Piaget

The progression from concrete to abstract

Piaget may be the education theorist that you will hear most about as you study to be a teacher. His thesis was that children pass through a series of stages of development (which are not optional) as they learn and grow. The first stages are most concrete, and the last stage is the most abstract. This is in broad terms a really good description of how people learn, and many researchers have studied this from different perspectives. Piaget has been very influential, and a lot of educational research is explained in terms of its relationship to Piaget's theories.

Piaget describes children's development in four stages:

• Sensorimotor: The stage in infancy characterized by early language learning, awareness of object permanence, and learning basic concepts in several domains. In mathematics some early concepts are the concept of number and of relationships (more, less, same, in, under, etc.)
• Preoperational: The young child stage characterized by an increase in language acquisition and generalizations of various sorts. A typical language generalization associated with this stage would be the generalization that -s means plural, leading to the formation of words such as mans that they might have previously used correctly. In mathematics, children at this stage are learning to count and sort objects (grouping objects that are similar). Children at this stage typically will focus on one aspect of something to the exclusion of other properties; a typical example of this is the example that when you pour water into a wider or a narrower container, the child will typically identify the container that is narrower as having more water because the water level is higher. This stage corresponds roughly with pre-school development.
• Conrete operational: The stage that roughly corresponds to early elementary ages, where children are developing a lot of cognitive ideas, and where their understandings are grounded in concrete examples. Children learn from and problem solve with concrete materials during this stage. Children at this stage develop ideas of ordering, classifying, and working with numbers (addition, subtraction, etc) through concrete reference points. Children at this stage can successfully compare things by several different criteria (heighth, width, weight,etc.).
• Formal operational: A stage that is generally considered to be reached by late elementary or early middle school years. Children at this level can reason using abstract concepts and representations. Children at this stage can make inferences, evaluate problems and solutions, and can apply their learning to new contexts.

People doing educational research since Piaget, have made several refinements to the original theory. These refinements aren't considered part of Piaget's theory, but they are related theories, each of which are useful in their own contexts.

• The stages of development are tied to childrens prior experiences and learning, and that learning experiences can accelerate development (this is included in Piaget's original work), and those learning experiences are often necessary for children's progression (this is different--it means that abstract thought in some areas has to be taught, you don't just grow into it). So the progression from concrete to abstract is based on learning experiences in a measurable way.
• Students can be able to think abstractly about some things, and still be at a concrete stage for other things. This is a significant deviation from Piaget in that he saw a capacity for abstract thought as something that one attained at a certain age, and could apply to thinking about anything, whereas the research on specific progressions of knowledge show that domain of thought can have its own path of progression from concrete to abstract. For instance, a middle school student might be at an abstract level with respect to number and algebra, but they might be at a concrete level with respect to geometry (or vice versa).

We're not going to pay much attention in this class to the specifics of Piaget's theory most of the time, but we will be using the idea of moving from concrete to abstract ways of understanding math ideas.

Key vocabulary and ideas:

Concrete: a way of thinking (about math) that relies on objects (manipulatives), drawings, or other specific, manipulable representations.

Abstract: a way of thinking (about math) that uses numbers, symbols, graphs and other representations that summarize important ideas.

Reasoning: defn 1: a logical way of progressing from some ideas to related ideas; defn 2: a way of thinking and figuring out something.