Thousand Flowers Program: Paradigm shift in Olympiad Training

The central theme of the thousand flowers program is: connected ideas and connected problems. We will illustrate the idea using some examples.

But before we do so, let's point out the theoretical motivation behind such a program. It is greatly borrowed from the pedagogical experiments of Rabindranath Thakur. (Reference: https://bn.m.wikisource.org/wiki/বিশ্বভারতী). One of his major criticisms of existing pedagogical methods is this (I won't try to translate this):

এই শিক্ষাপ্রণালীর সকলের চেয়ে সাংঘাতিক দোষ এই যে, এতে গোড়া থেকে ধরে নেওয়া হয়েছে যে আমরা নিঃস্ব। যা-কিছু সমস্তই আমাদের বাইরে থেকে নিতে হবে—আমাদের নিজের ঘরে শিক্ষার পৈতৃক মূলধন যেন কানাকড়ি নেই। এতে কেবল যে শিক্ষা অসম্পূর্ণ থাকে তা নয়, আমাদের মনে একটা নিঃস্ব-ভাব জন্মায়। আত্মাভিমানের তাড়নায় যদি-বা মাঝে মাঝে সেই ভাবটাকে ঝেড়ে ফেলতে চেষ্টা করি তা হলেও সেটাও কেমনতরো বেসুরো রকম আস্ফালনে আত্মপ্রকাশ করে। আজকালকার দিনে এই আস্ফালনে আমাদের আন্তরিক দীনতা ঘোচে নি, কেবল সেই দীনতাটাকে হাস্যকর ও বিরক্তিকর করে তুলেছি।

Opposing the piggy bank method of education (where there is a teacher who 'knows' and a student who 'does not know') we want to seek the students' input to solve problems with their own creativity. We are assuming that the student 'knows' and is 'creative'; that he/she can do stuff. While he/she explores that inner strength, we catalyze the process with some inputs (skills, interesting problems) from time to time.

(Note that this program is run by Cheenta for young students, usually of age 7/8 to 10/11. It is designed as a launching pad for advanced Olympiad programs. Our central goal is to expose the students to rigorous creative problem-solving. This cannot be achieved by simple formula-learning. We must allow young minds to be creative. It takes years of hard work).

Examples

(each of the themes presented below may span over 6 to 8 classes (of 90 minutes). They should be punctuated by exercises and software simulations):

Primes and Algorithm of the Sieve of Eratosthenes (Connecting Mathematics and Computer Science)

• We begin with a description of prime numbers and how they are useful to 'build' other numbers. Next, we try to find methods of checking if a number is prime. An elementary number theoretic investigation reveals that to check n is prime, it is sufficient to perform \( \sqrt n \) divisions.
• A natural follow up question would be: how many primes are there between 1 and n? This leads to Sieve of Eratosthenes. It is unnatural to compute the sieve by hand. Here comes the introduction to algorithms. A simple implementation using Python does the trick.This theme is usually spread over 4 to 5 sessions. It is a fantastic introduction to elementary number theory and computer programming cum algorithms.
• Exercises: Divisibility 1 from Mathematical Circles by Fomin // Simple algorithms

Area, Irrationals and Algebraic Identities (Connecting Algebra and Geometry)

• We begin with simple algebraic identities such as \( (a+b)^2 = a^2 + 2ab + b^2 \). We show the geometric implementation of these identities. This immediately brings us to the discussion of 'area'. We define the area of a unit square as 1 and follow up with a development of area formula as a product of length and width (students realize that we are actually counting the number of unit squares.
• We immediately define rational numbers (as ratios of integers) and show that geometrically we can chop off the unit square into smaller pieces. Next, we use the area to draw pictures of more intricate identities. Finally, we show that some numbers (like \( \sqrt 2 \) cannot be expressed as ratios of integers. We prove that using parity argument and also present a geometric construction of such numbers (compass-straight edge construction)
• Exercises: Compass -straight edge construction of rational and irrational numbers starting with integers, Geometric proof of intricate algebraic identities, parity -argument proof of irrationality.

Vectors, Angle and Motion (Connecting Mathematics and Physics)

• We begin with a simple description of points on the plane (Cartesian).  We clearly describe that points can be visualized as static objects or as a representation of motion. For example (1, 2) can be regarded as just a point or some physical phenomena with a magnitude of \( \sqrt {1^2 + 2^2} \) and direction \( \tan^{-1} 2 \).This is a great point to introduce the notion of 'angle'. We describe it as a measure of rotation (an isometry of the plane). We specify that angle is just a ratio of arc over radius (and geometrically why that ratio is important). Next, we go over some physical phenomena that can be described using points.
• We draw pictures of position-time graphs. We define velocity and acceleration draw graphs of several combinations of those quantities. We solve problems of kinematics and describe the physical and mathematical aspects of it. Simulations in Geogebra or other software may aid the process.
• Exercises: Prove that arc over radius is invariant in the rotation. Plot points and vectors and differentiate them. Plot position -time, velocity-time graphs, Kinematics problems from Irodov.

As students, we mostly want to be inspired. A closely followed (connected) second would be to get intellectually challenged. A holistic problem-oriented approach to mathematical science (mathematics + computer science + physics and part of natural sciences) may serve both purposes.

Note for teachers: It is useless to lecture for a large span of time. In fact, it is extremely important to throw clever problems every now and then, that puts all the beads of ideas together.

Note for students/parents: The world is not separated by subjects and classrooms. It is important to approach a problem in a holistic manner. This approach, in fact, provides room for creativity and experiments.

We want to create a futuristic program for our children who will grow in a world of artificial intelligence and advanced technologies. If children of today are not allowed to be creative, then they won't be able to respond to a world where most mundane tasks will be done by machines anyways.

I will add more ideas and themes here. Let me know your opinion in the comments section or at helpdesk@cheenta.in. We are still a development stage for this program.

Link to our YouTube Channel: https://www.youtube.com/channel/UCK2CP6HbbQ2V6gn8Xp_vM-Q