math facts through conceptual understanding
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Math Facts through Conceptual Understanding. Dawn Sparks Ann 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!. - PowerPoint PPT PresentationTRANSCRIPT
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