A Manhattan Exhibit With Antiquity on the Clock

In a Roman mosaic from antiquity, a man on a street studies the sundial atop a tall column. The sun alerts him to hurry if he does not want to be late for a dinner invitation.

Sundials were ubiquitous in Mediterranean cultures more than 2,000 years ago. They were the clocks of their day, early tools essential to reckoning the passage of time and its relationship to the larger universe.

The mosaic image is an arresting way station in a new exhibition, ”Time and the Cosmos in Greco-Roman Antiquity,” that opened last week in Manhattan at the Institute for the Study of the Ancient World, an affiliate of New York University. It will continue until April.

The image’s message, the curator Alexander Jones explains in the exhibition catalog, is clearly delivered in a Greek inscription, which reads, “The ninth hour has caught up.” Or further translated by him into roughly modern terms, “It’s 3 p.m. already.” That was the regular dinnertime in those days.

Dr. Jones, the institute’s interim director, is a scholar of the history of exact science in antiquity. He further imagined how some foot-dragging skeptics then probably lamented so many sundials everywhere and the loss of simpler ways, when “days were divided just into morning and afternoon and one guessed how much daylight remained by the length of one’s own shadow without giving much thought to punctuality.”

An even more up-to-date version of the scene, he suggested, would show a man or a woman staring at a wristwatch or, even better, a smartphone, while complaining that our culture “has allowed technology and science to impose a rigid framework of time on our lives.”

Jennifer Y. Chi, the institute’s exhibition director, said: “The recurring sight of people checking the time on their cellphones or responding to a beep alerting them to an upcoming event are only a few modern-day reminders of time’s sway over public and private life. Yet while rapidly changing technology gives timekeeping a contemporary cast, its role in organizing our lives owes a great deal to the ancient Greeks and Romans.”

The exhibition features more than 100 objects on loan from international collections, including a dozen or so sundials. One is a rare Greek specimen from the early 3rd century B.C. The large stone instruments typically belonged to public institutions or wealthy landowners.

A few centuries later, portable sundials were introduced. Think of pocket watches coming in as movable timekeepers in place of the grandfather clock in the hall or on the mantel. They were first mentioned in ancient literature as the pendant for traveling. The earliest surviving one is from the first century A.D.

Six of these small sundials are displayed in the exhibition. These were owned and used mostly as prestige objects by those at the upper echelons of society and by the few people who traveled to faraway latitudes.

A bronze sundial in the center of one gallery is marked for use in 30 localities at latitudes ranging from Egypt to Britain. Few people in antiquity were ever likely to travel that widely.

A small sundial found in the tomb of a Roman physician suggested that it was more than a prestige object. The doctor happened to be accompanied with his medical instruments and pills for eye ailment, as seen in a display. Presumably he needed a timekeeper in dispensing doses. He may have also practiced some ancient medical theories in which astrology prescribed certain hours as good or bad for administering meals and medicine.

Apparent time cycles fascinated people at this time. One means of keeping track of these cycles was the parapegma, a stone slab provided with holes to represent the days along with inscriptions or images to interpret them. Each day, a peg was moved from one hole to the next. The appearances and disappearances of constellations in the night sky yielded patterns that served as signs of predictable weather changes in the solar year of 365 or 366 days. Not to mention when conditions are favorable for planting and reaping. Not to mention good or bad luck would follow.

For many people, astrology was probably the most popular outgrowth of advances in ancient timekeeping. Astrology — not to be confused with modern astronomy — emerged out of elements from Babylonian, Egyptian and Greek science and philosophy in the last two centuries B.C. Because the heavens and the earth were thought to be connected in so many ways, the destinies of nations as well as individuals presumably could be read by someone with expertise in the arrangements of the sun, the moon, the known planets and constellations in the zodiac.

Wealthy people often had their complete horoscopes in writing and zodiacal signs portrayed in ornamental gems, especially if they deemed the cosmic configuration at their conception or birth to be auspicious.

It is said that the young Octavian, the later emperor Augustus, visited an astrologer to have his fortune told. He hesitated at first to disclose the time and date of his birth, lest the prediction turn out to be inauspicious. He finally relented.

Why Math Education in the U.S. Doesn’t Add Up

Research shows that an emphasis on memorization, rote procedures and speed impairs learning and achievement

By Jo Boaler, Pablo Zoido | SA Mind November 2016 Issue

In December the Program for International Student Assessment (PISA) will announce the latest results from the tests it administers every three years to hundreds of thousands of 15-year-olds around the world. In the last round, the U.S. posted average scores in reading and science but performed well below other developed nations in math, ranking 36 out of 65 countries.

We do not expect this year’s results to be much different. Our nation’s scores have been consistently lackluster. Fortunately, though, the 2012 exam collected a unique set of data on how the world’s students think about math. The insights from that study, combined with important new findings in brain science, reveal a clear strategy to help the U.S. catch up.

The PISA 2012 assessment questioned not only students’ knowledge of mathematics but also their approach to the subject, and their responses reflected three distinct learning styles. Some students relied predominantly on memorization. They indicated that they grasp new topics in math by repeating problems over and over and trying to learn methods “by heart.” Other students tackled new concepts more thoughtfully, saying they tried to relate them to those they already had mastered. A third group followed a so-called self-monitoring approach: they routinely evaluated their own understanding and focused their attention on concepts they had not yet learned.

In every country, the memorizers turned out to be the lowest achievers, and countries with high numbers of them—the U.S. was in the top third—also had the highest proportion of teens doing poorly on the PISA math assessment. Further analysis showed that memorizers were approximately half a year behind students who used relational and self-monitoring strategies. In no country were memorizers in the highest-achieving group, and in some high-achieving economies, the differences between memorizers and other students were substantial. In France and Japan, for example, pupils who combined self-monitoring and relational strategies outscored students using memorization by more than a year’s worth of schooling.

The U.S. actually had more memorizers than South Korea, long thought to be the paradigm of rote learning. Why? Because American schools routinely present mathematics procedurally, as sets of steps to memorize and apply. Many teachers, faced with long lists of content to cover to satisfy state and federal requirements, worry that students do not have enough time to explore math topics in depth. Others simply teach as they were taught. And few have the opportunity to stay current with what research shows about how kids learn math best: as an open, conceptual, inquiry-based subject.
To help change that, we launched a new center at Stanford University in 2014, called Youcubed. Our central mission is to communicate evidence-based practices to teachers, other education professionals, parents and students. To that end, we have devised recommendations that take into consideration how our brains grapple with abstract mathematical concepts. We offer engaging lessons and tasks, along with a wide range of advice, including the importance of encouraging what is known as a growth mindset—offering messages such as “mistakes grow your brain” and “I believe you can learn anything.”

The foundation all math students need is number sense—essentially a feel for numbers, with the agility to use them flexibly and creatively (watch a video explaining number sense here: https://www.youcubed.org/what-is-number-sense/). A child with number sense might tackle 19 × 9 by first working with “friendlier numbers”—say, 20 × 9—and then subtracting 9. Students without number sense could arrive at the answer only by using an algorithm. To build number sense, students need the opportunity to approach numbers in different ways, to see and use numbers visually, and to play around with different strategies for combining them. Unfortunately, most elementary classrooms ask students to memorize times tables and other number facts, often under time pressure, which research shows can seed math anxiety. It can actually hinder the development of number sense.

In 2005 psychologist Margarete Delazer of Medical University of Innsbruck in Austria and her colleagues took functional MRI scans of students learning math facts in two ways: some were encouraged to memorize and others to work those facts out, considering various strategies. The scans revealed that these two approaches involved completely different brain pathways. The study also found that the subjects who did not memorize learned their math facts more securely and were more adept at applying them. Memorizing some mathematics is useful, but the researchers’ conclusions were clear: an automatic command of times tables or other facts should be reached through “understanding of the underlying numerical relations.”

Additional evidence tells us that students gain a deeper understanding of math when they approach it visually—for instance, seeing multiplication facts as rectangular arrays or quadratic functions as growing patterns. When we think about or use symbols and numbers, we use different brain pathways than when we visualize or estimate with numbers. In a 2012 imaging study, psychologist Joonkoo Park, now at the University of Massachusetts Amherst, and his colleagues demonstrated that people who were particularly adept at subtraction—considered conceptually more difficult than addition—tapped more than one brain pathway to solve problems. And a year later Park and psychologist Elizabeth Brannon, both then at Duke University, found that students could boost their math proficiency through training that engaged the approximate number system, a cognitive system that helps us estimate quantities.

Brain research has elucidated another practice that keeps many children from succeeding in math. Most mathematics classrooms in the U.S. equate skill with speed, valuing fast recall and testing even the youngest children against the clock. But studies show that kids manipulate math facts in their working memory—an area of the brain that can go off-line when they experience stress. Timed tests impair working memory in students of all backgrounds and achievement levels, and they contribute to math anxiety, especially among girls. By some estimates, as many as a third of all students, starting as young as age five, suffer from math anxiety.
The irony of the emphasis on speed is that some of our world’s leading mathematicians are not fast at math. Laurent Schwartz—who won math’s highest award, the Fields medal, in 1950—wrote in his autobiography that he was a slow thinker in math, who believed he was “stupid” until he realized that “what is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant.”

A number of leading mathematicians, such as Conrad Wolfram and Steven Strogatz, have argued strongly that math is misrepresented in most classrooms. Too many slow, deep math thinkers are turned away from the subject early on by timed tests and procedural teaching. But if American classrooms begin to present the subject as one of open, visual, creative inquiry, accompanied by growth-mindset messages, more students will engage with math’s real beauty. PISA scores would rise, and, more important, our society could better tap the unlimited mathematical potential of our children.
This article was originally published with the title “Why Math Education in the U.S. Doesn’t Add Up”

Reference:  https://www.scientificamerican.com/article/why-math-education-in-the-u-s-doesn-t-add-up/