How to Take a Child From Concrete to Abstract: Early mathematics experiences using concrete materials form the foundation for all later learning in mathmatics. A child's early attitudes about math determine there interest and motivation towards learning and using math. Without a sufficient number of appropriate concrete math experiences that providesufficient modeling for math concepts, at the early stage in the child's development, the child will encounter many learning difficulties in the future. The learned cooncrete concepts should be PLANNED to avoid frustration.
A.Important Factors in Mathematics Learning
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Strong and appropriate connections between the learning experiences at different levels must be made.
C. How Long? How Much?
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When and how to remove munipulatives materials as instructional aids in teaching of mathematics cocepts?
E. Key Skills
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Students can rely on one manipulative to solve problems. This can cause an over reliance for the student. To avoid this a teacher needs to present a mathematical concept in more than one concrete material and model.
G. Learning Process
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Every new mathematics concept has its own cognitive complexity, but at the same time it is dependent on many of the earlier concepts because each concept is developmental and cumulative in nature.
J. Tactile and Logico-Mathematical Experiences
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A child's success in either of these concepts-visual clustering and estimation, is dependent on the extent of the child's concrete experiences. Thus, to help children in the abstract math activities, the teacher needs to assess the child's level of cognitive development and the extent of concrete experiences he has had.
L. Implications
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Teachers should never assume a student is at the formal level of math for that grade. Students need to demonstrate through REPEATED experiecnes their understanding of the abstract. It is important that the teacher regularly exposes children to the oral use of math terms and at the same time asks them to verbalize their concrete experiences.
B.Need for Manipulative Materials
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In most cases, as soon as the child has been provided with appropriate concrete manipulative experiences and effective ways of their use the child learns mathematics much more quickly and effectively.
D. Mathematics Concept-triad
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Every math concept consists of three components: Linguistic, Conceptual, and Skill.
F. Perception and Mathematics Learning
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The very nature of math isists that a person must perceive objects, their attributes and their relationships.
I. Estimation
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Ability to see the range of the possible answers.
K. The Five Step Spiral
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Step 1: Free PayStep 2: Primary Operations and Primary ConceptsStep 3: Manipulatives ActivitiesStep 4: Recording the Concrete ExperienceStep 5: Experience at the Abstract Level
M. Examples of Concrete Instructional Materials
Visual Clustering
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Ability to identify the number of objects in a collection by visually approximating it, which means counting in clusters rather than one to one.
