eBook Series: In search of Insight by Sajil C. K. · Sajil C. K. 1. Preface . The theme of this...

eBook Series: In search of Insight by Sajil C. K.

1


eBook Series

The eBook series of the University of Kerala, launched under the auspices of the Internal Quality Assurance Cell of University of Kerala, has multiple aims--to provide a platform for speedy publications of scholarly works; to promote dissemination of scholarly thoughts and new findings in a free manner, under a Copyleft license; and to promote outreach activities of the University of Kerala. The opinions expressed in this eBook are not opinions of the University of Kerala. The author is responsible for any copyright infringement and the University of Kerala's responsibility shall be limited to withdrawing the e Book.

This work is released under Creative Commons CC-BY-NC-ND (Attribution NonCommercial-NoDerivs) license, details of which are available athttps://creativecommons.org/licenses/ bync-nd/3.0 /

Internal Quality Assurance Cell University of Kerala

2015

2


In search of Insight

Sajil C. K.

eBook Series No.2 Internal Quality Assurance Cell

2015

3


In search of Insight

“The best brains of the nation may be found on the last benches of the classroom”

-Dr. A. P. J. Abdul Kalam

Dedicated to all those back benchers

4


1. Preface

The theme of this book is about the process of learning. How different are different people learn. I have been obsessed with such things for quite some time. In this book I am sharing some of these things interesting to me.

As far as I am concerned the problem with education is we expect everyone to perform same on a set of conditions. To me it feels like judging a fish with its ability to climb a tree. Since each one is different with different set of skills hence should be education too.

If one student is week on a specific skill then he should be strong on something else and the point is that we have to find out what it is. I am not claiming that I have a master solution for this; if it is then I will be contradicting myself.

The views and ideas in this book are based on my observation. These are less scientific and are more of personal experience. Take whatever ideas you think worthy and throw away those you think false. I would be really happy if it helps you in any way to take another look at the process of learning and make it more productive.

Best Wishes,

Sajil C. K.,

5


2. How high a building is?

“Knowing the name of something and knowing something is different” -RichardP.Feynman Once there was a student who was asked to find out the height of a building with a barometer. The student answered like this “Tie a long piece of string at one end of the barometer, go to the top of building, and lower the barometer till it touches the ground. Then measure the length of the string and barometer. Height of the building will be length of barometer plus length of the string”.

The examiner obviously expected the student to measure the height using the barometer reading and its equations. Since the student was able to solve the problem in his way the examiner gave the student another chance. This time he comes up with a much more interesting experiment. “Go to the top of the building, drop the barometer from there, measure the time it takes to smash on the ground, then find the height using the equation of acceleration due to gravity.

6

eBook Ser

The exaanotherideas. T “ Place tlength, shadow

This timstudentwent lik

ries: In search

aminer war method. TThen he cam

the baromefind the r, now use t

me also thes idea bag

ke this for q

h of Insight b

as not at aThe studenme up with

eter verticaatio betwethe ratio to

e examinerg seemed tquite some

by Sajil C. K

all happy nt seemed

h a non des

ally on theeen the two find out th

r was not ho be still f

e time.

.

7

K

about the d to have astructive id

e ground, mwo, then me

he height o

happy. He kfull with n

barometea magic badea.

measure itseasure theof the build

er and he ag of never

kept askinnever endin

s length ane length ofding”

asked himr ending c

m for crazy

nd it’s shadf the build

dow’s ing’s

g other solng brilliant

lutions andt ideas! An

d the nd it

eBook Ser

“Use ththen you

“Use thhim the And it kknowledyoung ait and h The phito convireally kapproacthat youpractice

3. Why

“Realize

Let’s talament”inside to

Please n

ries: In search

he barometu will get b

he baromete height”

kept going dge and awage. He knehe knew the

ilosophy of ince you anknow somech? How cru thought e and devel

y this app

e that every

ake an exam” by Paul Lo the area

note that le

h of Insight b

ter itself tobuilding he

ter to knoc

like this. Fwarded maew to solvee problem.

f this book nd give ansething. Whreativity ahard. How

lop these sk

proach? –

ything conn

mple to illLockhart. Yof the recta

engths of th

by Sajil C. K.

8

K

o measure eight in mu

the heighultiples of b

ht while gobarometers

oing up in s height”

the stair ccase,

ck the doorr of engine

Finally thearks. This e it in man

is centeredswerers to hy should and imaginw are diffekills? We w

– The ‘ah

nects to eve

― Leo

lustrate thYou were aangle outsi

he sides ar

e examinerstory is oft

ny different

d on this idquestions we do lik

nation will rent thing

will try to a

ha’ momen

erything els

onardo da

his from thasked to fiide.

re not given

eer who buuilt the buuilding andd ask

r was conviften attribut ways. In

inced abouuted to Nieshort he h

ut the studeels Bohr at

had insight

ent’s t his into

dea. This blike what

ke this? Whelp you

gs connecteanswer the

book is a hit means t

What’s the to break t

ed togetherse question

umble atteto say thatbenefit of through thr? How canns in this b

empt t you

this hings n we book.

nts

se.”

Vinci

e famous bind out the

book “A me ratio of a

mathematiciarea of tria

ian’s angle

n purposefufully.


I am sure all of you had similar experience at least once in your lifetime. You try to understand a topic or solve a problem very hard but simply didn’t get it. You start to bang your head against the wall. You are about to give up.

Then somewhere out of the blue you get some idea or hint, a light glows up in your brain. Then you start to understand the concept or solve the problem.

If you haven’t got the answer, before looking for the answer, think about the problem in as many different ways as possible. Look for similar shapes around you. Try measuring, folding or anything from your imagination.

If you are really fed up with it then go for a walk, come back and try to work it out. If you do so, chances are pretty high for you to come back with an answer.

The ratio is . Area of the triangle is exactly half of that of rectangle. Do you have any idea how it is ? To understand this lets draw a line on this figure. Now can you explain why the answer is ?

If you are seeing this problem first time in then you just had an “aha” moment in your life. Insightful learning will let you experience such lots of aha moments in your life. The primary purpose of this book is to convince you that how a different learning approach can bring you lot of aha moments, insights, pleasure of finding things out etc.

This is important because it will help you to see the world from another point of view. This will help you to connect seemingly different ideas together. Rather than learning anything abstract you can connect it to your own life and experiences which you will never forget. It will let you break apart complex ideas into simpler ones and arrive at a solution very fast.

Vantage point is a famous movie which depicts the assassination of a countries’ president. What remarkable about the movie is that the scene of assassination is shown from different perspective. Same is true is about learning too. More perspective you have for a particular topic the more you understand it.

9


Knowing a concept or topic from different perspective is very important. If you have hundred ways to reach a destination then it means less traffic. You can reach at the solution very fast. Moreover problems seemed complex in one perspective may be solved easily by looking it from another perspective.

This time I am going to give you a square. We all know for a square its length will be equal to its breadth. Now I divide it into four parts.

A B

B

A

I am sure we all have seen this before, of course not in this form. Do you see anything familiar to you? Think about it in terms of the area of the boxes.

A B

B

It is the famous-infamous algebra equation 2 . And this description of the formula makes much more sense right? Here you have made a connection between abstract algebra and geometry. Area of larger square is same as sum of areas of individual squares. Now go ahead and fill in these.

10


2 2 2

4. Connecting the dots

Once a great teacher showed me an image like this and asked me, do you see anything interesting? To me it looked like a natural scene I see every day. Then he asked do you see the spiral, curly movement of water? There is a very interesting thing about it. The spiral movement will always be in either in clock -wise or in anti-clock wise direction depending on which hemisphere on earth you are. There is science behind this.

We all see it in everyday in different places, in river, in tea cup, in kitchen sink.etc. But chances are rare for you to think why it rotating in one direction? Why not in

11


other direction? If you know the science behind it then you may think, what if we were at the equator? This was one of the greatest lessons I had so far in my life.

When we are children, we all used to wonder about such things and ask similar questions. But very soon we will lose such an incredible ability and curiosity. No matter how many times we see such things in front of us we won’t realize it.

If someone has insight into something then he will start to see similar connections between things he see, learn or experience. Try this funny and simple experiment. Open any random page in your dictionary; find a word new to you, understand its meaning. Now watch for it a week. Chances are pretty high for the word to come before you within one week either through a news paper, TV show, someone saying it or in some other form.

Everything we read, learn or study is coming in front of us in one form or other. The fact is that most of the times we are not realizing it. The same concept may come in another form in some other discipline. It’s just a matter of one’s imagination to connect different ideas together. Someone who is able to connect new things with existing things he already knew is more likely to retain that information. Another dream aim of this book is to let you realize and discover these connections.

5. Power of a mental model

“Everything we read would be translated as best we could into some reality “

-Richard P. Feynman

Let’s take another example from the famous book “One two three… infinity by George Gamow”.

It asks you to find out the sum of the odd number series 1, 3, 5, 7, 9…etc up to n. One can come up with a mental model for this. Suppose you have some pieces of chocolate bars. You put one first then group another three then another five and so on.

1 + 3 + 5 + 9

Now we can rearrange it little bit in an interesting fashion and we see a pattern emerging.

12


1 1+3 = 4 1+3+5 = 9 1+3+5+7 = 16

?

Draw the next pattern here

Here after at each step we get total number of bars =step number2. Or we can think in terms of total area. So after n terms sum will be n2.

Step No. 1 2 3 4

No. of bars/area 1 4 9 16

Now if I ask you what is the sum of the series ? Can you say the answer by thinking visually?

With the help of this mental model or metaphor one could visualize, learn a concept, or solve a problem without too much effort or mathematics. That’s why it is important to have such models.

13


Now so many other questions come up. How do I know if there is a beautiful analogy like this for any problem? Is everything we read and learn are transformable like this. Will the models be perfect always?

The important thing is that you should get into such a thinking habit. Your own version of analogy or story may be most appealing to you. You have to learn to do that. Also there are lots of outstanding resources and creative people who use these methods. I can suggest few of them.

6. Story Telling

“If you really know something then you can explain it to your grandmother”

- Albert Einstein

Try to remember the topic of the seminar you took first time way back in college or school. Chances are pretty high that you still remember that topic and the concepts. Can you say at least the names of all subjects you had in last semester confidently..?. At least the electives if you had any..?. If you are not able to remember, don’t’ worry it is not exactly your fault.

What makes the difference is, in seminar you were asked to explain it to someone else. Teaching is the best way to learn for most people. It’s a technique which works for a large majority of people. It’s called Feynman learning technique. Still it naturally comes with many people. You may find many people following this technique knowingly or unknowingly.

There is an easy way to test to know whether you know a concept properly. If you can explain a concept to a layman or a 10 year old kid then you understand the concept much better. The more you simplify the language the more you will understand.

While you teach them you will get a clear idea of how well you understand the topic, where are the gaps in your knowledge, which are the areas you have to brush up etc? Then you can go back and refer to materials to fill in the gaps in your knowledge.

Famous scientists who are considered to be gifted with high intelligence are all great teachers. They were able to explain complex concepts in simple terms. They all had insights.

14


Students today will be teachers of tomorrow, they will teach future students. If you miss to grasp a concept properly then your future students will also lose that. It will go like father to son...and to his son...etc. This may seem like an unsolvable problem. The solution is, if you have not got it from your teacher then you will have to earn it yourself.

As motivated scholars we should be in constant search for such insights and knowledge. The more you have the more rich you are. More tools you will have in your toolset and the more productive you will be in solving problems.

In the culture I grew up there was a great tradition of grandmothers telling little moral stories to very young children. We can use the same concept or reverse engineer it. You can make your own version of a story or model of a concept you learn. It’s just a matter of imagination and pretending as if you are teaching your grandmother or grandson the concept.

It may take some time to make up such a model or cooked up story. It may not be perfect. It may not work at all conditions. But still it’s extremely useful and powerful! The point is that over these years you have tried hard to learn lot of things. But those you understand or recollect are very less. Considering the hard work and time you put to by heart all those things which strategy you will prefer?

Once you made such an analogy it will be with you always. You will never find a concept hard to remember because it will have your own experiences, tastes and life in it. It will make your life a lot easier and above all it will be incredibly powerful.

Let’s try to learn the skill of storytelling. I am not going to tell you the topic in advance but I hope by the end of the story you will realize it. Once upon a time there was a bird really addicted to mountain hiking.

he was obsessed with breaking his own records every time. Keep in mind that the mountain here is a very simplified one with not many crusts and troughs. He had

15

.

16

y Sajil C. KKeBook Serries: In searchh of Insight bb

the hablook at i

bit of recorit later wit

ding slop th some sat

of the platisfaction a

ane at diffeand accomp

erent pointplishment.

ts in time so that hee can

He has point inpointing

an arrow pn the plang down slop

pointer to ne. If arrowpe is minim

measure tw is point mum and if

he slope thup then t

f it is horiz

hat he willthe slope

zontal then

l put in paris maximu

n the slope

rallel withum if arrois zero.

h any ow is

Maximum

In this c

He contto get m

But soomounta

case his gr

tinued to bemore and m

n he starteain by looki

aph looked

eat mountamore compli

ed to noticing at those

Minimum

d somethin

Zero

m

g like beloww.

ains one aficated.

ce somethine graph lat

fter anothe

ng. He founter. Say for

er. Each tim

nd he can r example i

me the mouuntains staarted

roughly sain the grap

ay the shapph below.

pe of


He found out that the more steep up the mountain is, the more high the slope value will be and the more shallow down the mountain is, the more low the slope is. He also noticed that if mountain is like a right-triangle then slope graph is rectangular!!!

He realized that almost any mountain shape can be decomposed into a sum of such tiny blocks.

Later some day he was confronted with another special mountain. It was of the following shape.

As usual he took the slope readings and went back to his nest. But while looking the slope graph at home he noticed something very interesting. The slope graph of the hill looked like the hill itself! But with a slight change! He had seen so many hills by this time in his life but none of them seemed to have this property. He also noticed the slope of the hill is zero both at the top and at the valley.

17


Hill

Slope

From all these experiences he is now able to roughly tell how the slope curve will be for any given mountain. Hopefully from this story you also able to do it.

What in essence he did by drawing slope curve is the process of taking derivative of mountain height.

Now we know

• The concept from multiple perspectives (more life, less abstract!!!)

• Why derivative of sin is cos (Eigen functions)and what special about these curves (The ‘Aha!!!’ moments)

• To roughly differentiate a given function visually! (Power of a mental model!!!!)

• A story to tell your students which you and they will never forget!!!

• Why a simple pendulum stops every time before changing its direction (Connecting the dots!!!)Displacement is sinusoidal so velocity or derivative has to be zero when displacement is maximum and minimum)

18


19

7. Age of Scholars

This is the century of scholars. All most any information is available at one’s finger tip but the problem is that often this information is in an abstract form. Here I introduce a few remarkable books and resources.

1. http://betterexplained.com/

2. Gilbert Strang. RES.18-005 Highlights of Calculus, Spring 2010. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu

3. Feynman, Richard P., Robert B. Leighton, and Matthew Sands. The Feynman Lectures on Physics, Desktop Edition Volume I. Vol. 1. Basic Books, 2013.

4. http://www.feynmanlectures.caltech.edu/

5. Stewart, James. Calculus: early transcendentals. Cengage Learning, 2015.

6. Mahajan, Sanjoy. Street-fighting Mathematics. MIT Opencourseware, 2008.

7. Gamow, George. One Two Three... Infinity: Facts and Speculations of Science. Courier Corporation, 2012.

8. Lockhart, Paul. A mathematician's lament. New York: Bellevue literary press, 2009.

http://betterexplained.com/

http://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010

http://www.feynmanlectures.caltech.edu/

eBook Series: In search of Insight by Sajil C. K. · Sajil C. K. 1. Preface . The theme of this...

Documents

Transcript of eBook Series: In search of Insight by Sajil C. K. · Sajil C. K. 1. Preface . The theme of this...