All I know is I always have them cut my pizza pies into six slices, because I can’t eat eight.
Well, it was a real life example, and the four people who took home the other pieces were very pleased. Everybody gets a portion that looks like a normal pie after some slices have been taken, without chopping it all into a mess.
Good enough! (I’ll volunteer to take the two small pieces, as punishment for not thinking this through.)
As the teacher, I occasionally write a junker of a word problem. My rule is, if you don’t understand what the question is asking, let me know, and I’ll be happy to rephrase it for you even if my wording isn’t at fault. If you point out an ambiguity, I’ll stop the class and praise your insight (assuming you’re the type of third-grader who appreciates that), and if it’s a test, you’re getting extra credit, and meanwhile given the ambiguity I’ll accept any answer with work shown. I mean, c’mon, I’m not in a competition against these kids; if they can show they understand the math, aren’t we all happy here? What kind of insecure douchebag doubles down?
Also, kaylasdad, are you aware that language is idiomatic?
Last week I went to the doctor. She told me to alternate Advil with Tylenol for my wretched feverish viral crud. “How often can I take it?” I asked her, and she told me I should limit myself to 2400 mg/day. “Okay,” I said, and since I remembered seeing it was a 200 mg tablet and that a dose was 2 pills, I had a word problem right there in front of me.
I almost gave it to my own students for morning work today :).
Example of a time I was super-impressed with a kid: another teacher (watch me pass the buck) wrote a problem something like this:
I told the kids that I knew of four possible answers (677, 697, 797, 897). They got to work.
Then one girl raised her hand. “Mr. Dorkness?” she said. “I think there are more than four answers.”
“Hmm. Show me what other answer you found.”
“Well, there are a lot. You didn’t say it was a three-digit number.”
God, how I love moments like that. If you don’t, I’m not sure you should be a teacher.
I don’t wish to pretend to more understanding than I actually have. Could you explain that, in the context of my position being, well, not *wrong, per se, * but also not authoritative?
Thanks. 
Sure. “Two time more”, taken strictly literally, does suggest to me what you said: n + 2n.
However, in a real-world setting, I have never heard anyone use such a construction with that intended meaning. Instead, the “more” would be used in order to indicate that the amount run is greater–as someone else said, greater by a factor of two.
Some phrases if understood strictly literally fail as communicative phrases: the audience ends up with an entirely different meaning from the meaning intended by the speaker. In these cases, an idiomatic understanding is preferable.
Interpreting “two times more” to mean “more by a factor of two” is much likelier to be the intended meaning of any speaker I’ve heard in my life.
Sorry, but when presented with a mathematical word problem, literally is the only way I know how to take the words. I value precision in language, as it encourages precision in thinking. Conversely, I am of the opinion that it DIScourages imprecise thinking.
As far as “intended meaning of the speaker,” the speaker has a perfectly serviceable idiom to express “(nx).” It is “n times as much as.” Speakers who wish to convey “(nx +x)” get the idiom “n times more than” because there isn’t any other idiom for those people to use.
Whereas I value accuracy in communication over precision in language. The emphasis you place on precision negatively affects the accuracy of communication when you’re the audience.
In the spirit of your appreciation of precision, however, I accept your apology!
In any event, I think we can all agree that the wording of the question is sloppy, and ought to be rephrased in a way that’s less likely to be misinterpreted, regardless of which interpretation is correct or intended.
My favorite example of that is from a simple electrical circuit. You know those switches at either end of a hallway, or maybe at the top and bottom of a flight of stairs, where you can toggle either one of them and toggle the light? Well, the problem showed the students a diagram of what the individual switches look like, and asked them how to wire the circuit.
There is one standard answer to this question, and most students get that one (mostly likely because they’ve either seen it before, or looked it up). But my first year teaching that lab, I had a student come up with a different circuit that also did the job (there are practical reasons why it’s not favored, but it still meets all of the requirements). Hm, I thought, I guess there are two valid solutions.
Then the next year, I knew to look for both of those solutions, and a student came up with yet another solution. Hm, I thought, I guess there are actually three solutions.
Then another year, and another, and so on. Finally, I wised up. I no longer think “There are N different solutions”. Now, I think “There are at least six and a half solutions, that I know of”. And I wouldn’t be particularly surprised if a student showed me a seventh.
Out of curiosity - would anybody’s answer be different for the following:
“Mary ran four laps. John ran an amount two times more than the amount run by Mary. How many laps did John run?”
“Mary ran four laps. John ran an amount two times greater than the amount run by Mary. How many laps did John run?”
Ambiguous, but I’d interpret 12.
For this one I would definitely answer 12.