Pariksha SaathiLesson 6 of 8 · Mathematics Pedagogy
Errors and Misconceptions in Arithmetic, Fractions, Decimals and Geometry
Drafted with AI assistance and fact-checked on 28 October 2026; not yet read line by line by a human editor. Spotted a mistake? Tell us. Editorial standards & corrections
Slips, errors and misconceptions
Not every wrong answer is of the same kind. A slip is a careless, random mistake, such as copying a digit wrongly, which the child corrects at once when asked to check. A systematic error recurs in the same way across many items because the child is following a consistent but faulty rule; Brown and Burton (1978) called such rules buggy algorithms. A misconception is the underlying belief that produces systematic errors, often a rule that worked for whole numbers stretched too far. Slips need checking habits; systematic errors and misconceptions need teaching aimed at the concept.
Errors in whole-number arithmetic
In addition, 47 + 38 = 715 shows the child writing each column's total in full instead of regrouping 15 as 1 ten and 5 ones, and 47 + 38 = 75 shows the carried ten forgotten (the answer is 85). In subtraction, besides taking the smaller digit from the larger in each column, children go wrong when borrowing across zero: 503 − 168 = 435 (the answer is 335) shows that the 0 in the tens place became 9 but the 5 hundreds were not reduced to 4. In multiplication, 23 × 14 = 115 shows the second partial product (23 × 10 = 230) written as 23 without its place value; the answer is 322. In division, 618 ÷ 6 = 13 (the answer is 103) shows a 0 in the quotient dropped because 6 does not go into 1. Each of these points to weak place value, not to weak tables.
Errors with fractions
Many fraction errors come from treating the numerator and the denominator as two separate whole numbers. 1/2 + 1/3 = 2/5 adds tops and bottoms; a quick check shows it cannot be right, since 2/5 is less than 1/2, one of the numbers being added. Saying 1/5 > 1/3 because 5 > 3 ignores that more equal parts means smaller parts. Calling any one of 4 unequal pieces one-fourth shows that the need for equal parts is not understood. Not seeing that 2/4 and 1/2 name the same amount shows that fractions are not yet seen as numbers with a place on the number line. Fraction strips, paper folding, sharing rotis equally and marking fractions on a number line help.
Errors with decimals and measurement
The longer-is-larger misconception (0.25 > 0.7, 0.125 > 0.5) carries whole-number thinking into decimals; the opposite shorter-is-larger misconception (0.3 > 0.35, because tenths are bigger than hundredths) also occurs. 0.3 × 0.2 = 0.6 (the answer is 0.06) shows that decimal places are not understood, and lining up the last digits instead of the decimal points when adding shows the point is treated as a separator rather than as a marker of place value. In measurement, children often start from 1 instead of 0 on the ruler or count the marks instead of the spaces between them, and many believe that two figures with the same perimeter must have the same area.
Errors in geometry and algebra
Children at the visual van Hiele level think that a square turned on its corner is a diamond and not a square, that a triangle must stand on a horizontal base, and that a square cannot be a rectangle. Many think an angle is bigger when its arms are drawn longer, because they look at length rather than the amount of turn. In early algebra, besides reading 3a as three apples, children write 2a + 3b = 5ab, think a letter must stand for one particular number, or assume that different letters cannot have the same value.
Error analysis
Error analysis is the teacher's systematic study of a child's mistakes to find the rule behind them. She collects several pieces of the child's work, looks for a pattern across items, forms a hypothesis ('he does not regroup'), tests it with a few designed items, and confirms it by asking the child to think aloud ('Tell me how you did this'), the clinical interview method Piaget used. Teaching then targets the misconception, often through cognitive conflict: the child meets a result that contradicts his rule, such as seeing that 2/5 is less than 1/2, and reorganises his idea (Piaget's disequilibrium and accommodation). Simply marking the answer wrong, re-teaching in the same way or setting more drill of the same type does not change the underlying idea.
What errors reveal
Errors show that children are actively making sense and applying rules they have constructed, often by overgeneralising from whole numbers. They are therefore information and a normal part of learning, not a sign of laziness to be punished. In CTET, an option saying that the error shows what the child has not yet understood, or that the teacher should find out how the child is thinking, is usually correct; an option saying the child is careless or needs more practice of the same kind usually is not.
Key facts
- Slip: careless, random, corrected on checking. Systematic error: a consistent faulty rule (a buggy algorithm, Brown and Burton 1978). Misconception: the belief behind it.
- 47 + 38 = 715: column totals written in full, no regrouping. 47 + 38 = 75: carried ten forgotten. The answer is 85.
- 503 − 168 = 435: borrowing across zero, the tens became 9 but the hundreds were not reduced; the answer is 335.
- 23 × 14 = 115: second partial product written as 23 instead of 230 (answer 322). 618 ÷ 6 = 13: zero in the quotient dropped (answer 103).
- 1/2 + 1/3 = 2/5 adds tops and bottoms; check: 2/5 is less than 1/2, so it cannot be the sum.
- A fraction needs equal parts: calling one of 4 unequal pieces one-fourth is a misconception.
- Decimals: longer is larger (0.25 > 0.7), shorter is larger (0.3 > 0.35), and 0.3 × 0.2 = 0.6 instead of 0.06.
- Measurement errors: starting from 1 instead of 0 on a ruler, counting marks instead of spaces, thinking equal perimeter means equal area.
- Geometry errors: a tilted square called a diamond, a square not accepted as a rectangle, longer arms thought to make a bigger angle. Algebra: 2a + 3b = 5ab.
- Error analysis: collect work, find the pattern, form a hypothesis, test with designed items, interview the child, then teach through cognitive conflict.
Quick revision
Slip vs systematic error vs misconception; buggy algorithms (Brown and Burton 1978); 47 + 38 = 715 and 503 − 168 = 435 show weak place value; 618 ÷ 6 = 13 drops a zero; 1/2 + 1/3 = 2/5; fractions need equal parts; longer-is-larger and shorter-is-larger decimals; ruler read from 1; tilted square; longer arms; 2a + 3b = 5ab; error analysis: pattern, hypothesis, test, interview, cognitive conflict; errors are information.