The Role of Manipulative Materials in the Learning of Mathematical Concepts
If the operator CO1-math fulfills m cO1-math for arbitrary functions we can derive the following relation: cO1-math which shows the and CO1-math can be more simply described by modifying the definition of. Math You Need > Rearranging Equations the world around you! See the Practice Manipulating Equations page for just a few examples. This is not a comprehensive dictionary of mathematical terms, just a quick addition, subtraction, multiplication and division (the more advanced manipulation of numbers . formula: a rule or equation describing the relationship of two or more.
The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes green triangles, blue thick rhombi, red trapezoids, yellow hexagons, orange squares, and tan thin rhombi are applied to make mosaics.
Interlocking cubes are also available in 1 centimeter size and also in one inch size to facilitate measurement activities. Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence, Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue.
Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring and graphing, perimeter, area and volume. Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. We see illustrations of this idea everywhere, but we do not see the idea itself.
In a similar way the symbol "2" is used to elicit a whole series of recollections and experiences that we have had entailing the concept of two, but the squiggly line 2 in and of itself is not the concept. How then do we teach children about the concept of number if as indicated it is a total abstraction? The answer is very much related to the concept of an isomorphism. For if a parallel structure that was more accessible and perhaps manipulable could be identified having the same properties as the set of whole numbers, then it would be possible to operate within this more accessible and isomorphic structure and subsequently to make conclusions about the more abstract system of number.
This is precisely what happens.
Manipulative (mathematics education) - Wikipedia
Sometimes these artificially constructed systems are called interpretations or embodiments of a concept. Some examples of these partial isomorphisms are using counters or sets of objects to represent the counting numbers a discrete modelusing lengths such as number lines or Cuisenaire rods to represent the set of real numbers a continuous modeland using the area of a rectangle to represent the multiplication of two whole numbers or fractions. Manipulative materials may now be viewed simply as isomorphic structures that represent the more abstract mathematical notions we wish to have children learn.
Each represents the cognitive viewpoint of learning, a position that differs substantially from the connectionist theories that were predominant in educational psychology during the first part of the twentieth century.
Modern cognitive psychology places great emphasis on the process dimension of the learning process and is at least as concerned with "how" children learn as with "what", it is they learn. Emphasis is placed, therefore, on the interrelationships between parts as well as the relationship between parts and whole. Each of these men subscribes to a basic tenet of Gestalt psychology, namely that the whole is greater than the sum of its parts.
Each suggests that the learning of large conceptual structures is more important than the mastery of large collections of isolated bits of information. Learning is thought to be intrinsic and, therefore, intensely personal in nature. It is the meaning that each individual attaches to an experience which is important.
It is generally felt that the degree of meaning is maximized when individuals are allowed and encouraged to interact personally with various aspects of their environment. This, of course, includes other people.
I t is the physical action on the part of the child that contributes to her or his understanding of the ideas encountered. Proper use of manipulative materials could be used to promote the broad goals alluded to above.
I will discuss each of these men more fully since each has made distinct contributions to a coherent rationale for the use of manipulative materials in the learning of mathematical concepts. Jean Piaget Piaget's contributions to the psychology of intelligence have often been compared to Freud's contributions to the psychology of human personality.
Piaget has provided numerous insights into the development of human intelligence, ranging from the random responses of the young infant to the highly complex mental operations inherent in adult abstract reasoning. He has established the framework within which a vast amount of research has been conducted, particularly within the past two decades. In his book The Psychology of IntelligencePiaget formally develops the stages of intellectual development and the way they are related to the development of cognitive structures.
His theory of intellectual development views intelligence as an evolving phenomenon occurring in identifiable stages having a constant order. The age at which children attain and progress through these stages is variable and depends on factors such as physiological maturation, the degree of meaningful social or educational transmission, and the nature and degree of relevant intellectual and psychological experiences.
Piaget regards intelligence as effective adaptation to one's environment. The evolution of intelligence involves the continuous organization and reorganization of one's perceptions of, and reactions to, the world around him.
This involves the complementary processes of assimilation fitting new situations into existing psychological frameworks and accommodation modification of behavior by developing or evolving new cognitive structures.
The effective use of the assimilation-accommodation cycle continually restores equilibrium to an individual's cognitive framework. Thus the development of intelligence is viewed by Piaget as a dynamic, nonstatic evolution of newer and more complex mental structures. Piaget's now-famous four stages of intellectual development sensorimotor, preoperational, concrete operations, and formal operations are useful to educators because they emphasize the fact that children's modes of thought, language, and action differ both in quantity and quality from that of the adult.
Piaget has argued persuasively that children are not little adults and therefore cannot be treated as such. This proposition, if followed to its logical conclusion, would substantially alter the role of the teacher from expositor to one of facilitator, that is, one who promotes and guides children's manipulation of and interaction with various aspects of their environment.
While it is true that when children reach adolescence their need for concrete experiences is somewhat reduced because of the evolution of new and more sophisticated intellectual schemas, it is not true that this dependence is eliminated. The kinds of thought processes so characteristic of the stage of concrete operations are in fact utilized at all developmental levels beyond the ages of seven or eight.
Manipulative (mathematics education)
Piaget's crucial point, which is sometimes forgotten or overlooked, is that until about the age of eleven or twelve, concrete operations represent the highest level at which the child can effectively and consistently operate.
Piaget has emphasized the important role that social interaction plays in both the rate and quality with which intelligence develops. The opportunity to exchange, discuss, and evaluate one's own ideas and the ideas of others encourages decentration the diminution of egocentricitythereby leading to a more critical and realistic view of self and others.
I t would be impossible to incorporate the essence of these ideas into a mathematics program that relies primarily or exclusively on the printed page for its direction and "activities. It is generally felt that the basic components of a theoretical justification for the provision of active learning experiences in the mathematics classroom are embedded in Piaget's theory of cognitive development. Dienes and Bruner, while generally espousing the views of Piaget, have made contributions to the cognitive view of mathematics learning that are distinctly their own.
The work of these two men lends additional support to this point of view. Dienes Unlike Piaget, Dienes has concerned himself exclusively with mathematics learning; yet like Piaget, his major message is concerned with providing a justification for active student involvement in the learning process. Such involvement routinely involves the use of a vast amount of concrete material. Rejecting the position that mathematics is to be learned primarily for utilitarian or materialistic reasons, Dienes sees mathematics as an art form to be studied for the intrinsic value of the subject itself.
He believes that learning mathematics should ultimately be integrated into one's personality and thereby become a means of genuine personal fulfillment. Dienes has expressed concern with many aspects of the status quo, including the restricted nature of mathematical content considered, the narrow focus of program objectives, the overuse of large-group instruction, the debilitating nature of the punishment- reward system gradingand the limited dimension of tile instructional methodology used in most classrooms.
Dienes's theory of mathematics learning has four basic components or principles. Each will be discussed briefly and its implications noted. The reader will notice large-scale similarities to the work of Piaget. This principle suggests that true understanding of a new concept is an evolutionary process involving the learner in three temporally ordered stages.
The first stage is the preliminary or play stage, and it involves the learner with the concept in a relatively unstructured but not random manner.
For example, when children are exposed to a new type of manipulative material, they characteristically 'play' with their newfound 'toy. This means less memorization! Manipulating equations can help you keep track of or figure out units on a number. Because units are defined by the equations, if you manipulate, plug in numbers and cancel units, you'll end up with exactly the right units for a given variable! Where is this used in the geosciences?
To be honest, equation manipulation occurs in almost every aspect of the geosciences.
Because equations can be used to describe lots of important natural phenomena, being able to manipulate them gives you a powerful tool for understanding the world around you! See the Practice Manipulating Equations page for just a few examples. You probably learned a number of rules for manipulating equations in a previous algebra course.