The role of a visual language in reconnecting a compartmentalized curriculum Shannon Holland Ctr. for Innovation in Engin. Educ. Arizona State University Tempe, AZ 85287 firstname.lastname@example.org http://ciee.eas.asu.edu/fc/microscope Matthias Kawski Department of Mathematics Arizona State University Tempe, AZ 85287 email@example.com http://math.la.asu.edu/~kawski This work was partially supported by the National Science Foundation: through the grants DUE 97-52453 (Vector Calculus via Linearization: Visualization …) and DUE 94-53610 (ACEPT), and through the Cooperative Agreement EEC 92-21460 (Foundation Coalition)
Visual language ? • Algebraic symbols are one, but not the only way to do mathematics, or to learn mathematics… • Why now, not at previous times? • Before the printing press? • Before the PC? • Before JAVA? • New technologies suggest to reevaluate old paradigms!
Disconnected Curriculum I • Common occurrence, cycles….. • Efficiency: establish standard syllabus with well-delineated courses • specialists perfect each syllabus • while communication with original customersfades away • sudden uproar asks to re-evaluate objectives… • courses adapt, or are replaced by new “courses”…..
Disconnected Curriculum II • Concerns: • Waste of resources, endless duplication • Not taking advantage of structural reinforcementthrough “cross-links” (c.f. A.Gleason, Samos 1998) • Poor public image w/ all undesired consequences… • Uninspired students, math is conceived as a collectionof unrelated facts, rules, algorithms,….. • …...
A specific case • VC and LA have often been combined…. • In 1995 ASU FC identified an integrated coursein VC - DE - Circuits as desirable from organiza-tional point of view (registration,….).Lots of colleagues/students wondered/asked:Do VC / DE share, have anything big in common?
How badly even our knowledgeis compartmentalized During presentation on vector calculus at professional meeting with very good mathematicians in audience: Which of the pictured vector fields is linear?………. Our tenet: Can’t talk about differentiation w/o first understanding “linear”!
How badly even our knowledgeis compartmentalized During presentation on vector calculus at professional meeting with very good mathematicians in audience: Which of the pictured vector fields is linear?………. Our tenet: Can’t talk about differentiation w/o first understanding “linear”! No answers -- until audience is prompted to think in terms of DEs -- there the pictures are familiar, everyone immediately answers!
Connections between VC and DE • Not much in terms of algebraic symbols(aside from the ubiquitous “x” and “d/dx”) • Vector fields (“arrows”) clearly are an obvious tie.But students/faculty don’t trust these “pictures”…WHY NOT?
Connections between VC and DE • Vector fields (“arrows”) clearly are an obvious tie. • But much more is true! Curl and divergence are very meaningful in DEs.Only via DEs do they really acquire meaning!HOW? -- Via interactive pictures, not via formulas!
If zooming is so compelling in calc Iwhy not zoom for curl, div in calc III? In the pre-calculator days limits meant factoring and canceling rational expressions; and secant lines disappeared to a point to reemerge as tangent lines……... Today every graphing calculator has a zoom button.The connection: Derivative <=> local linearity is inescapable Local approximability by linear objects underlies ALL notions of derivative -- yet in the past students often had trouble connecting calc 1, curl/div, Frechet deriv’s
Distinguish zooming for integrals / for derivatives For catalogue see fourth-coming book:“Zooming and Limits: From Sequences to Stokes’ theorem” Zooming in the domain only is appropriate for integrals and continuity Here the domain is the xy-planethe range is represented by arrows
Zooming for derivatives Derivatives always involve a difference:First step is to subtract the drift at point of interest Then magnify domain (xy-plane)and range (arrows) at equal ratesto observe convergence to linear part
Solid knowledge of linearity is critical Zooming for a derivative of a linear object returns the same object! Recognize linearity! 1st subtract drift Then center the lens. Linear objects appear the same on any scale! L(cP)=c L(p) L(p+q)=L(p)+L(q)
Decompositions of linear fields: Basic ideas The easiest case: Skew symmetric“rotation”, “curl” Multiple of the identity“divergence”, “trace”
Interactively visualizing continuity/integrals Zooming of zeroth kind magnifies only domain. Visual approach to “continuity” =“local constancy”needed for: solutions to systems of DEs (Euler, Runge Kutta), and for Riemann integrability (line/surface integrals).
Interactively visualizing curl/divergence In complete analogy to --- lines/slopes before calculus, --- linear functional analysis before convex analysisdevelop curl & divergence first in a linear setting -- almost linear algebra, images are compelling: It is as easy to SEE the curl and the divergence of a linear field as the slope of a line.As lens is dragged, curl and div change (if the field is nonlinear), are constant (if field is linear).
Irrotational is a local property Test case for understanding: Is pictured field “irrotational” ? Many students take a global view, say “NO”, i.e. do NOT understand that any derivative provides info about LOCAL properties. Tactile experience of dragging lens and changing the zoom-factor dramatically convey “local”,“limit”Lens shows that field irrotational (key property of magnetic field about straight wire w/ constant current, or of complex field 1/z, the origin of algebraic topology).
Individual integral curves • Regions evolving under various flows: • Full nonlinear flow • Linearized flow • Components of lin. Flow -- trace (divergence!) -- symmetric part (chaos!) -- skew symm. part (curl) • User draws polygonal region and chooses the flow -- each corresponds to a magn.lens Interactively visualizing various flows