The presentation describes an experimental interdisciplinary course in Multivariate Calculus and Newtonian Physics at the United States Military Academy in 2000.

1. U.S. Military Academy
West Point, New York
An Attempt to Combine Math and
Physics Courses at West Point
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

2. Context
Other schools:
– University of Nebraska-Lincoln
– Villanova University
– Dartmouth College
– Montana State University
– Rose-Hulman Institute of Technology
– Portland State University

3. 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

4. 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

5. 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)
WRR = Written Partial Review, TEE = Term End Exam

6. Two week schedule
Core course
Hr Mon Tue Wed Thu Fri
A PH201 PH201
B MA205 MA205 MA205 MA205
PM
A PH201 PH201 PH201
B MA205 MA205 MA205 MA205
PM PH LAB
(2 hr)
Combined course (e.g.first two weeks)
A MA205P
Parametric Eqns
MA205P
Vector Functions
B MA205P
Intro
PH251M
1 Dim. Motion
MA205P
Vectors
MA205P
Deriv Int of
Vector Fuctions
MA205P
Problem Solving
Lab
A
MA205P
Fundamental
Skills Exam
MA205P
Intersections
Collisions
B PH251M
2 Dim. Motion I
MA205P
Motion in Space
PH251M
2 Dim. Motion II
MA205P
Problem Solving
Lab
PH251M
LAB
Ballistic Motion

7. 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

8. Course Structure (Four WPR Blocks)
Vectors / Kinematics / Forces
Work / Energy / Momentum /
Multivariable Calculus
Polar Cylindrical Coordinates /
Rotational Motion
Series / Vector Calculus /Oscillations /
Waves

9. Combined Course Lessons
(e.g. first block of instruction)
Math
Course Intro Review
Parametric Equations
Vectors
Deriv. Int. of Vector Functions
Problem Solving Lab
Fundamental Skills Exam
Motion in Space
Eqns of Lines, Planes, Distance
Problem Solving Lab
Intersections Collisions
Dot Product
Cross Product
Problem Solving Lab Review
Major Exam
Physics
1 Dim. Motion
2 Dim. Motion
2 Dim. Motion
Ballistic Motion Lab
Newton’s Laws
Newton’s Laws
Newton’s Laws

10. Example material
Parametric equations
Linear motion
Rotational motion
2-D motion
Conservation of energy
0 10 20 30
0
1000
500
x
t
0 10 20 30
0
600
400
200
y
t
0 200 400 600 800
0
600
400
200
y
x

11. Context Rich Problems
Ravine
bridge
road
river
Mass of T64 = 38100 kg
Mass of bridge = 18500 kg
Maximum tension of each cable = 5.00 x 105 N
mk
= 0.400 (between bridge surface and tank pads)
ms = 0.500 (between bridge surface and tank pads)
SIDE VIEW
15 m
25 m
5 m
15 m
(enemy)
near
bank
(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

12. 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
Force Longbow
Compound Bow
Draw Length

13. ILAP #1 Resources
Bow arrows
Instructors
Handout
Web page
– Digital photos
– Digital videos
– Java applet

14. ILAP #2: Large Angle Pendulum
Develop formula for finding period of a “large
angle” pendulum
max
T r ò ÷ø ö
2 2 1
g d
q
q q
q
q
cos cos max
çè æ
÷ø ö
çè æ
-
=
Approximate solution to elliptic integral using
Taylor series
Design a physics lab experiment to test the theory
Oral / written reports

15. 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

16. Preliminary Results
Our 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.

17. Performance
Multiple Choice
100
80
60
40
20
0
1 2 3 4 5 6 7 8 9 10
regular
advanced
combined

18. Performance
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%
 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.

19. Performance
The math-physics course shows
stronger correlations than one year
earlier.
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
1.00
0.95
0.90
0.85
0.80
0.75
PH251M Final Grade
MA205P Final Grade
Grades not
hurt. Time is saved that can
be used for ???
MA205 Section 1
PH201
0.62
MA205X
PH251
0.65
MA205
PH251M
0.78

20. 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 1
PH251 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 0
PH251 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%

21. 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.

22. 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

23. 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.

24. PH251M Time Survey
480
420
360
300
240
180
120
60
0
0 5 10 15 20 25 30 35 40
Lesson Number
Minutes of Preparation
Daily
Average
Reset Avg.
Thanksgiving
no longer graded
Navy
Graded daily
WWW homework
gives us 30 – 40 min
more preparation
time
“I’m working
harder than CDT
Smith who is
taking PH201,
and he is getting
an A with less
work.”

25. 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

26. If We Were Kings...
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
MA205
Modeling real systems
Applications
Demos / virtual experiments
Analytic geometry
PH201
HHCs
Mathcad
Experiments
real virtual