6-8 Interventionists Training Session 3 November 12, 2013

Today’s Learning Target & Success Criteria We are learning to … Understand the meaning and importance of “explicit and systematic instruction.” We will be successful when we can… Identify what “systematic and explicit instruction” looks like and sounds like during small group intervention for fraction equivalence.

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., C  R  A, 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 areas Source: 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

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 Look back at your notes from our last meeting and gather 2 ideas to share regarding how you might explain “explicit and systematic instruction.” Jot those ideas down on a notecard.

Engaging tasks and clear problem-solving models (e.g., tape diagrams) Time for students to think Teacher modeling followed by guided and independent practice using carefully orchestrated examples and sequences of examples. Corrective feedback as needed Opportunities for students to participate and hear teachers thinking aloud Concrete objects to understand abstract representations and notation (C  R  A) Explicit and Systematic Instruction

Hiking Around The Lake Jason hiked 3/4 of the way around Devil’s Lake. Jenny hiked 3/5 of the way around the lake. Who hiked the farthest? Solve this task as a student in your intervention group might.

Hiking Around The Lake What’s the mathematics targeted in this task? Move the task through the 5 modes of representation. In what ways do these translations from one representation to the next surface the mathematics? How might we make the connections between representations explicit?

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

The Garden 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? Work this through with a partner. How might you approach this in your intervention group “systematically and explicitly.”

Building on An Understanding of Fraction as Number Fraction Equivalence

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. ---Petit, M., Laird, R., & Mardsen, E. (2010). A Focus on Fractions: Bringing Research into the Classroom. New York, NY: Routledge.

The Importance of Fraction As Number “Students who really do not understand what a fraction means will have a hard time finding another fraction equivalent to it.” (Bezuk & Bieck, 1993, p. 123)

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.

Place the whole fraction strip that represents 0 to 1 on a sheet of paper. Draw a line labeling 0 and 1. Lay out your fraction strips, one at a time, and make a tally mark on the line you drew. Write the fractions below the tally mark. Look for patterns to help you decide if two fractions are equivalent. Equivalency

Which fractions are equivalent? How do you know?

Understanding the Concept: Equivalent Fractions Two properties are central to the equivalence of fractions Saying that two fractions are equivalent is saying that the two fractions are different names (symbols) for the same number. There are an infinite number of different names for a give fraction. ---Petit, M., Laird, R., & Mardsen, E. (2010). A Focus on Fractions: Bringing Research into the Classroom. New York, NY: Routledge.

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

How might your intervention students approach this task? There are some candies in a dish. 2/5 of the candies are chocolate 3/10 of the candies are peppermint Are there more chocolate candies or more peppermint candies? Move this task through the 5 Modes of Representation.

What does this model highlight about this student’s understanding about fractions? Zero in on your targeted skill of equivalence: What questions might you ask this student to help her see the relationship between fifths and tenths?  Looking at Figure 1 and Figure 2 what relationships is this student ready to see that will help you hit the target of equivalence? Figure 1 Figure 2 ---Petit, M., Laird, R., & Mardsen, E. (2010). A Focus on Fractions: Bringing Research into the Classroom. New York, NY: Routledge.

How can the patterns students find in visual models lead to an efficient procedure for finding equivalent fractions? “Researchers suggest developing the connections between the concept and the procedure through interaction with models and manipulatives. Using the models and manipulatives helps to reveal patterns and relationships built on an awareness of the connections between the size and number of equal parts in a whole.” p. 136 A Focus on Fractions

 ---Petit, M., Laird, R., & Mardsen, E. (2010). A Focus on Fractions: Bringing Research into the Classroom. New York, NY: Routledge.

Missing Number Equivalencies Van de Walle, p. 223 Activity Find the missing number. Explain your solution. 1 st : Use a visual fraction model. 2 nd : Use numbers. 3 rd : Describe how the numbers apply to the models using accurate math language. Practice the language from the previous activity.

NCTM Illuminations Fraction Tracks & Equivalent Fractions Give these a try with your partner.

Systematically Moving Toward Fluency Researchers have found that instruction which helps students to move flexibly between representations (spoken and written words (two- fifths), pictorial representations, manipulatives, contexts, and symbols) and within representations (e.g., ¾ =6/8) will help students move toward equivalence reasoning that becomes free of the need to model. (Post, Wachsmuth, Lesh, & Behr, 1985).

Understanding equivalence and have an efficient procedure to find equivalent fractions are critical to success with problems involving comparing, ordering, and operating with fractions. ---Petit, M., Laird, R., & Mardsen, E. (2010). A Focus on Fractions: Bringing Research into the Classroom. New York, NY: Routledge.

Read these standards. With your partner agree on 2-3 big ideas that these standards address. How were they present in our work today? Standards 3NF3a-d, 4NF1 & 4NF2

Lunch!! 11:30- 12:30

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 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

Resources to Support Context Web Resource #1: Learn Zillion A good website to support using visual models to add and subtract fractions: addition-of-fractions-with-unlike-denominators Lessons 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 tools Accessing Thinking Blocks: Click on either “Adding Fractions with Like Denominators

Illustrative Mathematics

Howard County Math Wikis One resource for problem solving tasks ss.org/Grade+1+Home ss.org/Grade+1+Home