Cognitive Science 301

Applied Cognitive Science and Education
Fall 2006

Mathematics, Chapter 11

Mathematics requires: "precision, consistency, attention to detail, conceptual agility, problem-solving flexibility, speed of processing and recall, and cumulative learning" (p 398).  Therefore all of the neurodevelopmental constructs interact with adequate performance in mathematics.

Each of the following are "Major requisites for mathematical accomplishment during school years" (p 398)

• Number concepts

  • According to the famous psychologist, Piaget: classification into categories, ordering in a logical sequence, one-to-one correspondence, and conservation (the volume and/or quantity of objects remains the same regardless of how it is organized or presented) are present well before 9 years of age

• Basic operations

  • addition, subtraction, multiplication, division

  • place value (regrouping, monetary value, percentiles, fractions)

  • recognition of which procedure to use

  • solving multistep problems

  • solving word problems (use of linguistics and math vocabulary)

  • organizing the numbers within an addition, subtraction, multiplication, division problem

• Graphomotor implementation

  • writing the actual numbers legibly

  • writing the problem in the appropriate space with the appropriate organization

• Transfer of knowledge

  • applying math in the classroom to everyday life

  • abstract to concrete math knowledge

• Mathematical linguistics

  • vocabulary associated with specific mathematical functions: e.g., division: dividend, quotient, divisor

  • vocabulary associated with geometric shapes, fractions

  • complex syntax of word problems

  • See p 408 for many other examples of linguistic expertise that is necessary in mathematics

• Visualization (mental imagery and spatial appreciation)

  • is necessary to understand number concepts

  • is necessary to understand geometric concepts

• Problem solving

  • is different from computation and understanding of math concepts

  • Bloom and Broder (1950) studied good and poor problem solvers in college

    • need to know how to get started

    • need to know exactly what is asked for in a problem

    • need to know what information is irrelevant/relevant to solving the problem

    • strategy for breaking the task into different, necessary steps

    • memory for steps involved and knowing where one is in the problem solving process

  • Krutetskii (Driscoll 1982) studied gifted mathematics students (in addition to being strong in the above tasks)

    • all of the above stated by Bloom and Broder

    • ability to be an active learner (goal oriented and flexible cognitively)

• Capability for estimation: moving back and forth between actual concrete numbers and the concept of estimation

  • students with weak previewing have weakness in estimation tasks

• Active working memory and mental arithmetic

  • need for holding relevant information, while manipulating it within a problem, until its solution

  • multistep problems provide a challenge

  • attention to detail throughout solving a problem (especially a word problem)

  • rapid retrieval of math facts for fluency in solving problems

• Higher-order cognition: and abstraction, proportion, equation, proofs

  • abstraction:  working with numbers means using and manipulating symbols

  • proportion: understanding and using fractions, percentiles

  • geometric proofs: selecting the appropriate deductive pathway

    • this involves (p 413) recognition of the shape of the geometric figure and appropriate vocabulary

    • analysis of the properties of the geometric figure

    • categorization ("knowing that all squares are rectangles")

    • deducing the appropriate postulates and theorems and their relationships to the proof

    • rigor: precision in logic

• Depth of knowledge and access to knowledge

  • acquisition of skills builds progressively

  • inadequate acquisition of initial skills limits the capability of the individual to conceptualize abstractly

  • problems can occur on a variety of levels (pp 413-414)

    • from only a "tenuous grasp" of the information and facts to knowing the relevance of application of mathematical concepts to daily life

• Attention to detail and self-monitoring

  • previewing and planning

  • monitoring and assessing progress

  • flexibility to change strategies in problem solving

See Table 11-2 (p 415) for Common error patterns in mathematics computation related to neurodevelopmental constructs

Stages of Learning of Mathematics (Smith and Rivera, 1991).  Knowing these stages is important in assessing an individual's mathematical progress.

• acquisition of skills

• proficiency (fluency after adequate practice of acquired skills)

• maintenance (need for periodic practice)

• generalization (flexible application of skills)

• adaptation (application of skills and concepts for "problem solving, reasoning, and decision making", p 417)

Main Ideas

1. Acquisition of mathematics knowledge demands stepwise mastery of many different skills beginning with preschool and kindergarten.

2. In addition to understanding basic operations, solution of multi-step problems and application of math concepts to real world situations are necessary during the elementary school years.

3. Math language is highly specialized and its vocabulary is essential for progress in mathematics.

4. having a good understanding of computation is not adequate for application and understanding of mathematical concepts.

5. Review all of the "need to know" items of Bloom and Broder (1950) that contribute to an individual's strength in mathematics.

6. Know how all of the neurodevelopmental constructs affect acquisition of mathematics proficiency.

Questions

1. List the different major requisites for mathematical accomplishment during school years (p 398).  Explain why Levine hypothesizes that each is required.

2. How does graphomotor function affect all of the other areas of major requisites listed on page 398?

3. Chunking is useful in writing and also in mathematics, explain why it is useful.

4. Is higher order cognition involved in computation?  Explain why or why not.

5. Can someone have attention weaknesses, but be strong in mathematics?  Explain.