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
