Hajira Abdul Qayyoom, a fourth grader, was puzzled. She had just participated in an exciting discussion on divisibility rules. Rafia Aunty, her math teacher at Al Qamar Academy, a tiny alternative school in Chennai, India, was not a typical math teacher. She didn’t just tell the students how to solve problems. She didn’t just show the students how to solve the problems. She guided discussions in which kids figured math stuff out, and then explained their ‘A-Ha’ moment to the class. But Rafia Aunty had just let her down. Today, the class was learning about divisibility rules—that is, methods for figuring out whether one number is divisible by another. So, for example, the divisibility rule for “2” is that if the last digit of a number is even, it’s divisible by 2. The class sailed through the divisibility rules for 2, 4 and 5, and wrestled with the logic behind the rule for 3 (“a number is completely divisible by 3 if the sum of its digits is divisible by 3”—why does that work?). They connected the rule for 3 to the rule for 6; the class started up at 7. Rafia Aunty flatly said, “There is no divisibility rule for 7.” The flatness and finality of the statement didn’t sit well with Hajira. “Why?” she wondered aloud. “Well, why don’t you try and find a divisibility rule for 7?” said Rafia, lightheartedly.
That was the spark Hajira needed. She spent the next five days obsessing over the problem amidst reams of paper, half-chewed pencils and scattered notebooks. She sat in the classroom. She sat under the tree. Skipped lunch. And classes. She was looking for patterns. Rafia had seen what the girl was up to and requested the other teachers to excuse Hajira from their lessons. Rafia made herself available for Hajira whenever the child wanted to discuss something. Then, one day, Hajira came up with an elegant divisibility rule for 7. The rule laid out a pattern for ones, tens, hundreds, thousands and beyond: Here, in print for the first time, is Hajira’s divisibility rule for 7:
= 4*1 + 7*3 + 3*2 + 8*-1 + 6*-3 + 7*-2 + 8*1 + 5*3 +4*2 + 1*-1
= 4 + 21 + 6 + 8 +15 + 8 – 8 – 18 – 14 – 1
= 62 – 41 = 21
Since 21 is divisible by 7, the 1458768374 is also divisible by 7
Note: The ‘needs’ are the positive multipliers and the ‘doesn’t needs’ are the negative multipliers.
In most Indian schools, math teaching gives centrality to the curriculum, the textbook and the teacher. Math is taught as a set of techniques or procedures to be applied in a given problem. The class learns in a lock-step manner. A concept is taught usually as a set of procedures, sums are worked out and then forward, on to the next concept. Since time is of essence, deep learning is difficult. In a world of ‘correct answers’, there is little space for students’ creative or original thinking. Students are tested frequently, the notion being that testing reinforces the learning. However, this approach has been linked to student passivity, disengagement and higher math anxiety (Hunter, 2012; Williams, 2013). Anxious students develop negative attitudes towards math, are less likely to use problem solving strategies and perform poorly on math assessments (Lipnevich et al., 2016; Ramirez et al., 2016). A national level conducted study in India found that while students could easily answer rote learning based questions, conceptual understanding or application was sadly lacking (Quality Education Study, 2006). And anecdotally, a common complaint amongst students is that math is hard, boring or irrelevant. Al Qamar was not different in its early years. While younger students learned math using Montessori material, the thirty-odd, upper elementary/middle school students were taught math using traditional curricula. Despite the absence of exams and a presence of discussion in class, math was clearly not a hot favorite. This is a tale about how one innovative math teacher managed to transform the school’s approach to math teaching using powerful innovative pedagogical practices.
When Rafia Riaz signed on as a math teacher, she was determined to bring about a change in the students’ attitudes. A seasoned educator, Rafia intuitively knew that to change attitudes, kids had to develop a love for math. She knew that kids enjoyed working with each other in groups. Kids relish challenges. And she knew that all kids liked to have a say in their learning. What pedagogues called engagement, motivation, peer-to-peer learning and student autonomy. Rafia may not have known these terms, but she knew that these were required. To create a buy-in for her plan, Rafia first negotiated, advocated, and convinced the rest of the faculty at the school about the importance of inquiry based math, teamwork and fun in math teaching. Her enthusiasm was so infectious that we all joined her bandwagon. Rafia’s plan for the 4th-7th graders was simple. The younger ones still needed to use math manipulatives they had used in Montessori. For the older ones, Rafia adopted inquiry-based math textbooks as a resource. Supplementing the ‘formal’ learning for both groups were a range of math storybooks, puzzles, math games and other fun activities.
To start off, Rafia changed the physical structure of the classroom to align with her plan. Furniture was rearranged and sets of desks were joined together to create collaboration spaces. Rafia hijacked empty classrooms for her teams to wander off into – to have animated and undisturbed discussions. She encouraged the teams to find quiet spots in the school. There were several occasions when the teams were found sitting outside under trees, in the organic terrace garden and near bushes busy working on math.
Next, Rafia introduced teams in math classes. Her premise was that collaboration instead of competition would provide scaffolding and lead to better thinking. Teams got the resources they needed – calculators, chart paper, stationery and a whiteboard. Initially, Rafia planned the work for each team, but over time, student teams took up the mantle of creating their own work plans. Kids scheduled meetings with Rafia, to update her on their progress, thereby maintaining rigor and task focus. When the team faced difficulties, Rafia analyzed the causes and addressed them. When there were internal team issues – conflicts or “lone ranger” work, she counseled and advised. When issues were external – due to the frequent electricity shutdowns, lack of resources or noisy environments, Rafia worked with other faculty to minimize the problems.
The process of mathematical thinking was given high priority. Math concepts were taught as investigations. Students were encouraged to think for themselves and use different approaches including problem solving, modeling and experimentation. Googling was impossible given the nature of the problems. Asking parents was a no-no. Rafia had three clear ground rules: “respectful interaction”, “wrong answers are wonderful” and “the means are more important than the end”. These guidelines helped reduce the fear factor usually associated with math. Problems were a challenge and setbacks were a part of the deal. Honor lay in trying persistently and innovatively, and not in getting the right answer. Once the allotted teamwork time was over, teams regrouped to share the results of their investigations. Each team had to explain the thinking process they used to solve the math problem to the entire cohort. This, of course, led to animated and sometimes heated discussions amongst the teams – as each group had to justify its method using argument and evidence. Debate was an integral part of classroom discourse. The discussions made student thinking visible and added to the diversity of approaches that students used in problem solving. Over time, the inter-team competition evolved into inter-team collaboration, as students were heard discussing math during lunch break, in their free time and, to my feigned disgust, even during my English class.
The introduction of math puzzles, games and storybooks shifted the notion that a textbook is the only source of mathematical knowledge. These were arranged tastefully in the main hall which functioned as a library, hangout space and a computer lab. Children had free access to these resources. There was a whole shelf of math stories starting with the Murderous Math series, Life of Fred, Malba Talha, and others. Like the proverbial Pied Piper, Rafia would grab a Tower of Hanoi puzzle off the shelf and start ‘playing’ with it by herself. Seeing a teacher so deeply absorbed in a puzzle, the students would gravitate towards her with helpful hints and suggestions. Slowly, the group would become engaged as Rafia quietly put posers to them. “Which peg do you need to start with to make the tower in the last one?”, “Can you come up with a generic rule for the division of diamonds amongst the pirates?” Her questions would start animated discussions, with kids grabbing sheets of paper and pencils to prove their solution was the right one. The weekly puzzle card was a big attraction as children and teachers pored over a card and tried different solutions. It sometimes took days to crack a problem. Which was fine, because the class had learned about Iranian mathematician Maryam Mirzakhani, who was famed for her slow, deep thinking (and near constant-doodling) and who is the first woman ever to win the Field medal, known as the “Nobel prize of mathematics.”
Puzzles afforded children the opportunity to work by themselves at their own pace. Fifthth grader Sarah, who had terrible math anxiety when she joined Al Qamar, used to wait for Monday mornings when the Play with Your Math card was released (Play with Your Math is a website that puts out a digital “card” with an intriguing math problem every week). Sarah assiduously worked on each card through the week, sitting in a corner by herself, and conferring with Rafia when she got stuck. She would try different solutions, think of myriad ways to solve the puzzle. This set her up the kind of sustained thinking required when she encountered the Pirate treasure problem (See Box). Sarah was not content with solving the problem. She wanted to come up with a generic solution despite not having learned algebra.
Rafia shifted the role of the teacher from being a dispenser of knowledge to being a coach and mentor. She became a sounding board, facilitated discussions, provided feedback and gave helpful hints rather than answers. This helped the children construct their understanding using their own thinking rather than passively learning math techniques. Rafia used questions to elicit the student’s understanding, make their thinking visible and identify misconceptions. These interactions between students and the teacher added to the democratic culture of the classroom, by reducing hierarchy, fear and other barriers to innovative thinking.
Clearly the children were captivated by the cognitive challenge posed by the math problems, enjoyed the process of inquiry and deriving formulae from first principles. Math became fun and cool. Math was celebrated. They enthusiastically signed up for online math competitions. Kids thought in math terms “How should you angle a cricket drive to pass over the bowler’s head and for the boundary?” “How could you price a muffin so that the profit could be evenly split?”. Sure, the cohort didn’t cover the range of concepts which their grade level peers in other schools were racing through. We were nowhere close. We sacrificed breadth for depth. And were okay with all the twists and turns that students encountered in their math journey. We knew that through this process children were building a lifelong love of math. The goal of completing the syllabus was peanuts compared to this outcome.
What did the experiment in math teaching lead to in the three years that it took place? First, it clearly led to deeper learning. This showed up in the nature of math insights children developed, examples of which were the divisibility rule for 7 and the pirate problem. The engagement with math also fostered creativity. There was a shift in children’s attitude to math. The students who previously wouldn’t have had anything to do with math were now spotted reading math books and using math manipulatives in the classroom. Rafia recalls how Abi, who earlier would mentally shutdown in math class, realized he was good at estimations. So good at it, he found errors in math textbooks where their calculations/ estimations didn’t make sense. He was now approaching math as real-life situations and values and not just numbers off a textbook. An element of playfulness was introduced. Some children loved experimenting with math-art building a spatial understanding of geometry in real-time. Others started creating math related board games and their own puzzles. Children developed a sense of persistence as they struggled with challenging math problems for days. Kids like Shahid and Hawa who struggled with arithmetic, found they enjoyed geometry. And excelled at it. Rafia nurtured their sense of achievement with positive feedback. For all these kids, there were no test scores as a reward for their engagement with math and the accompanying struggle. No prizes. But a sense of curiosity and enjoyment of intellectual challenge drove them, as it drives mathematicians all over the world. The impact of the program continued even when the children went off to other schools after Al Qamar shut down. Rafia narrates her experience with Izzy,
“who slept over math worksheets for two years, only to wake up in Grade 6 and realize math is also about seeing patterns, and she’s good at it. That one observation changed her attitude and from there on it was a two-year journey till she got good at the subject, by choice, competing with kids who had already excelled at Math. Even today though she’s back in a traditional system and the Math makes no sense, she’s not all lost, and finds ways to understand and proceed. The liking for the subject which she developed in middle school helped her realize the purpose and logic of the subject and keeps her going. Else she feels she may have completely detached herself from it and hated it engulfed by a senseless fear of the subject.”
An interesting and unintended fallout of the experiment was the impact Rafia’s experiment had on the girls. Traditionally, girlsare not expected to excel at math and, because of the power of teacher expectations, they often don’t. However, when they saw their teacher- so passionate about math, these girls became math-mad. They loved math, enthusiastically solved puzzles and transferred their skills to other subjects like history and English. Their sense of self efficacy in doing math shot up. Let’s be clear. There were still kids who disliked math. But they were in the minority.
So how did this all happen? Hunter et al. (2016) list three key factors which lead to both math competency and positive math attitudes. The first is innovative and powerful math learning environments. These are built through classroom cultures where children take responsibility for their learning, use a range of practices like discussions, explanations and justifications using reasoning and develop self-identities as mathematicians. The cultures support risk-taking, persistence, and place a premium on students’ thinking process. Teachers communicate their high expectations to all their students, but provide support and scaffolding as needed. The second factor is the use of innovative math teaching practices which promote inquiry, student autonomy and relevance. Inquiry is central to good math learning. Marshall & Horton (2011) showed how time invested in exploring a math concept led to higher cognitive thinking as compared to time spent in teacher explanations. Collaboration is a key innovative math teaching practice. It acts as a form of scaffolding where the intra-mental activity between students leads to higher inter-mental activity for each child (Vygotsky, 1978). Teachers act as co-learners but also use their professional judgment to notice students’ thinking, ask probing, open-ended questions and respond to students’ reasoning to facilitate the process of student inquiry. Embedded in these innovative practices is a sense of democratic power-sharing, which brings students to the center of the classroom discourse. The third factor is the use of mathematical tasks which promote deeper learning. These tasks are open-ended, messy and allow for several solutions. The tasks do not necessarily emerge from textbooks but are curated from a wide set of resources. Moreover, the tasks are presented thoughtfully to encourage student ownership and inquiry, unlike traditional classrooms where even conceptual math tasks are often simplified and taught as procedures.
Rafia created exactly this kind of a math culture at Al Qamar. She showed kids that math could be fun, she flipped things around to support their learning, and experimented with new ideas. Her belief that “..at the end of day it was we can all do math, we are all good at it, even if only at parts of it. The teamwork helps better the other parts” communicated a sense of confidence to the children. There are other Rafias in a world inundated with unimaginative math teachers. These teachers are beacons of light amidst the metaphorical gloom that math teaching is associated with. However, these Rafias need supportive school cultures, a sense of derring-do among the management who thumb their nose at pervasive cultures of syllabus completion and testing.
Here’s to you, Rafia – may your tribe multiply.
Epilogue: Hajira’s elegant divisibility rule for 7 went unacknowledged by the wider community. In a country with millions of students, who notices one student’s breakthrough? How could she have informed them? Whom could she have informed? Retrospectively, I realized that we could have encouraged Hajira to write a paper and send it to a journal. But water flowed under the bridge, Al Qamar Academy closed down and Hajira moved on. She continued her work, trying to find divisibility rules for 11, 13 and 17. Clearly, she still retains her love for math, and who knows, may well be the next Mirzakhani all due to the spark provided in her early years by a passionate math educator. Hajira’s rule appears for the first time in print in this article.