Q&A #2642

Real life math

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From: Gail (for Teacher2Teacher Service)
Date: Nov 27, 1999 at 00:39:26
Subject: Re: Real life math

I searched he NCTM (National Council of Teachers of Mathematics) website, and
found some informaiton that might get you headed in the right direction...


"The standards address both mathematical content and mathematical processes.
Roughly speaking, the content standards represent what students should know;
the process standards represent ways of acquiring and using that knowledge.
This separation is artificial, however. In practice, what one can do depends
in important ways on what one knows and on how one can exploit that

 Five standards describe the mathematical content that students should learn:

                         Number and Operation
                         Patterns, Functions, and Algebra
                         Geometry and Spatial Sense
                         Data Analysis, Statistics, and Probability

 Five standards describe the mathematical processes through which students
should acquire and use their mathematical knowledge:

                         Problem Solving
                         Reasoning and Proof

 It should be recognized that this set of ten standards does not neatly
separate the content of school mathematics into nonintersecting subsets.
Because mathematics as a discipline is highly interconnected, the areas
overlap and are integrated. Some topics in data analysis, for example, could
be characterized as part of measurement. Patterns and functions appear
throughout geometry. Number pervades all the areas of mathematics. The
arrangement of the standards is designed solely as a way of organizing the
content and processes. Any method of dividing the content of mathematics
will, at the top level, highlight some areas and obscure others. Consider
discrete mathematics as an example. In this draft of Principles and
Standards, the main topics of discrete mathematics are included, but they are
distributed across the standards, rather than separated into a separate

 The process standards point to major aspects of student competency that are
essential to students' mathematical growth. As students learn mathematics,
they will develop an increasing repertoire of problem-solving skills, a wide
range of mathematical "habits of mind," and increasing sophistication in
mathematical argument. Also, students should become proficient at expressing
themselves mathematically, both orally and in writing, gain fluency in the
language of mathematics and able to make connections within mathematics and
from mathematics to other disciplines."

"Each of the standards presented in this draft document has formal beginnings
in the pre-K-2 years. The ten standards, rather than being separate topics
for study, are designed to support the development of important, connected
mathematical ideas and should be closely interwoven. At the core of
mathematics in grades pre-K-2 are the number and geometry standards. Each of
the other mathematical standards, including patterns, measurement, and data
and the process standards, contribute to and are learned through the number
and geometry strands.  Children's number development moves through a pattern
of increasingly sophisticated levels of constructing ideas and skills. In
particular, children learn to count, group, combine, and separate numbers,
sets, and shapes as they build their concepts of place value and the four
arithmetic operations. The big ideas in number include the many meanings of
numbers; systems of counting numbers (i.e., one-to-one correspondence,
increasing and decreasing by one more and less, skip counting); composing and
decomposing numbers, place value relationships, and part-whole relationships.
Children develop understanding of space and geometry through their own
movement and based on visual and tactile explorations. Two- and
three-dimensional shapes, properties of shapes (including symmetry) and
transformations are encountered and explored in the early primary grades.
The discussion of the ten standards in this chapter reflects the
inter-relatedness of content and process in young children's learning of
mathematics. Children recognize patterns of all sorts in the world around
them and gradually begin to use patterns as a strategy for solving problems.
Both number and geometry are used in measurement, for example, as children
answer questions such as "How heavy?" "How big?" "How long?" and "How tall?"
The ability to gather, organize, represent, and use data to answer questions
is likely to involve all of the process standards. Skills are acquired in
ways that make sense to children. Teachers must continually attend to
maintaining a balance between emphasis on conceptual and procedural (skill)
aspects of mathematics. Children whose skills and strategies are based on
understanding of fundamental mathematical concepts are more likely to retain
and be able to expand their knowledge and understanding in later years.

 The pre-K-2 mathematics program should take advantage of technology. Guided
work with calculators can enable children to explore number and pattern,
focus on problem-solving processes, and investigate realistic applications.
Through their experiences, and with teacher support, children come to
recognize when using a calculator is appropriate and when it is more
efficient to compute mentally. Computer environments also can make a powerful
and unique contribution to the mathematics program. They enable children to
manipulate virtual representations of numbers and see, for example, that "4
tens and 7 ones" is the same as "3 tens and 17 ones." Simple programming
environments such as Logo can be used to engage students in geometric and
spatial investigations.

 Children frequently possess greater knowledge than they are able to express
in standard symbolic notation, and teachers need to determine through
assessment what children already know and what they still have to learn.
Information from a wide variety of classroom experiences-classroom routines,
conversations, written work (including pictures), and observations-provides
the teacher with an assessment of what young children already know, are in
the process of learning, and still need to know. Quality assessments inform
instructional decisions and allow teachers to monitor individual children's
progress while focusing on how children are thinking about mathematics.
Mathematics learning for pre-K-2 children is active, rich in mathematical
language, and filled with thought-provoking challenges. Educational research
and classroom experience together challenge us to hold high expectations for
all children and look at what they can do with understanding rather than to
plan programs around preconceptions about children's limitations. This does
not mean abandoning children's ways of knowing and representing; rather, it
is a clear call for creating opportunities for children to learn new,
important mathematics in ways that make sense to them."

You might be better off going directly to the site to read it.  :-)

 -Gail, for the Teacher2Teacher service

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