An Attempt to Combine Math and Physics Courses at West Point U.S. Military Academy West Point, New York Department of Mathematical Sciences LTC Jeff Strickland LTC Joe Myers MAJ Jon Shupenus Prof. Duff Campbell Department of Physics COL Thomas Lainis LTC Dave Bedey MAJ Chris Lehner Prof. Robert Fuller
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MAPH - Math & Physics and Interdisciplinary Course
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An Attempt to Combine Math and Physics Courses at West Point
U.S. Military Academy
West Point, New York
Department of Mathematical SciencesLTC Jeff Strickland
LTC Joe MyersMAJ Jon ShupenusProf. 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
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 Sophomores from the top of the class – Excelled in Freshman 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
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 energy
0 10 20 300
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 ProblemsRavine
bridge
road
river
Mass of T64 = 38100 kgMass of bridge = 18500 kgMaximum tension of each cable = 5.00 x 105 Nµk = 0.400 (between bridge surface and tank pads)µs = 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 were designed to break at this force, the enemy tank would theoretically travel 15 meters across 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 of the 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
� 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
θθθ
θ
θ
dg
rT ∫
−=
max
maxcoscos122
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 ResultsOur quantitative results would be so singularly impressive and awe inspiring that all colleges and universities across America 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.
Performance
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
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 i f you are interested in really learning and understanding the physics and math principles taught in PH251 and MA205, this class is r ight 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 learning
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
Daily
Average
Reset Avg.
no longer graded
Thanksgiving
Navy
Graded 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 /