Mathematics: Error cataloguing

What you'll be able to do: Turn a term of scattered mistakes into a list of three or four wrong rules, which is what your mistakes actually are.

Why mathematics is the best subject for this

Errors here are unambiguous, locatable, and (the part that surprises people) extremely tightly clustered.

A student's mathematics log over a term typically reduces to three or four underlying misrules. Not thirty errors: three rules, generating thirty symptoms across algebra, calculus and trigonometry, which is exactly why they looked unrelated while they were happening. Each individual instance seemed like carelessness, and carelessness is the wrong diagnosis in almost every case.

The three-field format does the work:

WHAT I DID: the wrong thing, specifically
WHAT I BELIEVED: the rule I was applying that made it look right WHAT'S TRUE: the correct rule, in my own words

The middle field is the method. Everyone skips it and writes a log of corrections, which is a list of right answers you already have in the back of the book.

What mathematics misrules actually look like

These recur across thousands of students, and seeing them named is often enough:

• The minus sign belongs to the bracket, not to each term inside.
• Multiplication makes things bigger. Fine until fractions, then quietly wrong for years.
• sin⁻¹ returns the answer, rather than one solution in a periodic family.
• Cancelling terms rather than factors: (x+3)/(x+5) "simplifying".
• The derivative of a product is the product of the derivatives.
• Squaring both sides is reversible.
• A function is a formula, so a piecewise or arbitrary mapping isn't one.
• d/dx means "put a little mark on it": a notation habit rather than an operation.
• Failing to reject the null means accepting it.
• Matrix multiplication commutes, because multiplication does.

Every one is a rule, correctly applied. That's why they're stable, why they survive being marked wrong, and why "be more careful" never fixes them.

The weekly analysis

Here are this week's error log entries. Don't summarise them. Tell me:
(1) what single misunderstanding would explain the most of them, (2) which entries that explanation does NOT cover, (3) three problems that would test whether I've actually fixed it.

Part (2) keeps it honest, a pattern that explains everything has been stretched. And part (3) matters especially in mathematics because of the discrimination problem: most textbook exercises can be solved with either the right rule or a near-miss version of it. That's precisely why a misrule survives a hundred successful problems. You need problems where the two rules give different answers.

The mathematics-specific field worth adding

Add a fourth field: where on the page.

Errors that cluster at the end of long manipulations are a stamina and care problem, the remedy is shorter lines, more intermediate steps, and checking before continuing.

Errors that cluster at the setup are a comprehension problem, the remedy is prerequisite diagnosis.

Those are entirely different problems producing identical-looking wrong answers, and mathematics is one of the few subjects where the position of the error on the page tells you which you have.

Across the syllabus

Algebra. Sign handling and cancelling dominate. Usually two rules, not ten errors.

Calculus. Chain rule application and the difference between differentiating with respect to x and with respect to y. The implicit differentiation gap (article 03) shows up here as repeated log entries.

Trigonometry. Periodicity, almost always. A student losing marks across equations, graphs and identities frequently has one misrule about what an inverse trig function returns.

Statistics. Assumption violations, and the conflation of significance with size. Log by assumption violated rather than by question.

Linear algebra. Order and orientation, which index is which, which side you multiply on. Positional rather than conceptual, and the page-position field catches it.

Proof. Structural errors: assuming what you're proving, omitting the base case, proving the converse. Log these separately; they're not computational and they don't respond to computational practice.

Probability. Independence assumed where nothing established it. One misrule, many wrong answers.

Series. Applying a convergence test outside its conditions, a boundary error rather than a knowledge gap.

Pitfalls

1. Skipping "what I believed". The failure that makes the log worthless.
2. Logging arithmetic slips. Unless you make the same slip repeatedly, in which case it isn't a slip, it's a rule.
3. Non-discriminating practice. Problems both rules solve confirm nothing.
4. Letting the correct rule be written for you. Write it in your own words or you've copied a correction.
5. No re-test. Misrules reassert under time pressure. Book the re-check.
6. The tell: three weeks of neat entries and you can't name one misconception you've eliminated.

Try this today

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

Write the three fields for each, spending most of your effort on the middle one, Then ask what those three have in common.

In mathematics there's a good chance the answer is "one thing", and a better chance you didn't know it was.