Creating and Implementing 
Mathematica-based Courseware 
for K-12 Mathematics Classrooms

Kurt Peckman

Product Manager, Precollege Education

Wolfram Research, Inc.

Some Introductory Comments

Your input is important!

My presentation will emphasize 2 things

K-12 Education

Pedagogical issues of Mathematica use rather than elegant Mathematica code

"The views expressed by the speaker..."

Pedagocial Issues of Electronic Lesson Design

The number of necessary representations and

mental "translations" increases.

Cos[π]

Solve[{3  x - 4  y == 12, 2  x + 5  y == 10}, {x, y}]

DSolve[y''[x] + y'[x] - 6  y[x] == 0, y[x], x]

In response to this...

solveSystem[f_, s_] := Solve[{f, s}, {x, y}];

showMe[func_] := 

      Plot[func, {x, -5, 5}, 

        PlotStyle -> {RGBColor[1, 0, 0], Thickness[0.008]}, 

        TextStyle -> {FontFamily -> "Times"}]

showMe[(x^2)]

solveSystem[2  x + 3  y == 12, x - 3  y == 15]

2*9 + 3*-2 == 12

This is the case because...

solveSystem[f_, s_] := 

      Solve[{f, s}, {(f)[[1, 1, 2]], (s)[[1, 2, 2]]}];

solveSystem[2  a + 3  b == 12, a - 3  b == 15]

solveSystem::usage = 

      "solveSystem[first eq., second eq.] solves a system of equtions.";

?solveSystem

The learning modalities of the students are significantly affected.

\(f[x_] := 1\/\(x + 1\); 
NestList[f, 1, 10]\)

N[NestList[f, 1, 10]]

∫x Sin[x] \[DifferentialD]x

In response to this...

A component to notebooks which requires pencil and paper. We will see this 
in many of today's samples, namely Maths & Fun and Chaos with a Heart.

Using Mathematica requires exactness of language.

f[x_] := x + 1; g[x_] := (x^2);

f[g[x]]

g[f[x]]

Table[(1/(1 + i + j)), {i, 0, 5}, {j, 0, 5}]

In response to this...

Solve[3  x + 7 == 111, x]

a = 1

a == 1

The default settings of Mathematica may be radically 
different from the default settings of the student.

(Sqrt[24])

Sin[1]

Sin[45  °]

(((x + 1))^3) // Expand

Solve[(x^2) == Sin[x], x]

FullForm[(1/x)]

Factor[Inverse[(({{a, b}, {c, d}}))] ]

Clear[a]

In response to this...

(Sqrt[24])

(Sqrt[24.0])

Courseware example #1: Maths & Fun

Let's take a look at exp.nb

Interface Issues from the Student's Standpoint

Consistency of buttons and commands.

Example: the AlgebraicManipulation palette versus the BasicInput palette

∫x \[DifferentialD]x

1 + 3 x + 3 (x^2) + (x^3)

\[Placeholder]

In response to this...

When creating buttons, make them only paste buttons.

Requiring students to process mathematics.

Example 1: Maths & Fun

Example 2: Chaos with a Heart

Example 3: The Beginner's Guide to Mathematica version 3

Security.

Example: putting the Option Inspector to use

Appropriate Uses of Mathematica in the K-12 Classroom

As an aid to true comprehension.

TrigToExp[Sin[x]]

Plot[{Abs[x], (x^2)}, {x, -0.1, 0.1}, 

      PlotStyle -> {RGBColor[0, 0, 1], GrayLevel[0]}];

As a visualization tool.

Plot3D[Sin[x] Exp[y], {x, -2 π, 2  π}, {y, -3, 3}];

Knife notebook.

To lift the burden of mathematical "process" allowing students to focus on 
mathematical "product."

Solve[ 3  ageOfJim + 11 == 59, ageOfJim]

What is an "acceptable" range for ageOfJim given that the coefficent is 3? Is 4? Is n?

For use with real or messy data.

\(\(data = Table[i\^2 + \((\(-1\))\)\^i*i*Random[], {i, 0, 20}]; 


    ListPlot[data]; \ \)\)

data

\(\(Table[
	Random[Integer, {1, 6}], {i, 6}]\ \)\)

As an exploratory tool.

Expand[(((x + 1))^3)]

Solve[a (x^2) + b x + c == 0, x]

Solve[a (x^3) + b (x^2) + c x + d == 0, x]

To make conjectures and verify or disprove them.

Factor[(x^3) - 1]

Factor[(x^2) + 1]

Factor[(x^2) + 1, GaussianIntegers -> True]

To see cause and effect.

For bookkeeping.

Courseware example #2: Chaos with a Heart

Let's take a look at cHprog1.nb

Inappropriate Uses of Mathematica in the K-12 Classroom

Using Mathematica to verify solutions already calculated by hand.

\(\(NSolve[{
		x + y + z == 1, 
		2  x - 3  y + 4  z == 10, 
		

        x + 8  y - 14  z == 9
	}, {x, y, z}]\ \)\)

\(NSolve[{
		x + y + z == 1, 
		2  x - 3  y + 4  z == 10, 


        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \( == 8\)
		}, {

        x, y, z}]\)

Now do it such that x  ∈ ℤ, 

      y ∈ ℝ, 

      z ∈ ℂ.

Using Mathematica like paper and pencil.

Plot[Sin[x], {x, 0, 2  π}];

Needs["Graphics`SurfaceOfRevolution`"]

SurfaceOfRevolution[Sin[x], {x, 0, 2 π}, {θ, 0, π}];

Stand-Alone versus Supplemental Courseware

Predetermined objectives versus 

creation of an exploratory, educational environment.

kurtp@wolfram.com

Kurt Peckman

Product Manager, Precollege Education

Wolfram Research, Inc.

100 Trade Center Drive

Champaign, Illinois 61820