USING COMPUTABLE DOCUMENTS FOR
ONLINE MATHEMATICS INSTRUCTION

James Howard

University of Maryland University College

Problems teaching mathematics online

Existing problems teaching mathematics

Notation is intimidating for many students

Many students feel a certain anxiety at the subject

New problems teaching mathematics online

Asynchronous communications creates delay between question and answer

Static instructional material is unable to respond to student demands

Creating interactive examples in Mathematica

Advantages

Can show the student how the example responds to changes

Presents a clean and consistent interface

Has strong notation formatting capabilities

Disadvantages

CDF is not available by default on most computers

Response must be preprogrammed

Exponential Growth Example

Manipulate[Plot[{P*E^((k/100)*t), g}, {t, 0, 10}, PlotRange->{0, 100}, 
PlotLabel->Row[{"Time to goal: ",   AccountingForm[Log[g/P]/(k/100.0),3], " years"}],
AxesLabel->{"Years","Dollars"}, ImageSize->{500}], 
{{P, 35, "Principal ($)"}, .01, 100, Appearance->"Labeled"},
{{k, 4, "Interest rate (%)"}, 1, 10, Appearance->"Labeled"},
{{g, 50, "Goal ($)"}, 25, 100, Appearance->"Labeled"},
Deployed->t, ControlPlacement->Left, FrameLabel->"Savings Growth"]

Function Translation Example

Manipulate[Plot[{x^2+1,(x+xshift)^2+1+yshift},{x,-10,10},PlotRange->10,ImageSize->600, PlotLabel->Row[{"f(x) = (x + ",xshift, ")^2  + (1 + ",yshift,")\nf(x)= (x^2) + 1 with horizontal translation: ", xshift,  " vertical translation: ",yshift,""}]],
{{xshift,0,"Horizontal shift"},-5,5,Appearance->"Labeled"},{{yshift,0,"Vertical Shift"},-5,5,Appearance->"Labeled"},FrameLabel->"Function Translation"]

Sine/Cosine/Tangent Example

Manipulate[Framed@Graphics[{Thick, Black, Line[{{0,0},pt1,pt2, {0,0}, {0,1}, {1,1}, {1,0}}], 
Blue, Circle[pt1, 1, {(3/2)Pi,ArcSin[(( (Sqrt[(pt2).(pt2)]))/( (Sqrt[(pt1-pt2).(pt1-pt2)])))]+(3/2)Pi}],
Text["α", pt1 +{.25, -1.25}],Rotate[Text[Row[{"A = ", N[(Sqrt[(pt1).(pt1)])]}],{-.2,((Sqrt[(pt1).(pt1)])/2)}], (Pi/2)],
Red, Circle[pt2, 1, {Pi,Pi-ArcSin[((Sqrt[(pt1).(pt1)])/( (Sqrt[(pt1-pt2).(pt1-pt2)])))]}],
Text["β", pt2 +{-1.25, .25}],Text[Row[{"B = ", N[(Sqrt[(pt2).(pt2)])]}],{((Sqrt[(pt2).(pt2)])/2),-.2}],
Black, Rotate[Text[Row[{"C = ", (Sqrt[(pt1-pt2).(pt1-pt2)])}],{((Sqrt[(pt2).(pt2)])/2)+.2,((Sqrt[(pt1).(pt1)])/2)+.2}], ((3Pi)/2)+ArcSin[(( (Sqrt[(pt2).(pt2)]))/( (Sqrt[(pt1-pt2).(pt1-pt2)])))]],
Black, Text[Style[Column[{Row[{"∠α = ", N[ArcSin[(( (Sqrt[(pt2).(pt2)]))/( (Sqrt[(pt1-pt2).(pt1-pt2)])))]]}],
Row[{"sin α = ", N[(( (Sqrt[(pt2).(pt2)]))/( (Sqrt[(pt1-pt2).(pt1-pt2)])))], ", ", (( (Sqrt[(pt2).(pt2)]))/( (Sqrt[(pt1-pt2).(pt1-pt2)])))}],Row[{"cos α = ", N[(( (Sqrt[(pt1).(pt1)]))/( (Sqrt[(pt1-pt2).(pt1-pt2)])))],", ", (( (Sqrt[(pt1).(pt1)]))/( (Sqrt[(pt1-pt2).(pt1-pt2)])))}],Row[{"tan α = ", N[(( (Sqrt[(pt2).(pt2)]))/( (Sqrt[(pt1).(pt1)])))], ", ", (( (Sqrt[(pt2).(pt2)]))/( (Sqrt[(pt1).(pt1)])))}],
}], FontSize->20], {8.5,8.5}]
},
ImagePadding->20,ImageSize->500, PlotRange->{{0, 10}, {0,10}},
GridLines->If[gridLines,{Range[0, 10], Range[0,10]}, None]],
{{pt1,{0,10}},{0,1},{0,10}, {1,1}, Locator},
{{pt2,{10,0}},{1,0}, {10, 0},{1,1},  Locator},
{{gridLines, False, "Show grid?"}, {False, True}},
FrameLabel->"Sines, Cosines, and Tangents"]

Best practices

Create simple examples

Examples should be self contained

Examples should only cover one topic

Interactivity should benefit the student

Examples should not use interactivity that does not further the learning experience

Examples should enrich the learning experience

Use other material

Do not use CDF as an exclusive medium

Textbooks, videos, and other material can help students using their own learning methods

Mesh interactive examples with those textbooks and other media, such as videos

Conclusions

Acknowledgments

Peter Van Buren, Ray Board, Suzanne Sands, and Kat Duncan

See Also

Maria Andersen, “Wolfram Demonstrations and the Online Classroom” (presented at the International Mathematica User Conference, Champaign, Ill., October 2008).

Marcus McGuff, “Give Me Tools, Not Tutorials” (presented at the International Mathematica User Conference, Champaign, Ill., October 2009).

Richard Mercer, “Interactive Tools for Mathematics Instruction” (presented at the International Mathematica User Conference, Champaign, Ill., October 2009).