petrogradphilos

> You will never understand every detail of the implementation of every level on that stack; but you can understand all of them to some level of abstraction, and any specific layer to essentially any depth necessary for any purpose.

As David Deutsch argues in The Beginning of Infinity, this is true not just for computers, but for everything.

You could also replace % with 10⁻² and use scientific notation:

      0.72% ⋅ 0.03%
    = 0.72 ⋅ 10⁻² ⋅ 0.03 ⋅ 10⁻²
    = 7.2 ⋅ 10⁻³ ⋅ 3 ⋅ 10⁻⁴
    = 7.2 ⋅ 3 ⋅ 10⁻³ ⋅ 10⁻⁴
    = 21.6 ⋅ 10⁻⁷
    = 2.16 ⋅ 10⁻⁶
    ≈ 2 in 1 million

Never use a macro to do an inline function’s job.

It’s correct to regard trees, but not people, as products of their environment. Trees don’t make choices. People do.

> 1 Introduction

> The increased availability of data linking students to teachers has made it possible to estimate the contribution teachers make to student achievement.

There was some data available before (or the sentence would not have used the word "increased"). Why wasn't it possible to estimate with that?

> By nearly all accounts, this contribution is large.

It goes on to talk about what "large" means:

> Estimates of the impact of a one standard deviation (σ) increase in teacher “value-added” on math and reading achievement typically range from 0.10 to 0.30σ, which suggest that a student assigned to a more effective teacher will experience nearly a year's more learning than a student assigned to an less effective teacher (Hanushek & Rivkin 2010;...).

(Typo: "an less effective" should be "a less effective".)

A "range from 0.10 to 0.30σ" doesn't make sense. A Greek lowercase sigma (σ) is used to represent one standard deviation, but the sigma is used only on the upper end of the range. Should it have been from 0.10σ to 0.30σ?

And how are they measuring the impact on achievement of an increase in teacher "value-added", anyway? It says that estimates of the impact "typically range from 0.10 to 0.30σ", but it doesn't say what units those figures are in.

The sentence goes on to say that those unit-less estimates "suggest" that "a student assigned to a more effective teacher will experience nearly a year's more learning than a student assigned to an less effective teacher". Over what time period? That is, how long does a student have to study under a "more effective teacher" to get "a year's more learning"? 1 week? 12 years? It doesn't say.

And finally, how do those unit-less estimates "suggest" an impact measured in learning time? It doesn't say.