Mathematical Problem Solving in Grades 4 to 8: A Practice Guide John Woodward Dean, School of Education University of Puget Sound

What is This? Mathematical Problem Solving Skills Concepts Processes Metacognition Attitudes Singapore’s Mathematics Curriculum Framework

Improving Mathematical Problem Solving in Grades 4 Through 8 Panelists John Woodward (Chair; University of Puget Sound) Sybilla Beckmann (University of Georgia) Mark Driscoll (Education Development Center) Megan Franke (University of California, Los Angeles ) Patricia Herzig (Math Consultant) Asha Jitendra (University of Minnesota) Ken Koedinger (Carnegie Mellon University) Philip Ogbuehi (Los Angeles Unified School District)

Where Can I Find This Guide? Or Google: IES Practice Guides Problem Solving

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 What are Practice Guides? Fourteen practice guides currently exist on the WWC Web site.

Structure of the Practice Guide Recommendations  Levels of evidence  How to carry out the recommendations  Potential roadblocks & suggestions Technical Appendix

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

Recommendations and Evidence Ratings for the 5 Recommendations in the Guide

 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

 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 averages to 5. What are 6 other numbers that average to 5? Challenging Issues for the Panel

 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 Evaluation your answer Go to the next problem Challenging Issues for the Panel

 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 Challenging Issues for the Panel

 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? Recommendation 1

 This one is very significant for struggling students. –We need to have a clear purpose for problem solving –We need to determine how long we devote to problem solving (and what support is needed) –We need to modify the content and language of many problems Recommendation 1

 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 Recommendation 1 How many squares on a checkerboard?

Non-Routine Problems* Determine angle x without measuring. Explain your reasoning. *“non-routine” is “relative to the learner’s knowledge and experience

 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

Yacht? Slip? Harbor? Ensure that Students Will Understand the Problem 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?

Revised Problem for Struggling Students A boat sails at 5 miles per hour with no current. It sails at 8 miles per hour with the current. If the boat sailed for 2 hours with no current and 3 hours with the current, how far did it travel? Jasmine walks 4 miles per hour. She runs 7 miles per hour. If Jasmine walked for 2 hours and ran for 1 hour, how far did she go? OR

 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 1 or 2 problems Recommendation 1

Recommendation 2  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  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

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?

 What are the various ways I might approach the problem?  Is my approach working? If I am stuck, is there another way 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? Provide Prompts or Model Questions (continued)

Recommendation 3  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.

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

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?

Problem Representation Schematic 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?

Strip Diagrams as a Tool 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 coat She had 3/5 remaining after buying the coat The remaining money. The 3/5 is now 3/3 or the new whole.

Strip Diagrams as a Tool 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 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.

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 coat She spent 1/5 of her money on a sweater She had 2/5 remaining after buying the coat & 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 = $375 Eva started with $375 Strip Diagrams as a Tool (continued)

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.

You Saw This Problem Earlier Determine angle x without measuring. Explain your reasoning. Can you think of multiple solutions to this problem?

What Is the Measure of Angle X? 155° x° 110° 70°155° 95° 25° 85°

What Is the Measure of Angle X? 155° x° 110° 65° 95° °

What Is the Measure of Angle X? 155° x° 110° 25° 95° ° 65° 70° 20°

What Is the Measure of Angle X? 155° x° 110° 155° 95° 25° 70°

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.

How Many Squares on a Checkerboard? 2 x 2 squares

2 x 2 squares How Many Squares on a Checkerboard?

2 x 2 squares How Many Squares on a Checkerboard?

2 x 2 squares How Many Squares on a Checkerboard?

2 x 2 squares How Many Squares on a Checkerboard?

3 x 3 squares How Many Squares on a Checkerboard?

3 x 3 squares How Many Squares on a Checkerboard?

3 x 3 squares How Many Squares on a Checkerboard?

7 x 7 squares How Many Squares on a Checkerboard?

7 x 7 squares How Many Squares on a Checkerboard?

7 x 7 squares How Many Squares on a Checkerboard?

7 x 7 squares How Many Squares on a Checkerboard?

Size of squares Total 1 x 1 (8 2 ) 64 2 x 2 (7 2 ) 49 3 x 3 (6 2 ) 36 4 x 4 (5 2 ) 25 5 x 5 (4 2 ) 16 6 x 6 (3 2 ) 9 7 x 7 (2 2 ) 4 8 x 8 (1 2 ) 1 If we were studying squared numbers….. 204

INTHEENDINTHEEND The Virtue of Problem Solving the