MAXIMIZE ALL STUDENTS’ MATHEMATICAL LEARNING
THROUGH THE USE OF
POWERFUL
INSTRUCTIONAL
STRATEGIES & TECHNIQUES
JANUARY 18, 2017
FEBRUARY 16, 2017
MARCH 1, 2017
Paul J. Riccomini, Ph.D.
[email protected]
@pjr146 Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Components of Effective
Mathematics Programs
Mathematics
Curriculum &
Interventions
Assessment &
Data-Based
Decisions
100% Math
Proficiency
Teacher
Content &
Instructional
Knowledge
© Paul J. Riccomini 2017
[email protected] Components of Effective
Mathematics Programs
Mathematics
Curriculum &
Interventions
Assessment &
Data-Based
Decisions
100% Math
Proficiency
Teacher
Content &
Instructional
Knowledge
Instruction Matters
© Paul J. Riccomini 2017
[email protected] Topics for Math PD Series
• Designing instruction to help students of all skill levels achieve success in mathematics.
• NMAP, 2008 Final Report‐OVERVIEW
– Working memory and impact on learning
• Instructional Techniques & Strategies
1.
2.
3.
4.
Fractions linked to number line early and often
Interleave Worked Solution Strategy (IWSS)
Spaced Learning Over Time (SLOT)
Fluency & Automaticity
1.
5.
•
6.
Task ~ Material ~ Content Teaching Vocabulary
•
•
Computation and Procedural Fluency
Instructional Scaffolding Progressions
Frayer Model Conclusion & Wrap Up
© Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Breakout Activity (HO #1)
• Discuss challenges you see for students who struggle and students with disabilities in the area of mathematics
• Consider issues involving Instruction, Curriculum, Assessments, co‐teachers, & students—All levels
• Come up with a Top 4 List
4
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International comparisons
Low fractions of proficiency on NAEP
Falling proficiency at higher grades
Heavy remedial demand upon entry into college
• Achievement gap
•
•
•
•
Algebra as a gateway
5
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Translated to Real World Performance
• 78% of adults cannot explain how to compute interest paid on a loan
• 71% cannot calculate miles per gallon
• 58% cannot calculate a 10% tip
Mathematics Advisory Panel Final Report, 2008
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Our Own Fraction Foundation (1a)
• Shade 3/8 on the provided grid. © Paul J. Riccomini 2017
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• Low fractions of proficiency on NAEP
• Teachers anchor the development of whole numbers with the visual representation of a Number Line.
– Very important because the number line clearly demonstrated quantity
– Visually shows that numbers have specific locations on a number line
http://www.mathsisfun.com/number‐line.html
8
© Paul J. Riccomini 2017
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• Low fractions of proficiency on NAEP
• As teachers begin to introduce fractions...the number line vanishes and the concepts of fractions are anchored by a circle
– Circle does not demonstrate that a fraction is a number and that it has a location on a number line
– Circles do not clearly demonstrate the quantity of fractions compared to other fractions
• A number line displaying fractions is very important to the conceptual understanding
9
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Fraction and Magnitude
• Use a tick mark to indicate were the number ½ is located on the two number lines below:
0
1
0
2
Student Estimates
Whole Numbers
P e r c e n t s
D e c i m a l s
F r a c t i o n s
Results of a Recent Study
• 6th‐8th grade students estimated whole number and percentage magnitudes accurately, but on average, were very inaccurate when estimating fraction and decimal magnitude.
• 8th graders’ performance was not measurably stronger than 6th
graders’ performance– Instruction may not effectively target this skill.
• Fraction magnitude knowledge is strongly related to overall math achievement (PSSA‐M) and predictive of future math performance (5th grade high school).
– Magnitude knowledge = Number Sense and must extend from whole
numbers to rational numbers.
– Strong magnitude knowledge helps students (1) understand fractions
as numbers; (2) comprehend the meaning of fractions problems, and
(3) assess the reasonableness of their answers/calculations.
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Math Proficiency of U.S. Students
• Low fractions of proficiency on NAEP
– Use number line representations to help students recognize fractions are numbers and that they expand the number system
13
© Paul J. Riccomini 2017
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• Low fractions of proficiency on NAEP
– Use number line representations to help students recognize fractions are numbers and that they expand the number system
14
© Paul J. Riccomini 2017
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• Order the following fractions from least to greatest
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Our Own Fraction Understanding
• Robin and Jim took cherries from a basket. Robin took 1/3 of the cherries and Jim took 1/6 of the cherries. What fraction of the cherries remained in the basket? Breakout Activity‐(HO #1b)
Fractions and the Number line
• Discuss your current curricular materials in relationship to the use of the number line to represent fractions, decimals, and percents.
– Is the # line a common representation?
– Are fractions and decimals ever displayed together on a # line?
– How often are students exposed to the number line with fractions?
• During computation instruction???
– What types of activities involve the # line?
• Where can the # line be used more frequently in your math lessons in conjunction with fractions?
– Whole group, small group centers or stations, etc?
17
© Paul J. Riccomini 2017
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National Mathematics Advisory Panel
Final Report, March 2008
Select Slides taken from the NMAP-Final Report
Presentation available at: http://www.ed.gov/MathPanel
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Curricular Content‐NMAP 2008
Streamline the Mathematics Curriculum in Grades PreK‐8:
• Follow a Coherent Progression, with Emphasis on Mastery of Key Topics
• Focus on the Critical Foundations for Algebra
‐ Proficiency with Whole Numbers
‐ Proficiency with Fractions
‐ Particular Aspects of Geometry and Measurement
• Avoid Any Approach that Continually Revisits Topics
without Closure (pg 22)
19
© Paul J. Riccomini 2017
[email protected] Curricular Content‐NMAP 2008
Streamline the Mathematics Curriculum in Grades PreK‐8:
• Follow a Coherent Progression, with Emphasis on Mastery of Key Topics
**Enrichment & Reteaching Targets**
• Focus on the Critical Foundations for Algebra
‐ Proficiency with Whole Numbers
‐ Proficiency with Fractions
‐ Particular Aspects of Geometry and Measurement
• Avoid Any Approach that Continually Revisits Topics
without Closure (pg 22)
20
© Paul J. Riccomini 2017
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• To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, factual knowledge and problem solving skills. • Limitations in the ability to keep many things in mind (working‐memory) can hinder mathematics performance.
‐ Practice can offset this through automatic recall, which results in less information to keep in mind and frees attention for new aspects of material at hand.
‐ Learning is most effective when practice is combined with instruction on related concepts.
‐ Conceptual understanding promotes transfer of learning to new problems and better long‐term retention.
NMAP, 2008 © Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Instructional Practices‐NMAP‐2008
Instructional practice should be informed by high
quality research, when available, and by the
best professional judgment and experience of
accomplished classroom teachers.
• All‐encompassing recommendations that instruction should be student‐centered or teacher‐directed are not supported by research.
22
© Paul J. Riccomini 2017
[email protected] Instructional Practices‐NMAP‐2008
Research on students who are low achievers, have difficulties in mathematics, or have learning disabilities
related to mathematics tells us that the effective practice includes:
 Explicit methods of instruction available on a regular basis
 Clear problem solving models
 Carefully orchestrated examples/ sequences of examples.
 Concrete objects to understand abstract representations and notation.
 Participatory thinking aloud by students and teachers. © Paul J. Riccomini 2017
[email protected] For More Information
Please visit us online at:
http://www.ed.gov/MathPanel
• Read it! Fact Sheet SH#1c
• The report and Factsheet should be on the desk of every teacher responsible for teaching and planning math.
24
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Learner Characteristics
• Strategic Learners
– Able to analyze a problem and develop a plan – Able to organize multiple goals and switch flexibly from simple to more complicated goals – Access their background knowledge and apply it to novel tasks – Develop new organizational or procedural strategies as the task becomes more complex – Use effective self‐regulated strategies while completing a task – Attribute high grades to their hard work and good study habits – Review the task‐oriented‐goals and determine whether they have been met http://iris.peabody.vanderbilt.edu/srs/chalcycle.htm
© Paul J. Riccomini 2017
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• Non‐Strategic Learners
– Unorganized, impulsive, unaware of where to begin an assignment – Unaware of possible steps to break the problem into a manageable task, possibly due to the magnitude of the task – Exhibit problems with memory – Unable to focus on a task – Lack persistence – Experience feelings of frustration, failure, or anxiety – Attribute failure to uncontrollable factors (e.g., luck, teacher's instructional style) http://iris.peabody.vanderbilt.edu/srs/chalcycle.htm
© Paul J. Riccomini 2017
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1. Make sense of complex problems and persevere in solving them.
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
(CCSS, 2010)
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Learning Outcomes of CCSS‐MP
(McCallum, 2011)
© Paul J. Riccomini 2017
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These are BIG challenges for students with disabilities and those that are struggling.
It will only happen if it is purposefully facilitated through teacher INSTRUCTION!
(McCallum, 2011)
© Paul J. Riccomini 2017
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• Much of teaching is about helping students master new knowledge and skills and then helping students NOT to forget what they have learned.
• Work Smarter NOT Harder!
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Essential Question for Teachers
• Essential Question
– What did I do “instructionally different” to support learning for the struggling students?
• Asked during instructional planning and after instructional delivery!!!
© Paul J. Riccomini 2017
[email protected] Strategies & Techniques 1
• Interleaved Worked Solution Strategy
2
• Spaced Learning Over Time (SLOT)
3
• Fluency & Automaticity
4
• Instructional Scaffolding Progressions
5
• Teaching Vocabulary
© Paul J. Riccomini 2017
[email protected] Strategies & Techniques 1
• Interleaved Worked Solution Strategy
2
• Spaced Learning Over Time (SLOT)
3
• Fluency & Automaticity
4
• Instructional Scaffolding Progressions
5
• Teaching Vocabulary
© Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Excellent Resource: IES
• Institute of Education Science
– Review research to determine instructional strategies and techniques that are supported by high quality evidence
http://ies.ed.gov/
© Paul J. Riccomini 2017
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© Paul J. Riccomini 2017
[email protected] Breakout Activity
• Discuss the following questions:
1. What do you do to help students complete:
•
•
•
•
Homework Independent practice opportunities Study for tests & Quizzes (e.g., study guides)
Review important skills
2. What do you do to help students remember important information from previous lessons throughout the course of the academic year?
© Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Learner Characteristics
• Non‐Strategic Learners
– Unorganized, impulsive, unaware of where to begin an assignment – Unaware of possible steps to break the problem into a manageable task, possibly due to the magnitude of the task – Exhibit problems with memory
– Unable to focus on a task – Lack persistence – Experience feelings of frustration, failure, or anxiety – Attribute failure to uncontrollable factors (e.g., luck, teacher's instructional style) http://iris.peabody.vanderbilt.edu/srs/chalcycle.htm
© Paul J. Riccomini 2017
[email protected] Instructional Practices‐NMAP‐2008
Research on students who are low achievers, have difficulties in mathematics, or have learning disabilities
related to mathematics tells us that the effective practice includes:
 Explicit methods of instruction available on a regular basis
 Clear problem solving models
 Carefully orchestrated examples/ sequences of examples.
 Concrete objects to understand abstract representations and notation.
 Participatory thinking aloud by students and teachers. NMAP, 2008 © Paul J. Riccomini 2017
[email protected] Six Critical Features of explicit instruction
1.
2.
3.
4.
5.
6.
Daily Reviews
Presentation of New Content
Guided Practice
Explicit feedback and Correctives
Independent Practice
Weekly and Monthly Reviews
© Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Essential Question for Teachers
• Essential Question
– What do you do instructionally different” to support learning for the struggling students?
• Asked during instructional planning and after instructional delivery!!!
© Paul J. Riccomini 2017
[email protected] Interleave Worked Solution Strategy
• Interleave worked example solutions and problem‐solving exercise
• Literally, alternate between worked examples demonstrating one possible solution path and problems that the student is asked to solve independently
• This can markedly enhances student learning
IES Practice Guide, (2007, September)
© Paul J. Riccomini 2017
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• To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, factual knowledge and problem solving skills. • Limitations in the ability to keep many things in mind (working‐memory) can hinder mathematics performance.
‐ Practice can offset this through automatic recall, which results in less information to keep in mind and frees attention for new aspects of material at hand.
‐ Learning is most effective when practice is combined with instruction on related concepts.
‐ Conceptual understanding promotes transfer of learning to new problems and better long‐term retention.
NMAP, 2008 © Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Interleave Worked Solution Strategy
• Typical Math Homework assignment
– Pg. 155 #1‐21 odd
• Students are required to solve all problems.
IES Practice Guide, (2007, September)
© Paul J. Riccomini 2017
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• Interleaved Homework assignment
– Pg 155 1‐10 (all)
– Odd problems
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Solutions as a Strategy
• Providing students opportunities to reason and explain worked out problem solutions enhances learning.
• Studying solutions can focus students on STRUCTURE
of problem solutions
• Critical thinking and analysis are very difficult skills for students struggling and students with disabilities and providing solutions can help re‐engage students
• De‐emphasizing an answer will better focus (support & enhance) students on the underlying structure and improve critical thinking
© Paul J. Riccomini 2016
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• Other considerations:
1. The amount of guidance an annotation accompanying the worked out examples varies depending on the situation
2. Gradually fade examples into problems by giving early steps in a problem and requiring students to solve more of the later steps
3. Use examples and problems that involve greater variability from one example or problem to the next
• Changing both values included in the problem and the problem formats.
© Paul J. Riccomini 2017
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© Paul J. Riccomini 2016
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Interleave Worked Solution Strategy
•
During Whole Class instruction
1. Start off discussion around an already solved problem
•
Pointing out critical features of the problem solution
2. After discussion have students pair off in small groups or work individually to solve a problem (JUST ONE!) on their own
3. Then back to studying an example, maybe one students present their solutions and have others attempt to explain
4. Then after studying the solved example, students are given another problem to try on their own.
IES Practice Guide, (2007, September)
© Paul J. Riccomini 2017
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• Example of implementing the IWSS strategy into Algebra Homework.
• Students must be prompted to study the solution Leverage the Instructional Power of a Solution
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Questions to Guide Whole Class Discussion HO#4e
Problem
Student Response
Questions to Guide Discussion
© Paul J. Riccomini 2016
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Correct and Incorrect Comparisons HO #4f
© Paul J. Riccomini 2016
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Instructional Planning Breakout (SH #4g)
• Modify the Algebraic Equations Worksheet to include worked out examples following the IWSS guidelines.
• Remember there is no “EXACT” structure for IWSS. Base it on your experience, the students’ instructional needs, and the target content.
• Each group will exchange their IWSS sheets for review by another group.
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Developing IWSS Practice Sheets
Step 1
Identify Skill for practice
Step 2
Find practice sheet for identified skill
Step 3
Match Problems by similar attributes and arrange as needed
Step 4
Modify, change, add or delete problems as necessary
Step 5
Determine the type of solution that you will embed on the practice sheet:
• Steps‐only
• Steps plus annotation
• Other variation
Step 6
Determine progression for fading of solutions
Step 7
Embed Solutions in an alternating
format
Step 8
Create final IWSS practice sheet ready for use
© Paul J. Riccomini 2017
[email protected] Implementation Ideas‐IWSS
• Have students alternate between reading already worked solutions and trying to solve problems on their own
• As students develop greater expertise, reduce the number of worked examples provided and increase the number of problems that students solve independently IES Practice Guide, (2007, September)
© Paul J. Riccomini 2017
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HO #4h
• Teach learners how to use the solutions included with the IWSS.
– How will you teach the learners to use the solutions?
– How will you model the process of studying a solution?
– How will you provide feedback during the initial stages of learning the IWSS strategy?
– How will you provide opportunities to practice studying the solution?
• What are the essential steps when reviewing a solution?
• What are key questions that you want the learners to ask and answer when reviewing the solution?
© Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Classroom Application Activity‐IWSS
• Identifying materials that you will use in an upcoming chapter/unit that best fits IWSS.
• Modify or adapt those materials to incorporate the IWSS technique.
• If you have already developed materials using the IWSS technique, review those materials and check for fidelity of the materials.
• Each group will be asked to provide an example of materials with the IWSS technique.
© Paul J. Riccomini 2017
[email protected] Solutions as a Strategy
• Providing students opportunities to reason and explain worked out problem solutions enhances learning.
• Studying solutions can focus students on STRUCTURE
of problem solutions
• Critical thinking and analysis are very difficult skills for students struggling and students with disabilities and providing solutions can help re‐engage students
• De‐emphasizing an answer will better focus (support & enhance) students on the underlying structure and improve critical thinking
© Paul J. Riccomini 2016
[email protected] Strategies & Techniques 1
• Interleaved Worked Solution Strategy
2
• Spaced Learning Over Time (SLOT)
3
• Fluency & Automaticity
4
• Instructional Scaffolding Progressions
5
• Teaching Vocabulary
© Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Space Learning Over Time
• Arrange for students to have spaced instructional review of key course concepts through the SLOT Strategy
– At least 2 times/year
– Separated by several weeks to several months
• Why:
– Helps student remember key facts, concepts, and knowledge
IES Practice Guide, (2007, September)
© Paul J. Riccomini 2017
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• Make sure important and essential curriculum content is reviewed at least 3‐4 weeks after it was initially taught.
• Benefits of a delayed review is much greater than the same amount of time spent reviewing shortly after initial instruction (Rohrer & Taylor, 2006).
IES Practice Guide, (2007, September)
© Paul J. Riccomini 2017
[email protected] Space Learning Over Time (HO#5)
© Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Space Learning Over Time (HO#5)
Divide School year into 4‐6 week chunks
Using Scope & Sequence list out big ideas taught in each chunk
Drill down to more specific problem skills and concepts using any available data
Select 2 of the identified problem areas.
PRIORITIZE
Fast forward 4‐
6 weeks from when identified skills were taught & list date here to revisit
© Paul J. Riccomini 2017
[email protected] Space Learning Over Time (HO#5)
Important Consideration:
• Concepts and/or skills identified in the first two cells should be revised at least TWO times before the end of the year.
• For example, if you revisit a concept/skill beginning of October, it needs to be revisited at least one more time in January/February
Fast forward 4‐
6 weeks from when identified skills were taught & list date here to revisit
© Paul J. Riccomini 2017
[email protected] Implementation Ideas‐SLOT
• Identify key concepts, terms, and skills taught and learned during each 6 week unit
• Arrange for students to be re‐exposed to each Big Idea on at least two occasions, separated by a period of at least 4‐6 weeks.
• Arrange homework, quizzes, and exams in away that promotes delayed reviewing of important course content
IES Practice Guide, (2007, September)
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Maximize Students’ Mathematical Learning
3‐Day PD Series
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Classroom Application Activity‐SLOT
• Using the spaced instructional review sheet, divide the school year into 6 week units starting at the beginning of the year through the end of the year.
• List the Big ideas taught during that first 6 week unit and identify which are often problematic and very important for students
• Then, during the extended instructional planning time, complete the rest of the SLOT planning chart for the remainder of the school year.
© Paul J. Riccomini 2017
[email protected] Strategies & Techniques 1
• Interleaved Worked Solution Strategy
2
• Spaced Learning Over Time (SLOT)
3
• Fluency & Automaticity
4
• Instructional Scaffolding Progressions
5
• Teaching Vocabulary
© Paul J. Riccomini 2017
[email protected] Learning Processes‐NMAP‐2008
• To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, factual knowledge and problem solving skills. • Limitations in the ability to keep many things in mind (working‐memory) can hinder mathematics performance.
‐ Practice can offset this through automatic recall, which results in less information to keep in mind and frees attention for new aspects of material at hand.
‐ Learning is most effective when practice is combined with instruction on related concepts.
‐ Conceptual understanding promotes transfer of learning to new problems and better long‐term retention.
NMAP, 2008 © Paul J. Riccomini 2017
[email protected] © Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Why learn Facts?
1. Knowledge of simple facts is needed for proper use of calculators
2. Ability to estimate implies mastery of single digit facts
3. Students slow at facts are less likely to learn more complex math problem types
4. Students must know multiplication facts quickly to be able to master fractions
5. Algebra is not open to those who haven’t mastered fractions.
Crawford, 2002
© Paul J. Riccomini 2017
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• Math fact fluency is the ability to accurately and quickly recall basic addition, subtraction, multiplication, and division facts (Burns, 2005; McCallum, Skinner, & Turner, 2006; Poncy, Skinner, & Jaspers, 2006).
• Students who possess fluency can recall facts with automaticity, which means they typically think no longer than two seconds before responding with the correct answer.
© Paul J. Riccomini 2017
[email protected] Fluency and Automaticity
• Math fact fluency is the ability to accurately and quickly recall basic addition, subtraction, multiplication, and division facts (Burns, 2005; McCallum, Skinner, & Turner, 2006; Poncy, Skinner, & Jaspers, 2006).
• Students who possess fluency can recall facts with automaticity, which means they typically think no longer than two seconds before responding with the correct answer.
© Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Instructional Strategies
and Interventions
Progression to Fluency
Achieve
Automaticity
and Fluency
Long Term
Memory
Short-term
Working
Memory
Instructional Time
Progression to Fluency
• Learning Progression Stages 1. Understanding
• Manipulatives & Pictorial Reps
2. Relationship
• Making connections within & across
3. Fluency
• Strategy development for accuracy
4. Automaticity
• Practice to facilitate automaticity
© Paul J. Riccomini 2017
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• Operations & Algebraic Thinking K.OA.A.5
– Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
• Fluently add and subtract within 5
• Operations & Algebraic Thinking 2.0A.2
– 2. Fluently add and subtract within 20 using mental strategies. • By end of Grade 2, know from memory all sums of two one‐digit numbers
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Maximize Students’ Mathematical Learning
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Steps to Assessing Fluency
• Identify target skill (i.e., single digit facts)
• Determine a regular assessment scheduled – For example, every 4th day
• Develop/Create/Locate the assessments
– More problems than student can solve in the allocated time. (It’s NOT about Finishing)
– 1‐2 minute timings
– Standard Directions‐used all the time
• Score and data is charted on a graphic display
– Shared and discussed with student
• Main priority is to show an INCREASE Common Core State Standards
• Operations & Algebraic Thinking 3.0A.7
– Fluently multiply and divide within 100 using strategies such as the relationship between multiplication and division (e.g., knowing that 8x5=40, one knows 40/5=8 or properties of operations. • By the end of Grade 3: know from memory all products of two one‐digit numbers
• Number & Operation Base Ten 3.NBT
– Use place understanding to round numbers to the nearest 10 or 100
– Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
© Paul J. Riccomini 2017
[email protected] Common Core State Standards
• 5.NBT.5. Perform operations with multi‐digit whole numbers and with decimals to hundredths
– Fluently multiply multi‐digit whole numbers using the standard algorithm.
• 6.NS.2 Compute fluently with multi‐digit numbers and find common factors.
– Fluently divide multi‐digit numbers using the standard algorithm.
• 6.NS.3. Compute fluently with multi‐digit numbers and find common factors and multiples.
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Common Core State Standards
• 7.EE.4 Use variables to represent quantities in a real‐world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. • Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently.
Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
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• Algebra I students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing down the equation of a line given a point and a slope. Such fluency can support them in solving less routine mathematical problems involving linearity, as well as in modeling linear phenomena (including modeling using systems of linear inequalities in two variables). •
Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in algebra, as well as in their symbolic work with functions. Manipulation can be more mindful when it is fluent. • Fluency in transforming expressions and chunking (seeing parts of an expression as a single object) is essential in factoring, completing the square, and other mindful algebraic claculations
High School Fluency & PARCC
Fluency Recommendations
•
The high school standards do not set explicit expectations for fluency nor will the PARCC assessments address fluency, but fluency is important in high school mathematics.
•
For example, fluency in algebra can help students get past the need to manage computational details so that they can observe structure and patterns in problems.
– Such fluency can also allow for smooth progress beyond the college and career readiness threshold toward readiness for further study/careers in science, technology, engineering, and mathematics (STEM) fields. •
These fluencies are highlighted to stress the need for curricula to provide sufficient supports and opportunities for practice to help students gain fluency.
•
Fluency is not meant to come at the expense of understanding; it is an outcome of a progression of learning and thoughtful practice. Curricula must provide the conceptual building blocks that develop in tandem with skill along the way to fluency. http://www.parcconline.org/15‐resources/mathematics‐frameworks/98‐course‐specific‐analysis
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Instructional Strategies
and Interventions
Progression to Automaticity
Achieve
Mastery
Long Term
Memory
Short-term
Working
Memory
Instructional Time
Fluency/Automaticity
• Fluency and automaticity is needed for single digit facts as well as beyond the basic facts.
• Clearly articulated in the Core Standards
• Mass practice or Drill‐n‐Kill is NOT an efficient or targeted technique for many struggling students • Most practice efforts are NOT FLUENCY based…MORE ACCURACY BASED
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[email protected] What I See being used
• Fluency and Automaticity?
–
–
–
–
–
–
–
–
Drill and Kill
Games
Mad minutes
Flash cards
Parents‐ “Do it at Home”
Computers
Give calculator
others
???
Are these activities PURPOSEFUL
PLANNED &
TARGETED?
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Grade Level Planning Questions
• Fluency/Automaticity
– How many students in your current classroom AND school are not automatic with basic facts?
Grade 1‐Addition
Grade 2‐Subtraction
Grade 3‐4: Multiplication &Division
Grade 4‐12: Multiplication is Priority
– How are you currently helping student build fluency and automaticity with basic math facts…ALL 4 Operations • 4th Grade+ focus on Multiplication?
– How are these current efforts to facilitate fluency/automaticity working?
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[email protected] Steps to Fluency &Automaticity
Requires
1. Specific criterion for introducing new facts
2. Intensive practice on newly introduced facts (more than 1x)
3. Systematic practice on previously introduced facts
4. Adequate allotted time (5‐10 min/day)
5. Record keeping
6. Motivational procedures
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[email protected] Steps to Fluency &Automaticity
3-4 Facts
Requires
1. Specific criterion for introducing new facts
2. Intensive practice on newly introduced Each fact
facts (more than 1x)
on 4 cards
3. Systematic practice on previously Continuous
introduced facts
Cycle
4. Adequate allotted time (5‐10 min/day)
Regular
5. Record keeping
Schedule
6. Motivational procedures
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Practice Procedures Math Facts
• Peer to Peer or individual or small group
• Students must say the fact (4 x 5 is 20)
– “Four times five is twenty”
• Error Correction Procedure:
– The only correct response is the correct answer to the fact.
– All other responses should be corrected
• For example….Saying incorrect fact, hesitation, using a strategy
1. Stop student and say correct answer (I say)
2. Say correct answer with student (We say)
3. Have student say correct answer (You say)
4. Partner says correct answer (I say)
– Fact is placed three cards back to make sure student has opportunity to re‐practice the fact while the correction is still in short term memory
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[email protected] Breakout Activity
• Partner up with a peer and practice the error correction procedure. Each of you take a role as the player‐
practicing the fact and the coach‐leading the fact
– The only correct response is the correct answer to the fact.
– All other responses should be corrected
• For example….Saying incorrect fact, hesitation, using a strategy
1. Stop student and say correct answer (I say)
2. Say correct answer with student (We say)
3. Have student say correct answer (You say)
4. Partner says correct answer (I say)
– Fact is placed three cards back to make sure student has opportunity to re‐practice the fact while the correction is still in short term memory
© Paul J. Riccomini 2017
[email protected] Instructional Planning Activity
• Discuss the Big Ideas covered to this point:
– Number of students who are NOT fluent
– Discuss current activities that teachers are doing for fluency.
• How are they working?
– Discuss the 6 Steps and how to implement with flash cards in the currently used fluency activities.
• Is this how you are already practicing
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Steps to Monitoring Fluency
• Identify target skill (i.e., numerals, single digit facts)
• Set Goals or targets—Focus is growth
• Determine a regular assessment scheduled – For example, every 4th day
• Develop/Create/Locate the assessments
– More problems than student can solve in the allocated time. (It’s NOT about Finishing)
– 1‐2 minute timings
– Standard Directions‐used all the time • (NO SKIPPING AROUND)
• Score and data is charted on a graphic display
– Shared and discussed with student
• Main priority is to show an INCREASE © Paul J. Riccomini 2017
[email protected] Monitoring Assessments (PVSD Fluency)
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Monitoring Assessments
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[email protected] Graphing Fluency Progress (PVSD)
• Chart data as it is collected.
• Focus on improvement
• Reward growth
• It will vary widely within a class.
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[email protected] Graphing Fluency Progress (PVSD)
GOAL: 40 Facts in a minute
• Chart data as it is collected.
• Focus on improvement
• Reward growth
• It will vary widely within a class.
• DECISION:_________
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Graphing Fluency Progress
GOAL: 40 Facts in a minute
• Chart data as it is collected.
• Focus on improvement
• Reward growth
• It will vary widely within a class.
• DECISION:_________
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[email protected] Graphing Fluency Progress
GOAL: 40 Facts in a minute
• Chart data as it is collected.
• Focus on improvement
• Reward growth
• It will vary widely within a class.
• DECISION:_________
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[email protected] Graphing Fluency Progress
GOAL: 15 Facts in a minute
• Chart data as it is collected.
• Focus on improvement
• Reward growth
• It will vary widely within a class.
• DECISION:_________
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Instructional Planning Activity
• Discuss a plan that you will put in place to determine monitoring schedule, determine goals, charting progress, sharing progress with students, and reinforcing growth.
– Target Skill
– Set Goals
– Monitoring Schedule
– Charting and Sharing Progress
– Making Decisions
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[email protected] Fluency Practice Activities
Beyond basic arithmetic:
• Fluency/automaticity in other mathematic computations and procedures becomes important for the transition to Algebra and Geometry
– Complex Computational Fluency – Procedural Fluency
© Paul J. Riccomini 2014
[email protected] Fluency & Automaticity Activities
• The Math Dash Activity (MS‐HS) HO #7
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Fluency & Automaticity Activities
The Math Dash Activity
1. Explain fluency and the purpose of fluency to your students
2. Target a Skill that students have already learned & are accurate (Rule of thumb 85%‐90%)
3. Create a PowerPoint with 8‐10 problems in that specific skill area
4. Display problems one at a time at a predetermined rate appropriate for age and targeted skill (e.g., 3 seconds…). Vary rate as necessary.
5. Only ONE problem at a time is displayed and students are required to just right the answer
6. Display answers—check students for accuracy
7. Repeat 2‐3 times for several days © Paul J. Riccomini 2014
[email protected] Fluency & Automaticity Activities
• The Method of Repeated Calculation (HO #7a, 7b, 7c)
© Paul J. Riccomini 2014
[email protected] Fluency & Automaticity Activities
• The Method of Repeated Calculation
1. Explain fluency and the purpose of fluency to your students—especially that students are not expected to answer ALL of the problems •
Goal is to solve more each time—even if that is one more
2. Select target skill and create a worksheet with around 30 problems (depends on targeted skill). Remember—
Already learned skill
3. Students are given 1 minute to solve as many problems as they can. Students can skip problems they don’t know and go to the next.
• DO NOT ALLOW SKIPPING AROUND THE PAGE
4. Repeat this activity 2‐3 times (no more than 3) for several days—use the same problems
5. Students can track their progress
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Instructional Planning Activity
• Identify a target skill for fluency in the grade level content that you teach
• Develop either a Math Dash or Repeated Calculation activity for the identified skill.
– Develop the problems or locate a practice sheet online that is already developed
• Plan the 5 days of fluency practice following the guidelines for fluency and automaticity that we just learned.
© Paul J. Riccomini 2017
[email protected] Facilitating Fluency & Automaticity
• Automaticity of facts is vital, but instruction for conceptual understanding must occur first
• Automaticity activities must be cumulative
– Newly introduced facts receive intensive practice, while previously introduced facts receive less intensive, but still SYSTEMATICALLY PLANNED.
• Fluency/Automaticity building activities should NOT use up all of the allocated math time (less than 10 minutes).
• Fact automaticity instruction is often overlooked by most math programs or ill‐conceived. • Automaticity practice must be purposeful and systematic as well as carefully controlled by the teacher
© Paul J. Riccomini 2017
[email protected] Strategies & Techniques 1
• Interleaved Worked Solution Strategy
2
• Spaced Learning Over Time (SLOT)
3
• Fluency & Automaticity
4
• Instructional Scaffolding Progressions
5
• Teaching Vocabulary
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Instructional Scaffolding
Instructional scaffolding is a process in which a teacher adds supports for students to enhance learning and aid in the mastery of tasks. © Paul J. Riccomini 2017
[email protected] Instructional Scaffolding
Instructional scaffolding is a process in which a teacher adds supports for students to enhance learning and aid in the mastery of tasks. 3 Main Types:
Task ~ Material ~ Content © Paul J. Riccomini 2017
[email protected] Instructional Scaffolding
• Task Scaffolding
– Specify the steps in a task or instructional strategy
– Teacher models the steps in the task, verbalizing his or her thought processes for the students. – the teacher thinks aloud and talks through each of the steps he or she is completing – Even though students have watched a teacher demonstrate a task, it does not mean that they actually understand how to perform it independently © Paul J. Riccomini 2017
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Providing Structured Guidance HO #8
Who or what is
involved in the action
Math vocabulary
used
Paraphrase the
question / problem
Write equation to
obtain solution
Explain equation
reasoning
Explain solution
© Paul J. Riccomini 2017
[email protected] • Task Scaffold
– These lines are prompting the students to explain the steps involved in this process.
Instructional Scaffolding
• Material Scaffolding
– Material scaffolding involves the use of written prompts and cues to help the students perform a task or use a strategy. – This may take the form of cue sheets or guided examples that list the steps necessary to perform a task. – Students can use these as a reference, to reduce confusion and frustration.
– The prompts and cues should be phased out over time as students master the steps of the task or strategy. © Paul J. Riccomini 2017
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IWSS • Material Scaffold
– The worked out problems are providing a cue or guided example.
• Very helpful to ALL students
Instructional Scaffolding
• Content Scaffolding
– the teacher selects content that is not distracting (i.e., too difficult or unfamiliar) for students when learning a new skill.
– allows students to focus on the skill being taught, without getting stuck or bogged down in the content • 3 Techniques for Content Scaffolding
– Use Familiar or Highly Interesting Content
– Use Easy Content
– Start With the Easy Steps © Paul J. Riccomini 2017
[email protected] Instructional Scaffolding
• Math Word Problems Strategy Instruction
• For example:
– Robert planted an oak seedling. It grew 10 inches the first year. Every year after it grew 1 ¼ inches. How tall was the oak tree after 9 years?
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Instructional Scaffolding
• Math Word Problems Strategy Instruction
– Allows students to focus in process of strategy
– Remove irrelevant information
– Include answer in the problem (i.e., no question)
• For example:
– Robert planted an oak seedling. It grew 10 inches the first year. Every year after it grew 1 ¼ inches. How tall was the oak tree after 9 years?
– An oak seedling grew 10 inches in the first year. Every year after it grew 1 inch. After 9 years the oak tree was 18 inches tall.
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[email protected] Instructional Scaffolding (SH #9)
• Solve the more complex problem
– Robert planted an oak seedling. It grew 10 inches the first year. Every year after it grew 1 ¼ inches. How tall was the oak tree after 9 years?
10 + 1 ¼ + 1 ¼ + 1 ¼+ 1 ¼+ 1 ¼ + 1 ¼ + 1 ¼ + 1 ¼ =20 inches tall 10 + (1 ¼ )(8) = 20 inches tall
10 + (1 ¼ )(9‐1) = 20 inches tall
• Scaffolded Instructional Progression
– This is how teachers can help students progress from simple tasks to more complex problem solving tasks.
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[email protected] Instructional Scaffolding Progression HO #10
Original Word Problem‐Target
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Instructional Scaffolding Progression HO #10
Instructional Guidance Progression
• Heavily teacher guided and think aloud is modeled 3‐4 times. Then students write the number sentence with heavy teacher guidance. This is followed by students explaining the number sentence to a peer.
• Teacher guidance is gradually reduced and students do more of the think aloud and number sentence writing
• A question reintroduced into the problem further decreasing the amount of teacher guidance
• The full original word problem is presented
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[email protected] Original Word Problem‐Target
Instructional Planning Breakout (SH #11)
• Application of Content Scaffolding – Using the example scaffold for Robert’s Oak Tree develop an instructional sequence for a common word problem expectation for your grade level.
• Select a word problem type
• Re‐write problem using content scaffolding • Develop the sequence/chart 3‐5 problems.
Instructional Scaffolding
Key Information
• 3 Types of Scaffolding (Content, Task, Materials)
• Scaffolding Instruction can help students better focus on the problem solving process
• Many Different ways to scaffold student learning
• Scaffolding is a necessity for students with disabilities
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Instructional Scaffolding
• How much scaffolding is necessary?
• BOTTOM LINE:
As much as the students require to learn and be successful!
© Paul J. Riccomini 2017
[email protected] Strategies & Techniques 1
• Interleaved Worked Solution Strategy
2
• Spaced Learning Over Time (SLOT)
3
• Fluency & Automaticity
4
• Instructional Scaffolding Progressions
5
• Teaching Vocabulary
© Paul J. Riccomini 2017
[email protected] Mathematical Proficiency Defined
National Research Council (2002) defines proficiency as:
1. Understanding mathematics
2. Computing Fluently
3. Applying concepts to solve problems
4. Reasoning logically
5. Engaging and communicating with mathematics
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Breakout Activity (HO #12)
Make Vocabulary Part of Learning and Assessment
• Define and provide and example of the following vocabulary terms:
–
–
–
–
–
–
–
Formula
Integers
Quadrant
Rational Number
Simplest Form
Solution to an Equation
Variable
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Mathematical Vocabulary
•Mathematical vocabulary can have significant positive and/or negative impact on students’ mathematical performance
© Paul J. Riccomini 2017
[email protected] Mathematical Vocabulary
• Mathematics can be thought of as a language
that must be meaningful if students are to communicate mathematically and apply mathematics productively.
• Vocabulary development is crucial to any experience involving language
• Vocabulary is central to mathematical literacy
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Breakout Activity
• Define the following terms:
– Numerator
– Denominator
– Fraction
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[email protected] Mathematical Vocabulary
Challenges
• Reading mathematics is a complex task that includes:
– Comprehension
– Mathematical understanding
– Fluency and proficiency with reading:
• Numbers
• Symbols, and
• WORDS‐‐vocabulary
© Paul J. Riccomini 2017
[email protected] Teaching Vocabulary
• Students at‐risk and/or with learning disabilities:
– often use short sentences with poor word pronunciations
– have limited receptive and expressive vocabularies
– are poor readers and do not tend to read on their own
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Teach for Deeper Understanding
• 4 General Guidelines
1. Employ a variety of methods of teaching vocabulary
2. Actively involve students in vocabulary instruction
3. Provide instruction that enables students to see how target vocabulary words relate to other words
4. Provide frequent opportunities to practice reading and using vocabulary words in many contexts to gain a deeper and automatic comprehension of those words
(Foil & Alber, 2002)
© Paul J. Riccomini 2017
[email protected] Breakout Activity
• How do you currently teach math vocabulary?
– List the specific activities that you use to teach essential mathematical vocabulary
– How do you assess essential mathematical vocabulary?
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[email protected] Mathematical Vocabulary Instruction
General Guidelines for Math Instruction
1. Establish a list of vocabulary for each subject area or unit
2. Evaluate comprehension of mathematics vocabulary on a periodic basis
3. Probe students’ previous knowledge and usage of important terms before it is introduced during instruction
4. Frame the context for new mathematics vocabulary
Adams, 2003
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Mathematical Vocabulary Instruction
Guidelines for Math Instruction
5. Develop an environment where mathematics vocabulary is a normal part of instruction, curriculum, and assessment
6. Encourage students to ask about terms they don’t know
7. Teach students how to find meanings of vocabulary words (e.g., dictionary, internet, notes, etc)
Adams, 2003
© Paul J. Riccomini 2017
[email protected] Levels of Vocabulary Knowledge
• Level 1: Identification
– Is this a…………?
• Level 2‐Reproduction
– Draw me a ……..?
• Level 3‐ formal definition
– A blank is ………….?
• Level 4– Real world context
– When do you see/use a………?
• Level 5‐‐‐Real world Application
– Why is ………?
© Paul J. Riccomini 2017
[email protected] Instructional Planning Activity
• Essential Vocabulary
– Identify 2‐3 essential Vocabulary terms that from a particular Standard or chapter unit
– Select wisely‐‐‐You can’t directly teach ALL vocabulary
• We will mimic an excellent activity to be conducted in a PLC or common planning time or half day of PD—Instructional Planning
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Teachers set the Vocabulary Focus
• When vocabulary is not made a regular part of math class, we are indirectly saying it isn’t important!
– Require students to use mathematically correct terms,
– Teachers must use mathematically correct terms
– Classroom tests must regularly include math vocabulary
– Instructional time must be devoted to mathematically vocabulary
© Paul J. Riccomini 2017
[email protected] Vocabulary Impacts Proficiency
• Clearly, an individual’s vocabulary recognition and knowledge are vital components to become mathematically proficient
• When students struggle to learn important math vocabulary, teachers must employ evidenced‐based instructional practices to help ALL students become successful.
© Paul J. Riccomini 2017
[email protected] Instructional Strategies
1. Practice using & remembering essential vocabulary terms
2. Frayer Model
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#1 Practice Using & Remembering
• Key features to more effective practice
– Pronounce the word for the students
– State definitions, and have students repeat definitions
– Provide students with good and bad examples
– Review the new words along with previously learned words to ensure students have the words in long term memory
– Teach the word within it’s context
IRIS Center http://iris.peabody.vanderbilt.edu
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[email protected] #1 Practice Using & Remembering
• Practice is an important variable in the learning of mathematical vocabulary. Math maybe the only time students have the opportunity to hear and use math vocabulary.
2‐3 minutes is all that is needed
IRIS Center http://iris.peabody.vanderbilt.edu
© Paul J. Riccomini 2017
[email protected] #1 Practice Using & Remembering
Cue Cards
• Can helps students organize and more easily remember important math information
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VA DOE Math Vocabulary Resource
http://www.doe.virginia.gov/instruction/mathematics/resources/vocab_cards/
© Paul J. Riccomini 2017
[email protected] CCSS Math Vocabulary Resource
http://www.ncesd.org/Page/983
http://www.graniteschools.org/ma
thvocabulary/vocabulary‐cards/
© Paul J. Riccomini 2017
[email protected] #1 Instructional Planning Activity
• Develop a plan (or discuss how you are already use) for using the 3 x 5 index cards to help facilitate practice and remembering of essential math vocabulary.
– 3‐5 minutes a couple times a week
• Share any resources available online for flashcard like activities
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#2 Graphic Organizers for Math
• Concepts maps are visuals (sometimes called graphic organizers) that assist students in remembering information
• Use to help students learn important attributes, remember steps in a problem and important formulas
© Paul J. Riccomini 2017
[email protected] #2 Frayer Model (aka four corners)
• The Frayer Model is a graphical organizer
with 4 sections
• Helps students with word analysis and vocabulary building. • Helps students create and organized visual reference for vocabulary
• Produces a paper product that can be revisited easily and quickly throughout the year using a variety of activities
© Paul J. Riccomini 2017
[email protected] #2 Frayer Model (SH #13)
• The framework of the model prompts students to think about and describe the meaning of a word or concept in four parts
– Defining the term, – Describing its essential characteristics, – Providing examples of the idea, and – Offering non‐examples of the idea.
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#2 Frayer Model
• Make a bulleted list
• Add a picture that helps you understand the meaning of the “word”
• In own words
• Technical Definition
• List of illustrate at least 3 examples
• List of illustrate at least 3 non examples
• Make a list about what the “word” is not about
• Non‐examples should be similar to the examples
#2 Example Frayer Model
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[email protected] #2 Example Frayer Model
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#2 Using the Frayer Model
Four Steps :
1. Explain the Frayer model graphical organizer to the class. Provide a model using a familiar term/concept.
2. Select a list of key concepts from a math Chapter you just taught. 3. Divide the class into student pairs. Assign each pair one of the key concepts and have complete the four‐square organizer for this concept. 4. Ask the student pairs to share their conclusions with the entire class. Use these presentations to review the entire list of key concepts. © Paul J. Riccomini 2017
[email protected] Frayer Model & Peer Tutoring
• In class Activity—SO #14
– Are activities that include a set of instructional procedures where by students are guided by peers – Students work together through a series of structured activities to practice important skills during peer‐mediated instructional time © Paul J. Riccomini 2017
[email protected] Vocabulary Considerations
(Handout #16)
• Vocabulary Instruction Consideration
– How are you teaching math vocabulary?
– How are you facilitating the learning and remembering of math vocabulary?
– Does each teacher have an established list of essential vocabulary for the year or per unit?
– Are you testing math vocabulary regularly?
– Are you practicing using math vocabulary throughout the course of the year?
– Is there any preteaching of vocabulary occurring? Where? When? Who? How?
© Paul J. Riccomini 2017
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Maximize Students’ Mathematical Learning
3‐Day PD Series
NJPSA
Summary Math Vocabulary
• Understanding of essential mathematical vocabulary is essential for students to become proficient
• Some students will struggle with learning vocabulary; therefore, teachers must use evidenced‐based practices to teach vocabulary
• Mnemonics and the keyword strategy offer teachers an instructional practice that is effective for all students
© Paul J. Riccomini 2017
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• The learning needs of struggling students and students with disabilities in mathematics is extremely challenging for teachers.
• The research base addressing the specific instructional strategies and interventions clearly suggests the importance of Explicit Instructional Techniques
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Interleave Worked Solution Strategy‐IWSS
Spaced Learning overtime‐SLOT
Fluency & Automaticity
Content Scaffolding‐Word Problems
Task and Material Scaffolding
Mathematical Vocabulary Strategies
• Conclusion
© Paul J. Riccomini 2017
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Contact Information:
Paul J. Riccomini
[email protected]
@pjr146
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