Have you tried starting this kind of lesson with an assignment where you provide a large list of problems for students to organize into those (or other) types of error-types? I would think this could set the stage for further discussion when they write and submit their own problems.

@guest Do you think someone who can flawlessly calculate 2.4*1.3 by hand, but writes a question like the unintelligible one above, is going to be well equipped to teach? Would they even know when to use multiplication if it came up in their real life (say buying 1.3 pounds of beans from the bulk bin at a price of $2.40 per pound)?

@NickC I usually give such a problem to the class, collect anonymous responses, and then we discussions them and categorize them. I encourage a respectful classroom, where we do not ridicule errors, and this is usually an enjoyable and (seemingly) productive class period.

@StevenGubkin I was thinking of a situation where you provide a list of different responses to the prompt "Write a variety of word problems which would require the student to multiply 2.3 by 1.4" and have students (perhaps in groups) arrange and rank them by clarity/mistakes/etc. Then you're not setting anyone up to be ridiculed since you've created the set of responses.

@StevenGubkin I ask because I've had colleagues do this with other tasks, such as learning how to "show work" in an intermediate algebra class. The first assignment is to take a stack of 8-10 "solutions" to a word problem, read them with your group, and rank them by clarity and completeness. Then the class discussion revolves around lessons you can take from this when writing up your own solutions to word problems.

@NickC I could do that, and I will try it this semester. I view it as similar to what I am doing, only in my method the responses are being generated on the spot by the class, rather than pre-prepared. I forgot to mention, I tell them to write "both correct and incorrect" questions, which provides some shielding to the students who cannot write a correct question.

It feels to me like every one of your examples qualifies as "target the wrong operation". So it feels like maybe there needs to be additional exploration/testing on how different operations are applied. But for those who can't even grok that, then there's little hope.

Follow-up to my comment above: Is there direct instruction/testing in the class for what applications each operation is good for? Or would that be additional content?

@DanielR.Collins I think the first two do not have any operation which answers them: there is simply no way to combine the given numbers to produce an answer to the question which they ask. I give an example in my reply to guest, above, of what an appropriate question would be. We do directly address identification and construction of questions for each operation.

Steve: No. But (1), I think that people who screw that up, likely also screw up the calculations. Perhaps not as much since "word problems are hard", but some for sure. After all they pretty clearly don't track some basic concepts so I bet they'd mess up fractions for instance.

And 2, given they have such limitations, what is the benefit of spending time with that tomfoolery. When they are just bad at math overall.

And 3 in a non perfect world, where we don't pour water in the ground and grow teachers, perhaps, less capable teaches, who ARE going to teach anyhow, ought to be concentrating on something different than your stuff.

If they're not going to be good educators then fail them?

Do you know whether these students can write a coherent paragraph? In more detail, do you have enough information to tell whether their difficulty is with math, with English, with the connection between them, with thinking in general, or with something else?

It seems like a university that would admit such students should have its accreditation called into question. It would be a betrayal of the public trust to award a bachelor's degree to someone lacking an 8th-grade education.

@DanielMcLaury: Unfortunately, by that metric, likely every education program in North America would need to be shut down (which is not actually completely insane). For over 100 years those programs have attracted the weakest of all incoming college students. Many go into those programs precisely because they believe they can avoid math in that work. web.archive.org/web/20150204031027/qz.com/334926…

@StevenGubkin: Many problems in the examples, yes. But: Example #1, "How much all together?" is normally a cue for an addition operation. Example #2: "How much more money do I need to buy the ice cream?" is usually a cue for a subtraction operation. So in my diagnosis there's a root of not understanding applications of each operation in each example, which metastasizes to further nonsense in the stopgap attempt to ape standard questions. I'd be very interested in the results of a quiz with a whole bunch of short word problems, with multiple-choice options for which operations to use.

The easiest fix is homeschooling ;)

@LamarLatrell - see DanielR.Collins's comment - if all such students failed, there would be a severe shortage of schoolteachers in the US, and the universities would be blamed for failing to graduate future teachers.

@Alexander Woo, please read what you've just said and imagine how I might pick it apart. There are 2 hotspots, good luck 👍

@AlexanderWoo Having studied, it is normal that any subject attracts people that are not suited, and maybe shouldn't even study to begin with, or at least another subject. I studied CS and it was normal that about 50% drop out in the first semester, having figured out that this was not for them. Having them fail early instead of late allowed them to reorient quickly without having lost too much time.

@DanielR.Collins As you said, it's not completely insane.

Sure, one solution is to fail every student without a good understanding of elementary arithmetic and shut down education programs. A better solution would be to have math and science specialists available to handle those subjects from the very earliest grades.

@Thierry I agree with you, and would like to begin advocating for political reforms along these lines.

@Thierry Weirdly enough, most American public elementary schools already have specialists for art, PE, and music (at least, in places where they still exist). It has frustrated me for years that elementary schools put mathematics in the hands of non-specialists. :\

@Thierry: Also agree we need specialists. However: A few years ago I was talking to a former U.S. junior high school principal (now sociology professor); I said just that, and the response was, "No way! You can't find anyone who understands both science and can communicate with children". Which I'm still steamed about. (It somehow works in every other country.)

@kutschkem: The problem isn't for the students. The problem is that teaching is such an unattractive profession in the US that the only way we can have enough babysitters to satisfy the need is to have people who are unqualified to do the job try to do it. Plus the politicians want to hide this fact from the voters, so they say the teachers are qualified and blame everyone who says otherwise for lying. It doesn't help that many voters think of school mainly as free childcare.

Out of sheer curiosity of someone from Europe: don't you have some kind of examination (at a higher level - state, national, not at the institution they learn I mean) that would weed out such candidates before they get their diploma?

@WoJ There is a state level examination for licensure for educators, but it is multiple choice and laughably easy. There are no state or national tests in order to earn the degree.

@StevenGubkin: ah ok, thank you, this explains the issue. In France for instance to be a teacher in primary school you need to have a specialized MSc and then pass a nation-wide exam in order to get a teaching license. This probably limits such cases (though we have all kind of teachers of course, better and worse)

"would like to begin advocating for political reforms" — more power to you. According to what I know, very few uni profs ever wade in these waters, they prefer quiet haven of higher ed math, although they look with disdain at elementary teachers. I kept my son on home math/alg/geom/physics/some chem/lit diet in addition to school classes until high school. So far he was in 97-99 percentile, but now I dread his math classes using Core Plus Math — utter junk with or without knowledgeable teachers. The whole K-12 system must be overhauled, but this will never happen.

The talk of needing "math specialists" for simple multiplication makes me very sad.

The point being that the conditions for teaching in the USA MUST improve before we start expecting the people who are interested in the career to be "better." I had the same naive thoughts when I was graduating from my education program: That simply failing these students was the answer. But failing these students means that our elementary kids get long-term subs instead of permanent teachers due to the severe lack of people willing to even do the job in the first place.