“Please put away that pocket calculator, Billy.”.

“Please put away that pocket calculator, Billy.”

BASIC FACTSBASIC FACTS

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

3x 412

5x 630

0x 1

0

9x 981

1x 8

8

10x 440

12x 672

11x 11121

12x 0

0

10x 10100

III. What Is Basic About Basic III. What Is Basic About Basic Facts?Facts?

IV. Why Does a Child Need IV. Why Does a Child Need to Know Basic Facts?to Know Basic Facts?

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?

VIII. The Role of Drill in VIII. The Role of Drill in Learning Basic FactsLearning Basic Facts

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

Cover Up

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

Making the ConnectionsMaking the Connections

ModelsConcrete/Iconic

LanguageOral/Written

Symbols

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

6 6

2

4

2

4

+

6

2+

4

6

Make a Record

Activity: Horizontal Addition Sentence

• Helping HandsAdapted from “Math Their Way” by Mary Baratta-Lorton, Addison-Wesley, 1995

5

53 2 3 2 5

= 5

= 53 2+

+

3 + 2

Make a Record

Activity: Horizontal Subtraction Sentence

• Take-away hand


Activity: Multiplication

• Cups-n-Counters

Activity: Division (Cars & Boats)

Instructional Sequence and ActivitiesInstructional Sequence and Activities

BASIC FACTSBASIC FACTS

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”?

II. An Instructional II. An Instructional SequenceSequence

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

A. Operation conceptsA. Operation conceptsandand

B.B. Naive countingNaive counting

Activity: Vertical Addition SentenceActivity: Vertical Addition Sentence

BowlsBowls


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


Activity: MultiplicationActivity: Multiplication

Cups-n-CountersCups-n-Counters

Activity: Division (Cars & Activity: Division (Cars & Boats)Boats)

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

0 1 2 3 4 5 6 7 8 90123456789

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

“Please put away that pocket calculator, Billy.”.

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