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!
– 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 Card
5th Grade Migrant Math Data
Migrant Math Data 5th Grade Math
Wa KIDS
Kindergarten Readiness ESD 105
100
90
80
70
60
All
50
Hispanic
American Indian
40
30
20
10
0
Social Emotional
Physical
Language
Cognitive
Literacy
Mathematics
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
• 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
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?
Count On
Addition
Use Doubles
Bridge to 10
Building on three strategies…
Count on 1
Count on 2
Count On
Count on 3
Count on 0
Addition
Double
Use Doubles
Double Plus 1
Bridge to 10
Double Plus 2
Operation ~ Strategy Clusters ~ Sub Strategies ~ Teaching Strategies
Introduce
Reinforce
Count on 1
Practice
Count on 2
Count On
Extend
Count on 3
Count on 0
Addition
Double
Use Double
Double Plus 1
Bridge to 10
Double Plus 2
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 Lines
Tables
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