Finding \alpha and \beta of Beta-binomial model via method of moments I am looking for a laymen step by step of how the process of finding the 1st and 2nd sample moments located: 
http://en.wikipedia.org/wiki/Beta-binomial_distribution#Maximum_likelihood_estimation
Also it's my limited understanding that k-th sample moments are defined as $${\frac {\sum _{i=1}^{n}{x_{{i}}}^{k}}{n}}$$
For samples $x_1, x_2...x_n$ where $n$ = total number of samples. (source: http://en.wikipedia.org/wiki/Moment_(mathematics)#Sample_moments)
Given their example data:
Males       0   1   2   3   4   5     6     7     8    9    10  11  12
Families    3   24  104 286 670 1033  1343  1112  829  478  181 45  7

The first thing I don't understand is why they say $n=12$ when there are 13 data points. Wouldn't that imply $n=13$
I believe the sample moments are:
$m_1 = \frac{0+1+2+3+4+5+6+7+8+9+10+11+12}{13} = 6$
$m_2 = \frac{0^2+1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2+9^2+10^2+11^2+12^2}{13} = 50$
Yet they have 
$$m_1 = 6.23$$
$$m_2=42.31$$
Even If I use $n=12$ and cut off either the first or last record I am left with different values.
Despite that, even using their values of $$m_1 = 6.23$$$$m_2=42.31$$$$n=12$$ going by the equation for the method of moments estimates:
$$\alpha= \frac{( nm_{{1}}-m_{{2}} ) }{n ( {\frac {m_{{2
}}}{m_{{1}}}}-m_{{1}}-1 ) +m_{{1}}} = 33.59257915$$
$$\beta= \frac{( n-m_{{1}} )  ( n-{\frac {m_{{2}}}{m_{{1}}}}
 )}{n ( {\frac {m_{{2}}}{m_{{1}}}}-m_{{1}}-1
 ) +m_{{1}}} = 31.11222820 $$
which do not match his values of:
$$\alpha= 34.1350$$
$$\beta = 31.6085$$

Edit: Given this question was spawned from Rating system incorporating experience; For purposes of record keeping for later googlers, I decided to reword this question to better suit the answers. A detail explanatin of Beta-binomial model and the MLE method of finding $\alpha$ and $\beta$ are located there.
 A: The moments should be $$m_k = \frac{   \sum_{i=0}^{12} f_i \times i^k}{\sum_{i=0}^{12} f_i}$$ where $f_i$ is the number of families with $i$ males.
The calculation of $\hat{\alpha}$ and $\hat{\beta}$ require the use of $n=12$, as @did says.
Do both and you will get the stated values.
A: The method of estimation that you are describing is called method of moments.  It is not maximum likelihood estimation.  To do maximum likelihood you have to write down the likelihood function for your observed data based on the parametric model.  Then you search for a maximum value for that function (which is often unique).  When the partial derivatives of the likelihood with respect to the parameters can be computed you take them and solve the equation for the local extrema which should turn out to be the global maximum.
A: Just for historical purposes, here is how I got the $\alpha$ and $\beta$ using Maple 15:

Maple code:
males := [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];
fams := [3, 24, 104, 286, 670, 1033, 1343, 1112, 829, 478, 181, 45, 7];
n := 12; 

k := 1; 
m[1] := (sum(fams[i]*males[i]^k, i = 1 .. n+1))/(sum(fams[i], i = 1 .. n+1)); 

k := 2; 
m[2] := (sum(fams[i]*males[i]^k, i = 1 .. n+1))/(sum(fams[i], i = 1 .. n+1)); 

alpha := (n*m[1]-m[2])/(n*(m[2]/m[1]-m[1]-1)+m[1]);
beta := (n-m[1])*(n-m[2]/m[1])/(n*(m[2]/m[1]-m[1]-1)+m[1]);

printf("

n = %d
m_1 = %f 
m_2 = %f 
alpha = %f 
beta = %f

", n, m[1], m[2], alpha, beta)

Output:
n = 12
m_1 = 6.230581 
m_2 = 42.309403 
alpha = 34.135021 
beta = 31.608492

