Abstract
I’m reminded of Brian Harvey’s 1983 paper titled “Stop Saying ‘Computer Literacy’!” His lament was partly that the analogy to literacy-literacy is, at best, thin. I’ve recently been adopted onto a project that keeps talking about numeracy—another adaptation of the L-word. Though I keep referring to my focus as ‘mathematics’—which will guide me also in this talk—I’ve become curious about what people mean when they use these literacy-like terms. Googling didn’t help except to connect the varied and vague usage with the Real World. Whatever that is. I’ve struggled with the Real World for years. The real world of children or adults? They’re different. What about the real worlds of the barely-subsisting subsistence farmer, the fairly wealthy city-dweller, and the blue collar laborer? And is that what really catches peoples’ interest? What about the very real world of the mind? To take seriously the idea of serving people well and to avoid limiting or pre-judging their eventual paths, we might focus on the latter.
Many educational terms share a common problem: When you or I use the terms, we do know what we’re talking about. At least sort of. But whoever we’re talking with might well have a different understanding, because the terms have no universally shared definition. Without trying to declare what mathematical literacy should mean—that would be yet another usage, unshared except among us—I’ll punt. I’ll take our question “what is mathematical literacy?” to mean “what is a mathematics education that is ‘useful’ to people?” and will focus not just on the topics of mathematics but on the thinking, the real world of the mind.
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Notes
- 1.
Though this calls for logical grounding, the field’s response has been mixed. Even the National Council of Teachers of Mathematics recommended (in NCTM 1989) a reduction in the emphasis on proof. Though that statement may have been just a poor choice of wording—the same document clearly called for greater emphasis on reason than on rote—many in mathematics education hailed that reduction in emphasis.
- 2.
Quoted in National Research Council (1989).
- 3.
A focus on the pragmatic utility of mathematical results works against the development of mathematical sensibility. If a mathematical result’s value is its utility, one hardly needs to understand why it works or go through the effort to prove it, as long as an authority has approved the result. Proof becomes “academic.”
- 4.
Culture can exert strong local influences over an individual’s interests and efforts. Early tracking can lock in those effects long after experiences grow, tastes and interests mature, and an individual has developed the ego to break with conventions, expectations, and stereotypes. By creating distinct math/non-math tracks too early, we virtually guarantee that the non-math tracked students never make it back into the running.
- 5.
“The real world is overrated.” —Al Cuoco.
- 6.
Standard elementary school content is a legacy of methods optimized for large accurate computations by hand. The algorithms are particularly safe and easy for adding long columns of numbers, or performing large multiplications. But other methods, even other “basic facts,” would better serve today’s needs for algebra readiness and mental calculation. Consider the carry-method for, say, 39×65, that begins “five times nine is forty-five, write down the five and carry the four…” That algorithm camouflages the fact that four products are being found and added. If, instead, students learned “30×60 is 1800, write that down; 9×60 is 540, write that down,” and so on, they would get a better first approximation, see the four products (and thus a better model of the algebraic steps which have no “carry”) and would have a generally more accessible method for mental computation.
- 7.
Developing the real world of the mind might be easier in an alternative curriculum that slices knowledge a different way, replacing traditional subjects like mathematics with “courses” such as Communication (comprised of and needed in mathematics, poetry, politics, law, managing our health…), Reasoning Under Constraint (mathematics, personal budgeting, law, ecology, business management…), Troubleshooting and Problem Solving (mathematics, science, diagnosing a car, computer, or person), and so on. Mathematics can help teach these, but so can other subjects. Replacing current disciplines with such “courses,” however, is impractical. On the other hand, if each current discipline, with its own unique contexts and facts, were internally organized by the elements of reasoning that make it a discipline, the inevitable areas of overlap and commonality would surely become a mutually supportive theme making transfer among disciplines more natural.
- 8.
Certain logic puzzles that appear in recreational mathematics websites and magazines and books and that show up in math classes on rainy Thursday afternoons to kill time (but only for students who have finished all of their “real” work) are used as basic exercises in law, so important that an entire section of the Law School Admission Test (LSAT) is devoted to these “Logic Games”.
- 9.
In the United States, all control and authority over education must remain at State level; our Constitution does not allow the Federal government to establish central standards or control. The Balkanization of US, education eventually led States to organize against the chaos and establish a common set of standards for education.
- 10.
See http://thinkmath.edc.org/index.php/Difference_of_squares for one approach to developing the language of algebra “naturally” with elementary school students as they attempt to describe the pattern they’ve seen and applied.
- 11.
For more about how this approach is used in at least one curriculum, see http://thinkmath.edc.org/index.php/Addition_and_subtraction#All_about_10.
- 12.
“My fingers are tired, so I’ll just tell you how many I’d hold up, and you say how many you don’t see. OK? Seven. (three) Eight. (two) Two. (eight)” and so on. Playfully and at a lively pace.
References
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National Research Council (1989). Everybody counts. Washington: Nat. Academy Press.
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Goldenberg, E.P. (2014). “Mathematical Literacy”: An Inadequate Metaphor. In: Fried, M., Dreyfus, T. (eds) Mathematics & Mathematics Education: Searching for Common Ground. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7473-5_9
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