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I've recently run across a series of problems that didn't reflect reality.

For example -

  • An algebra problem with two teens on bicycles. The resulting times showed the bike was moving at 120MPH.
  • A quadratic equation, "The football follows a path of....." but the equation didn't reflect proper gravity, either in feet or meters, and the solution had the football hanging in the air for an absurd time.
  • A trig problem, the answer being that a man's height, calculated from his shadow, was 8 feet.

Disclosure - I am a tutor in a High School math center. These questions were from homework, and in one case, a quiz I proctored. The student pointed out to me that the answer made no sense, but had checked it twice. I'm not supposed to coach or comment during quizzes or tests, but I saw the answer was right, and told the student so.

As the title asked - Should word problems be reasonable? I often ask students to reflect on their answers to understand if it makes sense. These recent questions caught me off guard.

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    $\begingroup$ In short, yes. The classic example (originally in German, I think) is in Barsuk (1989): "There are 125 sheep and five dogs in a flock. How old is the shepherd?" Students typically consider the four different operations (addition, subtraction, multiplication, division) and decide only 125/5 = 25 could be a "reasonable" answer; many, many students will answer the above question despite it being somewhat ridiculous! To your question, the focus on sense-making is in keeping with CCSSM, especially SMP #1 (Make sense of problems) and the focus on math modeling. (Yes to being reasonable...) $\endgroup$ – Benjamin Dickman Nov 18 '14 at 20:52
  • $\begingroup$ Once I got "The cost of a metal bucket" to be ~Rs.20000 ($323)! $\endgroup$ – Kartik Nov 19 '14 at 9:45
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    $\begingroup$ I once obtained the result that the mass of an object was negative. However, this was not a word problem: this was the analysis of an actual experiment I did in chemistry class. And despite being unreasonable, it was the correct answer to give, based on the measurements recorded during the experiment. $\endgroup$ – Hurkyl Nov 19 '14 at 17:01
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    $\begingroup$ @Hurkyl that was an exception, we dont want exceptions to puzzle students, in that case we can have a man with 8feet height as well $\endgroup$ – shabby Nov 19 '14 at 18:05
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    $\begingroup$ For the football equation, you could just say that the football is on another planet (or the moon, sun, death star..), as gravitational motion IS parabolic $\endgroup$ – Mitch Nov 19 '14 at 18:36
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Absolutely!

In fact, in my opinion, the most important "math skill" that should be taught in conjunction with, and using, word problems is checking whether the answers make sense. This is an absolutely invaluable part of making any practical use of mathematics, as opposed to just blindly applying formulas for the sake of passing an exam.

There are several parts to this skill, which all should be explicitly taught and reinforced at every level, such as:

  • Dimensional analysis: A bicycle does not move at a speed of 15 kg/m². Any such answer is simply nonsense, and students need to recognize it as such.

  • Fermi estimation: The number of bicycles on Earth is probably of the same order of magnitude as the number of people, i.e. in the billions. There are several ways to guesstimate this, but the important point is that any calculation resulting in, say, less than a thousand or more than a trillion bicycles is almost certainly nonsense.

  • Physical and logical constraints: Little Timmy might own zero, one, two or even three bicycles. He most likely does not own 0.3675 bicycles, and he definitely does not own $\sqrt{-5}$ or infinitely many bicycles. That would just be ridiculous.

  • Just plain common sense: A bicycle does not travel faster than sound (unless fired from a gun or dropped from orbit). Any answer that implies this is nonsense. Any question that expects such an answer is equally nonsense.

Most children naturally learn these skills at an early age, but unfortunately, through the efforts of all too many early and middle school math teachers, a large fraction of them manage to forget them, or at least to acquire the regrettable belief that such sensible rules only apply outside the classroom. Thus, in order to give such students a practical level of everyday numeracy, and to help them see mathematics as a useful tool rather than just a pointless classroom exercise in memorization and rule-application, these skills need to be re-taught as an explicit part of any responsible math curriculum.

Exams and homework exercises — being, for better or worse, an unavoidable part of most math teaching methods — also need to support these lessons, or at least should not actively undermine them, as "word problems" like those quoted above do. Word problems as such are a great tool for teaching practical problem solving, but only when properly used. As a responsible teacher, you need to make sure that any problems you present have sensible answers, and to instill in your students the understanding that an answer is not complete until it has been verified, and that a nonsensical answer is worse than wrong — a wrong but plausible answer may still be worth partial credit, but a clearly nonsense answer, presented with no indication that the student understands that it must be wrong, is simply unexcusable.

If other teachers are undermining your teaching by giving your students nonsense questions like those quoted above, I'd suggest first having a friendly chat with those teachers about it, perhaps even pointing them to this thread. If that doesn't help, I'd go ahead and tell your students that, while other classes might include word problems that don't make any sense, yours doesn't (or, at least, shouldn't) — and that, even in other classes, they should still make as much use of the answer-checking skills you've taught them as possible. (Make sure to note that, even for nonsense problems, double-checking the answer still helps them catch mistakes in exams. That's always a good motivation.)

As tempting as it may be, though, you may wish to refrain from suggesting to your students that, when they answer word problems from other classes, they should point out any nonsensical assumptions made in the problem statement in their answers. Or maybe not — it depends on the situation.


Addendum: As noted by neminem in the comments, one possible exception to the rule of keeping word problems reasonable are problems deliberately meant to be funny and absurd, and clearly signaled as such. The last part is important — students need to clearly recognize that certain parts of those particular problems may violate common sense. One way of doing so, as noted in neminem's comment, is to pick an inherently funny animal or other being or object, and make clear (via your lecture examples, and/or simply by openly stating so) that, in your exercises, those particular creatures or things may frequently violate laws of physics or common sense. That gives your students an easy way to identify not-so-realistic exercises — "that's another one with a llama, it's probably silly" — while also setting up a running gag.

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The answers to a word problem should in my opinion make sense (within reasonable limits). The goal you mentioned that students should check their answers is one shared by many, as also witnessed by our recent question How to award points for sense-making at the end of a problem?

Now, if it is not a given any more that the solution does indeed make sense, this is completely moot.

Obviously, there can be simplifying assumptions, a word problem cannot be an example of or exercise in state-of-the-art mathematical modeling either. Yet, the results should at least not be glaringly implausible.

If one only wants to practice skills in manipulation and calculation, one should simply not do word problems but something else.

In brief, in my opinion, either do word problems that make some sense, or just don't do them.

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    $\begingroup$ I would argue yes, unless it's funny and it's clear that it's supposed to be funny. (We had one physics prof in college that was in love with sending rhinos at various nontrivial fractions of the speed of light at each other. Just how that rhino got to be moving that fast, and why it was not instantaneously reduced to individual rhino particles, was not part of the problem.) Gave his exams way more flavor. $\endgroup$ – neminem Nov 18 '14 at 23:56
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    $\begingroup$ This is a good point. I agree things like this can be useful. I too remember less colorful but still unrealistic things like trains moving at c/2 and things like that. These things might qualify as some sort of thought experiment. $\endgroup$ – quid Nov 19 '14 at 10:35
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Physically (or even economically) unreasonable answers are confusing. The student needs to be shown that math is a tool that can solve practical problems. Answers that are unreasonable send the wrong message. (That math is a logic game versus a commercially relevant skill.) Why do that, when there are so many easy problems to construct that don't make this mistake? There is no need.

Heck, I have to wonder if a problem has an unreasonable answer that the math teacher lacks basic science/engineering skills. I think for any service course, teachers should have basic competency in college physics, chemistry, and a smattering of engineering. This is not so they can teach those courses, but so they can understand the connection between the subjects while teaching students (the vast majority of whom will use the tools from math class in these subjects). I think this sort of basic understanding gives a little bit of sympathy, empathy and just understanding for what the students are doing. This ends up helping the instruction.

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  • $\begingroup$ There is some validity to your point, esp. regarding some competency in basic cognates (though where to acquire that is another question). My experience is that really obviously absurd problems end up being okay too. $\endgroup$ – kcrisman Sep 22 '17 at 0:57
  • $\begingroup$ I'm used to "necromancer" being a standard forum complaint about me, not something to get a badge for. ;-) $\endgroup$ – guest Sep 24 '17 at 22:40
  • $\begingroup$ I used to give my 6th grade students partial credit for getting the wrong answer if their method was good. However if the wrong answer was unreasonable they would lose points unless they pointed that out. I also had them make up word problems and told them only those with reasonable numbers would be accepted. $\endgroup$ – Amy B Sep 27 '17 at 10:48
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I think trying to make "real-world" word problems is often posing more problem than it solves. One expects using maths in real-world problem would make them more appealing and show their usefulness, but honestly in most cases the problems are either ridiculous, or artificial, or boring to students, or all of these. Then the whole idea backfires, and what you describe is a good example of a much more general issue.

So, this does not literally answer your question, but the best option might be to make "fantasy-world" word problem instead of "real-world" ones. They will be more fun, less misleading, and you won't have to care too much about the result being reasonable (if you compute the size of a Gargumash from its shadow, 8 feet seems just as reasonable as any other (positive) result given no one nows what a Gargumash is).

This goes further than maths: I know a biology teacher who did not want to use fake data for real species in an exam in genetics, but wanted to design it freely to tune the difficulty. She chose to make the test about unicorns.

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  • $\begingroup$ While I agree with the basic sentiment, perhaps some qualification is necessary. For example, in a vector calculus class one can very reasonable (should? must?) consider examples coming from electromagnetism. After all, modern vector calculus starts with Green and Maxwell and the like, who developed it to study electromagnetism ... What seems to me problematic are problems that are really caricatures of genuine "applications" in which the science/engineering/economics isn't treated with the same seriousness as the mathematics is. $\endgroup$ – Dan Fox Sep 27 '17 at 9:34
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Yes. Making sure a solution makes sense in the context of a real-life situation is an essential part of applied mathematics.

But it’s not just enough to have “real-life” problems with solutions that make sense in the context of real-life situations. When a ridiculous context is forced onto a pure mathematics problem, you reinforce the pervasive belief that mathematics is not actually useful in real life. Real-life tasks/problems have to be authentic.

I think the worst thing we do as mathematics teachers is invent real-life contexts to make pure mathematics questions seem more relevant. I think the results are lame 99% of the time.

Robin and Pat are planning a wedding banquet. The cost per guest, $G$ dollars, is modelled by the function $G(n)=95e^{(-0.02n)}+40$, for $20<n<200$, where $n$ is the number of guests.

From a real examination question!

“… and they say pure mathematics has no real-world applications.” (Professor Farnsworth, Futurama S06E10)

Robin and Pat are the sort of people who scare others away by applying mathematics indiscriminately and are unlikely to have friends numbering in the range given.

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