What makes an estimation question worth asking

Most estimation questions in circulation are recall questions wearing a scale. "What year did the Berlin Wall come down" has a number for an answer, but nothing about it is estimation: you know it or you do not, and sliding to a number instead of picking from four options changes only the input method.

A real estimation question asks students to get close to something nobody in the room knows, using things they already do know. The number is a by-product. The method is the content.

What the format is actually for

The classic example is Fermi's: how many piano tuners are there in Chicago. Nobody knows. Almost anyone can get within a factor of ten by chaining together things they can estimate. How many people live there, how many households own a piano, how often a piano needs tuning, how many a tuner can do in a day.

That chain is the skill. It is the same move as sanity-checking an answer in physics, sizing a problem before committing to it, or noticing that a number in a news article cannot possibly be right. It is worth teaching directly, and estimation questions are one of the few formats that teach it under time pressure.

Why the distribution is worth more than the answer

With multiple choice you learn how many students knew. With numeric estimates you get a shape, and the shape diagnoses the method rather than the knowledge.

• Tight cluster, wrong. The class shares a method and the method has a flaw in it. This is the most useful outcome available, and the flaw is usually one assumption you can name in a sentence.
• Scattered across the whole range. There was no method. Everyone guessed, which means the question did not give them anything to reason from.
• Two clusters. Two competing methods, and the discussion writes itself: one group from each explains their chain and the room works out where they diverged.

This is a different thing from showing a spread to build interest before a reveal. Here the spread is the diagnosis. You are reading which reasoning the room used, and none of it is visible in a mark scheme that records only how many were right.

1. Make it arrivable at, not lookup-able

The test is simple. Ask yourself whether a student could reason to within a factor of two using only things they already carry around. If the honest answer is that they either know it or they do not, it is a recall question and it belongs in a quiz.

"How many breaths do you take in a year" passes. "How deep is the Mariana Trench" does not.

2. Choose the anchor deliberately, because you cannot remove it

The first number anyone sees drags every estimate toward it. A scale running to a thousand and the same question running to a hundred thousand will produce different answers from the same class, and the effect is strong enough that you should assume it is operating every time.

You cannot switch it off, so pick it on purpose. Set the range wide enough that its midpoint is not a hint, because the midpoint is where every student starts and a scale centred near the answer rewards doing nothing.

3. Ask for the method before the number

Thirty seconds of "what would you need to know to work this out" before anyone commits changes the task from guessing to calculating. Students who had nothing now have a way in, and students who were about to guess confidently have to notice they were about to guess.

This is the single change that does the most, and it costs half a minute.

4. Score closeness, not correctness

Exact-answer scoring punishes good reasoning that landed one step away and rewards a lucky tap. It also teaches the wrong lesson, which is that estimation has a right answer you were supposed to already know.

Give it a tolerance band wide enough that a sound chain of reasoning gets credit. If you are using a tool with a numeric scale, this is usually a setting rather than a workaround; ours is described in the slider article. For anything genuinely open-ended, order of magnitude is the honest unit: within a factor of ten is a good estimate of the piano tuners, and scoring it any tighter is theatre.

5. Say where the real number came from

An estimation question dies if the answer is simply asserted at the end. Show the chain you would have used and the number it produces, then say what the measured figure is and where it comes from.

Students who reasoned well and landed far away need to see why, and the answer is often that one assumption in the chain was off by a lot while everything else was fine. That is a much better lesson than the number itself, and it is the one that transfers.

What to try next lesson

Take one question you already ask that has a number for an answer, and change two things: widen it so nobody in the room could simply know it, and ask what they would need to work it out before anyone commits.

Then look at the shape of what comes back rather than the score.

One caution worth stating plainly. Estimation questions are worse than quiz questions at the job quizzes do, which is telling you who has learned the material. They are a poor substitute for assessment and a good addition to it. If you replace your checks for understanding with these, you will have a livelier room and much less idea what anyone knows.