Liping studied the accounts of various math teachers of what and how they were trying to teach children, and then analyzed what these remarks showed about the teachers' own comprehension of mathematical topics.
In the first chapter, on subtraction with regrouping, one teacher explains the technique this way:
Whereas there is a number like 21 - 9, they would need to know that you cannot subtract 9 from 1, then in turn you have to borrow a 10 from the tens space, and when you borrow that 1 it equals 10, you cross out the 2 that you had, you turn it into a 10, you now have 11 - 9, you do that subtraction problem then you have the 1 left and you bring it down.
This explanation sounds a lot like what I remember from my own elementary math instruction. As Liping points out, though, it contains an outright factual misstatement and a misleading metaphor.
You certainly can subtract 9 from 1, even if young children don't yet know how to do that.
Ma (emphasis added):
Although second graders are not learning how to subtract a bigger number from a smaller one, it does not mean that in mathematical operations one cannot subtract a bigger number from a smaller number. In fact, young students will learn to subtract a bigger number from a smaller number in the future. Although this advanced skill is not taught in the second grade, a student's future learning should not be confused by emphasizing a misconception.
And to speak of the process of borrowing suggests that the digit in the tens column is somehow a separate number from the digit in the ones column. Some of the other teachers quoted speak in terms of going to a friend or neighbor to borrow something.
Another misconception suggested by the "borrowing" explanation is that the value of the number does not have to remain constant during computation, but can be changed arbitrarily—if a number is "too small" and needs to be larger for some reason, it can just "borrow" a certain value from another number.
In other words, the choice of metaphors is not innocent because people will reason on the basis of those metaphors in ways that may undermine the basic principles we're trying to teach.
Ma goes on to note that teachers who have a solid understanding of the underlying mathematical concepts use different language to talk students through the problem. For example, instead of saying, "you cannot subtract 9 from 1," the more aware teachers will say something like, "4 minus 6, are we able to do that?" The shift is a subtle one, but it's important. It no longer implies that we can't subtract the larger from the smaller number, we simply aren't able to do that with our current knowledge.
Grammar, like mathematics, is an immensely intricate field, and obviously we can't teach it all at once. But it would be immensely helpful if teachers gave some consideration to the consequences that their simplifications have for later understanding. Consider the issue of tense. It is completely uncontroversial among linguists that English has only two primary tenses (present and past—or if you prefer, preterite). There is no separate future tense in English. And yet every K-12 grammar book I've ever looked at teaches that will + an infinitive verb is the English future tense. This is an old, old misconception, dating to the earliest English grammars, which merely copied their categories straight from the preexisting Latin grammars.
I can't believe that every single writer of contemporary K-12 grammar books is ignorant of the linguistic consensus on this point. They have chosen to stick with the old account out of some combination of adherence to the familiar and an idea of what is pedagogically helpful to young children. I take it for granted that neither ignorance nor blind adherence to an error merely because it's traditional is a valid reason for persisting in the old way. But what about pedagogy? We certainly shouldn't try to teach elementary-school students the complete system of tense, mood, aspect, and voice in one shot.
The problem is that the simplification that justifies a future tense in English amounts to the equation tense = time. Making that connection is easy, and it will cover some prototypical cases, but it will also lead to real confusion down the line.
Don't believe me? If you have access to a group of high-school or college students, ask them what is the tense in a sentence like "My flight leaves at 10pm tomorrow." I have never asked this question without a significant fraction of the class responding "future." (In case there's any dobt, this leaves is a present-tense verb.)
Just as with the subtraction method, we've wound up confusing students' future learning about grammar by teaching a simplification that is only partially true.
Now I've spent all my teaching career working with high school and college students, so I'm probably not the best person to suggest what will work with younger kids, but surely a better way to go about introducing tense is to start with a discussion of different ways we have of indicating the time of the event and then talk about tense as one way (albeit an important one) of marking time.
I could go on and on about the various grammatical misconceptions that are embedded in schoolbook grammars (possessive nouns are adjectives, the subject is a noun, the passive means that the subject has an action done to it, etc.). All of these wind up down the line in it-must-be-so arguments about the way correct English should be. Because they are based on mistaken assumptions, those arguments are deeply flawed, yet teachers continue to foist these arguments on their students.
Teaching students this sort of nonsense puts students, to paraphrase Michael Dorris, not at point zero, but at minus ten. If they ever hope to gain real knowledge of the way language works, they will first have to unlearn all the garbage.