When Totonacú children in Mexico learn to count, they learn the numbers to count animals – one-two-three pigs, one-two-three chickens. Then they learn the numbers for counting wool. And earthen furrows. Those are different. And the numbers for grapes, and other things that grow in clusters? Different. And for round things? And long, straight things? Yep, different.

The Totonacú language has 71 different systems of counting, each for counting different things. Totonacú children have no trouble learning all 71, says Fanny Cruz Garcia, a Mixteco professor of culture and language at the Intercultural University of Puebla who is working with Indigenous teachers in Veracruz. But when those kids get to school, and they encounter the numbers in Spanish – *uno*,* dos*,* tres*, no matter what you’re counting – well, that seems alien. Then they start to learn math, with just those numbers – and the math seems entirely divorced from the real world. It’s one more obstacle in an education that is already of lower quality, by every measure, than that given to students in non-Indigenous communities.

“It’s a shock – kids bring one cultural understanding from their homes and arrive at school and have to learn in a different way,” Prof. Cruz said. “They start to adapt Totonacú to the form of counting that they learn at school – and they abandon the use of the 71 forms of counting.” And all of those words for numbers, too.

But some of Mexico’s Indigenous teachers are working together to try to bridge those worlds, collaborating on a research project to develop resources other Indigenous teachers can use to help their students understand and use the numeration system of their first language – to help them apply a 71-ways-of-counting system to math. They want to give math meaning – and at the same time, to keep those systems and those numbers alive.

There are 7.5 million Indigenous people in Mexico – nearly 7 per cent of the population – and they speak 69 languages. For decades the Mexican government, much like the Canadian one, had an explicit policy to eradicate Indigenous languages, seen as a barrier to consolidating the state and assimilating native people, and so education had the explicit goal of supplanting Indigenous languages with Spanish. Since 2003, it’s been the law here that Indigenous children must be given access to a bilingual and bicultural education – but, in practice, Prof. Cruz said, that rarely happens.

Many teachers don’t speak the language, and even when they do, they don’t have the words for the numbers – which, linguists will tell you, are always the first thing to be displaced from a threatened language: Commerce, the primary point of contact with a colonizing culture, relies on numbers, and so Spanish has replaced the original words in a great many of Mexico’s Indigenous languages. And even speaking the language doesn’t mean you can teach the number systems, she said.

“Few of them have had the opportunity to reflect on this mathematical aspect of their own language,” she said. “So they need training so they can develop that awareness and take it back and value their culture and their own system of counting more, and then be able to teach it to children and be able to say, ‘Look, our language has this richness and this other one counts differently’ – but the teachers haven’t had that chance.”

(Seventy-one is the most complex numbering system she’s run into so far, but she isn’t ruling out that research may yet uncover a system even more complex.)

The project originated at the National Pedagogic University in Mexico City, where José Luis Cortina, a white professor of instructional design, started talking to students in the Indigenous Education program about numbers and math as they used them. In a class one day, a Ch’ol student of his realized for the first time that the counting system he grew up with in his language was different.

Prof. Cortina was already somewhat familiar with the phenomenon: Back in 1995, he volunteered to be a human rights observer in an Indigenous village in Chiapas following the Zapatista uprising. He went with a couple of other professors, and while they were living in the village – where schools had been closed for three years because of the fighting – they decided to start teaching classes. The children were monolingual speakers of Tzeltal, a language that has more than 20 systems of counting, including one for water. He tried to learn Tzeltal, but never managed to master the numbers.

THE BODY NUMERALS

The body numerals were developed with the intention of providing students with opportunities to explore a system for representing numbers consistent with the quantitative rationale of many of the numeration systems of the indigenous languages that use 20 as a multiplicative base, instead of 10. Objects are interpreted by students as quantitative representations of body parts

Object

Interpretation

Value

CIRCLE

FINGER/TOE

1

STICK

HAND/FOOT

5

TRIANGLE

BODY

20

EXAMPLE

The number 63 is expressed in English as “sixty-three” (six, ten, three; 6x10+3) and in Tu’un savi as “uni xiko uni” (three, twenty, three; 3x20+3)

63

+

=

MURAT YUKSELIR / THE GLOBE AND MAIL, SOURCE:

JOSÉ LUIS CORTINA, MATHEMATICS EDUCATION

RESEARCH JOURNAL

THE BODY NUMERALS

The body numerals were developed with the intention of providing students with opportunities to explore a system for representing numbers consistent with the quantitative rationale of many of the numeration systems of the indigenous languages that use 20 as a multiplicative base, instead of 10. Objects are interpreted by students as quantitative representations of body parts

Object

Interpretation

Value

CIRCLE

FINGER/TOE

1

STICK

HAND/FOOT

5

TRIANGLE

BODY

20

EXAMPLE

The number 63 is expressed in English as “sixty-three” (six, ten, three; 6x10+3) and in Tu’un savi as “uni xiko uni” (three, twenty, three; 3x20+3)

63

+

=

MURAT YUKSELIR / THE GLOBE AND MAIL, SOURCE:

JOSÉ LUIS CORTINA, MATHEMATICS EDUCATION

RESEARCH JOURNAL

THE BODY NUMERALS

The body numerals were developed with the intention of providing students with opportunities to explore a system for representing numbers consistent with the quantitative rationale of many of the numeration systems of the indigenous languages that use 20 as a multiplicative base, instead of 10. Objects are interpreted by students as quantitative representations of body parts

Object

Interpretation

Value

CIRCLE

FINGER/TOE

1

STICK

HAND/FOOT

5

TRIANGLE

BODY

20

EXAMPLE

The number 63 is expressed in English as “sixty-three” (six, ten, three; 6x10+3) and in Tu’un savi as “uni xiko uni” (three, twenty, three; 3x20+3)

63

+

=

MURAT YUKSELIR / THE GLOBE AND MAIL, SOURCE: JOSÉ LUIS CORTINA,

MATHEMATICS EDUCATION RESEARCH JOURNAL

Years later, his students began to describe the number systems they grew up with, and a group decided to pursue the subject in their theses. From there, they worked to try to design educational materials that would work in other languages. (All of these languages have an oral numeric tradition, but not a graphic one.)

Gerardo Rojas López, a veteran teacher who has spent most of his career in six-grade, one-room schools in mountain communities in Oaxaca, has done some of the most advanced work adapting math curriculum. His people, the ñuu savi, or people of the rain, speak tu’un savi, and in this, as in many Indigenous Mexican languages, the first multiplicative base is 20, rather than 10, as it is in Spanish and English with their Indo-Arabic numeral system (the decimal system). This likely comes from a system that originally used people as the basic unit – each one with 20 fingers and toes.

“So the number 40 is expressed in English as ‘forty’ (four tens; 4 x 10), and in tu’un savi as ‘uu xiko’ (two twenties, 2 x 20); 63 in English is sixty-three (six tens, plus three; 6 x 10 + 3) and in tu’un savi as ‘uni xiko uni’ (three twenties, plus three; 3 x 20 + 3),” explained Prof. Cortina, picking up a pen and a bit of paper to try to draw a new initiate through the system, his eyes dancing with delight at the complexity.

“In Tzeltal, Tzotzil, Ch’ol and Tojolabal, all from Chiapas, the word they use for 20 is the same as the word for ‘human body’,” he added. “In Nahuatl, the words for ‘five’ and ‘10’ closely resemble the word they use for ‘hand.’ In Hñähñú, the word for 20 is the same as the word for ‘soul’.” Linguists, he said, have been of vital help to the mathematicians as they puzzle through this stuff.

Mr. Rojas, however, believes language, and the loss of it, is the potential downfall of this undertaking. “In schools we are plagued by this disease of mixing Spanish with Mixtec, and children don’t know how to speak either of them well,” he said. They get no help at home because their parents, who have often been economic migrants, don’t either. He estimates that just 10 per cent of his students can count in Mixtec (the larger language group to which they belong). In schools where he taught successive generations of students, he was able to instill a passion for rescuing the language – but now he’s retiring, and he fears the efforts will die out without him there.

The math part itself is not difficult: When the teachers working with this project piloted some of the activities in their classrooms, they said that children learned the basic number sequences, forward and backward, up to 20, in less than 20 minutes. They gave kids manipulatives (physical objects they can move around that represent numeric values) that they called “body numerals”: round wooden chips that represent a finger or a toe, for one; wooden rectangle bars, for hands and feet, worth five; and little triangles, representing the whole body, for 20. Children grasped those almost immediately, too, teachers said, and could add and subtract with them – although it was a whole new way of reasoning.

Prof. Cortina said the Totonacú children aren’t going to be isolated if the state fulfills its obligation to provide a truly bicultural curriculum. The Canadian idea of “additive bilingualism” where kids aren’t just taught to replace one language with the other, but learn to see a second way of doing things, is a good model, he said. “It becomes cognitively much richer.”

There is another initiative under way in Mexico’s Yucatán using the mathematics system of the Mayan civilization to teach children how to do math. It’s novel, Prof. Cortina said, but it serves a different purpose – those kids aren’t learning math that is more relevant to how they live today. “Most of the stuff that has been done with heritage math in Mexico, and also in Guatemala, has focused on the Mayan system to write numbers, but the Mayan written system was discovered by archaeologists and it was probably not used even when the Europeans arrived,” he explained.

There are also small-scale initiatives under way in Costa Rica and Guatemala to train teachers in Indigenous mathematical systems, and research projects by ethnomathematicians in Brazil and Chile, trying to figure out how math curricula might be adapted.

In Mexico, the project is complicated by the role of Indigenous teachers themselves. While the project has teachers participating from 14 different First Nations, not all Indigenous teachers support the idea of modifying the curriculum to include more Indigenous language or a new way of teaching math. In many of these communities, the job of teacher is one of the few secure sources of income. Teaching positions are often de facto hereditary. But a teacher’s child who went to high school and perhaps college will have studied in Spanish and is more likely to have lost their first language, explained Prof. Cortina – and so although the law obliges them to teach in it, they can’t and don’t do it, and they don’t want to see the way math is taught changed either.

“In Mexico to do these things you have a great advantage and a great disadvantage,” he said. “The state is weak, so teachers have great autonomy to initiate things, and so to get things started is much easier. But if you want to do large scale reform, it’s a problem.”

But their project, which has designed curriculum materials for just a handful of languages so far, is a long way from a nationwide effort, he said. “We haven’t got past how to give kids a fighting chance to learn the number system of their own language – there’s a ton more to do.”