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

– Math: 56 lessons, 8 problem solving labs– Physics: 40 lessons, 8 labs (x 2 hours)

WRR = Written Partial Review, TEE = Term End Exam

Two week schedule� Core course

� Combined course (e.g.first two weeks)

Hr Mon Tue Wed Thu Fri

A PH201 PH201B MA205 MA205 MA205 MA205PM

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 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

– 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

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

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 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 /

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 coursesCoordinated schedules by topicInterdisciplinary projects

separate but coordinatedcombined

Same section rostersBack to back classesCoordinated labsSame textbookWeekly meetings

PH201HHCsMathcadExperiments

real & virtual