Children's errors and remedial teaching in maths for CTET

When a child writes 52 − 27 = 35, the easy response is a red cross and "careless!" in the margin. But the answer is not random. The child has followed a rule, "take the smaller digit from the larger", and followed it perfectly: 7 − 2 = 5 and 5 − 2 = 3. Until someone finds that rule and replaces it, the child will keep getting every regrouping subtraction wrong, however many worksheets they do. That is the core of error analysis, and CTET asks about it in almost every maths pedagogy section.

The syllabus lists "error analysis and related aspects of learning and teaching" and "diagnostic and remedial teaching" for both papers. Expect a child's wrong answer followed by "the child's error shows…" or "the teacher should…". This guide gives you the types of error, a catalogue of the common ones, a method for diagnosis and the principles of remedial teaching.

Types of errors

A useful test: ask the child to check or explain. A slip is usually corrected at once. A conceptual error survives the checking, because the child's rule still seems right to them.

Systematic and random errors

Systematic errors follow a consistent wrong rule and appear again and again. Random errors are one-off slips. Systematic errors are the valuable ones for a teacher, because they point to a specific misconception that can be fixed.

A catalogue of common errors

Check any row yourself: 23 × 4 is 20 × 4 + 3 × 4 = 80 + 12 = 92, and 612 ÷ 6 = 102 because 6 × 102 = 612.

For the number errors see numbers and operations; for fractions and decimals see fractions and decimals; for algebra see algebra.

Diagnosing from a child's work

Diagnosis is detective work. The steps:

1. Collect several pieces of the child's work on the same skill.
2. Look for a pattern: which items are right, which are wrong, and what do the wrong ones share?
3. Talk to the child. Ask them to solve one aloud: "Tell me what you are doing." This kind of one-to-one interview often reveals more than the written work.
4. Form a hypothesis about the wrong rule or missing concept.
5. Test it with items designed to confirm or rule it out.

Worked example. A Class 3 child's subtraction:

• 75 − 32 = 43
• 41 − 17 = 36
• 62 − 38 = 36
• 85 − 29 = 64
• 90 − 45 = 55

The first is correct. The rest are wrong: the correct answers are 24, 24, 56 and 45. The pattern: every wrong item needs regrouping (the top ones digit is smaller than the bottom one), and in each the child has taken the smaller digit from the larger. In 41 − 17, 7 − 1 = 6 and 4 − 1 = 3, giving 36. In 90 − 45, 5 − 0 = 5 and 9 − 4 = 5, giving 55.

Diagnosis: the child can subtract when no regrouping is needed but has not understood regrouping; the "smaller from larger" rule fills the gap. This is a conceptual error, not carelessness.

Test: give 64 − 21 (no regrouping) and 64 − 28 (regrouping). A correct 43 and an answer of 44 for the second, instead of 36, confirm the hypothesis.

Remedial teaching

Remedial teaching is teaching planned to overcome a specific, diagnosed difficulty. It is not simply repeating the lesson louder, or giving more of the same sums.

Principles:

• Start from the diagnosis. Target the exact gap, such as regrouping, not "subtraction" in general.
• Use a different approach from the one that did not work the first time.
• Go back to concrete materials: bundles of ten sticks, base-ten (Dienes) blocks, place-value charts, fraction strips.
• Small steps with immediate feedback, moving from concrete to pictorial to symbolic.
• Individual or small-group work, including peer tutoring.
• Build confidence. Start with what the child can do; praise effort and strategy.
• Re-assess after teaching to check that the error has gone and not just gone quiet.

Remedial plan for the child above: (1) make 41 with 4 bundles and 1 loose stick; (2) try to take away 7 loose sticks and discover there are not enough; (3) open one bundle, so there are 3 bundles and 11 sticks; (4) take away 1 bundle and 7 sticks, leaving 2 bundles and 4 sticks, 24; (5) record each step on a place-value chart; (6) move to drawings, then symbols; (7) re-test with a mix of regrouping and non-regrouping items.

Causes of persistent difficulty

• Gaps in prerequisite concepts, especially place value.
• Procedures learnt by rote without meaning.
• Language barriers in word problems.
• Maths anxiety and a history of failure.
• A specific learning disability such as dyscalculia, which is listed among specific learning disabilities in the Rights of Persons with Disabilities Act, 2016. It needs careful identification, not a quick label. See learning disabilities.

Scenario questions with reasoning

1. A child writes 43 − 18 = 35. The teacher should: (a) give ten more sums (b) move on (c) tell the child to be careful (d) recognise a regrouping error and re-teach using bundles of sticks
2. Remedial teaching is based on: (a) the syllabus (b) the class average (c) diagnosis of the child's specific difficulty (d) the textbook exercises
3. A child writes 23 × 4 = 812. The child has difficulty with: (a) place value and carrying in multiplication (b) multiplication facts (c) addition (d) reading
4. Error analysis mainly helps a teacher to: (a) rank children (b) understand how the child is thinking (c) finish the syllabus (d) set harder papers
5. A child is asked to check an answer and immediately corrects 36 to 63, which is the number in the question. This is most likely: (a) a conceptual error (b) dyscalculia (c) a careless slip (d) a factual error
6. A child writes 612 ÷ 6 = 12. The best first step is to: (a) ask the child to multiply 12 by 6 and compare with 612 (b) scold the child (c) give a formula (d) skip division
7. Most children in a class make the same error in comparing decimals. This suggests: (a) the children are weak (b) a test is needed (c) decimals are too hard for this class (d) the teaching approach needs to change

Answers and reasoning

1. (d). 3 − 8 was turned into 8 − 3. The correct answer, 25, needs regrouping, which the child has not understood.
2. (c). Remedial teaching targets a diagnosed gap.
3. (a). 3 × 4 = 12 should be 2 ones and 1 ten carried; the child wrote both digits.
4. (b). Errors are windows into thinking.
5. (c). It was corrected at once on checking, so it was a slip.
6. (a). 12 × 6 = 72, far from 612. The child sees the problem and can find the missing zero: 102.
7. (d). A class-wide error is feedback on the teaching, so re-teach with a different model such as 10 × 10 grids.

What to do next

• Write the error catalogue from memory with the correct answer and the child's thinking for each row.
• Make a five-item diagnostic set for one skill, mixing items that do and do not need the key step.
• Write a seven-step remedial plan for 1/2 + 1/3 = 2/5 using fraction strips.
• Continue with maths teaching aids and evaluation in mathematics.