What is pi? We start our ‘Circle’ topic with that question, a fine example of a question a young child could ask that can stymie an unprepared teacher. The most common answers we get are, ’3.14’ or ‘22/7’, and sometimes the textbook answer, ‘the ratio of a circle’s circumference to its diameter.’ Most will not venture any answer at all, preferring instead to wait for us trainers to lay out the truth.
Of course, we don’t; instead we lay out three pratika for them to do:
1. Measure the circumference and diameter of 4 or 5 circles (rolls of tape, cans, CDs, buckets, rings, etc.) and find the quotient of each pair.
2. Take the diameter just measured for each circle, multiply it by 3.14, and see how close you get to the circumference you measured.
3. Cut out a paper circle, cut it like a pizza into small pieces, and use those approximate triangles form a parallelogram. Calculate the area of that parallelogram and compare the result to the area you get multiplying 3.14 by the circle’s radius squared.
Through the process of doing these pratika and discussing the results, we find teachers can start to get a feel for this magnificently mysterious number. Its intrigue stems from the fact that you can’t write it down exactly with numerals, and that it’s always the same, regardless of units used and regardless of whether you’re measuring a spherical star or a toroidal microbe. The teachers in this training were indeed intrigued, and had rousing discussions.
We made plenty of time for discussing because our goal is not only for teachers to learn more information. We’re trying to help them and their students build connections between concepts and organize the new ones together with the old ones. This pratika is a great example of the necessity of connections, because although it appears in the planar geometry unit, it is impossible to do without talking about place value, significant figures, rational and irrational numbers, ratios and fractions, algebra, equations and formulae, and the notion of infinity.
The question about pi shows how a 100% correct answer can be 100% useless if not linked to what the student already knows. Even after you memorize the textbook definition and use pi to solve problems about circles – like you are led to do in any decent textbook – you can be left with the vague idea that pi is a magical constant necessary to do calculations on circles. Your short term goal, if you are not careful, becomes getting your answer to match the one in the back of the book. In stark contrast, as our pratika progressed, we heard more and more questions from the teachers, more attempts to explain what they thought they understood, and much nodding and careful consideration on all parts.
The most pointed question came at the very end, after we’d wrapped up the pratika, gone over all the basic concepts involved, and asked if there were any lingering concerns. A teacher spoke up and said slowly that all was well, but it would be even better if he was able to point to pi: the exactly spot on the circles we had measured and drawn on the board that pi represented, so he could show his students that this, this right here, is pi.
So we delved back into the discussion, and out came more questions, each pointing toward productive answers: What quantity exactly is it: length, angle, size? Why doesn’t pi have units? How is it not like other irrational numbers, the square root of 2 for example, where you can choose your units, draw your isosceles right triangle and measure it straight away with a ruler? How can you say a number is a relationship? And most befuddling of all, how is it that this irrational number is formed by a ratio?
Each of these questions has an entirely correct answer that leaves one frowning unsatisfied, yearning for so much more than a snippet of accurate information. To get satisfaction takes time and effort and most often group work to gain multiple perspectives. In short, it takes pratika.
Our trainings consist in large part of accompanying the teachers to carry out the pratika lessons in the manuals we’ve created, one each in science and mathematics for 7th through 9th grade. These manuals are full of activities using simple stuff and situations from everyday life to apply and understand the mathematics concepts in the curriculum. The current textbooks were made in Portugal and have only a faint overlap with the Timorese experience. Our manuals were made in Timor by Timorese teachers using Timorese materials, so no cultural or geographic translation is necessary. They’re clearly written and professionally edited with plenty of graphics, and our presentations are getting to be pretty well polished as well.
But we find all this only sets the stage for learning. The final critical thing we offer the teachers is the time and space to do these pratika together, with support from ourselves and each other. Sometimes this process takes much longer than I anticipate, but Mestra Angela and the other SESIM trainers are rarely surprised. They can see well the path toward results, and are happy to be accompanying these teachers as they walk steadily down it.