Foundations for Learning Strategies

Share Cognitive Science Implications for Teaching Mathematics
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This blog post is longer than most because it seems to me the topics must be presented as a single unit.  Please study carefully.

As you study this essay, remember that every concept in mathematics begins with a definition and remember that all definitions in mathematics are stipulative definitions.

Every prominent cognitive scientist agrees that retrieval practice is the most powerful learning strategy available to a learner.  The list of scientists who currently make this claim include: Roediger, McDaniel, Willingham, Mayer, Willis, Bjork, Karpicke, Pashler, Weinstein, Sumeracki, and the list goes on.

Current research demonstrates that it is relatively easy to get information into long-term memory (see the diagram 5 Jul 18).  We have also discovered that if learning has failed the most likely cause is a failure to establish and exercise appropriate retrieval paths from long-term memory.  (see 12 Jul 18).

Proposed remedies are the eight techniques for strengthening retrieval paths listed on the above cited diagram. They warrant repeating here. With each technique I have also included comments relevant to teaching stipulative definitions in mathematics.

Whether you develop your own strategy for studying definitions in mathematics or adopt a strategy developed by someone else, the following should guide your development or selection.

  • Retrieval Practice 3 APR 18
    • Absolutely must be part of strategy
    • Very effective as a learning activity
    • Implemented with flash cards
      • Classic 3X5 cards
      • On-line implementations
        • I will discuss some of these programs in a future blog
    • Implemented with low-cost self-tests
  • Spaced Retrieval 5 APR 18
    • Absolutely must be part of strategy
    • Leads to stronger long-term retention than when it is massed
      • First observed by Ebbinghaus in 1885
      • Implemented online with Flashcard Programs
        • I will discuss some of these programs in a future blog
  • Interleaved Retrieval 10 APR 18
    • Absolutely must be part of strategy
    • Easily implemented with flashcards
    • Significant advantage of interleaving is learning how to:
      •  Assess context
      • Discriminate between problems
      • Selecting correct solution from a range of possibilities.
      • Intermingle a variety of mathematics topics
      • Intermingle mathematics with other subjects
  • Elaboration 12 APR 18  (see graphic at end of page)
    • Absolutely must be part of strategy
    • Extremely important step in learning a stipulative mathematics definition
    • Without extensive elaboration the memorized definition has no value
    • Elaboration is the process of finding layers of supportive material
    • Elaboration for a stipulative mathematics definition should be spaced over considerable time
    • Elaboration should include:
      • Illustrative examples
      • Illustrative exercises
      • Classification schemes
      • Illustrative images and diagrams
      • Relating to knowledge you already know
      • Relating to future knowledge and/or application
      • Position in a structure
        • Title of course
        • Textbook name
        • Chapter title
        • Section heading
        • Subsections headings
        • Named algebraic structures
  • Reflection 17 APR 18
    • Should be part of strategy
    • Review what is being learned
    • Ask yourself questions about what is being learned
      • Is this an object, an operation, or a relation? (14 JUN 18)
      • How does this material relate to previous knowledge?
      • Did the examples illustrate the concepts?
      • Can I create additional examples?
      • What kind of problems will this help solve?
      • What makes this concept important?
      • Is this a generalization of a known concept?
  • Generation19 APR 18
    • Inclusion as part of strategy is optional
    • An implementation might be fill-in-the-blank questions on low-cost self-tests
  • Calibration 24 APR 18
    • Should be part of the strategy
    • Aligning your judgments of what you know with objective feedback.
    • Objective feedback is obtained from tests
      • Implementation with low-stakes tests designed to determine mastery
      • Implementation with low-stakes tests designed to diagnose difficulties.
        • An implementation might be as simple as the Leitner procedure described on 19 JUL 19.
  • Mnemonics 26 APR 18
    • Do not include in a strategy for learning definitions
    • Not beneficial for learning mathematics definitions
  • Massed Practice 1 MAY 18
    • Do not include in a strategy for learning definitions
    • Probably the least effective study and learning technique
  • Rereading 3 May 18
    • Do not include in a strategy for learning definitions
    • Not beneficial for learning mathematics definitions
      • Flash cards provide any necessary review of wording
  • Underlining and Highlighting 8 MAY 18
    • Do not include in a strategy for learning definitions
    • Might be helpful in locating a definition in the text
    • Not beneficial for learning mathematics
Essential Elaboration to Understand Stipulative Definition

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