The Accuracy Principle

This is the way I described the Accuracy Principle months ago, in a post that has since been taken down:

The Accuracy Principle asks that text satisfy the standards of clarity and truthfulness. In isolation under this principle, the clarity of content is determined by the mutual agreement of producers and consumers for a certain grouping of students (usually groupings by grade level). The truthfulness of content, although not immune from debate, is less malleable. Operationally, truthfulness can refer to longer-term learning objectives, while clarity refers to short-term learning. In short, to be considered accurate, content must be understandable when presented and be fundamentally consistent with later learning.

As with the English Principle, the ideal of accuracy in text must be bounded by the desire to make content understandable. Certainly I could find an accurate presentation of fractions in a sixth-grade text, but this content would not necessarily be appropriate for third graders. There may be a lot of assumptions about prior learning made or the language may be too difficult, and so on.

There are also a number of different aspects to accuracy that should be explored—from the trifling to the severe. For a trifling example, consider that in the fifth and sixth grades, basal texts often introduce exponents in this way:

In the expression 2⁴, the number 2 is the base, and the number 4 is the exponent. The exponent tells how many times the base is used as a factor.

This statement, by the way, suffers from less severe accuracy problems than the equally typical statement referring to the exponent as indicating the number of times the base is multiplied by itself. By the logic of the first statement, the expression 2⁴ would be a simplification of many expressions, so long as they included four 2's being multiplied. By the second, the equation 10¹ = 10 × 10 would be true, since the expression 10 × 10 shows 10 being multiplied by itself exactly once. Also, neither of the statements above mention that their respective definitions apply only to indices that are positive integers. And, to be accurate, they should.

More severe accuracy problems can be unsurfaced only after a bit of analysis. I have mentioned before, here, that when fractions are introduced in elementary mathematics, they are introduced as ratios, not as division expressions, integers are modeled as quantities rather than as values indicating direction and magnitude, graphing is taught without regard to convention, operations are almost exclusively discussed as action statements rather than as descriptive statements, topics are taught inductively when a clear and simple deductive approach is available, the list goes on and on. In and of themselves, even these problems seem unimportant, but when taken all together, one begins to understand what Feynman was talking about:

The books were so lousy. They were false. They were hurried. They would try to be rigorous, but they would use examples (like automobiles in the street for "sets") which were almost OK, but in which there were always some subtleties. The definitions weren't accurate. Everything was a little bit ambiguous--they weren't smart enough to understand what was meant by "rigor." They were faking it. They were teaching something they didn't understand, and which was, in fact, useless at that time, for the child.

Content should be accurate and understandable. But we spend so much time focusing on getting the poor little dears to learn something, we often forget that what we're teaching them isn't really accurate. Given our current efforts in education, 100% of our students will one day understand all the mathematics to which they are exposed--and none of it will be true.