## Archive for the ‘Articles’ Category

*MAA Convergence* is both an online journal on the history of mathematics and its use in teaching and an ever-expanding collection of online resources to help its readers teach mathematics using its history. Founded in 2004 by well-known mathematics historians and educators Victor Katz and Frank Swetz, the Mathematical Association of America’s *Convergence* brings you a variety of interesting articles and teaching tools.

We highlight here some of our newest articles and resources for use in your classroom. Many of them use interactive features to help students understand and explore historical mathematical ideas.

In “Exploring Liu Hui’s Cube Puzzle: From Paper Folding to 3-D Design,” author Lingguo Bu offers history, classroom activities, and interactive applets to help you and your students explore Liu Hui’s 3^{rd} century dissection of the cube into three pieces with volumes 1/2, 1/3, and 1/6 of the volume of the cube. *The three puzzle pieces are shown above and below. The pieces in the image below were made using a 3-D printer.*

For a different kind of puzzle, try “Mathematicians from A to Z,” a *New York Times*-style crossword puzzle created by mathematics instructor Sid Kolpas and a crossword puzzle creator Stu Ockman.

The article, “Misseri-Calendar: A Calendar Embedded in Icelandic Nature, Society, and Culture,” by Kristín Bjarnadóttir, reviews the calendar’s long history from Viking times to the present, and offers animations and ideas for your classroom.

In “A Translation of Evangelista Torricelli’s ‘The Quadrature of the Parabola, solved by many methods through the new geometry of indivisibles,’” authors Andrew Leahy and Kasandra Sullivan provide plenty of history and helpful diagrams along with their translation.

In “A Series of Mini-projects from **TR**ansforming **I**nstruction in **U**ndergraduate **M**athematics via **P**rimary **H**istorical **S**ources” the TRIUMPHS team introduces the first of a collection of mini-Primary Source Projects (mini-PSPs), “The Derivatives of the Sine and Cosine Functions” (by Dominic Klyve), a classroom assignment in which Calculus I students learn how Leonhard Euler (1707-1783) obtained these derivatives via differentials. *Above, students work on a Primary Source Project under the supervision of Janet Barnett at a TRIUMPHS Site Tester Workshop in Denver, Colorado, in September of 2016.*

In “Illustrating *The Nine Chapters on the Mathematical Art*: Their Use in a College Mathematics History Classroom,” author Joel Haack shares how he used his experiences on an MAA Mathematical Study Tour to China to enrich his teaching. *The photo above is of a statue in the National Museum of China of a civil servant from the Sui Dynasty (581-618), an intended user of the *Nine Chapters.

“Moses ibn Tibbon’s Hebrew Translation of al-Hassar’s *Kitab al Bayan,” *by Jeremy I. Pfeffer (Hebrew University of Jerusalem) features the arithmetic of fractions as you’ve (possibly) never seen it before!

See fractions in the context of problem-solving using the method of double false position in the Arabic manuscript *Kitab al-nuzah* in “Mathematical Treasure: The Method of Scales in ibn al-Ha’im’s *Book of Delights,*” by Randy Schwartz and Frank Swetz. *Above: This diagram is used in this and other manuscripts to illustrate and carry out the “method of scales.”*

In “Mathematical Treasures at the Linda Hall Library,” author Cynthia Huffman highlights the mathematics collections available at this rare book library in Kansas City, Missouri. See images of mathematics books by Euclid, Pacioli, Cardano, Torricelli, Maria Agnesi, and Emilie du Chatelet.

Our “Index to Mathematical Treasures” includes hundreds of images for use in your classroom from dozens of libraries and sources.

See all of these articles and more at *MAA Convergence:*

http://www.maa.org/press/periodicals/convergence

Join us at the *Convergence* of mathematics, history, and teaching!

*Janet Beery*

Editor, *MAA Convergence*

University of Redlands, California

USA

** **

**An Example of Using the **

**History of Iranian Mathematics for the Math Classroom**

Dividing a right angle into five equal angles with only a straightedge and a compass is used to construct particular tiling patterns in Islamic arts, and one of these patterns is presented in figure 1 (10-petal rose construction). The methods and ideas that I explain in this paper were obtained from Iranian math history (Jazbi, S. A. (translator), *Applied Geometry*, appendix2. Soroush Press, ISBN 964 435 201 7, Tehran 1997). All figures have been created by the author using the Geometer’s SketchPad (GSP) software program. These samples have been used at the Isfahan Math House (IMH) in workshops for teaching math history to secondary students and mathematics teachers.

**Figure 1.** *10-petal rose construction (girih construction)*

** **1) **Task 1: Dividing a right angle into five congruent angles with only a straightedge and a compass **Construct arbitrary arc (figure 2).

**Figure 2**.

2) Construct *D* the midpoint of *OA* then find *D*‘ as *OD* = *OD*‘ (figure 3).

**Figure 3**.

3) Then construct a circle with center *D*‘ and radius *D*‘*B*. This circle cuts *OA* at point *E *(figure 4).

**Figure 4.**

4) Now construct segment *BE* (figure 5).

**Figure 5.**

5) Finally, construct a circle with center *B* and radius *BE*, and label the intersection point of the green circle and new circle (in magenta), *F* (figure 6).

**Figure 6**.

6) Construct segment *OF*, and then (figure 7). (Prove it!)

**Figure 7**.

7) Now divide into four equal angles (explain your work!). Now you have five angles.

**a)****Task 2: Dividing a right angle into six congruent angles with only a straightedge and a compass**

1) Construct a circle with center *O* and radius *OB* (figure 8).

**Figure 8**.

2) Construct a circle with center *B* and radius *BO*. Label the intersection point of the two circles, *D* (figure 9).

**Figure 9**.

3) Construct segment *OD*, then . Why? (figure 10)

**Figure 10**.

4) Construct the bisector, and repeat again for created angles. Construct bisector, then you have six angles (figure 11).

**Figure 11.** *Dividing a right angle into six * *angles.*

Dividing a right angle into six equal angles can be used to construct Islamic art patterns. One of them is named a 12-petal rose pattern like the one shown in figure 12.

**Figure 12**. *12-petal rose pattern*

** Narges Assarzadegan, **Math teacher, math history researcher, Isfahan Mathematics House (IMH) (Iran)

*MAA Convergence* is both an online journal on the history of mathematics and its use in teaching and an ever-expanding collection of online resources to help its readers teach mathematics using its history. Founded in 2004 by well-known mathematics historians and educators Victor Katz and Frank Swetz, *Convergence* brings you a variety of interesting articles and teaching tools. It is freely available from the Mathematical Association of America (MAA) website: http://www.maa.org/press/periodicals/convergence

We highlight here some of our newest articles and resources for use in your classroom. Many of them use interactive features to help students understand and explore historical mathematical ideas.

In **“Ancient Indian Rope Geometry in the Classroom,”** Cynthia Huffman and Scott Thuong offer information, activities, and applets to help you and your students explore the geometry of altar construction in ancient India. In the photograph, boys work on a model of the bird-shaped fire altar in an *Agnicayana* ritual in Panjal, Kerala, India in 2011. (Photo courtesy of Professor Michio Yano.)

In **“Geometrical Representation of Arithmetic Series,”** Gautami Bhowmik explores a geometric tradition in Sanskrit arithmetic texts from Medieval India and shares problems from these texts for your students.

**“Historical Activities for the Calculus Classroom,” **by Gabriela Sanchis, presents curve-sketching, tangent lines, and optimization in the context of historical problems, and is illustrated by 24 interactive applets and 10 animations.

In **“Descartes’ Method for Constructing Roots of Polynomials with ‘Simple’ Curves,”** Gary Rubinstein explains and derives Descartes’ methods from his 1637 *Geometry *and illustrates them using interactive applets. The diagram shows a step in the construction of roots of sixth degree polynomials using a ‘Cartesian parabola’ and circles (from GeoGebra applet by Gary Rubinstein).

In **“Pythagorean Cuts,”** Martin Bonsangue and Harris Shultz answer the question ‘Can Euclid’s proof of the Pythagorean Theorem be adapted to shapes other than squares?’ and encourage you to pose it to your students.

**“Some Original Sources for Modern Tales of Thales,”** by Michael Molinsky, features earliest known sources for stories about Thales, and applets illustrating methods attributed to him. The diagram shows how Thales might have measured the distance from ship to shore (from GeoGebra applet by Michael Molinsky).

**“A GeoGebra Rendition of One of Omar Khayyam’s Solutions for a Cubic Equation,”** by Deborah Kent and Milan Sherman, explains and illustrates how the 11th century Persian mathematician, philosopher, and poet *geometrically* determined a positive real solution to a cubic equation.

**“Edmund Halley, 1740”** is an historical poem in which Halley reflects on his role in publishing Newton’s *Principia,* by award-winning Oxford poet Andrew Wynn Owen.

**“D’Alembert****, Lagrange, and Reduction of Order,” **by Sarah Cummings and Adam Parker, offers two historical approaches, one familiar and one unfamiliar, to enrich your differential equations course.

In **“Euler and the Bernoullis: Learning by Teaching,”** author Paul Bedard reflects on lessons he has learned about mathematics teaching and learning from these great mathematicians.

In **“Can You Really Derive Conic Formulae from a Cone?”** Gary Stoudt uses 17 interactive applets to explain how attempts to double the cube led ancient Greek mathematicians to discover and develop the conic sections.

Finally our **“Index to Mathematical Treasures”** includes hundreds of images for use in your classroom, including photographs of **“The Cambodian (Khmer) Zero”** (of 683 CE) by Amir and Debra Aczel.

See all of these articles and more at *MAA Convergence:*

http://www.maa.org/press/periodicals/convergence

Join us at the *Convergence* of mathematics, history, and teaching!

*Janet Beery *

Editor, *MAA Convergence*

University of Redlands

(USA)

MAA Convergence is both an online journal on the history of mathematics and its use in teaching and an ever-expanding collection of online resources to help its readers teach mathematics using its history. Founded in 2004 by well-known mathematics historians and educators Victor Katz and Frank Swetz, Convergence brings you a variety of interesting articles and teaching tools. We highlight here some of our newest articles and resources for use in your classroom.

As a web-based publication, Convergence aims to take advantage of new technologies in order to present, explore, and better understand what are often very old ideas. The articles and resources featured here all exemplify this combination of old and new.

(1) “Euclid21: Euclid’s Elements for the 21st Century” introduces a dynamic, interactive version of Euclid’s classic circa 300 BCE geometry text organized via its logical structure.

Figure 1. Euclid’s first three uses of his Parallel Postulate, as illustrated in Euclid21 (Image from Euclid21 computer application created by Eugene Boman and his student team)

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(2) “Oliver Byrne: The Matisse of Mathematics” offers both the most complete biography of Byrne to date and ideas for using Byrne’s colorful Euclid’s Elements (1847) in the classroom.

Figure 2. Byrne’s illustration of Euclid’s “windmill” proof of the Pythagorean Theorem. Byrne’s color-coded Euclid was a marvel of Victorian printing and of Pestalozzian pedagogy. (Photo by author Sid Kolpas of his own copy of the book)

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(3) “Bridging the Gap Between Theory and Practice: Astronomical Instruments” shows how your students can design and build armillary spheres, astrolabes, quadrants, sextants, and sundials using such modern technology as 3D printers.

Figure 3. Student-built sundial from Toke Knudsen’s Ancient Mathematical Astronomy course at SUNY Oneonta (photo by T. Knudsen). See “Bridging the Gap Between Theory and Practice: Astronomical Instruments.”

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(4) “Problems for Journey Through Genius: The Great Theorems of Mathematics” celebrates the popular book’s 25th year in print with downloadable problem sets for each chapter by author William Dunham himself.

Figure 4. Students can explain how Archimedes wrote the area of an ellipse in terms of the area of a circumscribing circle. (Image created by Janine Stilt)

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(5) In “Pantas’ Cabinet of Mathematical Wonders: Images and the History of Mathematics,” Convergence’s chief treasure-hunter Frank Swetz showcases Convergence’s “Mathematical Treasures,” an ever-growing collection of hundreds of images of historical texts, manuscripts, and objects for classroom use. Search or browse “A Collection of Mathematical Treasures – Index.”

Figure 5a. A simple but compelling application of the Pythagorean Theorem from Robert Recorde’s Pathway to Knowledge (1551)

Figure 5b. Caption: Book VI of Euclid’s Elements (originally composed circa 300 BCE) begins with a definition of similar rectilinear figures. This copy of Euclid’s Elements was handwritten on vellum around 1294 CE. (Image courtesy of Columbia University Libraries)

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(6) “Online Museum Collections in the Mathematics Classroom” introduces 27 mathematical object collections from the Smithsonian Institution’s National Museum of American History and offers suggestions for using them with students of all ages.

Figure 6. Grunow’s circa 1860 spherometer (Photo courtesy of Smithsonian Institution)

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(7) “Jan Hudde’s Second Letter: On Maxima and Minima” contains a translation of the letter and explanation of Hudde’s pre-calculus optimization methods, including an early quotient rule.

Figure 7. Diagram added by Frans van Schooten when he published Hudde’s “second letter” in 1659. (Image courtesy of ETH-Bibliothek, Zürich, Switzerland)

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(8) “Alan Turing in America” focuses on the important projects in logic and computing Turing worked on during two visits to the U.S.

Figure 8. This photo of a young Alan Turing is believed to be from 1936-38 when he was at Princeton University. (Photo from Convergence Portrait Gallery)

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See all of these articles and more at MAA Convergence:

http://www.maa.org/publications/periodicals/convergence

Join us at the Convergence of mathematics, history, and teaching!

Janet Beery (USA)

Editor, MAA Convergence

University of Redlands

The Hans Freudenthal Medal for 2011 goes to Luis Radford, Université Laurentienne, Sudbury, Canada. Here is the official information from the ICMI website:

It is with great pleasure that the ICMI Awards Committee hereby announces that the Hans Freudenthal Medal for 2011 is given to Professor Luis Radford, Université Laurentienne, Canada, in recognition of the theoretically well-conceived and highly coherent research programme that he initiated and has brought to fruition over the past two decades, and which has had a significant impact on the community. His development of a semiotic-cultural theory of learning, rooted in his interest in the history of mathematics, has drawn on epistemology, semiotics, anthropology, psychology, and philosophy, and has been anchored in detailed observations of students’ algebraic activity in class. His research, which has already garnered several awards, has been documented extensively in a vast number of highly renowned scientific journals and specialized books and handbooks, as well as in numerous invited keynote presentations at international conferences. The impact of Luis Radford’s programme of research has been felt especially by the community of research in algebra teaching and learning where his theoretical and empirical work has led to significant new insights in this domain, and more broadly by the entire community of mathematics education research with his development of a groundbreaking, widely applicable theory of learning.

by Professor R. C. Gupta, Ph. D. (Hist. of Math.)

The recent excellent book by Jöran Friberg (see References at the end) clearly exposes some significant connections between mathematics of ancient Egypt and Babylonia. Earlier the cultural unity of the popular antique Surveyor’s Rule

…(1)

for the area of a quadrilateral of sides *a, b, c, d*, has been well illustrated by the present author (Gupta, 2002). The case of the bow-figure (circular segment up to the semicircle) is discussed in this article.

By Professor R. C. Gupta, Ph. D. (Hist. of Math.)

Ganita Bharati Academy

R-20 Ras Bahar Colony, P. O. Sipri Bazar

JHANSI-284003, India

The *Jiu Zhang Suan Shu* (JZSS), or “The Nine Chapters of Mathematical Art”), is the most important work of ancient China on mathematics. Although based much on earlier material, its present form is placed in the first century of our Common Era (CE). Its earliest extant version is that of Liu Hui (263 CE), whose commentary exerted profound influence on Chinese mathematics for well over a millennium.

As the title indicates, the *JZSS* is divided into nine chapters and has a total of 246 problems. The first chapter is entitled *Fang-tian* (Mensuration of Fields) and contains 38 problems. It is devoted to calculation of areas of various geometrical figures. Problems 31 and 32 are on the area of a circle of circumference c and diameter d and contains the popular exact formula

along with two approximate rules and , which imply the use of . Problems 33 and 34 are about the area of the surface of a spherical segment. However, there is some disagreement on this issue and great divergence in interpreting the prescribed rule. The purpose of the present article is to briefly mention the various views and give some new interpretations.

I completed a review of papers published in the proceedings for the HPM 2000 conference in Taipei and the HPM 2008 conference in Mexico City with the intent of answering three questions:

- What kinds of empirical sources are we utilizing in HPM research?
- What school levels are we doing research for?
- What kinds of teaching with history do we advocate?

I will discuss several ideas from the analysis in this short report.

I analyzed 49 papers from the proceedings of the HPM 2000 conference proceedings and 71 papers from the HPM 2008 proceedings. Without reading all 120 papers in full, I tried to establish what kinds of empirical sources they were based on, the age level of the students for which the research is relevant, and the teaching methods that were either advocated or referred to in the article. Often, no age level is mentioned; in such instances I tried to estimate an age range based on my own experience.

Unsurprisingly, almost all articles were based on empirical sources (primary or secondary) from the field of history of mathematics. The whole field of HPM is meaningless unless the history we teach is founded in historical sources – and indeed, the HPM group has a long tradition of including articles giving historical accounts of the development of concepts that are interesting to mathematics educators.

Mathematical Treasures

One of the major features in the MAA’s online magazine in the history of mathematics Loci: Convergence is the article entitled “Mathematical Treasures”. This article contains annotated copies of various book pages chosen from the George Arthur Plimpton and David Eugene Smith collections at Columbia University, one of the best collections of rare books and manuscripts in the country.

During the first half of the twentieth century, David Eugene Smith (1860-1944) was a moving force in the world of mathematics education. As the chairman of the mathematics education department at Columbia University’s Teachers College, Smith led the way in teaching reforms attuned to the Progressive Education Movement. He firmly believed that the teaching of mathematics should be closely associated with the history of the subject. As an historian of mathematics, he wrote and lectured widely on the subject and also collected historical mathematical materials: texts, documents and artifacts. Smith befriended the wealthy New York lawyer and publisher, George Arthur Plimpton (1855-1936), who was also a bibliophile and avid collector. Under Smith’s influence, Plimpton enriched his collection with mathematical manuscripts and many early Renaissance texts on arithmetic. When Plimpton died in 1936, he bequeathed his collection to Columbia University. Similarly, beginning in 1931, David Eugene Smith began donating his extensive collection of mathematical memorabilia: historical texts; correspondence; portraits of famous mathematicians; signatures and concrete artifacts to the Columbia University Library.

There are a few important changes to the HPM Newsletter now.

Chris Weeks is stepping down as co-editor after working on the newsletter since 2004. With his knowledge of the field and the group, he has been essential to the newsletter for more than six years. He is being replaced by no less than three new co-editors, which is a clear suggestion of his value to the HPM community.

The new newsletter co-editors are Kathy Clark, Snezana Lawrence and Helder Pinto, whom will be presented more fully in time.

The other important change is that we from today is publishing the newsletter online as a supplement to the paper version. There will be new articles online every month, and these articles will go into the issues that will be published three times a year as before.

The online version offers opportunities for publishing content more quickly and to have feedback on articles – as well as making it even simpler for new people to find the HPM group. Any suggestions and input is welcome.