Friday, February 12, 2010

Insight is a good thing

San Diego Times article "Insight is a good thing" (February 9th, 2010) appears to cast a weak comment against the practice of "ability-grouping." Corrections does not understand why.

As long as teachers approach students with different expectations and rely on dangerous practices such as ability-grouping – blue birds, red birds etc. – in order to make those classrooms more manageable, then we shouldn’t be surprised that students continue to perform at different levels.

The sentence appears quite correct, save for the phrase "dangerous." Ability-grouping is the tactic of putting groups of students at similar skill levels together. If other-student knowledge is complementary to student learning, then putting groups of students together is a good idea. Students help one another learn. Everyone would be better off with a higher-achieving individual in their reading group. However, in terms of efficiency, it may be better to match high individuals with high individuals if higher knowledge individuals help other individuals with higher knowledge more than they help lower knowledge individuals.

That is to say, if a sixth-grade reading level individual boosts another sixth-grade reading level individual's reading level up two grades, and would only boost a fourth-grade reading level up one, then we might say that it creates more total learning to match the two highest--a concept called positive assortative matching in economics. This is more easily seen in the table below (click to enlarge). The table displays the joint returns to schooling. We see that higher reading level individuals create more surplus when matched with higher reading level individuals than they would with lower reading level individuals.

In matching theory, our graph displays supermodularity. Corrections displays this graph in colored 2-d and from two perspectives in 3-d to give an idea of why we might want to positively assort individuals. Click to enlarge 2-d. Click to enlarge 3-D #1. Click to enlarge 3-D #2. Note that colors match between the two 3-D graphs, but not between these graphs and our 2-D contour graph. In all cases, the "surplus" might be imagined to be total extra points on reading tests created from a combination. It ranges from zero to twenty-five (without loss of generality).

While the article is correct that this practice will cause more inequity, it may also cause a higher total amount of learning.

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