Teachers’ biggest affordance is their ability to problematize material continuously, with their students, which lets them start with more ambiguous, more interesting, and less linguistically demanding material.

Nice pithy sentence here. Well put, my man.

I would also add it is not only an affordance but a challenge. As one of those who both writes and teaches, I’m not sure I can recall the last time I launched a task from the book. Oh, the task is often in the book, but the book isn’t where our attention is as a class.

Jason: In many other cases, so as to be absolutely precise. As already pointed out, even the problem above is ambiguous: what if the birds aren’t hungry? what if one bird hogs all the worms? To account for that the question would have to be reworded into something like “Given each bird attempts to get a worm and gets at most one worm, how many birds won’t get a worm?”

It makes so much sense to handle those exceptions verbally, in conversation, after they’ve occurred to students, rather than in text, before anyone has thought about birds who aren’t hungry. For one, you’re helping students problematize life rather than problematizing it for them. For the other, some large fraction of your students were already assuming each bird got a worm. Those students are burdened with extra text if you lock down all the exceptions in writing.

Jason: A textbook author in particular can’t expect to be around to make clarifications, so they have to go for the more exact language (or more usually, resorting to mathematical vocabulary which is more exact to begin with, hence the “how many more’).

Right. The constraints on teachers and authors are so different. A lot of authors I work with would modify their own problems if they were using them live, in the classroom. I wonder how many teachers treat textbook problems with the same flexibility. Teachers’ biggest affordance is their ability to problematize material continuously, with their students, which lets them start with more ambiguous, more interesting, and less linguistically demanding material.

What is the reason why math problems have to be worded in the most complicated manner, leaving students only to guess at what the question is really asking, rather than be straight forward and ask the intended question like “how many birds won’t get worms?”

In this case, so as to be general, so it applies to other problems where you ask “how many more X than Y?”

In many other cases, so as to be absolutely precise. As already pointed out, even the problem above is ambiguous: what if the birds aren’t hungry? what if one bird hogs all the worms? To account for that the question would have to be reworded into something like “Given each bird attempts to get a worm and gets at most one worm, how many birds won’t get a worm?” The piling on of sub-clauses can make ghastly prose that would make my just-learned-English-last-year students have their heads explode. Hence it is worthwhile to make sacrifices for simplicity, but I have had test questions before where my attempt at linguistic simplicity backfired and I had a student interpret a problem in a different, perfectly valid way.

A textbook author in particular can’t expect to be around to make clarifications, so they have to go for the more exact language (or more usually, resorting to mathematical vocabulary which is more exact to begin with, hence the “how many more’).

I think you bring up a good point. The immediate question that comes to mind for me is why is the answer more intuitive for primary students when asked “How many birds won’t get a worm”? Is it a listening comprehension issue? When worded the second way, do the students kind of zone out mid-sentence and forget what they’re being asked? Or is there something else going on there?

5 – 3 =

That, too, tests something different than “how many birds won’t get worms?” One hasn’t improved students’ understanding of math by simply changing the nature of the question.

If a question is vague or ambiguous, that’s one thing, but there’s nothing wrong with a question that requires a little more analysis and thought. Those kinds of questions just test something different than 5 – 3 = ?.