February 27, 2014 - Much of today’s curriculum encourages students to “explore” and “discover” mathematical concepts on their own. Vern Williams, a Virginia middle school math teacher whose student teams have been state champions for 24 of 29 years in a MathCounts competition, thinks such approaches are ineffectual. “It took very smart adults thousands of years to develop modern mathematics, so it makes sense to teach it to students rather than get them to ‘discover’ it themselves”, said Williams.
Many teachers, students, and parents are frustrated by today’s mathematics curriculum. Canada has dropped below the Top Ten in International Math Education Standings, a change that seems to coincide with increasing parental concern that too few students are learning basic math skills.
In my personal career, I was a mathematics teacher, but I was also one of the early advocates for the use of computers in education. However, it is clear to me now, thirty years later, that the introduction and use of various technologies in classrooms has not been the panacea that it promised to be. Children still need strong teachers who are comfortable with their subject material, who care about their students, who value good effort, and who demand excellence.
I have witnessed the introduction of calculators into math classrooms and the subsequent changes to the curriculum that saw the elimination of the teaching of long division by traditional algorithms. Schools installed computer labs, and math teachers were told to take classes into the rooms to work with subject-related software – but they were not given sufficient training to develop their own expertise and confidence with the products.
A 2012 report from the University of Michigan showed that K through 8 teachers, with no math specialization, got only half the questions right on a base-line test aimed at determining whether they knew the material they were required to be teaching. Most teachers were aware of their own limitations. Only 10% of the non-math specialization K-8 teachers said they were “confident to teach all topics” in math. Now, think about the problem of making such teachers use technology to teach chunks of the curriculum! Add unfamiliarity with computers and software to a lack of confidence in mathematics topics and we have a recipe for disaster.
Problem-solving strategies sometimes seem like aimless wanderings. Students who were never taught basic arithmetic skills pull out a calculator in senior high school classes to add 8 + 13! Instead of learning basic computational skills to add, subtract, multiply and divide, they rely on calculators to do that for them – and are told to use “estimation skills” to determine if the calculator’s answers are in the “right ballpark”.
In my opinion, it’s harder to learn competent estimation skills than it is to learn how to add in the first place! I am happy to see that many provinces are re-examining the methodologies that are currently being used in our math classrooms in the hopes of correcting a situation that is unacceptable. There is a place for basic computational skills, conceptual understanding, problem-solving skills, the use of technologies, and multiple experiences: the problem is to find the correct balance. Perhaps we’ll get there yet. John Dewey, the father of American education reform, wrote, “Growth depends upon the presence of difficulty to be overcome by the exercise of intelligence.” To me, that means focus, effort, work, and excellence.
Johann Carl Friedrich Gauss, a noted German mathematician, physicist and astronomer (1777-1855) stated, “Mathematics is the queen of the sciences…”
“The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it.”
I am going to try to demonstrate the beauty of mathematics by writing a series of articles for CHASA’s E-Mag, but in a manner, that reveals this beauty without the need of an in-depth understanding of the subject. Many young people today only see mathematics as numerical computations and problem solving, so to them it seems rather dry, perhaps boring, and often difficult; they may not see the wonders that lie further down the road in this area. My aim is not to try to explain the various fields of study in mathematics but to introduce and discuss some of its interesting topics and stories.
Galileo Galilei (1564-1642), a world famous Italian astronomer and physicist who defended the theory that the earth circled the sun stated, “If I were again beginning my studies, I would follow the advice of Plato and start with mathematics.”
We all begin our lives by learning to walk, to talk and to count. Our first steps are the beginning of mobility and the freedom to explore our physical world. Our first words may just be the imitations of familiar sounds, but eventually the association is made that words describe things – and soon afterwards, reading opens new worlds to us through the understanding of the printed word. Counting, at first, may also just be the mindless repetition of words – but when the association is made that a number describes a quantity then the world of mathematics begins to open doors of understanding.
Albert Einstein remarked, "Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
Mathematics is a tool for working in the sciences, as is imagination and creativity. I hope to open a window to many of the wonders of mathematics and generate some interest and excitement.
There is so much mathematics in the universe: the shape of coastlines and clouds, the spirals of snail shells and galaxies, the paths of planetary orbits, the formulas that govern motion, and the equations that describe natural forces… Sometimes it seems as if the universe was constructed using mathematical principles and equations, and that humans are working to pry out its secrets by using mathematical skills.
Sir Isaac Newton, a remarkable English physicist and mathematician (1643-1727), said, “God created everything by number, weight and measure.”
Paul Dirac, a famous British physicist (1902-1984) who shared the Nobel Prize for Physics in 1933 said something similar: “God is a mathematician of very high order…” “He used beautiful mathematics in creating the world.”
An artist tries to capture the beauty of nature by painting a glorious landscape or an intricate still life. Mathematicians are artists too – using numbers and geometries to describe natural structures and processes. Sometimes a mathematical topic is developed that appears to have no practical application in the real world. Seemingly, just a mental exercise or diversion at first, it later turns out to be very practical and useful. The following story will help to illustrate this point.
The Binary Number System
The concept of “yin and yang” (which dates from the 9th century B.C.) in Chinese philosophy is used to describe how opposite forces are interconnected and interrelated in nature. Life and death, hot and cold, up and down, black and white, and on and off are some examples of this duality. Everything has yin and yang aspects. A shadow cannot exist without light, for example.
Joachim Bouvet (1656 – 1730), a French Jesuit missionary who visited China in 1685, saw Christian concepts in yin and yang – such as right and wrong, or good and evil – and used just the two digits ‘0’ and ‘1’ to help him express some of the pictorial hexagram representations that appear in the I-Ching, one of the oldest classical Chinese texts. The text of the I-Ching is written with figures called hexagrams, each of which consists of 2 three-line arrangements called trigrams. There are eight possible trigram arrangements. Each of the three lines in a trigram is either a Yin (a broken line) or a Yang (a solid line). Bouvet, by representing a Yin as ‘0’ and a Yang as ‘1’, could express the eight possible trigrams shown below as 000, 100, 010, 110, 001, 101, 011, and 111.
He introduced the I-Ching to his mathematical friend, Gottfried Leibnitz (1646 – 1716) and Leibnitz extended Bouvet’s numerical representations into a complete number system (the binary system), which he published in 1703. At the time, there was no useful application of the binary system. People tend to use the decimal system (also known as the Base Ten System), which is based on ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), for their counting and calculating. Who would use a system of counting and calculating based on just two digits (0, 1)? In 1703, the Base Two System was purely an academic exercise.
In another unrelated area of human activity – logic and reasoning – progress was made over the centuries by numerous philosophers to develop a structured form of logic. Symbolic logic was one of those areas of study that evolved from the early work of Aristotle (384 – 322 B.C.) in Greece. Many mathematicians worked on applying the techniques of formal logic to mathematics. In 1854, British mathematician George Boole (1815 – 1864) published a landmark paper detailing an algebraic system of logic that came to be known as Boolean algebra.
In 1937, Claude Shannon (1916 – 2001) wrote his master's thesis "A Symbolic Analysis of Relay and Switching Circuits" at MIT on the subject of electronics using both binary arithmetic and Boolean algebra to explain the basic operation of electronic devices. His thesis was the foundation of digital circuit design and an essential part of subsequent work on digital computers.
Digital computers operate on electrical and magnetic principles, in two states: on or off, open or closed, magnetized or not magnetized, electrical pulse or no pulse – any of which may be represented by a ‘0’ or a ‘1’ – a binary digit (a bit). Combinations of those bits form bytes, the basic units of digital information in modern computing and telecommunications. Binary arithmetic is the basis for the computations performed in the circuitry of today’s computers. Who would have thought that ancient philosophies from the Chinese and the Greeks, studied and symbolized by mathematicians over the centuries, would eventually become the foundations for the world’s most complex computational and communication tools? Complex tools – but based on a simple system of 1’s and 0’s developed over 300 years ago! People count with their ten fingers and use the decimal system; computers count with just "two fingers” and use the binary system.
E-Mag Mathematics Articles
You don’t have to choose to study mathematics in greater detail in order to appreciate the elegance, wonder, beauty, and excitement in the topics and stories to come in this series of articles. People begin the study of mathematics by counting and doing arithmetic – but few know where these studies may lead. Let me take you on a trip that will bring you to some of those destinations. My trip will take us into a look at Geometry, where patterns and shapes play key roles. Euclidean and non-Euclidean geometries will be touched on, as well as fractals. We may look at other special topics such as The Golden Rectangle, the numbers Pi and e, and ‘packing problems’. We may delve into the world of computers and binary arithmetic in greater detail – or examine some topics in Algebra.
Sit back and enjoy the marvellous ride of mathematics!
# 1 Euclidean Geometry - Issue 23 - April 1, 2014
# 2 Non-Euclidean Geometry - Issue 24 - May 1, 2014
# 3 Rational Numbers – Fractions, Decimals and Calculators - Issue 25 - June 1, 2014
Dave Didur is a retired secondary school mathematics teacher with a B. Sc. degree from the University of Toronto majoring in Mathematics and Physics. He was Head of Mathematics for over twenty years, as well as the Computer Co-ordinator and consultant for the Board of Education for the City of Hamilton. He served with the Ontario Ministry of Education for three years as an Education Officer.