When my students examine data and questions such as the one shown in the box below they are introduced to the four goals of the criticalmathematical literacy curriculum.
2 Understanding the mathematics of political knowledge.
3 Understanding the politics of mathematical knowledge.
4 Understanding the politics of knowledge.
In the United States, the unemployment rate is defined as the number of people unemployed, divided by the number of people in the labor force. Here are some figures from December 1994. (All numbers in thousands, rounded off to the nearest hundred thousand).
. In your opinion, which of these groups should be considered unemployed? Why?
. Which should be considered part of the labor force? Why?
. Given your selections, calculate the unemployment rate in 1994.
1. 101,400: Employed full-time
2. 19,000: Employed part-time, want part-time work
3. 4,000: Employed part-time, want full-time work
4. 5,600: Not employed, looked for work in last month, not on temporary layoff
5. 1,100: Not employed, on temporary layoff
6. 400: Not employed, want a job now, looked for work in last year, stopped looking because discouraged about prospects of finding work
7. 1,400: Not employed, want a job now, looked for work in last year, stopped looking for other reasons
8. 60,700: Not employed, don’t want a job now (adults)
For discussion: The US official definition counts 4 and 5 as unemployed and 1 through 5 as part of the labor force, giving an unemployment rate of 5.1%. If we count 4 through 8 plus half of 3 as unemployed, the rate would be 9.3%. Further, in 1994 the Bureau of Labor Statistics stopped issuing its U-7 rate, a measure which included categories 2 and 3 and 6 through 8, so now researchers will not be able to determine "alternative" unemployment rates (Saunders, 1994).
In this article, I will develop the meaning of each of these goals, focusing on illustrations of how to realize them in their interconnected complexity. Underlying all these ideas is my belief that the development of self-confidence is a prerequisite for all learning, and that self-confidence develops from grappling with complex material and from understanding the politics of knowledge.
I start lessons with a graph, chart, or short reading which requires knowledge of the math skill scheduled for that day. When the discussion runs into a question about a math skill, I stop and teach that skill. This is a non-linear way of learning basic numeracy because questions often arise that involve future math topes. I handle this by previewing. The scheduled topic is formally taught. Other topics are also discussed so that students’ immediate questions are answered and so that when the formal time comes for them in the syllabus, students will already have some familiarity with them. For example, if we are studying the meaning of fractions and find that in 1985, 2/100 of the Senate were women, we usually preview how to change this fraction to a percent. We also discuss how no learning is linear and how all of us are continually reviewing, recreating, as well as previewing in the ongoing process of making meaning. Further, there are other aspects about learning which greatly strengthen students’ understanding of mathematics:
(B) considering the interactions of culture and the development of mathematical knowledge; and
(C) studying even the simplest of mathematical topics through deep and complicated questions.
These are explained in more detail below.
Students gain greater control over mathematics problem-solving when, in addition to evaluating their own work, they can create their own problems. When students can understand what questions it makes sense to ask from given numerical information, and can identify decisions that are involved in creating different kinds of problems, they can more easily solve problems others create. Further, critical mathematical literacy involves both interpreting and critically analyzing other people’s use of numbers in arguments. To do the latter you need practice in determining what kinds of questions can be asked and answered from the available numerical data, and what kinds of situations can be clarified through numerical data. Freire’s concept of problem-posing education emphasizes that problems with neat, pared down data and clear-cut solutions give a false picture of how mathematics can help us "read the world". Real life is messy, with many problems intersecting and interacting. Real life poses problems whose solutions require dialogue and collective action. Traditional problem-solving curricula isolate and simplify particular aspects of reality in order to give students practice in techniques. Freirian problem-posing is intended to reveal the inter-connections and complexities of real-life situations where "often, problems are not solved, only a better understanding of their nature may be possible" (Connolly, 1981). A classroom application of this idea is to have students create their own reviews and tests. In this way they learn to grapple with mathematics pedagogy issues such as: what are the key concepts and topics to include on a review of a particular curriculum unit? What are clear, fair and challenging questions to ask in order to evaluate understanding of those concepts and topics?
Example: When we are learning the algorithm for comparing the size of numbers, I ask students to think about how culture interacts with mathematical knowledge in the following situation:
Example: In the text below, Sklar and Sleicher demonstrate how numbers presented out of context can be very misleading. I ask students to read the text and discuss the calculations Sklar and Sleicher performed to get their calculation of the US expenditure on the 1990 Nicaraguan election. ($17.5 million - population of Nicaragua = $5 per person.) This reviews their understanding of the meaning of the operations. Then I ask the students to consider the complexities of understanding the $17.5 million expenditure. This deepens their understanding of how different numerical descriptions illuminate or obscure the context of US policy in Nicaragua, and how in real-life just comparing the size of the numbers, out of context, obscures understanding.
b. using calculations to follow and verify the logic of someone’s argument, to restate information, and to understand how raw data are collected and transformed into numerical descriptions of the world. The purpose underlying all the calculations is to understand better the information and the arguments and to be able to question the decisions that were involved in choosing the numbers and the operations.
It may be lonely at the top, but it can’t be boring - at least not with all that money. Last week the federal government released figures showing that the richest 1 percent of American households was worth more than the bottom 90 percent combined. And while these numbers were widely reported, we found them so shocking that we thought they were worth repeating. So here goes: In 1989 the top 1 percent of Americans (about 934,000 households) combined for a net worth of $5.7 trillion; the bottom 90 percent (about 84 million households) could only scrape together $4.8 trillion in net worth.
Example: Students are asked to discuss how numbers support Helen Keller’s main point and to reflect on why she sometimes uses fractions and other times uses whole numbers. Information about the politics of knowledge is included as a context in which to set her views.
Open side drawer of standard desk 0.014
Open center drawer 0.026
Close side drawer 0.015
Get up from chair
Sit down in chair 0.033
Turn in swivel chair 0.009
When used in textbooks and other media, combined with the general (mis)perception that size relates to various measures of so-called "significance", the Mercator map distorts popular perceptions of the relative importance of various areas of the world. For example, when a US university professor asked his students to rank certain countries by size they "rated the Soviet Union larger than the continent of Africa, though in fact it is much smaller" (Kaiser, 1991), associating "power" with size.
Political struggles to change to the Peter’s projection, a more accurate map in terms of land area, have been successful with the United Nations Development Program, the World Council of Churches, and some educational institutions (Kaiser, 1991). However, anecdotal evidence from many talks I’ve given around the world suggest that the Mercator is still widely perceived as the way the world really looks.
As Wood (1992) emphasizes:
Example: Political struggle/choice over which definitions should guide how data are counted. In 1988, the US Census Bureau introduced an "alternative poverty line", changing the figure for a family of three from $9453 to $8580, thereby preventing 3.6 million people whose family income fell between those figures from receiving food stamps, free school meals and other welfare benefits. At the same time, the Joint Economic Committee of Congress argued that "updating the assessments of household consumption needs ... would almost double the poverty rate, to 24 percent" (Cockburn, 1989). Note that the US poverty line is startlingly low. Various assessments of the smallest amount needed by a family of four to purchase basic necessities in 1991 was 155% of the official poverty line. "Since the [census] bureau defines the [working poor] out of poverty, the dominant image of the poor that remains is of people who are unemployed or on the welfare rolls. The real poverty line reveals the opposite: a majority of the poor among able-bodied, non-elderly heads of households normally work full-time. The total number of adults who remain poor despite normally working full-time is nearly 10 million more than double the number of adults on welfare. Two-thirds of them are high school or college-educated and half are over 33. Poverty in the US is a problem of low-wage jobs far more than it is of welfare dependency, lack of education or work inexperience. Defining families who earn less than 155% of the official poverty line as poor would result in about one person in ever four being considered poor in the United States" (Schwartz and Volgy, 1993).
Example: Political struggle/choice over which ways data should be disaggregated. The US Government rarely collects health data broken down by social class. In 1986, when it did this for heart and cerebrovascular disease, it found enormous gaps:
There are many aspects of the politics of knowledge that are integrated into this curriculum. Some involve reconsidering what counts as mathematical knowledge and re-presenting an accurate picture of the contributions of all the world’s peoples to the development of mathematical knowledge. Others involve how mathematical knowledge is learned in schools. Winter (1991), for example, theorizes that the problems so many encounter in understanding mathematics are not due to the discipline’s "difficult abstractions", but due to the cultural form in which mathematics is presented. Sklar (1993), for a different aspect, cites a US study that recorded the differential treatment of Black and White students in math classes.
And, of course, Freire (1970) theorizes about the politics of "banking education", when teachers deposit knowledge in students’ empty minds.
Underlying all these issues are more general concerns I argue should form the foundation of all learning, concerns about what counts as knowledge and why. I think that one of the most significant contributions of Paulo Freire (1982) to the development of a critical literacy is the idea that:
In a non-trivial way we can learn a great deal from intellectual diversity. Most of the burning social, political, economic and ethical questions of our time remain unanswered. In the United States we live in a society of enormous wealth and we have significant hunger and homelessness; although we have engaged in medical and scientific research for scores of years, we are not any closer to changing the prognosis for most cancers. Certainly we can learn from the perspectives and philosophies of people whose knowledge has developed in a variety of intellectual and experiential conditions. Currently "the intellectual activity of those without power is always labelled non-intellectual" (Freire and Macedo, 1987). When we see this as a political situation, as part of our "regime of truth", we can realize that all people have knowledge, all people are continually creating knowledge, doing intellectual work, and all of us have a lot to learn.
(2) This situation has changed in Massachusetts, which now has a flat rate structure, and my reference did not contain real data for Michigan. So although the context-setting data are real, the numbers used to understand the concept of declining block rates are realistic, but not real.
(3) Grossman (1994)
argues that "Europe has always been a political and cultural definition.
Geographically, Europe does not exist, since it is only a peninsula on
the vast Eurasian continent." He goes on to discuss the history and various
contributions of geographer’s attempts to "draw the eastern limits of ‘western
civilization’ and the white race" (p.39).
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Cockburn, A. (1989) December 11, Calculating an end to poverty, The Nation.
Frankenstein, M. (1983) Critical mathematics education: An application of Paulo Freire’s Epistemology, Journal of Education, 165.4:315-340.
Frankenstein, M. (1989) Relearning mathematics: A different third r radical maths, London, Free Association Books.
Frankenstein, M. (1990) Incorporating race, class, and gender issues into a critical mathematical literacy curriculum, Journal of Negro Education, 59.3:336-347.
Frankenstein, M. (1994) Spring, Understanding the politics of mathematical knowledge as an integral part of becoming critically numerate, Radical Statistics, 56:22-40.
Frankenstein, M. (1997) Breaking down the dichotomy between teaching and learning mathematics, in Freire, P. J., Fraser, D., Macedo, et al (Eds.) Mentoring the mentor: a critical dialogue with Paulo Freire, pp59-88, New York, Peter Lang.
Frankenstein, M. (1997) In addition to the mathematics: Including equity issues in the curriculum, in Trentacosta, J. (Ed.) Multicultural and gender equity in the mathematics classroom: The gift of diversity, pp10-22, Reston, VA, National Council of Teachers of Mathematics.
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