improving mathematical problem solving in grades 4 through 8

Improving Mathematical Problem Solving in Grades 4

through 8

REL Appalachia Bridge EventNovember 6, 2013

Louisville, KY

Welcome & Overview

Dr. Justin Baer

Director, REL Appalachia

CNA

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What is a REL?

• A REL is a Regional Educational Laboratory. • There are 10 RELs across the country.• They are administered by the U.S. Department of

Education, Institute of Education Sciences (IES).• A REL serves the education needs of a designated

region.• It works in partnership with the region’s school

districts, state departments of education, and others to use data and research to improve academic outcomes for students.

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What is a REL?

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REL Appalachia’s Mission• Meet the applied research and technical assistance needs

of Kentucky, Tennessee, Virginia, and West Virginia.• Conduct empirical research and analysis.• Bring evidence-based information to policy makers and

practitioners: – 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.

www.relappalachia.org

http://www.relappalachia.org/

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Welcome• Thanks to our partners:

– Kentucky Association of School Administrators

– Kentucky Center for Mathematics

– Kentucky Department of Education

• Please follow @REL_Appalachia on Twitter, and use the hashtag #RELAP when tweeting about this event!

• Make sure to pick up your certificate of attendance before you leave at the end of the day!

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Agenda 9:40–10:30 Introduction to the Practice Guide

10:30–11:00 Teaching Mathematical Problem Solving

11:00–11:10 BREAK

11:10–12:00 Identifying Challenges: Table Discussion and Group Work

12:00–12:20 Lunch & Keynote Address

12:20–12:50 Lunch & Networking

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Agenda12:50–1:40 Applying the Practice Guide’s Five Key

Recommendations: Breakout Sessions (Block 1)

1:40–1:50 BREAK

1:50–2:40 Applying the Practice Guide’s Five Key Recommendations: Breakout Sessions (Block 2)

2:40–2:45 BREAK

2:45–3:15 Next Steps in Improving Students’ Problem Solving / Taking It Back to the Classroom

3:15–3:30 Closing Remarks

3:30–3:45 Stakeholder Feedback Survey

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Goals• Build participants’ knowledge of the importance of math

problem solving in grades 4 through 8. (Why does it matter so much?)

• Engage participants in a meaningful and practical discussion of the evidence-based research in the Practice Guide. (How do we know what works?)

• Provide participants with evidence-based strategies that can be implemented in their schools. (How can we use this material in our classrooms to help students?)

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Goals• Plan for making use of the research-based practices to

improve instruction, student achievement, and students’ preparation for more rigorous mathematics courses in middle and high school.

• Facilitate networking/idea exchanges among participants, especially around improving mathematics instruction.

• Provide participants with materials that can inform their efforts in improving students’ mathematical problem-solving ability.

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Introduction to the Practice Guide

“Improving Mathematical Problem Solving in Grades 4 through 8”

Dr. John WoodwardDean, School of EducationUniversity of Puget Sound

Where Can I Find This Guide?

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• http://ies.ed.gov/ncee/wwc/PracticeGuide.aspx?sid=16• Doing What Works website:

http://dww.ed.gov/Math-Problem-Solving/topic/index.cfm?T_ID=41

http://ies.ed.gov/ncee/wwc/PracticeGuide.aspx?sid=16



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Practice guides provide practical research-based recommendations for educators to help them address the everyday challenges they face in their classrooms and schools.

• Practice guides include:• Concrete how-to steps• Rating of strength of evidence• Solutions for common roadblocks

• The What Works Clearinghouse currently has 14 Practice Guides

What Are Practice Guides?

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Evidence Rating

Each recommendation receives a rating based on the strength of the research evidence.

Strong: high internal and external validity

Moderate: high on internal or external validity (but not necessarily both) or research is in some way out of scope

Minimal: lack of moderate or strong evidence, may be weak or contradictory evidence of effects, panel/expert opinion leads to the inclusion in the guide

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Evidence RatingRecommendation Level of Evidence

1. Prepare problems and use them in whole-class instruction.

Minimal

2. 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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• One definition of problem solving– Common agreement:

• Relative to the individual• No clear solution immediately (it’s not

routine)• It’s strategic

– Varied frameworks• Cognitive: emphasizing self-monitoring• Social Constructivism: emphasizing

community and discussions

Challenging Issues for the Panel

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• How much time should be devoted to problem solving (per day/week/month)– It’s not a “once in a while” activity– Curriculum does matter– Sometimes it’s a simple change

• 4 + 6 + 1 + 2 + 9 + 8 averages to 5. What are 6 other numbers that average to 5?


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• A script or set of steps describing the problem-solving process.– What we want to avoid:

• Read the problem.• Select a strategy (e.g., draw a picture).• Execute the strategy.• Evaluate your answer.• Go to the next problem.


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• The balance between teacher-guided/modeled problem solving and student-generated methods for problem solving.– Teachers can think out loud, model, and prompt.– Teachers can also mediate discussions, select and

re-voice student strategies/solutions.


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Recommendation 1

• Prepare problems and use them in whole-class instruction. Include both routine and non-routine problems

in problem-solving activities.

What are your goals?

Greater competence on word problems with operations?

Developing strategic skills?

Persistence?

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• There are many kinds of problems.– Word problems related to operations or topics:

• I have 45 cubes. I have 15 more cubes than Darren. How many cubes does Darren have?

– Geometry/measurement problems.– Logic problems, puzzles, visual problems.

How many squares on a checkerboard?

Recommendation 1

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• Prepare problems and use them in whole-class instruction.

Ensure that students will understand the problem by addressing issues students might encounter with the problem’s context or language.

Linguistic issues are a barrier.

Cultural background is a big factor.

Recommendation 1

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Ensure That Students Will Understand the Problem

yacht? slip? harbor?

A yacht sails at 5 miles per hour with no current. It sails at 8 miles per hour with the current. The yacht sailed for 2 hours without the current and 3 hours with the current and then it pulled into its slip in the harbor. How far did it sail?

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• Prepare problems and use them in whole-class instruction.

Consider students’ knowledge of mathematical content when planning lessons.

Sometimes it’s appropriate to have students practice multiple problems in the initial phase of learning.

Concept of division, unit rate proportion problems.

Sometimes it is appropriate to have a more inquiry-oriented lesson with only one or two problems

Recommendation 1

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Assist students in monitoring and reflecting on the problem-solving process.

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.

Recommendation 2

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This is what we want to AVOID. Read the problem (and read it again).

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

Solve the problem.

Evaluate the problem.

Recommendation 2

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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?

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What are the various ways I might approach the problem?

Is my approach working? If I am stuck, is there another way I can think about solving this problem?

Does the solution make sense? How can I check the solution?

Why did these steps work or not work?

What would I do differently next time?

(continued)

Provide Prompts or Model Questions

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Teach students how to use visual representations.

Select visual representations that are appropriate for students and the problems they are solving.

Use think-alouds and discussions to teach students how to represent problems visually.

Show students how to convert the visually represented information into mathematical notation.

Recommendation 3

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Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with?

Cognitive Load: Problem Solving Through Words Alone

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Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with?

Draw a Picture?

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Problem Representation

Tape Diagrams vs. Pictures

Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater.

She had $150 remaining. How much did she start with?

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Tape Diagrams as a ToolEva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with?

She spent 2/5 of her money on a coat

She had 3/5 remaining after buying the coat

The remaining money. The 3/5 is now 3/3 or the new whole.

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Tape Diagrams as a ToolEva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with?

She spent 2/5 of her money on a coat

She had 3/5 remaining after buying the coat

She spent 1/3 of what was left on a sweater. This is the same as 1/5 of the original amount.

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Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with? She spent 2/5 of her

money on a coatShe spent 1/5 of her money on a sweater

She had 2/5 remaining after buying the coat and the

sweater. This portion is $150.

$150 = 2/5 of the money. That means 1/5 = $75

5 x 1/5 = 5/5, or the whole amount, so 5 x $75 = $375Eva started with $375.

Tape Diagrams as a Tool

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Recommendation 4

Expose students to multiple problem-solving strategies.

Provide instruction in multiple strategies.

Provide opportunities for students to compare multiple strategies in worked examples.

Ask students to generate and share multiple strategies for solving a problem.

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Recommendation 5

Help students recognize and articulate mathematical concepts and notation.

Describe relevant mathematical concepts and notation, and relate them to the problem-solving activity.

Ask students to explain each step used to solve a problem in a worked example.

Help students make sense of algebraic notation.

Practice Guide—Classroom Application

Ms. Karen CampbellDr. Tim Gott

Ms. Kelly Stidham

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Karen Campbell

Math Consultant

Kentucky Center for Mathematics

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Kentucky Core Academic Standards• Where do we stand in implementing

the new standards?• Is everyone on the same page?


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• Teachers know that it’s good practice to teach mathematical problem solving, but actually being able to implement this concept is another story.

• Teachers need to learn how to combine the topic and skills into deeper problem-solving units.

But how do you get from point A to point B?


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Teaching a problem versus completing a task:• Problem solving is less about the problem and more

about the solving. You have to teach the solving part.• Problem solving teaches us to have multiple options

for finding our way to a solution.


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• Teachers need to also teach analysis!• Teachers need to use a variety of visual

representations!• Teachers know what they need to do (for the most

part), but reinforcing these ideas is critical.• Connecting both sides of the brain…


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Math is a foreign language.


Tim GottDirector, The Carol Martin Gatton Academy of Mathematics

and Science in Kentucky

Bowling Green, Kentucky

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• Mathematics is more than multiplication tables and percentages.

• We know what problem solving looks like. It’s exploration, it’s using tools, it’s inquiry based.

• We must integrate problem solving into the mathematics curriculum.


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Problem solving is the conduit for making things relevant.


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Flow of Mathematical Concept Development*Experiential

Kinesthetic, Tactile, Hands-OnManipulatives, Experiments, Physical tasks, Field trips, Measurements, Demonstrations

PictorialVisual, Spatial

Pictures, Drawings, Graphs, DiagramsConcrete representations, Sketches

VerbalAuditory, Linguistic

Words, Critical vocabulary, Sentences, PhrasesBoth written and spoken

SymbolicAbstract, Generalized

Formulas, Equations, ExpressionsVariables, Symbols, Structures

* Courtesy of Dr. Tim Gott

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The more strategies a student has for finding the answer, the more that student can choose the right strategy.


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Math doesn’t have to be entirely symbolic. Problem

solving conceptualizes these equations into real-world,

tangible ideas.


Kelly Stidham

Math Strategist

Northern Kentucky Cooperative for Education

@kastidham #RELalgebra

[email protected]

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mailto:[email protected]

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Statewide shift to Kentucky Core Academic Standards• It’s exciting, but difficult.• Strategies are aligned with standards of effective

teaching, but the challenge comes in implementation.


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Changing Perspective:Memorization Strategy-based fluency


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How can teachers translate the standards from an outcome to the vehicle through which they are solved?


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Problems create critical thinking, but where can teachers find quality content?• Working together.• Sharing resources.


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What does it mean to be a “Problem Solver”?• Students need time to get deeper into the content,

strategies.• A number of students believe they “aren’t good at

math” – but problem solving can change that.• Redefine the role of the student, focus on personal

growth through the mathematical process.


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• Visual representations are an incredibly important part of learning how to do math.

• Being able to see mathematical concepts in different ways helps break down the procedures.


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The solution isn’t about just addressing these modalities (area and length x width visualizations pictured above), but much more about giving students the space to find meaning and understand how things have the same relationship.


Identifying ChallengesTable Discussion & Group Work

Lydotta TaylorREL Appalachia

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Identifying Challenges

Please assign a table leader who will report out to the larger group at the end of this session.

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Discussion Question

• What role do you play in your school?– Teacher.– Administrator (Principal, etc.).– Curriculum Coach.

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Discussion Question

• How do you use problem solving outside the classroom?

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Work with your tablemates through each question. At the end of this session, you will be asked to identify the top three challenges your table faces when teaching problem solving.

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• How does teaching problem solving fit into your mathematics curriculum?

• What obstacles do you face in bringing the recommendations in the Practice Guide to your classroom to help students?

• Do you feel you have time to dedicate to whole classroom instruction of problem solving? How can you spend more time on these areas?

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Identifying Challenges• What challenges do your students face when they

work through the context of specific problem-solving exercises?

• What barriers do you face in teaching your students how to represent problems visually?

• How can you work with your colleagues – other teachers, administrators, curriculum coaches – to strengthen your own teaching of problem solving in mathematics?

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Identifying Challenges• Where do you find the best mathematics resources?

Are they proven, or aligned with the research in the Practice Guide?

• Teachers: Does your administration support the professional development you need?

• Administrators: Do you support the professional development your teachers need?

• How can you use the recommendations from the Practice Guide to improve students’ problem solving?

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Discussion Question

• What are the top three challenges you’ve faced in teaching problem solving?

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Connecting the Dots: Responses from Content Experts

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LUNCHPlease grab a boxed lunch and return to your tables!

• Keynote Address

• Networking

Keynote Address

Ms. Felicia Cumings-SmithKentucky Department of Education

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A Statewide Perspective

Felicia-Cumings Smith

Associate Commissioner,Office of Next Generation Learners,Kentucky Department of Education

@KyDeptofEd

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NetworkingGet to know your tablemates!• Where do you work? What’s your role and your specific

challenges in teaching, school leadership, or other role in a school or district?

• Are you on Twitter? Follow @REL_Appalachia and share your own Twitter handle with your tablemates!

• Were you familiar with the Practice Guide before today?• What resources do you use to support mathematics instruction?

Can you share resources that align with the evidence-based recommendations in the Practice Guide?

• How do you plan to use the Practice Guide moving forward in your classroom or school?

• What is your favorite success story from teaching math?

Applying the Practice Guide’s Five Key

RecommendationsBreakout Sessions

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Applying the Recommendations

Please report to your breakout session (chosen during registration).

• Algebra, Kelly Stidham – Wilson Room.

• Fractions, Karen Campbell – McCreary Room.

• Statistics, Sue Peters – Taylor Room.

• Students Struggling with Mathematics, John Woodward – Jones Room.

Next Steps in Improving Student’s Problem Solving

Taking It Back to the Classroom

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Where Do We Go from Here?

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Discussion Question

• Who can be your ally to support your implementation of the recommendations from the Practice Guide?

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Discussion Question

• How can you involve parents to continue students’ learning at home?

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Discussion Question

• How can you use examples of problem solving in the classroom?

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Where Do We Go From Here?

We encourage you to set specific goals about how you will use the recommendations from the Practice Guide to improve problem solving in your classrooms and schools!

Closing Remarks &

Stakeholder Feedback Survey

Dr. Justin BaerDirector, REL Appalachia

CNA

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Closing Remarks• Thank you for spending your day with REL

Appalachia!• Join us on Wednesday, December 4, at 4 p.m. EST

for a follow-up webinar, Identifying and Assisting Students Struggling with Mathematics. For more, please visit http://www.relappalachia.org/news-events/events.

• Before you leave, please complete the Stakeholder Feedback Survey. This allows us to produce better future events, catering to your specific needs!

http://www.relappalachia.org/news-events/events

http://www.relappalachia.org/news-events/events

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Stakeholder Feedback Survey

In return for your completed Stakeholder Feedback Survey, we will give you a certificate of attendance from the Kentucky Department of Education, demonstrating that you have earned professional development credit.

Thank you!

improving mathematical problem solving in grades 4 through 8

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