### mathematics

Luisa Rosu just sent me a link to this clever video on how to connect the dots. We often think of mathematics as a tool to help us understand the world better. In this case we see how we can also use the world to understand mathematics better.

Happy Tau Day! Today is June 28, or 6/28 in the US date format, and τ = 6.283185307179586…

There’s a movement of sorts to use τ (which is equal to 2π), instead of the more familiar π. π is the ratio of the circumference, C, of a circle to its diameter, D.

π = C / D

τ proponents say that since a circle is defined as the set of points a fixed distance (the radius, r) from a given point, a more natural definition for the circle constant would use r:

τ = C / r

This of course makes τ equal exactly to 2π. One might well ask, “What’s the big difference?” τ advocates say simply, “π is wrong.” By this, they don’t mean that it’s wrong to say:

C / D = 3.14

That equation is wrong mathematically, just as any calculating device is wrong because it has to approximate π. Neither π, nor τ can be expressed as a simple fraction (or as a terminating or repeating decimal). But the argument here is not about approximations. It’s that π is wrong pedagogically: π is a confusing and unnatural choice for the circle constant. The confusions run through all sorts of statements about circles, angles, statistical and physical relationships.

The full reasons for using τ are explained in Bob Palais’s article P Is Wrong and in Michael Hartl’s The Tau Manifesto, which is

dedicated to one of the most important numbers in mathematics, perhaps the most important: the circle constant relating the circumference of a circle to its linear dimension. For millennia, the circle has been considered the most perfect of shapes, and the circle constant captures the geometry of the circle in a single number. Of course, the traditional choice of circle constant is π—but…π is wrong. It’s time to set things right.

Michael Blake has created a musical interpretation of τ up to 126 decimal places. It maps τ to musical notes, and sounds quite nice:

Yalikavak, Turkey

Our last stop in Istanbul was at the İstanbul Islâm Bilim ve Teknologi Tarihi Müzesi (Museum for the History of Science and Technology in Islam) in the Eminönü district. It’s a wonderful museum, displaying centuries of achievements in geography, navigation, astronomy, mathematics, music, optics, chemistry, chronometry, historiography, medicine, military, civil engineering, and other disciplines.

I was told in school that the period when most of these discoveries and creations occurred (9th-16th century CE) was called the Dark Ages, a time of fear, superstition, lack of progress, even regression from the Classical era. Then the Classical learning was miraculously rediscovered and expanded during the Renaissance. And of course, like many things I learned in school, it was partly true.

But the fact is that while great scientific and technical accomplishments were happening in the Islamic world, much of Christian Europe languished in these areas, not completely, but to a large extent in comparison. Islamic scholars maintained and extended the Classical learning, and incorporated additional ideas from Greek, Byzantine, Indian, Judaic, and other traditions. They not only advanced learning considerably, but did so by listening to and learning from other cultures.

Long before Roger Bacon, they articulated and promoted experimental science, and they wrote about inductive or scientific methods long before Francis Bacon. They studied the circulation of blood before William Harvey, and made many other medical advances. But the exhibits do not take a “who did it first?” approach; instead, they emphasize the continuity of learning, across time and across cultures.

The Ages were not Dark everywhere, and the Renaissance in Europe was not autonomous; it was dependent upon and grew organically from an Islamic culture that valued learning in all its forms.

The Museum displays fascinating astrolabes, glassware, maps, globes, medical instruments, ships, an elephant clock, and many other artifacts. I’ve never seen such an assemblage anywhere, and these are beautifully presented and explained with multilingual text and video. Scientists, mathematicians and other scholars, such as Muhammad ibn Mūsā al-Khwārizmī, Ibn Sena (Avicenna), Geber, Al-Jazari are featured.

That’s why it was both surprising and a bit depressing to see how few people came to the exhibits. I counted five total visitors during the entire time that we were there, Make that seven if I include Susan and me. There were about fifteen staff and guards. Of course, it’s been open only two years.

İstanbul offers tough competition for any museum. World Heritage sites like the Topkapi Palace nearby, the Sultanahmet Cami (Blue Mosque), and Hagia Sophia are just a few of those within easy walking distance, each offering jaw-dropping sights. But those also offer long lines and crowded viewing. It’s difficult to fully appreciate the Topkapi dagger while being shoved along in a crowd. The Museum for the History of Science and Technology in Islam offers a different and equally important view of Islamic culture, one that I suspect is not well known by many within or outside of Islam,

The Museum is housed in the Has Ahırlar (Imperial Stables) complex now in Gülhane Park. This area was once the outer garden for the Topkapı Palace during the days of the Ottoman Empire.

The working instruments and other objects were constructed by the Institute for the History of Arabic-Islamic Science at the Johann Wolfgang Goethe University, Frankfurt, based on illustrations and descriptions in textual sources, and to some extent, on surviving original artifacts. There is a similar exhibit there under the direction of Fuat Sezgin. The İstanbul museum is a joint project of the Ministry of Culture and Tourism, the Scientific and Technological Research Council of Turkey (TÜBİTAK), the Turkish Academy of Sciences (TÜBA), the İstanbul Metropolitan Municipality, and Goethe University.

A gravity-defying illusion has won the 2010 Best Illusion of the Year Contest, held yesterday in Naples, Florida.

Koukichi Sugihara, from the Meiji Institute for Advanced Study of Mathematical Sciences, Japan, developed the illusion, which you can see in the video below. Wooden balls appear to roll up the channels, as if they are pulled by a magnet. We’re fooled by assuming that each supporting column is vertical, and that the longest column in the center is the tallest.

This is is similar to the Adelbert Ames’s illusions, such as the Ames room or Ames window, all of which demonstrate that perception is an interaction of the perceiver with the environment.

The Best Visual illusion of the Year Contest is a celebration of the ingenuity and creativity of the world’s premier visual illusion research community. Contestants from all around the world submitted novel visual illusions (unpublished, or published no earlier than 2009), and an international panel of judges rated them and narrowed them to the TOP TEN. At the Contest Gala in the Naples Philharmonic Center for the Arts, the top ten illusionists presented their creations and the attendees of the event voted to pick the TOP THREE WINNERS! via Results of the 2010 Contest now announced!

The other finalists are all worth watching, too.

Happy Pi day!  It’s March 14, or 3/14, the first three digits in the decimal expansion of Pi.

This only works for those of us living in Belize, Micronesia, Palau, Philippines, the US, and sometimes, Canada. The 95% of the world that more logically puts the day first thinks of today as 14/3. They’ll have to wait until July 22, but will have the consolation of knowing that 22/7 is a better approximation of Pi than 3.14.

As a gift for today, New Scientist offers five tasty facts about the famous ratio “We did consider giving you 3.14 facts but alas we had five…”

Jack Easley was an professor at the University of Illinois from 1962 until his retirement in 1989. His research on cognitive development in the learning of science and mathematics across various cultures influenced educators around the world. He co-founded the Dialogues in Methods of Education group, which continues to this day. He was also a much loved friend, who died December 10, 1994.

I recently came across some insightful email messages from Jack. Here’s one that I’m certain he would like to have shared more widely, even though they were simply rough notes related to a project:

There is a lot of attention given over to kits and manipulative materials for inquiry. Since these are not always available, it is worthwhile looking at what can be done without the kits, the manipulative blocks, etc.

Math

The Japanese schools use cardboard replicas of plastic tiles, and several teachers in the US have found that these can be cut out of file folders with a paper cutter. It is not necessary to have one set for each child, but the following sizes would be appropriate for each team:

• 5 square units (half-inch squares are usually fine, but 1in or 1 cm can be used.)
• 2 oblongs, 5 units long (e.g., .5 in by 2.5 in)
• 5 oblongs, 10 units long (e.g., .5 in by 5 in)
• 2 fifties (e.g., 2.5 in by 5 in)
• 10 hundreds (e.g., 5 by 5 in)

With rulers, children can mark one side of the oblongs, fifties and hundreds into ways that show how they all fit together. Other sizes ( 20s, 40s, 25s, etc.) are often convenient, depending on the story problems (going to the bank, etc.) children are solving with these cardboard tiles.

Using bulletin board paper, scrolls of 500 or 1,000 units can also be cut and rolled up (e.g., 5 in by 25 in, or 5 in by 50 in). To make representations of even larger numbers is not much of a problem with the smaller sized units, but if you use 1 sq in as a unit, it begins to get out of hand.

The size of unit can be chosen not only with the fine motor coordination of children in mind, but with the fact that place value and round numbers upwards of 99 are much easier to talk about than those between 9 and 100. Smaller unit sizes (.5 in or 1 cm) should permit more meaningful work with scrolls for numbers like 5,000 or 10,000.

In my opinion, and that of a minority of mathematics educators, the word “ten” is one of the least often suspected but most often confused among number names. The problem may be that “ten” is not a word that easily takes adjectival modification as in “Two tens, three tens, etc.” Ten is most often used as an adjective itself as in “ten fingers, ten hundred, etc.” Research suggests that it takes children until about fourth grade to realize that ten can be a unit instead of just a counting number or the cardinal number of a collection (Cobb & Wheatley, 1988; Steffe, 1983; Steffe & Cobb, 1988.) Informal observations suggest that 100, 1,000, and 1,000,000 are treated as abstract units quite naturally by most 6-year-olds. The debate is whether or not young children can plausibly attach concrete representations to those units.

There are other troubles with the names of numbers greater than 9 and less than 100, e.g., 18 and 81 sound too much alike, both beginning with the word, “eight,” and there are few people who would think that “twenty” was originally pronounced, and possibly spelled, “twain tens.” (Some have tried introducing new number names, onety, twoty, threety, fourty, fivety, and doing that seems to help in regrouping, but teachers and parents complain that children don’t know how to translate them into standard English.) Saying how many tens there are in 11, 22, 35, etc. is no longer a part of English speech today. Instead, everyone learns to rattle off the counting numbers 1 to 100 without pausing to think that there are ten cycles in that series. It may work like telling time or money. (With digital timepieces, we count minutes from 01 to 59 and then hours. We count cents from 01 to 99 and then dollars.) Starting over, which is the essence of place value, is something we don’t seem to think about naturally with those funny two-digit number names. (In the orient, and many native American languages, number names are much more sensible than in European languages.) However, all is well when we get to a hundred and we have three digits. A great deal of regrouping in arithmetic, which is the real advantage of understanding place value, can be learned by working with cardboard tiles and scrolls, without adding and subtracting those peculiarly named numbers from 10 through 99. Adding and subtracting hundreds and thousands, multiplying and dividing by hundreds and by thousands teach place value well and provide ample practice for first and second graders on basic, one-digit addition and subtraction facts.

Cutting templates for drawing the cm size tiles and scrolls in coffee can lids permits children in first and second grade to represent numbers by drawings on paper instead of actually manipulating the tiles themselves. The Japanese have found that drawings of tiles to represent an operation is a valuable intermediate step between manipulation of tiles with number sentences and writing numerical algorithms without manipulations, for it helps children invent and test their own algorithms.

Geometrical forms can be cut out of folders or paper. Also, it is instructive to draw circles, squares, triangles, and other regular figures six or seven inches across and measure their circumferences in various ways. One way to measure a circumference is to set the compass for an inch or a cm of separation and count how many steps it takes to walk around the figure and back to the starting point.

Place a pencil across your hand near the tips of your fingers. Put the heel of your other hand on top of it. Predict, Observe, Explain (POE) where the pencil will be when you have moved the heel of your top hand back until it is over the heel of the bottom hand. Do this motion several times without the pencil, then POE where the pencil will be.

Architecture

Tiling patterns that repeat endlessly can be made on a flat surface. One interesting challenge is to design and cut-out a piece of paper that folds up to make a box, a prism, a pyramid, or some other shaped three-dimensional object.

Columns can be made from rolled or folded construction paper and tested for load bearing by piling textbooks on top. The number of science books, or math books, that a column can hold is something to predict, observe, and explain (POE). One can even measure (POEM) the length, diameter, and circumference of such columns and figure out some kind of graph that represents how those quantities relate to the load a column will carry. Applications (POEMA) of what has been learned can be found, in studying the structure of buildings, bridge supports, street light and traffic light posts, and in making models of buildings. (This is also a good use for science and mathematics books which children and teachers find boring.)

Making designs for stained glass windows with a compass is an intriguing activity. A six-pointed rose window is one goal, but many other designs are possible. Of course coloring one’s design in the most attractive way possible is an added challenge, which assumes everyone has some crayons, or whatever to color them with.

Optics

Punching a pencil through the middle of a dark piece of construction paper 8-11 inches wide and laying it down on a white piece of paper on a flat desk in a well-lighted classroom raises the following question: Looking at the white spot (after making the edges neat by tearing off or folding back the torn pieces the pencil left), try to predict (P) what shape and size that white spot will become when the dark paper is raised an inch or two. (Of the hundreds of people I have asked that question, only one 3rd grade girl, who must have tried it before and one physics Ph.D. could come close.) Observe and Explain (POE) what has been observed. Measure (POEM) how high the dark paper is raised above the white paper and measure what you can of the pattern of light you can see when looking underneath the dark paper (POEM). Is there a relation between the two measurements? What is the best way to make such measurements as you gradually raise the dark paper higher and higher? Plot a graph.

Apply (POEMA) this phenomenon to other sources of light besides schoolroom lights. E.g., tape the dark paper to the window, and cover the rest of the window(s) and turn out the lights. If you hold a thin piece of white paper near the pencil hole, can you see any pattern on the white paper? Substitute a magnifying glass or hand lens for the pencil hole? How does that change the way things look? the graph? Go outdoors on a sunny day with a piece of dark paper in which you have carefully cut three or four different shaped holes about the size of a dime or less. Hold the dark paper so it casts a shadow over a white paper. What is the shape of the light spots going through the holes? How do they change as you move the dark paper higher? (POEM)

Put some water in the plastic cup or glass bottle. Put a pencil in the water. How does it look? Why? If you can find a straight soda straw, put it in and compare it’s shape with the pencil. POE what you will see when you look through the soda straw into the water.

Air

• Blow through a piece of tubing or soda straw into a jar or cup of water. What is the smallest bubble you can blow? What is the biggest bubble you can blow? Can you blow a bubble and suck it back in before it leaves the end of the tube or straw? What is inside the bubbles you blow? How is it different from the air in the room? Where does the air in the bubble come from? Where does it go when a bubble pops?
• Put a wad of tissue or paper towel in the bottom of the plastic cup or glass bottle, big enough so it won’t fall out when you turn it upside down. (Use tape if necessary to hold it.) POE what will happen to the paper when you push it carefully up-side down into a coffee can, plastic tub, acquarium, or other large container half full of water. (POEM) Measure how much water goes into the cup or jar. If possible, make measurements at different depths under the water. Plot a graph of how much water goes into the jar for each depth under the water. POEMA What use can you think of for the air trapped in an open container under water? Can you arrange for a cricket or other small animal to breathe that air while under water? Pour out the air trapped in a container while it is under water. Do you think you could catch it in another container under the water, pouring it from one to the other under water? Borrow another container and try.
• Put a soda straw into water and place your finger or thumb over the open end. Raise it out of the water. What is inside? Can you do that with a piece of hose? (POE) What makes the water run back when you let go? (POEA) Homework (with parental consent and assistance): Can you do it with a wide tube like a cardboard tube waterproofed with rubber cement or melted wax?
• If you can get a box that a drink (milk or juice) was in, and put the hose over the straw, can you blow and suck on the tube to make the sides of the box go out and in? What does it take to make a tight fit? What happens when the air can leak around the straw? What happens to the tube when you blow or suck on it?

Social Studies
Graphs

• Sample people in your class to find out how many live with grandparents, aunts and uncles, with one parent, two parents, etc.
• Find out who knows where various foods are produced, what kind of people produce them, etc.
• Find out what children think about where adults get the money they need for food and rent if they work at a bank, a store, a restaurant, a post office, a police station, a school, as a house cleaner, a nurse, a doctor, a care giver, a university, a power company, etc. What do such people have to spend money for to do their work?

Science

For the following science activities, certain other things like wax paper, a mirror, a soda straw, a milk carton, a large bowl, etc. are mentioned as needed. Other things in the generic kit may be used, and POEMA may be used also. They come from: Science Games & Puzzles, by Laurence B. White, Jr. drawings by Marc T. Brown, Addison Wesley, 1975

• Racing drops of water on wax paper.
• Stand sideways against a wall. Push the side of your foot against the wall. Now try to lift your other foot.
• Dip one end of a drinking straw in dishwashing liquid. Take it out. Blow in the other end. Keep blowing. Try cut ting your straw end like a cross.
• Blow bubbles on a very cold day. Your warm breath makes them very light.
• Push a thumbtack into a pencil eraser. Touch the thumbtack on your lip. Rub the tack hard 20 times on your sleeve and touch it to your lips again.
• Try to drop a coin into a glass under water in the middle of a big bowl.
• Collect and taste rain water. Does it taste different from other water?
• Try printing your name while looking at the pencil and paper in the mirror.
• Roll a little piece of foil in a ball and drop it in a funnel. You cannot blow it out unless you stop up the funnel.
• Balance a ruler on your finger. with & without a ball of clay on top.
• Have your friend lay his (her) head on a table or desk while you tap softly on the bottom.
• Hold a pencil in your teeth while scraping on it.
• Is your pet right or left pawed? Put some food in a jar. Which paw is used?
• Can you freeze a penny in the middle of a piece of ice?
• Can you turn yourself upside down with a teaspoon?
• Can you eat an apple without tasting it?
• Can you tie your arms in a knot? Cross them and hold the two ends of a tube while uncrossing.
• Write ‘A BOX’ on a card and look at it in a mirror several different ways.
• Punch three holes in a paper cup or milk carton. Which hole will squirt best?
• Can water stick to itself? Punch two holes side by side.
• Can you separate pepper and salt that have been mixed?
• Roll down a slope a full can, an empty can, a hollow ball, a base ball, etc. Which one wins?
• Tie a string around a nail, then tie the string around another nail, and another. This is how to make a string nail xylophone, which you can play with another nail.

Norma Scagnoli referred me to a wonderful podcast by LeAnn Erickson, Associate Professor of Film and Media Arts at Temple University. Erickson is an independent video/filmmaker, whose work has appeared on public television, in galleries, and has won national and international awards.

Entitled, Hidden Her-story: The Top-Secret “Rosies” of World War II, it was recorded in January at the EDUCAUSE 2009 Mid-Atlantic Regional Conference in Philadelphia. I expected to listen for a minute and then go on to more pressing things, but after listening a little I decided that those things weren’t so pressing after all. It’s a fascinating story for anyone who has an interest in history, computers, women, education, mathematics, warfare, politics, Philadelphia, science, workplace equity, morality, or life in general.

In 1942, only months after the United States entered World War II, a secret military program was launched to recruit women to the war effort. But unlike recruiting “Rosie” to the factory, this search targeted female mathematicians who would become human “computers” for the U.S. Army. These women worked around-the-clock shifts creating ballistics tables that proved crucial to Allied victory. “Rosie” made the weapons, but the female computers made them accurate. When the first electronic computer (ENIAC) was invented to aid ballistic calculation efforts, six of these women were tapped to become its first programmers. “Top Secret ´Rosies’: The Female ‘Computers’ of WWII” is a documentary project currently in postproduction that will share this untold story of the women and technology that helped win a war and usher in the modern computer age.

Controls for the podcast appear beneath the description on the EDUCAUSE page.

The Mathematics Genealogy Project and its cousins, the AI [artificial intelligence] Genealogy Project, and the Philosophy Family Tree are attempts to compile information about scholars in various fields, including where they received their degrees and the titles of their dissertations. The information is organized in an academic family tree, in which one’s adviser is one’s parent.

Here’s the mission statement for the Mathematics Genealogy Project:

The intent of this project is to compile information about ALL the mathematicians of the world. We earnestly solicit information from all schools who participate in the development of research level mathematics and from all individuals who may know desired information.

Please notice: Throughout this project when we use the word “mathematics” or “mathematician” we mean that word in a very inclusive sense. Thus, all relevant data from statistics, computer science, or operations research is welcome.

I’m actually in all three of these trees. My PhD is in Computer Sciences, specifically in AI; the core of the dissertation is in mathematical logic; and my adviser, Norman Martin, was a philosopher. His work was in the area of logic, as was that of a committee member, Michael Richter, a mathematician.

One of the best Christmas presents I received was a depiction of this tree made by Emily and Stephen (above, click to enlarge). There is so much detail, that you need to see the full-scale poster to read it all, but you may be able to make out the names of my adviser, and co-adviser, Robert F. Simmons, as well as early ancestors, Copernicus and Erasmus. It’s fun to explore the connections, which ultimately show how interconnected we all are.

After being inspired by George Reese’s work with Buffon’s needle, then seeing the movie, Le Pacte des loups (Brotherhood of the Wolf), then Buffon’s statue in the Jardin des Plantes, I’ve kept an eye out for Georges-Louis Leclerc, Comte de Buffon. He did amazing work in natural history, mathematics, biology, cosmology, translation, and essays. He also examned alleged specimens for the beast of Gévaudan, which provided the basis for the movie.

On Sunday we saw his house in Dijon (where he lived from 1717 to 1742), on naturally, rue Buffon.

Reasoning Under Uncertainty was a project funded by the National Science Foundation, under its EHR/Applications of Advanced Technology program, during the years 1985-91. The project led to a variety a publications and presentations (e.g., Rosebery & Rubin, 2007; Rubin & Bruce, 1991). Andee Rubin and I were the PIs, but the project eventually involved many other colleagues at Bolt Beranek and Newman, in Cambridge, Massachusetts.

A centerpiece of the project was the Reasoning Under Uncertainty curriculum, a 285-page guide for teachers at middle-school, high-school, and college levels. We developed the curriculum through extensive work with students at Belmont High School and Cambridge Rindge and Latin High School.

The curriculum conceived reasoning involving probability and statistics as central to both science and everyday life. One touchpoint was the daily Boston Globe, which never failed to have articles about elections, the economy, sports, fashion trends, and more, that could not be well comprehended without an understanding of concepts such as margin of error. Thus, we sought to expand the concept of literacy to include quantitative reasoning, or more generally, inquiry (Bruce & Davidson, 1996).

Using both online and offline materials, students could work with existing datasets, such as one demonstrating the relationship between cricket chirps and temperature, or another from Boston magazine showing income, education, and health patterns for Boston-area neighborhoods. They could also collect and analyze their own data, such as license plate data from cars passing through a traffic rotary, which could be used to determine town location of the cars. (more…)

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