So I’ve spent some more time thinking about this, and I think my position is that on the one hand, the technique is of highly dubious value and there’s an argument that time shouldn’t be spent teaching it, but on the other I maintain that it’s alarming that neither you nor your husband could understand the technique.
First, the technique itself. On three-digit summands it catches errors outside of a range of 200, and where that range is centered relative to the correct value varies based on the second and third digits of the summands. I’ll admit that I no longer have any but the vaguest memory of what it’s like to have a third-grader’s understanding of math (or a sixth-grader’s, for that matter), but my guess is that most errors on these problems are likely to be small relative to 200—forgetting to carry, or double-carrying, or incorrectly performing addition on two single-digit numbers. Since you have to be able to add the leading digits correctly to get the estimate correct, it effectively only works for errors that make the result too large. It looks like it’s really mostly useful when the last two digits of both numbers are relatively large—which admittedly is probably the circumstance in which elementary schoolers would most likely be making mistakes, but since it has so little value the student won’t have a good reason to use it for its own sake, which means it will quickly be forgotten. Unless the entire point of teaching this is to reinforce principles like two two-digit numbers summing to less than 200, I can’t see any point to it.
That said, on to understanding:
“Under what math EVER would 291 be “estimated” or “rounded off” to 200?”
Well, let’s look at what happens when you make different rounding/truncation choices. You seem to want to use the standard unqualified “rounding” rules ( 0, ≥5 –> 10); the third option is to always round up (we’ll ignore other options like round-to-even). Since the given example makes standard rounding and rounding up the same, we’ll instead use 344 and 291.
If you round down, you get an estimate of 500, so you know the answer is between 500 and 700.
If you round up, you get an estimate of 700, so you know the answer is between 700 and 500.
If you round-on-5s, you get an estimate of 600. What’s the answer between? There are ways of recovering the interval, but I’m pretty sure they’re all more difficult than simply adding the original two sums, which entirely defeats the purpose.
So that’s why we don’t want to round-on-5s. As we see, rounding up and down are in one sense equivalent, but keeping in mind that we’re talking about third-graders the fact that rounding down simply involves taking the first digit is pretty compelling. Then the upper bound is calculated with addition instead of subtraction, which I seem to remember as being easier at that age.
The question then is why you two didn’t see this. In the post, you repeatedly conflate the estimate with the result despite the problem clearly saying that the estimate is “to check that each answer is reasonable”. Although my initial reaction was to question your math ability, and I still think that’s part of the issue, I wonder now if the problem isn’t your assumptions going into the worksheet. Given your stated strong negative views on the current math curriculum, I’m thinking that when you encountered something that at first glance seemed counterintuitive or nonsensical you concluded that it was nonsense rather than examining it further to see if your conclusion was justified and maybe running some numbers.
Anyway, I can’t say I know enough about the Common Core to have an opinion on it (I don’t have a dog in the fight, as it were), but it seems to me that by failing to understand the technique being taught you weaken your criticism—instead of a strong argument that time and effort is being wasted on an unenlightening and marginally-useful technique that will probably never be used again and will be promptly forgotten, you have an easily-rebuttable argument that the technique is “fuzzy math” and “completely and utterly wrong”.
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