Integrating mathematics and 1st/2nd year engineering Experiences and insights from 5 years teaching in ASU

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Integrating mathematics and 1st/2nd year
engineeringExperiences and insights from 5
years teaching in ASUs integrated engineering
curriculaof the Foundation Coalition
Matthias Kawski, Department of
Mathematics Arizona State University Tempe, AZ
85287, USA http//math.la.asu.edu/kawski
kawski_at_asu.edu
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Overview

• Who is Matt Kawski?
• Background NSF Engineering Coalitions
• ABET Engineering Criteria 2000
• The thrusts of the Foundation Coalition
• The FC 1st and 2nd year programs at ASU
• Projects as integrating theme
• Guiding theme Shaping the Future
• Technology
• Coordination of engineering/math/phys/English
• Communication
• Projects - a closer look
• Conclusion Reflections and advice

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Matthias Kawski

• Differential Geometric Control Theory Ph.D.
  1986 at U Colorado, Boulder (H.Hermes)
• At ASU since 1987 (after 1/2 yr at Rutgers)
• Taught in FC-programs for 5 years
• Current interests
• Chronological algebras lt-gt connections, optimal
  control
• Interactive visualization in undergrad courses,
  currently funded for Vector Calculus via
  Linearization
  Visualization and Modern Applications

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The NSF Engineering Coalitions
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ABET 2000 http//www.abet.org/eac/eac2000.htm
CRITERIA FOR ACCREDITING PROGRAMS IN ENGINEERING
IN THE US

• Criterion 3. Program Outcomes and Assessment
• Engineering programs must demonstrate that
  their graduates have
• (a) an ability to apply knowledge of
  mathematics, science, and engineering
• (b) an ability to design and conduct
  experiments, as well as to analyze and interpret
  data
• (c) an ability to design a system,
  component, or process to meet desired needs
• (d) an ability to function on
  multi-disciplinary teams
• (e) an ability to identify, formulate, and
  solve engineering problems
• (f) an understanding of professional and
  ethical responsibility
• (g) an ability to communicate effectively
• (h) the broad education necessary to
  understand the impact of engineering solutions
  in a global and societal context
• (i) a recognition of the need for, and an
  ability to engage in life-long learning
• (j) a knowledge of contemporary issues
• (k) an ability to use the techniques,
  skills, and modern engineering tools necessary
  for engineering practice.
• Criterion 4. Professional Component
• The Professional Component requirements
  specify subject areas appropriate to engineering
  but do not prescribe specific courses. The
  engineering faculty ..
• The professional component must include
• (a) one year of a combination of
  college level mathematics and basic sciences
  (some with experimental experience) appropriate
  to the discipline

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ENGINEERING CRITERIA 2000 PROGRAM CRITERIA

• PROGRAM CRITERIA FOR
• ELECTRICAL, COMPUTER, AND SIMILARLY NAMED
  ENGINEERING PROGRAMS
• Submitted by The Institute of Electrical and
  Electronics Engineers, Inc
• These program criteria apply to engineering
  programs which include electrical, electronic,
  computer, or similar modifiers in their titles.
• 1. Curriculum
• The structure of the curriculum must provide both
  breadth and depth across the range of engineering
  topics implied by the title of the program.
• Graduates must have demonstrated
• knowledge of probability and statistics,
  including applications appropriate to the program
  name and objectives knowledge of mathematics
  through differential and integral calculus, basic
  sciences, and engineering sciences necessary to
  analyze and design complex devices, and systems
  containing hardware and software components, as
  appropriate to program objectives.
• Graduates of programs containing the modifier
  "electrical" in the title must also have
  demonstrated a knowledge of advanced mathematics,
  typically including diffe-rential equations,
  linear algebra, complex variables, and discrete
  mathematics.
• Graduates of programs containing the modifier
  "computer" in the title must have demonstrated a
  knowledge of discrete mathematics.

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ABET 2000 http//www.abet.org/eac/eac2000.htm

• GONE are dozens of pages of specific
  requirements
• New, very brief, outcomes-oriented criteria
• do not require any courses in a math department
• do not prescribe specific syllabi and manual
  skills
• emphasize teamwork, technology, applications
• emphasize assessment - improvement cycles
• ABET 2000 looks a lot like MSE reform -- there is
  a major difference ABET has teeth that bite,
  NAS-MSEB does not, NSF DUE carrots are small
  compared to ABETs teeth.

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The Foundation Coalition
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The thrusts of the FC

• An improved human interfaceteam-based/cooperati
  ve learning.
• Curriculum IntegrationLess segregation of
  subjects. More emphasis on ties bet-ween
  subjects. Provide more realistic, contextual
  settings.
• Technology-enabled problem solving
• Diversity. Increase proportion of traditionally
  underrespresented groups in engineering.
• Assessment, evaluation,dissemination.(defining
  desired outcomes, establishing measurement tools,
  closing the feedback loop).

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FCs 1st-year prgm at ASU

• Since Fall 1994, 32 students in 1st pilot, now
  start w/ 80
• Intro to Engineering, English composition I and
  II, Physics I and II, Calculus I and II
• Faculty slowly rotating in and out, e.g. 2-3
  years each
• Retention and assessment data positive (see the
  experts)

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FC 2nd-year program at ASU

• 1995 to 1998 different packages of courses
  tried
• recently ElecCircuits, DiffEquns, VectorCalculus
• scheduling and other difficulties lead to low
  enrollment
• success / more emphasis on active learning,
  technologyintegration of math and engineering
  less successful
• next year pilot a new DE course for engineering
  studs

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Projects as umbrella

• Two or three month-long integrated team projects
  each semester (e.g.
  bungee omelet)
• Integrated final team exams (sometimes
  additional
  shortened
  exams in
  individual
  subjects)

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Bungee-omelette project
A 1st semester team project, due in week
13Model the free-fall / elastic stretch
including damping calculate, optimize, design
release mechanism Objectives Longest possible
free-fall , as close to the ground as possible,
constraints on max acceleration Engineers
INTEGRATE the nonlinear, 2ndorder, only
piecewise smooth, DE no matter whether math
delivers or not -- use EXCEL in the 1st semester
for what math usually barely delivers in 4th
semester.
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My (our?) guiding philosophyShaping the Future
of SMET(1997 NSF-report, Mel George)
http//www.ehr.nsf.gov/EHR/DUE/documents/review/96
139/summary.htm
The goal indeed, the imperative deriving
from our review is that All students have
access to supportive, excellent undergraduate
education in science, mathematics, engineering,
and technology, and all students learn these
subjects by direct experience with the methods
and processes of inquiry. America's
undergraduates all of them must attain a
higher level of competence in science,
mathematics, engineering, and technology.
America's institutions of higher education must
expect all students to learn more SMET, must no
longer see study in these fields solely as narrow
preparation for one specialized career, but must
accept them as important to every student.
America's SMET faculty must actively engage
those students preparing to become K-12 teachers
technicians professional scientists,
mathematicians, or engineers business or public
leaders and other types of "knowledge workers"
and knowledgeable citizens. It is important to
assist them to learn not only science facts but,
just as important, the methods and processes of
research, what scientists and engineers do, how
to make informed judgments about technical
matters, and how to communicate and work in teams
to solve complex problems.
inquiry based learning
problem solving
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Inquiry-based learning in math?
Computer technology as a vehicle towards an
inquiry-based approach to math
Dont just tell the answers if the questions
have not even been asked yet
Theory Physics, engineering, and math
computer-experimentsgenerate the need for
analysis --- in turn math providestools for
efficient problem solving in phys and engineering
Typical example (calculus reform standard)

• Calculus a la Bourbaki
• Define a sequence (as subset of N x R)
• Define convergence of sequences
• Define series
• Define convergence via partial sums
• Develop battery of convergence tests
• Define power series
• Analyze radii of convergence
• Apply to Taylor series

• Start w/ problem that demonstrates the for need
  better approximations
• Go from linearization to polynomial approximation
• Discover convergence as order increases
• Formalize convergence
• Discover finite intervals of convergence,
  establishing a need for new tools
• Analyze geometric series as special case of power
  series
• Develop ratio test, and formalize comparison
  criteria
• Develop error-bounds that allow a-priori
  determination of required order

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Technology in the classroom
Calculus II Naïve Fourier approximations

• Work with a real signal (sound)
  supplied by physics

• Fourier decomposition to be utilized by
  engineering (inputforcing) of linear
  circuits..
• Hands-on in-class use of technology, first
  EXCEL, (data is 2 x 3000 table), then MATLAB
  (if we get that far)
• Ultimate collaboration in classroom-- different
  teams work with different base frequencies
  (signal is NOT periodic. Studs did not agree
  what to use for base frequency due to drift..)

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Technology in the classroom

• Technology not an add-on lab, but fully
  integrated (compare RPI studio,
  workshop-physics, .).
• Everyday computers are used, often only a
  little.
• Syntax problems, need for experimentation
  almost forbid an environment of students working
  alone. Teamwork is natural!
• Students have access to all computer-software
  in all examinations (except a few
  basic-skills gateway tests that are taken
  on-line). Consequently we need to pose more
  intelligent problems on the tests -- often
  these are inverse problems (that are less
  likely to be trivialized by computer
  technology) -- thereby getting again close to
  engineering design

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Technology -- articulation

• Need to agree among disciplines which software to
  use.This is VERY PAINFUL, and VERY HEALTHY!Need
  to compromise, is opportunity to learn from
  others.Also sharing technology overhead makes
  all lives easier!
• MBL and Vernier sensors used throughout.
• EXCEL
• amazingly powerful (all the way to elliptic
  PDEs)
• most suitable for concept-development,
  tangible
• ideal for experimentation with live
  graphs/tables
• Determination to use professional tools
• graphing calculators are out
• AZ-software (Lomen-Lovelock) was given up
• keeping MAPLE required a making a good case
• managed to keep MATHCAD out
• managed to postpone intro of MATLAB
• PSPICE 2nd half of 2nd semester.
• More compromises
• LaTeX, ScientificWorkplace, ghostview, .ps
  Linux did not make the cut, have to live
  with MS-equation editor
• Powerpoint considered essential by engineers.
• Word (Wordperfect) as far as English would go.
  

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Technology -- daily practice

• Timing of introduction Not each instructor
  can introduce her/his pet-software in
  week one. Need to compromise when to introduce
  what
• Timely reinforcement by cross-utilization once
  software has been introduced (are we
  willing to use PSPICE, MBL in calculus???
  do we expect physicist to learn MAPLE ??)
  e.g. real data (e.g. EXCEL data table) from
  physics experiments are basis for analysis
  in mathematics.
• Consistent policies / agreements needed also
  for WWW-surfing during class? Which
  software is allowed on exams? Electronic /
  internet cheat-sheets? How much are students
  expected to learn on their own? (e.g.
  detailed MAPLE/MATLAB help keystroke by keystroke
  - or learning on the job as in.ppt,
  .doc, WWW, printing?)

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Curriculum integration

• Why? Many, but not everyone, subscribe to
  this concept. Before starting
  anything make sure that
  fundamental philosophies are compatible
• It is MUCH harder than anyone expects! Be
  proud of any, whatever small achievement.
• We started MONTHS before any class -- exchange
  texts, tests, old syllabi, made presentations
  to each other of the central objectives of
  respective classes..
• Agree right away that certain things are simply
  not doable e.g. Bungee-DEs in week 9,
  Line integrals and divergence theorem in
  week 13 Linear algebra, derivatives,
  integrals, vector calc and
  differential equations (resonance) all in 1st
  year?
• Never underestimate the severity of the impacts
  of different notation and language
  function, solve,

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Day-by-day integration

• After meetings long before we start, lots of
  reading, e-mail, and planning, faculty team
  meets every week throughout the academic
  year. (Team-training, how to hold
  effective meetings is just as important
  for faculty as it is for students!!!)
• Typical agenda items Forming new student
  teams, monitoring student teams. Early
  attention to possible problem cases --gt
  retention!!! Sharing progress made and
  concerns of items not yet mastered by
  students at desired level.. Fine-tuning
  timing of exercises (who goes how far on
  which day -- what is students responsibility
  what can each faculty team member expect how
  far others went) Common minor changes in
  schedules Lots of small things, like common
  notation r ? R, t ? T, d for distance??
  (dd/dt)?

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Communication

• A major, real benefit
• In the integrated program it is much easier to
  enforce high standards for presentation of
  student, incl punctuation, grammatically
  correct complete sentences, spelling. No
  longer acceptable are scratch paper like
  collections of half-finished equations with a
  boxed numerical answer With ABET and united
  faculty team students appear to be much
  more willing to accept the standards, do not
  just consider them harassment.
• Personally, I include at least one substantial
  writing assignment on every test -- the
  results very well illuminate the real
  level of understanding acquired Explain in
  your own words what it means
  for a Taylor series to converge.
  Compare and contrast Taylor and Fourier
  approximations. What is a derivative?
  What is calculus about? (still hard after
  three semesters!)

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Team projectRolling races
Experiment and analyze objects rolling down an
inclined plane.Design a rolling object that will
win a competition.Engineering Modelling and
design process. Teamwork.Physics Rotational
kinetic energy. Integrate DEs of motion.
Overcoming major misconceptions.Mathematics
Set-up and use definite integrals to calculate
moments of inertia (of rotationally
symmetric bodies). Applied optimization.
The traditional physics problem analyzes
rolling objects on an inclined plane. It goes as
far as asking which object will win the race
(compare D. Druckers Mathematical Roller
Derby in CMJ 11/1992). The calculus link are
moments of inertia, i.e. iterated integrals,
and a simple separable DE.The problem solution
never goes beyond the level of analysis. The
engineering problem goes one CRITICAL STEP
further We ask thestudents to apply the
knowledge gained by DESIGNING and BUILDING a
rolling object that will win a race in the class!
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details, details,.
Final competition as head-2-headrace versus
time-trials w/ profes-sional timing equipment???
PROUD winners of competition
One instructor helps students w/ one tricky part
and therebygives away the solution that
students were supposed todevelop in the other
subject .need ever more communication
Design specifications set byengineer trivialized
mathoptimization -- 40cm max???more
communication
The hands-on BUILDINGis essential to get a
completewell-rounded project -- donot stop w/
computer simu- lations ..
The results are amazingly fast Further useof
calculus yields an optimal design with J0.02
ma2 as opposed to J0.40 ma2 for a solid
billiards queue ball!!!
Open ended problemExtreme slippage .???
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Aiming high - my favorite A third semester
project
(So far done only as a math-project in the
FC-sophomore program -- hopefully this will form
basis of Mechanics-CalcIII integrated course..)
A compelling, non-EM project for Stokes
theorem.Perfect match Need for new problems
that are not trivializedby modern software ?
inverse questions ? engineering design
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Use 3D-reorientation problem for motivation. --
Student are intrigued by a sophomore class that
connects to NASA andcurrent research -- rather
than just covering 300-year-old stuff!
But play smart Projectis 2-D model that
whiledoable at this level, still...
exhibits the fundamental features that make the
3D-models work, andthat coincides with core-math
topic of the sophomore class ((R.Murray also
discussed only the 2D-model at NAS workshop.))
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Objective Reorient the three-body assembly via
internal motions

• Mathematical (vector calculus) content
• traditional emphasis, physics point of view
  conservative ( integrable ) vector fields,
• closed loops lift to a potential surface

Modern emphasis, engineering point of view
Controllable ( nonintegrable ) vector fields,
design the closed loop in base so that the
vertical gap of the lifted curve is as desired.
q2
Traditional Given F and C find Da (boring w/
computer algebra system)Modern Given F and Da
find C (intelligent, ubiquitous applications)
q1
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Concluding remarks
Integrating math into the engineering curriculum

• It can be done.
• It is VERY hard work.
• There may be no viable alternative.
• It is much less painful than it looks.
• We dont really have to give up much.
• It may improve our own programs.Learn from the
  engineers how to cope with new demands.
• Start right now! Actively shape the future,
  instead of just being shaped by it.
• Work togetherlots o/ meetings/e-mail.