What Are The Characteristics of a Good Group Problem?

Good Group Problems
What Are The Characteristics of a Good Group Problem?


Group problems should be designed to encourage students to use an organized,
logical problem-solving strategy instead of their novice, formula-driven, "plug-and-
chug" strategy. Specifically, they should encourage students to (a) consider physics
concepts in the context of real objects in the real world; (b) view problem-solving as a
series of decisions; and (c) use their conceptual understanding of the fundamental
concepts of physics to qualitatively analyze a problem before the mathematical
manipulation of formulas.
Group problems should be more difficult to solve than easy problems typically given
on an individual test. But the increased difficulty should be primarily conceptual, not
mathematical. Difficult mathematics is best accomplished by individuals, not by
groups. So problems that involve long, tedious mathematics but little physics, or
problems that require the use of a shortcut or "trick" that only experts would be likely
to know do not make good group problems. In fact, the best group problems involve
the straight-forward application of the fundamental principles (e.g., the definition of
velocity and acceleration, the independence of motion in the vertical and horizontal
directions) rather than the repeated use of derived formulas (e.g., v 2
2
f - vo = 2ad).

There are twenty-one characteristics of a problem that can make it more difficult to
solve than a standard textbook exercise:

Approach
1 Cues Lacking
A. No explicit target variable. The unknown variable of the problem is not
explicitly stated.
B. Unfamiliar context. The context of the problem is very unfamiliar to the
students (e.g., cosmology, molecules).
2 Agility with Principles
A. Choice of useful principles. The problem has more than one possible set of
useful concepts that could be applied for a correct solution.
B. Two general principles. The correct solution requires students to use two
major principles (e.g., torque and linear kinematics).
C. Very abstract principles. The central concept in the problem is an abstraction
of another abstract concept. (e.g., potential, magnetic flux).
3 Non-standard Application
A. Atypical situation. The setting, constraints, or complexity is unusual
compared with textbook problems.
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 Unusual target variable. The problem involves an atypical target variable when
compared with homework problems.
Analysis of Problem
4 Excess or Missing Information
A. Excess numerical data. The problem statement includes more data than is
needed to solve the problem.
B. Numbers must be supplied. The problem requires students to either
remember or estimate a number for an unknown variable.
C. Simplifying assumptions. The problem requires students to generate a
simplifying assumption to eliminate an unknown variable.
5 Seemingly Missing Information
A. Vague statement. The problem statement introduces a vague, new
mathematical statement.
B. Special conditions or constraints. The problem requires students to generate
information from their analysis of the conditions or constraints.
C. Diagrams. The problem requires students to extract information from a spatial
diagram.
6 Additional Complexity
A. More than two subparts. The problem solution requires students decompose
the problem into more than two subparts.
B. Five or more terms per equation. The problem involves five or more terms in a
principle equation (e.g., three or more forces acting along one axes on a single
object).
C. Two directions (vector components). The problem requires students to treat
principles (e.g., forces, momentum) as vectors.
Mathematical Solution
7 Algebra Required
A. No numbers. The problem statement does not use any numbers.
B. Unknown(s) cancel. Problems in which an unknown variable, such as a mass,
ultimately factors out of the final solution.
C. Simultaneous equations. A problem that requires simultaneous equations for
a solution.
8 Targets Math Difficulties
A. Calculus or vector algebra. The solution requires the students to sophisticated
vector algebra, such as cross products, or calculus.
B. Lengthy or Detailed Algebra. A successful solution to the problem is not
possible without working through lengthy or detailed algebra (e.g., a messy
quadratic equation).
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Good Group Problems
BEWARE! Good group problems are difficult to construct because they can easily be
made too complex and difficult to solve. A good group problem does not have all of
the above difficulty characteristics, but usually only 2- 5 of these characteristics.
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How to Create Context-rich Group Problems


One way to invent group problems is to start with a textbook exercise or problem, then
modify the problem. You may find the following steps helpful:
1. If necessary, determine a context (real objects with real motions or interactions)
for the textbook exercise or problem. You may want to use an unfamiliar
context for a very difficult group problem.
2. Decide on a motivation -- Why would anyone want to calculate something in
this context?
3. Determine if you need to change the target variable to
(a) make the problem more than a one-step exercise, or
(b) make the target variable fit your motivation.
5. Determine if you need to change the given information (or target variable) to
make the problem an application of fundamental principles (e.g., the definition
of velocity or acceleration) rather than a problem needing the application of
many derived formulas.
4. Write the problem like a short story.
5. Decide how many "difficulty" characteristics (characteristics that make the
problem more difficult) you want to include, then do some of the following:
(a) think of an unfamiliar context; or use an atypical setting or target variable;
(b) think of different information that could be given, so two approaches (e.g.,
kinematics and forces) would be needed to solve the problem instead of one
approach (e.g., forces), or so that more than one approach could be taken
(c) write the problem so the target variable is not explicitly stated;
(d) determine extra information that someone in the situation would be likely
to have; or leave out common-knowledge information (e.g., the boiling
temperature of water);
(e) depending on the context, leave out the explicit statement of some of the
problem idealizations (e.g., change "massless rope" to "very light rope"); or
remove some information that students could extract from an analysis of
the situation;
(f) take the numbers out of the problem and use variable names only;
(g) think of different information that could be given, so the problem solution
requires the use of vector components, geometry/trigonometry to eliminate
an unknown, or calculus.
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Creating Context-rich Problems
6. Check the problem to make sure it is solvable, the physics is straight-forward,
and the mathematics is reasonable.
Some common contexts include:
¢" physical work (pushing, pulling, lifting objects vertically, horizontally, or up
ramps)
• suspending objects, falling objects
• sports situations (falling, jumping, running, throwing, etc. while diving, bowling,
playing golf, tennis, football, baseball, etc.)
• situations involving the motion of bicycles, cars, boats, trucks, planes, etc.
• astronomical situations (motion of satellites, planets)
• heating and cooling of objects (cooking, freezing, burning, etc.)


Sometimes it is difficult to think of a motivation. We have used the following
motivations:
• You are . . . . (in some everyday situation) and need to figure out . . . .
• You are watching . . . . (an everyday situation) and wonder . . . .
• You are on vacation and observe/notice . . . . and wonder . . . .
• You are watching TV or reading an article about . . . . and wonder . . .
• Because of your knowledge of physics, your friend asks you to help him/her . .
. .
• You are writing a science-fiction or adventure story for your English class about .
. . . and need to figure out . . . .
• Because of your interest in the environment and your knowledge of physics, you
are a member of a Citizen's Committee (or Concern Group) investigating . . . .
• You have a summer job with a company that . . . . Because of your knowledge of
physics, your boss asks you to . . . .
• You have been hired by a College research group that is investigating . . . . Your
job is to determine . . . .
• You have been hired as a technical advisor for a TV (or movie) production to
make sure the science is correct. In the script . . . ., but is this correct?
• When really desperate, you can use the motivation of an artist friend designing a
kinetic sculpture!


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Judging Problem Difficulty
Decision Strategy for Judging Problems

Outlined below is a decision strategy to help you decide whether a context-rich
problem is a good individual test problem, group practice problem, or group test
problem.

1. Read the problem statement. Draw the diagrams and determine the equations needed
to solve the problem (through plan-a-solution step).

2. Reject if:
• the problem can be solved in one step,
• the problem involves long, tedious mathematics, but little physics; or
• the problem can only be solved easily using a "trick" or shortcut that only experts
would be likely to know. (In other words, the problem should be a straight-
forward application of fundamental concepts and principles.)

3. Check for the twenty-one characteristics that make a problem more difficult:

Approach
Analysis
Mathematical Solution
• Cues Lacking
• Excess or Missing Info.
• Algebra required
___ A. No target variable
___ A. Excess data
___ A. No numbers
___ B. Unfamiliar context
___ B. Numbers required
___ B. Unknown(s) cancel

___ C. Assumptions
___ C. Simultaneous eqns.
• Agility with Principles


___ A. Choice of principle
• Seemingly Missing Info.
• Targets Math Difficulty
___ B. Two principles
___ A. Vague statement
___ A. Calc/vector algebra
___ C. Abstract principle
___ B. Special constraints
___ B. Lengthy algebra

___ C. Diagrams
• Non-Standard Application

___ A. Atypical situation
• Additional Complexity
___ B. Unusual target
___ A. >2 subparts
___ B. 5+ terms
___ C. Vectors


4. Decide if the problem would be a good group practice problem (20 - 25 minutes), a
good group test problem (45 - 50 minutes), or a good (easy, medium, difficult)
individual test problem, depending on three factors: (a) the complexity of
mathematics, (b) the timing (when problem is to be given to students), and (c) the
number of difficulty characteristics of the problem:


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Judging Problem Difficulty



Type of Problem
Timing
Diff. Ch.
Group Practice Problems should be just introduced to concept(s)
2 - 3
shorter and mathematically easier than just finished study of concept(s)
3 - 4
group test problems.
Group Test Problems can be more
just introduced to concept(s)
3 - 4
complex mathematically.
just finished study of concept(s)
4 - 5

Type of Problem
Timing
Diff. Ch.
Individual Problems can be easy,

medium-difficult, or difficult:
Easy
just introduced to concept(s)
0 -1
just finished study of concept(s)
1 - 2
Medium-difficult
just introduced to concept(s)
1 - 2
just finished study of concept(s)
2 - 3
Difficult
just introduced to concept(s)
2 - 3
just finished study of concept(s)
3 - 4

There is considerable overlap in the criteria, so most problems can be judged to be
both a good group practice or test problem and a good easy, medium-difficult, or
difficult individual problem.














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Judging Problem Difficulty

Examples of how to judge context-rich problems

Example Problem #1:
You are helping your friend prepare for her next skate board exhibition. For her program, she plans to
take a running start and then jump onto her heavy duty 15-lb stationary skateboard. She and the
skateboard will glide in a straight line along a short, level section of track, then up a sloped concrete
wall. She wants to reach a height of at least 10 feet above where she started before she turns to come
back down the slope. She has measured her maximum running speed to safely jump on the skateboard
at 7 feet/second. She knows you have taken physics, so she wants you to determine if she can carry
out her program as planned. She tells you that she weighs 100 lbs.

Assume that students have just started to study the conservation of energy and
conservation of momentum.

The approach to solve this problem involves using conservation of energy and
conservation of momentum.

Should we reject it?
• The problem can be solved in one step. NO.
• The problem involves long, tedious mathematics, but little physics. NO.
• The problem can only be solved easily using a "trick" or shortcut that only
experts would be likely to know. (In other words, the problem should be a
straight-forward application of fundamental concepts and principles.) NO.

Which of the characteristics does this problem involve?

Approach
Analysis
Mathematical Solution
1. Cues Lacking
4. Excess or Missing Info.
7. Algebra required
_v_ A. No target variable
___ A. Excess data
___ A. No numbers
___ B. Unfamiliar context
___ B. Numbers required
___ B. Unknown(s) cancel

_v_ C. Assumptions
___ C. Simultaneous eqns.
2. Agility with Principles


___ A. Choice of principle
5. Seemingly Missing Info.
8. Targets Math Difficulty
_v_ B. Two principles
___ A. Vague statement
___ A. Calc/vector algebra
___ C. Abstract principle
___ B. Special constraints
___ B. Lengthy algebra

___ C. Diagrams
3. Non-Standard Application

___ A. Atypical situation
6. Additional Complexity
___ B. Unusual target
___ A. >2 subparts
___ B. 5+ terms
___ C. Vectors
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Judging Problem Difficulty

This has a difficulty rating of 3, two of which are in the approach. The mathematics
involved is easy. This would make a decent group practice problem or a medium-
difficult individual test problem. It is too easy for a group test problem.

If teaching this as a group practice problem, you could expect students to spend more
time on the setup of the problem, and less time on the math.










Example Problem #2:

Electric and Gravitational Force: You and a friend are reading a newspaper article about nuclear
fusion energy generation in stars. The article describes the helium nucleus, made up of two protons and
two neutrons, as very stable so it doesn't decay. You immediately realize that you don't understand why
the helium nucleus is stable. You know that the proton has the same charge as the electron except that
the proton charge is positive. Neutrons you know are neutral. Why, you ask your friend, don't the
protons simply repel each other causing the helium nucleus to fly apart? Your friend says she knows
why the helium nucleus does not just fly apart. The gravitational force keeps it together, she says. Her
model is that the two neutrons sit in the center of the nucleus and gravitationally attract the two protons.
Since the protons have the same charge, they are always as far apart as possible on opposite sides of
the neutrons. What mass would the neutron have if this model of the helium nucleus works? Is that a
reasonable mass? Looking in your physics book, you find that the mass of a neutron is about the same
as the mass of a proton and that the diameter of a helium nucleus is 3.0 x 10-13 cm.

Assume that students have just finished studying electric forces.

The approach to solve this problem involves using the idea of electric force and
gravitational force.


Should we reject it?
• The problem can be solved in one step. NO.
• The problem involves long, tedious mathematics, but little physics. NO.
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Judging Problem Difficulty
• The problem can only be solved easily using a "trick" or shortcut that only
experts would be likely to know. (In other words, the problem should be a
straight-forward application of fundamental concepts and principles.) NO.

Which of the characteristics does this problem involve?

Approach
Analysis
Mathematical Solution
1. Cues Lacking
4. Excess or Missing Info.
7. Algebra required
_v_ A. No target variable
___ A. Excess data
___ A. No numbers
_v_ B. Unfamiliar context
___ B. Numbers required
___ B. Unknown(s) cancel

___ C. Assumptions
___ C. Simultaneous eqns.
2. Agility with Principles


___ A. Choice of principle
5. Seemingly Missing Info.
8. Targets Math Difficulty
_v_ B. Two principles
___ A. Vague statement
___ A. Calc/vector algebra
___ C. Abstract principle
___ B. Special constraints
___ B. Lengthy algebra

_v_ C. Diagrams
3. Non-Standard Application

_v_ A. Atypical situation
6. Additional Complexity
___ B. Unusual target
___ A. >2 subparts
___ B. 5+ terms
___ C. Vectors


This has a difficulty rating of 5, four of which are in the approach. The mathematics
involved is easy, but the difficulty is all in the setup. Students probably have not
studied gravitational force lately, which makes the problem more difficult. This would
be a difficult group test problem.
If teaching this as a group test problem, you could expect groups to spend most of their
time on the setup of the problem, so don’t worry if they haven’t gotten to the math by
the middle of the hour.
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