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Math, Maps, and Misrepresentation

By Eric Gutstein

As the basis of my curriculum, I used Mathematics in Context (MiC) a new, innovative mathematics curriculum developed in accordance with the National Council of Teachers of Mathematics Standards. I found that MiC helped students learn to use mathematics to understand the world, as it developed many critical mathematical reasoning skills, but by itself, it was not enough. Therefore, I developed 17 real-world mathematics projects that my seventh- and eighth-grade students completed over the almost two years I was their teacher. (I moved up to eighth grade with my class.) This story, from the spring of 1999 when they were in eighth grade, is about one of those projects.

One significant lesson I learned is that going beyond mathematics is important in helping middle-school students read the world with mathematics. Teachers need to develop a classroom culture that incorporates reading the world and examining injustice and oppression. An important part of going beyond mathematics is to try to normalize politically taboo topics. For example, my students and I had many conversations about race and racism, and they were central to a number of our classroom projects. I found that such an orientation is vital for students to appreciate and be more interested in mathematics, because they begin to see that mathematics can help them make sense out of their surroundings.

ANALYZING MAP PROJECTIONS

The project I describe here was called Analyzing Map Projections-What Do They Really Show? Maps are two-dimensional representations of the earth that we often take for granted. Few of us think that our standard maps might be woefully inaccurate, and we do not often consider how the images students see everyday on classroom walls shape their perceptions of the world. Mathematics is central to map making, and different mathematical ways of representing the world produce very distinct maps. A goal of the map project was for students to use mathematics to analyze diverse map projections and to raise questions about what the various maps showed - and why. A larger goal of this project was to help students develop a more critical outlook towards knowledge in general.

I used two very different projections: the Mercator projection (developed in 1569 in Germany), the traditional map in U.S. schools (including Rivera); and the Peters projection (developed in 1974 by Arno Peters). The sizes, shapes of land masses, and coloring schemes of the maps are quite distinct. The Mercator map was developed during European expansion when colonial exploitation required that maps be used to navigate accurately (so as not to repeat Columbus' blunder) and was used successfully to find new territories. All maps unfortunately are misleading because they are two-dimensional projections of a sphere - and the Mercator suffers from serious visual distortion by altering the relative size of land masses. This is because the scale changes as you move away from the equator. Thus countries far from the equator (e.g., Greenland) appear much larger than they are. (Some Mercator maps have, in fine print, an explanation of this distortion and the mathematical information necessary to find the actual areas.)

For example, Mexico is about 760,000 square miles, Alaska is about 590,000, but Alaska looks two to three times larger because it is farther from the equator. The representations of Greenland and Africa are more distorted. Greenland, at 840,000 square miles, appears roughly comparable to Africa, which, at 11,700,000 square miles, is about 14 times larger. In addition, Germany is near the center of the map, which may have made sense from the perspective of European expansionism. However, since Germany is in the northern quadrant of the earth (Berlin is 52¡ north), the only way to make it the center is to push the equator approximately two thirds of the way down the map. This compresses the Southern Hemisphere and enlarges the Northern.

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