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A Model of Homeschooling Growth

Instructions: drag the scroll bar thumbs back and forth (or click on the arrow buttons or to the left and right of the thumbs) to adjust the model parameters to what you think represents current reality, and the model output will change accordingly.

About This Model

This model is based upon the following assumption: the probability that a child will be rescued from school is determined solely by the percentage of school-age children that are already being homeschooled. In other words, most parents are "following the crowd" and will continue to force their children to attend school until they see that "enough" other parents are homeschooling.

Of course, this is a very simple model that's mostly for fun and edification and cannot be relied on for accurate predictions. Some of the model's most important simplifications are the following:

Children in private school are not considered. It is assumed that the private school population will grow at the same rate as the general school-age population.
In the growth of the school-age population, no distinction is made between internal growth and immigration. It is possible that immigrants would be less likely to homeschool (even more subject to conventional wisdom).
The model of growth for the school-age population is very simple. It starts with the given rate of growth for year zero and then (depending on a setting) reduces this growth rate by a fixed amount each year.

Due to the rapid and accelerating pace of technological and social change, it is absurd to expect that a model that spans five decades would bear any resemblance to reality. In only one or two decades, we will see astounding changes in technology, especially in computers and communications. I believe that these changes will increase the pressure on the conventional wisdom to the breaking point. We may be headed for a fall-of-the-Berlin-wall scenario, rather than the smooth transition predicted by the model. On the other hand, perhaps a large percentage of parents will never consider homeschooling under any circumstances, so that homeschooling growth will eventually level off at some fraction of the population.

And now, for those who are interested, here is the math behind the model.

Math

Definitions:

H[n]	=	# of children being homeschooled in year n, n = 0 .. 50.
S[n]	=	# of children in public school in year n.
T[n]	=	Total number of school age children (excluding private school).
K[n]	=	growth rate of school-age population in year n.

Initialization for year zero:

H[0]	=	# of children being homeschooled in year zero (first slider).
G	=	growth rate of homeschooling this year (second slider).
S[0]	=	# of children in public school in year zero (third slider).
K[0]	=	growth rate of school-age population in year zero (fourth slider).
L	=	annual decrease in growth rate of school-age population (fifth slider). It is assumed that population growth is gradually slowing.
T[0]	=	S[0] + H[0]
R	=	growth parameter for model, adjusted so that homeschooling growth in year zero is equal to G (see below).

Iteration for each year of the model (n = 1 .. 50):

Annual increase in school age population, and decrease in rate of increase of school age population:

    T[n] = T[n-1] (1 + K[n-1])
    K[n] = K[n-1] (1 - L)

The heart of the model. Next year's school age population is this year's plus school-age population growth, minus children lost to homeschooling. Note that the percentage of the school population lost to homeschooling is proportional (factor R) to the ratio of the school age population that is already being homeschooled.

    S[n] = S[n-1] (1 + K[n-1] - (R H[n-1] / T[n-1]))
    H[n] = T[n] - S[n]

The R parameter is calculated by setting G, the homeschooling growth rate for year zero, equal to the growth rate as calculated from H[0] and H[1]:

    G = (H[1] - H[0]) / H[0]

When this is solved for R, the result is:

    R = (T[0] / S[0])(G - K[0])