“please put away that pocket calculator, billy.”
TRANSCRIPT
I. Objectives for this I. Objectives for this LessonLesson
Define basic facts for each of the operations, discuss Define basic facts for each of the operations, discuss their role and importance in a mathematics program, their role and importance in a mathematics program, and identify reasonable outcomes for students.and identify reasonable outcomes for students.
Identify and discuss factors which contribute to a Identify and discuss factors which contribute to a child’s success with basic facts.child’s success with basic facts.
Identify desired student outcomes with basic facts. Identify desired student outcomes with basic facts. Justify your decision.Justify your decision.
Define Brownell’s stages for the successful learning Define Brownell’s stages for the successful learning of the basic facts and elaborate on examples and of the basic facts and elaborate on examples and techniques at each level. List techniques Brownell techniques at each level. List techniques Brownell identified as being inappropriate in learning the basic identified as being inappropriate in learning the basic facts. What happens if the children begin the drill facts. What happens if the children begin the drill stage too soon?stage too soon?
II. What Is a Basic Fact?II. What Is a Basic Fact?
3+ 4
7
2+ 7
9
2+ 911
9+ 918
0+ 0
0
7+11
18
3+10
13
20+30
50
12+ 012
10+ 111
V. What Does a Child Need V. What Does a Child Need to Know About Basic Facts?to Know About Basic Facts?
VI. What is Known About VI. What is Known About Learning Basic Facts?Learning Basic Facts?
Understanding first - drill laterUnderstanding first - drill later Experiences with concrete materialsExperiences with concrete materials There are specific stages of developmentThere are specific stages of development A careful instructional sequence yields A careful instructional sequence yields
successsuccess Use effective drill techniques after earlier Use effective drill techniques after earlier
stages are developedstages are developed Attitudes are critical!Attitudes are critical!
Brownell’s Stages of Learning Brownell’s Stages of Learning the Basic Factsthe Basic Facts
Effective TechniquesEffective Techniques1.1. Naive countingNaive counting2.2. Sophisticated Sophisticated
countingcounting3.3. StrategiesStrategies
(solution (solution techniques)techniques)
4.4. Immediate recallImmediate recall(meaningful (meaningful habituation)habituation)
Ineffective TechniquesIneffective Techniques1. Guessing1. Guessing2. Recitation of tables2. Recitation of tables3. Rote recall without3. Rote recall without understandingunderstanding
Comparison of StagesComparison of Stages
StageStage
1.1.
2.2.
3.3.
4.4.
AdditionAddition
RecountingRecounting
Counting-Counting-OnOn
StrategiesStrategies
DrillDrill
Multiplication
Counting
Skip Counting
Strategies
Drill
VII. Do All Children Go VII. Do All Children Go Through All of the Stages?Through All of the Stages?
IV. Uses of NumberIV. Uses of Number
Cardinal (Discrete) - How many?Cardinal (Discrete) - How many? Cardinal (Continuous) - How much?Cardinal (Continuous) - How much? Ordinal - Which one?Ordinal - Which one? Nominal - A name or codeNominal - A name or code
Counting-onCounting-on
Four and three more makes: four, five, Four and three more makes: four, five, six, sevensix, seven
This is a critical counting skill!This is a critical counting skill! ActivitiesActivities
Counting-backCounting-back
Rote counting backwardsRote counting backwards Why is counting-back important?Why is counting-back important?
Invariance of NumberInvariance of Number
The most critical “hurdle” of the The most critical “hurdle” of the primary grades is conservation primary grades is conservation (invariance) of number.(invariance) of number.
Can we teach invariance of number?Can we teach invariance of number? Can we create an environment where Can we create an environment where
invariance is learned?invariance is learned? ActivitiesActivities
Skip CountingSkip Counting
Two, four, six, eight, ten,...Two, four, six, eight, ten,... A prerequisite to place-value, basic A prerequisite to place-value, basic
facts, multiplication, division, time, facts, multiplication, division, time, money,...money,...
Seldom mentioned directly in Seldom mentioned directly in textbooks other than as sequences to textbooks other than as sequences to be completedbe completed
2 4 6 __ 102 4 6 __ 10 ActivitiesActivities
Extending to TwentyExtending to Twenty
The child’s first work beyond ten should The child’s first work beyond ten should not emphasize place-value. Rather, it not emphasize place-value. Rather, it should be an extension of the should be an extension of the familiarfamiliar work with numbers of ten or less. One work with numbers of ten or less. One ten vs. ten ones is very confusing.ten vs. ten ones is very confusing.
There are significant problems with the There are significant problems with the number names between ten and twenty.number names between ten and twenty.
ActivitiesActivities
III. Operation “SituationsIII. Operation “Situations””
Almost all uses of the four basic Almost all uses of the four basic operations may be organized into operations may be organized into just eight “situations”just eight “situations”
There is no reason to assume that There is no reason to assume that instruction in one “situation” will instruction in one “situation” will generalize into another “situation”. generalize into another “situation”. Instruction must be provided in Instruction must be provided in each.each.
Operation “Situations”Operation “Situations”Addition and SubtractionAddition and Subtraction
Join ProblemsJoin Problems Separate ProblemsSeparate Problems Part-Part-Whole Part-Part-Whole
ProblemsProblems Compare ProblemsCompare Problems
IV. Stages in Developing IV. Stages in Developing thethe
OperationsOperationsStep Presentation Solution Repor t
1 Oral Concrete Oral
2 Oral Concrete Pictorial
3 Oral Concrete Pictorial/Symbolic
4 Oral Pictorial Symbolic
5 Symbolic Pictorial Symbolic
6 Symbolic Mental Image Symbolic
Operation “Situations”Operation “Situations”Multiplication and DivisionMultiplication and Division
Equal Groups Equal Groups (partition (partition and measurement)and measurement)
Multiplicative Multiplicative ComparisonComparison
CombinationsCombinations Product of MeasuresProduct of Measures
Activity: Vertical Addition Sentence
• Bowls
Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison-Wesley, 1995
Activity: Horizontal Addition Sentence
• Helping HandsAdapted from “Math Their Way” by Mary Baratta-Lorton, Addison-Wesley, 1995
Activity: Horizontal Subtraction Sentence
• Take-away hand
Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison-Wesley, 1995
I. Objectives for this I. Objectives for this LessonLesson
Identify appropriate instructional Identify appropriate instructional techniques for each of Brownell’s stages.techniques for each of Brownell’s stages.
Identify inappropriate instructional Identify inappropriate instructional techniques for helping children learn the techniques for helping children learn the basic facts.basic facts.
Identify a wide range of strategies for each Identify a wide range of strategies for each of the operations. Give examples of of the operations. Give examples of instructional activities for each technique.instructional activities for each technique.
Identify an overall plan for teaching the Identify an overall plan for teaching the basic facts. What does it mean to teach basic facts. What does it mean to teach “strategy selection”?“strategy selection”?
Ages and StagesAges and Stages
Stage Add Subtract Multiply Divide
Operation Concepts Pk, K, Pk, K K, 1 K, 1
Naive Counting K, 1 K, 1 1, 2 1, 2
Sophisticated Counting 1,2 1,2 2, 3 2, 3
Solution Techniques 2 2 3, 4 3, 4
Immediate Recall 2 2 4 4, 5
Activity: Vertical Addition SentenceActivity: Vertical Addition Sentence
BowlsBowls
Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison-Wesley, 1995
Activity: Horizontal Addition SentenceActivity: Horizontal Addition Sentence
Helping HandsHelping HandsAdapted from “Math Their Way” by Mary Baratta-Lorton, Addison-Wesley, 1995
ActivityActivity: : Horizontal Subtraction Horizontal Subtraction SentenceSentence
Take-away handTake-away hand
Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison-Wesley, 1995
C. Sophisticated C. Sophisticated CountingCounting
Use counting activitiesUse counting activities ““Generate” the Basic FactsGenerate” the Basic Facts Begin to use tables to record Begin to use tables to record
solutions solutions Consider the multiple Consider the multiple
situations for each operationsituations for each operation
D. StrategiesD. Strategies
Introduce the strategyIntroduce the strategy Practice the strategyPractice the strategy Add new strategiesAdd new strategies IndividualizeIndividualize Practice strategy selectionPractice strategy selection
E. DrillE. Drill
Make sure students are successful in Make sure students are successful in previous stages first.previous stages first.
Use small drills, distributed over time.Use small drills, distributed over time. Provide immediate feedback at first.Provide immediate feedback at first. Use “disguised practice” to maintain Use “disguised practice” to maintain
motivationmotivation Continue to focus on success and Continue to focus on success and
positive attitudes.positive attitudes.
III. Other IssuesIII. Other Issues
Fingers vs. concrete Fingers vs. concrete materialsmaterials
Upper grade remediationUpper grade remediation ““Help” from parentsHelp” from parents