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Day 5

Philosophical Remarks

When did we first fall in love with mathematics? For me, it was in class 6.

My father exposed me to a problem from Euclidean geometry. We were traveling in Kausani.

After days of frustration and failed attempts, I could put together the ‘reason’ that made ‘everything fit together perfectly’. The problem was solved and beauty of ‘pure reason’ revealed itself. It was breathtaking. I fell in love!

This has been the guiding principle in my teaching efforts. **At the core of mathematics is ‘reason’.** We definitely draw inspiration from real world observations. **However, one does mathematics because he or she adores ‘reason’ itself** **and not the observations**.

Years later, I was exposed to Kant’s ‘Critique of Pure Reason’ (thanks to my wife). I was tempted to switch to Philosophy. The promise to go beyond reason was alluring.

Methodology

The philosophical foundation of this eight - week course in beautiful mathematics is therefore well-grounded in these personal experiences. I hope to expose the students to the enchanting beauty of ‘reason’. This is planned in the following manner:

- Begin with some observations of objects
- After repeated observations, find a pattern. A pattern is, roughly speaking, Plato’s ‘form’. It is something that reveals itself when you threw away all the ‘unimportant elements’ from your observations.
- Once observation and pattern recognition are accomplished, one employs ‘reasoning’ to see why the pattern could be relevant in broader contexts.

Hence we have the following recipe:

**Observation —> Pattern Recognition —> Generalization**

Here is a concrete example:

**Observe**platonic solids (try drawing them, draw 2-dimensional projections, etc.)- Recognize pattern by counting vertices, faces, and edges (Euler’s number)
- Generalize that Euler number is, in essence, an ‘invariance phenomena’. Invariance is omnipresent. Employ that to understand golden ratio (that is see the application of invariance principle in a completely different context)

There are two other things, that I would love to try in this course

- Dialectical investigation in the line of Tarasov. Roughly speaking, this appeals to two steps
- deconstruction of a big idea into fundamental pieces
- reconstruction of the big idea from those pieces.

- Rabindranath’s experiments with pedagogy, especially relating to the objects of observation. Rabindranath recognized that if one stays close to nature and social fabric at the
**observation**stage, then the**pattern recognition**and**generalizations**are fundamentally altered. Though I have not experimented or studied this claim in detail, it seems plausible.

Cheenta - Filix Level 1 Math Olympiad Starter module.

**Day 0** - Warm up with beautiful problems and drawings.

**Day 1** - Platonic Solids (Cube, Tetrahedron, Octahedron, projections)

**Day 2** - Platonic Solids (Icosahedron, Dodecahedron, projections)

**Day 3** - Counting the simplexes

**Day 4 **- Invariance principle (Euler number)

**Day 5 **- Invariance principle (Golden ratio)

**Day 6 **- Algorithms (Fibonacci number generator)

**Day 7** - Algorithms (Fibonacci number generator)

**Day 8** - General problems from invariance principle

How the sessions are designed?

- Each session begins with a
**‘Motivation problem sheet’**. Students are expected to try these problems on their, possibly even before attending the class. They are allowed and encouraged to discuss amongst themselves. - The discussion kickstarts with a
**big problem or big idea**. Lectures are limited to 15 minute slots. Students will need to ‘do’ mathematics after each such 15 minute slot. - The session ends with a ‘
**Follow up problem sheet**’. These problems are ‘collaborative homework’. They are most effective when students discuss them in groups. - The faculty may recommend some
**additional reading**!

Key Points

**Faculty:**Ashani Dasgupta, founder-faculty at Cheenta, pursuing Ph.D. at University of Wisconsin Milwaukee, USA, in Geometric Group Theory**Who may attend the course:**Students of class 3 to 7 or anyone who wants to fall in love with mathematics**When are the sessions:**Friday 6:30 PM I.S.T. to 8 PM I.S.T.**When is this starting:**November 22, 2019**What is the cost:**Rs. 4899 for two-month enrollment (free for existing students at Cheenta)**How are the classes conducted:**Online, live, interactive**How to enroll:**Email us at support@cheenta.com to apply or click on the following link to join.

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