No matter how complex or convoluted the problem was, it was almost a given that students would stay up – until they had solved it. ‘Fun’ as part of the learning process was not a concept that teachers  in India were familiar with.

While I was not happy with the teaching methods, it did help me in two ways :

1. It promoted a deeper understanding of the underlying concepts – some problems could only be solved by re-grasping the basics.
2. It helped build ‘tenacity’ in problem-solving. This was a skill that would help me in any and every profession.

Math is a lot like golf in this regard – you can only learn by doing. You can read all you want – but your understanding will only be superficial. The ‘doing’ in math consists of solving as many problems as possible.

Computer Simulations and the Learning Process

I used to think that Physics was the same way. Solve a bunch of problems  – and you will gain insight into the underlying physical concepts. It wasn’t until I got into computer programming that I discovered another way to gain insight into physics.

Write a computer simulation – as detailed as you can make it. A computer program requires every step to be meticulously spelt out – leave out one step – and the entire program fails. This helps expose any ‘gaps’ in understanding of a problem.

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I posed this to several grown-ups. No one got it. Next, I posed it to my 5th grade class. They spent exactly two minutes – trying different answers – before one girl said ‘I know – the bag is against the wall’

She was right of course, but what is more important is why most adults don’t get the answer.

We grown-ups overcomplicate the problem.

It’s just a bag on the floor. There has to be a simple explanation as to why no one can jump over it. Nothing fancy inside the bag (nails sticking out), nothing wrong with the people (legs are fractured), nothing wrong with the room – just a regular room and a regular bag and regular people to jump over it.

This wasn’t the end of it.

I posed some harder riddles to the 5th graders.

This is something that comes before or after – but never in-between

It cannot be heard, seen. felt, smelled or touched.

What is it?

Additional clue – it also hides in caves and behind stars.

Most grown-ups spend hours and still cannot answer this one. Another 5th grader in my class (not the same one who got the first riddle), suddenly blurted out ‘Dark’! And she looked at me expectantly. I replied ‘Who said Dark?’ – She raised her hand. I congratulated her – she had somehow arrived at the correct answer.

The answers that grown-ups give for this puzzle range from just plain wrong to ridiculous. When grown-ups tackle the question, they expect – nay – DEMAND – that their brain come up with the answer! With kids, they have no just expectations. They know it has to be figured out – and they get right to it. 

Summary

Is it true that kids are better problem solvers than grown-ups? In some sense, yes. They have an unobstructed view of the problem – whereas grown-ups ‘insert’ too many obstructions of their own.

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Physics

A law encompasses a specific area – e.g. mechanics is fully described by Newton’s ‘laws’ – not Newton’s principles. A principle, on the other hand, encompasses ALL natural phenomenon – and is applicable to multiple domains. E.g. – Conservation of energy is a principle and can be applied to mechanics as easily as it can be applied to electricity and magnetism.

Mathematics

In mathematics, every theorem is essentially a principle. Theorems are universally valid – regardless of the domain in which they are originally defined. In that sense, mathematics doesn’t really have laws – just underlying principles that are applicable throughout the domain of mathematics as well as other sciences.

Summary

Laws only apply to specific scientific sub-domains  (laws of mechanics , laws of thermodynamics etc.) – whereas principles are applicably across domains to all of nature (principle of energy conservation,  principle of symmetry invariance etc.)

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• Areas of interest, individualized personal objectives
• Five hardest questions in different areas of calculus
• Informal class discussion

Day 2 – Graphing

• General equation of a line, Physical meaning of slope and y-intercept
• Parallel and Perpendicular lines
• Applications of lines and slopes (water usage, F to Celsius conversion…)

Days 3 – 9 – Functional Relationships – between 2 variable quantities

Functions

• linear
  • Examples from daily life – water usage bill, gas mileage
• quadratic and polynomial
  • throwing stones, parabolas, polynomial division
• exponential
  • Compounded interest versus Simple interest
  • Continuous compounding versus daily compounding
  • Present Value
• logarithmic
  • Calculating ‘how long’ instead of ‘how much’
  • Real world examples
• trigonometric
  • Quadrants, terminal points, sine, cosine
  • Law of Sines, Law of Cosines, Half-angle identities
  • Unit circle, rotating radius
• rational functions
  • asymptotes, roots
• Irrational Numbers, Square root of 2

Day 10 – Mathematical Induction

• 2^2^n + 1 = prime – failure. counter example found
• All horses are the same color – Logical reasoning fallacy
• Useful proofs – infinity of primes, more examples

Days 11 – 14

Mathematical Models

• Material decay, half-life
  • radioactive decay
• Population Growth
  • Algae growth
• Sound Waves and Trigonometry
• Physics Equations – Waves, Newton’s laws…
• Contest – Derivatives, Optimization – Soda can , Area of Material vs. Cost

Day 15

Evaluation, Future direction/interests for each student

The post “Deriving” into Pre Calculus, Course S68–S.I.G., University of Texas, Austin appeared first on Anuj Varma, Hands-On Technology Architect, Clean Air Activist.

Yes – you need a 4.0 GPA. Yes – you need killer recommendations and very unique and humbling work experience. Yes – you need to make the GMAT look like child’s play and display your mastery over it. Guess what?

Every applicant to these top 10 schools meets ALL OF THE CRITERIA above !

And this is why it boils down to the dreaded essays.

The essays are a true window into your soul – something that exam scores and GPAs do not reveal. And it is also where most applicants stumble. Not because they do not have great life stories to share – but because they do not know how to share them. Being a great story teller is a very different skill than being a management genius.

There was one common factor in all the applicants that I know who succeeded in getting into business schools. They all used an essay editing service. One such service – essaytopia.com – was used by two of my close relatives/friends. They have nothing but ‘genius’ to attach to the editing service. What came out of the process was an essay unlike anything they had ever written. The ideas and initial drafts were entirely theirs. Those ideas were like the ‘paper napkin’ scribble of Silicon Valley engineers. What emerged was like the iPod – a killer product set to take the users by storm.

Don’t believe me? Go ahead and give it a try yourself. You have nothing to lose (except your coveted spot in a top MBA school if your essay doesn’t make the grade).

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