What happens when students begin to question what they have been taught in school? What other questions does this help them raise, and how does it help them to better understand the world? How can teachers create these kinds of experiences?
I considered such questions as I taught my eighth-grade mathematics class at Rivera, a public school in a Mexican immigrant community in Chicago. As part of my job as a faculty member at DePaul University (where I teach several sections a year of Teaching and Learning Elementary Mathematics), I have taught one middle-school math class at Rivera on a daily basis for the past few years. My goal as a teacher has been influenced by the work of Paulo Freire and other critical educators. I want to help my students learn to read (i.e., understand) the world – through learning and using mathematics – as a way for them to begin to write (i.e., change) the world.
GOING BEYOND MATHEMATICS
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.
These distortions remain true of today’s Mercator maps, even though we now have better navigational means than the Mercator to sail across the Atlantic. Whether or not Mercator meant consciously to diminish the South, it is worthwhile to focus on the effect of its widespread use today.
In contrast, the Peters projection was developed to fairly and accurately portray the earth. As Ward Kaiser writes in an explanation of the Peters projection, “Peters is … clearly focused on justice for all peoples, recognizing the values and contributions that all nations and all cultures can bring to the emerging world civilization.”
The Peters map distorts shapes somewhat, but unlike the Mercator, it accurately presents the relative size of land masses.
As part of our project, I gave each group of three students a large Mercator map, borrowed from classrooms in the building. I gave them Mexico’s size to use as a unit of measure and turned them loose to measure and compare several areas on their Mercator map (like Greenland and Africa; Mexico and Alaska). They used a variety of mathematical means to estimate the areas. For example, some traced Mexico and estimated how many Mexicos fit into other countries. They then multiplied the area of Mexico by the number (of Mexicos) that fit into other countries. Others overlaid centimeter grids on top of Mexico to determine the area per square centimeter, then measured other countries and found their areas. And some students used a combination of methods, including reallocating areas to make rectangular shapes whose sizes were easier to estimate. Afterwards, they dug through our almanacs to find the real areas of the places they had measured and compared those to their estimates. Finally, I had students respond in writing to several questions, both in their groups and individually.
QUESTIONING PAST LEARNING
My students determined for themselves that the Mercator map did not show equal areas equally. This was further confirmed when they examined the Peters map. They were astonished and upset to discover the Mercator’s visual distortions, and the majority felt they had been “lied to.” Many were quite concerned about the implications of this misinformation, and some questioned other sources of knowledge and things they had learned.
In particular, many wanted to know why they had been mistaught. As one of my students, Rosa, commented, “The questions raised in my mind are why teachers never told us how wrong the map was.”
Another student, Lupe, took a position of advocacy: “I think it’s sad that we’ve all been taught this way. We should make our analysis public and let it be known. I just want to understand what [is] the point exactly of Mercator’s map. What did he want us to believe, to see?”
One of the questions I had asked was, “Knowing we were all raised on the Mercator Map, how does that make you feel?” In response, Marisol commented that it “makes me think what other wrong information we have been given since childhood. It makes you doubt your social studies book, history written by the white people.” And Gloria summed up the feeling of many. “Thinking that we were all raised with the Mercator map makes me feel kind of miseducated,” she said, “because all these years we’ve been using the wrong map.”
RACE AS A CENTRAL ISSUE
As politically taboo topics became normalized in our class, race and racism became central discussion topics. Students talked and wrote about race and racism as easily as about field trips and homework. A number of my projects focused more closely than this one on race (see article below). However, because race was an ordinary topic of conversation and a political issue about which students were concerned, and because I asked which races (in general) lived in the North and the South, students also analyzed the project using race as a point of inquiry.
Several found the shrinking of the South relative to the North and the skewing of the relative sizes of Alaska and Mexico particularly troublesome. Sandra had a particularly cogent analysis. “Doing this project has opened my eyes in different ways,” she wrote. “I am learning how small details like maps, etc., have a lot to do with racism and power. Even though these kinds of things are small it can make a big difference in a person’s view after learning what’s really going on.”
Another question I asked students to reflect on why was the Mercator map is used so much in schools. One student, Elena, wrote in response, “I guess that’s so because they wanted to teach [that] all Americans [are] superior and that all whites (that color and race) are better and superior than us (brown or lightly toasted, hardly white and Mexican). We were always taught that we were a minority and didn’t deserve anything.”
Issues of race and racism also surfaced in students’ responses to the last question I asked: “In your opinion, is this in any way connected to anything else we’ve studied over the last two years?” Among those who answered affirmatively, Javier wrote, “It all has to do with white being the superior … because the map maker was from Europe, he put Europe in the ‘middle of the world’ and on top of other countries. But as we know, there is no such thing as a middle of the world.”
Marisol also commented on race. “I think this is connected to what we studied over the past two years about equality, minorities, etc.,” she wrote. “These maps seem to make minority-filled countries smaller than white-filled countries. To make us smaller and more insignificant? Yes, definitely.”
A principal purpose of this and other projects was to get my students to think deeply about their educational experiences and what they had learned both in and out of school. I wanted my students to discover for themselves that “what you see is not always what you get.” I did not want my students to take my word, nor anyone else’s, without questioning. I wanted them to both use mathematics to read their worlds and to see the value of doing so – that is, to also develop a particular disposition towards using mathematics for social analysis.
My students did begin to read their world with mathematics, but clearly they were still developing in this area; for a variety of reasons, their development was uneven. They were still struggling to form their ideas about the world, and their contradictory beliefs often surfaced. Some at times tried to explain away certain things, for example, hypothesizing that we still used the Mercator because perhaps the earth had changed since 1569 and mapmaking was just catching up. This type of accommodation repeatedly appeared in class, sometimes even from students who at other times took strong positions against various forms of injustice.
It is important to note that beyond beginning to analyze society using mathematics, my students also succeeded in mathematics on conventional academic measures. All passed their standardized tests and graduated eighth grade on time, and several made it to academic magnet high schools. Although these are necessary accomplishments under the current high stakes testing policies, they are not enough. Educator Gloria Ladson-Billings’ perspective on educating African-American students also applies to working-class Latinos and other marginalized students. In her 1994 book The Dreamkeepers, she writes that “parents, teachers, and neighbors need to help arm African-American children with the knowledge, skills, and attitude needed to struggle successfully against oppression. These, more than test scores, more than high grade-point averages, are the critical features of education for African Americans. If students are to be equipped to struggle against racism, they need excellent skills from the basics of reading, writing, and math, to understanding history, thinking critically, solving problems, and making decisions.”
A broader definition of educational success incorporates the vision of working for social justice. Mathematics can be an appropriate tool to realize this vision. The words of my student Lupe exhibit the sense of purpose needed in this struggle.
In commenting on our map project she wrote, “This is definitely connected to all we’ve done during the past two years. This goes back to why South America is so small in the Mercator’s Projection Map, to injustice and some sort of propaganda with false information. This relates to not just accepting what we have, but to search for answers to our questions. You have taught us to do that in many ways, and that only makes us grow. Who knows? Maybe we can someday prove things wrong and show the right way!”
A New View of the World: A Handbook to the World Map: Peters Projection, W. L. Kaiser (Amherst MA: Friendship Press, 1987). Call 800-889-5733 for ordering information.