maximize students’ mathematical thinking january 22, 2018

Maximize Students’ Mathematical Thinking Part 3

January 22, 2018IU #11

DEVELOPING, SUSTAINING, & SUPPORTING HIGH QUALITY

MATHEMATICS INSTRUCTION:MAXIMIZE ALL

STUDENTS’ LEARNINGTUSCARORA INTERMEDIATE UNIT 11

JANUARY 22, 2018

PART 3

Paul J. Riccomini, Ph.D.

pjr146@psu.edu

@pjr146

Topics for Math PD Series• Designing instruction to help students of all skill levels

achieve success in mathematics.• Instructional Techniques & Strategies Part 1‐2

1. Content Scaffolding Progression• Problem Solving Tasks

2. Fluency and Automaticity3. Spaced Learning Over Time (SLOT)

• Retention Strategy

• New Strategies Part 3– Interleaving Practice Format (IFP)– Practice Test and Retrieval (PTR)– Fraction Understanding– Supporting Open Constructed Responses through ACE

• Conclusion & Wrap Up

Developing, Sustaining, & Supporting Mathematics Learning

Mathematics Standards,

Curriculum & Interventions

Assessment & Data-Based Decisions

Teacher Content &

Instructional Knowledge

Performance Growth ALL

Students

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)

http://iris.peabody.vanderbilt.edu/srs/chalcycle.htm

Essential Question for Teachers

• Essential Question–What did I do to “intensify instruction” to support the struggling students (and/or most advanced students)?

• Asked during instructional planning and after instructional delivery!!!

Core Mathematics Program • Serves as the foundation for the learning of mathematics

• Not all created equal or created with all students in mind!

• Have inherent strengths and weakness– Fluency Practice, Retention emphasis, practice in general, problem solving instruction, engagement in reasoning and explaining, language/vocabulary development, Written reasoning and explaining opportunities

• Supplementing is used to strengthen the identified core component weaknesses for struggling students

• Regular reflection on core program is essential

Core Mathematics Program • Serves as the foundation for the learning of mathematics

• Not all created equal or created with all students in mind!

• Have inherent strengths and weakness–Fluency Practice, Retention emphasis, practice in general, problem solving instruction, engagement in reasoning and explaining, language/vocabulary development, Written reasoning and explaining opportunities

• Supplementing is used to strengthen the identified core component weaknesses for struggling students

• Regular reflection on core program is essential

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.

• Spaced Learning Over Time (SLOT)• Spaced Learning Over Time (SLOT)1

• Content Scaffolding Progression (CSP)• Content Scaffolding Progression (CSP)2

• Fluency and Automaticity• Fluency and Automaticity3

Strategies & Techniques

Retention

• Major problem for students with disabilities, but a challenge for ALL students.

• Forgetting previous learned information is normal, but particularly problematic in mathematics.

• Typical review and practice techniques are ineffective for many students

Intensive Retention Strategies

Two Specific Practice Techniques–Does not require any change to instruction

High Leverage Practices (HLP)

1. Spaced Learning Over Time (SLOT)

2. Interleaving Practice Format (IPF)

3. Practice Testing and Retrieval (PTR)

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

Space Learning Over Time

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

Space Learning Over Time

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

Spaced Learning Over Time

• List the Non‐Negotiable Key Ingredients

• Negotiable Key Ingredients

Steps to Implement SLOT1. Using the spaced instructional review sheet,

divide the school year into 4‐6 week units starting at the beginning of the year through the end of the year.

2. List the Big ideas taught during each 4‐6 week unit

3. Identify areas that are often problematic and very important for students based on data

4. Plan to revisit each identified topic at the specified date (4‐6 weeks after it was taught)

Practice Structure in Mathematics• Practice in Mathematics is important to promote initial acquisition development, proficiency, fluency, and retention.– It is a critical feature to mathematics instruction

• Typically mathematics practice is facilitated through– Homework

– In class activities

– Computer practice opportunities

– Game like situation both low tech and high tech

• Discuss the structure of the practice in terms of how the math problems are SEQUENCE (the order in which they are presented?

Interleaving Practice Format (IPF)

• Interleaving practice format (IPF) is a structure applied to practice activities to improve long term learning outcomes. – Produces “durable” learning

• IPF involves the intentional mixing‐up of items within the same practice session– abc abc abc

– Much more effective than blocking practice

– Blocking design groups similar problems together (aaaaaaa bbbbbbbb ccccc)

Rohrer (2012); Rohrer, Dedrick, and Stershic(2014); Taylor and Rohrer (2010)

Massed or Blocked Practice compared to Interleaved Practice.

Good initially….but a HUGE Difference on TEST

• Interleaving practice format (IPF) results in much longer learning– Students retain information and actually improve as time passes

• Why does the mixed sequence of problems produce much longer retention?

Interleaving Practice Format (IPF)Why does IFP work?• The blocking structure does not require deeper and careful processing– Students recognize the strategy and just blindly apply it because they know in advance

– No retrieval is necessary, the solution strategy is only held in Short term working memory

• IPF requires the students to think more deeply about each problem because it is different– Requires students retrieve different strategies from long term memory

– Forces students to “pay attention” to the problem features at same time they are retrieving the appropriate solution strategy

Considerations1. Blocking is important in the initial stages

of learning a new concept/skill1. Begin with blocking practice, but then move to IPF

once students have “some” familiarity with new topic

2. Students prefer blocking structured practice even though the research results are crystal clear in terms of improved performance with IPF.

Blocked Practice• Problems are grouped

according to type: 2x2 Multiplication, 1x2 division, and word problems (aaa bbbcc)

• Requires minimum thought because students recognize the process in advance and are simply applying it to consecutive problems

• Initial learning benefits, but decreased long term retention.

Hughes, Riccomini, & Morris, in press

Interleaving Practice Format

• Problems involve multiplication and division computation and word problems.

•• Problem types are alternating in

an abc abc sequence•• Requires students think more

deeply about the process by alternating problem types

•• Initial learning may feel delayed,

but produces better long term learning.

Classroom Application Activity‐IPF

• Discuss the current practice structure that is prevalent in your math curriculum

– Blocking structure v. IPF

• Brainstorm ideas for planning, designing, and providing regular opportunities for students to practice in an interleaving structure

– Study guides, review days, etc etc

• Provide students at least 2‐3 IPF opportunities each month from September‐May

Interleaving Practice Format (IPF) See HO#1

Planning and Developing IPF Opportunities1. Identify 2‐3 different problem types

2. Arrange the problem types in an alternating sequence

• Abc abc abc or abc bca cba

3. Inform students of the new homework structure• Students will actually “sense” the new sequence and

that it slows initial learning

4. Provide regular IPF activities across the school year• Homework, study guides, review sheets, etc

5. Provide IPF opportunities 2 to 3 times per month

Interleaving Practice Format

• Non‐Negotiable Key Ingredients

Practice Test and Retrieval (PTR)

• One of the most effective study techniques• Practice testing improves student learning• Practice testing provides opportunities for students to retrieve target information which requires to student to activate related knowledge– This strengthens the learning of the target information

– Promotes long term retention (durable learning)

• Practice testing helps students organize information in ways that help with retrieval– Learning is highly supervised in the early grade levels‐‐‐i.e. teachers tell students exactly what they need to study

– By middle school (definitely high school), the responsibility of the learning shifts to the student

• Knowing what to study, how to study, what they know and don’t know, how long to study, recognizing good study habits

• Think Strategic Learning vs Non‐Strategic Learner• Often called self‐regulation

– Low achievers and students with disabilities DO NOT DO THIS WELL

Implementation• Practice test activities must include corrective

feedback– Delayed or immediate– Without corrective feedback, the errors will continue

to remain

• Practice test activities must include Production responses– Short answer, fill in the blank, solve the problem– NOT EFFECTIVE WITH MULTIPLE CHOICE ITEMS

• Multiple choice decreases RETRIEVAL requirements

– Practice tests with production responses significantly improves multiple choice test performance

Practice Test and Retrieval (PTR)Activity Description Frequency

Flash Card Q&AShort answer questions or vocabulary terms are placed on flash cards. Students are paired and ask each other the questions and then discuss the answer. Answer is provided on the back so feedback is embedded in the activity

First 10 minutes of class every 4th day

Self‐Check QuizzesA self‐check activity is not graded and includes short answer questions that students have to answer. The answers are immediately corrected through a class discussion where students share answers and discuss both incorrect and/or correct answers.

Every 6th day or as much as appropriate

Jeopardy Type GamesAny type of game that encourages unassisted recall and equal opportunities for students to answer questions. Questions could be drawn from recent chapters.

Every 2nd Friday of the month

Throw‐Back‐Thursday Ticket Out the Door

(less Guided than SLOT)

Students are given a question related to the topics in the previous week’s lesson and write down everything that they can remember about the question. Feedback is provided the next day in the first 5 minutes of class.

Every Thursday or when appropriate based on sequence of content

Bottom‐line– Regular activities spread‐out across the school

year that require students to Recall information from memory helps solidify that information resulting in longer retention rates.• Closed book, no notes, etc

– Production response items and feedback are a key ingredient to PTR

– Low stakes is also a key ingredient• Not graded

Classroom Application Activity‐PTR

• Discuss ways that you can incorporate more opportunities for “low stakes” Practice test activities on a regular bases across the course of the year

– Consider this within chapters (or strands of knowledge)

– Consider in the context of the end of year exams

– Are there computer applications available that allow for production responses

• Develop a general framework for systematically implementing the PTR strategy.

Math Proficiency of U.S. Students

• Low fractions of proficiency on NAEP– Failure to master fractions contributes to the

mathematics achievement gap between students with and without disabilities (Hecht, Vagi, & Torgesson, 2007; Hecht & Vagi; 2010;).

– Gateway to algebra and higher-level mathematics (Booth, Newton, & Twiss-Garrity, 2014).

– Algebra I failure limits access to post-secondary education and career-training opportunities (NMAP, 2008).

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

Fraction Understanding

• Why do students struggle with fractions?

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

• 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:

Importance of FM Knowledge

Number Line– Demonstrating fraction equivalence on the number line models magnitude.

– Successful fraction MK interventions use number line representations

Fraction Magnitude Knowledge• Measurement (magnitude) and part‐whole interpretations are most important for

early fraction learning (Hecht & Vagi, 2010), but over‐emphasis on part‐whole interpretation may contribute to whole number bias (e.g., Vamvakoussi & Vosniadou, 2004, 2010).

• Common error associated with WNB:

Number line estimation

Why is this skill important?• Performance on fraction number line estimation

tasks measure magnitude knowledge

• Number sense essentially refers to a student's “fluidity and flexibility with numbers,” (Gersten & Chard, 2001). He/She has a sense of what numbers mean, understands their relationship to one another, is able to perform mental math, understands symbolic representations, and can use those numbers in real world situations.

Number LinesHow do we address this skill in instruction?‐ Number lines

‐ NMAP, NCTM, CCSS, Standardized Tests

1. Estimate fractions on number lines‐• Teach strategies:

– Benchmark– Halving/Doubling

2. Order and compare fractions on number lines3. Relate concepts (equivalency) and computation

(adding/subtracting) back to the number line4. Fluency activities

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

Locating a fraction on the number line

1. Identify the denominator. Split the unit into equal‐length segments (denominator = number of segments).

• If the denominator is 2, 4, or 8:

– Split the unit in half, then halve each segment until the number of segments = the denominator.

Locating a fraction on the number line

1. Identify the denominator. Split the unit into equal‐length segments (denominator = number of segments).

– If the denominator is 3, 6, or 12:

» Split the unit into thirds, then halve each segment until the number of segments = the denominator.

Breakout Activity‐(HO #4)Fractions and the Number line• Discuss your current curricular materials in relationship to the

use of the number line to represent fractions (but also decimals, and percents):– Is the # line a common representation?

– Estimate fractions on number line activities?

– Teaching strategies such as benchmark fractions and/or halving /doubling strategies

– Order and comparing fraction activities

– Relate concepts (equivalency) and computation (add/subtract) back to number line

– Fluency type activities involving fractions

• Where and how can you intensify instruction with the # line in your math lessons in conjunction with fractions, decimals, percents?

Content Scaffolding Supports• 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.

Content Scaffolding Progression (CSP)

• 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

– Excellent for Problem Solving Tasks

• 3 Techniques for Content Scaffolding– Use Familiar or Highly Interesting Content– Use Easy Content– Start With the Easy Steps

• 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.

Content Scaffolding Progression (CSP)

Worked Solution Strategy• Non‐Negotiable Key Ingredients

Open Construction Response Support

Academic Vocabulary

• Many students in grades 7 and below struggle with the following academic vocabulary common in open construction responses

• Trait

• Evidence

• Qualities

• Sequence

• Infer

• Point of view

• Support

Teach Academic Vocabulary

Strategy #1 Using a Frayer Model• 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

Teach Academic Vocabulary• 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.

• 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

Frayer Model

Example Frayer Model

Teaching Academic VocabularyHO #6 & 7

• List common academic vocabulary for the PA open construction response math assessment questions.

• Develop a Frayer model for one academic Term

• Develop a Frayer model for 2 ‐3 important math terms coming up in the next chaper/unit

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.

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. Identify 2‐5 terms per unit or chapter to target for Frayer Model development and activities1. Create and give to students for discussion

2. During Initial instructions, teacher leads the creation/development (guided notes) of Frayer Model

3. After instruction, students create a Frayer model with a partner

4. Peer to peer activities that use Frayer Models

Using the Frayer Model

Frayer Model Puzzles1. After a vocabulary term has been taught and students

have developed a solid understanding.

2. Take 2‐3 completed Frayer Models, laminate the Frayer models and then cut into puzzle pieces.

3. Working a partner or small group, students are given an envelope with 2‐3 Frayer models cut into pieces and mixed together.

4. Students work to put the Frayer Models back together.

Bulletin Board Frayer Model1. Design a bulletin board in a Frayer Model Format

Breakout Activity

Vocabulary Activities

• Discuss your current curricular materials and activities related to vocabulary.

• Where and how can you intensify vocabulary instruction and activities using the Frayer Model?

– Or other structure for vocabulary

Open Construction Response Support

Academic Vocabulary

• Many students in grades 7 and below struggle with the following academic vocabulary common in open construction responses

• Trait

• Evidence

• Qualities

• Sequence

• Infer

• Point of view

• Support

Strategy #1: Teach academic vocabulary and the specific terminology used on the assessments.Strategy #2: Make sure students are familiar with the rubric

PA Open Construction Rubric

In Grade Level Groups

• Go to the PSSA Item and Scoring Sampler– Find your grade level

– K‐2 go to the 3rd grade

• Go to Open Ended Question Section– Item specific scoring guideline

• Review Questions and Responses

Task Scaffold

• Task Scaffolding (Most Common)

– 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

Task Scaffold

• CUBES is a form of a Task scaffold

– It prompts students to work through a series of specified steps or tasks while solving word problems

Two TASK Scaffolds Identified• RACE (ELA)

– Reword/Restate the questions

– Provide and Answer

– Cite using Evidence from the text

– Explain how the evidence supports your answer

• ACE (Math)

– Already know (highlight key terms, identify what you already know)

– Compute your work (show your work, label, draw a model)

– Explain in writing how got your answer (step by step details, Mathematical terminology used‐how and why?)

(Bennett, 2017)

ACE Application HO #8Solve this problem by applying ACE Task Scaffold

Jennifer wants to take piano lessons that cost $15 each. She plans to take 10 lessons, for which she has $85 saved. How much more money does she need in order to pay for the lessons?

ACE Implementation• ACE is a Task Scaffold:

– 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

– Distributed opportunities to practice using the steps is critical

– Continual revisiting of the meaning and parts of the strategy

– Continuous self‐monitoring and evaluating the writing based on the strategy

ACE Planning and Implementation

• Considerations

1. Does ACE match/correspond to the rubric used on the LA assessments

• Does it make sense?

2. If No, Boost ACE so it better aligns to the rubric used in LA

3. Discuss ideas for teaching ACE to your students as it applies to Math open construction responses.

4. Plan to provide a Task Scaffolded Open Construction Opportunity a minimum of two times per month?

5. Share out ideas

ACE Task Scaffold

Prompted Writing

Challenges• Writing in mathematics is a complex task that includes:– Comprehension– Mathematical understanding– Spelling– Vocabulary– Fluency and proficiency with reading:

• Numbers• Symbols, and• WORDS‐‐vocabulary

Prompted Writing

• 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

Prompted Writing

• 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)

Prompted Writing (HO#9)

• Write about the term FRACTION using the following terms: Number, whole, numerator, and denominator

• Describe the term CIRCLE using the terms: chord, diameter, radius, center

Summary Math Vocabulary

• Understanding of essential mathematical vocabulary is essential for students to become proficient – Facilitates writing

• Some students will struggle with learning vocabulary; therefore, teachers must use more intensive vocabulary instruction and activities

Summary• The learning needs of struggling students and students with disabilities in mathematics is extremely challenging for teachers.

• Instructional Strategies and Techniques1. Content Scaffolding Progression (CSP)

2. Spaced Learning Over Time (SLOT)

3. Interleaving Practice Forma (IFP)

4. Practice Test and Retrieval (RTP)

5. Fraction and the Number line

6. Supported Open Construction Response • ACE and Prompted Writing

• Frayer Model for Academic Vocabulary

pjr146@psu.edu

maximize students’ mathematical thinking january 22, 2018

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