maximize all students mathematical learning in …

MAXIMIZE ALL STUDENTS’ MATHEMATICAL LEARNING IN

INCLUSIVE ELEMENTARY

CLASSROOMS THROUGH THE

USE OF POWERFUL

INSTRUCTIONAL STRATEGIES& TECHNIQUES

DAY 2 OF 2MAY 9, 2016

Paul J. Riccomini, Ph.D.

[email protected]

@pjr146

Maximize K‐5 Students’ Math LearningPart 2 of 2

May 9, 2016

© Paul J. Riccomini [email protected] 1

Topics for Today• 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. Spaced Learning Over Time (SLOT)2. Fluency & Automaticity

• Computation and Procedural Fluency

3. Fractions represented through the number line4. Instructional Scaffolding Progressions

• Content Scaffolding Progressions for Problem Solving• Task Scaffolding• Material Scaffolding‐Interleaved Worked Solution Strategy (IWSS)

5. Frayer Model to help frame Math Vocabulary

• Conclusion & Wrap Up

© Paul J. Riccomini [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


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.



May 9, 2016


Learning Outcomes of CCSS‐MP

(McCallum, 2011)

1. SLOT2. Fluency &

Automaticity3. Rational #s4. Scaffolding5. Worked

Examples6. Mathematical

Language



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)

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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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.


Progression to Fluency

• Learning Progression Stages

1. Understanding• Manipulatives & Pictorial Reps

2. Relationship• Making connections within & across

3. Fluency• Strategy development for accuracy & Practice for fluency

4. Automaticity• Practice to facilitate automaticity


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



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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

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

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]


May 9, 2016


Instructional Planning Activity• Identify a target skill for fluency in the grade level content that you teach

• Develop a plan for fluency practice activities using either Math Dash or Repeated Calculation.

• Develop 5 days of fluency practice following the guidelines for fluency and automaticity that we just learned.

• Each group will present their 5 day plan to whole group.

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


•Rational Numbers•Rational Numbers1

•Instructional Scaffolding•Instructional Scaffolding2.

•Worked Solutions•Worked Solutions3

•Mathematical Language•Mathematical Language4

Strategies & Techniques



May 9, 2016



16

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


17


• 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

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


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19


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


20


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


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


May 9, 2016




• 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– Remove irrelevant information

– Include answer in the problem (i.e., no question)

– Allows students to focus in process of strategy

• 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.


Content Scaffolding Summary• Changes the focus initially from an answer driven process to more of a reasoning and explaining process.

• Identifying the underlying general structure of the word problem to promote transfer or generalization to other similar problem types.

• Content progression gradually releases more responsibility to the learner through the word problem attributes and fading of teacher guidance/support.



May 9, 2016


Instructional Planning Breakout (SH #18)

• 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.




• Task Scaffolding (HO #19)

– 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


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.


May 9, 2016



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



• 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



May 9, 2016



• 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.



• 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.



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




May 9, 2016


Classroom Application Activity‐IWSS

• 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?


Planning Sessions‐IWSS• Identify materials that you will use in an upcoming chapter/unit that best fits IWSS.

• Modify or adapt those materials to incorporate the IWSS technique.

• Each group will be asked to provide an example of materials with the IWSS technique.

• Prepare the IWSS Sheets as if you are using them next week.



Key Information

• 3 Types of Scaffolding: Content, Task, Material

• Chunking across problems

• Interleaved Worked Solution Strategy (IWSS)

• 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



May 9, 2016


•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


Common Core State Standards

6. Attend to precision.

• They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.

• In the elementary grades, students give carefully formulated explanations to each other.

• By the time they reach high school they have learned to examine claims and make explicit use of definitions.



May 9, 2016



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


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?



May 9, 2016


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


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

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



May 9, 2016


Frayer Model (HO #21)• 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.


#2 Example Frayer Model


#2 Example Frayer Model



May 9, 2016


#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.


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

• Frayer Model is an excellent graphic organizer for math vocabulary


Conclusion• Instructional Strategies Covered

1. Spaced Learning Over Time (SLOT)

2. Fluency and Automaticity Techniques

• Computational Fluency/Automaticity

• Procedural fluency

3. Fractions represented through the number line

4. Instructional Scaffolding Progressions• Content Scaffolding Progressions for Problem Solving

• Task Scaffolding

• Material Scaffolding‐Interleaved Worked Solution Strategy (IWSS)

5. Frayer Model to help frame Math Vocabulary

• Conclusion and Wrap‐Up© Paul J. Riccomini 2016

[email protected]

maximize all students mathematical learning in …

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