Euler's Eyes
2026-10-26
Street mathematics and school mathematics: mathematics off the page
In the early 1980s three researchers at the Federal University of Pernambuco, in Brazil, Terezinha Nunes Carraher, David Carraher and Analúcia Schliemann, went to the markets of Recife to talk to children and teenagers aged 9 to 15 who sold coconuts, corn and popcorn. They posed as customers and set sums at the moment of purchase. Later they came back with the same sums written on paper, school-style.
The result, published in 1985 under the title Mathematics in the streets and in schools, became famous. In the market, the children got about 98% of the sums right. With the same sums presented as word problems, about 74%. As written computation exercises with no story, about 37%. The book the authors published later, in Brazil, summed up the situation in its title: Na vida dez, na escola zero (1988), “ten out of ten in life, zero at school”.
Two ways of calculating
What changed was not the sum, it was the notation. In one of the study’s dialogues, a customer asks how much four coconuts cost, at 35 cruzeiros each. The boy answers something like: “three is one hundred and five; plus thirty, one hundred and thirty-five; plus five, one hundred and forty”. He starts from a price he already knows by heart (three coconuts), adds piece by piece and, at each step, knows what each number is: money, coconuts.
On paper, the same sum becomes $35 \times 4$, set out in columns, with carrying. The written algorithm is powerful precisely because it forgets meaning: it works the same for coconuts, metres or years. But with meaning forgotten, the pupil also loses control: if they get a column wrong, they do not notice that $1400$ cruzeiros for four coconuts is absurd. In the street, that control is never lost.
Ethnomathematics
The Brazilian mathematician Ubiratan D’Ambrosio (1932–2021) gave a name, from the 1970s on, to a field of study of these mathematics that live outside school: ethnomathematics, the ways (tics) of explaining and dealing with (mathema) reality in each cultural setting (ethno). Market sellers, builders, seamstresses, fishermen, indigenous peoples: all count, measure, compare and calculate, with their own notations and procedures, often oral.
In Mozambique, the mathematician Paulus Gerdes studied the sona, drawings made in the sand by the Chokwe peoples of Angola, with a continuous line that winds around a grid of dots without lifting the finger. Behind them lie regularities that can be written with greatest common divisors and symmetries. Ethnomathematics does not say that these practices are “the same” mathematics as school’s in other clothes; it says they are real knowledge, and that school gains when it starts from them rather than ignoring them.
Ethnomathematics is the mathematics practised by cultural groups, such as urban and rural communities, groups of workers, professional classes, children of a certain age group, indigenous societies, and so many other groups that identify themselves by objectives and traditions common to the groups.
— Ubiratan D'Ambrosio, Etnomatemática: elo entre as tradições e a modernidade (2001)
The bridge
The lesson for notation is twofold. School writing is not the only way of thinking mathematically, and the pupil who fails on paper may be a skilled calculator in life. But school writing gives something oral calculation does not: generality, a record, the possibility of checking and of passing on. Good teaching builds the bridge between the two: it starts from “three is one hundred and five” and shows that $4 \times 35 = 3 \times 35 + 35$ is the same thing written another way.
Exercise: ten coconuts
Work out, the market seller’s way, the price of ten coconuts at 35 cruzeiros, starting from the fact that three cost 105.
Three make $105$; six, $210$; nine, $315$; one more, $350$. In school writing, $10 \times 35 = 350$ comes from a rule (add a zero). The seller’s path is longer, but each step is a real price, and he would know at once that $3500$ is wrong.
Exercise: the change
A customer buys three popcorns at 1.50 and pays with a 10 note. Work out the change in two ways: “the street way”, counting up, and with a written sum.
The street way: three popcorns, $4.50$; from $4.50$ to $5$, $0.50$ is missing; from $5$ to $10$, another $5$; change of $5.50$. With a written sum: $3 \times 1.50 = 4.50$ and $10.00 - 4.50 = 5.50$. The first method, counting up from the price to the amount paid, is the one cashiers used before machines, and it avoids subtraction with borrowing.
Chapter index · ← The names on theorems: Stigler’s law · Notation for machines: proofs the computer checks →