Why I hated the New Math (published 22 Feb 1984)

Published in the Arkansas Gazette, on the op-ed page, February 22, 1984.

If a can of light beer contains 96 calories and this amount is one-third fewer calories than the amount in a can of regular beer, how many calories are in a can of regular beer?

Light beer is too light for my taste, and so is most American beer, but the label on a can of light beer recently provided me with the above problem.  It is a simple algebraic problem.  It can be solved without setting up an equation, but the basic thought process involved is nevertheless a kind of implicit way of doing algebra.

Before readers with a distaste for algebra start scattering like a covey of quail taking flight, I should say that I too once loathed any implicit or explicit form of algebra.  After I loathed it until I could loathe no more, I started ignoring it.  Unfortunately for me, I was enrolled in an Algebra II class in high school at the time.

About three years after that fiasco, spelled with a capital ‘F,’ I enrolled in a basic electronics class at a vocational-technical school.  The night I first opened the book I’d bought for that class was the first night I’d ever wanted to stay home with a textbook instead of going out with friends.  The book had some simple equations in it, but they were equations that not only made sense to me, they also made want to learn more about the subject.

Since then, that is what I’ve been doing.  And learning more about electronics at the theoretical level, where my main interest lies, has meant learning a lot of math and physics, both of which I now enjoy doing.  (I think “doing” is the proper word to use.  I’m not just studying math and physics at school—I’m using them every day and learning more about them on my own.)

So the question I am now pondering is:  Why did I not become at least slightly interested in algebra when I was first exposed to it?

My answer to that question may be of interest to those people who are now trying to make some improvements in the teaching of mathematics.  My answer is that  I could not swallow the so-called “new math.”

My problem with “modern” algebra was my inability to accept the abstract concepts presented by the textbooks.  The concepts were presented as postulates, from which proven statements were later extracted.  I could not muster any interest in the postulates because to me they were dogmatic proclamations with no immediate relevance.  My fellow students and I were, in effect, being told not to do any thinking for ourselves until all the laws had been laid down—until, in other words, we had been told how to think.

We were, however, supposed to be impressed with what we were told.  My high school algebra book, at the end of the first paragraph of the first chapter, told me that it would be my “privilege” to learn about  sets and about the “postulational basis of algebra,” both of which had in the recent past been studied only in graduate courses at the great universities!

These are not words that impress a high school student.  They more likely impress only the textbook writing team that wrote them.

I don’t think high school students should even have to put up with words like “postulational,” but the new math textbooks are, or at least were, full of such graduate math course verbiage.

The new math was not much of a success with my fellow high school students either.  I seem to remember them complaining about having to do  an occasional “word problem” or two.  I now think we were being taught badly, mainly as a result of the textbook’s emphasis on abstract algebra, because word problems are the only real problems.  The rest of algebra is learning and applying rules for manipulating numbers and learning to solve equations.  Learning the rules is certainly necessary, but if a person cannot apply those rules to real problems, what is the use of teaching him or her the rules?

In my case, if someone had presented me with a practical problem and allowed me to try to solve it, and then had helped me understand how algebra could be used to solve it, I would have developed an appreciation for math much earlier in my life.

I hope that now, with a renewed interest in improving science and math teaching, students will b e taught how to use mathematics.  Indeed, the only way to learn it is to use it, and to practice using it, and the same thing applies to learning physics.  What is needed now is not an emphasis on “computer literacy,” but an emphasis on mathematical literacy.  A computer is a very easy tool to learn to use; it gives immediate response to one’s programming errors.  Developing mathematical problem-solving skills takes years.  There is no way to do it other than to work a lot of problems, and that is not as much fun as computer programming.

For the most part, working problems is done with a pencil and paper, and a good eraser.

So now we are left with the big question:  how many calories are in a can of regular beer?  My answer is 144.

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Let's hope Common Core Math is better than the New Math. It seems to be better from what I saw when I looked at the website.  By the way,"one-third fewer calories" on the can of Lite beer is a sort of trick.  People I asked about it often thought it meant one-third the calories, but the word "fewer" should tell you differently.  What it means is more commonly stated in percentage terms, as in 33% fewer calories. This is like a piece of clothing in a store on sale for 33% off.  Which means you pay 2/3 of the original price.  And so with Miller Lite:  it has 2/3 the calories of regular Miller.

The algebra way, however, is the way I thought of it.  Let x be the unknown regular beer calories.  Then x minus one-third of itself is equal to 96,

x - x/3 = 96  or taking out common x factor,   x(1 - 1/3) = x(2/3) = 96, and there's your 2/3 amount. 

Review of A Brief History of Time 1988

This is the manuscript copy of my review (see scanned pages below). I'm not sure if I ever got a copy of the printed version. It was published sometime in late 1988, in Spectrum, in Little Rock, Arkansas.  I was living in Austin, Texas at the time. Maybe I'd let my subscription to Spectrum expire, since money was difficult to come by after I got fired from my technical writing job in September of '88.  See other posts below for more info on Spectrum and some of my other articles that appeared in it. 

Update, July 2014:

In my review, I mention an error I found on page 21 of A Brief History of Time.  (See pages 3 and 4 of my manuscript.)  I checked the Pine Bluff library's copy of A Brief History of Time and the error had been corrected in it, although it isn't a newer edition than my copy or even a later printing, as far as I can tell.  But in the process of comparing the two copies, I noticed mine is a Book-of-the-Month club printing, given to me in April 1988 by Karen Jo (we were married at the time). So it was probably printed even earlier than the normal first printing and thus had that uncorrected error (and others too, probably) in it. The book can now be read in its entirety on the Web.  Here's the uncorrected sentence from page 21 in my copy of the book:  "The time it has taken is, after all, just the light's speed--which the observers agree on--multiplied by the distance the light has traveled--which they do not agree on." The problem: time is equal to distance divided by speed, no matter what speed you're talking about.  Here's the corrected sentence, as it appears on page 12 of the link I provided above:  "The time taken is the distance the light has traveled--which the observers do not agree on--divided by the light's speed--which they do agree on."  Hawking, as is his way, likes to joke about the publication date of the book being April 1st 1988, which I guess was his choice.  "Black Holes Ain't So Black" is chapter 7 of the book, not chapter 4 as I say in my review.  Hey, we all make mistakes.  The big question is will they be discovered, or admitted to, and corrected?

 

"The Power of Crystals"

By Elizabeth F. Shores and David Trulock

Shamaan Ochaum is a dream therapist from Austin, Texas who brings groups of clients to Mt. Ida. Together, they sleep outside on Fisher Mountain, where Ochaum says the underground veins of quartz crystals emanate powers which enable them to "increase the dream experience." The place has "a very intense life charge," which fosters a sense of peace and well-being, she said.

Ochaum claims to be the daughter of a Shoshoni medicine man, and said she is carrying on the tradition of using quartz crystal as a tool in healing.

Quartz crystal is unique, she believes, because it emanates electromagnetic energy in an unchanging pulse. This energy has a healing effect on physical ailments, Ochaum believes, as when she carefully moves a crystal upward along a person's spine. It can also be used to gain a "vision" of the interior of the human body, she said, in order to locate and identify illnesses such as ovarian cysts, and can even help a person move into an altered state of consciousness.

Whether or not one believes in quartz crystals as sources of supernatural power, there is no question that the stones do possess scientifically proven natural powers, governed by the laws of physics.

Many crystals, including quartz, are piezoelectric, and it is this property that makes them so useful. 'Piezo" means pressure, so piezoelectricity means "pressure-electricity." Putting pressure on a piezoelectric crystal generates a voltage; conversely, applying a voltage generates internal pressure, causing the crystal to change its physical shape very slightly.

Because sound is a variation in air pressure, piezoelectric crystals have been used for many years in the recording and reproduction of music. Microphones, phonograph cartridges and high-frequency "tweeters" in loudspeakers are examples of devices that have exploited the piezoelectric effect.  Improved magnets or magnetic fluids have replaced piezo crystals in some applications, however, since the dynamic range and frequency response of crystals are not ideally suited to audio.

It is, in part, this physical quality of crystals that prompts some to believe that they possess special powers. Ochaum described these benefits of quartz while attending the Fourth Annual Quartz Crystal Festival at Mt. Ida October 24. Crystal dealers came from around the country, and the festival offered $1,000 in cash prizes for big crystals in a Championship Quartz Crystal Dig competition. There was also a quilt show.

Ochaum grants that these mystical effects of crystals are not empirically measurable.

"Science is skittish about doing research because it is so subjective," she said of crystals' alleged powers. And understanding their power is something of a balancing act, because some of the power may come from the stone itself, while some may come from the person's faith in it.

Don Owens, a geologist at the University of Arkansas at Little Rock, does not reject the claims of quartz's mystical powers, but, he added, "I disbelieve more than I believe."

"There are obviously some people who have an inner power, such as clairvoyants," he said. "I don't totally disbelieve it."

But others are more skeptical. "I'm a Christian. I put my faith and trust in the Lord. I don't need a durn rock," said Jimmy Reynolds of the National Forest Service. "I think they're kooks."

Mike Howard, a geologist with the Arkansas Geological Commission, has heard "a never-ending series of wild claims" about the powers of quartz. "I don't believe it, but that's neither here nor there.... A placebo in any form can get results."

But apart from faith-healing and audio reproduction, there are other uses for crystals as well. Quartz in particular is suited for very high-frequency applications, in the range of millions of vibrations per second. The ubiquitous "quartz watch" is one such application. A tiny sliver of quartz, with its high degree of geometric order, has certain natural vibrational frequencies called resonant frequencies. When an electrical frequency corresponding to a resonant frequency is applied to a piece of quartz, the vibrations are extremely stable over long periods of time and are relatively unaffected by changes in temperature.

The stability of the quartz time-base is used in computer circuits, providing the clock signal that determines how fast the computer operates and how well all the operations are synchronized. Quartz resonant frequencies are used by radio and TV stations to maintain their assigned frequencies, and constitute the basic tuning standard in most newer radios and televisions.

Also, the high-frequency vibrational modes of some piezoelectric crystals are ideal for generating and detecting ultrasonic underwater sound, an activity that has become more important in this age of ultra-quiet, nuclear-armed submarines that can avoid conventional sonar.

Although large single crystals are rare and valued for their beauty, microscopic crystalline structure itself is quite common. All rocks and metals exhibit a microscopic crystal structure. Salt, sand, snow and ice are examples of different forms of crystalline structure. The silicon computer chips produced and designed in California's Silicon Valley provide an example of the practical importance of artificially grown crystals, and modern electronics at its most fundamental level is nothing more than the study of how electrons behave in crystals.

(2014 notes:  I had recently moved to Austin when this article was published in December 1987 in Little Rock (in the weekly alternative newspaper Spectrum).  I wrote only the physics-related parts, at the request of the editor. I had never heard of Shaaman Ochaum, and still haven't. In the summer of 1997 I made copies of this article and gave them to students in a second semester conceptual physics class I was teaching at Southwest Texas State University in San Marcos, which is now called Texas State University-San Marcos.  Several students asked me after class if I knew how to get in touch with Shaaman Ochaum.  They were disappointed when I told them I didn't. This article was a side-bar to a larger, front-page article by Elizabeth Shores on crystal hunting and mining in the Ouachita National Forest near Hot Springs.)

Tickling the Dragon's Tail at Los Alamos

Nuel Pharr Davis describes Slotin's job: 

Slotin sat at his desk amid a clutter of radiation counters. He laid the bomb-metal pieces on his desktop far enough apart to be out of range of each others' neutrons. Then, with a screwdriver, he slowly pushed them together. As he did so, each piece responded to neutrons from the other by emitting more neutrons. The effect was a progressive multiplication of neutrons that, if charted, would appear as an ascending curve. Slotin did not push the pieces all the way together. He stopped as soon as in his mind's eye he could extrapolate the ascending curve into the vertical straight line that would stand for neutron multiplication in a detonating bomb.

 

Interested only in Dragon

Davis describes Slotin as "A thin, short, blue-bristled man who usually looked in need of a shave, [who] at age thirty-one had dark circles under his eyes, which contrasted oddly with the heavy tan of his sunken cheeks. He seldom spoke with much animation except about the Dragon and about his other interest, the extrapolation of blast and radiation casualties."

The extrapolation of radiation casualties apparently was something that not many of the men or women at Los Alamos thought about very much. They knew there would be radiation casualties, but they severely underestimated the extent of the casualties, according to some recent reports.

Perhaps some insight can be gained from Davis' book, although one must tread lightly: the book was criticized by two knowledgeable reviewers because of some wrongly-credited and other possibly-invented quotations.

Davis writes that most of the scientists at Los Alamos, the weapons lab several hundred miles north of the Jornada del Muerto, let Robert Oppenheimer, their leader, take "protective custody of their emotions." Others may have had no emotions that needed custody. At any rate, before the Hiroshima born was dropped, only a few Los Alamos scientists showed any emotional reaction to the consequences of the use of the bomb. Slotin was one who showed emotion,  but not much in the way of sympathy.

In The Uranium People, a sort of anecdotal history of the beginning of the nuclear age, published in 1979, Leona Marshall Libby, a Manhattan Project scientist, wrote about Slotin. She had known him from their days together at the University of Chicago:

He and I were laboratory assistants in a second year physics course that went on several hours two afternoons a week.  There were hours at a time with nothing for us to do except be there, so we spent the time talking, leaning against the warm steam radiators looking out at the snowy campus or watching the tiny spring leaves growing.

He had spent several years in Spain as member of the Abraham Lincoln Brigade [in the Spanish Civil War].  It had been for him a marvelous adventure as well as a crusade.

 

Slotin Angered Fermi  

Davis contrasts Slotin with Enrico Fermi, who sometimes took breaks from his theoretical calculations by working on a different experiment in the same building with Slotin. "Fermi had a hard, clear mind," David writes, "which liked to wrestle with only limited questions, such as the behavior of slow and fast neutrons and high-energy particles. Slotin's behavior angered him by raising questions of a different order."

Fermi, who died of cancer in 1954 at the age of 53, seemed to be one whose emotions needed no protective custody. Slotin, on the other hand, might have benefitted from some emotional assistance, but would not have accepted such an abrogation of responsibility, if Davis is correct.

Nonetheless, Oppenheimer seemed to understand Slotin, perhaps because he felt much the same way. The following paragraph from Lawrence and Oppenheimer offers a vignette of Slotin and also some insight into the reason for Oppenheimer's success with the atomic bomb:

In its frustration at having become the physicists' test animal, the human race should find a certain comfort in the thought that Slotin knew what he was doing. He could not have passed the most elementary personality-profile test of the kind now routinely used in government and industry. By giving Slotin responsibility and by going to relax in spiritual rapport with him in Omega [the building where the Dragon test was performed], Oppenheimer too outraged present-day administrative standards. His reason, of course, was that he wanted to build the bomb. If he had employed only sound, wholesome organization men for his project, Los Alamos would still be designing impressive remote- control machines with which to check its first implosion assembly.

Cursed Already

After the Hiroshima bomb was dropped, things did not go well at Los Alamos. On October 16, 1945, Oppenheimer's last day as director of the laboratory, he accepted an award from the Army for the laboratory's contribution to ending the war and said, in a speech reproduced in Weapons and Hope (see previous post below), "If atomic bombs are to be added to the arsenals of a warring world or to the arsenals of nations preparing for war, then the time will come when mankind will curse the names of Los Alamos and Hiroshima."

Davis takes his cue from Oppenheimer and writes:  

The men in Omega showed they knew they were cursed already. Early in August, one low-grade Omega technician died in a chemical explosion and two were blinded. No further uranium bombs could be made for months, but a third plutonium bomb had been readied for the Dragon check. Slotin, sulking because Oppenheimer would not let him go see Japanese casualties at first hand, took a holiday, leaving his chief assistant Harry Daghlian, to undertake it. Daghlian, also thin, dark and morose, got an overdose of radiation and died on September 15. Slotin performed the check and several others, then next spring on May 21 gained what he seemed to long for. Poking the segments of the Bikini test bomb a little too close together, he set up a blue ionization glow in the room. Lunging from his chair, he covered the segments with his body until a half dozen observers could file out.

By that time it was too late for Slotin, but the most talented woman physicist on the Mesa, Elizabeth Graves, was asked by telephone to compute the chances [for survival] of a man who had watched with his hand on Slotin's shoulder ... Elizabeth Graves was a no-nonsense type -- Hiroshima, she used to say, was no worse than napalm. Methodically she began punching a calculator, then learned from another telephone call that the subject was her husband. 'My mind went blank,' she said. 'I couldn't do the simplest sums in arithmetic.' Graves survived with cataracts. Something like superstitious terror halted further necessary Dragon checks until entirely different methods could be devised from those Oppenheimer had kept going with complete safety during the war.

Slotin died within days of acute radiation sickness.

Notes: The caption on the photo should say the Bikini Atoll atomic bomb test of 1946, not 1948. Slotin made up the story about fighting in the Spanish Civil War. Info about Slotin, Daghlian, radiation sickness, and the "demon core" that killed both Daghlian and Slotin can be found at Wikipedia's Louis Slotin webpage.