Math Facts through Conceptual Understanding
description
Transcript of Math Facts through Conceptual Understanding
![Page 1: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/1.jpg)
Math Facts through Conceptual Understanding
Dawn SparksAnn Sipe
![Page 2: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/2.jpg)
Math Assessment Probe
• Starting from what you know…– Fold assessment in
half– Respond to one side
only
![Page 3: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/3.jpg)
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
![Page 4: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/4.jpg)
Washington State Report Card5th Grade Migrant Math Data
![Page 5: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/5.jpg)
Migrant Math Data 5th Grade Math
![Page 6: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/6.jpg)
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
![Page 7: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/7.jpg)
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…
![Page 8: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/8.jpg)
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.
![Page 9: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/9.jpg)
Developmental Nature of Basic Fact Mastery
Counting Reasoning Mastery
![Page 10: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/10.jpg)
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
![Page 11: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/11.jpg)
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
![Page 12: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/12.jpg)
Phase 3:
• Producing answers efficiently– just knows it– 4 + 7 = 11
• Retrieval within 3 seconds• Grade appropriate strategy• Efficient• Scalable
Mastery
![Page 13: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/13.jpg)
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.
![Page 14: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/14.jpg)
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
![Page 15: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/15.jpg)
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
![Page 16: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/16.jpg)
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.
![Page 17: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/17.jpg)
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.
![Page 18: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/18.jpg)
Resources
• John Van de Walle–Research
Origo The Box of Facts• Addition• Subtraction• Multiplication• Division
![Page 19: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/19.jpg)
Learning Facts: Addition
Addition
![Page 20: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/20.jpg)
How many strategies?
Addition
Count On
Use Doubles
Bridge to 10
![Page 21: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/21.jpg)
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
![Page 22: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/22.jpg)
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
![Page 23: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/23.jpg)
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
![Page 24: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/24.jpg)
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
![Page 25: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/25.jpg)
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
![Page 26: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/26.jpg)
Week One: Introduce Concept Through Real Objects Transitioning to Model
• Counting bottles• Cubes in a cup• Addition stories• Count on 1 Cards
![Page 27: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/27.jpg)
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)
![Page 28: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/28.jpg)
Week Three: Practice through Games
• Spin, Count On 1, Record Fact• Reinforce with Count On Flash Cards• Count On 1 Bingo – Total and Expression
![Page 29: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/29.jpg)
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
![Page 30: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/30.jpg)
Supporting Migrant Students Through Family Support…
• Take Games Home– Instructions– Materials
• Send home strategies in newsletters– Language
![Page 31: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/31.jpg)
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
![Page 32: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/32.jpg)
Count On Turnarounds…
Week One: Introduce Concept Through Real Objects Transitioning to Model• T-shirt• Count On Strategy Cards• Blending Realia and
Strategy Cards
![Page 33: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/33.jpg)
Count on Turnarounds…
Week Two: Reinforce Through Models• Count on Strategy Cards• Using a Number Line• Graphic Organizers• Count on Flash Cards
![Page 34: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/34.jpg)
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
![Page 35: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/35.jpg)
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
![Page 36: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/36.jpg)
Repeating with new strategies and operations
• Same structure for subtraction-opposite of addition
• Same format for multiplication and division
![Page 37: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/37.jpg)
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
![Page 38: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/38.jpg)
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
![Page 39: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/39.jpg)
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
![Page 40: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/40.jpg)
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
![Page 41: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/41.jpg)
Introduce: Five Facts
• What are our experiences with fives?• Using students to model with hands• Clocks• Use Ten Strategy Cards
![Page 42: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/42.jpg)
Reinforce: Fives Facts
• Use 10 Strategy Cards• Number Lines• Finding Unknown Numbers
![Page 43: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/43.jpg)
Practice: Five Facts
• Multiplication Flash Cards• Spinning Five Facts• Tally Game• Assess and Monitor
![Page 44: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/44.jpg)
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
![Page 45: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/45.jpg)
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
![Page 46: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/46.jpg)
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
![Page 47: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/47.jpg)
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.
![Page 48: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/48.jpg)
Math Assessment Probe
• Starting from what you know…– Fold assessment in
half– Respond to one side
only
![Page 49: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/49.jpg)
Resources
• ESD Fact Fluency Courses
• Migrant Fluency-
• Origo: The Box of Facts• Elementary and Middle School Mathematics
by John Van de Walle
![Page 50: Math Facts through Conceptual Understanding](https://reader036.fdocuments.us/reader036/viewer/2022081422/5681668b550346895dda4c08/html5/thumbnails/50.jpg)
• 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