Philosophy of the Math Department

Mathematical Literacy  All students must be mathematically literate  They must perform in the workplace  They will be lifelong learners  They must be problem solvers

Mathematical Literacy for Engineers  Used to learn engineering concepts  Apply concepts in real life situations  Be a lifelong learner in chosen profession Mathematics is the tool that makes these possible

Demands of Advancing Technology  Today’s engineer needs a working knowledge of Patterns Functions Algebra Spatial relationships Geometry Measurement Data analysis Probability Competent use of technology

Use of Technology in the Classroom  We are riding on a wave of change  It is not going away  We cannot reject it or ignore it  ABET requires it  We must find a balance of how best to use these new technologies Without sacrificing basic mathematical skills Engineering programs must demonstrate that their students attain: … k) an ability to use the techniques, skills, and modern engineering tools necessary for engineering practice. Engineering programs must demonstrate that their students attain: … k) an ability to use the techniques, skills, and modern engineering tools necessary for engineering practice.

Concerns  Students who cannot envision basic functions  Inability of students to evaluate the reasonableness of a calculator answer  Lack of basic skills with Algebra Derivatives Integrals Differential Equations “students have become less familiar with basic algebra and trigonometry. This change has been coincident with their heightened usage of calculators” Bill Graff “students have become less familiar with basic algebra and trigonometry. This change has been coincident with their heightened usage of calculators” Bill Graff

Addressing Concerns  Stressing basic functions, drilling recognition  Repeatedly discussing whether an answer given by technology is reasonable  Requiring “Gateway” tests to demonstrate/review basic derivative and integration skills without a calculator

Addressing Concerns  Repeated reminders that a calculator can be used both to solve a hard problem and make a very bad mistake  Learning to use the calculator as a tool Remember that misuse of the tool is not the tool’s fault A hammer can be used to build a mansion or break a window

Good Uses of Technology  Discovery teaching  Making connections  “Messy” problems  Using a variety of solution strategies

Discovery Teaching  Example: Pose the problem of finding the derivative of ln(x) using the limit definition for a derivative

Messy Problems  Consider  Is the decomposition what we want the student to learn?  Or is it to be able to use it to do something else with (inverse Laplace transform)?

Shift of Teaching Strategies  Our teaching goals are shifting from Performance of mathematical operations To the use of mathematical concepts.  Assessment methods Two tiered exams  Without the calculator to assess basic understanding of the material  With the calculator to assess problem solving skills

Shift of Teaching Strategies  Use various ways of looking at a problem Formulas Tables of values Graphs Textual descriptions  This aids all learning styles

Shift of Teaching Strategies  Consider the classical parachute problem We must ask more than the usual “after how many seconds will the parachutist hit the ground?”  We give students direction by asking more detailed questions  Have them analyze the motion of the falling body Geometrically, Numerically, and Analytically.

Varieties of Solution Strategies  Try alternate methods to find a solution  Look at the graph  Manipulate the formula  View the table of values, use regression to come up with a mathematical model

Implications of Available Technology  Our role as guides in the learning process is more important than ever  Must decide when to use/not use technology  The challenge for all of us … Take advantage of the symbolic computation possibilities and do more mathematics … more engineering

Philosophy of the Math Department

Problem  Consider f(x) = log a x  What if we try to use the definition for derivative using the limit  No way to break up this portion of the expression to let h → 0

Possible Solution  We know that the derivative is the "slope function"  What if we graph y=ln(x) and check the slopes … plotting them

Slope Results  The table at the right shows the values of the slopes at various x values  What function might this be?  Appears to be x slope of ln(x) at x

Derivative of the Log Function  For the natural logarithm ln(x)  For the log of a different base log a (x)