Developmentally Appropriate Practices

Understanding how children's development at specific ages affects the teaching and learning of mathematics

Bank Street
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 This notion has gradually been replaced by one that matches children's observable behavior more closely, that children think in a different way from adults, that they draw distinct conclusions from the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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