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Key ideas about information processing & math learning
Don’t confuse “knowledge” with “memory”
“Knowledge” = understanding & relating
“Memory” = retaining & recalling knowledge
However, the way information is processed by the learner has a DRAMATIC affect on BOTH!
EVERYTHING WE TEACH BOILS DOWN TO…
The learner…
FactsBasic Concepts
Associated Concepts
ProcessesStrategies
What would be an example from math curriculum of…
Fact Dichotomous information (right or wrong)
Basic concept
Idea that must be experienced to understand – cannot be defined with words
Associated concept
Requires a explanation using words
Process Series of consistent steps to follow
Strategy Loosely structured plan of attack that requires on-going decision making about the application of different processes, monitoring, & reflection
The learner…
FactsBasic Concepts
Associated Concepts
ProcessesStrategies
So which kind of knowledge is Calvin struggling with?
Fact
Basic conceptAssociated concept
Process
Strategy
The learner…
FactsBasic Concepts
Associated Concepts
ProcessesStrategies
Fact
Basic conceptAssociated concept
Process
Strategy So which kind of knowledge does this picture address?
The learner…
FactsBasic Concepts
Associated Concepts
ProcessesStrategies
Which kind of knowledge is illustrated in this cartoon?
Fact
Basic conceptAssociated concept
Process
Strategy
The learner…
FactsBasic Concepts
Associated Concepts
ProcessesStrategies
Which kind of knowledge is illustrated in this cartoon?
Fact
Basic conceptAssociated concept
Process
Strategy
The learner…
FactsBasic Concepts
Associated Concepts
ProcessesStrategies
Fact
Basic conceptAssociated concept
Process
Strategy
The learner…
FactsBasic Concepts
Associated Concepts
ProcessesStrategies One more time…
Fact
Basic conceptAssociated concept
Process
Strategy
The learner…
Makes many connections to other ideas
Can explain how idea is affected by various
phenomenaCan explain impact of idea on the
world
Makes effective comparisons with other ideas
Has sufficient knowledge of relevant facts
Can identify different manifestations of idea
Can summarize or explain gist of idea in own words
SEMANTIC KNOWLEDGE
FactsBasic Concepts
Associated Concepts
ProcessesStrategies
PROCEDURAL KNOWLEDGE
CONDITIONAL KNOWLEDGE
how to do it
what it is about & why it’s relevant
when to use / not use
3 BASIC KINDS OF KNOWLEDGE
different ways to do it
UNDERSTANDING A MATH ASSOCIATED CONCEPT
The learner…
Can perform the process smoothly and accurately
Can explain why the process is necessary to perform certain kinds of problems
Makes effective comparisons with other processes
Has sufficient knowledge of relevant mathematical facts
Knows WHAT the steps to the process are
SEMANTIC KNOWLEDGE
FactsBasic Concepts
Relational Concepts
ProcessesStrategies
PROCEDURAL KNOWLEDGE
CONDITIONAL KNOWLEDGE
how to do it
what it is about & why it’s relevant
when to use / not use
3 BASIC KINDS OF KNOWLEDGE
different ways to do it
UNDERSTANDING A MATH COMPUTATION
Recognizes when it is appropriate and not appropriate to apply the process
The learner…
SEMANTIC KNOWLEDGE
FactsBasic Concepts
Relational Concepts
ProcessesStrategies
PROCEDURAL KNOWLEDGE
CONDITIONAL KNOWLEDGE
how to do it
what it is about & why it’s relevant
when to use / not use
different ways to do it
UNDERSTANDING A MATH COMPUTATION
How do you get students to have good computation skills & knowledge?
The learner…
SEMANTIC KNOWLEDGE
FactsBasic Concepts
Relational Concepts
ProcessesStrategies
PROCEDURAL KNOWLEDGE
CONDITIONAL KNOWLEDGE
how to do it
what it is about & why it’s relevant
when to use / not use
different ways to do it
UNDERSTANDING A MATH COMPUTATION
How do you get students to have good computation skills & knowledge?
The learner…
SEMANTIC KNOWLEDGE
FactsBasic Concepts
Relational Concepts
ProcessesStrategies
PROCEDURAL KNOWLEDGE
CONDITIONAL KNOWLEDGE
how to do it
what it is about & why it’s relevant
when to use / not use
different ways to do it
UNDERSTANDING A MATH COMPUTATION
How do you get students to have good computation skills & knowledge?
PERFECT PRACTICE
= PERFECT LEARNINGIMPERFECT PRACTICE
= IMPERFECT LEARNING
SCAFFOLDEDI DO IT
WE DO ITY’ALL DO IT
YOU DO IT
Instruction must be…DEVELOPMENTALLY APPROPRIATE
The learner…
SEMANTIC KNOWLEDGE
FactsBasic Concepts
Relational Concepts
ProcessesStrategies
PROCEDURAL KNOWLEDGE
CONDITIONAL KNOWLEDGE
how to do it
what it is about & why it’s relevant
when to use / not use
different ways to do it
UNDERSTANDING A MATH COMPUTATION
How do you get students to have good computation skills & knowledge?
SCAFFOLDED
Instruction must be…DEVELOPMENTALLY APPROPRIATE
SUFFICIENT TO CREATE BRAIN PATHWAYS
PERIODICALLY REVIEWED
Most appropriate math homework is practicing previously learned skills to build fluency….
…NOT working problems that address new concepts or skills just introduced in class!
EVERYTHING WE TEACH BOILS DOWN TO…
The learner…
SEMANTIC KNOWLEDGE
FactsBasic Concepts
Relational Concepts
ProcessesStrategies
PROCEDURAL KNOWLEDGE
CONDITIONAL KNOWLEDGE
how to do it
what it is about & why it’s relevant
when to use / not use
different ways to do it
UNDERSTANDING A MATH COMPUTATION
How do you get students to have good computation skills & knowledge?
ELABORATION
ELABORATION
ELABORATION
ELABORATION
ELABORATION
!
Remember
Short-term memoryHold on to the information long enough for working memory to process the information
Role
How Rote – no meaning given to info; sub-vocalizing
Affected by Fatigue, stress, multi-tasking, brain functions
Improved by
Medicine; no evidence that training/education helps
There is nothing teachers can do to strengthen short-term memory
There are plenty of things teachers can do to make the situation worse!
There are some things teachers can do help circumvent the problem
e.g., writing instructions on board rather than telling them orally
SHORT TERM
WORKING
LONG TERM
So how does memory fit into this?
Brain elaborates informationto develop “relational understanding”
Generative strategy training helps a LOT! Medications; meaningful brain-exercise
Quality of short-term memory; degree of acquired knowledge; language development, emotions, time to process, stress; intentional behaviors & intentional learning, complexity of infoImproved
by
7 Key ELABORATION Strategiessummarizing
organizingvisualizingcomparingpredicting
questioningevaluating
NOTE! Many students with LD or related learning language problems experience working memory deficits
Make meaning; connect/integrate ideas to existing knowledge; evaluate info; monitor comprehension
Role
How
Affected by
Working memory
Short-term memory
Unclear; stagnant “filing” metaphor is too simplistic; evidence that information in memory is in constant state of reconstruction.
Unclear whether medicine improves long term memory, (may be due to positive impact of drug on short-term, & working memory processes, as well as executive functions)
Quality of elaboration that occurred when info processed by working memory; the more info is patterned via neural pathways, more accessible info is for retrieval; the more semantic information is linked to episodic memory, the better the long-term memory
Improved by
Store & retrieve informationRole
How
Affected by
Working memory
Long-term memory
Short-term memory
Characteristics of students with learning disabilities that affect math learning
Do not know basic math facts, thus hugely struggle learning sophisticated processes that require this knowledge
Lack of reading skills
Tend to be rote learners, use verbal rehearsal rather the visualization of math concepts
Characteristics of students with learning disabilities that affect math learning
Typically fail to link new learning with background knowledge or previously learned skills
IMPLICATIONS: Advance organizers,
Review perquisite knowledge & skills,
Cue students to make links
Cue real-world connections
Characteristics of students with learning disabilities that affect math learning
Typically non-strategic; very poor at monitoring
IMPLICATIONS: Explicit instruction in strategies;
Write the steps on board & handout. Touch step, say out load, then do it.
Using think-alouds to model monitoring
Characteristics of students with learning disabilities that affect math learning
Most are very poor at using elaboration as a learning tool; poor relational understanding
IMPLICATIONS: Employ “low-risk” elaboration activities
Cue use of specific elaboration strategies…“Picture in your mind…”“How are these different?” “Explain the steps to me.”
Use concrete analogies
Use think-sheets & graphic organizers
Characteristics of students with learning disabilities that affect math learning
Difficulty maintaining attention
IMPLICATIONS: Employ direct instruction that results in high frequency correct responding
Provide explicit feedback frequently
Characteristics of students with learning disabilities that affect math learning
Some experience difficulty with short-term memory and/or auditory processing =dramatically affects board to paper note-takingIMPLICATIONS:
Reduce the amount of language students must process.
Pre-teach vocabulary
Employ think-sheets & graphic organizers.
LOTTERY:A tax on people who are bad at math
Specific settings/contexts the student should use the strategy to be more successful
Strategy This strategy is designed to enable students to ...
Likely background knowledge and/or experiences of students
Strategy Unit Plan
Key Points to Make when motivating students to learn the strategy
Essential prerequisite skills to review and ensure students have mastered How prerequisite skills will be reviewed
Advanced Practice Practice MaterialsControlled Practice
Practice Tasks / Materials Practice Tasks / Materials
Generalization
Generalization Tasks
Generalization PromptsWho Prompt type When
Specific behaviors to be assessed How behaviors will be assessed
Generalization Evidence
Use this think-sheet to plan a unit about a math problem-solving
process or strategy
The Strategy Unit Plan
Skills to reviewStudents Objectiveor re-teach Materials OutcomeLearning Level
OPENING - Anticipatory SetSet goals and/or plot goal on progress chart
Information about relevance of strategy
Re-visit what students did best on last practice activity
Steps to performing the strategy Situations where the strategy can be used
Review student progress toward strategy
generalization
Course of Study Standard
Best at…
Needs to learn …
IntroductorySimple practiceApplied practiceGeneralization
I do itWe do itY’all do itYou do it
Best at…
Needs to learn …
IntroductorySimple practiceApplied practiceGeneralization
I do itWe do itY’all do itYou do it
Best at…
Needs to learn …
IntroductorySimple practiceApplied practiceGeneralization
I do itWe do itY’all do itYou do it
BODY
Forecast future learning & generalization
Facilitate student reflection about strengths & weakness
associated with performing the strategy CLOSING
Use this think-sheet to plan specific lessons that address a math problem-
solving process or strategy
The Strategy Lesson Plan
