The Effect of Limited Working Memory on Reading Proofs John Selden Mathematics Education Resources Company
Annie Selden Tennessee Technological University
[email protected]
Special Session on Research in Undergraduate Mathematics Education
SOUTHEASTERN SECTION MAA ATLANTA, MARCH 1997
Proofs are written in a distinctive style (amongst general deductive arguments) This style minimizes readers errors due to working memory overload (as opposed errors due to misconceptions)
MEMORY Except for process associated with perception there are three main kinds of memory systems: short-term long-term working
SHORT-TERM MEMORY Half-life: about 15 seconds Capacity: about 7 'chunks' of information A 'chunk' consists of any information that can be thought of as a unit, e.g., words familiar patterns of chess pieces Pythagorean Theorem
One is aware of the contents of short-term memory The contents are available for reasoning
LONG-TERM MEMORY Long-term memory is memory that is not short-term Has a very large (essentially unlimited) capacity It is not readily available for reasoning One has no awareness of it It can be 'activated,' i.e., brought into short-term memory
WORKING MEMORY Short-term memory (including memory activated from long-term memory) Reasoning and control mechanisms that swap information between short-term and long-term memory
SUBCATEGORIES OF LONG-TERM MEMORY Knowledge base: lasting at least several years, e.g., the Pythagorean Theorem for most mathematicians Local memory: more ephemeral, partly activated, e.g., memory arising from 'validating' (reading and checking) a proof. One might remember, for a short time, that a proof began with 'Let x be a number.'
WHAT KIND OF PROOFS? The kind found in published papers, reference books, and advanced textbooks Those seen by and expected of students taking advanced courses Their main purpose: to determine whether a theorem is true. (Reading for this purpose we call 'validation.')
Errors due to working memory overload are errors of omission, perhaps due to distractions, e.g., One forgets to check an inference. One forgets to check a case, e.g., for real numbers, one checks x
