Matthias Kawski , Department of Mathematics Arizona State University

1. Integrating mathematics and 1st/2nd year engineeringExperiences and insights from 5 years teaching in ASU’s 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@asu.edu

2. 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 Overview

3. “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 <-> connections, optimal control Interactive visualization in undergrad courses, currently funded for “Vector Calculus via Linearization: Visualization and Modern Applications” Matthias Kawski

4. The NSF Engineering Coalitions

5. 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 ABET 2000 http://www.abet.org/eac/eac2000.htm CRITERIA FOR ACCREDITING PROGRAMS IN ENGINEERING IN THE US

6. 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. ENGINEERING CRITERIA 2000 PROGRAM CRITERIA

7. 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 ABET’s teeth. ABET 2000 http://www.abet.org/eac/eac2000.htm

8. The Foundation Coalition

9. An improved human interface:“team-based/cooperative 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). The thrusts of the FC

10. 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) FC’s 1st-year prgm at ASU

11. 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 stud’s FC 2nd-year program at ASU

12. 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)

13. Bungee-omelette project A 1st semester team project, due in week 13:Model 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.

14. My (our?) guiding philosophy:“Shaping the Future of SMET”(1997 NSF-report, Mel George) http://www.ehr.nsf.gov/EHR/DUE/documents/review/96139/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 SME&T, 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 SME&T 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”’

15. Computer technology as a vehicle towards an inquiry-based approach to math Inquiry-based learning in math? Don’t 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

16. 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 (input=forcing) 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. Stud’s did not agree what to use for base frequency due to drift……..)

17. 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”

18. 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.

19. 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?)

20. 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”,

21. 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 --> 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 student’s 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)?

22. 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!)

23. Team project:Rolling 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. Drucker’s “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!

24. 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 thereby“gives 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 J=0.02 ma2 as opposed to J=0.40 ma2 for a solid billiards queue ball!!! Open ended problem:Extreme slippage ….???

25. 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-E&M project for Stokes’ theorem…….Perfect match: Need for new problems that are not trivializedby modern software  inverse questions  engineering: design

26. 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….))

27. 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 findDa (boring w/ computer algebra system)Modern: Given F and Dafind C (intelligent, ubiquitous applications) q1

28. It can be done. It is VERY hard work. There may be no viable alternative. It is much less painful than it looks. We don’t 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 together:lots o/ meetings/e-mail. Concluding remarks Integrating math into the engineering curriculum