MAXIMIZE ALL STUDENTS MATHEMATICAL LEARNING …njpsa.org/documents/pdf/MathInstructionalStrategies... · · 2017-01-19100% Math Proficiency ©PaulJ ... Fractions linked to number

MAXIMIZE ALL STUDENTS’

MATHEMATICAL LEARNINGTHROUGH THE USE OF

POWERFUL

INSTRUCTIONALSTRATEGIES & TECHNIQUESJANUARY 18, 2017FEBRUARY 16, 2017

MARCH 1, 2017

Paul J. Riccomini, Ph.D.

[email protected]

@pjr146

Maximize Students’ Mathematical Learning3‐Day PD Series

NJPSA

© Paul J. Riccomini [email protected] 1

Components of Effective Mathematics Programs

Mathematics Curriculum & Interventions

Assessment & Data-Based Decisions

Teacher Content &

Instructional Knowledge

100% Math Proficiency

© Paul J. Riccomini [email protected]

Components of Effective Mathematics Programs

Mathematics Curriculum & Interventions

Assessment & Data-Based Decisions

Teacher Content &

Instructional Knowledge

100% Math Proficiency

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 & Strategies1. Fractions linked to number line early and often2. Interleave Worked Solution Strategy (IWSS)3. Spaced Learning Over Time (SLOT)4. Fluency & Automaticity

1. Computation and Procedural Fluency

5. Instructional Scaffolding Progressions• Task ~ Material ~ Content

6. Teaching Vocabulary• Frayer Model

• Conclusion & Wrap Up© Paul J. Riccomini 2017

[email protected]


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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


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Math Proficiency of U.S. Students

• 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

Mathematics Performance

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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Our Own Fraction Foundation (1a)

• Shade 3/8 on the provided grid.



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Math Proficiency of U.S. Students• 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


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Math Proficiency of U.S. Students

• 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


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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

0 2

1

Student Estimates

Whole NumbersP e r c e n t sD e c i m a l sF r a c t i o n s

• 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.

Results of a Recent Study


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–Use number line representations to help students recognize fractions are numbers and that they expand the number system


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–Use number line representations to help students recognize fractions are numbers and that they expand the number system

Our Own Fraction Understanding

• Order the following fractions from least to greatest


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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?


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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?

Foundations for SuccessNational 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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Curricular Content‐NMAP 2008Streamline 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)


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Curricular Content‐NMAP 2008Streamline 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)

**Enrichment & Reteaching Targets**

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



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• All‐encompassing recommendations that instruction should be student‐centered or teacher‐directed are not supported by research.

Instructional practice should be informed by high quality research, when available, and by the best professional judgment and experience of accomplished classroom teachers.

Instructional Practices‐NMAP‐2008

Instructional Practices‐NMAP‐2008

Research on students who are low achievers, have difficulties in mathematics, or have learning disabilitiesrelated 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.



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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.


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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


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)


CCSS for Mathematical Practices1. 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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Learning Outcomes of CCSS‐MP

(McCallum, 2011)


Learning Outcomes of CCSS‐MP

(McCallum, 2011)

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!



Facilitate Learning through Instruction

• 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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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!!!


• Interleaved Worked Solution Strategy• Interleaved Worked Solution Strategy1

• Spaced Learning Over Time (SLOT)• Spaced Learning Over Time (SLOT)2

• Fluency & Automaticity• Fluency & Automaticity3

• Instructional Scaffolding Progressions• Instructional Scaffolding Progressions4

• Teaching Vocabulary• Teaching Vocabulary5

Strategies & Techniques










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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/


Excellent Resource: IES


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?



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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)


Instructional Practices‐NMAP‐2008Research on students who are low achievers, have difficulties in mathematics, or have learning disabilitiesrelated 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

1. Daily Reviews

2. Presentation of New Content

3. Guided Practice

4. Explicit feedback and Correctives

5. Independent Practice

6. Weekly and Monthly Reviews

Six Critical Features of explicit instruction



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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!!!



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)







NMAP, 2008



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• Typical Math Homework assignment

– Pg. 155 #1‐21 odd

• Students are required to solve all problems.




• Interleaved Homework assignment

– Pg 155 1‐10 (all)

– Odd problems


Typical Practice with Fractions



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• Providing students opportunities to reason and explain worked out problem solutions enhances learning.

• Studying solutions can focus students on STRUCTUREof 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

Solutions as a Strategy


• 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.


Another Example of IWSS Practice



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• During Whole Class instruction1. 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.



IWSS #4c

• 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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Questions to Guide Whole Class Discussion HO#4e

Problem Student Response Questions to Guide Discussion


Correct and Incorrect Comparisons HO #4f


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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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 alternatingformat

Step 8 Create final IWSS practice sheet ready for use


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



Classroom Application Activity‐IWSSHO #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?



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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.



• Providing students opportunities to reason and explain worked out problem solutions enhances learning.

• Studying solutions can focus students on STRUCTUREof 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

Solutions as a Strategy









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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



Spaced Learning Over Time

• 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).



Space Learning Over Time (HO#5)


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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



Fast forward 4‐6 weeks from when identified skills were taught & list date here to revisit


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

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




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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.















NMAP, 2008



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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

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.


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.



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Instructional Time

Inst

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Short-term Working Memory

Achieve Automaticity and Fluency

Progression to Fluency

Long Term Memory

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


Common Core State Standards

• 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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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

– Fluentlymultiply 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.



• 5.NBT.5. Perform operations with multi‐digit whole numbers and with decimals to hundredths

– Fluentlymultiply 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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• 7.EE.4Use 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?


Fluency Recommendations‐PARCC• 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 & PARCCFluency 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 Time

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Short-term Working Memory

Achieve Mastery

Progression to Automaticity

Long Term Memory

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


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 PURPOSEFULPLANNED &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?

– 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?


Grade 1‐Addition Grade 3‐4: Multiplication &DivisionGrade 2‐Subtraction Grade 4‐12: Multiplication is Priority


Steps to Fluency &Automaticity

Requires1. 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


Steps to Fluency &Automaticity

Requires1. 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

3-4 Facts

Each fact on 4 cards

Continuous Cycle

Regular Schedule


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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


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


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


Monitoring Assessments (PVSD Fluency)


Monitoring Assessments



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Monitoring Assessments


Graphing Fluency Progress (PVSD)

• Chart data as it is collected.

• Focus on improvement

• Reward growth• It will vary widely

within a class.


Graphing Fluency Progress (PVSD)




within a class.

• DECISION:_________

GOAL: 40 Facts in a minute



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Graphing Fluency Progress





within a class.







within a class.





within a class.








within a class.




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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


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


Fluency & Automaticity Activities• The Math Dash Activity (MS‐HS) HO #7



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Fluency & Automaticity ActivitiesThe 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 accuracy7. Repeat 2‐3 times for several days


Fluency & Automaticity Activities• The Method of Repeated Calculation

(HO #7a, 7b, 7c)


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© Paul J. Riccomini 2014

[email protected]


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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.


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










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Instructional scaffolding is a process in which a teacher adds supports for students to enhance learning and aid in the mastery of tasks.

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




• 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


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Providing Structured Guidance HO #8

Who or what is involved in the action

Math vocabulary usedParaphrase the question / problem

Write equation to obtain solution

Explain equation reasoning

Explain solution

• 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.


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IWSS

• Material Scaffold

– The worked out problems are providing a cue or guided example.

• Very helpful to ALL students



• 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


• 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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• 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.



• 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.

Instructional Scaffolding (SH #9)

Original Word Problem‐Target

Instructional Scaffolding Progression HO #10



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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

Original Word Problem‐Target© Paul J. Riccomini 2017

[email protected]

Instructional Scaffolding Progression HO #10

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.


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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• How much scaffolding is necessary?

• BOTTOM LINE:

As much as the students require to learn and be

successful!








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

•Mathematical vocabulary can have significant positive and/or negative impact on students’ mathematical performance

Mathematical Vocabulary



• Mathematics can be thought of as a languagethat 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


Challenges

• Reading mathematics is a complex task that includes:– Comprehension

– Mathematical understanding

– Fluency and proficiency with reading:• Numbers

• Symbols, and

• WORDS‐‐vocabulary


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)


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?


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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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


Mathematical Vocabulary Instruction

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 ………?


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


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.


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 learnedwords to ensure students have the words in long term memory

– Teach the word within it’s context

IRIS Center http://iris.peabody.vanderbilt.edu



• 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


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/


CCSS Math Vocabulary Resource

http://www.ncesd.org/Page/983

http://www.graniteschools.org/mathvocabulary/vocabulary‐cards/


#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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• 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

#2 Graphic Organizers for Math


#2 Frayer Model (aka four corners)

• The Frayer Model is a graphical organizerwith 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


#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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• In own words

• Technical Definition

• Make a bulleted list• Add a picture that helps you

understand the meaning of the “word”

• 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 Frayer Model

#2 Example Frayer Model


#2 Example Frayer Model



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#2 Using the Frayer ModelFour 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.


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


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?



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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


Conclusion• 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 Techniques1. Interleave Worked Solution Strategy‐IWSS2. Spaced Learning overtime‐SLOT3. Fluency & Automaticity4. Content Scaffolding‐Word Problems5. Task and Material Scaffolding6. Mathematical Vocabulary Strategies

• Conclusion


QUESTIONS?


Contact Information:Paul J. [email protected]

@pjr146

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