Webinar:Improving Mathematical Problem

Solving in Grades 4 through 8

September 30, 2014

WebEx Instructions

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WebEx instructions

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WebEx instructions

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WebEx instructions

Attendees should utilize the “Q&A” feature to pose questions to the speaker, panelists, and/or host.

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panelists until the end, when they will be answered during a Q&A session.

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Welcome and Overview

Lydotta Taylor, Ed.D.Research Alliance Lead, REL Appalachia

Regional Educational Laboratory (REL) Program

• U.S. Department of Education, Institute of Education Sciences.• RELs provide regional support for:

– Applied research and evaluation.

– Technical support and information sharing to build capacity to use data for improved education outcomes.

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REL Appalachia’s mission

• Meet the applied research and technical support needs of Kentucky, Tennessee, Virginia, and West Virginia.

• Bring evidence-based information to policymakers and practitioners:– Provide support for a more evidence-reliant education system.

– Inform policy and practice for states, districts, schools, and other stakeholders.

– Focus on high-priority, discrete issues and build a body of knowledge over time.

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How we work: Research alliances

• What is a research alliance?– A partnership between education stakeholders and REL Appalachia.

• What is the purpose of a research alliance?– As partners, REL Appalachia and alliance members develop and carry out a

research and analytic technical assistance agenda on priority topics.

• Who are the education stakeholders in an alliance?– May include representatives from one or more schools, divisions, state

education agencies, and other organizations (e.g., colleges and universities).

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Goals of today’s webinar

• Understand the importance of incorporating mathematical problem solving in daily instruction.

• Learn about the links between the NCTM’s 5 process standards, the Common Core Standards for Mathematical Practices, and the Virginia Standards of Learning.

• Describe the research base for the standards and illustrate how they can be put into practice in the middle grades.

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Connect with us!

www.relappalachia.org

@REL_Appalachia

Lydotta TaylorLydotta@levation.com

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Improving Mathematical Problem Solving in Grades 4 through 8

John Woodward, Ph.D.Dean, School of EducationUniversity of Puget Sound

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Why is problem solving important for today’s math classroom?

Why problem solving is important

• Problem solving has tended to play a minor role in most math classes.– Independent seatwork.

– Homework.

– “End of chapter” problem sets.

• Problem solving is one of the defining features of math instruction in successful countries.– Singapore makes problem solving central to its most recent curriculum.

• Data, technology, and the world of work.

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Links between the NCTM Process Strands, the CCSS Mathematical Practice Standards, and the Virginia

Mathematics Standards of Learning

Links between NCTM, CCSS, and the Virginia SOLs

• National Council of Teachers of Mathematics (NCTM) Process Strands• Common Core State Standards (CCSS)• Virginia Mathematics Standards of Learning (SOL)

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Making sense of different standards for problem solving

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5 strands with common titles

NCTM PROCESS STRANDSVIRGINIA MATHEMATICSSTANDARDS OF LEARNING

PROBLEM SOLVING MATHEMATICAL PROBLEM SOLVING

REASONING AND PROOF MATHEMATICAL REASONING

REPRESENTATIONS MATHEMATICAL REPRESENTATIONS

COMMUNICATION MATHEMATICAL COMMUNICATION

CONNECTIONS MATHEMATICAL CONNECTIONS

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5 strands with common titles

NCTM PROCESS STRANDSVIRGINIA MATHEMATICSSTANDARDS OF LEARNING

PROBLEM SOLVING• “persistence”• “strategies”

MATHEMATICAL PROBLEM SOLVING• “strategies”

REASONING AND PROOF• “conjectures”• “evaluating arguments”• “justifying results”

MATHEMATICAL REASONING• “inductive and deductive

reasoning”• “use logical reasoning to analyze

arguments”

REPRESENTATIONS• “use a variety of methods”

MATHEMATICAL REPRESENTATIONS• “use a variety of methods”

Continued20

5 strands with common titles

NCTM PROCESS STRANDSVIRGINIA MATHEMATICSSTANDARDS OF LEARNING

COMMUNICATION• “reflect, refine, discuss ideas”• “clear, convincing, and precise”

MATHEMATICAL COMMUNICATION• “use the language of mathematics,

including specialized vocabulary and symbols”

• “express precisely”

CONNECTIONS• “mathematics is an integrated

field”

MATHEMATICAL CONNECTIONS• “mathematics is an integrated

field”• “make connections to science”

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Foundation for CCSS-M Mathematical Practice Standards

• “These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education.”—Introduction to the CCSS Mathematical Practice Standards

• The CCSS Mathematical Practice Standards draw from:– NCTM’s Process Strands.– National Research Council’s Adding It Up: Helping Children Learn

Mathematics (2001).

Sources: CCSS Mathematical Practice Standards,

http://www.corestandards.org Adding It Up (2001), http://www.nap.edu

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The 8 CCSS-M Mathematical Practice Standards

1. Make sense of 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.

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Virginia SOL and CCSS Standards

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

Mathematical Problem Solving 1. Make sense of problems and persevere in solving them.

5. Use appropriate tools strategically.

Mathematical Reasoning 2. Reason abstractly and quantitatively. 3. Critique the reasoning of others. 8. Look for and express regularity in

repeated reasoning.

Continued

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Virginia SOL and CCSS Standards

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

Mathematical Representations 4. Model with mathematics.

Mathematical Communication 3. Construct viable arguments.

Mathematical Connections 6. Attend to precision. 7. Look for and make use of structure

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From research to application: Putting the standards into practice in the middle grades

What kind of empirical support for these practices?The Problem Solving Guide

(PSG)

Source: Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., Koedinger, K. R., & Ogbuehi, P. (2012). Improving mathematical problem solving in grades 4 through 8: A practice guide.

The RtI Guide (RtI)

Source: Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools.

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PSG: Relevant recommendations and evidence

Recommendation Level of Scientific Evidence2. Assist students in monitoring and reflecting

on the problem-solving process. Strong

3. Teach students how to use visual representations.

Strong

4. Expose students to multiple problem-solving strategies.

Moderate

5. Help students recognize and articulate mathematical concepts and notation.

Moderate

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RtI: Relevant recommendations and evidence

Recommendation Level of Scientific Evidence4. Interventions should include instruction on

solving word problems that is based on common underlying structures.

Strong

5. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.

Moderate

Recommendations

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

PSG RECOMMENDATIONS(IES)

Mathematical Problem Solving

1. Make sense of problems and persevere in solving them.

Rec 2 – Assist students in monitoring and reflecting.

Mathematical Reasoning

2. Reason abstractly and quantitatively.

Rec 5 – Help students recognize and articulate mathematical concepts and notation.

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Continued

Recommendations

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

PSG RECOMMENDATIONS(IES)

Mathematical Communication

3. Construct viable arguments and critique the reasoning of others.

Rec 4 – Expose students to multiple problem solving strategies.

Mathematical Representations

4. Model with mathematics.

Rec 3 – Teach students to use visual representations.

Mathematical Problem Solving

5. Use appropriate tools strategically.

Continued

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Recommendations

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

PSG RECOMMENDATIONS (IES)

Mathematical Connections

6. Attend to precision. Rec 5 – Help students recognize and articulate mathematical concepts and notation.

Mathematical Connections

7. Look for and make use of structure.

Mathematical Reasoning

8. Look for and express regularity in repeated reasoning.

Rec 5 – Help students recognize and articulate mathematical concepts and notation.

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Recommendations

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

RTIRECOMMENDATIONS (IES)

Mathematical Reasoning

2. Reason abstractly and quantitatively.

Rec 4 – Solve word problems based on common underlying structure.

Mathematical Representations

4. Model with mathematics.

Rec 5 – Use visual representations.

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Virginia SOL and CCSS Standards

Example 1

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

PSG RECOMMENDATIONS(IES)

Mathematical Problem Solving

1. Make sense of problems and persevere in solving them.

Rec 2 – Assist students in monitoring and reflecting.

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PSG Recommendation 2

• Provide students with a list of prompts to help them monitor and reflect during the problem-solving process.

• Model how to monitor and reflect on the problem-solving process.

• Use student thinking about a problem to develop students’ ability to monitor and reflect.

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PSG Recommendation 2

This is what we want teachers to AVOID:

– Read the problem (and read it again).

– Find a strategy (usually “make a drawing”).

– Solve the problem.

– Evaluate the problem.

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PSG Recommendation 2

Instead, provide prompts or model questions:

– What is the story in this problem about?

– What is the problem asking?

– What do I know about the problem so far?

– What information is given to me?

– How can this help me?

– Which information in the problem is relevant?

– Is this problem similar to problems I have previously solved?

– Why do (or did) these steps work or not work?

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Virginia SOL and CCSS Standards

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

RTIRECOMMENDATIONS(IES)

Mathematical Reasoning

2. Reason abstractly and quantitatively.

Rec 4 – Solve word problems based on common underlying structure.

Example 2

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RtI Recommendation 4

• Problem: “It is summer and my job is to weed the garden. Last Friday I counted 45 weeds in the garden. I didn’t do any work on the weekend. Now it is Wednesday, and there are 72 weeds in the garden. How many more weeds grew in the garden?”

• Type of problem: Subtraction Compare Problem• Other types of subtraction problems:

– Separate (“I had 10 balloons, and I gave 4 to my sister ...”)

– Part-Part-Whole (“Sarah has 12 coins; 3 are nickels, 4 are dimes, the rest are

pennies. How many …”)

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Virginia SOL and CCSS Standards

Example 3

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VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

PSG/RTI RECOMMENDATIONS(IES)

Mathematical Representations

4. Model with mathematics.

(PSG) Rec 3 – Teach students to use visual representations.

(RTI) Rec 5 – Use visual representations.

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PSG Recommendation 3, RtI Recommendation 5

Simple ratio and proportion problems are just that.• Problem: “Apples are 3 for $2. How much would you pay for 12 apples?”

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PSG Recommendation 3, RtI Recommendation 5

• Problem: “Apples are 5 for $2. How much would you pay for 9 apples?”

It costs $3.60 for 9 apples.

Virginia SOL and CCSS Standards

Example 4

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

PSG RECOMMENDATIONS(IES)

Mathematical Communication

3. Construct viable arguments and critique the reasoning of others.

Rec 4 – Expose students to multiple problem solving strategies.

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PSG Recommendation 4

• Problem: “Find the measure of angle x.”

PSG Recommendation 4

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PSG Recommendation 4

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PSG Recommendation 4

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PSG Recommendation 4

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PSG Recommendation 4

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PSG Recommendation 4

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PSG Recommendation 4

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Virginia SOL and CCSS Standards

Example 5

VIRGINIA MATHEMATICS STANDARDS OF LEARNING

CCSS MATHEMATICAL PRACTICE STANDARDS

PSG RECOMMENDATIONS(IES)

Mathematical Reasoning

2. Reason abstractly and quantitatively.

Rec 5 – Help students recognize and articulate mathematical concepts and notation.

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PSG Recommendation 5

• Problem: “It is a very cold day at the North Pole. It started at ‒17 degrees. Later in the day it was 13 degrees. How much did the temperature change?”

• Visual representations (PSG Rec 3 and RtI Rec 5) help.

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PSG Recommendation 5

• Problem: “It is a very cold day at the North Pole. It started at -17 degrees. Later in the day it was 13 degrees. How much did the temperature change?”

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PSG Recommendation 5

• Problem: “It is a very cold day at the North Pole. It started at -17 degrees. Later in the day it was 13 degrees. How much did the temperature change?”

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Final thoughts

The IES problem solving guide cannot tell teachers “how long” or “how many times per week.”• Some curricula are easier to adapt than others.• Sometimes problem solving is a slight change in the task.

– “Find the average of 6, 8, 4, 9, 5, and 10.” (routine)

– “Find the same average with six numbers where one number is 14.” (less routine)

• Instructional planning is key: Make sure it is there and it happens.

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Q&A Session

Closing Remarks

Lydotta Taylor, Ed.D.Research Alliance Lead

REL Appalachia

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