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. 

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 with Leitner boxes
    • 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 mathematic