6-8 Interventionists Training Session 2

September 25, 2013

Today’s Learning Targets

We are learning to …• Understand the meaning and importance of

“explicit and systematic instruction.”• Deepen our understanding of the Number and

Operations: Fractions Domain

Success Criteria

We will be successful when we can…• Identify what “systematic and explicit

instruction” looks like during PIE time.• Clearly explain the mathematical content in

selected Grade 3-5 CCSSM standards and be able to provide examples of the mathematics.

A Research Based Approach to Intervention

Emerging Key Research Themes• Increased instructional time in addition to core

mathematics taught in Tier 1.• Small-group instruction utilized in all tiers• Explicit methods of instruction (e.g., CRA, Talk Moves)• Use of concrete and pictorial representations to facilitate

conceptual understanding• Strategy instruction for problem solving (e.g., Think Aloud)• Focus on problem solving skills (not just computation)• Careful alignment of instruction and content in Tier 1 and

Tier 2 • Screening and progress monitoring to target deficit areasSource: Adapted from Newman-Gonchar, R., Clarke, B., & Gersten r. (2009). A summary of nine key studies: Multitier intervention

and response to interventions for students struggling in mathematics.Retrieved from www.centeroninstruction.com

IES Practice Guide: Assisting Students Struggling with Mathematics: Response to

Intervention for Elementary and Middle School Students

• US Department of Education• Research-based education practices

Committee Chair: Russell Gersten

Published by:What Works Clearinghouse (April 2009)

Explicit and Systematic Instruction

What is it?How might it look with my PIE group?

Explicit and Systematic Instruction

Read and highlight Recommendation 3

With your group come to consensus on 3-4 important points you would like to share with the group.

Explicit & Systematic Instruction

Summarize key aspects of “explicit and systematic instruction” as defined by this reading.

What do we need to know as teachers to do this well in our intervention groups?

A structure for explicit and systematic instruction

Symbols

Give a context: tell a story

Explain orally and/or in writing

Make a picture

Use concrete models:manipulatives

Laying the FoundationInitial Fraction Concepts

Supporting the Transition From the Whole Number World

Read the grade level focus for 3rd through 5th Grade

•3rd Grade: Read 3rd Grade Critical Area #2 p. 21 Review 3rd grade Cluster Statements p. 24•4th Grade: Read 4th Grade Critical Area #2 p. 30-31 Review 4th Grade Cluster Statements p.48•5th Grade: Read 5th Grade Critical Area #1 p. 33 Review 5th Grade Cluster Statements p. 36-37

Be prepared to summarize the Standards’ progression of Number and Operations: Fractions.

3rd through 5th Progression

Where does the difficulty with fractions begin? Research says…

• Premature experience with formal procedures may lead to symbolic knowledge that is not based on understanding impeding students’ number and operation sense.

• Some students “have a continuing interference from their knowledge of whole numbers.”

• Difficulty reasoning with fraction symbols as quantities.• Knowledge that is dependent primarily upon memory, rather

than anchored with a deeper understanding of foundational concepts, contributes to incorrect use and misunderstanding of formal algorithms.

---A Focus on Fractions

Learning Target:Deepen our understanding of the Number and Operations: Fractions domain

Examine fractions as numbers using modelsDeepen understanding of partitioningUnderstand and use unit fraction reasoningRead and interpret the cluster of CCSS standards related to fraction concept development.

So let’s step back into the elementary world for a minute….

Fractions as Numbers

3 4

What are ways we want students to “see” and “think about” fractions?

CCSSM 1.G.3 and 2.G.3

How does fraction work begin in Grade 1 and Grade 2?

Examining Partitioning

....“early experiences with physically partitioning objects or sets of objects may be as important to a child’s development of fraction concepts as counting is to their development of whole number concepts” (Behr and Post, 1992)

Stages of Partitioning

Read pgs. 71 – 75 (through stages of partitioning)

Highlight key phrases from the reading

* Star the important ideas

? Question mark the confusing thoughts

Table group discussion summarizing and clarifying thoughts

Work through problem number 1 on pages 77 & 78

Whiteboard Work

Take a slate and divide it in thirds

On each 1/3 draw a model; a area model, a set model, and a number line.

Discuss the features of each model.

Features of Models

What is the whole?

How are equal parts defined?

What does the fraction indicate?

Area Model

The whole is determined by the area of a defined region

Equal areaThe part covered of whole unit area

Set Model

The whole is determined by definition (of what is in the set)

Equal number of objects

The count of objects in the subset of the defined set of objects.

Number LineUnit of distance or length (continuous)

Equal distance

The location of a point in relation to the distance from zero with regard to the defined unit.

Fractions Composed of Unit Fractions

• Fold your fraction strip to show ¾

• How do you see this fraction as

‘unit fractions’?

Making Fraction Strips

White: whole

Green: halves, fourths, eighths

Yellow: thirds, sixths, ninths

?: twelfths

Note relationships among

the fractions as you fold.

Remember – no labels.

Looking at a Whole

• Arrange the open fraction strips in front of you.• Look at the thirds strip. How do you see the

number 1 on this strip using unit fractions?• In pairs, practice stating the relationship between

the whole and the number of unit fractions in that whole (e.g., 3/3 is three parts of size 1/3).

CCSSM 3.NF.1

• Understand a fraction1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

• How do you make sense of the language in this standard connected to the previous activity?

Extension of Unit Fraction Reasoning

Jason hiked 3/7 of the way around Devil’s Lake. Jenny hiked 3/5 of the way around the lake. Who hiked the farthest?

Use fraction strips and reasoning to explain your answer to this question.

Extension 2

Jim and Sarah each have a garden. The gardens are the same size. 5/6 of Jim’s garden is planted with corn. 7/8 of Sarah’s garden is planted with corn. Who has planted more corn in their garden?

Use fraction strips and reasoning to explain your answer to this question.

Whiteboard Work

• write on your whiteboard the reasoning that you used to explain your answer.

• Be sure your reasoning is connected to unit fractions and fraction strips.

Reflect

Think of the work you have done with your PIE group up to this point…

• How will this knowledge help you be more explicit with your instruction?

• Where might you step in with explicit instruction?

A structure for explicit and systematic instruction

Symbols

Give a context: tell a story

Explain orally and/or in writing

Make a picture

Use concrete models:manipulatives

Planning for Instruction

What is instructional supports are avaialble?

• MTSD RtI Math Resources• Conceptua Fractions• Targeted Problem Solving Tasks

– Howard County Math Wikis

MTSD RtI Math Resources

http://www.mtsd.k12.wi.us/schools/staffaccess.cfm

• General Documents• Grades 3-7 Intervention Guides Organized by

Standard• Conceptua “Homework” – Paper and Pencil

Tasks

Conceptua Fractions• Opener• Guided Practice & Skills Check• Closer• Remediation Lessons• Tool Investigations• Teacher & Student Dashboard

LOVE the visual models!LOVE the opportunity for translational teaching!NEED context for Big Ideas 1-7

Resources to Support ContextWeb Resource #1: Learn ZillionA good website to support using visual models to add and subtract fractions:http://learnzillion.com/lessons/1051-use-a-model-to-solve-word-problems-involving-addition-of-fractions-with-unlike-denominatorsLessons 3 through 8 offer 3-4 minute “mini-lessons” showing how to solve addition and subtraction of fraction story problems using different models including number lines and tape diagrams. The links are found on the left side of the screen.

Web Resource #2: Thinking Blocks – will give you story problems and modeling toolsAccessing Thinking Blocks:http://www.mathplayground.com/NewThinkingBlocks/thinking_blocks_fractions.html

• • • • • •

Click on either “Adding Fractions with Like Denominators• • • • • • •

Illustrative Mathematics

http://www.illustrativemathematics.org/

Howard County Math Wikis

One resource for problem solving tasks https://grade1commoncoremath.wikispaces.hcpss.org/Grade+1+Home