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An Attempt to Combine Math and Physics Courses at West Point
U.S. Military AcademyWest Point, New York
Department of Mathematical SciencesMAJ Jon ShupenusLTC Jeff Strickland
LTC Joe Myers Prof. Duff Campbell
Department of PhysicsCOL Thomas Lainis
LTC Dave BedeyMAJ Chris LehnerProf. Robert Fuller
AN ATTEMPT TO COMBINE MATH AND PHYSICS COURSES
AT WEST POINT
An Attempt to Combine Math and Physics Courses at West Point
U.S. Military AcademyWest Point, New York
Department of Mathematical SciencesMAJ Jon ShupenusLTC Jeff Strickland
LTC Joe Myers Prof. Duff Campbell
Department of PhysicsCOL Thomas Lainis
LTC Dave BedeyMAJ Chris LehnerProf. Robert Fuller
Context
Other schools:– University of Nebraska-Lincoln– Villanova University– Dartmouth College– Montana State University– Rose-Hulman Institute of Technology– Portland State University
College EngineeringProgram (5)1950s
1960s
1970s
1980s
College EngineeringProgram (5)
College EngineeringProgram (5)
ReformMovement
TechniquesApproach
1-Year CalculusSequence
SpaceSupremacy
ColdWar
College EngineeringProgram (4)
1950s
1960s
1970s
1980s
College EngineeringProgram (5)
College EngineeringProgram (5)
ReformMovement
College EngineeringProgram (4)
TechniquesApproach
1-Year CalculusSequence
1-Year CalculusProgram
IncreasedEnrollment
LessSelectiveSpectrum
MoreDiverse
Spectrum
1-Year CalculusProgram
1-Year CalculusProgram
Technology
1950s
1960s
1970s
1980s
ComputerScience
Need forDiscrete Math
DecliningMath Interest
College EngineeringProgram (5)
College EngineeringProgram (5)
ReformMovement
1-Year CalculusProgram
1-Year CalculusProgram
1-Year CalculusProgram
College EngineeringProgram (4)
TechniquesApproach
1-Year CalculusSequence
“Calculus Instruction:Crucial but Ailing”
1950s
1960s
1970s
1980s
What did it look like?
Core-Mathematics
Using Symbolic Algorithms
MinimalApplications
Recitation &Memorization
TraditionalPedagogy
Techniques Approach
emphasizes
Scientific Reasoning
results in
contributes to
InterdisciplinaryInteraction
involves
Technology
via
ActiveLearning
employs
MAPH
led to
ILAPs
ConceptualUnderstanding
Problem-solvingStrategies
Real-worldApplications
Exploration &Discovery
Change inPedagogy
What does it look like now?
Why We Tried This Approach
For the students:– Improve comprehension– Minimize redundancy– Emphasize problem solving
For the instructors:– Seek connections and efficiencies– Standardization and consistency– Teach / learn new tools
Organization
Courses– MA205(P): Calculus II (follows DDS & Calc. I)
– PH251(M): first semester Advanced Physics
Students– 64 Yearlings from the top of the class – Excelled in Plebe math and chemistry– “Mostly” volunteers, selected by physics department
Organization
Three WPRs & a TEE in each course– First WPR: one page counts towards the other course– Second WPR: combined two-hour mid-term – Third WPR: separate– TEE: separate (used core-course TEEs)
Both classes in one room with adjacent lab 120 classroom hours
– Math: 56 lessons, 8 problem solving labs– Physics: 40 lessons, 8 labs (x 2 hours)
Two week schedule Core course
Combined course (e.g.first two weeks)
Hr Mon Tue Wed Thu Fri
A PH201 PH201B MA205 MA205 MA205 MA205
PM
A PH201 PH201 PH201B MA205 MA205 MA205 MA205
PMPH LAB
(2 hr)
A MA205PParametric Eqns
MA205PVector Functions
B MA205PIntro
PH251M1 Dim. Motion
MA205PVectors
MA205PDeriv & Int of
Vector Fuctions
MA205PProblem Solving
Lab
AMA205PFundamentalSkills Exam
MA205PIntersections &
Collisions
B PH251M2 Dim. Motion I
MA205PMotion in Space
PH251M2 Dim. Motion II
MA205PProblem Solving
Lab
PH251MLAB
Ballistic Motion
Redesigning The Courses
Physics sequence of topics was relatively fixed. Math sequence was reordered to support physics. Scheduling Philosophy:
– Schedule math topics first so that students learn the mathematical tools before they need them in physics.
– Schedule physics immediately after the applicable math topics to reinforce learned skills.
– Schedule problem-solving labs, in-class exercises, and projects last in order to go beyond the normal limits of either course
Course Structure (Four WPR Blocks)
Vectors / Kinematics / Forces Work / Energy / Momentum /
Multivariable Calculus Polar & Cylindrical Coordinates /
Rotational Motion Series / Vector Calculus /Oscillations /
Waves
Combined Course Lessons (e.g. first block of instruction)
MathCourse Intro & Review
Parametric EquationsVectors
Deriv. & Int. of Vector FunctionsProblem Solving Lab
Fundamental Skills ExamMotion in Space
Eqns of Lines, Planes, DistanceProblem Solving Lab
Intersections & CollisionsDot Product
Cross ProductProblem Solving Lab & Review
Major Exam
Physics
1 Dim. Motion
2 Dim. Motion
2 Dim. MotionBallistic Motion LabNewton’s Laws
Newton’s Laws
Newton’s Laws
Example material
Parametric equations Linear motion Rotational motion
2-D motion Conservation of
energy0 10 20 30
0
500
1000
x
t
0 10 20 300
200
400
600
y
t
0 200 400 600 8000
200
400
600
y
x
Context Rich Problems
Ravine
bridge
road
river
Mass of T64 = 38100 kgMass of bridge = 18500 kgMaximum tension of each cable = 5.00 x 105 N
k = 0.400 (between bridge surface and tank pads)
s = 0.500 (between bridge surface and tank pads)
SIDE VIEW
15 m
25 m
5 m
15 m
(enemy)nearbank
(friendly)far bank
a. Calculate the maximum tension permitted in each cable such that if the cables weredesigned to break at this force, the enemy tank would theoretically travel 15 metersacross the bridge (measured from the near bank and shown by a dotted line in the figure)before the bridge collapsed.
b. Assuming that the cable strength is directly proportional to the cross sectional area ofthe cable, calculate how deep you should cut each cable.
cut line
ILAP #1: Bow & Arrow Analysis
Predict and measure speed of arrow Based on physical characteristics of bow and arrows Determine range, engagement range, and max effective range Concepts:
– Vector functions– Extrema / Lagrange multipliers– Total differential– Intersections
– Work-Energy theorem– Newton’s 2nd Law– Ordinary differential equations– Non-constant acceleration
Draw Length
Fo
rce
Longbow
Compound Bow
ILAP #1 Resources
Bow & arrows Instructors Handout Web page
– Digital photos– Digital videos– Java applet
ILAP #2: Large Angle Pendulum
Develop formula for finding period of a “large angle” pendulum
Approximate solution to elliptic integral using Taylor series
Design a physics lab experiment to test the theory Oral / written reports
T 2rg
0
max1
cos ( ) cos max( )d2
Hoped to find...
… improvement in math skills and retention … improvement in physics skills and understanding … improvement in problem solving skills … increased appreciation for power of math & physics … better attitude towards math and physics … synergistic areas to improve core courses
Preliminary Results
Our quantitative results wouldbe so singularly impressiveand awe inspiring that allcolleges and universities acrossAmerica would adopt it.
This was not the case.Results are mixed.
Performance
Multiple Choice
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10
regular
advanced
combined
Course Size Grade FSE TEE
MA205P 64 91.8% 0.896 90.9%
MA205 Section 1 100 92.1% 0.821 91.9%
MA205 core 752 79.4% 0.596 76.1%
MA205X (99-2) 111 85.9% n/a n/a
PH251 34 87.7% n/a 83.3%
PH251M 64 87.0% n/a 78.1%
PH201 919 76.30% n/a 67.70%
Performance
The combined course did not hurt math grades. From initial examination of qualitative feedback, the students felt that
the math course helped them in their physics course, but the physics course did little to assist them in their math course.
1.000.950.900.850.800.750.700.65
1.00
0.95
0.90
0.85
0.80
0.75
PH251M Final Grade
MA
205P
Fin
al G
rade
Grades not hurt.
Time is saved that can be used for ???
The math-physics course shows stronger correlations
than one year earlier.
MA205 Section 1 & PH201
0.62
MA205X & PH251
0.65
MA205 & PH251M
0.78
Course End Survey
Strongly Agree Agree Neutral Disagree Strongly Disagree
My fellow students contributed to my learning in this course.
USMA 29% 43% 18% 8% 2%D/Math Sci 31% 45% 16% 7% 1%D/Physics 32 41 19 7 1
MA205 30% 47% 15% 6% 1%PH201 28 44 19 8 1PH251 55 18 21 3 3
MA205P 47% 40% 8% 3% 2%PH251M 77 14 6 3 0
In this course, my critical thinking ability increased.
USMA 33% 44% 16% 5% 1%D/Math Sci 32% 48% 16% 4% 1%D/Physics 43 42 11 3 0
MA205 37% 49% 11% 2% 1%PH201 45 42 10 2 0PH251 33 52 2 9 0
MA205P 44% 50% 6% 0% 0%PH251M 50 31 14 5 0
I am more comfortable turning physical problems into math problems and interpreting results.
MA205P 19% 65% 15% 0% 2%
Student Comments
“I am confident that I learned more from this class than I could have from the regular math or physics classes. If your grade is your main focus, regular classes would be better for you. But if you are interested in really learning and understanding the physics and math principles taught in PH251 and MA205, this class is right for you.”
“Because I had a better grasp of the math, it helped me to see the physics in a different light.”
“This course is not for everyone.”
“I am extremely relieved that I am through with these two classes.”
I would not wish this course on my worst enemy.
Instructor Comments
Instructors can assist the students by referring to the vocabulary of the mathematicians. As an example, “intersection and collision.”
A spaceship captain is enroute to space station zebra (coordinates...) along the parametric equation shown. The captain wishes to shut down the engine so that the ship can “coast” to the station. Where should the engines be shut down?
Sum of the instructorship is greater than its parts.• flexibility of instructors • student centered learningSaved time
We decided to cover more
Extra topics•Special relativity•Many context rich problems•Bow and Arrow interdisciplinary project•Large angle interdisciplinary laboratory•Terminal speed and coffee filter laboratory•Large angle pendulum•Two to four briefings
WELL THE EFFORT.
PH251M Time Survey
0
60
120
180
240
300
360
420
480
0 5 10 15 20 25 30 35 40
Lesson Number
Min
ute
s o
f P
rep
ara
tio
n
DailyAverageReset Avg.
no longer graded
Thanksgiving
NavyGraded dailyWWW homeworkgives us 30 – 40 minmore preparation time
“I’m working harder than CDT Smith who is taking PH201, and he is getting an A with less work.”
Lessons Learned
Students synthesis < instructor synthesis Professional growth for faculty Student confidence Coordination -- expensive but valuable Common topics (e.g., total derivative /
uncertainty), notation Technology / software (JAVA, Mathcad, wireless
keyboards) Combined exams
If We Were Kings...
MA205 Modeling real systems Applications Demos / virtual experiments Analytic geometry
Both core courses– Coordinated schedules by topic– Interdisciplinary projects
separate but coordinated combined
– Same section rosters– Back to back classes– Coordinated labs– Same textbook– Weekly meetings
PH201 HHCs Mathcad Experiments
– real & virtual
Move both departments into the same building with windows.Add an eighth day to each week.
Questions & Discussion
It’s up to you.