text:
User:Egm6321.f09.Team1.andy/HW6

= Problem 13: Legendre Polynomials' Orthogonality = From lecture slide 34-2

Given
Consider the boundary condition
 * {| style="width:100%" border="0" align="left"

\psi(1,\theta) = f(\theta) = T_0 \cos \theta $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

f(\mu) = T_0 \sqrt{1 - \mu^2} $$ $$ is even, and
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

\mu = \sin \theta $$ $$ Eq. 5 from lecture slide 33-2
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

A_n = \frac{2n + 1}{2}  $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 4)
 * }
 * }

Find
1. Without calculation, find property of $$\displaystyle A_n$$, i.e.
 * {| style="width:100%" border="0" align="left"

\begin{cases} A_{2k} = 0 \\ A_{2k+1} = 0 \end{cases} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 5)
 * }
 * }

2. Compute 3 non-zero coefficients of $$\displaystyle A_n$$

2a. Analytical, using either $$\displaystyle \theta$$ or $$\displaystyle \mu$$ as integrating variable

2b. Numerically, using Gauss-Legendre, accurately within $$\displaystyle 5\%$$

Solution
2.

2a.

The expression for $$\displaystyle A_n $$ is given by,

 * {| style="width:100%" border="0" align="left"

\begin{align} A_n & = \frac{2n+1}{2} \int_{-\pi/2}^{\pi/2} f(\theta)P_n(\sin\theta)d(\sin\theta) \\ & = \frac{2n+1}{2} \int_{-1}^{1} f(\mu)P_n(\mu)d\mu \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

where $$\displaystyle \mu = \sin\theta $$ and $$\displaystyle f(\mu) = T_0 \sqrt{1-\mu^2} $$.

$$\displaystyle A_n = 0 $$ for odd values of $$\displaystyle n $$.

(i.) When $$\displaystyle n = 0 $$, we know $$\displaystyle P_n(\mu) = 1 $$ and so,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

A_0 = \frac{2(0)+1}{2} \int_{-1}^{1} T_0 \sqrt{1-\mu^2} d\mu

$$ $$
 * $$\displaystyle (Eq. 12)

 * }.
 * }.

Using definite integrals in Matlab, we get,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\int_{-1}^{1} \sqrt{1-\mu^2} d\mu = \left[ \frac{\pi}{2} \right]

$$ $$
 * $$\displaystyle (Eq. 13)

 * }.
 * }.

Substituting equation (13) in (12),

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow A_0 = \frac{T_0}{2} \left[ \frac{\pi}{2} \right]

$$
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

$$\displaystyle
 * style="width:2%; padding:10px; border:2px solid #8888aa" |
 * style="width:2%; padding:10px; border:2px solid #8888aa" |

\Rightarrow A_0 = \frac{\pi T_0}{4}

$$ $$
 * style= |
 * $$\displaystyle (Eq. 14)
 * }.
 * }.

(ii.) When $$\displaystyle n = 2 $$, we know $$\displaystyle P_n(\mu) = \frac{1}{2} (3

\mu^2 - 1) $$ and so,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

A_2 = \frac{2(2)+1}{2} \int_{-1}^{1} T_0 \sqrt{1-\mu^2} \left[\frac{1}{2} (3\mu^2 - 1) \right]d\mu

$$ $$
 * $$\displaystyle (Eq. 15)

 * }.
 * }.

Using definite integrals in Matlab, we get,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\int_{-1}^{1} \sqrt{1-\mu^2}\left[\frac{1}{2} (3\mu^2 - 1) \right] d\mu = \left[ \frac{-\pi}{16} \right]

$$ $$
 * $$\displaystyle (Eq. 16)

 * }.
 * }.

Substituting equation (16) in (15),

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow A_2 = \frac{5T_0}{2} \left[ \frac{-\pi}{16} \right]

$$
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

$$\displaystyle
 * style="width:2%; padding:10px; border:2px solid #8888aa" |
 * style="width:2%; padding:10px; border:2px solid #8888aa" |

\Rightarrow A_2 = \frac{-5\pi T_0}{32}

$$ $$
 * style= |
 * $$\displaystyle (Eq. 17)
 * }.
 * }.

(iii) When $$\displaystyle n = 4 $$, we know $$\displaystyle P_n(\mu) = \frac{35}{8} \mu^4 - \frac{15}{4} \mu^2 + \frac{3}{8} $$ and so,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

A_4 = \frac{2(2)+1}{2} \int_{-1}^{1} T_0 \sqrt{1-\mu^2} \left[\frac{35}{8} \mu^4 - \frac{15}{4} \mu^2 + \frac{3}{8} \right]d\mu

$$ $$
 * $$\displaystyle (Eq. 18)

 * }.
 * }.

Using definite integrals in Matlab, we get,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\int_{-1}^{1} \sqrt{1-\mu^2}\left[\frac{35}{8} \mu^4 - \frac{15}{4} \mu^2 + \frac{3}{8} \right] d\mu = \left[ \frac{-\pi}{128} \right]

$$ $$
 * $$\displaystyle (Eq. 19)

 * }.
 * }.

Substituting equation (19) in (18),

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow A_4 = \frac{9T_0}{2} \left[ \frac{-\pi}{128} \right]

$$
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

$$\displaystyle
 * style="width:2%; padding:10px; border:2px solid #8888aa" |
 * style="width:2%; padding:10px; border:2px solid #8888aa" |

\Rightarrow A_4 = \frac{-9\pi T_0}{256}

$$ $$
 * style= |
 * $$\displaystyle (Eq. 20)
 * }.
 * }.

2b. The expression for $$\displaystyle A_n $$ is given by,

 * {| style="width:100%" border="0" align="left"

\begin{align} A_n & = \frac{2n+1}{2} \int_{-1}^{1} f(\mu)P_n(\mu)d\mu \\ & = \frac{2n+1}{2} \sum_{j=1}^{m} w_jf(x_j)P_n(x_j) \end{align} $$
 * $$\displaystyle
 * $$\displaystyle

 * }
 * }

where,

 * {| style="width:100%" border="0" align="left"

w_j = \frac{-2}{(n+1)P'_m(x_j)P_{m+1}(x_j)} $$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

(i.) For n = 0 and m = 20 we get,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow P_0 = 1

$$

 * }
 * }

and,

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow A_0 & = \frac{2(0)+1}{2} \sum_{j=1}^{20} T_0 w_j \sqrt{1-x_{j}^{2}}\\ & = \frac{T_0}{2} \left[ 1.570892146048340 \right] \end{align} $$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

$$\displaystyle
 * style="width:2%; padding:10px; border:2px solid #8888aa" |
 * style="width:2%; padding:10px; border:2px solid #8888aa" |

\Rightarrow A_0 = 7.854460730241699\cdot 10^{-1} \cdot T_0

$$ $$
 * style= |
 * $$\displaystyle (Eq. 21)
 * }.
 * }.

(ii.) For n = 2 and m = 20 we get,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow P_2 = \frac{1}{2} (3x^2-2)

$$

 * }
 * }

and,

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow A_2 & = \frac{2(2)+1}{2} \sum_{j=1}^{20} T_0 w_j (\frac{1}{2}) (3x_{j}^{2}-2) \sqrt{1-x_{j}^{2}}\\ & = \frac{5T_0}{2} \left[ -1.962526165998847\cdot 10^{-1} \right] \end{align} $$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

$$\displaystyle
 * style="width:2%; padding:10px; border:2px solid #8888aa" |
 * style="width:2%; padding:10px; border:2px solid #8888aa" |

\Rightarrow A_2 = -4.906315414997119\cdot 10^{-1} \cdot T_0

$$ $$
 * style= |
 *
$$\displaystyle (Eq. 22)
 * }.
 * }.

(iii.) For n = 4 and m = 20 we get,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow P_4 = \frac{35}{8} x^4 - \frac{15}{4} x^2 + \frac{3}{8}

$$

 * }
 * }

and,

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow A_4 & = \frac{2(4)+1}{2} \sum_{j=1}^{20} T_0 w_j \left(\frac{35}{8} x_{j}^{4} - \frac{15}{4} x_{j}^{2} + \frac{3}{8} \right) \sqrt{1-x_{j}^{2}}\\ & = \frac{9T_0}{2} \left[ -2.444412881080017 \cdot 10^{-2} \right] \end{align} $$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

$$\displaystyle
 * style="width:2%; padding:10px; border:2px solid #8888aa" |
 * style="width:2%; padding:10px; border:2px solid #8888aa" |

\Rightarrow A_4 = -1.099985796486008 \cdot 10^{-1}\cdot T_0

$$ $$ These numerical solutions are all less than 1 % from the exact solution.
 * style= |
 *
$$\displaystyle (Eq. 23)
 * }.
 * }.

 Matlab Code:

Please refer and download matlab files from Gauss-Legendre Matlab Central. Below is the file functions.m that will be called by gausslege.m function.

Below is the function call to find the numerical integrals every time after modifying the functions.m file for corresponding n values [This is done at the command line or can be done as script].

= Problem 14: Gauss-Legendre Quadrature = From lecture slide 35-3

Given
From lecture slide 31-3, general expression for Legendre Polynomial is given by,

 * {| style="width:100%" border="0" align="left"

P_n(x) = \sum_{i=0}^{\lfloor n/2 \rfloor} \frac{1 \cdot 3 \dots \left(2n - 2i - 1\right)}{2^i i! (n-2i)!} (-1)^i x^{n-2i} $$
 * $$\displaystyle
 * $$\displaystyle

 * },
 * },

and from lecture slide 35-2, weights $$\displaystyle w_j $$ of the Legendre Polynomial $$\displaystyle P_n(x)$$ are given by,

 * {| style="width:100%" border="0" align="left"

w_j = \frac{-2}{(n+1)P'_n(x_j)P_{n+1}(x_j)} $$ \left[\forall j=1,2,...,n \right] $$ where each $$\displaystyle x_j $$ are the roots of $$\displaystyle P_n(x) = 0 $$.
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.
 * }.

Find
1. Verify the expressions on Wikipedia for the Legendre polynomials $$\displaystyle P_n$$ with $$\displaystyle n = 0, 1,..., 6$$. (See homework on lecture slide 31-3)

2. Verify the table for Gauss-Legendre quadrature in Wikipedia, analytical expressions of $$\displaystyle \{x_j\}$$ and $$\displaystyle \{w_j\}$$, $$\displaystyle j=1,...,5$$, and $$\displaystyle n=1,...,5$$, where $$\displaystyle n$$ is the number of integration points.

3. Evaluate numerically $$\displaystyle \{x_j\}$$ and $$\displaystyle \{w_j\}$$ and compare results with Abramovitz & Stegum. See [http://en.wikiversity.org/wiki/User:Egm6321.f09/Lecture_plan lecture plan].

Solution
1. The general expression for Legendre Polynomial is given by,

 * {| style="width:100%" border="0" align="left"

P_n(x) = \sum_{i=0}^{\lfloor n/2 \rfloor} \frac{1 \cdot 3 \dots \left(2n - 2i - 1\right)}{2^i i! (n-2i)!} (-1)^i x^{n-2i} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 1)

 * }.
 * }.

Please refer Problem 6 Solution above, for $$\displaystyle P_0(x),\ P_1(x),\ P_2(x),\ P_3(x),\ P_4(x) $$.

Substituting $$\displaystyle n=5$$ into Eq. 1 gives,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow P_5(x) = \sum_{i=0}^{\lfloor 5/2 \rfloor} \frac{1 \cdot 3 \dots \left(2n - 2i - 1\right)}{2^i i!(n-2i)!}(-1)^i x^{n - 2i}

$$

 * }.
 * }.

The sum is then expanded,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow P_5(x)  = \frac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9} {1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} x^5 + \frac{1 \cdot 3 \cdot 5 \cdot 7 } {2 \cdot (3!)} (-1)x^3 +  \frac{1 \cdot 3 \cdot 5} {2^2 \cdot (2!)} (-1)^2 x

$$ $$
 *
$$\displaystyle (Eq. 2)
 * }.
 * }.

Equation (2) is then,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \Rightarrow P_5(x) = \frac{63}{8} x^5 - \frac{35}{4} x^3 + \frac{15}{8} x $$ $$.
 * style="width:30%; padding:10px; border:2px solid #8888aa" |
 * style="width:30%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 3)
 * }
 * }

Substituting $$\displaystyle n=6$$ into Eq. 1 gives,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow P_6(x) = \sum_{i=0}^{\lfloor 6/2 \rfloor} \frac{1 \cdot 3 \cdot \cdot \cdot \left(2n - 2i - 1\right)}{2^i i!(n-2i)!}(-1)^i x^{n - 2i}

$$

 * }.
 * }.

The sum is then expanded,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

\Rightarrow P_6(x)  = \frac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9 \cdot 11} {1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6} x^6 + \frac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9} {2 \cdot (4!)} (-1)x^4 +  \frac{1 \cdot 3 \cdot 5 \cdot 7} {2^2 \cdot (2!) \cdot (2!)} (-1)^2 x^2 +  \frac{1 \cdot 3 \cdot 5} {2^3 \cdot (3!)} (-1)^3 x^0

$$ $$
 *
$$\displaystyle (Eq. 4)
 * }.
 * }.

Equation (4) is then,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \Rightarrow P_6(x) = \frac{231}{16} x^6 - \frac{315}{16} x^4 + \frac{105}{16} x^2 - \frac{5}{16} $$ $$
 * style="width:30%; padding:10px; border:2px solid #8888aa" |
 * style="width:30%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 5)
 * }.
 * }.

The equations (3), (5) and equations (10), (13), (16), (19), (22) of Problem 6 match the equations in Wikipedia.

2. The weights $$\displaystyle w_j $$ of the Legendre Polynomial $$\displaystyle P_n(x)$$ are given by,

 * {| style="width:100%" border="0" align="left"

w_j = \frac{-2}{(n+1)P'_n(x_j)P_{n+1}(x_j)} $$ \left[\forall j=1,2,...,n \right] $$ $$ where each $$\displaystyle x_j $$ are the roots of $$\displaystyle P_n(x) = 0 $$.
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 6)
 * }.
 * }.

(i). When $$\displaystyle n = 1 $$,

 * {| style="width:100%" border="0" align="left"

P_1(x) = x;\ P'_1(x) = 1; \ P_2(x) = \frac{1}{2}(3x^2-1);when\ P_1(x) = 0 \Rightarrow\ x_1 = 0; $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 7)
 * }
 * }

Substituting values of Equation (7) into (6)

 * {| style="width:100%" border="0" align="left"

w_1 = \frac{-2}{(1+1)(1)(1/2)(-1)} = 2 $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

And so,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle x_1 = 0;\ w_1 = 2 $$ $$
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 8)
 * }.
 * }.

(ii). When $$\displaystyle n = 2 $$,

 * {| style="width:100%" border="0" align="left"

P_2(x) = \frac{1}{2}(3x^2-1);\ P'_2(x) = 3x; P_3(x) = \frac{1}{2}(5x^3-3x);when\ P_2(x) = 0 \Rightarrow\ x_{1,2} = \bigg\{\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}} \bigg\}; $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 9)
 * }
 * }

Substituting values of Equation (9) into (6),

 * {| style="width:100%" border="0" align="left"

w_1 = \frac{-2}{3(3)(\frac{-1}{\sqrt{3}})(\frac{1}{2})\left[5(-1/\sqrt{3})^3 - 3(-1/\sqrt{3})\right]} = \frac{4}{-5+9} = 1 $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

 * {| style="width:100%" border="0" align="left"

w_2 = \frac{-2}{3(3)(\frac{1}{\sqrt{3}})(\frac{1}{2})\left[5(1/\sqrt{3})^3 - 3(1/\sqrt{3})\right]} = \frac{-4}{5-9} = 1 $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

And so,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle x_{1,2} = \bigg\{\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}} \bigg\};\ w_{1,2} = \bigg\{1,1 \bigg\} $$ $$
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 10)
 * }.
 * }.

(iii). When $$\displaystyle n = 3 $$,

 * {| style="width:100%" border="0" align="left"

P_3(x) = \frac{1}{2}(5x^3-3x);\ P'_3(x) = \frac{1}{2}(15x^2-3);\ P_4(x) = \frac{35}{8}x^4 - \frac{15}{4}x^2 + \frac{3}{8}; when\ P_3(x) = 0 \Rightarrow\ x_{1,2,3} = \bigg\{-\sqrt{\frac{3}{5}},0,\sqrt{\frac{3}{5}} \bigg\}; $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 11)
 * }
 * }

Substituting values of Equation (11) into (6),

 * {| style="width:100%" border="0" align="left"

w_1 = w_3 = \frac{-2}{4(1/2)\left[15(3/5) - 3 \right]\left[(35/8)(9/25) - (15/4)(3/5) + (3/8) \right]} = \frac{5}{9} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

w_2 = \frac{-2}{2(-3)\frac{3}{8}} = \frac{8}{9} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

And so,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle x_{1,2,3} = \bigg\{-\sqrt{\frac{3}{5}},0,\sqrt{\frac{3}{5}} \bigg\};\ w_{1,2,3} = \bigg\{\frac{5}{9},\frac{8}{9} ,\frac{5}{9}\bigg\} $$ $$
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 12)
 * }.
 * }.

(iv). When $$\displaystyle n = 4 $$,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

P_4(x) = \frac{35}{8}x^4 - \frac{15}{4}x^2 + \frac{3}{8}

$$ $$
 *
$$\displaystyle (Eq. 13)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

P'_4(x) = \frac{35}{2}x^3 - \frac{15}{2}x; $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 14)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

P_5(x) = \frac{63}{8} x^5 - \frac{35}{4} x^3 + \frac{15}{8} x; $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 15)
 * }.
 * }.

Substituting (13), (14), (15) into equation (6),

 * {| style="width:100%" border="0" align="left"

w_j = \frac{-2}{(5)(\frac{5{x_j}^2}{16})(7{x_j}^2-3)(63{x_j}^4-70{x_j}^2+15)}; \forall j = 1,2,3,4 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 16)
 * }.
 * }.

When $$\displaystyle P_4(x) = 0 $$,

 * {| style="width:100%" border="0" align="left"

\Rightarrow\ x_1 = - \sqrt{\frac{3+2\sqrt{6/5}}{7}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 17)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

\Rightarrow\ x_2 = - \sqrt{\frac{3-2\sqrt{6/5}}{7}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 18)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

\Rightarrow\ x_3 = \sqrt{\frac{3-2\sqrt{6/5}}{7}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 19)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

\Rightarrow\ x_4 = \sqrt{\frac{3+2\sqrt{6/5}}{7}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 20)
 * }.
 * }.

Substituting equations (18) and (19) into (16),

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \Rightarrow\ w_{2,3} = \frac{18+\sqrt{30}}{36} $$ $$
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 21)
 * }.
 * }.

Substituting equations (18) and (19) into (16),

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \Rightarrow\ w_{1,4} = \frac{18-\sqrt{30}}{36} $$ $$
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 22)
 * }.
 * }.

(v). When $$\displaystyle n = 5 $$,

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle
 * $$\displaystyle

P_5(x) = \frac{63}{8}x^5 - \frac{35}{4}x^3 + \frac{15}{8}x

$$ $$
 *
$$\displaystyle (Eq. 23)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

P'_5(x) = \frac{315}{8}x^4 - \frac{105}{4}x^2+\frac{15}{8}; $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 24)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

P_6(x) = \frac{231}{16} x^6 - \frac{315}{16} x^4 + \frac{105}{16} x^2 - \frac{5}{16}; $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 25)
 * }.
 * }.

Substituting (23), (24), (25) into equation (6),

 * {| style="width:100%" border="0" align="left"

w_j = \frac{-2}{6(\frac{315}{8}x^4 - \frac{105}{4}x^2+\frac{15}{8})(\frac{231}{16} x^6 - \frac{315}{16} x^4 + \frac{105}{16} x^2 - \frac{5}{16})}; \forall j = 1,2,3,4,5 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 26)
 * }.
 * }.

When $$\displaystyle P_5(x) = 0 $$,

 * {| style="width:100%" border="0" align="left"

\Rightarrow\ x_1 = - \sqrt{\frac{35+\sqrt{280}}{63}} = - \frac{1}{3}\sqrt{5+2\sqrt{\frac{10}{7}}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 27)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

\Rightarrow\ x_2 = - \sqrt{\frac{35-\sqrt{280}}{63}} = - \frac{1}{3}\sqrt{5-2\sqrt{\frac{10}{7}}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 28)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

\Rightarrow\ x_3 = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 29)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

\Rightarrow\ x_4 = \sqrt{\frac{35-\sqrt{280}}{63}} = \frac{1}{3}\sqrt{5-2\sqrt{\frac{10}{7}}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 30)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

\Rightarrow\ x_5 = \sqrt{\frac{35+\sqrt{280}}{63}} = \frac{1}{3}\sqrt{5+2\sqrt{\frac{10}{7}}} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle (Eq. 31)
 * }.
 * }.

Substituting equations (29) into (26),

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \Rightarrow\ w_3 = \frac{128}{225} $$ $$
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 32)
 * }.
 * }.

Substituting equations (28) and (30) into (26),

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \Rightarrow\ w_{2,4} = \frac{322+13\sqrt{70}}{900} $$ $$
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 32)
 * }.
 * }.

Substituting equations (27) and (31) into (26),

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \Rightarrow\ w_{1,5} = \frac{322-13\sqrt{70}}{900} $$ $$
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style="width:15%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle (Eq. 32)
 * }.
 * }.

3.

(i). The numerical values when $$\displaystyle n = 1 $$ are given by equation (8).

(ii). The numerical values when $$\displaystyle n = 2 $$ are,

 * {| style="width:100%" border="0" align="left"

x_{1,2} = \bigg\{\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}} \bigg\};\ w_{1,2} = \bigg\{1,1 \bigg\} $$
 * $$\displaystyle
 * $$\displaystyle

 * }
 * }

 * {| style="width:100%" border="0" align="left"

\Rightarrow
 * $$\displaystyle
 * $$\displaystyle

\begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix} = \begin{bmatrix} \frac{-1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \end{bmatrix} = \begin{bmatrix} 5.773502691896258e-001 \\ 5.773502691896258e-001 \\ \end{bmatrix}

$$ $$
 *
$$\displaystyle (Eq. 33)
 * },
 * },

and,

 * {| style="width:100%" border="0" align="left"

\Rightarrow
 * $$\displaystyle
 * $$\displaystyle

\begin{bmatrix} w_1 \\ w_2 \\ \end{bmatrix} = \begin{bmatrix} 1.0000000000000000 \\ 1.0000000000000000 \\ \end{bmatrix} $$ $$
 *
$$\displaystyle (Eq. 34)
 * }.
 * }.

(iii). When $$\displaystyle n = 3 $$,

 * {| style="width:100%" border="0" align="left"

w_j = \frac{-2}{(4)(\frac{1}{2})(15x^2-3)(\frac{35}{8}x^4 - \frac{15}{4}x^2 + \frac{3}{8})} $$
 * $$\displaystyle
 * $$\displaystyle

$$
 *
$$\displaystyle (Eq. 35)
 * },
 * },

the numerical values are calculated to be,

 * {| style="width:100%" border="0" align="left"

\Rightarrow
 * $$\displaystyle
 * $$\displaystyle

\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} = \begin{bmatrix} -7.745966692414834e-001 \\ 0.000000000000000 \\   7.745966692414834e-001 \\ \end{bmatrix}

$$ $$
 *
$$\displaystyle (Eq. 36)
 * },
 * },

and,

 * {| style="width:100%" border="0" align="left"

\Rightarrow
 * $$\displaystyle
 * $$\displaystyle

\begin{bmatrix} w_1 \\ w_2 \\ w_3 \\ \end{bmatrix} = \begin{bmatrix} 5.555555555555554e-001 \\ 8.888888888888888e-001 \\ 5.555555555555554e-001 \\ \end{bmatrix} $$ $$
 *
$$\displaystyle (Eq. 37)
 * }.
 * }.

Matlab source code:

.

(iv). When $$\displaystyle n = 4 $$, the expression for $$\displaystyle w_j $$ is given by equation (16) and the numerical values are,

 * {| style="width:100%" border="0" align="left"

\Rightarrow
 * $$\displaystyle
 * $$\displaystyle

\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ \end{bmatrix} = \begin{bmatrix} -8.611363115940526e-001 \\ -3.399810435848563e-001 \\ 3.399810435848563e-001 \\ 8.611363115940526e-001 \\ \end{bmatrix}

$$ $$
 *
$$\displaystyle (Eq. 38)
 * },
 * },

and,

 * {| style="width:100%" border="0" align="left"

\Rightarrow
 * $$\displaystyle
 * $$\displaystyle

\begin{bmatrix} w_1 \\ w_2 \\ w_3 \\ w_4 \\ \end{bmatrix} = \begin{bmatrix} 3.478548451374542e-001\\ 6.521451548625461e-001\\ 6.521451548625461e-001\\ 3.478548451374542e-001\\ \end{bmatrix} $$ $$
 *
$$\displaystyle (Eq. 39)
 * }.
 * }.

Matlab source code:

.

(v). When $$\displaystyle n = 5 $$, the expression for $$\displaystyle w_j $$ is given by equation (26) and the numerical values are,

 * {| style="width:100%" border="0" align="left"

\Rightarrow
 * $$\displaystyle
 * $$\displaystyle

\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ \end{bmatrix} = \begin{bmatrix} -9.061798459386641e-001 \\ -5.384693101056831e-001 \\ 0 \\   5.384693101056831e-001 \\ 9.061798459386641e-001 \\ \end{bmatrix}

$$ $$
 *
$$\displaystyle (Eq. 40)
 * },
 * },

and,

 * {| style="width:100%" border="0" align="left"

\Rightarrow
 * $$\displaystyle
 * $$\displaystyle

\begin{bmatrix} w_1 \\ w_2 \\ w_3 \\ w_4 \\ w_5 \\ \end{bmatrix} = \begin{bmatrix} 2.369268850561896e-001 \\ 4.786286704993663e-001 \\ 5.688888888888889e-001 \\ 4.786286704993663e-001 \\ 2.369268850561896e-001 \\ \end{bmatrix} $$ $$
 *
$$\displaystyle (Eq. 41)
 * }.
 * }.

Matlab source code:

 * {| style="width:100%" border="0" align="left"

The equations (8), (33), (34), (36), (37), (38), (39), (40) and (41) match the numerical values given by Abramovitz & Stegum's table.
 * style="width:78%; padding:10px; border:2px solid #8888aa" |
 * style="width:78%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }.
 * }.

title: User:Egm6321.f09.Team1.andy/HW6


license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Ervin1


text:
Wikiversity:Meetings/Learning on Wikiversity/03June2008/log

This is a log of a meeting on IRC on 3rd June, 2008 about 'learning in Wikiversity' - see Meetings/Learning on Wikiversity.

A short summary you can find here.

intro

 * [5:05pm] cormaggio: ok, maybe we can start and people can join in as they arrive (McCormack might be here)
 * [5:07pm] cormaggio: I'm hoping to get some discussion going around 'learning on Wikiversity' - how it works, what problems exist, what we've experienced, and what we could do about it..
 * [5:08pm] cormaggio: this is partly towards my research - hence this meeting being logged - and I'd like to use some of this discussion as part of my PhD
 * [5:09pm] cormaggio: but I'd like to make clear that I will check this with everyone afterwords to see whether it is ok that I use something, or whether it should be anonymized etc
 * [5:09pm] cormaggio: is that ok, or any questions so far?
 * [5:09pm] assassingr: I'm ok with that
 * [5:10pm] Erkan_Yilmaz: what is the topic of your PhD ? and why the discussion now ? did there anything special happen?
 * [5:10pm] cormaggio: (that's not a request for consent, btw - just to make sure)
 * [5:10pm] Erkan_Yilmaz: with using the info: ok
 * [5:10pm] cormaggio: the topic of my PhD is 'Developing Wikiversity through action research'
 * [5:11pm] cormaggio: (but you don't need to read that!)
 * [5:11pm] Erkan_Yilmaz: no worries :-)
 * [5:12pm] Erkan_Yilmaz: so and now you are at the end of your PHD and want some more info?
 * [5:12pm] cormaggio: why the discussion now - I'm hoping to get an *action plan* together for improving our understanding of how learning works on Wikiversity, and implementing some actions to improve learning on Wikiversity
 * [5:12pm] cormaggio: no - I still see this PHD as in process
 * [5:12pm] cormaggio: very much so
 * [5:13pm] Erkan_Yilmaz: and final question:
 * [5:13pm] cormaggio: there has been a lot of discussion - and action - but I'm not sure I can fit it all into my research question
 * [5:13pm] Erkan_Yilmaz: do you also agree that we can publish the chat afterwards?
 * [5:14pm] Erkan_Yilmaz: (well we can talk also later about that)
 * [5:14pm] cormaggio: oh yes, i agree of course - but I think it's only fair to check *afterwards* just in case someone isn't comfortable with something they said earlier
 * [5:14pm] Erkan_Yilmaz: of course
 * [5:14pm] Erkan_Yilmaz: you mean like: Osama bin Laden?
 * [5:15pm] cormaggio: that's what i meant earlier about feeling comfortable
 * [5:15pm] cormaggio: well, I can't see how Osama fits in here, but.. :-)
 * [5:15pm] Erkan_Yilmaz: :-)
 * [5:16pm] assassingr: and a question from me if Erkan finished with his
 * [5:16pm] _Lou: hi you all !
 * [5:16pm] Erkan_Yilmaz: temporarily I am finished :-)
 * [5:16pm] cormaggio: and another thing I'd like to discuss is the research itself - how you see it and your own relation to it; whether it's been of any influence to you; whether you feel like you're involved ..
 * [5:16pm] assassingr: You are mostly focusing in en.wv?
 * [5:17pm] cormaggio: yes assassingr - that's a good question - it's mainly on en (even though I'm very interested in other projects)
 * [5:17pm] Erkan_Yilmaz: well that would mean you would have to participate there also :-)
 * [5:18pm] Erkan_Yilmaz: or get the knowledge from there to here
 * [5:18pm] cormaggio: yes, exactly - though it;s interesting to hear other projects' participants' reports of their experiences
 * [5:18pm] cormaggio: well, you're both here - so you've brought some knowledge ;-)
 * [5:19pm] Erkan_Yilmaz: another question: how long will this meeting now go ? is there a time limit?
 * [5:19pm] cormaggio: ok - i had thought about 1.5 hours - is that ok, or do you have a cut off point?
 * [5:19pm] Erkan_Yilmaz: that is ok, as long I have coffee
 * [5:19pm] cormaggio: maybe 1 hour could be enough - we'll see
 * [5:20pm] assassingr: we can see that as the conversation rolls

what is the major obstacle to learning in Wikiversity?

 * [5:20pm] cormaggio: so maybe let's start with a problem ;-) - what is the major obstacle to learning in Wikiversity?
 * [5:21pm] cormaggio: what difficulties have we observed people having?
 * [5:21pm] cormaggio: or what difficulties have we had ourselves?
 * [5:21pm] Erkan_Yilmaz: depends on the person - how he learns - one point can be: the used means for learning
 * [5:21pm] Erkan_Yilmaz: WV is mostly just a collection of pages, static
 * [5:21pm] Erkan_Yilmaz: they can, but must not contain multimedia
 * [5:22pm] cormaggio: so, it's a case of not having stimulating enough materials, or..?
 * [5:23pm] Erkan_Yilmaz: yes, that is one - I mean take for example vocabulary learning programs like Rosetta they use many things which stimulate the senses
 * [5:23pm] Erkan_Yilmaz: such things is a little rare at WV
 * [5:24pm] Erkan_Yilmaz: the more senses you use - e.g. when eating - you get a better experience
 * [5:24pm] cormaggio: multimedia has been a long source of frustration for me (and many other people) - our lack of an easy to embed media player is annoying
 * [5:24pm] assassingr: well, at the current stage I think that the lack of material is one problem, but that's not due to the learning model of WV
 * [5:24pm] Erkan_Yilmaz: media upload: yes, that is really bad from commons side to not allow more formats :-(
 * [5:24pm] cormaggio: Erkan - right, so we could be making more interactive materials..
 * [5:25pm] cormaggio: assassingr - that's great - I'd like to know what you think of the 'learning model" of WV - is it one model, or..?
 * [5:25pm] Erkan_Yilmaz: lack of material: that depends what you are searching for I guess - I think stuff like mathematics, IT and so were (e.g. on de.WV) the first things to come - which category/topics you mean assassingr?
 * [5:26pm] assassingr: Erkan_Yilmaz: well, the math, physics and engineering departments which I have checked are not so developed I think
 * [5:27pm] cormaggio: Erkan - the formats issue is a long-standing and very complex one - this is a discussion that's been around the Foundation for a long time - there is a sliding scale from ultra-hardcore people to semi-hardcore people to people that want any content as long as it's free...
 * [5:27pm] Erkan_Yilmaz: depends who is reading it - e.g. most people think the materials must be for university level, since WV contains 'versity'
 * [5:27pm] assassingr: but I think that's about to change in 1 or 2 years from now - take WP as an example
 * [5:27pm] Erkan_Yilmaz: but in general assassingr you are right: the things at WV are less then elsewhere
 * [5:28pm] Erkan_Yilmaz: I also think that material will increase with time: people just think of the image of WP and arrive here at WV and will extend the WV pages
 * [5:28pm] cormaggio: well, on 'not much material', we can take for granted that it's "early days" at WV, right?
 * [5:28pm] Erkan_Yilmaz: yup
 * [5:28pm] Erkan_Yilmaz: WP is now also about 7 years old or?
 * [5:29pm] cormaggio: yeah - Jan 15 2001
 * [5:29pm] assassingr: cormaggio: for starters, I don't believe than any learning model should be static or that there should exist just one
 * [5:29pm] cormaggio: I'm with you there assassingr :-)
 * [5:30pm] assassingr: we could try to develop more learning models and try to categorize them on how people can learn
 * [5:31pm] cormaggio: back to the not much material question - I wonder if there is something holding back content development, as well as it being early days (why, here's McCormack now!)
 * [5:31pm] assassingr: (i don't know if that does make any sense in English, I'm not familiar with education vocabulary so much)
 * [5:31pm] cormaggio: I understand assassingr - and I agree again
 * [5:32pm] Erkan_Yilmaz: any task requires time
 * [5:32pm] Erkan_Yilmaz: and to give for a task x time that task x must have prio for person y
 * [5:33pm] Erkan_Yilmaz: I think people are not so much motivated probably because WV is still young and not like WP or they just don't want to share or didn't learn to share
 * [5:33pm] McCormack: Nothing is holding back content development except lack of contributor retention.
 * [5:33pm] Erkan_Yilmaz: because when they press the save button the things don't belong to them anymore
 * [5:33pm] Erkan_Yilmaz: anyone can theoretically edit it
 * [5:34pm] ¥ McCormack says "contributor retention" again.
 * [5:34pm] cormaggio: well, that's interesting McCormack - that begs the question why we're not retaining contributors..
 * [5:34pm] McCormack: Possibly go back a step and ask first why it is that contributor retention is the problem and not something else?
 * [5:35pm] cormaggio: hold on Erkan - are you saying that this dynamic is different on WV than WP?
 * [5:35pm] McCormack: Successful projects have exponential development - which is what we would like to see.
 * [5:35pm] Erkan_Yilmaz: of course
 * [5:35pm] assassingr: also - when we were about to begin el.wv there was someone that told me that it wouldn't be easy to approach academic people. That way "expert knowledge" wouldn't be freely available and still it would be a prerogative for some few people
 * [5:35pm] Erkan_Yilmaz: one point is: WV and WP have different goals
 * [5:35pm] McCormack: WV has linear development, upwards, steady, but a straight line.
 * [5:35pm] assassingr: I don't know if I agree with that, but it's something to think about
 * [5:35pm] darkcode: it's probably because people try to treat Wikiversity like any other classroom/college/university oriented website when it's not
 * [5:36pm] cormaggio: Erkan - I just want to make sure I understand - the fact that anyone can edit affects contributors on WV more than those on WP?
 * [5:36pm] cormaggio: hi darkcode
 * [5:37pm] ¥ assassingr waves at McCormack and darkcode
 * [5:37pm] Erkan_Yilmaz: yes, because on WP people try to be objective because they are creating an encyclopedia article, but on WV it is about learning in all aspects. That means also non-objective things could be posted
 * [5:37pm] darkcode: I think it's the mix of the fact anyone can edit and people trying to treat Wikiversity like other university oriented websites that is the problem
 * [5:37pm] Erkan_Yilmaz: which then could be edited immediately because someone doesn't like it
 * [5:37pm] cormaggio: assassingr - what was the perceived problem with academic people again?
 * [5:37pm] Erkan_Yilmaz: in WP they have some kind of "guarantee" that when they post neutral enough that their contribution stays
 * [5:38pm] darkcode: people contributing to university oriented websites are probably use to writing papers that aren't edited without there approved review
 * [5:39pm] Erkan_Yilmaz: yeah, people have when thinking of university some kind of fixed/structured/hierarchical viewpoint
 * [5:39pm] assassingr: Academic people wouldn't be interested in doing research in Wikiversity, that was the problem
 * [5:39pm] cormaggio: ok Erkan - I think I see better what you're saying - so have you any thoughts on how this subjectivity can be harnessed in a productive and perhaps wiki way?
 * [5:39pm] cormaggio: (or anyone else for that matter!)
 * [5:40pm] darkcode: Wikiversity probably needs to work on encouraging more NPOV if that's an issue, I think original research can be done in NPOV way, by also including the fact that people disagree with said original research or interpretations, and by including what other conclusions are possible
 * [5:40pm] cormaggio: assassingr - was this because it could be edited, or..?
 * [5:40pm] Erkan_Yilmaz: by having a common goal: that we welcome anyone and any edit
 * [5:40pm] Erkan_Yilmaz: even if it may be not neutral
 * [5:40pm] ¥ Erkan_Yilmaz sees it now more form an individual POV
 * [5:41pm] Erkan_Yilmaz: because most of the time think when thinking of WV on a group/ group of student
 * [5:41pm] cormaggio: ok - just to be provocative - would we accept a Nazi POV?
 * [5:41pm] Erkan_Yilmaz: s
 * [5:41pm] darkcode: nothing is learned when everyone agrees
 * [5:41pm] assassingr: cormaggio: no, because there are interested in doing research inside a bricks-and-mortar university, and research there could make more profit
 * [5:42pm] Erkan_Yilmaz: I: yes, because it brings then people together, people interact then, and they learn on the topic and share their viewpoints
 * [5:42pm] Erkan_Yilmaz: because: when people do such things in private you can not bring them to others viewpoints
 * [5:42pm] Erkan_Yilmaz: you could not convince them perhaps
 * [5:42pm] Erkan_Yilmaz: when they do it openly here: it attracts others (pro + contra)
 * [5:42pm] cormaggio: darkcode - that's interesting - but surely a wiki is a form of peer review - different from academia sure, but no less/more susceptible to conflict :-)
 * [5:43pm] darkcode: Wikiversity shouldn't be about trying to convince anyone of there personal POV, but rather about including all POVs in order for people to make up there own conclusions
 * [5:44pm] cormaggio: assassingr - oh sure, the profit motive will play its role - but there are others who see research as something which *must* be in the public domain. indeed, such research is being increasingly funded..
 * [5:45pm] cormaggio: darkcode - that's a good point (or at least that's how I see it) - to allow the possibility to recognize and co-critique different perspectives on any given subject
 * [5:45pm] assassingr: sure, but from personal experience I could tell that there would be may professors that would mock wiki research
 * [5:45pm] cormaggio: oh yes assassingr - I know plenty myself :-)
 * [5:46pm] darkcode: there are professors who use to mock Wikipedia as well and now they tell there students to use Wikipedia to research stuff for there class
 * [5:46pm] cormaggio: but darkcode's idea of NPOV research is interesting - how would that work?
 * [5:47pm] cormaggio: for me the issue of "neutrality" is always problematic - but I see an analogy between WP's method of NPOV and academia's method of 'academic/scholarly practice'
 * [5:47pm] darkcode: neutral in this sense would mean including conflicting perspectives for any research done, and letting the audience make up there own conclusions from the research
 * [5:47pm] assassingr: (about the Nazis, there are some guidelines about fringe research at betawikiversity:Wikiversity:Research_guidelines/En - just to make sure everyone knows)
 * [5:48pm] cormaggio: mmm, interesting
 * [5:48pm] darkcode: even in academic/scholarly research not everyone agrees with conclusions made
 * [5:49pm] cormaggio: yes, the 'Nazi' issue was the main question over whether/how we would allow for research on WV - one of the main questions at its set-up
 * [5:49pm] cormaggio: (not just Nazi - I'm just using it as an extreme example)
 * [5:49pm] Erkan_Yilmaz: so that means that some people (perhaps a majority) agrees to suppress a smaller group's idea?
 * [5:49pm] assassingr: oh, I know, I'm sure you don't have anything personal with Nazis :)
 * [5:49pm] cormaggio: so where are we - how to allow for subjectivity while promoting collaboration - yes?
 * [5:50pm] cormaggio: I'm just trying to keep track of multiple threads
 * [5:50pm] darkcode: by including all conflicting views/perspectives the research can be made less bias, also including any supporting facts/data from the research or from previous research, could booster its reliability
 * [5:50pm] cormaggio: does this relate to contributor retention perhaps?
 * [5:51pm] wknight8111: and we could all sit around the campfire and sing kumbaya
 * [5:51pm] cormaggio: Erkan - you make a good point - how do you validate who decides what is ok and what is not..?
 * [5:51pm] cormaggio: wknight8111 (!!)
 * [5:52pm] wknight8111: there are going to be an infinite number of conflicting views, and some of them are always going to be obvious quackery
 * [5:53pm] Erkan_Yilmaz: you forget one thing: one person has now a viewpoint x (which may be wrong), but by allowing him/her to post it, a process of learning starts, who says that x months later the person doesn't realize the viewpoint was wrong? by not allowing to post/stay you take away this chance
 * [5:53pm] darkcode: using Nazi as an example, could include the Nazi point of view, include information like historical info that may have influenced the Nazi perspective, include any scientific data that might be available that supports there view, as well as any data that is in conflict with the Nazi perspective
 * [5:53pm] Erkan_Yilmaz: people should not limit others freedoms right from the begin, we don't know what will happen with something in the future
 * [5:53pm] cormaggio: I'm going to semi-answer the question I just raised - if it is a community-consensus-driven process, doesn't that eliminate some of the ethical problems with majority rule?
 * [5:54pm] assassingr: I think Erkan's answer is different than yours
 * [5:54pm] wknight8111: and include the perspective that there were no Nazis, no holocaust. Include the idea that Nazis are aliens. Include the idea that Nazis are demons that return to earth every 666 years, include the idea that Jews are made of candy and need to be killed to release the candy, etc
 * [5:55pm] darkcode: cormaggio: not really because the majority can be equated with the most popular POV, and the minority with the least popular POV
 * [5:55pm] assassingr: wknight8111: o.O
 * [5:55pm] cormaggio: ok - I seem to have pressed the 'Nazi button' :-)
 * [5:55pm] Erkan_Yilmaz: well, even if someone posts false data, it is something they have fun with are interested in, they can use WV as their sandbox to spend their time here - we give them an atmosphere to live their dreams
 * [5:56pm] wknight8111: I'm just saying, there are a million points of view, and you can't include them all because all of them aren't serious
 * [5:56pm] assassingr: well, Erkan_Yilmaz, you are extremely liberal for my taste
 * [5:56pm] Erkan_Yilmaz: assassingr probably I also will behave different from situation to situation
 * [5:56pm] Erkan_Yilmaz: but when talking abstract, we should also be open for others ideas
 * [5:56pm] assassingr: that would be better
 * [5:57pm] cormaggio: I'm sure Erkan also has boundaries - like he says
 * [5:57pm] darkcode: well assume good faith plays some role in that, we can assume there serious until reason is given to believe there not being serious
 * [5:57pm] Erkan_Yilmaz: abstract is a high level and when you eliminate right there something you lose many other things below
 * [5:57pm] assassingr: don't take me wrong, I'm a liberal too, but using WV as a sandbox, that's.... new
 * [5:57pm] darkcode: well Wikiversity does have a sandbox server
 * [5:57pm] Erkan_Yilmaz: well it depends: JWS e.g. had some idea with a WV for vandals
 * [5:58pm] darkcode: vandals.wikiversity.org ;p
 * [5:58pm] Erkan_Yilmaz: WV should be for anyone - people should not hinder others form being here
 * [5:58pm] assassingr: darkcode was very right about that - the assume-good-faith part
 * [5:58pm] cormaggio: can I reframe this in pure terms of learning? I think Erkan is raising the point of how a person learns and participates as a *process* - I quite like that lens...
 * [5:58pm] Erkan_Yilmaz: yup, because I speak personally also
 * [5:58pm] Erkan_Yilmaz: when I joined the wiki verse:
 * [5:58pm] Erkan_Yilmaz: I saw it just as a medium to make ads
 * [5:58pm] Erkan_Yilmaz: but since then:
 * [5:59pm] Erkan_Yilmaz: I have had process changes
 * [5:59pm] Erkan_Yilmaz: and now as you all know: devote my time for the health of WV
 * [5:59pm] Erkan_Yilmaz: when someone right at begin had blocked me, I wouldn't be here
 * [5:59pm] Erkan_Yilmaz: and I guess you all would say: that is a loss
 * [5:59pm] Erkan_Yilmaz: you never know how someone develops over time
 * [5:59pm] cormaggio: absolutely! (loss)
 * [6:00pm] Erkan_Yilmaz: and WV is for me besides doing academic research/learning a place for the individual to learn and to develop
 * [6:00pm] cormaggio: thanks Erkan
 * [6:00pm] Erkan_Yilmaz: it goes a little more in the direction like: it is a virtual world which allows people to be home
 * [6:00pm] assassingr: well, I'm sure that everyone is acting the same way as regards ads - leave a message in the talk page
 * [6:01pm] Erkan_Yilmaz: but that is just my POV :-)
 * [6:01pm] cormaggio: (I'm not sure I understand "ads", btw)
 * [6:01pm] Erkan_Yilmaz: ads: I explain
 * [6:02pm] Erkan_Yilmaz:
 * [6:03pm] cormaggio: ah yes - ok
 * [6:04pm] cormaggio: so yes exactly - people who come in for purposes alien to the spirit of the project can become acculturated etc and become great contributors
 * [6:04pm] darkcode: one way I guess a page could be evaluated is by asking the question "is this a serious page that I can learn/study and allows me to easily make up my own conclusions?"
 * [6:04pm] cormaggio: thanks for bringing that up darkcode
 * [6:05pm] cormaggio: I was going to return to the question of what Wikiversity should include - as we were discussing - which is one part of this
 * [6:05pm] Erkan_Yilmaz: rehi assassingr :-)
 * [6:06pm] cormaggio: are there ways that we can help a random clicker evaluate a page perhaps?
 * [6:06pm] cormaggio: could/should we incorporate a feedback mechanism into every page?
 * [6:06pm] Erkan_Yilmaz: evaluate in regards to ? design/content quality/understandability?
 * [6:07pm] cormaggio: Erkan - right - could/should we ask all of these questions? Or would it put people off?
 * [6:07pm] Erkan_Yilmaz: feedback mechanism would be nice, e.g.a simple one: make a template on the main page asking: did you like the content ? click here and share your view - and the here goes to the talk page
 * [6:07pm] darkcode: well a random clicker is already going to evaluate it by different things, like by its appearance, if it catches there interest, if it sounds plausible, if it includes information they need or if it's lacking in information to backup any claims, etc.
 * [6:07pm] Erkan_Yilmaz: someone who is willing to contribute will add his opinion anyway
 * [6:08pm] cormaggio: hmm, surely "willing to contribute" spans many types of people in many different 'states'..
 * [6:08pm] darkcode: different people will evaluate it in different ways, depending on what there needs are and purpose in looking for it on Wikiversity
 * [6:09pm] Erkan_Yilmaz: I think any kind of feedback is good even if it's vandalism (because vandalism shows, that the page got known)
 * [6:09pm] Erkan_Yilmaz: because so far you don't know besides edits on talk or main page that users saw it
 * [6:09pm] cormaggio: sure darkcode - perhaps simply a "tell us what you think" link which is a new section on the talk page..
 * [6:09pm] Erkan_Yilmaz: we don't have access to server logs :-(
 * [6:10pm] Erkan_Yilmaz: we could go over search engine hits which appear at begin of a search or so, but still this is not reliable
 * [6:10pm] cormaggio: Erkan - yes, this is an issue - but WP has it and presumably so will we eventually..
 * [6:11pm] Erkan_Yilmaz: implementing an easy to use and simple feedback mechanism which rewards people in some kind of form would be something that also non-altruistic people would then use
 * [6:11pm] McCormack is now known as Sleeping.
 * [6:11pm] Erkan_Yilmaz: the reward can happen e.g. already: by increasing edit count, because some only work for that :-)
 * [6:12pm] darkcode: well its been discussed before about having some kind of comments extension that acts similar to the cite extension, adding a comments section to the bottom of the page when say is used on it
 * [6:13pm] Erkan_Yilmaz: interesting: I just thought: the talk page takes the user from the page where he is and takes him out of his thoughts probably - he doesn't see the relation so much more?
 * [6:13pm] Erkan_Yilmaz: I mean if you read the page and you are on the page your mind is still "there"
 * [6:13pm] cormaggio: that's another interesting point Erkan :-)
 * [6:13pm] Erkan_Yilmaz: and not radically - by your eyes - split
 * [6:14pm] Erkan_Yilmaz: comment on the main page sounds ok
 * [6:14pm] cormaggio: and I hadn't noticed that proposal darkcode
 * [6:14pm] Erkan_Yilmaz: but then: would that people like ? e.g. the authors?
 * [6:14pm] darkcode: its only been discussed before on IRC cormaggio
 * [6:14pm] cormaggio: ok
 * [6:15pm] cormaggio: but feedback is only one part of this, surely
 * [6:16pm] cormaggio: if we're discussing how to make learning resources more easily usable for people, there are other considerations..
 * [6:16pm] darkcode: just as the Classes Clock idea has only been discussed on IRC
 * [6:17pm] ¥ Erkan_Yilmaz thinks we have lost some of the participants and should try to bring them in again?
 * [6:17pm] Erkan_Yilmaz: IRC participants now
 * [6:18pm] darkcode: I think some of the problems are that people try too hard to anticipate what people want and divide people into imaginary stereotypes
 * [6:18pm] Erkan_Yilmaz: hi Devourer
 * [6:18pm] cormaggio: how/where do you see that, darkcode?
 * [6:19pm] Erkan_Yilmaz: perhaps when they try to make a 100% perfect page instead of developing it slowly with errors - everybody can see?
 * [6:19pm] darkcode: well for instance I see in the Schools vs Topics vs Portals namespaces, it's assumed that people are only interested in a certain thing and would not want to be distracted by other things
 * [6:20pm] ¥ Erkan_Yilmaz has successfully avoided that discussion :-)
 * [6:20pm] darkcode: it's also in the page about the different type of Wikiversity participants, in calling people teachers, learners, whatever
 * [6:20pm] cormaggio: I wonder if "having errors" is seen as the key to other people jumping in and collaborating? Is that our model for participation?
 * [6:20pm] Erkan_Yilmaz: darkcode: yes, how they title others :-)
 * [6:20pm] Erkan_Yilmaz: but people just need such stereotypes to make the world look easier, simpler to understand
 * [6:21pm] Erkan_Yilmaz: seeing errors and jumping could mean 2 things: either I want to help or I want to show the other has made an error
 * [6:21pm] cormaggio: well, I think there's benefit in identifying uses of Wikiversity - because people will have clear needs to use Wikiversity for
 * [6:22pm] darkcode: it encourages the attitude that people are just learners, just students or just teachers, rather than encouraging people to teach what they know, while trying to learn what they don't know
 * [6:22pm] cormaggio: but I think there's always been the attitude that people's roles and needs will always be flexible in Wikiversity - changing from a teacher to a learner to a browser, etc
 * [6:23pm] atglenn: I kind of like the approach of collective learning, where there is not someone teaching to an audience but, while someone learns a topic they are recording their experiences learning it and each participant is...
 * [6:23pm] atglenn: ... adding content to the others' contributions
 * [6:23pm] darkcode: well that is a problem too, people's roles don't necessarily change either, people may need to browse in order to learn or teach something
 * [6:24pm] cormaggio: atglenn - that's something that I'd like to see more of actually - and develop a style of working that might encourage people to take control of their learning
 * [6:24pm] darkcode: someone way want to build on existing work and avoid duplicating what someone else has already done
 * [6:25pm] atglenn: with the "record after you learn it" model, (thanks Erkan) people have to wait for one person to
 * [6:25pm] atglenn: have complete knowledge of the topic (and then be dependent on that person's knowledge)
 * [6:25pm] atglenn: with a collective learning process there is no waiting and each person can contribute as they experiment,
 * [6:26pm] atglenn: so that we benefit from their struggles during the learning process
 * [6:26pm] darkcode: ya
 * [6:26pm] atglenn: rather than seeing only the finished product
 * [6:26pm] cormaggio: atglenn - I'd really like to start to define that model - that would be a great action step for this meeting
 * [6:26pm] darkcode: everyone should be learning while mentoring those who want to learn, what they already have learned, as they've learned it
 * [6:26pm] assassingr: for me teacher/learner attributes are just to indicate if someone works towards providing resources or he wants to get more resources, nothing more than that
 * [6:26pm] atglenn: and since none of us actually learns bu producing a finished product straight away...
 * [6:27pm] atglenn: we all learn by encountering obstacles and working through them... this seems like an added benefit.
 * [6:27pm] Erkan_Yilmaz: yup, learning is a mental change
 * [6:28pm] ¥ assassingr darkcode confused me...
 * [6:28pm] assassingr: oh, I give up
 * [6:29pm] cormaggio: well, there are probably a few things I haven't understood fully myself
 * [6:29pm] Erkan_Yilmaz: only a few?
 * [6:29pm] darkcode: well like say your learning C++, while your learning C++, you could be mentoring someone on how to declare variables in C++ because you've learned that concept, but your still learning, and there learning from you, so your both learning and a mentor for another learner
 * [6:29pm] * Erkan_Yilmaz understands many things not
 * [6:29pm] cormaggio: but I'm aware that this meeting has been going a while now and I think we might want to put our heads together for some action
 * [6:30pm] Erkan_Yilmaz: darkcode: yes
 * [6:30pm] Erkan_Yilmaz: by seeing how someone else does it, you have another viewpoint instead just learning yourself alone
 * [6:31pm] darkcode: you can also explain what worked for you, with many people explaining what worked for them, people have more methods of learning
 * [6:32pm] Erkan_Yilmaz: darkcode - by any chance your example was in relation to pair programming?
 * [6:32pm] cormaggio: I'm not sure atglenn's model and darkcode's model are the same...
 * [6:33pm] darkcode: also I was thinking of an alternative to primary school, secondary school, etc. that might be a better way to organize resources
 * [6:33pm] cormaggio: perhaps they are..
 * [6:33pm] atglenn: I think my model is different. they are not mutually exclusive, certainly.
 * [6:33pm] cormaggio: ..but my point is: maybe we should write up this model explicitly on-wiki?
 * [6:34pm] darkcode: organize them by more general category like Resources for young children, Resources for older children, Resources for teens, Resources for young adults, Resources for adults, Resources for the elderly
 * [6:35pm] darkcode: those are more easily recognized categories that don't rely on as much on geographical location, but still rely some though on culture
 * [6:35pm] darkcode: might be a little less confusing
 * [6:35pm] darkcode: could also have Resources for everyone
 * [6:36pm] cormaggio: darkcode - yeah, but there's always going to be problems
 * [6:36pm] cormaggio: there are plenty of adult learners who don't know how to read advanced texts (or at all)..
 * [6:37pm] cormaggio: (but I'm just being my usual 'problematic self')
 * [6:37pm] darkcode: well the way I see it, these names make it easier for people to tell what kind of resources they might understand
 * [6:38pm] darkcode: with the educational levels approach more guesswork is involved I think
 * [6:39pm] darkcode: but I still think A depends on B, so B is categorized in Resources depending on A, is the best approach
 * [6:39pm] atglenn: can someone scour JWS' pages to see if he hasn't written up something like this already?
 * [6:39pm] cormaggio: right - well, we're not going to solve this one now perhaps..
 * [6:39pm] atglenn: (sorry, was occupied)
 * [6:40pm] Erkan_Yilmaz: about the categorizing there is hot discussion atm at en.wv
 * [6:40pm] cormaggio: ..but I'd like to start to bring this to a conclusion of 'what next'
 * [6:40pm] Erkan_Yilmaz: just browse through the colloquium
 * [6:41pm] atglenn: *already
 * [6:41pm] cormaggio: I definitely like the idea of clarifying how collective learning would work - without the need for 'waiting' for others
 * [6:41pm] Erkan_Yilmaz: what means collective learning ? learning together or creating a collective mind which knows all what was learned?
 * [6:42pm] cormaggio: atglenn - I'm pretty sure JWS hasn't written it up quite like this - but I'll send on some relevant links
 * [6:42pm] atglenn: one downside with that model is that it requires a group that learns together (ideally)... and with our current numbers this is difficult. but this is the model we could strive for.
 * [6:42pm] atglenn: I'll try to write something, where should it go?
 * [6:42pm] cormaggio: Erkan - that's an interesting question - one for the wiki :-)
 * [6:42pm] atglenn: learning together, Erkan, and leaving a record of that learning process
 * [6:42pm] atglenn: in my head, anyways
 * [6:42pm] cormaggio: I think we should set up a page Collective learning?
 * [6:42pm] Erkan_Yilmaz: leaving a record: like in a blog?
 * [6:43pm] cormaggio: or we can get a better name if we like
 * [6:43pm] Erkan_Yilmaz: page name: it is a wiki it can be changed anytime
 * [6:43pm] atglenn: not necessarily a blog. the on-wikt writings of the participants would be that record
 * [6:43pm] cormaggio: but there's also some other pages, two are linked in the channel info
 * [6:43pm] cormaggio: "learning model" and "discussion group"
 * [6:44pm] atglenn: well, I vote for the collective learning page and it can be linked to from anywhere else that's appropriate.
 * [6:45pm] cormaggio: I'd also like to pick up on this theme of subjectivity, personal development, and wiki inclusion/exclusion..
 * [6:45pm] cormaggio: Erkan - do you have a place for that, or a new page required?
 * [6:45pm] atglenn: please elaborate
 * [6:45pm] Erkan_Yilmaz: place for?
 * [6:46pm] cormaggio: Erkan - the notion of how to include people into the process of learning, when they might first appear to be radical POV's or vandals etc
 * [6:46pm] cormaggio: atglenn - that was the theme in a nutshell ;-)
 * [6:46pm] atglenn: thanks!
 * [6:46pm] Erkan_Yilmaz: well isn't there - assume good faith and don't bite newcomers?
 * [6:47pm] cormaggio: right

PLE

 * [6:47pm] cormaggio: I also think the notion of a PLE is relevant here..
 * [6:47pm] Erkan_Yilmaz: :-)
 * [6:47pm] Erkan_Yilmaz: well PLEs as the name PLE are a new wave at WV
 * [6:47pm] Erkan_Yilmaz: it was started by you :-)
 * [6:47pm] cormaggio: http://en.wikiversity.org/wiki/Wikiversity:Personal_learning_environment
 * [6:48pm] cormaggio: yes :-)
 * [6:48pm] Sleeping is now known as McCormack.
 * [6:48pm] Erkan_Yilmaz: rehi McCormack
 * [6:48pm] Erkan_Yilmaz: well I have a PLE
 * [6:48pm] Erkan_Yilmaz: but Cormac - I don't see your PLE or?
 * [6:48pm] Erkan_Yilmaz: or it is a little hidden or not declared as PLE?
 * [6:49pm] cormaggio: but I think it's relevant to the question of how the individual interacts with the community/space, and how community facilitators can help people on their personal learning curves
 * [6:49pm] cormaggio: hmm, interesting observation Erkan!
 * [6:49pm] Erkan_Yilmaz: well every interaction/edit in WV is a step in the PLE
 * [6:49pm] cormaggio: I have to admit - a lot of my reflections are in a diary on my desktop - not much/all of it goes onto my blog :-(
 * [6:50pm] Erkan_Yilmaz: so, why are they private?
 * [6:50pm] Erkan_Yilmaz: just lazy?
 * [6:50pm] cormaggio: that's a long discussion - but basically so I don't have to think about what I'm writing
 * [6:51pm] cormaggio: whether it's ok to be read etc
 * [6:51pm] cormaggio: but really, that's another discussion..
 * [6:51pm] Erkan_Yilmaz: ok
 * [6:52pm] cormaggio: I think contributor retention brought up by McCormack is an interesting point to take up - but it was left unexplored..
 * [6:52pm] cormaggio: maybe next meeting, or on-wiki
 * [6:53pm] Erkan_Yilmaz: could someone explain the meaning of c.r. ? because when I search the translator there is much words :-(
 * [6:53pm] atglenn: are these discussions only about models for en wikt?
 * [6:53pm] cormaggio: anything else that anyone wants to focus on - arising from this meeting?
 * [6:53pm] atglenn: or are they meant to be cross-wv models?
 * [6:53pm] atglenn: *discussions
 * [6:53pm] cormaggio: atglenn - it would be great if ideas could cross-pollinate between projects
 * [6:54pm] atglenn: I wonder if we might want to have these discussions in the main channel, rather than here
 * [6:54pm] cormaggio: ok - I'll bear that in mind for the next meeting - thanks atglenn
 * [6:54pm] atglenn: ok
 * [6:55pm] cormaggio: oh yes, NPOV was a big theme here - and whether/how it's useful..
 * [6:56pm] cormaggio: what's c.r. Erkan? where did you see it?
 * [6:56pm] Erkan_Yilmaz: c.r. = contributor retention
 * [6:57pm] cormaggio: ok - that would be 'how to keep (active) contributors (actively) contributing'
 * [6:57pm] Erkan_Yilmaz: I see
 * [6:57pm] Erkan_Yilmaz: making them feel comfortable and valued
 * [6:58pm] Erkan_Yilmaz: but we can elaborate next meeting also?
 * [6:58pm] cormaggio: that would be great, yes
 * [6:58pm] cormaggio: so, I'm not sure of the action coming from this meeting..
 * [6:58pm] Erkan_Yilmaz: you could make some points where we could talk for the next meeting?
 * [6:58pm] cormaggio: I see the page on collective learning as a great step - but what else?
 * [6:58pm] Erkan_Yilmaz: and everybody can suggest also themselves
 * [6:58pm] cormaggio: yes, I'll do that
 * [6:58pm] cormaggio: exactly
 * [7:00pm] Erkan_Yilmaz: well one action is: we have laid the basics for the next meeting
 * [7:00pm] Erkan_Yilmaz: people can see what it is we talk about and then from that they can bring more input - we were just 4-5 persons now
 * [7:01pm] McCormack: So did you reach any conclusions?
 * [7:01pm] cormaggio: yes - I'll have to read this log and see what actions/problems we've defined - and it would be great if people could add to the list
 * [7:01pm] cormaggio: I'll post a summary of the meeting as well as the log on-wiki
 * [7:02pm] cormaggio: conclusions McCormack? well, we at least have something to define and something to talk about - that was the purpose here :-)
 * [7:02pm] cormaggio: so, unless someone has something burning to say, maybe we should call this (meeting) to a close for now..?
 * [7:03pm] Erkan_Yilmaz: 2h is enough
 * [7:03pm] cormaggio: it's been 2 hours - sorry Erkan :-)
 * [7:03pm] Erkan_Yilmaz: :-)
 * [7:04pm] cormaggio: well, thanks very much to everyone - it's been very interesting, and, I think, useful
 * [7:04pm] atglenn: thanks for having us.
 * [7:04pm] cormaggio: but the test of that is in the action to come of course
 * [7:04pm] cormaggio: atglenn :-)
 * [7:05pm] cormaggio: oops - phone
 * [7:06pm] cormaggio: sorry everyone! thanks again!
 * [7:06pm] Erkan_Yilmaz: bye then for this meeting, now IRC life begins


text:
User:Egm6321.f09.Team1.sallstrom/HW7

= Problem 1: Even and Oddness of 2nd Solution to Laplace Equation = From lecture note slide 37-1.

Given
The general form of the Legendre polynomials can be written
 * {| style="width:100%" border="0" align="left"

P_n(x) = \sum_{i=0}^{[n/2]} (-1)^i \frac{(2 n - 2 i)! x^{n - 2i}}{2^n i! (n-i)! (n-2i)!} $$ $$ where $$\displaystyle [n/2]$$ means the integer part of $$\displaystyle n/2$$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

The second solution to Laplace equation is
 * {| style="width:100%" border="0" align="left"

Q_n(x) = P_n(x) \tanh^{-1}(x) - 2 \sum_{j=1,3,5}^J \frac{2n - 2j + 1}{(2n - j + 1)j} P_{n-j}(x) $$ $$ where $$\displaystyle J := 1 + 2 \left[ \frac{n-1}{2} \right]$$, and the square bracket means the integer of.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

Find
Use Eq. 2 to show when $$\displaystyle Q_n$$ is even or odd, depending on $$\displaystyle n$$.

Solution
= Problem 2: Plot of Solutions to the Legendre Differential Equation = From lecture note slide 37-1.

Given
The general form of the Legendre polynomials can be written
 * {| style="width:100%" border="0" align="left"

P_n(x) = \sum_{i=0}^{[n/2]} (-1)^i \frac{(2 n - 2 i)! x^{n - 2i}}{2^n i! (n-i)! (n-2i)!} $$ $$ where $$\displaystyle [n/2]$$ means the integer part of $$\displaystyle n/2$$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

The second solution to Laplace equation is
 * {| style="width:100%" border="0" align="left"

Q_n(x) = P_n(x) \tanh^{-1}(x) - 2 \sum_{j=1,3,5}^J \frac{2n - 2j + 1}{(2n - j + 1)j} P_{n-j}(x) $$ $$ where $$\displaystyle J := 1 + 2 \left[ \frac{n-1}{2} \right]$$, and the square bracket means the integer of.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

Find
Plot $$\displaystyle \{P_0, P_1, ..., P_4 \}$$ and $$\displaystyle \{Q_0, Q_1, ..., Q_4 \}$$.

Solution
= Problem 3: Orthogonality of Legendre Functions I = From lecture note slide 37-2.

Given
The general form of the Legendre polynomials can be written
 * {| style="width:100%" border="0" align="left"

P_n(x) = \sum_{i=0}^{[n/2]} (-1)^i \frac{(2 n - 2 i)! x^{n - 2i}}{2^n i! (n-i)! (n-2i)!} $$ $$ where $$\displaystyle [n/2]$$ means the integer part of $$\displaystyle n/2$$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

The second solution to Laplace equation is
 * {| style="width:100%" border="0" align="left"

Q_n(x) = P_n(x) \tanh^{-1}(x) - 2 \sum_{j=1,3,5}^J \frac{2n - 2j + 1}{(2n - j + 1)j} P_{n-j}(x) $$ $$ where $$\displaystyle J := 1 + 2 \left[ \frac{n-1}{2} \right]$$, and the square bracket means the integer of.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

Find
Show that
 * {| style="width:100%" border="0" align="left"

< P_n, Q_n > = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
 * }
 * }

Solution
= Problem 4: Orthogonality of Legendre Functions II = From lecture note slide 37-2.

Given
The following expression is given
 * {| style="width:100%" border="0" align="left"

\int_{-1}^1 L_m \left[ (1-x^2) L'_n \right]' \, {\rm d} x+ n(n+1) \int_{-1}^1 L_m L_n \, {\rm d} x = 0 $$ $$ where $$\displaystyle L_n$$ and $$\displaystyle L_m$$ are solutions to the Legendre equation.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

Find
Show that, by integrating the first term of Eq. 1, you obtain the expression
 * {| style="width:100%" border="0" align="left"

-\int_{-1}^1 (1-x^2) L'_n L'_m + n (n + 1) = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

Solution
= Problem 5: = From lecture note slide 38-3.

= Problem 6: = From lecture note slide 38-4.

= Contributing Team Members = Egm6321.f09.Team1.sallstrom 16:56, 18 November 2009 (UTC)

title: User:Egm6321.f09.Team1.sallstrom/HW7


text:
Wiki Resources/Learning Theories

Home | Learning Theories | Instructional Design & eAuthoring | ePortfolio | Storyboard | Wiki Analysis | Communication Tools | Resources Analysis | Group Work | Teaching Philosophy | Accessibility | Glossary

Introduction
This wiki project can be considered as a laboratory for several learning theories, especially those most relevant in regards to collaborative knowledge creation. The Wiki Resources/Learning Theories section will give you a short brief review of these theories and a summary of its main characteristics. It also will allow you to reflect on learning theories which has been an underpinning for development of this project.

Learning Outcomes
At the end of this chapter you will be able to answer the following questions:
 * 1) Which learning theories are most relevant to collaborative knowledge creation?
 * 2) What the main characteristics of each of these theories?
 * 3) Why the knowledge of the learning theories is essential for development of a learning object?
 * 4) Which learning theories that most has influenced the instructional design concepts?

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Overview
The major theme of Vygotsky's theoretical framework is that social interaction plays a fundamental role in the development of cognition. Vygotsky (1978) states:
 * "Every function in the child's cultural development appears twice: first, on the social level, and later, on the individual level; first, between people (inter-psychological) and then inside the child (intra-psychological). This applies equally to voluntary attention, to logical memory, and to the formation of concepts. All the higher functions originate as actual relationships between individuals." (Vygotsky, 1962).

A second aspect of Vygotsky's theory is the idea that the potential for cognitive development depends upon the "zone of proximal development" (ZPD): a level of development attained when children engage in social behaviour. Full development of the ZPD depends upon full social interaction. The range of skill that can be developed with adult guidance or peer collaboration exceeds what can be attained alone.

Vygotsky's theory was an attempt to explain consciousness as the end product of socialization. For example, in the learning of language, our first utterances with peers or adults are for the purpose of communication but once mastered they become internalized and allow "inner speech".

Scope/Application
This is a general theory of cognitive development. Most of the original work was done in the context of language learning in children (Vygotsky, 1962), although later applications of the framework have been broader (see Wertsch, 1985).

 * Example:
 * Vygotsky (1978, p56) provides the example of pointing a finger. Initially, this behavior begins as a meaningless grasping motion; however, as people react to the gesture, it becomes a movement that has meaning. In particular, the pointing gesture represents an interpersonal connection between individuals.

Principles

 * Cognitive development is limited to a certain range at any given age.
 * Full cognitive development requires social interaction.

Overview
This theory stipulates that there are several different types or levels of learning. The significance of these classifications is that each different type requires different types of instruction. Gagne identifies five major categories of learning: verbal information, intellectual skills, cognitive strategies, motor skills and attitudes. Different internal and external conditions are necessary for each type of learning. For example, for cognitive strategies to be learned, there must be a chance to practice developing new solutions to problems; to learn attitudes, the learner must be exposed to a credible role model or persuasive arguments.

Gagne suggests that learning tasks for intellectual skills can be organized in a hierarchy according to complexity: stimulus recognition, response generation, procedure following, use of terminology, discriminations, concept formation, rule application, and problem solving. The primary significance of the hierarchy is to identify prerequisites that should be completed to facilitate learning at each level. Prerequisites are identified by doing a task analysis of a learning/training task. Learning hierarchies provide a basis for the sequencing of instruction.

In addition, the theory outlines nine instructional events and corresponding cognitive processes. These events should satisfy or provide the necessary conditions for learning and serve as the basis for designing instruction and selecting appropriate media (Gagne, Briggs & Wager, 1992).

Scope/Application
While the nine instructional events covers all aspects of learning, the focus of the theory is on intellectual skills. The theory has been applied to the design of instruction in all domains (Gagne & Driscoll, 1988). In its original formulation (Gagne, 1962), special attention was given to military training settings. Gagne (1987) addresses the role of instructional technology in learning.

 * Example:
 * The following example illustrates a teaching sequence corresponding to the nine instructional events for the objective, Recognize an equilateral triangle:
 * Gain attention - show variety of computer generated triangles
 * Identify objective - pose question: "What is an equilateral triangle?"
 * Recall prior learning - review definitions of triangles
 * Present stimulus - give definition of equilateral triangle
 * Guide learning- show example of how to create equilateral
 * Elicit performance - ask students to create 5 different examples
 * Provide feedback - check all examples as correct/incorrect
 * Assess performance- provide scores and remediation
 * Enhance retention/transfer - show pictures of objects and ask students to identify equilaterals (1985, chapter 12) provides examples of events for each category of learning outcomes.

Principles

 * Different instruction is required for different learning outcomes.
 * Events of learning operate on the learner in ways that constitute the conditions of learning.
 * The specific operations that constitute instructional events are different for each different type of learning outcome.
 * Learning hierarchies define what intellectual skills are to be learned and a sequence of instruction.

Overview
A major theme in the theoretical framework of Bruner is that learning is an active process in which learners construct new ideas or concepts based upon their current/past knowledge. The learner selects and transforms information, constructs hypotheses, and makes decisions, relying on a cognitive structure to do so. Cognitive structure (i.e., schema, mental models) provides meaning and organization to experiences and allows the individual to "go beyond the information given".

As far as instruction is concerned, the instructor should try and encourage students to discover principles by themselves. The instructor and student should engage in an active dialogue (i.e., Socratic learning). The task of the instructor is to translate information to be learned into a format appropriate to the learner's current state of understanding. Curriculum should be organized in a spiral manner so that the student continually builds upon what they have already learned.

Bruner (1966) states that a theory of instruction should address four major aspects:
 * Predisposition towards learning
 * The ways in which a body of knowledge can be structured so that it can be most readily grasped by the learner
 * The most effective sequences in which to present material
 * The nature and pacing of rewards and punishments. Good methods for structuring knowledge should result in simplifying, generating new propositions, and increasing the manipulation of information.

In his more recent work, Bruner (1986, 1990, 1996) has expanded his theoretical framework to encompass the social and cultural aspects of learning as well as the practice of law.

Scope/Application
Bruner's constructivist theory is a general framework for instruction based upon the study of cognition. Much of the theory is linked to child development research (especially Piaget). The ideas outlined in Bruner (1960) originated from a conference focused on science and maths learning. Bruner illustrated his theory in the context of mathematics and social science programs for young children (see Bruner, 1973). The original development of the framework for reasoning processes is described in Bruner, Goodnow & Austin (1951). Bruner (1983) focuses on language learning in young children. Note that Constructivism is a very broad conceptual framework in philosophy and science and Bruner's theory represents one particular perspective.

 * Example:
 * This example is taken from Bruner (1973):
 * "The concept of prime numbers appears to be more readily grasped when the child, through construction, discovers that certain handfuls of beans cannot be laid out in completed rows and columns. Such quantities have either to be laid out in a single file or in an incomplete row-column design in which there is always one extra or one too few to fill the pattern. These patterns, the child learns, happen to be called prime. It is easy for the child to go from this step to the recognition that a multiple table, so called, is a record sheet of quantities in completed multiple rows and columns. Here is factoring, multiplication and primes in a construction that can be visualized."

Principles

 * Instruction must be concerned with the experiences and contexts that make the student willing and able to learn (readiness).
 * Instruction must be structured so that it can be easily grasped by the student (spiral organization).
 * Instruction should be designed to facilitate extrapolation and or fill in the gaps (going beyond the information given).

Overview
The social learning theory of Bandura emphasizes the importance of observing and modelling the behaviours, attitudes, and emotional reactions of others. Bandura (1977) states: "Learning would be exceedingly laborious, not to mention hazardous, if people had to rely solely on the effects of their own actions to inform them what to do. Fortunately, most human behaviour is learned observationally through modelling: from observing others one forms an idea of how new behaviours are performed, and on later occasions this coded information serves as a guide for action." (p22). Social learning theory explains human behaviour in terms of continuous reciprocal interaction between cognitive, behavioural, and environmental influences.

The component processes underlying observational learning are:
 * Attention, including modelled events (distinctiveness, affective valence, complexity, prevalence, functional value) and observer characteristics (sensory capacities, arousal level, perceptual set, past reinforcement)
 * Retention, including symbolic coding, cognitive organization, symbolic rehearsal, motor rehearsal)
 * Motor Reproduction, including physical capabilities, self-observation of reproduction, accuracy of feedback
 * Motivation, including external, vicarious and self reinforcement.

Because it encompasses attention, memory and motivation, social learning theory spans both cognitive and behavioural frameworks. Bandura's theory improves upon the strictly behavioural interpretation of modelling provided by Miller & Dollard (1941). Bandura’s work is related to the theories of Vygotsky and Lave which also emphasize the central role of social learning.

Scope/Application
Social learning theory has been applied extensively to the understanding of aggression (Bandura, 1973) and psychological disorders, particularly in the context of behaviouur modification (Bandura, 1969). It is also the theoretical foundation for the technique of behaviour modelling which is widely used in training programs. In recent years, Bandura has focused his work on the concept of self-efficacy in a variety of contexts (e.g., Bandura, 1997).

 * Example:
 * The most common (and pervasive) examples of social learning situations are television commercials. Commercials suggest that drinking a certain beverage or using a particular hair shampoo will make us popular and win the admiration of attractive people. Depending upon the component processes involved (such as attention or motivation), we may model the behaviour shown in the commercial and buy the product being advertised.

Principles

 * The highest level of observational learning is achieved by first organizing and rehearsing the modelled behaviour symbolically and then enacting it overtly. Coding modelled behaviour into words, labels or images results in better retention than simply observing.
 * Individuals are more likely to adopt a modelled behaviour if it results in outcomes they value.
 * Individuals are more likely to adopt a modelled behaviour if the model is similar to the observer and has admired status and the behavior has functional value.

Overview
This theory suggests that learning happens best under conditions that are aligned with human cognitive architecture. The structure of human cognitive architecture, while not known precisely, is discernible through the results of experimental research. Recognizing George Miller's research showing that short term memory is limited in the number of elements it can contain simultaneously, Sweller builds a theory that treats schemas, or combinations of elements, as the cognitive structures that make up an individual's knowledge base. (Sweller, 1988)

The contents of long term memory are "sophisticated structures that permit us to perceive, think, and solve problems," rather than a group of rote learned facts. These structures, known as schemas, are what permit us to treat multiple elements as a single element. They are the cognitive structures that make up the knowledge base (Sweller, 1988). Schemas are acquired over a lifetime of learning, and may have other schemas contained within themselves.

The difference between an expert and a novice is that a novice hasn't acquired the schemas of an expert. Learning requires a change in the schematic structures of long term memory and is demonstrated by performance that progresses from clumsy, error-prone, slow and difficult to smooth and effortless. The change in performance occurs because as the learner becomes increasingly familiar with the material, the cognitive characteristics associated with the material are altered so that it can be handled more efficiently by working memory.

From an instructional perspective, information contained in instructional material must first be processed by working memory. For schema acquisition to occur, instruction should be designed to reduce working memory load. Cognitive load theory is concerned with techniques for reducing working memory load in order to facilitate the changes in long term memory associated with schema acquisition.

Scope/Application
Sweller's theories are best applied in the area of instructional design of cognitively complex or technically challenging material. His concentration is on the reasons that people have difficulty learning material of this nature. Cognitive load theory has many implications in the design of learning materials which must, if they are to be effective, keep cognitive load of learners at a minimum during the learning process. While in the past the theory has been applied primarily to technical areas, it is now being applied to more language-based discursive areas.

 * Example:
 * In combining an illustration of blood flow through the heart with text and labels, the separation of the text from the illustration forces the learner to look back and forth between the specified parts of the illustration and the text. If the diagram is self-explanatory, research data indicates that processing the text unnecessarily increases working memory load. If the information could be replaced with numbered arrows in the labeled illustration, the learner could concentrate better on learning the content from the illustration alone. Alternatively, if the text is essential to intelligibility, placing it on the diagram rather than separated will reduce cognitive load associated with searching for relations between the text and the diagram (Sweller, 1999).

Principles
Specific recommendations relative to the design of instructional material include:

 * Change problem solving methods to avoid means-ends approaches that impose a heavy working memory load, by using goal-free problems or worked examples.
 * Eliminate the working memory load associated with having to mentally integrate several sources of information by physically integrating those sources of information.
 * Eliminate the working memory load associated with unnecessarily processing repetitive information by reducing redundancy.
 * Increase working memory capacity by using auditory as well as visual information under conditions where both sources of information are essential (i.e. non-redundant) to understanding.

Overview
When preparing eLearning materials, you should consider how users like to take in and process information (learning styles). Learning style inventories are related to personality indicators derived by Jung. The Myers-Briggs Type Indicator (MBTI) assessment is a psychometric questionnaire developed during WWII by designed to measure psychological preferences in how people perceive the world and make decisions. The initial questionnaire grew into the Myers-Briggs Type Indicator, which was first published in 1962. The classic learning styles inventories are based on Kolb’s Learning Inventory which flows from Kolb’s experiential learning cycle. These essentially have two components, an active experimentation (AC) - reflective observation (RO) continuum and a concrete experience (CE) - abstract conceptualisation (AC) continuum leading to four poles of learning preferences.
 * AC-RO Assimilating
 * AC-AE Converging
 * CE –AE Accommodating
 * CE-RO Diverging

title: Wiki Resources/Learning Theories