Mathematica Across the Curriculum: A case study in use throughout the Mathematics major.

Dr. Chris Lee
Associate Professor of Mathematics
Roanoke College

Abstract

At Roanoke College our Mathematics department has always had a strong committment to incorporating technology in our courses.  As most schools can attest, the incorporation of technology brings with it a continual struggle as to what technology should be used.

Our previous model had students using advanced calculators for courses during the first few years, and then as the complexity of needed calculations and graphics increased, Mathematica would be incorporated.  As we revisited our curriculum over the last few years we identified an important goal of continuity,  of not just teaching a check-list of courses, but focusing on the interrelations between the courses.  We were struck by how we were not applying this same goal to our use of technology.

Our solution was simple: incorporate Mathematica throughout our Mathematics curriculum.

The focus of this presentation will be to share the decisions and deliberations we went through with this choice, and most importantly to highlight the advantages we are already seeing.  For instance, an advantage of using Mathematica in the very first course that our majors encounter is that the students ability to use the software  will grow with the complexity of the problems they are solving.  This is a great improvement over a student having their first experience with Mathematica require them to visually analyze the solution to a complicated partial differential equation in an upper level course.  That presents a dauntingly steep learning curve.

Additionally, specific applications of how we are incorporating Mathematica will be presented.  Our first two calculus courses include a 90 minute weekly lab.  These labs are completely written in Mathematica and students work through notebooks as they explore ideas.  As an example, our lab on projectile motion includes the following exercise:

A golf shot going downhill.
Now, suppose that the fairway is not level and slopes  downward at a constant tilt.  In the manipulative below, α measures the angle from horizontal of the fairway.
In the manipulative below, α measures the angle from horizontal of the fairway.

Manipulate[Module[{v0,maxX,impacttime,xpos,ypos,distance,grassdepth,v0Vector,golfball,ball,anglelabel,fairwaylabel,x,y,grass,axis,trajectory},
v0 = 180*5280 /3600;
maxX = 2800;
impacttime = (v0 Sin[θ Degree]-v0 Tan[α Degree]Cos[θ Degree])/16;
xpos[t_]:= v0 Cos[θ Degree] t;
ypos[t_]:=v0 Sin[θ Degree] t - 16(t^(2  ));
distance = EuclideanDistance[{0,0},{xpos[impacttime],ypos[impacttime]}];
grassdepth = 500;
v0Vector=Graphics[{RGBColor[1,.47,0],Arrow[{{0,0},{(v0/3)5Cos[θ Degree],(v0/3)5Sin[θ Degree]}}]}];
golfball = ;
ball = Graphics[{Translate[Scale[golfball[[1]],{1.2,1.2}],{xpos[impacttime]-30,ypos[impacttime]+5}],
Text[Style[Row[{NumberForm[N[distance ],{5,1}]," ft." }],FontSize->12],{xpos[impacttime]+145,ypos[impacttime]+75}]
}];
anglelabel=Graphics[{Black,Circle[{0,0},.8 v0,{0,θ Degree}],{Dashed,Line[{{0,0},{350,0}}]},
Text[Style[Row[{"θ = ",NumberForm[N[θ ],{5,1}] }],FontSize->12],{1.5 v0 Cos[(θ Degree)/2],1.1 v0 Sin[(θ Degree)/2]}]}];
fairwaylabel = Graphics[Text[Style[Row[{"Fairway angle α = ",NumberForm[N[α ],{5,1}]," degrees" }],FontSize->12],{800,-4grassdepth/5}]];
grass=Graphics[{ColorData["HTML","LightGreen"],Polygon[{{0,-grassdepth},{0,0},{maxX,maxX*Tan[α Degree]},{maxX,-grassdepth}}]}];
axis= Graphics[{
Line[{{0,0},{maxX,maxX*Tan[α Degree]}}],
Table[Line[{{i Cos[α Degree],i Sin[α Degree]-20},{i Cos[α Degree],i Sin[α Degree]}}],
{i,500,maxX,100}],
Table[Line[{{i Cos[α Degree],i Sin[α Degree]-50},{i Cos[α Degree],i Sin[α Degree]}}],
{i,500,maxX,500}],
Table[Text[Style[i,FontSize->10],{i Cos[α Degree],i Sin[α Degree]-100}],
{i,500,maxX,500}]
}];
y= Tan[θ Degree]* x-(32/(2(v0^2) ((Cos[θ Degree])^2))) (x^2);
trajectory = Plot[y,{x,0,maxX},PlotRange->{{0,maxX},{-grassdepth,1100}},Axes->False,ImageSize->{400,250},
PlotLabel->Row[{"Golf Drive on a Sloping Fairway"}]];
Show[trajectory,v0Vector,grass,anglelabel,ball,axis,fairwaylabel]],
{{θ ,25  ,"Set the angle of the shot   θ"},1 ,89  ,.5  ,ControlPlacement->Left,ControlType->VerticalSlider},
{{α,-6 ,"Set the angle of the fairway   α"},-10 ,0 ,.1 ,ControlPlacement->Right,ControlType->VerticalSlider}]

Questions: If the fairway is sloping down at 6 degrees, what is the furthest the golfer can hit the ball?  Are you surprised at how this number compares to earlier answers?  At what angle does the ball need to be hit for maximum distance at this angle? If the fairway is sloping down at 10 degrees and the pin is 2100 feet away, at what angle should the golfer hit the ball to go for a hole-in-one?

A separate technological issue that will be addressed is the use of tablet PC's to run Mathematica live during class.  Currently all of my class lecture notes are done live on a tablet. This allows me to pull in graphics generated by Mathematica and mark them up with inking and highlighting.  This adds an element to the classroom that is impossible to acheive with calculators and a chalkboard.

Finally, the importance of support during development will be discussed.  Members of our department received a curriculum development grant to support the task of rewriting all of our calculus labs as we transitioned from calculators to Mathematica.  Additionally,  Wolfram research provided support  in our transition to a new license structure that allowed for student-computer use, critical to any wide scale adoption of the software.