The
following syntheses are based on "Estimation for Grades K-4" and
on "Statistics for Grades 9-12" from Curriculum and Evaluation Standards
for School Mathematics by the National Council for Teachers of Mathematics.
Order from NCTM, 1900 Association Drive, Reston, VA 22091, 1-800-235-7566.
The syntheses are accompanied by the instructional activities Pumpkin
Exploration and Choosing a Company Site
as well as short commentaries on the teacher's role as the
facilitator of each activity.
Estimation: Building Mathematical Power
Children
enter school with estimation skills. Jill knows she is "about 6,"
Daniel is "a little shorter" than his sister, and the entire class
knows when in it is "almost" playtime. This is knowledge based on
experience and it provides a foundation for learning to estimate
quantities.
Good
estimating skills introduce students to another dimension of mathematics.
Terms such as about, near, closer to, between, almost, and a little
less than illustrate that mathematics is more than exactness or
computation. Estimation interacts with number sense and spatial
sense to help children understand concepts and procedures. It encourages
flexibility in working with numbers and measurements, and gives
the student a way to check for reasonable results.
It
is important to learn a variety of estimation methods. For example,
a student who needs to know the value of 243 + 479 might estimate
by thinking, "200 and 400 is 600, 43 and 79 is more than 100, so
the sum is a little more than 700." This is "front-end estimation."
Another way of estimating is: "243 is just under 250, 479 is just
under 500, so the sum is less than 750." This flexible use of rounding
provides numbers that are easy to work with. Someone adept at mental
computation could estimate 243 + 479 in another way: " 24 (tens)
+ 48 (tens) is 72 (tens) so the sum is about 720." Discuss various
strategies and help students develop their own methods for solution.
Young
children can estimate large numbers-the number of blades of grass
in the yard, the number of candies in a jar-or small numbers. Shown
a cluster of ten dots, have students quickly estimate several other
clusters of dots as more than ten, fewer than ten, about ten. Talk
about "good" estimates-how close to the exact number must an estimate
be, and emphasize that for some situations the exact answer is no
better than the estimate.
Estimation
is especially important when children use calculators. Rough estimates
will give them enough information to decide whether the correct
keys were pressed and whether the calculator result is reasonable.
Such uses of estimation reduce the incidence of errors with calculators,
decrease the inappropriate use of calculators for simple computation,
and contribute to children's development of number sense, operation
sense and mathematical power.
In
later grades, mathematical instruction should concentrate on a variety
of problem-solving methods. It is not necessary to spend large portions
of instructional time on routine computations by hand, and students
must learn to choose between mental calculations, paper-and-pencil
computation, or use of calculators and computers. Estimation should
be a part of the students' repertoire of skills, to be used as a
problem-solving method as well as a way of checking the reasonableness
of results.
Statistics:
A Reflection of Our World
Collecting,
representing, and processing data are important activities in today's
society. Through the media, in the natural and social sciences,
in advertising claims and legal proceedings we are confronted with
data that has been summarized, analyzed, and transformed. To function
in the modern world, students should learn to apply such techniques
as simulations, sampling, fitting curves, testing hypotheses and
drawing conclusions. They will need such tools to solve problems
and evaluate the statistical claims they encounter daily.
The
study of statistics in grades 9-12 should build on understandings
of data analysis methods begun in the elementary and middle grades.
Students should learn the qualified nature of statistical analysis
and the role statistics plays in straddling the exactness of mathematics
and the subjective world of individual opinion. They should be encouraged
to apply statistical tools to other academic subjects through student-opinion
polls for social studies, word or letter counts for English, or
plant-growth records for biology. Such out-of-school activities
as athletics provide further opportunities for immediately relevant
data analysis.
Computing
technology allows quick and precise calculation and presentation
of data. The study of statistics should add an understanding of
the appropriateness of measures for a given problem and what such
measures as mean, variance, and correlation can tell about a problem.
Students must learn to interpret results intelligently. The notions
of randomness, representativeness, and bias in sampling will enhance
their ability to evaluate statistical claims. Students headed for
college should also be able to apply their understanding of sampling
in designing their own experiments to test hypotheses.
Students
should be aware that bias can arise in the interpretation of results
as well as in sampling: the interpreter's predisposition or expectation
may strongly affect the message derived from the statistical results.
Such bias often occurs in the presentation and interpretation of
data gathered for political purposes and advertising.
College-bound
students should be familiar with such distributions as the normal,
Student's t, Poisson, and chi square. They should be able to determine
when it is appropriate to use these distributions in statistical
analysis (e.g. to obtain confidence intervals or to test hypotheses).
Instruction should focus on the logic behind the process in addition
to the test itself.
Statistical
data, summaries, and inferences appear more frequently in the work
and everyday lives of people than any other form of mathematical
analysis. All high school graduates must acquire the capabilities
identified in this standard. This expectation will require that
statistics be given a more prominent position in the high school
curriculum.
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