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 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**- 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?

- Is this an object, an operation, or a relation? (

- Generation
**19 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**.

- An implementation might be as simple as the Leitner procedure described on

- 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