Specific Learning Disability Math Webinar Series
Webinar #2: Learning Whole Number Operations
Presented by Dr. Brad Witzel, Ph.D.
March 6, 2017
4:00-5:00 pm
Sponsored by The Exceptional Student Services Unit
Vision
All students in Colorado will become
educated and productive citizens
capable of succeeding in society, the
workforce, and life.
Every student
every step of the way
Mission
The mission of the CDE is to ensure
that all students are prepared for
success in society, work, and life by
providing excellent leadership,
service, and support to schools,
districts, and communities across the
state.
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2017 SLD Math Webinar Series
3/30/2017
CDE‐SLD website: http://www.cde.state.co.us/cdesped/SD‐SLD
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Zoom Webinar Info
• Your microphones will not be activated during this webinar.
• Please sign into the chat box with your name, role, grade level(s) you
work with.
• Please type any questions into the Question and Answer window or chat
box and we will address them at the end of the session.
• If you have difficulty accessing the webinar, or have technical issues,
please call Amanda Timmerman at 303.866.6969 or email her at
[email protected];
• Or contact Jill Marshall at [email protected]
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Webinar Evaluation
• There will be brief event evaluation survey at the end of this webinar.
• After you complete the evaluation survey your certificate of attendance
for one CDE training hour will open automatically. Please print your
certificate for your records.
• The recording of this webinar will be made available to participants in
the future. Please contact Jill Marshall or Amanda Timmerman for
viewing requests.
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Our Presenter
• Brad Witzel, Ph.D.
Office: 803-323-2453
Fax: 803-323-2585
Email: [email protected]
http://coe.winthrop.edu/witzelb/
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Learning Whole Number Operations
For Colorado Educators
Dr. Brad Witzel, Ph.D.
[email protected]
[email protected]
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Instruction Matters!
Even to calculator generation
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Our Path
March
Learning Whole Number Operations
April
Let’s Be Rational: Learning Integers, Fractions, Decimals
February
Focusing on the Nonstrategic Learner
May
Mathese: The Language of Mathematics
June:
Bridging the Arithmetic to Algebra Gap
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Computation to Mastery
Witzel & Little (2016)
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Computation
• Arrays, Number Line
• CRA
• Fluency
• Accommodations
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Linear Models
• While growing an understanding of magnitude
Ex. 3+2
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Linear Models
• Eventually working from unlabeled and open number lines
• Strategic counting Ex. 3+2
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Student concludes, “Three plus two is five.”
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Linear Models
• Extending to other operations develops directionality
• Strategic counting Ex. 3 – 2
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Student concludes, “Three minus two is one.”
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Linear Models
• Extending to more advanced operations
Ex. Multiples: 4x12
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48
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Including arrays in order to develop language
74 x 28
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70
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1400
80
Seven tens x two tens Four ones x two tens
7x2 = 14; 10 x 10= 100 4x2 = 8; 1 x 10= 10
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Seven tens x eight ones Four ones x eight ones
7x8 = 56; 10 x 1 = 10
4x8 = 32; 1 x 1 = 1
1400+560+80+32 = 1000 + 900 + 170 + 2 = 2072
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Use place value to show long division: 6th grade
250 remainder of 1
3 751
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What is difficult about long division?
Rearrange 751 to work with multiples of 3
600 + 151 = 600 + 90 + 61= 600 + 90 + 60 + 1
600 + 90 + 60 + 1
3 3 3 3
= 200 + 30 + 20 + 1/3 = 250 1/3
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Computation
• Arrays, Number Line
• CRA
• Fluency
• Accommodations
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CRA approach
• CRA is the Concrete to Representational to Abstract sequence of instruction. • Three stages of learning
• C = Learning through concrete hands‐on manipulative objects • R = Learning through pictorial forms of the math skill
• A = Learning through work with abstract (Arabic) notation
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Beginnings
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Comparative
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CRA Example
Use place value to add within 100
26 + 18
(Witzel, et al, 2013)
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Place Value Progressions
(Fuson & Beckmann, 2013; Witzel, Riccomini, & Herlong, 2013)
Place value language, such as 65, usually referred to as sixty‐five, as six tens, five ones.
74
+ 28
90
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102
seven tens
+ four ones
+ two tens + eight ones
nine tens
+ twelve ones
nine tens +one ten+two ones
one hundred + two ones
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CRA Example
Use place value to subtract within 100
33 ‐ 18
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(Witzel, et al, 2013)
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Abstract Outcomes from CRA:
Options for computation based on place value
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Partitive
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Computation
• Arrays, Number Line
• CRA
• Fluency and Automaticity
• Accommodations
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Assess Accuracy before Teaching Fluency
Expect Fluency before Automaticity
Accuracy before Speed
Know where you’re going before you ask for speed
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Why Develop Fluency and Automaticity
• “Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (NRC, 2001, p. 121)
• “…students should understand key concepts, achieve automaticity as appropriate …develop flexible, accurate, and automatic execution of the standard algorithms, and use these competencies to solve problems” (NMP, 2008, p. 17).
• Computational speed builds confidence and a positive self‐identity (Berry, Thunder, & McClain, 2011).
• Students who expend too much of their cognitive capacity performing basic operations may have insufficient capacity to apply toward complex mathematics (Parkhurst et al., 2010; Woodward, 2006).
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Scope and Sequence of Multiplication
Kindergarten – Numbers and early strategic counting (Number Sense)
1st grade – Counting by 10s, 2s, 5s, (multiples)
2nd grade – Completing the multiples; Introduction to “times” and groups; Missing factors
3rd grade – Single digit multiplication to automaticity; one‐digit x two‐
digit strategies
4th grade – multi‐digit multiplication to 2x3 strategies
5th grade – multi‐digit multiplication to 3x3 strategies
6th grade – Rational number strategies
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Practice operational facility to gain fluency and automaticity
• The RtI Panel (Gersten, Beckman, Clarke, Foegen, Marsh, Star, and Witzel, 2009) concluded that all students (K‐8) receiving interventions should receive at least 10 minutes of practice per day in fact fluency.
• K‐5 should focus on whole numbers
• 4‐8 should focus on rational numbers
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Typical means
http://www.mathfactcafe.com/home/
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Multiple Types of Fluency and Automaticity Three elements to fluency
a) Verbal and Written speed of recall
b) Reasoning explanations
c) Embedded
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Reasoning meets fluency
• Understandings and relationships between facts and properties
• Explanative approach to problem solving
See CCSS Math Practices
• Reason abstractly and quantitatively
• Construct viable arguments
• Look for and make use of structure
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Embedded Fluency: Consider multiple approaches to teaching fact memorization
Simplify the expression 6x (3y – 6x) – 5 (7y + 1y) = ‐4 (8 – x), solve for y
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Develop multiple ways to improve recall of facts
• Flash cards
• Games
• Worksheet
• Computer format (e.g., Mad Minutes)
• Daily Five
• Home practice packs
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Incremental Rehearsal: Don’t shred your flash cards just yet
Ratio
• Flash cards can be effective at establishing fluency and automaticity when using incremental rehearsal (Burns, 2005; Burns et al., 2014)
• Students are presented known to unknown material in a ratio of 9:1 (90% to 10%)
• In a ten card stack, this means that 9 of the answers are known and only one is yet to be learned. Presenting unknown problems
• Students must build momentum and motivation by answering several correct in a row before an unknown problem is presented.
• When the unknown problem is presented, the answer is immediately provided.
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Incremental Rehearsal
Organizational structure of flash cards
Each unknown • unknown (given)
• known, known, known, known, unknown (given)
• known, known, known, known, known, unknown (not given)
• known, known, known, known, known, known, unknown (not given)
• known, known, known, known, known, known, known, unknown (not given)
• known, known, known, known, known, known, known, known, unknown (not given)
• known, known, known, known, known, known, known, known, known, unknown (not given)
Fluency charts can be developed from this approach
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Incremental Rehearsal (IR) Steps 1. Print flash cards for visual and auditory practice
2. Assess the student to determine which facts are known (K) and which are still unknown (U).
3. Present one unknown fact (U)
4. Practice known to unknown material at a ratio of 9:1. In a ten‐card stack, this means that 9 of the answers are known and only one is yet to be learned.
5. Immediately provide the answer when the unknown problem is presented
6. Build momentum and motivation by having the student answer several questions correctly in a row before an unknown problem is presented.
Hold IR sessions over several days, so that not too many answers are learned per day (three to five). Long‐term recall is benefitted by short intervention sessions rather than long ones.
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Drill sandwich (Browder & Roberts, 1993)
1) Make or obtain a set of flashcards for the information to be learned.
2) Go through the complete stack of cards once, separating the list into two piles –
knowns and unknowns. 3) Build a “sandwich” using seven knowns (K) and three unknowns (U). Be sure to follow the pattern (K‐K‐K‐U‐K‐K‐U‐K‐K‐U)
4) Have the child/children practice identifying all ten items in the order above.
5) As the unknown items are learned so that the response is immediate and automatic, move them into a known section of the sandwich by removing repeated accurate knowns. 6) Add new unknowns and repeat
This approach adds 3 new facts daily for a total of 12 by a summative assessment on Friday
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Cover‐Copy‐Compare (Skinner, McLaughlin, & Logan, 1997)
https://www.youtube.com/watch?v=WjlkAi44qTY
Preparation:
The teacher selects up to 10 math facts for the student to practice during the session and writes those facts (including number sentence and answer) as correct models into the left column of the Cover‐Copy‐
Compare Worksheet.
The teacher then pre‐folds the sheet using as a guide the vertical dashed line ('fold line') dividing the left side of the student worksheet.
ebi.Missouri.edu
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Cover‐Copy‐Compare Steps
1) Study the correctly completed math fact (model) that appears in the left column of the sheet.
2) Fold the left side of the page over at the pre‐folded vertical crease to hide the original math fact ('Cover').
3) Copy from memory the math fact and answer, writing it in the first response blank under the 'Student Response' section of the Cover‐Copy‐Compare worksheet ('Copy').
Teacher uncovers the original correct model and compares it to the student response ('Compare'). If CORRECT, the student moves to the next item on the list and repeats these procedures. If INCORRECT, the student draws a line through the incorrect response, and repeats earlier steps and again checks the correctness of the copied item.
Continue until all math facts on the sheet have been copied and checked against the correct models.
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Detect, Practice, Repair (Parkhurst et al., 2010)
Detect, Practice, Repair (DPR) is a multi‐component, class‐wide procedure that focuses on enhancing fluency by allowing students to practice those math facts that they have not developed to the point of automaticity (Poncy et al. 2006). Parkhurst et al (2010) improved multiplication facts with 5th grade students in 6‐10 trials.
Detect = During a detect phase, Poncy et al. used a metronome to pace a group of students through a series of math facts, with the metronome signaling 1.5 s intervals to respond to each fact. Practice = After this paced assessment, each student circled those problems that he/she did not answer and then applied the practice phase to those identified problems by performing the Cover, Copy, Compare (CCC) procedure. Repair = Corrective feedback and repeat practice
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Weekly routine for interventions
• Set‐up at least 10 minutes of class time when fluency will be the focus, 4‐5x per week
• If the need applies to a small group, set up a space for the practice to occur
• Vary the output throughout the week
• Vary the type of fluency as the student develops proficiency
Monday
Tuesday
Multiplication
‐ Isolated and silent
Multiplication
‐ Conceptual
and verbal with partner
Wednesday
Thursday
Multiplication Multiplication
– isolated and ‐ Isolated and silent
oral
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Friday
Multiplication
‐ Isolated and oral game
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Computation
• Arrays, Number Line
• CRA
• Fluency
• Accommodations
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What do we do in the meantime for students who haven’t mastered their facts?
Calculator?
“The Panel cautions that to the degree that calculators impede the development of automaticity, fluency in computation will be adversely affected” (NMAP, 2008, p. xxiv)
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Hundreds Table Accommodation
• Mix the accommodation with fluency intervention
• Slowly fade the utility of the table by covering what has been “mastered”
• Make the table more cumbersome to use as the student progresses
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Hundreds Table
A Modification Accommodation © Witzel, 2017
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Hundreds Table Accommodation:
Step 1
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Hundreds Table Accommodation: Step 2
The student learned 1x and 10x
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Hundreds Table Accommodation: Step 3
The student learned 1x, 10x, and 5x
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Hundreds Table Accommodation: Step 3+
The student learned 1x, 10x, 5x, and others
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Whole Number Operations ‐ What stood out?
• Arrays, Number Line
• CRA
• Fluency
• Accommodations
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3‐2‐1 Take Home
3 things you learned
2 things you can implement
with ease
1 question you still have
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Thank you!
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2017 SLD Math Webinar Series
CDE‐SLD website: http://www.cde.state.co.us/cdesped/SD‐SLD
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“This material was developed under a grant from the
Colorado Department of Education. The content does
not necessarily represent the policy of the U.S.
Department of Education, and you should not assume
endorsement by the Federal Government.”
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Contact Information
Jill Marshall
SLD Specialist
Colorado Department of Education
[email protected]
http://www.cde.state.co.us/cdesped/SD‐SLD
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Event Evaluation Link
Webinar evaluation:
https://www.surveymonkey.com/r/SLD_Math_Web2
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