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Developmentally Appropriate Practices
Understanding how children’s development
at specific ages affects the teaching and learning of mathematics
What is child development?
There was a time, as recently as the past
century, when it was generally accepted that children appeared
to be (and interacted with the world as) miniature adults,
that the children’s thoughts were less complex, less
sophisticated, but not qualitatively different. This notion
has gradually been replaced by one that matches children’s
observable behavior more closely, that children think in a
different way from adults, that they draw distinct conclusions
from the same data. Child development is the study of the
way children see the world, the way their thinking, moving,
relating, and expressing themselves changes and evolves as
they become young adults.
How does an understanding of child development
affect how one might teach mathematics?
When child development and mathematics are
mentioned in the same breath, Jean Piaget (1896-1980) and
Lev Vygotsky (1896-1934) come to mind. Jean Piaget observed
his own children very carefully, noting their conversations,
explanations, and questions. He developed an articulated theory
about how children’s thinking develops. Piaget posited
that children evolve gradually through characteristic stages
of thinking, known familiarly as the pre-operational, concrete
operations, and formal stages of thinking. He wrote about
the ways in which cognitive growth takes place, a model that
allows for a continual “folding in” of more complex
understandings. Vygotsky contributed significant insight into
the way in which we learn from those around us, in context,
in connection with those who have more skill. Vygotsky was
intrigued not only by the skills and understanding that children
possess, but also by the skills and understanding they’re
on the verge of possessing, also known as the zone of proximal
development. (This discussion skims only the topmost layer
of their work. Please pursue the following Web sites for more
complexity and accuracy: http://www.piaget.org,
http://www.bestpraceduc.org/people/LevVygotsky.html
and http://www.massey.ac.nz/~ALock/virtual/colevyg.htm
.)
The theories and insights of Piaget and
Vygotsky provide practical guidelines for teaching children:
Look and listen very carefully to understand what children
are thinking. Try to develop and maintain the discipline
necessary to truly hear what children mean. This means not
making assumptions about what one may suppose children intend.
It includes the art of asking open ended questions, sometimes
asking questions to which you yourself don’t already know
the answer, and formulating questions that convey and include
a crucial curiosity in the child’s point of view. (For a wonderful
guide to this process, see Herbert P. Ginsburg’s, Entering
the Child’s Mind, ISBN 0-521-49803-1.)
The development of children’s cognition more clearly resembles
a spiral than an arrow. A thorough grasp of mathematical
concepts often requires repeated visits over many years. For
example, while kindergartners and first graders can count
in base ten, it may take until middle school until they are
able to comprehend that base ten is not the only counting
system that exists, and that there was a time when it didn’t
exist! And if that older child (or adult) is not afforded
the opportunity to look again, the understanding will remain
at the level of the first grader, even though the capacity
for more sophisticated understanding is now available.
Children’s minds grow in response to challenge. Just
as the process of teething involves chewing on dense objects
in order to acquire teeth, so learning mathematics includes
mentally gnawing on dense problems as a way to acquire the
tools necessary to penetrate and to absorb complex concepts.
For example, well before children are ready to manipulate
numbers and to understand their somewhat abstract meanings,
they learn to count. The counting song is random to begin
with, and only gradually evolves to be a source of wonder
about recurring patterns, written correlates, and unflinching
quantities. But, none of this would happen without the counting
sequence.
There are two brief discussions to have
before looking at age and stage recommendations. They include
the distinction between arithmetic and mathematics and formal
and informal mathematics.
Distinction #1: Arithmetic Vs, Mathematics
There was a time when school children were
introduced only to arithmetic, to the rules and practice of
computation. As children grew, so the breadth of their mathematics
grew, and other branches of mathematics were gradually included:
algebra, geometry, data analysis & probability, and measurement.
One of the central convictions of the mathematics reform movement
is that children be introduced to the full range of mathematics
(which of course includes arithmetic) starting at kindergarten.
Arithmetic, the study of number and operations, is a segment
of what is presented, but it is presented as a related piece
of the whole body of mathematics. Keith Devlin, a very well
published mathematician, defines mathematics as the search
for patterns. Mathematics includes the search for patterns
whether or not numbers are involved. Within the discipline
of mathematics, one searches for patterns in shapes and space
(geometry), within and between data sets (statistics), over
the course of many events (probability), between functions
(algebra), and within number systems (computation and number
theory).
Distinction #2: Informal vs. Formal Mathematics
Herbert Ginsburg, a noted contemporary scholar
of Piaget, has made the distinction between informal and formal
mathematics. Informal mathematics is the math one learned
on one’s own, through daily interactions, through real-life
experiences. It’s the math one learns without being “told
how.” Formal mathematics, is the grounding and extension of
informal mathematics. Formal mathematics joins the thinker
to what other people have figured out, what the culture has
contributed over time. In one sense, it is reasonable to say
that informal mathematics lives in a child’s stomach, whereas
formal mathematics resides in the brain.
Regardless of the age of the child, formal
mathematics must be built on informal mathematics. If a child
is not on “speaking terms” with an informal math concept,
through everyday personal experiences (some of them deliberately
facilitated by teachers) formal math has no anchor. Formal
math without an informal foundation is invariably memorized
and taken on faith. The science of mathematics becomes a disconnected
collection of facts, which requires great effort to memorize
and apply. Over time, children become anxious about their
ability to retrieve the right piece at the right time and
they become overwhelmed by a body of knowledge that floats
on a stressful continent of its own.
Constructivist learning, where children make active sense
of the formal mathematics they are adding to their repertoires,
decreases the stress and increases the depth of thought, the
joy, and the creativity in mathematics study.
Why would a child’s development affect learning
mathematics?
As a child’s world expands and her capacities
shift, so what fascinates her changes, too. When a child is
learning to count, there is nothing so absorbing as counting
buckets of little “thingies.” For a child at a later stage,
counting the items is no longer intriguing, but articulating
and sharing strategies for estimating without counting
may be interesting. For a youngster at an even more advanced
stage, a bucket of small objects might represent a point in
exponential growth.
As children grow, the types of questions they find intriguing
develops. How many dimensions they can absorb widens, too.
The abstractness of mathematics increases as children age,
though manipulatives and models retain their role as very
important. As children grow up, they use mathematics for different
purposes. Mathematics can become an important tool for making
sense of the world. For example, it helps support an understanding
of how the world’s resources are shared, whether a particular
practice is equitable, to find trends in data reflecting social
policy, and to analyze patterns in demographics.
The National Council of Teachers of Mathematics has very thoroughly
described the range of mathematics that may be expected and
encouraged of children across the grades. The electronic version
of the Curriculum and Evaluation Standards is the definitive
outline for curriculum guidelines across the age range.
Grades Pre-K-2
http://standards.nctm.org/document/chapter4/index.htm
Grades 3-5
http://standards.nctm.org/document/chapter5/index.htm
Grades 6-8
http://standards.nctm.org/document/chapter6/index.htm
Grades 9-12
http://standards.nctm.org/document/chapter7/index.htm
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