Mathematics: Predict then verify

What you'll be able to do: Say what the answer will roughly be before you compute it, which catches most sign, magnitude and setup errors before they propagate through six lines of correct arithmetic.

Why this is worth five seconds in mathematics

Mathematics is the subject where a small early error costs the most. A sign lost in line two makes every subsequent line wrong-but-consistent, so nothing downstream complains. You finish, you get a number, and the number looks like a number.

Prediction is the cheap guard. Before computing, commit to:

• the sign: positive or negative;
• the rough magnitude: near zero, order one, large;
• the behaviour: increasing, converging, bounded.

Then compute. When the answer contradicts the prediction, one of the two is wrong and you know to look, which is the entire benefit, because otherwise you'd have had no reason to.

The mechanism is prediction error: the gap between expectation and result is exactly the information your model was missing, and it's encoded far better than a confirmation. But the practical case in mathematics is simpler than the cognitive one: it's the only cheap error-detection you have on work that gives no other feedback until it's marked.

What to predict, by topic

Integration. Will the answer be positive? Is the area above or below the axis? Does the integrand look like it came from a product, a composition, or a quotient?

Differentiation. Is the function increasing at the point in question? Should the derivative be larger or smaller than the function there?

Limits. Does it converge, diverge, or oscillate? To roughly what? Students compute limits mechanically and are unsurprised by nonsense.

Series. Converge or diverge, commit before applying a test. If your test disagrees with your intuition, one of them is teaching you something.

Probability. Roughly what fraction? A probability above 1 or below 0 is caught instantly by a prediction and not at all by careful arithmetic.

Statistics. Will this be significant? Which direction is the effect? Predict before computing and you'll notice when you've swapped a numerator.

Matrices. What should the determinant's sign be? Is this transformation going to expand or contract?

Geometry. Sketch it and estimate. An answer that disagrees with your sketch by a factor of three is a setup error, not an arithmetic one.

The prompt

Before you tell me anything: ask me to predict the sign, rough magnitude and behaviour of the answer, and to say why. Wait. Then tell me the actual answer and (if I was wrong) which part of my reasoning was wrong, not just that my number was.

The last clause matters. "You were wrong" is a result. "You predicted positive because you assumed the function was above the axis throughout the interval" is a lesson.

The four kinds of miss

Classifying is where the value is:

• Wrong sign: your model of the situation is inverted. The most valuable miss; stop and find the inversion.
• Right sign, wrong magnitude: you have the structure and not the scaling, Usually a missing square, a missing factor, or an ignored constant.
• Right answer, wrong reason: the most dangerous, because it passes unnoticed. This is exactly why you state the reasoning, not just the number; without it you'd have scored this a hit and carried a broken model forward.
• No prediction possible: you genuinely have no model yet. Not a failure: a finding. This is material to learn, not to test.

The version for exams

Predict, then compute, then check the answer against the prediction before writing it down. Thirty seconds across a paper, and it catches the category of error that costs the most: a correct method applied to a wrong setup.

It's also the only self-checking that works under time pressure, because re-doing the computation takes as long as the computation and usually reproduces the same error.

Pitfalls

1. Predicting silently. Say it or write it. An unspoken prediction is revised the instant you see the answer and you'll believe you knew.
2. Predicting the number rather than the reasoning. "About 40" tells you nothing when you're wrong.
3. Only predicting when unsure. The confident cases are where a wrong model hides.
4. Abandoning the prediction instead of investigating. When the two disagree, find out which is wrong. That's the whole event.
5. The tell: you're always right. Either you're predicting things you already know cold, or you're not committing.

Try this today

On the next ten problems you do: before each, write the sign and rough magnitude you expect, and one clause of why.

Count how many times the prediction and the answer disagreed. Each of those was an error you'd otherwise have found out about when it was marked.