REACHING ALL STUDENTS: Teaching Quantitative Reasoning
By Christopher R. Wolfe and Carolyn Haynes
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One of the most enduring illustrations of ingenuity in investigation occurred before the year 200 B.C. Using simple geometry, Eratosthenes--the ancient Greek geographer and mathematician--accurately calculated the circumference of the earth. His result, 24,650 miles, is remarkably close to the current estimate (in this era of satellite technology) of 24,901.55 miles at the equator. All too often we miss opportunities to link quantitative reasoning explicitly with formal science and even with our daily experiences. Like any way of thinking, quantitative reasoning is not simply a natural gift; it can, and should, be cultivated and taught purposefully.
It is sad to see so many students turn away from numbers due to a few bad experiences with mathematics. To let your students begin to see the power of this way of thinking, adopt a broad approach to the many ways we make use of numbers. Consider four components of quantitative reasoning. All are integrated into the inquiries we feature in Dragonfly.
I. Measurement and Data Collection
Students ask intriguing questions about the world and collect data naturally as
a part of inquiry-based learning. Help students learn how to collect data
appropriately. As the example of Eratosthenes illustrates, the combination of
careful measurement and mathematics can produce amazing results. They need not
calculate the circumference of the earth; finding the area of your schoolyard
may be just as instructive.
II. Quantitative Expression
Students lose their fear of numbers when they become more confident in their
ability to express ideas with numbers. Effective investigators need to
understand various units of measurement (e.g., centimeters and grams) and to
express quantitative concepts visually (through pie charts, histograms, scatter
plots). Try asking students to trace and graph the distribution of maple seeds
and seedlings across the schoolyard. What do we learn when we make a plot to
scale, then use those data to make a bar chart indicating distance intervals
from the tree?
III. Evaluating Evidence
Numbers alone tell us nothing about the world. They must be interpreted by
humans. To facilitate the ability to evaluate evidence, students should learn
not to ignore contrary evidence. Care should be taken to avoid forming
conclusions based on insufficient evidence or atypical cases. Viewed in this
context, thinking about numbers is not only a part of mathematics, but a core
element of critical thinking.
IV. Quantitative Intuition
Quantitative intuition refers to a sense of comfort with quantitative concepts,
including the sense that an answer "feels" right or wrong. Everyone can improve
their quantitative intuition with the right kinds of practice. Try grounding
quantitative concepts in everyday experiences. For example, when teaching the
metric system, students could measure the length of their fingers in
centimeters. Regularly ask students for ballpark estimates of daily
occurrences. For instance, for how many seconds does the school bell ring? How
many clovers are growing in the courtyard lawn? Find ways to test their
estimates.
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Eratosthenes knew that the ancient cities of Syene and Alexandria were on the same meridian. At midday on the solstice, he measured the shadow cast by a pointer set in the middle of a hemisphere-shaped bowl in Alexandria. He found the shadow covered 1/25 of the hemisphere, and thus 1/50 of a complete circle.
Assuming that the sun's rays are parallel lines, he reasoned that the angle of the shadow at Alexandria (ABC in figure 1) is equal to the alternate angle BCD intersecting the arc BD. Official pacers measured the distance between Syene and Alexandria as 493 miles. Knowing that this angle is contained in a complete circle (360 degrees) 50 times the Alexandria-Syene arc length, he multiplied this distance by 50 to obtain the circumference of the earth.
Lively debates continue to this day about Eratosthenes. To learn more about him, see Kathryn Lasky's The Librarian Who Measured the Earth. Boston: Joy Street Books, 1994.
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