Error analysis, diagnosis and remedial teaching in mathematics

What this heading covers

This is the centre of the mathematics pedagogy paper. More questions come from it than from any other pedagogy heading, and they are rarely about the definition of a diagnostic test. They give you a child's wrong answer and ask what it shows — which means the examinable skill is reading an error, not naming a method.

The position the syllabus takes, and which every question is written from: a child's wrong answer is usually the correct application of a wrong rule. It is not carelessness, not a gap in attention, and not a lack of practice. The child has constructed a rule that works somewhere and is applying it where it does not.

Everything else in this lesson follows from that one sentence. If errors were careless, the remedy would be care. Because they are systematic, the remedy is to find the rule and break it.

Why errors are systematic

Three pieces of evidence the syllabus expects you to be able to give:

1. The same error repeats. A child who writes 1/2 + 1/3 = 2/5 will write 1/4 + 1/5 = 2/9. A careless slip does not reproduce itself with that fidelity.
2. The error is reasonable. Each one is a rule that genuinely worked earlier. Adding across worked for whole numbers; "bigger number means bigger value" worked before fractions.
3. The child can explain it. Asked how they got the answer, a child with a systematic error gives a confident, coherent account. A child who was careless cannot.

That last point is also the diagnostic instrument, and it is free.

The errors, and the rule behind each

Read this as a diagnosis table rather than a list of wrong answers. The right-hand column is what has to be taught; the left is only the symptom.

The child writes	The rule they are using
1/2 + 1/3 = 2/5	Add the tops, add the bottoms — whole-number addition carried into fractions
1/3 > 1/2	A bigger denominator means a bigger number
0.5 < 0.25	More digits means a bigger number
3 × 0.5 > 3	Multiplying always makes things bigger
6 ÷ 0.5 < 6	Dividing always makes things smaller
42 − 17 = 35	In each column, take the smaller digit from the larger, whichever is on top
205 written as 2005	Place value not secure — "two hundred and five" written as heard
7 + 0 = 0	Zero makes things disappear, carried over from 7 × 0
4 + 3 × 2 = 14	Work left to right; order of operations not yet established
Perimeter of a 4 × 6 rectangle = 24	Perimeter and area confused — the one formula remembered is multiply
3 cm + 4 cm² = 7	Units ignored as decoration rather than read as meaning
1/2 of 8 = 16	"Of" read as a signal to multiply by the denominator
A square called "not a rectangle"	A shape belongs to exactly one category
2.5 rounded to 2	Rounding down whenever the digit is not 6 or more

Each of these is a generalisation that overreached. That phrase is worth having, because a question asking why children make errors in mathematics wants it: a rule learned in one domain is being extended to a domain where it fails, which is ordinarily how learning works and is here producing a wrong answer.

The counter-case: how each rule is broken

A wrong rule is not removed by being told it is wrong. It is removed by meeting a case it cannot survive, which the child can see for themselves. This is the part most answers leave out, and the part a longer question asks for.

Wrong rule	The case that breaks it
Add tops and bottoms	Half a chapati and a third of one. Is that two-fifths of one? Show it with paper.
Bigger denominator, bigger number	A cake cut into 3 against the same cake cut into 12. Which piece is bigger?
More digits, bigger number	₹0.50 against ₹0.25 in coins on the desk.
Multiplying makes bigger	Half of 8 is 4, and half of 8 is 8 × ½.
Column subtraction either way	42 − 17 with bundles of ten sticks; the child has to break a bundle.
Perimeter is multiply	Walk the edge of the classroom, then tile its floor. Two different jobs.
A square is not a rectangle	Sorting hoops: four right angles and four equal sides overlap.

The pattern is the same each time: a physical or monetary situation where the child's rule gives an answer they themselves can see is wrong. Money and paper folding do most of this work, because a child will not accept that half a chapati plus a third is less than half.

Diagnostic testing

A diagnostic test locates a difficulty. That purpose decides everything about its design, and the paper tests whether you can tell it apart from an achievement test.

	Diagnostic test	Achievement test
Purpose	Find where and why a child is failing	Find out how much has been learned
Coverage	Narrow — one skill, many items	Broad — the whole unit
Items	Graded in small steps, each isolating one sub-skill	Sampled across the content
Timing	When difficulty appears	At the end of teaching
Scored for	The pattern of what is missed	A total mark
Followed by	Remedial teaching	A report, a grade, the next unit

The characteristic design point: a diagnostic test on subtraction does not ask twenty different subtractions. It asks several with no regrouping, several with regrouping in the tens, several with a zero in the minuend, and so on — so that the pattern of failure names the sub-skill. A total mark is almost useless for this; what matters is which block the child failed.

Other diagnostic instruments the syllabus recognises, which are not tests at all:

• Asking the child to explain their method aloud. The single most informative, and the cheapest.
• Watching the work, not the answer — where the pencil hesitates, what is counted on fingers.
• Observing over time, since one answer is not a pattern.
• Looking at the child's own written work, where the method is visible.

Remedial teaching

Remedial teaching is not more of the same. A child who has a wrong rule and is given thirty more exercises will apply the wrong rule thirty more times and finish more confident in it. That sentence is examinable almost verbatim.

The sequence:

1. Identify the pattern, across several pieces of work rather than one.
2. Find the rule behind it — ask the child to explain.
3. Break the rule with a case it cannot survive, as above.
4. Rebuild from the concrete, then pictorial, then symbolic; and only then practise.

Two further principles the syllabus names:

• Remedial work is individual or small-group, because the rule differs from child to child. Two children with the same wrong answer may have different rules behind it.
• It should be immediate. A misconception left to run gets built upon, and the repair grows.

And a point about attitude that questions do ask: the child must not be made to feel the error was stupidity. The error was reasoning. Treating it as reasoning that went one step too far is both true and the only way the child will tell you what they actually did next time.

A question of the kind the paper sets

A child writes: 1/2 + 1/3 = 2/5. The teacher should first: (a) tell the child the correct answer and the rule for adding fractions (b) give the child twenty more sums on adding fractions (c) ask the child to explain how the answer was reached (d) explain that fractions must have the same denominator before adding

The answer is (c).

Work through why the others fail, because the question is designed so that each is attractive:

• (a) gives the right answer without removing the wrong rule. The child now has two rules and no reason to prefer one.
• (b) is practice of an error. Twenty more sums produce twenty more wrong answers, more firmly held.
• (d) is the correct mathematics, and it is the second step. Taught before the child's own rule has been surfaced, it arrives as another instruction to be remembered rather than as a correction to something they believe.

The stem's operative word is first. The pedagogy paper is testing sequence, not content: diagnosis precedes instruction. If the option list contains "ask the child how they did it", that is almost always the answer.

Mistakes that cost marks here

• Treating errors as carelessness. The syllabus's position is that they are systematic.
• Choosing "more practice" as remediation. It is practice of the error.
• Choosing "tell them the rule" over "ask them to explain". Diagnosis comes first.
• Describing a diagnostic test as a shorter achievement test. It is narrow and graded, and it is read as a pattern rather than a total.
• Saying remedial teaching is for weak students generally. It is for an identified difficulty.
• Forgetting that remediation is individual. The same wrong answer can have different causes.
• Leaving out the counter-case when asked how to correct a misconception — telling is not correcting.

In one line

A wrong answer is usually the right application of a wrong rule, so the teacher's first move is to ask the child how they got it, the second is to give a case the rule cannot survive, and the last is practice — never the first.

In the classroom

The most useful routine here costs nothing: ask for the method before marking the answer. A class in which "how did you get that?" is asked of right answers as well as wrong ones will tell a teacher what it is thinking, because the question has stopped being an accusation.

Keeping a page of the errors a class actually makes, with the date, turns this heading from theory into the teacher's own data. After a term it shows which misconceptions are the class's, which are one child's, and which arrived with the way something was first taught — and that last category is the one worth most, because it is the only one the teacher can prevent rather than repair.