Teaching Math has been my primary responsibility for the last ten months. After all this time, I recently had my first encounter with a Math educator and researcher visiting the organization. Sneha, who has been working extensively on Mathematics education at the University of Iowa, discussed task-oriented teaching in a Math classroom with our team at Aavishkaar. I was blown away. We’ve had a plenty of reflective conversations on our content, pedagogy and classroom management but I had never heard of a task-oriented teaching approach.
We started off by describing what a ‘good task’ means to us and what are its characteristics. Based on my understanding, a good mathematical task creates a space for learners to implement their knowledge and develop new approaches while they’re thinking mathematically in small groups. Instead of the facilitator, it brings the learner to the centre stage, under the spotlight. The task has to be designed well enough with multiple points to enter and should pose just the right level of challenge to excite students for cracking it instead of them getting frustrated, eventually giving up. It caught my fancy when we unfurled the fine intricacies of a task. Undeniably, it was daunting to implement. “How do we take it to our classrooms”, some of us thought out loud.
“Just like swimming. You throw yourselves into deep waters and learn because you do not want to drown,” Sneha said with her wide toothed smile.
Her words and support motivated me to try it in the upcoming camp for high school students, PiSafari. Here’s one of the tasks that I took up. A jailer is in-charge of 100 prisoners. He has to set one prisoner free and is unable to decide who it should be. Then, he comes up with the following strategy – Assemble the prisoners in a circle and makes them count aloud from 1 to 100 in a sequence. The person at the hundredth count dies. Who will be the lone survivor that will be set free in this case? Now can you tell who would survive if there are n number of prisoners in the cell? This question is a slightly modified take on the popular Josephus Problem.
Apparently, in the original representation of the story, every soldier kills the one adjacent to him. The last person to survive kills himself. Josephus did not agree to this idea. He stood at the 9th position and survived.Josephus problem has many variants but the central idea is the same. Whether the story is about Christians and Turks throwing each other overboard, a selfish step-mother manipulating her husband to let her sons inherit his property or a king trying to find a suitor for his daughter, violence remains. Sounds simple? This is what happened when I tried to solve it.
When we were posed this question, we were a group of ten at the table and decided to simplify the problem with 10 prisoners before we worked with 100. The first 2 people to die in the game did not surprise anybody. Things got interesting when everyone at the table started guessing who would die next and who would be the last one standing. But in the short duration that we had, we could not find a solution. I decided to solve it by myself later. At first, I did not care as much for the answer as to find different approaches to think mathematically.
It might seem pretty straightforward at the first glance. But then you might realize that the pattern that you were betting on, isn’t consistent. Why would that happen? It is also a matter of discipline. Do you get frustrated after the first few attempts or do you find your own strategies? I even resorted to brute force where I scribbled all the numbers from one to hundred on my palm sized diary and started striking them off with desperate hope that finding the answer could help with seeing the process in reverse. Spoiler alert: You thought I got it? No, there’s no answer here.
It was well past well midnight now. But my math frenzy knows no bounds. I called up a friend and asked for assistance. Verbalizing what I was thinking and considering somebody else’s approach would help me see the obvious. We scratched our heads over it some more time only to accept that we were far too tired to come up with bright ideas at 3 AM and decided to call it a day, or better a night. I slept over it.
I intended to take it to the classroom the following day. My worst nightmare had come to life. “What if I don’t figure it out by tomorrow?”, I had asked Sneha panic stricken the previous day. “That will be the most realistic situation of conducting a class and having no option but to figure it out with the students”, she said as if that would soothe my nerves. We created an alternate plan for the class and in the end, it worked out well. I looked at my previous night’s ordeal and realized that while I struggled with the problem on my own, it made me understand how I learn better. Every child has a different approach and a learning style. It’s important for everyone to figure out what works for them and pursue it with confidence.
Spending time with a problem is imperative. Yeah, when I say spending time with the problem, imagine it is just you and the problem sitting in the park, surrounded by birds and bees with a soft summer breeze and nothing else matters. When I was told that somebody else had cracked the problem, it did not matter to me. It did not provide any incentive. I kept trying at my own pace. During this process, I thought of questions in abundance. My biggest concern was that a group of teenagers may not find this problem as enticing as I did. Only because I found value in solving it did not mean they would too.
This taught me how to stay with a problem without jumping to a solution. It reminded me of one of the first things I was cautioned against, as I started my fellowship journey. When you notice a problem that’s not imaginary and exists in our real world, affects people’s daily lives, it is hard to refrain the instantaneous urge to solve it. Even harder, when you feel it deeply within.
It is important to sit with a problem no matter how uncomfortable it may seem. Decision making and problem solving are essential skills no matter whether they are developed through Math or something else. What might speak to you, may not to others. If you wish to find your tribe that connects with your problem as much as you do, then the purpose must be clear.
In the social sector, the problem you choose to pursue might be the one you’ve undergone yourself or you may have related to it from others’ plight. But indigenous ideas arise when you have been through it yourself. You will also be more wary of your steps when you can empathize. Less than 5 weeks ago, my co-fellow Simran made me unlearn every trick I had learnt in school for solving fractions. She made me visualize those portions instead of multiplying numbers. The entire exercise of unlearning was infuriating. Was this what I did to my students when I taught them a concept that they had mastered solving using the bookish procedure
However I had faith in her objective and I believed in the merits of conceptual understanding. I pushed until I grasped the concept and exclaimed how rewarding it was. Working alone and researching on your own is important but collaborating with the right people will lead you further. Working in any field must always define the minimum that has to be achieved while leaving a plenty of space to meet the broader vision (similar to a low threshold, high ceiling problem in Math). When your heart is no longer involved and you’ve gone astray from your purpose, is when you choose to perish. Would you dare to join the circle of 100 people now?
Gratitude to Sneha Bhansali for her positive vibes and unintentionally infecting people with love for mathematics wherever she goes.
- Selecting and designing groupworthy tasks: https://www.nctm.org/Handlers/AttachmentHandler.ashx?attachmentID=CgLU0b2ctnw%3D
- Josephus Flavius game: http://www.cut-the-knot.org/recurrence/flavius.shtml
- Other versions of the game: https://www.cut-the-knot.org/recurrence/ancient.shtml
- What are ‘rich tasks’ in maths, and why are they important? https://calculate.org.au/2018/07/12/what-are-rich-tasks/