CM . . .
. Volume XIII Number 17 . . . .April 13, 2007
The Great Number Rumble: A Story of Math in Surprising Places.
Cora Lee & Gillian O'Reilly. Illustrated by Virginia Gray.
Toronto, ON: Annick Press, 2007.
104 pp., pbk. & cl., $9.95 (pbk.), $18.95 (cl.).
ISBN 978-1-55451-031-3 (pbk.), ISBN 978-1-55451-032-0 (cl.).
Grades 4-7 / Ages 9-12.
Review by Thomas Falkenberg.
So, what is math? The real question is: what isn't. (p. 2)
"Math is nothing special," he's always saying. "It's everywhere and in everything, and we all use it, not just me." Well, one day, he had to prove it. (p. 3)
"Look at the beauty of this shell," she exclaimed. "You can't tell me this has anything to do with math."
"This shell," he said, "is a great example." (p. 48)
Well, everybody laughed, and the math program stayed on the curriculum. Ms. Kay started a math club, which is pretty popular. Sam, me, Emily, Oscar, Jen, Ralph, Natasha Rosa, and some other kids joined. Even Mrs. Norton came in to help! You wouldn't believe the stuff we got into: mazes and puzzles, higher dimensions, knot theory, logic and paradoxes, calculus, statistics, game theory, geometries I never even knew existed, and tons more. (p. 85)
I can show kids the cool side of math. Like when some new kid starts complaining that math is too hard or that it's only for geniuses or geeks when that happens, I tell them this story. And you know what they always end up saying? That's math? Wow go figure! (p. 88)
As the subtitle promises, The Great Number Rumble: A Story of Math in Surprising Places is a book about the surprising places in which mathematical ideas can be found in nature, sport, the arts, music, and other places of human living. These discoveries are embedded in a fictional story of a contest between the Director of Education, Lawrence Lake, and Sam, the new friend of Jeremy, who is the narrator of the story. Sam is in Jeremy's words "a regular guy who saw the world differently as numbers, shapes, and patterns" (p. 2), and who calls himself a "mathnik". So when one day Lawrence Lake, as the Director of Education, declares that "mathematics will be removed from the school curriculum, effective immediately", Sam suggests a debate between him and the Director in which Sam plans to challenge the Director's view that mathematics has very little use, value and meaning in students' and, for that matter, adults' lives beyond basic arithmetic. Sam is so convinced of his case that he bets with the Director that he is "going to convince [him] and everyone [in the audience] that math is not only important, but exciting too, and part of everything we do. And if [he] can't make [the Director] change [his] mind, [he] promise[s] to work for [him] every day after school for the whole year" (p. 10), for which he asks the Director to "pay [him] one cent a day for the first day, then double it to two cents the second, four cents the third, eight cents the fourth, and so on". The Director agrees to these conditions and claims that Sam "certainly won't get rich that way", a move that the Director will bitterly regret because he did not see the mathematics behind the bet.
In the first of the eight chapters of the book, the reader is introduced to Sam and Jeremy. Jeremy tells the book's story and is Sam's best friend, but, unlike Sam, does not like mathematics very much, and thus is not very sad about Lawrence Lake's decision to eliminate mathematics from the school curriculum. It is also through Jeremy's eyes that the reader witnesses a change of conviction in Sam's audience about the usefulness, value and meaning of mathematics in people's lives as Sam provides example after example of surprising places where mathematics can be found.
In the second chapter, which starts out with the setting of the stage of the debate and the bet, Sam provides his first example of such a "surprising place" to the unsuspecting audience: mathematics in the gymnasium. First, Sam points to the geometry in the frames of (mountain) bicycles, where triangles are the core shapes of the frame for stability because "that's the strongest shape there is" (p. 12). Then Sam grabs a basketball and explains that the success of getting the ball into the basket depends on the angle in which the ball is launched and "that a medium angle 45 degrees gets you the best distance, if that's what you want" (p. 18).
Through subsequent chapters (3-7), Sam points to other "surprising places" where mathematical ideas and concepts can be found but are often unrecognized as such. In chapter three, it is mathematics in the art room with tessellations, digital 3-D animation, and perspective to which Sam draws the attention of his audience. In chapter 4, Sam looks at mathematics in music where he explores for the audience the patterns in a song and the fractional relationship between the values of notes. Here, he also makes a historical and a modern-day reference. He tells his audience about the Pythagoreans in Ancient Greece, who were quite taken by the strict relationship that exists between whole-number ratios of the length of a string and the harmonic relationships between the tones those strings produce when struck. He also makes reference to MIDI-equipped keyboards that record and 'translate' numbers into different types of music.
In chapter five, Sam points to the mathematics that can be found in nature which he explores with his audience in the teaching garden by the science room. Here he points to ants' mathematical skills in 'dead reckoning' which they use to find the shortest way back to their nest after they have searched for food, zigzagging all over the place. Sam also explains how a bee encodes distance and place of a food source through a dance for the other bees, where the speed of the bee's waggling and the relative angle in which the bee dances forward tells the other bees where and how far away the food source is located. Sam also points to the 'logarithmic spirals' that make up the shapes of shells, hurricanes, galaxies and sunflowers, and in which the famous golden ratio can be found, a ratio that has been considered particularly esthetically pleasing to the eye when used in architecture and art, as has been done in Renaissance art and buildings. He also explores the relatively new mathematical concept of fractals with his audience and their (partial) occurrence in nature (tree branches, ferns) and computer graphics.
In chapter six, Sam shows his audience how many 'tricks' have their roots in mathematical ideas. For instance, Sam surprises his audience when he makes good on his claim that he can fit through a post-card size piece of cardboard, or when he explains how he can design a piece of paper that has only one side and, thus, introduces the audience to a fundamental idea of the relatively new mathematical field of topology.
Sam starts to convince members of his audience of the important role that mathematics plays in art, nature and so on, and his audience acknowledges that "this stuff is pretty cool" (p. 70). But in chapter seven, Sam faces his greatest challenges, namely the claim that numbers, which are so much at the core of the mathematics curriculum, are simply "lame": "numbers are so blah. 1, 2, 3, 4
1+1=2, 2x3=6. No surprises, no mystery" (p. 70). In response, Sam explores patterns in the number system. He explores the 'Sieve of Erasthosthenes' to find smaller prime numbers and talks about the challenge of finding ever larger prime numbers. He then quite extensively looks at different patterns in the famous 'Pascal's Triangle.'
Sam seems to be successful in his attempt to convince his audience of the role mathematics plays in our understanding of the world human-made as well as the natural world. As Jeremy tells the reader: "You know, for my part, I thought this was a no-brainer. It might have taken me a while to come around, but then I'm not an easy guy to convince. And I don't think I'm too far off thinking all the kids, even Oscar, were onside now. How could the man [Mr. Lake, the Director of Education] not agree?" (p. 82). But he does not: "'This has been a very entertaining hour,' said Mr. Lake slowly, 'but I'm not changing my mind. The ban stays'" (p. 82). But rather than being devastated, Sam quietly submits to his loss of the bet and, thus, using the bet itself to demonstrate to Mr. Lake the power of understanding mathematics in a way that will have the Director of Education pay for his ignorance in a literal sense. Sam declares that he will start to work for Mr. Lake after school starting that very day, but he asks him whether he can have an advance of his first month's pay, pay for his work that is part of the bet arrangement. "Mr. Lake nodded, with that smirky smile on his face again. Yeah, he was in a good mood now, good enough to agree to anything" (p. 83). Sam proposes to quickly calculate how much Mr. Lake is owing him for the first-month work after school. The bet arrangement was that Mr. Lake would give him one cent for the first day and then double the amount of the previous day for each following day. As the table that Sam creates shows, the pay each day, as well as the total pay, increases slowly at first, being at a total pay of about $41 after day 12, but with each higher number, the total increases more rapidly and is at about $84,000 in total on day 23 and at a month-end total of $10,737,418.23! This surprising result had the Director of Education change his mind: "'I've been thinking it over while you were doing your, er, calculation,' said Mr. Lake. Got him! 'And perhaps I was a bit hasty in making my decision,' he continued. Warming up, he went on. 'A little hard work doesn't hurt anybody, and math can be useful, no matter what other people say
yes, I've always thought so, underneath it all'" (p. 85). And so, mathematics stayed in the school curriculum.
In addition to the story, the book also has quite a number of 'featurettes' (on average one per double-page) that take a third to a whole page. Those featurettes extend on the respective mathematical topic of the respective chapter. So, for instance, in chapter five where Sam explores mathematics in nature, one of the featurettes investigates the question whether animals can do mathematics the way we do; another one suggests to see from a mathematical point of view the phenomenon that "when you pitch a ball at an angle into the water, your dog streaks partway down the beach, before plunging in and swimming for it" (p. 47).
The book also includes seven full-page brief historical notes on mathematicians. The good news here is that only two are the traditionally listed old Greek guys, Pythagora and Archemides. Of the remaining five, two are women (Hypatia of Alexandria and Sophie Germain), one is from a non-Western country (Srinivasa Ramanujan from India) and one is a still living mathematician (Andrew Wiles). Also of interest is that the short biographies of the women mathematicians make it clear to the reader how challenging, even dangerous, 'mankind' has been making it for women to participate in the professional activity of doing mathematics. The recognition of women mathematicians is I would assume dear to the heart of Cora Lee, one of the two authors, who "coordinates the Vancouver chapter of the Canadian Association for Girls in Science" (p. 103).
The book is rich in illustrations, several on each page, and also includes some photos. The illustrations are well designed to help the young reader to 'see the surprising places' where mathematics can be found. However, it is also because of the illustrations that I need to raise some caution about the book. One illustration shows a dead Greek in the sea with blood on his chest and a 'square-rooted' sword that is still sticking in his wound. This is an illustration for one of the featurettes in which the reader is told that the (Ancient Greek) Pythagoreans killed one of theirs after he had proven to them that square root of two is an irrational number. The other illustration that might be of concern portrays Sam with his hands separated from his underarms, showing a clear, blood-red cut at both wrists with the hands separated from them. The image is to illustrate the idiom used by Sam: 'before my hands drop off.' Poor choice, I would say.
A glossary for mathematical terms used in the book, an index, and a "Further Reading" section with annotated references to nonfiction and fiction books for young readers on mathematical topics as well as a reference to a television series "featuring math as a crime-solving tool" are also part of the book.
To help students see the world (sometimes) in mathematical terms is not an easy endeavor. I highly recommend The Great Number Rumble as an excellent resource for this endeavor. It is conceptualized as and I am sure will be experienced by the curious student reader as a gold mine for places in our lives in which mathematical ideas are 'surprisingly hidden.' The experiential areas where those places can be found are clearly within the experiential realm of late early and middle year students. The fictitious framing of this goldmine will be an additional bonus for the student reader, especially since the narrator, Jeremy, provides a good figure with which to identify: he is critical, funny, loyal to his friend, and comes out at the end on the side of 'the winner.' Although many of the discussed topics are generally not part of what students experience in the school mathematics classroom, the way in which those topics are discussed should allow students to make connection to their classroom learning of mathematics depending on their grade-level. Not all students within the recommended age-range might be able to follow Sam's explanations fully, but, as it is with the rich language the book uses (parabola, calculus, logarithmic spirals, dead reckoning, Pascal's Triangle, and so on), the book immerses the reader into mathematical ideas and terminology whose understanding and use by the reader can evolve over time the seed is planted. The 'places' explored for their mathematics are so rich that I would also recommend the book to teachers as their resource to enrich their teaching and to help them do for their students what Sam has done for his audience.
Thomas Falkenberg teaches mathematics education in the Faculty of Education at the University of Manitoba.
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