An Attempt to Combine Math and Physics Courses at West Point U.S. Military Academy West Point, New York Department of Mathematical Sciences MAJ Jon Shupenus

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)


ReformMovement

TechniquesApproach

1-Year CalculusSequence

SpaceSupremacy

ColdWar


1950s

1960s

1970s

1980s



ReformMovement


TechniquesApproach


1-Year CalculusProgram

IncreasedEnrollment

LessSelectiveSpectrum

MoreDiverse

Spectrum



Technology

1950s

1960s

1970s

1980s

ComputerScience

Need forDiscrete Math

DecliningMath Interest



ReformMovement





TechniquesApproach


“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.

Documents

An Attempt to Combine Math and Physics Courses at West Point U.S. Military Academy West Point, New York Department of Mathematical Sciences MAJ Jon Shupenus