CHAPTER OBJECTIVES
1. Describe how children learn mathematics.
2. Define the strands of mathematics, as articulated by the National Council of Teachers of Mathematics.
3. Plan appropriate mathematics experiences for young children.
4. Recognize the qualities of materials that are appropriate for mathematics instruction.
5. Assess the mathematical understanding of young children.
6. Plan for connections between mathematics and literacy.
7. Adapt mathematics instruction for children with special needs.
8. Celebrate the diversity of mathematics learners in your classroom.
CHAPTER OUTLINE
1. Mathematics does not mean arithmetic. Mathematics is an understanding of numbers that allows experimentation and problem solving, and is therefore a helping discipline rather than a discrete subject area.
2. Piaget describes three aspects of learning:
A) Concepts from the physical world (e.g. hot, cold, & rough)
B) Concepts from the social world (e.g. religion, language, & superstition)
C) Logico-mathematical construction of mental relationships
i. Logico-mathematical thinking requires understanding of hierarchical relationships, and the differentiation of characteristics that are inherent in an object and characteristics that are imposed by the mental set.
ii. Simple abstraction, or empirical abstraction, means ignoring many properties of an object to focus on the property of interest.
iii. Reflective abstraction involves mentally manipulations that are based on past physical experiences but no longer require physical representation.
iv. Perceptual numbers are small sets that can be compared by even young children without counting the items in each set.
v. With larger sets, children must use a system of numbers with a stable order and one-to-one correspondence.
D) Vygotskian theory stresses the interaction between the child's experiences and the social worlds. Children can experience number and manipulation of sets, but must be supplied with the language of mathematics. Teachers can model language and structure situations to facilitate concept discovery.
3. Mathematics Instruction
A) A primary goal should be for children to feel like capable learners of mathematics and that they can use their math skills to solve real-life problems.
B) Math in the Classroom
i. The NCTM believes that their five content standards (e.g. numbers, operations and patterns) and five process standards (e.g. problem solving and measurement) should be interconnected and interwoven throughout the curriculum.
a) Content Standards:
(1) Number and operation
(2) Patterns, functions and algebra
(3) Geometry and spatial sense
(4) Measurement
(5) Data analysis, statistics and probability
b) Process standards
(1) Problem solving
(2) Reasoning and proof
(3) Communication
(4) Connections
(5) Representation
ii. Research suggests that children, whose skills and strategies are based on understanding these concepts, are more likely to retain and expand their knowledge.
4. Instruction of Mathematics
A) In accordance with Vygotsky, the NCTM advocates teaching mathematics through real-world activities covering a broad range of content areas and emphasizing problem-solving skills. Teachers can do this effectively by explicitly using math throughout the day as part of classroom management activities as well as in children's own activities.
B) It is always better to integrate math into activities than to allocate a separate subject time for math, but in primary grades some subject specific lessons may be necessary.
C) Many typical classroom activities, such as calendar, can be used to scaffold hierarchical organization and sequencing skills.
5. The Strand Model is designed to help teachers see that all aspects of mathematics are interrelated and go beyond arithmetic to include diverse problem solving skills.
A) NCTM uses this model to emphasize the importance of communicating the utility and flexibility of mathematics to students.
B) Process strands (communication, connection, representation, reasoning and proof, and problem solving) cross the entire model, because knowledge is only useful when it can be applied and explained.