When the pahadi-walla school in Sangam Vihar closed for summer, Reena Jha realised that she needed a maths tutor to complete her holiday homework, and that her family couldn't afford one.
"YouTube," Reena thought to herself. "I'll study from YouTube."
Her father was sick and unable to work, so her mother had to cancel the monthly TV subscription in order for Reena to buy an unlimited data plan on the family's beat-up smartphone.
"I'd heard YouTube has everything," said Reena, "so I searched for Class 12 CBSE Maths."
The reception is poor in this working-class settlement on Delhi's southern flank. Call drops are frequent and download speeds are slow. Reena still managed to find a YouTube channel called "CBSEClass Videos". It features a balding man in a bulky sweatshirt standing before an old-school chalkboard and methodically solving every question in her maths textbook.
"My phone screen's broken, so it is hard to see very much," Reena said. "I listen very carefully and solve the questions myself."
Very few students in government schools opt for higher mathematics as they lack the resources to pay for expensive tuitions and reference books.
One of 11 students studying higher mathematics in her school of 5,700 pupils, Reena is a rarity in Delhi's government school system. Though maths is essential for many professional courses, most students drop the subject right after Class 10, once it is no longer mandatory.
Maths isn't easy: it's the biggest reason why over 100,000 children do not pass Class 9 in Delhi every year. After failing their annual maths examinations, a majority of students quit the school system entirely. The difficulty of maths is such a severe problem throughout India that last month, the Bombay High Court asked educational boards to consider making mathematics optional in Class 10.
Yet numbers percolate at every street corner of Sangam Vihar in the routine calculations performed by vegetable vendors, the complex measurements of carpenters and masons, and the profits racked up by the local water-tanker mafia. A wide body of academic research from across the world has emphasised how children can intuitively perform complex mental calculations, but freeze the moment they are presented with simple word problems.
Anita Rampal, a Professor of Elementary and Social Education at Delhi University, who wrote sections of the NCERT's elementary school textbooks, said that her research shows that young kids tend to be more comfortable with mental maths than with written language.
In India, maths is still seen as a rare talent that must be found and nurtured rather than recognising it as something everyone does all the time"
The problem, according to Rampal, is not mathematics as a discipline, but the way the subject is taught in a needlessly difficult and rigid manner.
In its current form, mathematics perpetuates India's gaping inequality. It prematurely closes off opportunities for children who cannot afford tuitions and guidebooks, which are necessary to crack the exam. As a result, kids lose the chance to become architects, pilots, sailors, and computer programmers, all professions that require passing marks in Class 12 maths.
Reena, for instance, wants to become a chartered accountant, which calls for proficiency in arithmetic. Her syllabus, however, demands a mastery of calculus, inverse trigonometric functions and three-dimensional geometry.
How did mathematics come to be this way?
What motivates children in government schools to persevere against the odds? Watch Reena's story to find out.
Reena was seven years old when she first felt maths was slipping from her grasp.
"I couldn't understand division and fractions," she said.
Division, experts around the world agree, is one of the hardest concepts to teach young minds. In 2012, researchers at the American university Carnegie Mellon found that a child's grasp of division in elementary school predicted her overall maths achievement in high school, even after controlling for differences in income, age, gender and I.Q.
Reena's elementary school teacher almost never held a maths class, so her mother asked a neighbour's college-going daughter to explain this crucial concept as best she could. The neighbour's daughter zeroed in on what she thought was the problem: Math-Magic, Reena's maths textbook, which was developed by the National Council of Educational Research and Training.
"She said, 'Math-Magic is a rubbish book. This is not maths, this is full of shapes, stories and poems,'" Reena recalled. "She gave me another book to study from."
Yet Math-Magic was specifically designed by a committee of teachers to make studying maths easier and more enjoyable. The reason Reena felt the textbook was inadequate provides a glimpse into why maths came to seem so inscrutable in the first place, and how difficult it is to change India's convoluted education system.
Researchers are constantly seeking new ways to teach maths better. But the Indian classroom remains more or less unchanged.
For Jyoti Sethi, a primary school teacher who co-wrote Math-Magic and trained teachers to use the book, Reena's discomfort with the textbook is not surprising.
"There is a misconception amongst teachers and students that maths is only arithmetic," said Sethi, who is finishing a PhD in education, of primary school.
Playing with shapes, she said, is maths as well. "It's geometry, and shape-play is an important step towards learning algebra."
In 2006, Sethi was one of the few primary school teachers invited to rewrite the NCERT's textbooks. Till then, subject syllabi were designed by deciding what students should know in their first year college, and then working backwards to set the syllabus for senior school, middle school, and elementary school.
Taking together, a child's syllabus across five subjects assumes that he or she will go on to pursue an advanced degree in each one. Reena's middle and senior school maths syllabus, for instance, imagines that she will do Maths honours in college — a path she has no intention of pursuing.
This problem was flagged as early as 1993, when the Yashpal committee, formed by the ministry of human resource development to improve the quality of teaching and examine the workload of schoolchildren, concluded that "the syllabi and textbooks are evidence to say that the experts involved in preparing them have little knowledge of school and classroom realities."
But decades would pass before the import of its findings sunk in.
The Math-Magic textbooks sought to break this pattern by teaching math from the bottom up. The books tried to incorporate the diverse life experiences of India's children. An exercise in the Class V textbook uses journalist P Sainath's award-winning reportage on farmer suicides to illustrate how a small loan can multiply into a huge debt.
"Multiplication is about the power of a rapid increase in something," said Sethi, who wrote that section in the book, "so when you do it with a farmer's loan, it gives you goosebumps."
But when the NCERT asked her to train teachers to use the new text, she encountered resistance. "Teachers said, 'This is all very well, but this is not maths. Where are the sums?'"
In the classroom, sceptical teachers ignored the new book and taught maths the way they always had. Children like Reena picked up on their teacher's misgivings and began to distrust the text as well.
"We couldn't change the deep beliefs that teachers already had about mathematics," Sethi said. "A teacher is used to having a traditional textbook with a set of sums with a set algorithm to solve the sums."
Children are pushed to use these algorithms to solve each sort of question in the same ways. Yet one standard method often doesn't make for the easiest or the most intuitive way to learn — it is simply the fastest.
Standard algorithms, Sethi concluded, "are standard only because mathematicians felt this way will take less time."
Mathematics curricula in India remains wedding to standardised forms of problem solving and discourages students from seeking innovative solutions.
The algorithmic approach has its supporters. They feel that following set methods is the best way to drill a difficult concept into impressionable minds and, most importantly, to efficiently solve the Class 12 board exams.
Studying for the maths board exam is a bit like preparing for a marathon: students start slowly by solving a few sums every day and gradually scale up in intensity and complexity to build a sort of algorithmic muscle-memory that kicks into action the moment a question paper is placed before them.
Such an exhaustive training regimen requires an endless supply of practice questions. Most students turn to the "RD Sharma", a commercially published textbook that has acquired mythic status as a mathematical bible.
Ravi Dutt Sharma, the author of the eponymous book, is an embodiment of maths instruction in post-independence India. Legend has it that by age 9, Sharma had mastered both square roots and cube roots up to 20 and multiplication tables up to 40.
"For each type of problem, I give an algorithm on how to approach and how to solve the problem," Sharma told me over the telephone, explaining that the best way to introduce any concept was to provide an example of it in a mathematical problem.
In his book, each concept comes with its own set of exercises that become progressively harder to solve. "If a student does all the solved examples, in the sequence I have given, she will never face any problem," Sharma said.
Sharma sees no real need to change the existing mathematics syllabus. If anything, he feels that the NCERT erred by removing logarithms, a notoriously abstract concept, from its syllabus.
"It should be brought back as logs are required for Class 11 and Class 12 science," Sharma said. "Everyone needs to study maths. It makes a person more analytical."
A decade after she first went for tuitions, Reena has turned tutor herself. Here she helps two children from the neighbourhood with their maths homework.
On most days during her summer break, Reena follows a set routine.
"I wake up early, help ma with housework and then study till lunch," she said. In the afternoons, she tries to clear her doubts online, and sets aside particularly tough questions to ask her teacher when she returns to school.
Before her evening accountancy tuitions, she has a special appointment.
"Two children, in Class 1 and 3, come to me for tuitions," Reena said. "I help them with their homework and teach them basic concepts.'
Almost a decade after her neighbour's daughter taught her division, Reena is reprising that same role for another set of children. It is the nature of a place like Sangam Vihar where knowledge is too prized to keep to oneself, and must be shared almost as quickly as it is gained.
The children go to a private school, so they don't use the Math-Magic textbook.
Has she taught them division?
"The older one can divide with a single digit divisor," Reena said. "We haven't yet started division with 2 digits."