by Sharon Proctor
Photograpy by Curtis Trent

"She's awesome."
"She's incredible at explaining things."
"She truly enjoys teaching math and wants every student to succeed."
"Prior to her course I was terrible at math; now it's easy and I don't mind it."

The subject of this student praise is SFU's award-winning mathematics lecturer Malgorzata Dubiel. She is a 2011 YWCA Woman of Distinction, recipient of the 3M National Teaching Fellowship, an SFU Excellence in Teaching Award, and the Pacific Institute for the Mathematical Sciences Education Prize.

She has a special gift – the ability to convince people who dislike math that it can be a source of joy and success. Not only is familiarity with math vital to our society's future, but most new jobs will require either math skills or the problem-solving abilities that are stimulated by studying math. Dubiel has been working tirelessly to convert teachers and students to her way of thinking.

The fact is our entire modern lifestyle arose out of counting, measuring, algebra, geometry, trigonometry, and higher mathematics. Computers, food production, electric power, cars, condo towers, movies, TV, iPods, medicine, election results, lotteries, calendars, roller coasters, highways, airplanes, cars, stock-market reports – you name it and math was involved in its development. So the student who hates or fears math, or who avoids taking it in high school (or fails to take it seriously at any level), will lose access to countless career opportunities. For Dubiel this is unacceptable.

What is mathematics?
Math is about numbers, shapes, and patterns. It's also a way of thinking – and a language. We use math as a tool to solve problems and puzzles, to analyze trends, and to think logically. And math invariably helps us see the "bigger picture." Using statistics, for instance, we can see patterns in weather, bird migration, voter registration, TV viewing, electricity use, shopping, disease incidence, and other phenomena.

Then there are those formulas and equations students find in textbooks. Think of them as "recipes," akin to those in cookbooks. Just as a recipe describes the steps and ingredients needed to make a chocolate cake, so does each math formula or equation describe the steps and measurements needed to solve a problem. Both recipes and formulas are shortcuts to achieving specific results. We're lucky to have them! They save us a lot of trial and error to "get it right."

"Another area is geometry," says Dubiel. "It's important in fields like design, art, architecture, engineering, and computer science. Today's animated movies and games, for instance, are developed using computers and geometry."

The familiar adding, subtracting, multiplying, and dividing, plus fractions, decimals, and algebra, are the basic building blocks of math. But the higher one goes in math, the more creative, fun, and interesting it becomes. To convince young people of this, Dubiel first helps them overcome their anxieties.

Why are so many of us "math averse"?
Negative feelings about math seem to be a societal trend in North America. In fact, many people brag about their poor math skills, and parents and schools often pass on this attitude to children and teenagers. For example, at some schools girls may get the message that math is "too hard for girls," and "unfeminine." Boys, meanwhile, learn that math is for "nerds." And if a student does excel in math, he or she can be ridiculed by peers. "In our society," says Dubiel, "we're quick to see mathematicians as 'geeks' or weird people."

Then there's the modern trend toward instant gratification. Placed in front of a TV set as toddlers (to occupy their time), a lot of young people never learn self-discipline and end up with little curiosity about the world, let alone the skills to satisfy that curiosity. They prefer video games, texting, loud music, MPEG devices, and television. Add to this the current trend of ensuring that kids should "feel good" about themselves – the "every child wins" philosophy. Mathematics doesn't lend itself to that view because when you solve a math problem, you're either right or wrong. There's no in-between "feel good" result. Two more influences are parents who don't encourage their kids to study, and teachers with little training or interest in the subject.

Today's world stands in stark contrast to Dubiel's own childhood. Born in the south of Poland and raised in Warsaw, she loved geometry as a youngster. Her father was an aerospace ("rocket") engineer who always made time to spend with her. He loved mathematics and engineering, and he taught his young daughter geometry and how to do technical drawings. Moreover, math was an important subject in the schools she attended. "In Poland," she explains, "the schools didn't steer girls away from math the way some of our Canadian schools do."

Math is about numbers, shapes, and patterns. It's also a way of thinking  – and a language.

What will inspire young people to pursue math?
There's no question that learning mathematics can be hard. Very hard. It takes immense concentration, practice, and dedication to master algebra, geometry, trigonometry, analytical geometry, and calculus, for instance. But in fairness, the same can be said about hockey, football, teaching, figure skating, medicine, dentistry, acting, architecture, chess, or any other human endeavour. So how do we get kids to think positively about mathematics?

"There's no easy answer," explains Dubiel. "Such intervention takes time. What I try to do initially is let the students experience the unexpected – interesting puzzles, challenging problems, beautiful designs, stories, and anecdotes about people connected to mathematics and the ways mathematics is connected to our lives. I try to help people discover that they can do math and feel good about it. Interesting challenges are important." She likes to give students two types of problems to solve: those originating in real life, to which students can relate, and those that don't pretend to be real life, but are simply fun to solve.

Real-life problems
Dubiel suggests teachers take students outside the classroom to experience math. All around are buildings, walls, fences, grassy areas, sidewalks, trash cans, trees, gardens, to name just a few things. Students can walk around the school grounds, stroll in a park or forest, or explore the local neighbourhood, looking for patterns, shapes, and numbers. They can count, measure, or estimate quantities, distances, areas, and volumes. They can discuss how signs, buildings, garbage cans, flowers, leaves, and other things are constructed.

Older kids might then design a new park, a better school playground, or a vegetable garden, complete with dimensions, soil quantities, cement, grass seeds, plant types, spacing, and equipment. Or they can design a swimming pool. Younger kids can measure tree trunk circumferences, count bushes or birds, estimate tree heights, and document the different shapes they see. The very young can search for items smaller than their feet, find things that are one-centimetre long, count benches and sidewalk squares. There's a whole world of opportunities for counting, measuring, geometry, and trigonometry.

"Fun" problems
Fantasy puzzles and riddles can be a lot of fun to solve. Young children especially love stories and fairy tales, many of which offer settings for wonderful math exercises. For instance, this is one of Dubiel's favourite riddles, inspired by the folk tale 1001 Nights:

• One day a man brought in 59 jewels to sell to Abdul. Some were emeralds and some were rubies. The emeralds were carried nine to a bag, and the rubies four to a bag, with no bag containing both emeralds and rubies. If all the bags were full, how many of the jewels were rubies?

Can you figure out the answer?

Here's another one she likes:

• One thousand and one pennies are arranged in a row on a table. Every second coin is replaced with a nickel. Then every third coin is replaced with a dime. Finally every fourth coin is replaced with a quarter. What is the total value of the coins on the table? (See end of story for answers.)

Sometimes Dubiel uses familiar animated cartoons to illustrate math concepts. One of her favourites features The Simpsons. In one episode Homer Simpson falls into a three-dimensional world filled with spheres, cones, and other shapes. He sees himself changing from a flat cartoon character to a round three-dimensional one. He exclaims, "What's going on here? I'm so bulgy!" The cartoon is filled with math concepts and jokes. "Oh, there's so much I don't know about astrophysics," laments Homer. He goes on to say that if he knew more math, he could explain the frightening situation he's in.

Some final thoughts
"Actually the human brain is wired to do simple math, to see patterns and symmetry," explains Dubiel. "Babies can tell the difference between one, two, or three objects. And they can tell when one is missing. Even crows notice when an object is missing from a set.

"Our distant ancestors gradually expanded their mathematical abilities because they needed to. This occurred independently in different groups, on different continents. They started keeping track by using strokes; then a number system was developed. Geometry arose from the need to divide lands and build structures. Traders needed a way to determine quantities."

Every field of human endeavour has rewards. But one would be hard pressed to find one that can match the sense of accomplishment that comes from solving an intriguing math problem – be it a practical problem or a fantasy puzzle. Thanks to Malgorzata Dubiel, more and more young people are experiencing this wonderful sense of accomplishment. aq

Answers to puzzles:
Jewel puzzle: 32 rubies
Coin puzzle: $99.19

The Joy of Puzzles

"People have always loved puzzles," explains Malgorzata Dubiel. "We love the challenge and the intellectual effort of trying to solve an engaging problem. Plus, we love the feeling we get when we solve a difficult one – the more difficult the problem, the better we feel! Many puzzles are, in fact, related to math concepts. For example, the dragon riddle below is related to algorithms; the golden rings puzzle relates to the binary number system."

Prince Igor and the dragon
Prince Igor has to fight a dragon that has three heads and three tails. With each blow of his magic sword, Prince Igor can cut off either one or two heads, or one or two tails of the dragon. But when he cuts off one dragon head, a new head grows in its place. When he cuts off one tail, two new tails appear. When he cuts off two tails, one new head appears. If he cuts off two heads, nothing new grows. The dragon dies only if he has no head and no tail. Can Igor win if he has enough strength for only 10 blows of the sword?

Golden rings
You arrive at a hotel with three sets of golden rings. The first set has four rings all joined together. The second set has two joined rings, and the third set has only one ring. You cannot take the joined rings apart, nor can you exchange them for a different form of currency. Alas, the hotel clerk has no change. You stay at the hotel for seven nights, and you have to pay one golden ring for each night that you stay. You cannot pay in advance, nor can you pay all at once at the end of your stay. How do you pay for your seven nights at the hotel? aq

ANSWERS
Prince Igor
Yes. Here is just one of several valid strategies:
Two heads
Two tails
Two heads
One tail
One tail
One tail
Two tails
Two tails
Two heads

Can you find a better solution?

Golden Rings
On the first day, give one ring. On the second day, give two rings and get one ring back in change. On the third day, give one ring. On the fourth day, give four rings and get the first three back. Continue paying in this way.

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