Teaching Problem Solving in Large
Introductory Classes: The View from Physics
Ken Heller
School of Physics and Astronomy
University of Minnesota
20 year continuing project to improve undergraduate education with contributions by:
Many faculty and graduate students of U of M Physics Department
and the U of M Physics Education Group
Details at http://groups.physics.umn.edu/physed/
Supported in part by Department of Education (FIPSE), NSF,
and the University of Minnesota
A Guide for Discussion
 Problem Goals
• Why Solve Problems?
• What are Problems?
• Experts and Novices
 Teaching Problem Solving?
• Modeling a Framework
• Coaching
• Supporting Real Problem Solving
 Designing Problems
• What is Context-Rich?
• Why?
 How Well Does It Work
Private
Sector
Employment
Gov’t
Labs
High
Schools
Problem Solving
Interpersonal Skills
Technical Writing
Management Skills
Adv. Computer Skills
Spec. Equip. & Proc.
Business Principles
Statistical Concepts
Knowledge of Physics
Advanced Mathematics
0
Survey of Physics Batchelors, 1994-AIP
50
0
50
0
50
Percent Reporting Frequent Use
Survey of Faculty in Majors Requiring Introductory Physics
Algebra-based Course (24 different majors) 1987
4.7
4.2
4.2
4.0
4.0
Basic principles behind all physics
General qualitative problem solving skills
Overcome misconceptions about physical world
General quantitative problem solving skills
Apply physics topics covered to new situations
Highest Rated Goals
scale 1 - 5
Calculus-based Course (88% engineering majors) 1993
4.5
4.5
4.4
4.2
4.2
Basic principles behind all physics
General qualitative problem solving skills
General quantitative problem solving skills
Apply physics topics covered to new situations
Use with confidence
Biology Majors Course 2003
4.9
4.4
4.3
4.2
4.1
4.0
4.0
Basic principles behind all physics
General qualitative problem solving skills
Use biological examples of physical principles
Overcome misconceptions about physical world
General quantitative problem solving skills (*3)
Real world application of mathematical concepts and techniques
Know the range of applicability of the principles of physics
Lowest Rated (Biology Faculty)
Prepare students for the MCAT
20
25
40
5
5
0
2.4
10
25
50
15
0
0
2.7
0
30
45
25
0
0
3.0
Use computers to solve problems within the
context of physics
10
10
35
30 10
0
3.1
Formulate and carry out experiments
10
10
40
25 15
0
3.3
Express, verbally and in writing, logical,
qualitative thought in the context of physics
5
0
35
45 10
0
3.4
Understand and appreciate the historical
development and intellectual organization of
physics
Understand and appreciate 'modern physics'
(e.g. nuclear decay, quantum optics,
cosmology, quantum mechanics, elementary
particles,...)
Free Faculty Responses - Goals (Biology Faculty)
1. In your opinion, what is the primary reason your department
requires students to take this physics course?
Underlying Principles
Application
Problem solving/math
• To understand the basic laws of physics; to be able to apply physical
principles to other problems; to overcome fear of math, quantitative
approach to science.
• General understanding of how 1st & 2nd order linear differential
equations explain behavior of various physical systems (mechanics,
thermodynamics, electricity).
• Living things rely on a number of physical principles. Concepts we
cover in lecture & techniques/equipment used in the laboratory require an
understanding of physics. Physics is fundamental to many biological
processes, & develop skills in problem-solving & modeling.
• Provide basic concepts in physics as applied to biological functions; learn
how to think quantitatively about these applied physics concepts.
What Is Problem Solving?
“Process of Moving Toward a Goal When Path is Uncertain”
• If you know how to do it, its not a problem.
Problems are solved using general purpose tools
Not specific algorithms
“Problem Solving Involves Error and Uncertainty”
A problem for your student is not a problem for you
Exercise vs Problem
M. Martinez, Phi Delta Kappan, April, 1998
Some General Purpose Tools
External Representations
pictures, diagrams, mathematics
Means - Ends Analysis
identifying goals and subgoals
Working Backwards
step by step planning from desired result
Successive Approximations
range of applicability and evaluation
General Principles of Physics
Students’ Misconceptions About Problem Solving
You need to know the right formula to solve a problem:
Memorize formulas
Memorize solution patterns
Actions that reinforce the misconception
Test requires students to remember
important equations
Allow students to bring in "crib" sheets
It's all in the mathematics:
Manipulate the equations as quickly as possible
Plug-and-chug
Numbers are easier to deal with
Plug in numbers as soon as possible
Actions that reinforce the misconception
Single step problems.
Multi-part problems.
Remembering
311941483526616430678538799514282739
random
106614921620177618121860194120002006
pattern
1066 1492 1620 1776 1812 1860 1941 2000 2006
246810121416182022242628303234363840
One step rule based
011235813213455891442333776109871597
Two step rule based
Student Difficulties Solving Problems
• Lack of an Organizational Framework that links previous
experiences, knowledge, and procedures
• Physics Misknowledge
– Incomplete (lack of a concept)
– Misunderstanding (weak misknowledge)
– Misconceptions (strong misknowledge)
• No Understanding of Range of Applicability
–
–
–
–
Always True
True under a broad range of well-defined circumstances
True in very special cases
True in this situation
• Lack of internal monitoring skills (reflection on what they did and
why, asking skeptical questions about their actions)
Students need instructional support to solve problems
Cowboy Bob is camped on the top
of Table Rock. Table Rock has a
flat horizontal top, vertical sides,
and is 500 meters high. A band of
outlaws is at the base of Table
Rock 100 meters from the side
wall. Cowboy Bob decides to roll a
large boulder over the edge and
onto the outlaws. Determine how
fast Bob will have to roll the
boulder to reach the outlaws.
Algebra-based Physics
(second of four tests - 1989)
Circled statements
from evaluator
Components of Course
• Teach Students an Organizational Framework
– Emphasize decisions using physics
– Rule-based mathematics
• Use Problems that Require
– An organized framework
– Physics conceptual knowledge
– Connection to existing knowledge
• Use Existing Course Structure
– Lectures (given by Professors) MODELING
– Discussion Sections (run by TAs) COACHING
– Labs (run by TAs) COACHING
General Problem Solving Skills
(i.e. Polya 1957)
How to solve a problem when you don’t know how
Recognize the Problem
What's going on?
Describe the problem in
terms of physics
What does this have
to do with physics ?
Plan a solution
Can I use what I know
to get an answer?
Execute the plan
Get an answer
Evaluate the solution
Can this be true?
Problem-solving Framework
Used by experts in all fields
Chi, M., Glaser, R., & Rees, E. (1982)
Problem Solving Worksheet used at the beginning of the course
Focus the Problem
Picture and Given Information
Plan the Solution
Execute the Plan
Construct Specific
Equations
Solve the Equations
Question:
Approach:
Describe the Physics
Diagram and Define Quantities
Describe the Answer
Does it answer the question?
Target Quantity(s):
Does it have correct units?
Quantitative Relationships:
Page 1
Is it unreasonable?
Page 2
Teaching Students to Solve Physics Problems
Solving Problems Requires Conceptual Knowledge:
From Situations to Decisions using internal knowledge
• Visualize situation
• Determine goal
• Choose applicable principles
• Choose relevant information
• Construct a plan
• Arrive at an answer
• Evaluate the solution
Students must be taught a problem solving
framework that does this explicitly
The Dilemma
Start with simple problems to
learn expert-like framework.
Success using novice strategies.
Why change?
Start with complex problems
so novice strategy fails
Difficulty using new framework.
Why change?
What Using Cooperative Groups
Does for Teaching Problem Solving
1. Following a logical problem
solving framework seems too
long and complex for most
students.
Cooperative-group problem
solving allows practice until the
framework becomes more
natural.
2. Complex problems that need a strategy are initially difficult.
Groups can successfully solve them so students see the
advantage of a logical problem-solving framework
early in the course.
What Using Cooperative Groups
Does for Teaching Problem Solving
3. The group interactions externalize the planning,
connection, and monitoring skills needed to solve
problems allowing students to observe them in others.
4. Students practice using the language of the field,
"talking physics“, and explicitly connecting it to
their existing knowledge base.
5. Students must deal with and resolve their misconceptions.
6. Coaching by instructors is more effective
External clues of group difficulties
Group processing of instructor input
Having Students Work Together in Structured Groups
Cooperative Groups
Email 8/24/05
 Positive Interdependence

Face-to-Face Interaction

Individual Accountability

Explicit Collaborative Skills

Group Functioning Assessment
I was reading through your
'typical objections'. Another
good reason for cooperative
group methods: this is how we
solve all kinds of problems in
the real world - the real
academic world and the real
business world. I wish they'd
had this when I was in school.
Keep up the great work.
Rick Roesler Vice President,
Handhelds Hewlett Packard
Student Reaction to Learning Problem Solving
Changing a deep held way of thinking is traumatic
Death of your beloved ideas and way of doing something.
Death of a loved-one (Elisabeth Kubler-Ross)
• denial
• anger
• bargaining
• depression
• acceptance
5 stages to a common traumatic event : Problem Solving!
DENIAL --- I don’t really have to do all that? Try it again my own way! And
again. Read the book or ask someone and then..., try again.
ANGER --- "%$@^##& professor!", "I shouldn’t have to take this course. I
should wait until someone else teaches it. This has nothing to do with what I
need." Crumple up the paper and throw it away! “These problems are tricky,
unclear, and just weird."
BARGAINING --- "Oh please help me pass. Can I do extra work for extra credit.
Just for once give us enough time to solve the problems.”
DEPRESSION --- “What am I going to do. I'm going to fail. I give up. I’ll never be
able to pass the course with this rotten professor. What's the use".
ACCEPTANCE --- "Ok. I really need to have a logical and organized process to
solve these problems. These problems really are the kind of thing I need to be able
to solve. I can actually use this technique in my other classes."
Why Group
Problem Solving
May
Not Work
1. Inappropriate Tasks
2. Inappropriate Grading
3. Poor structure and management of Groups
The Monotillation of Traxoline
(attributed to Judy Lanier)
It is very important that you learn about traxoline.
Traxoline is a new form of zionter. It is montilled in
Ceristanna. The Ceristannians gristerlate large
amounts of fevon and then brachter it to quasel
traxoline. Traxoline may well be one of our most
lukized snezlaus in the future because of our zionter
lescelidge.
Answer the following questions.
1.
2.
3.
4.
What is traxoline?
Where is traxoline montilled?
How is traxoline quasselled?
Why is it important to know about traxoline?
A Complex Process
The procedure is quite simple but you may have to go somewhere else if the
facilities are not adequate. Before the process begins, you form different
groups. Of course, one group may be sufficient depending on how much
there is to do.
Next you get started. Be careful, a mistake can be costly. It is important not
to overdo things. It is usually better to do too few things than too many. This
is especially important when issues of compatibility arise. At first, the whole
procedure might seem complicated since timing can be crucial. With
practice, it can all become routine.
After the procedure is completed, form groups again to complete the
process. This whole cycle will need to be repeated often.
Answer the following questions.
1.
2.
3.
4.
What is the process being discussed?
What facilities are needed?
What are some compatibility issues?
Why is it important to form groups?
Household task
Laundry
Appropriate Problems for Problem Solving
The problems must be challenging enough so there is a real
advantage to using a problem solving framework.
1. The problem must be complex
enough so the best student in the
class is not certain how to solve it.
The problem must be simple enough
so that the solution, once arrived
at, can be understood and
appreciated.
2. The task must be designed so that
• the major problem solving heuristics are required
(e.g. physics understood, a situation requiring an
external representation);
• there are several decisions to make in order to do the
task (e.g. several different quantities that could be
calculated to answer the question; several ways to
approach the problem);
• the task cannot be resolved in a few steps by copying
a pattern.
3. The task problem must connect to each student’s
mental processes
• the situation is real to the student so other
information is connected;
• there is a reasonable goal on which to base decision
making.
The Form of the Question
Your task is to design an artificial joint to replace arthritic elbow joints in
patients. After healing, the patient should be able to hold at least a gallon of
milk (3.76 liters) while lower arm is horizontal. The biceps muscle is attached to
the bone at the distance 1/6 of the bone length from the elbow joint, and makes
an angle of 80o with the horizontal bone. For how strong a force should you
design the artificial joint? (The weight of the bone is negligible.)
• Gives a motivation – allows some students to access their mental connections.
• Gives a realistic situation – allows some students to visualize the situation.
• Does not give a picture – students must practice visualization.
• Uses the character “you” – allows some students to visualize the situation.
• Cannot be solved in one step by plugging numbers into an equation – students
must practice organized quantitative decision making and use mathematical skills.
Called Context-Rich Problem
Context-rich Problems
• Each problem is a short story in which the major
character is the student. That is, each problem
statement uses the personal pronoun "you."
• The problem statement includes a plausible
motivation or reason for "you" to calculate
something.
• The objects in the problems are real (or can be
imagined) -- the idealization process occurs
explicitly.
• No pictures or diagrams are given with the
problems. Students must visualize the situation by
using their own experiences.
• The problem requires the student to make
decisions. It can not be solved in one step by
plugging numbers into a formula.
Grading
EVERYTHING WE WANT STUDENTS TO DO IS GRADED
“If you don’t grade it, they don’t learn it!”
•
Always write physics principles and a logical,
organized problem solving procedure.
•
•
Only basic equations given on test are allowed .
Small, but significant part of grades is for group
problem solving.
•
During lecture, answers to questions are occasionally
collected and graded (can be done electronically).
•
Predictions for lab problems are graded.
ABSOLUTE SCALE
“If you win, I do NOT lose.”
X
Scaffolding 4
PHYSICS 1201.200
Grading Rubric for Students
Final Exam
December 19, 2005
This is a closed book, closed notes quiz. Calculators are permitted. The ONLY
formulas that may be used are those given below. Define all symbols and justify all
mathematical expressions used. Make sure to state all of the assumptions used to solve
a problem. Credit will be given only for a logical and complete solution that is clearly
communicated with correct units. Partial credit will be given for a well communicated
problem solving strategy based on correct physics. MAKE SURE YOUR NAME, ID
#, SECTION #, and TAs NAME ARE ON EACH PAGE!! START EACH
PROBLEM ON A NEW PAGE. Each problem is worth 25 points: In the context
of a unified solution, partial credit will be awarded as follows: a useful picture,
defining the question, and giving your approach is worth 6 points; a complete
physics diagram defining the relevant quantities, identifying the target quantity,
and specifying the relevant equations with reasons is worth 6 points; planning the
solution by constructing the mathematics leading to an algebraic answer and
checking the units of that answer is worth 7 points; calculating a numerical value
with correct units is worth 3 points; and evaluating the validity of the answer is
worth 3 points. The 30 multiple choice questions are each worth 1.5 points.
Scaffolding 5
Control of Equations that are Allowed
Equation sheet on the final exam
Structure and Management of Groups
1. What is the "optimal" group size?
• three (or occasionally four)
2. What should be the gender and performance
composition of cooperative groups?
• heterogeneous groups based on past test performance.
:
- one from top third
- one from middle third
- one from bottom third
• two women with one man, or same-gender groups
Structure and Management of Groups
3. How often should the groups be changed?
For most groups:
• stay together long enough to be successful
• enough change so students know that success is
due to them, not to a "magic" group.
• about four times first semester, twice second
semester
Structure and Management of Groups
4. How can problems of dominance by one student and
conflict avoidance within a group be addressed?
• do not use groups of two.
• assign and rotate roles:
- Manager
- Skeptic
- Checker/Recorder
- Summarizer
Structure and
Management
of Groups
5. How can individual accountability be addressed?
• assign and rotate roles, group functioning;
• seat arrangement -- eye-to-eye, knee-to-knee;
• individual students randomly called on to present group results;
• group work is practice for individual tests;
• each student submits an individual lab report.
Course Structure
LECTURES
RECITATION
SECTION
LABORATORY
TESTS
Four hours each week, sometimes with
informal cooperative groups. Model
constructing knowledge, model
problem solving framework.
One hour each Thursday – cooperative
groups practice using problem-solving
framework to solve context-rich
problems. Peer coaching, TA coaching.
Two hours each week -- same groups
practice using framework to solve
concrete experimental problems. Same
TA. Peer coaching, TA coaching.
Friday lecture -- problem-solving quiz &
conceptual questions (2 problems, 10
multiple choice) (1 group problem in
previous discussion section) every 3 weeks.
existing
ideas
new
ideas
Cognitive
Apprenticeship
Instruction
Learning in the environment
of expert practice
sights
and
sounds
• Why it is important
• How it is used
• How is it related a person’s
existing knowledge
model
coach
Collins, Brown, & Newman (1990)
fade
Scaffolding
Additional structure used to support the
construction of a complex structure.
Removed as the structure is built
Examples of Scaffolding in teaching Introductory Physics
• An explicit problem solving framework
• A worksheet that structures the framework
• Cooperative group structure that encourages productive group interactions
Grouping rules
Group roles
Group reflection
• Limit use of formulas by giving an equation sheet (only allowed equations)
• Explicit grading rubric for problem solutions to encourage expert-like behavior
• Problems that discourage novice problem solving
• Explicit grading rubric for lab problems to encourage expert-like behavior
Math pre vs FCI pre – Biology Students
30
25
20
math pre
R2 = 0.14
15
10
5
0
0
5
10
15
fci pre
20
25
Correlation between a math skills test and a physics concept test
General test taking (preparation, IQ, …)
30
FCI pre vs Math pre – Engineering students
30
25
R2 = 0.20
FCI pre
20
15
10
5
0
0
5
10
15
20
25
Math pre
Correlation between a math skills test and a physics concept test
General test taking (preparation, IQ, …)
Problem-Solving vs. Math pre – Engineering Students
Eng, PS grade vs. Math pre
100
90
PS grade (final)
80
70
60
50
40
30
R2 = 0.1173
20
10
0
0
5
10
15
Math pre
20
25
Problem-Solving vs. Math pre – Biology Students
120
100
final problems
R2 = 0.09
80
60
40
20
0
0
5
10
15
math test pre
20
25
30
PS vs FCI pre – Engineering students
120
Problem Score
100
80
R2 = 0.2966
60
40
20
0
0
5
10
15
20
FCI Pre
25
30
35
PS vs FCI pre – Biology Students
100
90
Problem Score
80
R2 = 0.0453
70
60
50
40
30
20
10
0
0
5
10
15
FCI pre
20
25
30
The End
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for more information:
http://groups.physics.umn.edu/physed/