Math Facts through Conceptual Understanding

Math Facts through Conceptual Understanding

Dawn SparksAnn Sipe

Math Assessment Probe

• Starting from what you know…– Fold assessment in

half– Respond to one side

only

Making the Case…Focus on number sense!

– Research indicates that early number sense predicts school success more than other measures of cognition like verbal, spatial or memory skills or reading ability. Jordoan, Kaplan, Locuniak, and Ramineni, 2007

Washington State Report Card5th Grade Migrant Math Data

Migrant Math Data 5th Grade Math

Wa KIDS Kindergarten Readiness ESD 105

Social Emotional Physical Language Cognitive Literacy Mathematics0

10

20

30

40

50

60

70

80

90

100

AllHispanicAmerican Indian

What is Mastery of Basic Facts:

• Addition and subtraction by the end of 2nd grade

• Multiplication and division by 3rd grade

• Fact fluency leads to other math fluent skills…

Guided Invention:

• Fact mastery will not magically happen. – “Time is a poor intervention.”

• Teacher’s job is to design tasks and problems that will promote invention of effective strategies, that these are shared in the classroom, thinking aloud.

Developmental Nature of Basic Fact Mastery

Counting Reasoning Mastery

Phase 1:

• Counting strategies-– using concrete objects – verbal counting

Example includes: • 5 + 1 = count on starting at 5, 6• 5 + 2 = 5, 6, 7• 4 +7 = 7 then 8, 9, 10, 11

Counting Strategies

Phase 2:

• Reasoning Strategy: – using known information to logically determine an

unknown combination– 4 + 7, student knows 3 +7 = 10 so just adds on 1

more. – 10 x 8 is 80 half of that is 5 x 8 is 40

Reasoning Strategy

Phase 3:

• Producing answers efficiently– just knows it– 4 + 7 = 11

• Retrieval within 3 seconds• Grade appropriate strategy• Efficient• Scalable

Mastery

Knowing Facts from Memory-“Passive Storage View”

Thinking that students will learn their facts if they just practice enough

• What does this mean-

– 100 isolated addition facts– 100 isolated multiplication facts– Also having to memorize subtraction and

division- well over 300 pieces of information to remember

– Not only having to memorize, but to keep practicing as well.

Effects of not knowing facts fluently…

• Lack of basic fact automaticity has been shown:– Limit participation in math class discussions– Impede successful problem solving– Severely impair the development of the standard

algorithms for multiple-digit addition and subtraction, long division and fractions

– Misapply facts and not seeing reasonableness of answers

– Inflexible thinking skills

Struggling Learners and Students with Disabilities:

• Have difficulty memorizing so many isolated facts (but can be successful with strategies)

• Drill creates, in a majority of students, unnecessary anxiety

• Undermine student interest and confidence in mathematics

ELL and Migrant Supports

• Scaffold the Language, but not the mathematic strategies

• Use Realia and Models• Use Graphic Organizers• Use Student Interactions for additional

practice• Use gestures for together, take apart, groups

etc.

Explicit Strategy Instruction:

• Strategies can be effective to learning math facts.

• Supports students thinking rather than give the students something new to remember.

• Key: Help students see possibilities and let them choose strategies that help them get to a solution without counting.

Resources

• John Van de Walle–Research

Origo The Box of Facts• Addition• Subtraction• Multiplication• Division

Learning Facts: Addition

Addition

How many strategies?

Addition

Count On

Use Doubles

Bridge to 10

Building on three strategies…

Addition

Count On

Count on 1

Count on 2

Count on 3

Count on 0

Use Doubles

Double

Double Plus 1

Double Plus 2Bridge to 10

Operation ~ Strategy Clusters ~ Sub Strategies ~ Teaching Strategies

Addition

Count On

Count on 1

Introduce

Reinforce

Practice

Extend

Count on 2

Count on 3

Count on 0

Use Double

Double

Double Plus 1

Double Plus 2Bridge to 10

Addition Strategies

• Count On– Count On 1– Count On Turnarounds– Count On 2– Count On 3– Count On 0

• Use Doubles– Double– Double Plus 1– Double Plus 2

• Bridge to 10• All Facts

Count On: Addition Strategy

Week One: Introduce Concept Through Real Objects Transitioning to Model

Week Two: Reinforce Through Models

Week Three: Practice through Games

Week Four: Extend or Reteach

Prepare…Before teaching strategies students need to know…

• Subitizing • Subitizing • Subitizing • Subitizing • The ability to instantly recognize the total

quantity of objects in a group without counting

Week One: Introduce Concept Through Real Objects Transitioning to Model

• Counting bottles• Cubes in a cup• Addition stories• Count on 1 Cards

Week Two: Reinforce Through Models

• Count on Cards• Number Tracks• Cube Trains• Count on Cards with Numeral Cards• Does This Make Sense?

• Moving from physical models to semi-abstract models to symbolic models (with numbers)

Week Three: Practice through Games

• Spin, Count On 1, Record Fact• Reinforce with Count On Flash Cards• Count On 1 Bingo – Total and Expression

Week Four: Extend or Reteach

• ___ + 1 = ___ Any Number-What’s the Rule• Moving to the teen numbers• Investigate Number Patterns– 8 + 1– 18 + 1– 28 + 1– 38 + 1

• Target Number

Supporting Migrant Students Through Family Support…

• Take Games Home– Instructions– Materials

• Send home strategies in newsletters– Language

Count On Turnarounds: Addition Strategy

Week One: Introduce Concept Through Real Objects Transitioning to Model

Week Two: Reinforce Through Models

Week Three: Practice through Games

Week Four: Extend or Reteach

Count On Turnarounds…

Week One: Introduce Concept Through Real Objects Transitioning to Model• T-shirt• Count On Strategy Cards• Blending Realia and

Strategy Cards

Count on Turnarounds…

Week Two: Reinforce Through Models• Count on Strategy Cards• Using a Number Line• Graphic Organizers• Count on Flash Cards

Count on Turnarounds…

Week Three: Practice Through Games• Flash Cards with Turn

Around Strategy• Bingo– Total Version– Expression Version

• Assessment and Monitoring

2 7 2 6 8

5 3 8 3 7

9 6 4 9 4

8 10 7 5 10

5 4 9 8 6

Count on Turnarounds…Week Four: Extend or Reinforce• Write 1 + ___ = ___ Develop a Rule• Use same models with larger

numbers• Use Frames

___ + ___ = 18___ + ___ = (numbers between 10-30)18 = ___ + ___ (after practice change position)

• Using a number line to show strategy

Repeating with new strategies and operations

• Same structure for subtraction-opposite of addition

• Same format for multiplication and division

Multiplication Strategies

• Use Tens– Five Facts

• Doubling– Two’s Facts– Four’s Facts– Eight’s Facts

• Use a Rule– One’s Facts– Zero’s Facts

• Build Down and Build Up– Nine’s Facts– Six’s Facts

• Last Facts

Use Tens: Prepare

• Assumptions-– Students should already

know tens facts– Be familiar with the

turnaround strategy– Turnarounds are an

application of the commutative property

• Realia-cubes in a bag• Suggesting stories

involving food• Sentences reflecting

multiplication (groups)• Chocolate Bars and

turnarounds-yum yum!• Connecting with

language

Use Tens: Prepare

• Assumptions-– Students should already

know tens facts– Be familiar with the

turnaround strategy– Turnarounds are an

application of the commutative property

• Making Models using Dice and Grid Paper

• Everyday situations involving 10

• Unifix Cubes Modeling 10’s

Count On Turnarounds:Multiplication Strategy

Week One: Introduce Concept Through Real Objects Transitioning to Model

Week Two: Reinforce Through Models

Week Three: Practice through Games

Week Four: Extend or Reteach

Introduce: Five Facts

• What are our experiences with fives?• Using students to model with hands• Clocks• Use Ten Strategy Cards

Reinforce: Fives Facts

• Use 10 Strategy Cards• Number Lines• Finding Unknown Numbers

Practice: Five Facts

• Multiplication Flash Cards• Spinning Five Facts• Tally Game• Assess and Monitor

Extend: Fives Facts

• ___ x ___ = 40– Using 5’s and 10’s write four number sentences

that would be true.– Repeat using numbers 80, 90, 110, 120 and so on.

Number LinesTables

Multiplication Strategies

• Use Tens– Five Facts

• Doubling– Two’s Facts– Four’s Facts– Eight’s Facts

• Use a Rule– One’s Facts– Zero’s Facts

• Build Down and Build Up– Nine’s Facts– Six’s Facts

• Last Facts

What to Do When Teaching Basic Math Facts

• Ask students to self-monitor• Focus on self-improvement• Drill in short time segments• Work on facts over time• Involve families• Make drill enjoyable• Use technology• Emphasize the importance of quick recall

What Not to Do When Teaching Basic Facts

• Don’t use lengthy timed test• Don’t use public comparisons of mastery• Don’t proceed through the facts in order from 0-9• Don’t work on all the facts at once• Don’t move to memorization too soon• Don’t use facts as a barrier to good mathematics• Don’t use fact mastery as prerequisite for

calculator use.

Math Assessment Probe

• Starting from what you know…– Fold assessment in

half– Respond to one side

only

Resources

• ESD Fact Fluency Courses

• Migrant Fluency-

• Origo: The Box of Facts• Elementary and Middle School Mathematics

by John Van de Walle

• Identify fact fluency is a developmental process.• Identify research based strategies that help

develop fact fluency • Understand strategies that are ineffective for

fact fluency development• Learn specific strategies that help develop fact

fluency