Imagine Magazine - Johns Hopkins - March/April 2011 - (Page 4)

big picture BIG IDEAS FOR BRIGHT MINDS EDITOR ™ Melissa Hartman ASSISTANT EDITOR The Unity of Mathematics If linguists study languages, and chemists study chemicals, what do mathematicians study? Children might identify numbers as belonging to the realm of mathematics. Students of algebra might observe that mathematicians study equations and solutions to equations. More advanced students might expand this view, identifying sets and functions as major objects of mathematical study. As students continue to study mathematics, they may encounter an even greater number of mathematical objects—groups, rings, fields, vector spaces, operators, metric spaces, manifolds, and so on. But what do all of these objects have in common? As a math major in college, I became acquainted with a wide array of mathematical objects much in the manner of a child visiting a zoo—intrigued by each species in succession, but rarely stopping to consider what features distinguish amphibians from reptiles, or what characteristics all mammals share. But in graduate school, as my mathematical vision widened and deepened, I began to appreciate the unity in the menagerie of mathematical structures. by Ashley Reiter Ahlin, PhD Amy Entwisle CONTRIBUTING WRITERS & EDITORS Kristi Birch Carol C. Blackburn Linda E. Brody Amy Dusto Michelle Muratori Michael Powell PROOFREADER Melissa House DESIGN & PRODUCTION Bonotom Studio, Inc. ART DIRECTOR Abigail Noonan ISSN 1071-605X Vol.18 No.4 | Mar/Apr 2011 Copyright © 2011 by The Johns Hopkins University All rights reserved. No portion of this journal may be reproduced by any process or technique without the formal consent of The Johns Hopkins University Center for Talented Youth. Published five times a year: September/October November/December January/February March/April May/June ADDRESS SUBSCRIPTION INQUIRIES TO: Imagine, JHU Press Journals Division, P.O. Box 19966 Baltimore, MD 21211-0966 800-548-1784, 410-516-6968 (fax) or order online:cty.jhu.edu/imagine ADDRESS EDITORIAL CORRESPONDENCE TO: Learning by Abstracting One of the most important unifying ideas of mathematics is abstraction. You use abstraction when you first understand 5 as a concept, independent of any particular five objects, such as five fingers or five apples. You then have the ability to abstract further, to perform arithmetic operations, such as 5+5, without referring to a particular set of five things. Furthermore, once you learn Melissa Hartman, CTY/Imagine 5801 Smith Avenue, Suite 400 Baltimore, MD 21209 mhartman@jhu.edu that 5+5=10, you never need to repeat the process of counting the fingers on each of your hands to see that the total number of fingers is 10. Your discovery—that 5+5=10—is independent of all objects and can apply to all objects. As you progress through elementary and high school mathematics, you continuously abstract from what you’ve learned earlier to understand new mathematical structures. In algebra, for example, you need not test all possible values for x to determine a solution to the equation x+5=10; rather, you can solve the equation by using your knowledge about the properties of addition that hold for all real numbers. Next, you can learn to solve all equations of the form x+5=? by considering a new mathematical structure, the function f(x)=x+5. At each step, the new structure you encounter is not an entirely new idea but a generalization of those studied earlier. Abstracting makes the original mathematical concepts easier to understand by showing how both old and new pieces fit together into one big picture. For example, when you learn to solve both linear and quadratic equations, you might notice differences between them: a linear equation has at most one solution, while a quadratic equation has two. But as you generalize from linear (first-degree) and quadratic (second-degree) functions to study solutions of higher-degree polynomial equations, you see that the degree of the polynomial reveals the number of solutions. That quadratic equations have exactly two solutions is just a specific case of this more general truth. 4 imagine Mar/Apr 2011

Table of Contents for the Digital Edition of Imagine Magazine - Johns Hopkins - March/April 2011

Imagine Magazine - Johns Hopkins - March/April 2011
Contents
Big Picture
In My Own Words
Problem Solving
Orange County Math Circle
Number Theory
Count Me In
National Mathematics Competitions
Math at the Science Fair
Just My Speed
MathPath
When Origami Meets Rocket Science
Selected Opportunities & Resources
Making a Difference
High School Options for Gifted Students
Off the Shelf
Word Wise
Exploring Career Options
One Step Ahead
Planning Ahead for College
Students Review
Mark Your Calendar
Knossos Games

Imagine Magazine - Johns Hopkins - March/April 2011