You don’t need to be a baseball fan to know that the distance from third base to first base is a lot less than three miles. Yet that’s what several students came up with (16,200 feet to be exact) in a class I recently visited. And though it may seem like their answer came way out of left field, these students were actually just a calculator button away from hitting a home run on a pretty challenging problem. In other words, they knew they were dealing with a right triangle whose legs were both 90 feet long (the distance between home plate and first base, and home plate and third base), and they applied Pythagorean Theorem accordingly. They just stopped a step short—failing to take the square root of 16,200—and were thus off by some 16,073 feet.
What bothered me most about this wasn’t that students skipped a step, but that they accepted 16,200 feet as the correct answer. Until, that is, I said to them, “So what you’re telling me is that it’s 90 feet from third base to home plate, and 16,200 feet from third base to first base. No wonder some of those guys take steroids.” To which a couple students replied, “That can’t be right.” A minute or two of troubleshooting later—with no help from me—students had correctly changed their answer to 127 feet.
This experience illustrates why we should teach students to treat every problem as an estimation problem. Model this before you work through a problem with students by asking, “What would you expect the answer to be?,” and then recording their estimates. Do this until students start doing it themselves as they solve problems on their own. And if they’re resistant or forgetful, you might want to make it a required first step in the problem solving process. Whatever it takes to get students in the habit of estimating before computing so that they have a reliable ballpark answer to compare their actual answer to for reasonableness. And so that they can then recognize when they’ve committed an error—before it’s too late to correct it.
(With standardized tests coming up, this is a particularly good time to stress this idea to students. You can be sure, for example, that 16,200 feet would have been a “trap” choice had the baseball field problem appeared on a multiple choice test.)
Coach G's Teaching Tips 
I totally agree!! We do a problem of the week school wide and have the same issues. “Does this make sense?” ought to be a natural last question in the process.
I sometimes feel like a broken record reminding students to check whether or not their solutions seem reasonable. Modeling the use of estimation first to encourage that ‘reasonableness check’ is a great idea. Thank you!
I loved this post and am sharing it with colleagues. Two comments:
(1) If you use the Pólya-inspired four-step problem-solving process (often rendered “understand, plan, solve, check”), predicting or estimating the answer is a great idea to incorporate into the “plan” stage.
(2) Before I scrolled down to your last paragraph, I was actually thinking “And you’d never catch this with most multiple-choice questions!” Test-builders generally try to ensure “parallel distractors” – that no answer choice sticks out from the other three or four options. But this is a common error, and when you’re building your own multiple-choice items, you’re free to capture it (and, as Coach G has pointed out before, to ask students to explain their answer).
No matter how many times I explain to my statistics students, some poor mook inevitably comes up with a probability value that’s either much larger than one or negative. AND HE DOESN’T UNDERSTAND WHY IT’S INCORRECT! Maybe he’s giving 110% effort….
Sendhil is right, put “common error” incorrect responses in your multiple-choice questions. I use the instant feedback “scratch off” quiz forms for review sessions, and this helps many of my students become more careful with their calculations.
This is a big pet peeve of mine as well. I think it’s especially important when students are using calculators — they need an estimate to know if the answer the calculator just gave them is unreasonable. Students think the calculator can never be wrong, but they forget that they might have typed something in wrong! See my calculator rant for more of my feelings on that.
My students are always quick to ask “is this right”? after they’ve tried a problem. I try to encourage them to find ways to check their work without just re-doing the problem. (Some types of problem make this easier than others, of course.) But even if they can’t find a way to totally check it on their own, I ask them to check and see if their answer makes sense. I call this the “sanity check” though I’m sure that I did not coin the term. (I personally don’t consciously estimate an answer before starting the problem, but do “sanity check” the answer when I’m done.) For combinatorics problems, there’s almost always a second whole way of looking at the problem, so I like doing it two different ways and seeing if I get the same answer. 🙂
When I was a student doing multiple choice tests, if I had extra time I’d try to figure out what mistake the test-makers were predicting with each distractor answer. I disagree that test-makers usually avoid the outlying distractor, but students do learn that if one answer stands out too far from the others, it’s almost always a wrong choice. 😉
I think every math teacher has encountered this, and you’re right on. However, in my experience, it is not enough to model. This becomes the most apparent to me in our trig unit. By the time we get here, we have learned a lot about triangles, particularly that the largest angle is always opposite the longest side. So I constantly ask whether answers make sense and even have students try to predict the answer before they start. Somehow, this still does not translate for many students into actually doing this themselves. This year, I will be tacking these questions onto some problems, actually requiring that students participate in this error-checking process.
And that’s exactly why I make them estimate – in writing – before starting the real work. A lot of them do not have the number sense to realize their answer doesn’t make sense. I also make them do the real work before using a calculator for the same reason – they trust the calculator even when they get a ridiculous answer.
I love it though when they do the real work and round the answer, then tell me that was their estimate! *sarcasm* That defeats the whole purpose!