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327 On a Clear Day . . . Doris Glass You can see seven-pointed star-fish floating under the pier if you look hard enough. Ordinarily, they rest on the bottom, but when they turn over so their white, light side is up they will slowly float to the surfaceif you wait long enough. It takes a lot of patience, but the reward is great. Sir John Tenniel Artist ’The time has come.’ the Walrus said, ’To talk of many things: Of shoesand shipsand sealing wax Of cabbagesand kings Through the Looking Glass by Lewis Carrol! It was this kind of patience that enab- led Isaac Newton to time the swing of the censer by his pulse during the service at Canterbury Cathedral. He noticed that the swing was in phase with the pace of the priest carrying it. Suddenly, it occurred to him that the length of its chain was about equal to the length of a leg. Sir Isaac’s pulse rose from its usual 60 per minute as he suddenly realized that here was the long sought for connection between length and time needed to complete the new international scientific metric system of units. Shortly thereaf- ter, as Warden of the Royal Mint, he established the standard length of the rod of a pendulum hung in the Tower of London that resulted in a period of exactly one second to be 39.37 inches, which he named the meter (Latin metron). The important principle here is that this space-time relationship is independent of the mass of the bob. The French Academy, in the spirit of international competition, immediately defined this length in terms of the diameter of the earth, and named it the metre. After Newton’s death in 1727, it was discovered that his great grandfather had made a clock at the ancestral castle in Scotland that had a pendulum of exactly this length, but it is doubtful that his grandfather was that great. All this led me to observe at dawn one April morning as we started our beach w^lk, "We can still see the white face of the spring full moon overhead. Have you ever won- dered why the moon is exactly the right size to cover the sun during an eclipse?" School Science and Mathematics Volume 86 (4) April 1986

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On a Clear Day . . .Doris Glass

You can see seven-pointed star-fishfloating under the pier if you look hardenough. Ordinarily, they rest on thebottom, but when they turn over so theirwhite, light side is up they will slowlyfloat to the surface�if you wait longenough. It takes a lot of patience, butthe reward is great.

Sir John TennielArtist

’The time has come.’ the Walrus said,’To talk of many things:

Of shoes�and ships�and sealing wax�

Of cabbages�and kings�

Through the Looking Glass

by Lewis Carrol!

It was this kind of patience that enab-led Isaac Newton to time the swing ofthe censer by his pulse during the serviceat Canterbury Cathedral. He noticedthat the swing was in phase with thepace of the priest carrying it. Suddenly,

it occurred to him that the length of its chain was about equal to the length of aleg. Sir Isaac’s pulse rose from its usual 60 per minute as he suddenly realizedthat here was the long sought for connection between length and time needed tocomplete the new international scientific metric system of units. Shortly thereaf-ter, as Warden of the Royal Mint, he established the standard length of the rodof a pendulum hung in the Tower of London that resulted in a period of exactlyone second to be 39.37 inches, which he named the meter (Latin metron). Theimportant principle here is that this space-time relationship is independent of themass of the bob. The French Academy, in the spirit of international competition,immediately defined this length in terms of the diameter of the earth, and namedit the metre.

After Newton’s death in 1727, it was discovered that his great grandfather hadmade a clock at the ancestral castle in Scotland that had a pendulum of exactlythis length, but it is doubtful that his grandfather was that great. All this led meto observe at dawn one April morning as we started our beach w^lk, "We canstill see the white face of the spring full moon overhead. Have you ever won-dered why the moon is exactly the right size to cover the sun during an eclipse?"

School Science and MathematicsVolume 86 (4) April 1986

328 On a Clear Day

"No, I haven’t. That’s just an example of Cole’s Law�if the moon were big-ger, it would be farther away, and if it were smaller, it would be closer. The ratioof size and distance of sun and moon is a constant�400 to one."

"I see. Butwhy400?""In the metric system the right angle is divided into 100 parts, called gradients,

w^hich means a circle has 400 of them.""The man-in-the-moon looks sleepy. Could \\e hear him if he starts to

snore?""No, because there is no air to transmit the sound waves. No matter how loud

he might snore, our audiometer would read zero decibels."We continued walking in silence. My thoughts returned to the historical devel-

opment of systems of units. For example, we are becoming aware that 55 milesper hour is about 88 kilometers per hour, but how many know that the basis forthe mile kilometer ratio is Fibonacci’s Golden Mean? And wouldn’t a more ra-tional unit of mass be one cubic centimeter of a silver alloy having specific grav-ity 9.8, so that the acceleration of gravity is one meter per second per second, andthe confusion between weight and mass avoided?

Yes, we live in interesting times. Zero times any number no longer equals zero,and division by zero is now defined. For example, zero favorable events in asmall sample is larger than zero favorable events in a large one�0/100 < 0/10,and 100/0 < 10/0. But much remains to be done. Define has not yet been de-fined�for the moment, those who contend that define cannot be defined until itis�(a knife cannot cut itself) seem irrefutable. The difference between zero andinfinity becomes more important than ever�zero is approachable and attain-able; infinity is approachable and ^attainable.A plot of the Pythagorean theorem looks like this:

Ax+Ay

\Tl����4 a^b^c2+

-(p���������Ax

In education strange things are happening. SAT and ACT scores are rising;college grades are falling. Physics textbooks, completely new, and new editionsof the old, appear like dandelions in a neglected lawn. Teacher supply goes fromboom to bust. There are many educators who have talent, but few possess geniuswhere those with talent can do hard things, and those w4th genius can make itlook easy to do hard things. There is more and more talk of teaching about

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On a Clear Day 329

thinking, and less and less talk of thinking about teaching. What would I do?1. I would do for the teaching profession what our founding fathers did for

our nation when they established age requirements for members of Congress. Isthe qualification for maturity to be any less? Thus teachers K-12 would be notless than 25 years old (the requirement for members of the House is age 25 ormore), and for beyond 12th not less than 30 (the Senate requirement is 30).School principals and college Presidents�35.

2. The four years between graduation from college w4th a Bachelor’s Degreewould be spent about one-half doing practice teaching (with some substitutetime), and one-half in industry. The goal of this requirement is to forestall twounfortunate developments, which occur all too frequently under present proto-cal: a) Staying in teaching but continuing to wonder about missing out on thewonderful opportunities for interesting wwk and big pay "out there’*; b) Leav-ing teaching�and regretting it!

3. All textbooks would be banished to the reference section of the school li-brary. A leading educator tells us:

One can deplore some aspects of the dominant role that the textbook occupies. In-structors, particularly in introductory courses, find that their freedom to choose orconstruct a syllabus is powerfully influenced by whether or not there is available atext that roughly corresponds to it. (And, if not, they may end up by writing theirown!) Students, for their part, tend to have an excessive dependence on, and a crav-ing for, a single book that will present all that they are required to know in a givencourse. Giving them a list of several books that they might refer to tends to be met

with perplexity and even resentment.

So I’d have teachers (professors too) teach from their own notes, taken whenthey recently took the course they are teaching, with a 2?+ or better\. Studentswould take notes to use during tests.

Present realities and future prospects? Do you recall being told by leading edu-cators that: Textbooks are terrible. Publishers are polluting the minds of ourchildren with dumbed down instructional materials designed to appeal to thelowest common denominator of teacher and pupil. As long as purchasers of text-books continue to buy thinly disguised, unchallenging, and uninteresting intel-lectual junk food, the publishers w^ill continue to produce it.

Any plan for meaningful educational change must affect a whole spectrum ofeducational power. There must not only be alternative educational models, but alsoa legal campaign to change educational laws, a fight for a different system of educa-tional funding, an understanding of the need for children’s rights, an emphasis onwomen’s rights and changing the structure of the family, and a campaign to changethe nature and direction of research in the schools of education of major univer-sities.

Such important educational issues as curriculum, content of textbooks, and re-quirements for teacher certification are decided within an interlocking educational

School Science and MathematicsVolume 86 (4) April 1986

330 On a Clear Day

bureaucracy which includes professional organizations, state officials, universities,and publishing companies�not to mention the new learning corporations like IBMand Educational Testing Services, which represent the most important and rapidlygrowing parts of this bureaucracy.The concept of the school is out of date in modern technological society. The

school in the nineteenth century was viewed not only as a source of social control butalso as a center where all the materials of learning, books and teachers, could beconcentrated. With mass media and urban living there is no reason why a personshould not be able to learn the basic skills of reading, writing, and arithemetic justby growing and interacting within the community.One thing seems reasonable to predict. There will be a revolution. The school

mathematics curriculum, after some changes in the sixties, has now reverted to whatis almost a pre-1960 curriculum. There is no way that such an old curriculum canrepresent the needs and realities of today’s mathematical problem solving. It will beeven more out-of-date tomorrow.

What do you think?

Doris Glassc/o Editor, School Science and Mathematics

Editor’s Note:The clear day for this column was April First, a date traced back to ancient Indiawhen friends sent each other on false errands. If you have difficulty separatingfact from fiction, the author will be happy to provide details.

Aluminum Look To The 80’s

Aluminum, with a wide range of alloys, finishes and fabrication processes,looks to the 1980s as a period of design challenge and opportunity. As majormarkets shift their requirements and react to new technology and the need forgreater design efficiency, aluminum is striving to position itself for maximum ac-ceptance among designers.Aluminum is engaged in countering specific threats in some of its key markets

and aggressively attempting to further penetrate others. Competitive develop-ments and design directions, notably with advanced fiber composites, could sig-nificantly chip away at aluminum’s extensive long-term dominance in, for exam-ple, commercial aircraft. While 81 percent of the new Boeing 757 and 767 con-sists of aluminum, including new alloys such as 2224, 2324 and 7150, there is apotential for substantial reduction of the material in the next generation of sub-sonic commercial airliners. The aluminum industry, however, is not sitting backand waiting for this to happen. Much of this projection of aluminum replace-ment is based on drastic, yet unrealized, reductions in composite costs and on thealuminum industry doing essentially nothing to counter the threat. On the con-trary, specific developments from R&D could be a strong response to limit theimpact of composites on aluminum’s growth.

School Science and MathematicsVolume 86 (4) April 1986