Transcript The Problem with Math May Be the Problems Unsolved

Mathematical Problem Solving in
Grades 4 to 8: A Practice Guide
John Woodward
Dean, School of Education
University of Puget Sound
What is This?
Singapore’s
Mathematics
Mathematical
Problem
Curriculum
Framework
Solving
Concepts
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?
http://ies.ed.gov/ncee/wwc/PracticGuide.
Or
Google: IES Practice Guides Problem Solving
What are Practice Guides?
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
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
Recommendation
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
Challenging Issues for the Panel

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
• 4 + 6 + 1 + 2 + 9 + 8 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

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?
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
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
Recommendation 1

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
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?
Yacht?
Slip?
Harbor?
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?
OR
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?
Recommendation 1

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 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 problemsolving 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?
Provide Prompts or Model Questions (continued)
 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?
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.
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?
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?
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
The remaining money. The 3/5 is
now 3/3 or the new whole.
She had 3/5 remaining
after buying the coat
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 spent 1/3 of what was left on a
sweater. This is the same as 1/5 of
the original amount.
She had 3/5 remaining
after buying the coat
Strip Diagrams as a Tool (continued)
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
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°
95°
x°
110°
85°
70°
25°
155°
What Is the Measure of Angle X?
155°
90°
65°
x°
95°
90°
110°
65
90
+ 110
265
360
- 265
95
What Is the Measure of Angle X?
155°
25°
90°
65
+ 20
85
65°
95°
x°
20°
110° 70° 90°
180
- 85
95
What Is the Measure of Angle X?
155°
25°
x°
95°
70°
110°
155°
110°
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?
2x2
squares
How Many Squares on a Checkerboard?
2x2
squares
How Many Squares on a Checkerboard?
2x2
squares
How Many Squares on a Checkerboard?
2x2
squares
How Many Squares on a Checkerboard?
2x2
squares
How Many Squares on a Checkerboard?
3x3
squares
How Many Squares on a Checkerboard?
3x3
squares
How Many Squares on a Checkerboard?
3x3
squares
How Many Squares on a Checkerboard?
7x7
squares
How Many Squares on a Checkerboard?
7x7
squares
How Many Squares on a Checkerboard?
7x7
squares
How Many Squares on a Checkerboard?
7x7
squares
If we were studying squared numbers…..
Size of squares
Total
1 x 1 (82)
2 x 2 (72)
3 x 3 (62)
4 x 4 (52)
5 x 5 (42)
6 x 6 (32)
7 x 7 (22)
8 x 8 (12)
64
49
36
25
16
9
4
1
204
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The Virtue of Problem Solving
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