20 year continuing project to improve undergraduate education with contributions by:Many faculty and graduate students of U of M Physics Departmentand the U of M Physics Education Group

Ken HellerSchool of Physics and Astronomy

University of Minnesota

Teaching Problem Solving in Large Introductory Classes: The View from Physics

Supported in part by Department of Education (FIPSE), NSF,and the University of Minnesota

Details at http://groups.physics.umn.edu/physed/

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

Employment Private Sector

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

Percent Reporting Frequent Use0 50 0 50 0 50

Survey of Physics Batchelors, 1994-AIP

Highest Rated Goalsscale 1 - 5

Biology Majors Course 20034.9 Basic principles behind all physics4.4 General qualitative problem solving skills4.3 Use biological examples of physical principles4.2 Overcome misconceptions about physical world4.1 General quantitative problem solving skills (*3)4.0 Real world application of mathematical concepts and techniques

4.0 Know the range of applicability of the principles of physics

Calculus-based Course (88% engineering majors) 19934.5 Basic principles behind all physics

4.5 General qualitative problem solving skills

4.4 General quantitative problem solving skills

4.2 Apply physics topics covered to new situations

4.2 Use with confidence

Algebra-based Course (24 different majors) 19874.7 Basic principles behind all physics4.2 General qualitative problem solving skills4.2 Overcome misconceptions about physical world4.0 General quantitative problem solving skills4.0 Apply physics topics covered to new situations

Survey of Faculty in Majors Requiring Introductory Physics

Lowest Rated (Biology Faculty)

Prepare students for the MCAT 20 25 40 5 5 0 2.4

Understand and appreciate the historical development and intellectual organization of physics

10 25 50 15 0 0 2.7

Understand and appreciate 'modern physics' (e.g. nuclear decay, quantum optics, cosmology, quantum mechanics, elementary particles,...)

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

Underlying Principles

Free Faculty Responses - Goals (Biology Faculty)

1. In your opinion, what is the primary reason your department requires students to take this physics course?

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?

M. Martinez, Phi Delta Kappan, April, 1998

• If you know how to do it, its not a problem.

A problem for your student is not a problem for you

“Process of Moving Toward a Goal When Path is Uncertain”

Not specific algorithms

Problems are solved using general purpose tools

“Problem Solving Involves Error and Uncertainty”

vs ProblemExercise

Some General Purpose Tools

General Principles of Physics

Means - Ends Analysisidentifying goals and subgoals

Working Backwards step by step planning from desired result

Successive Approximationsrange of applicability and evaluation

External Representationspictures, diagrams, mathematics

Students’ Misconceptions About Problem SolvingYou need to know the right formula to solve a problem:

It's all in the mathematics:Manipulate the equations as quickly as possible

Memorize formulasMemorize solution patterns

Allow students to bring in "crib" sheets

Numbers are easier to deal withPlug in numbers as soon as possible

Plug-and-chug

Actions that reinforce the misconceptionTest requires students to remember important equations

Actions that reinforce the misconceptionSingle step problems.Multi-part problems.

Remembering

311941483526616430678538799514282739

246810121416182022242628303234363840

106614921620177618121860194120002006

random

One step rule based

011235813213455891442333776109871597 Two step rule based

1066 1492 1620 1776 1812 1860 1941 2000 2006

pattern

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.

Circled statements from evaluator

Algebra-based Physics(second of four tests - 1989)

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

Recognize the ProblemWhat's going on?

Describe the problem in terms of physics

What does this have to do with physics ?

Plan a solutionCan I use what I know to get an answer?

Execute the planGet an answer

Evaluate the solution

Can this be true?

General Problem Solving SkillsHow to solve a problem when you don’t know how

Problem-solving Framework

Used by experts in all fields

(i.e. Polya 1957)

Chi, M., Glaser, R., & Rees, E. (1982)

Focus the ProblemPicture and Given Information

Question:

Approach:

Describe the PhysicsDiagram and Define Quantities

Target Quantity(s):

Quantitative Relationships:

Plan the SolutionConstruct Specific Equations

Execute the PlanSolve the Equations

Describe the AnswerDoes it answer the question?

Does it have correct units?

Is it unreasonable?Page 1 Page 2

Problem Solving Worksheet used at the beginning of the course

Teaching Students to Solve Physics Problems

Solving Problems Requires Conceptual Knowledge:Solving Problems Requires Conceptual Knowledge:From Situations to Decisions using internal knowledge

Students must be taught a problem solving framework that does this explicitly

• Determine goal

• Choose relevant information• Choose applicable principles

• Construct a plan• Arrive at an answer

• Evaluate the solution

• Visualize situation

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

6. Coaching by instructors is more effective

4. Students practice using the language of the field, "talking physics“, and explicitly connecting it to their existing knowledge base.

3. The group interactions externalize the planning, connection, and monitoring skills needed to solve problems allowing students to observe them in others.

5. Students must deal with and resolve their misconceptions.

External clues of group difficultiesGroup processing of instructor input

Cooperative Groups

Positive Interdependence

Face-to-Face Interaction

Individual Accountability

Explicit Collaborative Skills

Group Functioning Assessment

Having Students Work Together in Structured Groups

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

Email 8/24/05

• denial• anger• bargaining• depression• acceptance

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 --- 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."

5 stages to a common traumatic event : Problem Solving!

Why Group Problem Solving

MayNot Work

1. Inappropriate Tasks

2. Inappropriate Grading

3. Poor structure and management of Groups

The Monotillation of Traxoline

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. What is traxoline?2. Where is traxoline montilled?3. How is traxoline quasselled?4. Why is it important to know about traxoline?

(attributed to Judy Lanier)

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. What is the process being discussed?2. What facilities are needed?3. What are some compatibility issues?4. Why is it important to form groups?

Laundry

A Complex Process

Household task

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.

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.)

The Form of the Question

• 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.

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.

“If you win, I do NOT lose.”

ABSOLUTE SCALE

Grading

X

Grading Rubric for StudentsScaffolding 4

PHYSICS 1201.200 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.

Control of Equations that are AllowedScaffolding 5

Equation sheet on the final exam

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

• do not use groups of two.

• assign and rotate roles:- Manager- Skeptic- Checker/Recorder- Summarizer

Structure and Management of Groups

4. How can problems of dominance by one student and conflict avoidance within a group be addressed?

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.

Structure and Management

of Groups

LECTURES

Four hours each week, sometimes with informal cooperative groups. Model constructing knowledge, model problem solving framework.

RECITATIONSECTION

LABORATORY

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.

TESTSFriday lecture -- problem-solving quiz & conceptual questions (2 problems, 10 multiple choice) (1 group problem in previous discussion section) every 3 weeks.

Course Structure

Cognitive Apprenticeship

Instruction

sights andsounds

new ideas

existing ideasinflu

ence

development

of

filter givemeaningto

INSTRUCTION

coach

model

fadeCollins, Brown, & Newman (1990)

Learning in the environment of expert practice

• Why it is important• How it is used• How is it related a person’s

existing knowledge

Scaffolding

Additional structure used to support the construction of a complex structure.

Removed as the structure is built

• An explicit problem solving framework• A worksheet that structures the framework• Cooperative group structure that encourages productive group interactions

Grouping rulesGroup rolesGroup 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

Examples of Scaffolding in teaching Introductory Physics

0

5

10

15

20

25

30

0 5 10 15 20 25 30fci pre

mat

h pr

e

R2 = 0.14

Math pre vs FCI pre – Biology Students

Correlation between a math skills test and a physics concept test

General test taking (preparation, IQ, …)

FCI pre vs Math pre – Engineering students

0

5

10

15

20

25

30

0 5 10 15 20 25

Math pre

FC

I p

re

Correlation between a math skills test and a physics concept test

General test taking (preparation, IQ, …)

R2 = 0.20

Problem-Solving vs. Math pre – Engineering Students

Eng, PS grade vs. Math pre

R2 = 0.1173

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25

Math pre

PS

gra

de (

fin

al)

R2 = 0.09

0

20

40

60

80

100

120

0 5 10 15 20 25 30

math test pre

final

prob

lems

Problem-Solving vs. Math pre – Biology Students

R2 = 0.2966

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35

FCI Pre

Pro

ble

m S

core

PS vs FCI pre – Engineering students

R2 = 0.0453

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25 30

FCI pre

Prob

lem

Sco

re

PS vs FCI pre – Biology Students

The End

Please visit our websitefor more information:

http://groups.physics.umn.edu/physed/