Stohl, H. (2005). Probability in teacher education and development. In G. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 345-366). Dordrecht, The Netherlands: Kluwer. Watson, J. M., Callingham, R. A., & Kelly, B. A. (2007). Studentsâ&#x20AC;&#x2122; appreciation of expectation and variation as a foundation for statistical understanding. Mathematical Thinking and Learning, 9, 83-130.

REVIEWERS FOR THE MATHEMATICS EDUCATOR, VOLUME 21, ISSUE 1

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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