Mathematics: Immediate targeted feedback

What you'll be able to do: Find out, within seconds of finishing a problem, not just that you're wrong but which rule you were following that made the wrong answer look right.

Why this is the first method for mathematics

Mathematics gives feedback the quality no other subject can. The answer is definitely right or definitely wrong. The error is at an identifiable line. And the reasoning that produced it is written down in front of you.

Everything else in the subject's toolkit depends on exploiting that. What has always been missing isn't the possibility of feedback, it's someone with the time to read thirty students' working, every night, and say you're treating the negative sign as belonging to the term rather than the operation.

The move that makes it work

Never paste the problem alone. Paste the problem and your working, including the part you weren't sure about, and ask for three things:

1. The first line that's wrong, not all of them.
2. What you appear to believe that made it seem right.
3. A question that would let you see it yourself.

Item 2 is the one that converts a correction into learning. Every consistent mathematical error is a rule, correctly applied, that happens to be the wrong rule. "You made a sign error" is marking. "You're distributing the minus sign over only the first term, which suggests you're treating it as attached to the bracket rather than to each term inside" is teaching.

Here's a problem and my full working. Do NOT solve it. Tell me only: (1) the first line where I go wrong, (2) what rule I appear to be applying that made it look right, and (3) one question that would help me see it. Then stop.

Worked examples across the syllabus

Algebra: expanding brackets. Student writes -(2x - 3) = -2x - 3. Marking: "sign error." Targeted: "Line 2. You appear to believe the minus sign attaches to the bracket as a whole rather than distributing over each term. What is −1 × (−3)?" The student now has a rule to fix, not an instance.

Calculus: the chain rule in reverse. Student integrates ∫2x·cos(x²)dx as sin(x²)·x². Targeted: "First error is your substitution step. You seem to believe substitution requires you to multiply by the inner function, rather than recognising that its derivative is already present in the integrand. What is du if u = x²?"

Trigonometry: solving equations. Student solves sin θ = 0.5 and gives only θ = 30°. Targeted: "Your working is correct and incomplete. You appear to be treating sin⁻¹ as a function that returns the answer, rather than one solution in a periodic family. How many solutions does the interval you were given contain?" This is the most common lost-marks pattern in trigonometry and it isn't an error of method.

Linear algebra: matrix multiplication. Student computes AB as if it were commutative. Targeted: "Line 1. You're applying a property that holds for numbers and not for matrices. Compute BA and compare." The student generates the counterexample themselves, which is the correction that survives.

Probability: conditional probability. Student computes P(A|B) as P(A)·P(B). Targeted: "You're using the independence formula. What in the question told you these are independent?" The answer is usually "nothing", and that realisation is the whole lesson.

Statistics: hypothesis testing. Student concludes "the null hypothesis is true." Targeted: "Your calculation is right and your conclusion doesn't follow. You appear to believe failing to reject is the same as accepting. What would your test have looked like if the effect were real but small?"

Sequences: induction. Student proves the inductive step and omits the base case. Targeted: "Your inductive step is valid. Without the base case, what have you actually proved?" Often the first time a student sees that the structure of a proof is load-bearing rather than ceremonial.

Geometry: similar triangles. Student sets up a ratio with corresponding sides mismatched. Targeted: "Line 1, check which vertex maps to which. You appear to be matching sides by position in the diagram rather than by correspondence in the similarity statement."

Finding the pattern rather than fixing instances

After a week, batch them:

Here are five problems I got wrong this week with my full working. Do not correct them one by one. Tell me what single misunderstanding would explain the most of them, and give me three new problems that test exactly that.

In mathematics this reliably collapses. Five errors across algebra, calculus and trigonometry frequently turn out to be one thing, usually sign handling, order of operations under a bar or radical, or treating a function as a value.

Scaling

• School: one problem at a time, adult spot-check. A confidently wrong correction is undetectable to a beginner.
• A-level / late high school: batch weekly and hunt the common cause. Start the error log here.
• Undergraduate: feedback on proofs rather than computations, ask for the weakest step in a chain that reaches a correct conclusion.
• Research level: feedback on whether a lemma is doing the work you think it is.

Pitfalls specific to mathematics

1. Pasting the problem without working. Converts the method into answer- getting, and mathematics is the subject where that's most tempting.
2. Accepting a correction you can't follow. The tool can mark correct work wrong. Demand the concrete case: "show me a number where my method fails."
3. Fixing the last error instead of the first. Later errors are usually consequences.
4. Not re-attempting before seeing the solution. Reading the right answer after being told where you went wrong produces strong fluency and little else.
5. The tell: you have a corrected problem set and couldn't state a single rule you'd been getting wrong.

Try this today

Take three problems you got wrong recently. Don't look at the corrections.

Paste each with your full working and ask only for the first error and what you appear to believe. Write the three beliefs down.

There's a good chance they're one belief.