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HOME There's a lot of debate about choice in education - pulblic school, private school, home school. Who is best qualified to decide for a child? |
Parental Choice: A Simple Mathematical Model
Question: Do Parents, or the School System, have the greatest
probability of making good decisions regarding a child's education? This is a 'back of the envelope' sort of model with numbers estimated on my own experiences, not based on any sort of legimate or rigorous study. I'll try to list all assumptions, debatable numbers, and what I consider reasonable limits on those numbers. I feel it is important to establish biases up front. No one is free from them, however, one can try to make them both apparent and also to be as objective as possible. My own bias is toward parents rather than institutions making decisions. The Categories First, I'm going to divide children into one of three groups based on their parents: Good Parents (GP), Mediocre Parents(MP), and Evil Parents(EP). Good parents try their best, and usually make good decisions for their child's future. Mediocre parents also do their best, but may not make good decisions despite good intentions. Evil parents knowingly and maliciously undermine their child's future. Evil parents are rare, but they do exist. In addition, I will also divide children into one of two groups based on the school system they would attend. These groups are Good Institution (GI) and Mediocre Institution (MI). Since social institutions aren't consciously or maliciously evil, I am not including that as a category. Good Parents are those who make good decisions regarding their child's education options between 80 and 100 percent of the time. A child with parents trying their best and succeeding. While I think this group should be considered quite large (80% or better), I do acknowledge that it could be much lower. I think reasonable boundaries for the percentage of children with this type of parent are 50% to 90%. I'll choose the most conservative value (i.e. worst for my bias), and use 50% for my model. Evil Parents will deliberately make poor decisions about their children's education. I think this type of parent is rare, less than 1%, but I could be wrong. I hope I'm not. I set an upper value of 5% on a child having this type of parent. The lower limit is 0. Again, to be conservative, I will use the highest value I consider remotely possible - 5%. In the third category, Mediocre, I would lump the remaining parents. This means for the purposes of my model, I will set this percentage at 45%. This category contains all the well meaning parents who just can't manage to get it right, the stupid parents, the busy ones, the lazy ones, the incompetent ones. You know, all those parents everybody always complains about. Not involved enough or intelligent enough to make the right decisions reliably. Good Institutions are able to reliably produce well-educated people at the end of the school years. Mediocre Institutions do so as well, but have a higher failure rate and in addition have lower standards in defining a successful student. Now, these percentages are basically made up out the air. They are my subjective assessment of what our society is like based on my own view of reality. I could be mistaken. You can debate the values with me if you like, but I doubt I'll be convinced that I'm way off base unless you have some sort of facts that could tie in with those categories and give better estimates. Otherwise, it's just our different opinions, and I'm entitled to use mine. On the other hand, I would welcome the chance to improve the model, so if you feel that you can make a good case for different proportions of children in those three categories, please do. The categories (GP, EP, MP, GI, MI) have selected based on my preferences. They could certainly be changed to make the model more realistic. I am willing to entertain other ideas about how to sort out parental and institutional decision-making ability and the what proportions the different categories might have. The Scenario Every model needs a scenario on which to run. I am assuming that there exist 10 possible educational choices for our hypothetical child. In reality there may be more or less for any individual. We will arbitrarily define 3 of the possibilities a good choice for the child, 5 are a mediocre choice, and 2 are bad choices. What is the probability that his/her parents will make a "good" decision for their child? Based on random chance alone, the probability would be 0.3. Decisions and their Probabilities: Since I have made the GP category those parents who make good decisions 80 to 100% of the time, I will assume that these parents make good decisions 90% of the time on average. The remaining 10% of the time decisions are equally likely to be bad or mediocre. The mediocre parents group would range in percentage of good decisions from 80% (over 80% would place them in the GP group) to 5% (Below 5% is such a bad record, they fall into the EP group.) Given that these are people who are well-intentioned, it seems reasonable to assume that their performance is no worse on average than random chance would allow, that is 30% good, 50% mediocre, and 20% bad. For the evil parent (EP) category I will assume that these people on average make a bad decision 95% of the time. The remaining 5% of the time, their decisions are merely mediocre. They avoid good decisions, and I am presuming they are able to do so successfully - again a conservative estimate. The Utility Ratings In any model, the utility of each possible choice must be defined and given a value. They can take on values from 0 to 1.0, with 1.0 having the greatest possible utility, and 0 having none. I am setting utility values as follows: a good choice has a utility value of 1.0, a mediocre choice has a utility value of .5, and a bad choice has a utility value of 0.0. This is somewhat arbitrary, and is meant to represent a variety of possible situations in which a decision may be made. These values could be changed to reflect a different scenario. Or the utility value of each choice could be set separately. Or the classification into good, bad, and mediocre could be designed differently. These are the choices I have made for the purpose of constructing a model. They are all subject to debate, but one must set the values at something in order to build the model. To calculate the utility value for each category, for both parents and institutions, we need to multiply the utility value for each type of decision by the probability of making that decision and then sum them up. This will give us a utility rating for each group. The GP group has a utility of: Good Choices (.9*1.0) + Mediocre Choices (.05*.5) + Poor Choices (.05 * 0) = .925 The MP group has a utility of: Good Choices (.3 * 1.0) + Mediocre Choices (.5 * .5) + Poor Choices (.2 * 0) = .550 The EP group has a utility of: Good Choices (0*1.0) + Mediocre Choices (.05*.5) + Poor Choices (.95 * 0) = .025 Now to calculate the overall utility score for parents. We do that by multiplying the utility value for each category by the proportion of children in that category. Then we sum up the utility for all 3 groups. GP(.5 * .925) + MP(.45 * .550) + PP(.05 * .025) = .71125 This means, that if you accept the assumptions and numbers outlines above, the chances that the parent makes a 'good' decision for their child's education is 71.125% And Now for the Social Institutions... If parents are not making choices for their children, then we can presume the decision is being made by a social institution of some kind. Presumably the representatives of the school system. I don't have any good way of determining the actual percentages, but since I have eliminated the category of Evil Institution, that leaves us with Good and Mediocre Institutions. Now my opinion is that institutions are terrible at making decisions about what is best for individuals. They are perhaps good at making decisions that will affect large numbers of people, but they are notorious for being uncaring to the individual. I think, therefor that their overall success rate would be small. A good rule of thumb is the 80/20 rule. I will set the proportion of Good Institutions at 20%, Mediocre Institutions at 80%. I am willing to discuss other possible values, especially if you have any ideas about what data might apply to help make the determination. The Utility Value for the Social Institution's Decision Making
To compute the total utility for the Social Institution, The GI category: (.9*1) + (.05 * .5)
+ (.05 * 0) = .925 The MI category: (.3*1) + (.5*.5) +
(.2 * 0) = .55 Total utility = (.20 * .925) + (.80
* .55) = .185 + .44 = .625 This means, that if you accept all of the assumptions and numbers outlines above, the chances that the Institution makes a 'good' decision for the child's education is 62.5%. Thus, the probability of parents making a good decision is higher than the probability for institutions. 71.125% > 62.5% This difference does not seem great, but for the given assumptions, it is the minimum improvement we would expect if parental choice were a reality. It could lead to tremendous improvement of our school system if we allowed parents to act on their greater knowledge of the child and improve their children's education to the best of their ability. The gain would snowball after a while as well, increasing the overall gain to society. This model, incidently, does not contain a factor to allow for the quicker response time that parents would have to correct bad decisions as well. Institutions are notoriously slow to recognize and then to correct mistakes. Parents who make a bad decision, on the other hand, are first to see the results and recognize an error. They are also much quicker to correct the mistake. Thus, a mediocre or bad choice or their part will be fixed much sooner than a similar mistake by an institution. This should increase the utility of their choices and the probability of making a good decision. If I can think of a way to add this into the model, it should provide more accurate probabilities. |
