Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University The role of a visual language in reconnecting a compartmentalized curriculum Matthias Kawski Department of Mathematics Arizona State University Tempe, AZ Shannon Holland Ctr. for Innovation in Engin. Educ. Arizona State University Tempe, AZ This work was partially supported by the National Science Foundation: through the grants DUE (Vector Calculus via Linearization: Visualization …) and DUE (ACEPT), and through the Cooperative Agreement EEC (Foundation Coalition)

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University 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!

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University Disconnected Curriculum I Common occurrence, cycles….. –Efficiency: establish standard syllabus with well-delineated courses –specialists perfect each syllabus –while communication with original customers fades away –sudden uproar asks to re-evaluate objectives… –courses adapt, or are replaced by new “courses”…..

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University Disconnected Curriculum II Concerns: –Waste of resources, endless duplication –Not taking advantage of structural reinforcement through “cross-links” (c.f. A.Gleason, Samos 1998) –Poor public image w/ all undesired consequences… –Uninspired students, math is conceived as a collection of unrelated facts, rules, algorithms,….. –…...

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University A specific case VC and LA have often been combined…. In 1995 ASU FC identified an integrated course in 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?

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University How badly even our knowledge is compartmentalized Our tenet: Can’t talk about differentiation w/o first understanding “linear”! During presentation on vector calculus at professional meeting with very good mathematicians in audience: Which of the pictured vector fields is linear?……….

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University How badly even our knowledge is compartmentalized Our tenet: Can’t talk about differentiation w/o first understanding “linear”! During presentation on vector calculus at professional meeting with very good mathematicians in audience: Which of the pictured vector fields is linear?………. No answers -- until audience is prompted to think in terms of DEs -- there the pictures are familiar, everyone immediately answers!

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University 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?

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University 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!

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University If zooming is so compelling in calc I why 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

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University Distinguish zooming for integrals / for derivatives Zooming in the domain only is appropriate for integrals and continuity Here the domain is the xy-plane the range is represented by arrows For catalogue see fourth-coming book: “Zooming and Limits: From Sequences to Stokes’ theorem”

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University 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 rates to observe convergence to linear part

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University Solid knowledge of linearity is critical L(cP)=c L(p) L(p+q)=L(p)+L(q) Zooming for a derivative of a linear object returns the same object! Recognize linearity! 1 st subtract drift Then center the lens. Linear objects appear the same on any scale!

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University Decompositions of linear fields: Basic ideas The easiest case: Multiple of the identity “divergence”, “trace” Skew symmetric “rotation”, “curl”

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University 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).

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University Interactively visualizing curl/divergence In complete analogy to --- lines/slopes before calculus, --- linear functional analysis before convex analysis develop 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).

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University 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).

Visual language / reconnect compartmentalized curriculum RUMEC SouthBend 9/98 Shannon Holland and Matthias Kawski, Arizona State University Interactively visualizing various flows 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