Pariksha SaathiLesson 3 of 8 · Mathematics Pedagogy
Language of Mathematics, Word Problems and Community Mathematics
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Mathematics as a language
Mathematics has its own symbols (+, −, ×, ÷, =, <, >, %), its own vocabulary (numerator, perimeter, quotient, parallel) and its own grammar, in which order and position carry meaning: 52 is not 25, 8 − 5 is not 5 − 8, and 3a means 3 × a. A symbol compresses an idea, so a child who reads symbols without the idea behind them is reading a language she does not understand. Symbols should therefore come after experience and talk, not before: first the action (putting 3 marbles and 4 marbles together), then everyday words ('3 and 4 more make 7'), then mathematical words ('the sum'), and only then 3 + 4 = 7.
Words that mean something else
Many mathematical words are ordinary words with a special meaning, such as table, product, difference, volume, odd, mean, face, degree, power and right (as in right angle). Some phrases reverse the order in which the symbols are written: '5 less than 8' is 8 − 5, and 'subtract 3 from 10' is 10 − 3. Children who have learnt key-word tricks ('more means add, left means subtract') go wrong when the words are not a reliable guide: 'Ravi has 12 marbles. He has 5 more than Sita. How many does Sita have?' contains 'more', yet the answer is 12 − 5 = 7. The remedy is to make sense of the situation, by acting it out or drawing it, rather than hunting for key words.
Why word problems are hard
In word problems the difficulty is often in reading and understanding rather than in calculating. Newman's error analysis (Anne Newman, 1977) follows the stages a child must pass through: reading the problem, comprehending it, transforming it into a mathematical form, using process skills to compute, and encoding the answer back in the terms of the problem. Many errors happen before any arithmetic is done. The teacher helps by reading the problem aloud, asking children to retell it in their own words, drawing or acting it out, using familiar names and contexts, and asking children to make up a story for a number sentence such as 20 ÷ 4, which shows whether they understand the operation.
Home language and school mathematics
Children come to school with a home language that may differ from the language of the textbook, and their number words may follow different patterns. Most Hindi number names from 11 to 99 say the ones before the tens: सत्तावन (57) is literally seven and fifty, while the numeral is written tens first, which can lead to reversals such as writing 75. A name such as उनतीस (29) means one less than thirty, and the English names for 13 to 19 also say the ones first. A child who handles quantities well in her mother tongue may look weak only because the school words are new. NCF 2005 treats multilingualism as a resource: let children reason and explain in the language they know, accept their informal words, and then bridge to the standard terms.
Ethnomathematics
Children also bring mathematics from home and community. The Brazilian mathematician Ubiratan D'Ambrosio gave the name ethnomathematics to the mathematics practised by cultural groups: weavers counting threads for a border pattern, masons and carpenters measuring and setting right angles, farmers estimating land and harvest, women drawing symmetric kolam and rangoli on a grid of dots, potters and tailors working with shapes and sizes, and traditional units such as the hand span (बालिश्त), the cubit (हाथ) and the bigha. A well-known study of child street vendors in Brazil by Carraher, Carraher and Schliemann (1985) found that children who calculated correctly while selling often failed the same sums when they were written in school form: the knowledge was real, but school had not connected with it.
Using community mathematics in class
The teacher treats this knowledge as a starting point, not as wrong or unscientific. She can set up a classroom shop with real price lists, ask children how their families measure land, grain or cloth, invite a mason or tailor to show their methods, compare a local unit with a standard unit, and accept children's own mental methods before teaching the standard written algorithm. India's heritage helps too: the Lilavati of Bhaskaracharya (12th century) taught arithmetic through verse problems about everyday situations. When school mathematics respects the mathematics of the home and the market, children see that they are already mathematical thinkers.
Exam angle
When a child solves sums orally in the market but fails in written form, the right step is to build on the informal method and connect it to written symbols. When a child struggles with word problems, the right step is to work on language and comprehension (retelling, drawing, acting out), not to give more computation practice. Rejecting a child's home language, local units or own method is always the wrong option.
Key facts
- Order carries meaning in mathematical language: 52 is not 25, 8 − 5 is not 5 − 8, and '5 less than 8' is written 8 − 5.
- Ordinary words with special meanings in mathematics: table, product, difference, volume, odd, mean, face, degree, power, right.
- Key-word trap: 'Ravi has 12 marbles, 5 more than Sita. How many has Sita?' says 'more' but needs 12 − 5 = 7. Teach children to understand the situation, not to hunt for key words.
- Sequence for symbols: action with objects, then everyday words, then mathematical words, then symbols (3 + 4 = 7).
- Newman's error analysis (1977): reading, comprehension, transformation, process skills, encoding; many word-problem errors happen before any calculation.
- Problem posing: asking children to write a story for 20 ÷ 4 checks whether they understand division as equal sharing or grouping.
- Hindi number names say the ones first: सत्तावन (57) is seven and fifty, which can cause reversals like 75; उनतीस (29) means one less than thirty.
- Ethnomathematics: name given by the Brazilian mathematician Ubiratan D'Ambrosio to the mathematics of cultural groups such as weavers, masons, farmers and kolam or rangoli makers.
- Carraher, Carraher and Schliemann (1985): Brazilian child vendors computed correctly in the market but often failed the same sums in written school form.
- Bhaskaracharya's Lilavati (12th century) taught arithmetic through verse problems on everyday situations.
Quick revision
Mathematics has symbols, vocabulary and grammar; order matters (8 − 5 is '5 less than 8'); symbols after action and talk; key-word tricks fail; Newman: reading, comprehension, transformation, process skills, encoding; problem posing; Hindi names say ones first (सत्तावन 57); multilingualism is a resource; ethnomathematics (D'Ambrosio); Brazilian street vendors study 1985; build on home and market mathematics.