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A Mundane Ode to To Kill a Mockingbird

Rujuta picked this book as part of her list of books to read in her ninth grade. I hadn’t read it before and, as a result, I too read it with her.

We’d read it just like the protagonist, Atticus, would read it with his daughter, Scout, but with a small change: Rujuta too read it to me. For a while now we have our homeschool conversation sessions (I hesitate calling them “classes”) done remotely. But that does not deter us from communicating effectively and joyfully. I wish sometimes that I were where she is, but I solemnly accept it like a serenity prayer practitioner would.

This is not a review of the book. Just a spontaneous (I just finished the book) reaction. This is a story about the drama that takes place in a fictional town of the “deep south” state of Alabama in the United States. It tells us about a white man’s struggle against a deeply racist white society. Harper Lee has masterfully narrated it and I have admired her no-fluff, rational writing a lot. She uses words that have connotations from a postwar, post-depression era of the second half of the twentieth century. That takes some getting used to, but once the reader does that, the true nature of the novel is slowly revealed.

Some modern interpretations make this anti-racist novel, somewhat surprisingly, a racist book. One reason for it might be that the protagonist is a white man who gives the appearance of a savior. Such an interpretation may have enough backing, but I didn’t think it was racist simply because it did not occur to me that the author committed the error of premature generalization — deriving a conclusion about a group from a few individuals’ behaviors. So, for me, it is not a racist novel. The book, for some reason, turned out controversial at least in the first couple of decades after it first came out.

Of course the protagonist is an idealist. I’d compare him with a saint rather than a human being. His ideals are perhaps too much to take for many of us especially in a society whose values are strongly influenced by materialism, conformity, and stability because of one’s positional strength.

Many a time, I wept and sobbed as I reflected on Lee’s words; perhaps because the story was gripping, the drama felt real, and sentiments overflowed. I visualized the situation in that small town of many characters and their value systems. I am born and brought up in a completely different world, but I could still understand and appreciate it because, you know, although human behavior has changed, human nature has changed very little.

Most of all, there were so many moments when I felt that the power of love trumps the love of power. It reminded me of Charles Bukowski:

“We’re all going to die, all of us, what a circus! That alone should make us love each other but it doesn’t. We are terrorized and flattened by trivialities, we are eaten up by nothing.”

It is a gem of a novel.

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A Michigan Dad Now

I am still preparing the remaining parts of my ongoing series on our homeschooling experiment. But it is worthwhile to report that Apoorv is officially an undergrad at the University of Michigan, Ann Arbor (UM).

I share this news not with pride, but with gratitude and joy. I am aware that many deserving students get into great colleges all the time, so I could hardly feel proud anyway. I humbly wear the Michigan Dad T-shirt — Thanks to this fine young man, the supporting family and encouraging friends, the welcoming city of Pune, and a great school that is UM. Go Blue!

Although I got to actually take him to UM, I wholeheartedly acknowledge the superwoman part that Deepa played. Without her support and contribution, this would not have been possible.

An outlier does not need to go to a college to study, but usually colleges are where many young people study, and, more importantly, find themselves. Of course, as the adage goes, in college, some pursue learning, and others learn pursuing!

US universities still hold sway worldwide when it comes to serious and fun higher education. Even the great Don Knuth has said (somewhere in an interview; I couldn’t find a link to it) that he gets a kick out of the US university atmosphere; universities take him back in time to Greece.

I have shared my Letter to a Young Undergrad with Apoorv!

He is in the school of engineering. They offer 18 majors. Although Computer Science is what he wants to do, the school encourages several courses (and minors) to increase the intellectual breadth of students. I browsed the CS curriculum (and Data Science and Computer Engineering curricula) and it looks pretty good. I was excited to visit UM and felt like I would enjoy the milieu a lot (if I were somehow associated with the university …) The campus is beautiful (especially in summer) with its three main campuses at Ann Arbor. Of course, one has got to prepare to face the Michigan winter.

He is also interested in Sanskrit and I know that Professor Madhav Deshpande founded the school of Asian studies at UM. I hope he gets to study at least some Sanskrit there.

I understand that all of this comes at a cost and higher education in the US is a (financial) mess according to many, but even if one is less privileged in their early youth, one can still dream to be at a college later. Enough perseverance should let them study at a coveted school. I wish everyone wanting to get college education luck and patience.

At some point, students feel like getting out of that cozy atmosphere and enter the real world. But for now, I hope that he and other freshers around the world find a seat at their colleges. I understand that personal care, the weather, the dorm, the food, social aspects, loneliness, self-control and self-regulation, tough courses and tough graders etc. need time to get used to; it takes time to belong to a place.

So, my suggestion to everyone: Hang in there!

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The DARK Homeschooling Experiment Recap – 1

We1 have been lucky!

It has been more than four and a half years since we relocated to India from California. Our friends and acquaintances in particular and the society at large have been very helpful and genial through our journey. Also, we have been steadfast about our decision to be a part of the independent learning project that we started in June 2018 in the Bay Area and carried it out through the intercontinental move.

Some acquaintances were interested in knowing more about our experience. I decided to write it down in the hope that it will help them. I have written in three parts:

  1. A Product Perspective (only results matter)
  2. A Process Perspective (process matters; results will follow)
  3. A Summary Perspective (what seems to have worked for us; is independent learning for you in this time and age?)

In this part, I shall cover the results (1) that can be objectively assessed. To a lot of people this matters the most. They want to focus more on the destination, not as much on the journey. Although I love processes and journeys that are fun, I have nothing against result-orientation, because I do think that goals and results are important. Obsession of anything, however, is fraught with issues.

We have also created a society that rewards good results and penalizes bad ones. We often say, “Nothing succeeds like success.” or (especially in the US,) “It’s either Yale or McDonald’s.” When there is a tremendous apparent competition, such polarization is understandable, but is it necessary?

The definitions of good and bad [results] are not clearly specified, but an invisible majority decides them. My definition is rather easily stated: good results are the ones that help us make progress (which is, although subjective, perhaps easier to understand objectively.) For example, a lot of us believe that graduating from high school and going to a college is progress. A regular college admission is therefore a good result.

Here are the results of the journey so far:

  1. We all navigated our way through an extremely challenging pandemic. As a result, we are better prepared.
  2. We learned to embrace the change and thrive while experiencing it amid difficulties. This is a life lesson learned well. Our kids’ social lives were expectedly different from that of a typical student going to a traditional school, but they were not adversely affected. They made good friends at both the gymnastics club and our housing complex.
  3. Rujuta (14 years old) graduated from the middle school with some very good progress on Gymnastics, Arithmetic, Algebra, Geometry, Chemistry, Problem-solving, Hindustani Classical Music, and English. She continues to learn from excellent books such as Euclid’s and Callahan’s Elements Redux, Gelfand and Chen’s Algebra, and PSSC Physics. For a lack of standard exams, she was assessed independently. She has spontaneously and enthusiastically documented her work (see here, here, and here). Rujuta will be relocating to the Bay Area because she wants to pursue gymnastics more seriously. She has seriously considered going to a traditional high school, but wants to continue homeschooling because of the clear flexibility it offers.
  4. Apoorv (18 years old) graduated from high school. I prefer to say that he is a self-taught student who preferred to homeschool. He made excellent progress on Computer Science, Mathematics, Chemistry, Biology, Physics, Sanskrit, and Gymnastics. He documented his work with great interest (see here, here, here, and here). He learned from the OpenStax textbooks, Inquiry Based Learning, Khan Academy, and numerous other resources. He also carried out a calculus co-learning project for younger students. All the participants were very happy with the sessions. He wrote several standard examinations (1580/1600 on SAT, 5/5 on 5 out of the 6 AP exams) and applied to US colleges through the College Board. Many colleges rejected him (more on that in the next article), but quite a few showed interest. He won a scholarship admission (CS) to the University of Massachusetts, Amherst. But finally he committed to the University of Michigan, Ann Arbor, Michigan (UMich). He will be majoring in either Computer Engineering or Data Science.
  5. Deepa and I (the full-time steachers at our homeschool) documented the experience not only in the text format (see the links above) but also in audio (there are literally dozens of recordings) and video formats. Personally this is a treasure simply for its nostalgic value. I have frequently listened to our discussions and, as a result, I am a better facilitator and steacher. In doing so, I have experienced what Russell has said about teaching: As an educator, being kind [especially to students] is what really matters.

In my view, these results speak for themselves. We all grew in one way or the other.

Given the amount of uncertainty, nobody could have asked for more. I may write about applying to US science, engineering, or arts colleges in a separate article, but Apoorv’s success at securing an admission to a reputed college is remarkable. It has not come at the cost of lost enthusiasm of learning something (unlike it usually happens for several students in some high-pressure school districts of America and India). Rujuta’s interest in mathematics (a subject she dreaded in her elementary school) and problem-solving was born and cultivated.

I am not going to boast about or endorse homeschooling, but frankly admit that it worked for us. Not because it resulted in a thriving high school student and a budding undergrad, but because the interest and flexibility in learning was preserved. It rarely felt like a drudgery even when things were challenging.

We wanted ourselves to experience the cultural and social medley that India is. In 2023, India is also doing exceedingly well on the economic front. In 2018 we were not so sure of it. While India has always positively amazed me (except, of course, the road traffic), moving to India was far from easy. Kids experienced familial values and traditional setup firsthand. They found meaning and discrepancies in it. They appreciated how such a big country was effectively vaccinated through a pandemic. They became a part of the mix that added unforgettable colors to their childhood fabric.

Apoorv and Rujuta were of course strangers when they came. They had an American accent. Their English sounded different. Road traffic and pollution were unbearable. The crowd was maddening. An utter lack of professionalism couldn’t often be ignored.

But the strangers slowly became acquaintances and then friends. The acceptance was mutual. Apoorv and Rujuta embraced an ever-changing India and their slice of India embraced them. They taught their parents (who were brought up in the India of the 80s and 90s) the Marathi slang of the gen-Z. They loved the food, restaurants, and grocery shops that sold pepsi colas and Cadbury eclairs and whatnot.

Above all, in a society that has its very evident classes, they liked its apparent practicality, frugality, simplicity, affordability, and security. Admittedly, a few things were just weird at times, but they didn’t get too much in their way. Though the two represent a financially stable middle class family which is typically a family of knowledge workers, they developed a deep respect for the working class members (the housekeeping staff and security guards at the apartment complex, the Zomato, Dunzo, Amazon couriers, the auto rickshaw drivers, and so on) that they ran into every now and then.

In short, we all experienced that Shift Happens!

The likelihood of having such an enriching experience in the Bay Area is rather low. Knowing your native language and being able to fluently express yourself in that language makes you feel good. It also makes your grandparents feel good for they can at least communicate (in their native language) well with their grandchildren (without a need for parental interpreters). I am not suggesting that every immigrant family must somehow make their children learn their native tongue, but the experience is pleasant when such learning happens organically. There is no better place to experience the beginning Marathi than the streets and houses of Maharashtra.

Of course, some of this could have still happened had they been to traditional schools. But homeschooling gave us the required flexibility. I still believe that not having to go through a traditional school where compliance2 is held in high regard was a boon in disguise. Homeschooling was a lot of work, but it was liberating. Quite understandably, Apoorv’s and Rujuta’s friends were curious about what they had been “learning”, what this homeschool thing was all about, and whether they were going to go back to the US.

Acceptance of homeschooling by universities is still low. Their documentation about homeschooling is sparse, hidden, and changing. US Colleges want all applying students to complete their admission requirements, especially English. Understandably, they want homeschooling applicants to showcase their abilities as demonstrated at external agencies (internship, projects, exams). But there are no avenues for this that we could find. Other than the SAT (which has been made optional or even needless by several US colleges) and the AP exams, there are very few (almost no) standard exams that are accredited by US universities (Please comment below if you know any accredited options for exams, but note that as of this writing, the SAT subject tests are a thing of the past.). Of course, if you are, say, a gold-medalist at an Olympiad, none of this matters, because then you are already an outlier. But a lack of clear documentation hurts people who have embarked on a sincere and serious, although uncommon, educational experiment.

I find it odd that US colleges are SAT-blind or SAT-optional for homeschooled students. Colleges of course have their reasons to go test-blind or test-optional, but homeschooled students are negatively affected as a result. Perhaps not many colleges care because homeschooled students are such a tiny minority that you don’t even consider their existence!

Even so, I thank US universities for the fact that perhaps at least a few of them paid attention to Apoorv’s college application. I suppose at least some of them acknowledged the unusual nature of his journey so far and UMich offered the highly competitive admission to Computer Engineering or Data Science. I don’t think a homeschooler could have secured an admission to a decent college (e.g. an IIT) in India (in spite of the alternatives like NIOS3).

In spite of the unknowns, we just kept feeling that we are doing the right thing because the time was being spent in peace and joy and we were all actively learning something new. I know that there is an unspoken precedent against homeschooling (on Reddit, you can read sad first-hand experiences of homeschoolers) and its acceptance is low perhaps for a reason, but, for us it just worked. That is, if you were to look at it from a Product Perspective.

I consider these results deeply satisfying.

(Part 2 and Part 3 are coming soon.)

Footnotes:

  1. We are a DARK (Deepa, Apoorv, Rujuta, Kedar) family! ↩︎
  2. To them scoring well in exams is the only indication of liking a subject! ↩︎
  3. We were told that he’d have to take the 10th standard NIOS exam and couldn’t take the 12th standard exam directly. ↩︎
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SICP: A Synonym for Computer Programming

Choosing a cryptic title for your blog-post has a polarizing effect. There is some novelty promise associated with an unfamiliar title. The promise is fulfilled if the blog-post is a good one. Let me try.

SICP stands for “Structure and Interpretation of Computer Programs.” If you are already familiar with computer programming, then there is a chance (sadly slim) that you have heard about this initialism. In my experience, few computer programmers (even experienced and professional) have heard of it. If you are unfamiliar with computer programming, then it is even less probable that you have heard of SICP.

SICP is, like the title of this blog-post proclaims, how programming should be introduced to people who have some interest in or inclination toward mathematics. If you have ever been attracted to the idea of a mathematical function, learning to program with SICP enhances your understanding of both mathematics and computer programming.

SICP is a university-level course given by MIT and it has a long history. But something very useful happened when people like Martin Henz, Tobias Wrigstad, and many others at Source Academy came together and provided a beautiful, self-paced course that ported SICP to a popular programming language, JavaScript. This course is called SICPJS and it is open-source and offered at no charge. As an introduction to programming as a problem-solving technique, nothing could be better than this for a budding computer programmer, and, dare I say, for a seasoned programmer. This, ladies and gentleman, is pure gold. These kind people have put in so much effort that I can’t even begin to thank them for it.

You don’t have to listen to me. Just look at what Guy Steele, Alan Perlis, Sussman and Abelson, and Henz and Wrigstad have to say (in their forewords and prefaces to this book) and you’ll perhaps be hooked to this course.

For the last three months, I have been doing SICPJS with a high school student, Apoorv Mhaswade, at our homeschool. Apoorv prepares to enter a college to pursue computer science — SICPJS could not have come at a better time.

The programming environment for SICPJS is not full-fledged JavaScript, but a growing subset of JavaScript, called Source (Source-1, Source-2, etc.). You write your programs in Source. The syntax is clean, the interpreter is well-made, and there is a thriving support community. I have cried, laughed, thrown my hands up in admiration and joy while reading this book and solving every programming problem in it. Apoorv has also thoroughly enjoyed this course. We have had invaluable discussions on many little topics and he appreciated every minute we spent on it. I am glad that we picked SICPJS for Apoorv’s high school senior year.

Don’t misunderstand that SICP is “programming for kids only”, or toy stuff, however. The ideas are timeless and although you may not agree with all of it, doing the course will enlighten you. The course is difficult. It makes you think. It makes you uncomfortable because seemingly simple questions trick you. If you are an experienced programmer, you even feel ashamed when you are not able to solve a simple-looking problem.

A programming language should have only as many tools as a programming craftsman needs to create programs. Being trained in imperative programming style may sometimes make you brand Source-1, or Source-2 programming languages (the ones used by SICPJS) as insufficient, but you don’t have to have plethora of tools. One needs to learn to think differently when learning new paradigms. Indeed, like the great Alan Perlis once said, “There is no point in learning a programming language if it does not change the way you think.”

My detailed log about our this course is here. But you forget about it and just head over to SICPJS. Keep calm and do SICPJS!

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A Fun Math Quiz

Rujuta wanted a math quiz. She is now in the eighth grade and has studied arithmetic, some algebra, and some geometry. She enjoys the process of discovery. Of course, like everyone else, she has some blind spots. But one thing that I really admire about her is that she gives it her everything. She is upset with herself when she makes a silly mistake or when she doesn’t get a (difficult) question. Her evolution as a “math person” is heartwarming.

In my view, she enjoys mathematics just the way the great mathematicians Hans Rademacher and Otto Toeplitz have said in their amazing book, The Enjoyment of Math.

I am wary of giving a quiz to young children. If the quiz is too tough, like most of us, they may lose confidence and if the quiz makes students plainly apply known algorithms, they may be bored or become overconfident. It is a delicate balance.

I have tried to apply sound principles. Here are the instructions to the quiz:

  1. Sit in a comfortable place with all the things you need: mostly a pencil and a lot of paper. This is an open-book exam in that you can refer to your notes and use a calculator. Just don’t Google the problem statement. 
  2. Be honest.
  3. Read the problem carefully. Draw diagrams. Use Geogebra if you like. No need to curb creativity. Don’t hesitate to ask if something is unclear.
  4. Try to stick to the time, but it is okay if you run over.
  5. The correct answer to each problem carries certain points (indicated against the problem number), but that’s not why we are writing this exam. The goal for me (as a facilitator) is to understand if I need to change my method of instruction and reinforce certain topics, whereas the goal for you should be to just enjoy solving the problem. Show your work.
  6. Take the quiz right here on this doc (provide answers inline). Of course, you can take a picture of your rough paper-pencil work and stick it here. Neatness matters but content is more important than form.

And here is the quiz in its entirety. Take a look. I hope you enjoy it. It contains Rujuta’s solutions and our discussion: https://docs.google.com/document/d/1xHOM5MSP9OFMPDUTruQ6ty1LZbKHwvjex3LcMo9rHkI/edit?usp=sharing

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Choosing Physics Resources for the First Year of a Self-paced Study

Rujuta is now willing and prepared to study Physics! She would have been in the eighth grade in a traditional school and we were thinking of how to study beginning physics. She asked me to come up with a “curriculum of sorts”. I gave it a try and here is how we are going to spend some of our time (this is an image, see the link below it for the entire article):

Here is the link to the entire article. It is written as an introduction to her class notes. It reflects our thinking at the moment and how we are collaborating.

Physics is entertaining, all-encompassing, and, in most cases, a pursuit of a lifetime. Of course, a lifetime pursuit of anything is for a thorough professional of a trade, but its enjoyment in its formative stages is for everyone.

I hope we enjoy this exciting journey!

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How to Do Research …

Paul R. Halmos had been a great teacher, educator, author, and, of course, mathematician. He wrote eloquently about his life’s journey as a mathematician in what he called his “automathography”: I want to Be a Mathematician. It is a book with a lot of mathematics (higher) but it also has a high literary value. He writes forcefully. His writing is quite original, thought-provoking, and effective. I might write about the book later, but, for now, I want to draw your attention to an essay, How to Do Research, from his book. Here are the opening lines (it is an image, emphasis mine):

Beautiful, isn’t it? True of most of us, isn’t it? What is also interesting about it is that it not only applies to formal, professional “research”, but also to everyday problem-solving, investigation of things that intrigue you. In other words, it is about researcher’s attitude.

AMS (American Mathematical Society) has been kind enough to publish this essay in its entirety. Here is the link to it in PDF: https://www.ams.org/notices/200709/tx070901136p.pdf

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MINDSTORMS

That is the main title of a life-changing book, at least for some. The subtitle is Children, Computers, and Powerful Ideas. Its author is Seymour Papert, who was a mathematician and educator.

I am still reading the book and will shortly do a longer post. But if you can’t wait to experience some powerful writing, please read the book. The MIT media lab and Seymour Papert’s family have made this book available for us to read for no charge. Making such a wonderful book (that is still in print) available for the general public for no charge is a truly generous gift. Here is a specimen of Papert’s writing (from Chapter 2: Mathophobia) —

PLATO WROTE over his door, “Let only geometers enter.” Times have changed. Most of those who now seek to enter Plato’s intellectual world neither know mathematics nor sense the least contradiction in their disregard for his injunction. Our culture’s schizophrenic split between “humanities” and “science” supports their sense of security. Plato was a philosopher, and philosophy belongs to the humanities as surely as mathematics belongs to the sciences. 

This great divide is thoroughly built into our language, our worldview, our social organization, our educational system, and, most recently, even our theories of neurophysiology. It is self-perpetuating: The more the culture is divided, the more each side builds separation into its new growth. 

This is marvelous writing, even life-changing, especially if you are an educator!

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Installing Ubuntu on a new laptop

Recently, I got a new laptop and I had one main job: install the Ubuntu operating system. My dad had given me the USB Ubuntu boot drive, so I went to Windows’ BIOS setup. At first, I didn’t know how to start. But then, I simply searched how to get to the BIOS setup and everything became quite clear.

When I went to the BIOS setup, it asked me what I wanted to do, I wanted to use the boot drive, so I simply clicked on the “Use a device” option. After I clicked on that, it showed me six different options: two USB ports (as in devices connected to the USB ports) and four other device options which did not make much sense to me. I realized I had inserted the drive into the first USB port, so clicked on that option.

After that, the system started installing, and soon I was looking at a screen that asked me whether I wanted to try Ubuntu or directly install it. I selected the “Install Ubuntu” option and along came a series of questions asking how I wanted my setup to be. These questions asked about my preferred keyboard layout, updates and software preferences, my preferred installation type, my current location, and my login details. (This part was fairly easy for me because I upgraded another device the same way.)

Next, all the Ubuntu packages were downloaded and installed, and I restarted my laptop in order to use the new installation. There it was! The newest version of the Ubuntu operating system.

The last thing I checked was if my Windows setup had been affected. I restarted my laptop, and this time selecting the Windows option, checked if everything was alright. I went to my Windows login, and could see no errors or major changes. The Ubuntu installation was a success.

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Calculus: Of the Students, By the Students, and For the Students

We are happy to announce that we are doing a calculus group discussion on the Internet! I have not seriously researched if this the first ever such attempt, but it looks like an uncommon one.

Apoorv and I have written about it in detail here. Please take a look and let people know.

We expect to start in the first week of October 2022!

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Sums of Consecutive Integers

Number theory problems can be some of the most difficult problems, but they are usually the most interesting and satisfying to solve. Recently, I was tasked with finding the solution to this problem: Find out which positive numbers cannot be expressed as the sum of two or more consecutive positive integers.

The first thing we can do is to try some small numbers. After trying a few, a pattern starts to emerge. All the powers of 2 seem to be the only numbers unable to be expressed as the sum of two or more consecutive positive integers. Although this is a pattern, we can only say that this pattern holds for small numbers. Now this question arises: Can we prove that this holds for all numbers, that only powers of two cannot be expressed as the sum of two or more consecutive positive integers?

Initially, I was unsure how to go about this. So I decided to find the numbers that could be expressed as a sum of two or more consecutive positive integers to help find the numbers that could not.

Since any odd number can be represented as 2n+1, and 2n+1 = (n)+(n+1), all odd numbers greater than one can be expressed as the sum of two consecutive positive integers. Let us call this statement A.

However, I could not figure out what to do after that. After thinking about it for a while, I remembered an old trick I had discovered. Any number i that is divisible by an odd number n can be represented as the sum of n consecutive integers centered around i\div n. For example, the number 40 is divisible by the odd number 5 (i = 40, n = 5). Then, the trick states that 40 is equal to the sum of five integers centered around 8. Indeed, we see that 6+7+8+9+10 = 40 The proof for this is quite simple (omitted here), but it suffices to say that it works because the sum of every pair of numbers around i\div n is equal to n. Hence, all numbers divisible by an odd number can be expressed as the sum of consecutive integers.

Combining statement A and statement B, we find that all numbers that are divisible by an odd number can be represented as the sum of two or more consecutive integers.

To find out whether a number is divisible by an odd number, we find its prime factorization. Since 2 is the only even prime number, if the prime factorization contains any number besides 2, the number is divisible by an odd number, and it can be represented as the sum of consecutive integers. The only numbers that do not fit into this category are the numbers whose prime factorizations contain only 2. In other words, the only numbers that cannot be expressed as the sum of two or more consecutive positive integers are the powers of 2 (1, 2, 4, 8, …).

This proof was the result of thinking by myself as well as fruitful discussions with my father. He was trying to solve the problem too, and at a point when we were both stuck, we sat down together and shared our separate insights, and soon after, we figured it out. This was a great example of how working together can help both parties because I would have taken much longer to solve this problem without his input (if I would have solved it at all!).

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A Function Composition Problem

A topic that has resurfaced during my undertaking of calculus this year has been function composition. During the year, I have been drawn toward the concept of function composition, something that was evident to my father. One day, as we sat down to begin our class, he posed this problem to me:

f(x)=3x+2=\underbrace{(g\circ g\circ g\circ ... \circ g)}_\text{100 times}(x).
Find g(x).

I believe this is a very interesting problem. You should take a pencil and paper and sit down to solve this problem—it is slightly difficult to solve in your head. Here is my solution:

First, I realized that g(x) must be a linear function. Since the functions are composed, the degrees of the polynomials will multiply because the function is raised to the degree of the polynomial. For example, if g(x) = x^3, g(g(x)) = (x^3)^3 = x^{3\cdot 3} = x^9. Hence, the function must be a first degree polynomial. As such, the function g(x) can be represented by the expression ax+b, where a and b are real numbers. Next, I listed g(x), g(g(x)), and so forth, a few times, to try and find a pattern. To clarify, the notation g^n(x) means g(g(g(...(g(x) n times.

g^1(x) = ax+b

g^2(x) = a(ax+b)+b = a^2x+ab+b

g^3(x) = a(a^2x+ab+b)+b = a^3x+a^2b+ab+b

g^4(x) = a(a^3x+a^2b+ab+b)+b = a^4x+a^3b+a^2b+ab+b

From these four iterations, a pattern started to emerge:

g^n(x) = a^nx+a^{n-1}b + a^{n-2}b+...+ab+b = a^nx+b(a^{n-1}+a^{n-2}+...+a+1) = a^nx + b(\frac{a^n}{a-1})

We can equate g^{100}(x) to 3x+2 and find a and b.

3x+2 = g^{100}(x) = a^{100}x+b(\frac{a^{100}}{a-1})

We can equate the degree 1 terms to find a:

3x = a^{100}x

3 = a^{100}

a = 3^{1/100}

Using a, we can find b.

2 = b(\frac{a^{100}}{a-1}) = b(\frac{{(3^{1/100})}^{100}}{3^{1/100}-1})

b = \frac{2}{3}(3^{1/100}-1)

Hence, g(x)=ax+b=3^{1/100}x+\frac{2}{3}(3^{1/100}-1).

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Bertrand Russell’s Gem

The functions of a Teacher compared to that of the Propagandist

No one can be a good teacher unless they have feelings of warm affection toward their pupils and a genuine desire to impart to them what they believe to be of value.

This is not the attitude of the propagandist. To the propagandist the pupil is a potential soldier in an army. They are to serve purposes that lie outside their own lives, not in the sense in which every generous purpose transcends self, but in the sense of ministering to unjust privilege or to despotic (meaning: of a cruel and oppressive ruler) power. The propagandist does not desire that the pupil should survey the world and freely choose a purpose which to them appears of value. The propagandist desires, like a topiarian (meaning: ornamental gardening) artist, that the pupil’s growth shall be trained and twisted to suit the gardener’s purpose. And in thwarting the pupil’s natural growth the propagandist is apt to destroy in them all generous vigor, replacing it by envy, destructiveness, and cruelty.

There is no need for human beings to be cruel; on the contrary, I am persuaded that most cruelty results from thwarting in early years, above all from thwarting what is good.

Bertrand Russell, Unpopular Essays (1953), Ch. VIII: The Functions of a Teacher, p . 118-9 (emphasis mine)

Russell is of course a great philosopher-mathematician of the 20th century. His honest commentary on various issues is always a joy to read. The above quote is no different. As I reflect upon my teaching and learning sessions with my children, I couldn’t agree more with what he has said. In the 21st century, knowledge should really be free. I know that this is not true for many people yet, but I am hopeful that the barriers to entry will be reduced as the Internet becomes more and more ubiquitous and accessible.

But if the so-called “programmers of the society” in the form of disciplinarian teachers, despotic generals, selfish politicians keep coming in the way of an obvious freedom, then we have a long way to go. Circumstances, a personal make-up, behavioral traits themselves are so challenging to individuals that we do not want the societies to be programmed.

And a selfless affection without any hidden motif (even that of imparting knowledge of certain kind) is the only thing we need in capable teachers.

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When Daddy Was a Little Boy …

Children’s literature has a special place in our hearts. In a way, recognizing it as a separate genre and branding a book as a children’s book is not right because many adults are just grown-up children. The true passion, the apparent silliness, alleged lack of knowledge and timelessness, the adults’ perception of having already experienced it — all of these are applicable to all literature. Age appropriateness may be a concern, but the associated risk can be mitigated at the source, that is, by producing it meaningfully (for which, again, the onus is on adults).

Even so, we are fortunate to have so much of wonderful children’s literature. One such book was written by Alexander Raskin: When Daddy Was a Little Boy. It has stories of a daddy when he was a little boy. They are told by a daddy himself to his young daughter when she used to fall sick. Here is an audio recording of how Raskin introduces his book:

When Daddy Was a Little Boy: Introduction (5 minutes)

Rujuta and Apoorv enjoyed almost all the stories. We have read them many times. Nothing fancy or grandiose, but simple, dramatic, and powerful. The stories are each less than 8 pages long and can be read in any order. Delicate humor and drama is a cornerstone of many stories and it brings out many aspects of the human psyche quite simply. Here is one of my favorite stories: Little Daddy Makes a Mistake.

Little Daddy Makes a Mistake (4 minutes)

We highly recommend this book to everyone, not only for a pandemic season, but for any season.

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Starting a Composting System!

I decided to start composting! It definitely needs a lot of work, but I guarantee you it’s a lot of fun. Previously when I tried it, it turned to a big, smelly mess and I ended throwing out the whole thing. Clearly, I was missing a few things, so I turned to a friend (Gayatri Mavashi) who has 2 composting systems and over 50 plants. Here is a dated log:

12/17/2021 (17/12/2021)
Today, I visited a friend who told me what I need to do to start a composting system. She showed me her own garden and her two types of composting. The first type she showed me was actually her own invention! The second type she showed me included a composting pit. She also showed me her collection of plants and told me that I could directly throw the waste in the plants once I got the hang of it.

Any green waste like onion peels, lemon rinds, tea leaves can be put in the pot. She told me to add the waste and mix it every day and also out some soil every 2-3 days.
To get some actual results from this project, it takes around 6 months. When you start, you are going to need some time to get the hang of it. It may fail the first few times, so do not despair! I needed a few things to start this project like extra soil and a good gardening set. I decided to continue on Tuesday, December 7th.
According to Wikipedia, compost is a mixture of ingredients used to fertilize and improve the soil. It is commonly prepared by decomposing plant and food waste and recycling organic materials. You can read more about it here: Compost.
Here are some photos of her garden and compost:

12/21/2021 (21/21/2021)
Today, I actually started my composting! I already had an urban pot, the perfect size to start. I started by taking a newspaper, folding it into four parts, and securing it at the bottom of the pot. Next, I took the green waste I had, cut it into small parts, and dumped it in the pot. After that, I took some soil in which there were quite a few earthworms and dumped them into the part. Lastly, I mixed the compost with either a small rake or two small shovels.

Here are some photos:

Mix Composting
Mixing the green waste and soil is very important in this type of composting. As you can see in the pictures, I put the layers of paper, the green waster, the soil, and then I finally mixed it. The last picture shows the mixed product. It does look a little messy, but that is how the compost is going to look; it’s normal.
After a few months, take a look under all the compost and you might just find something.

Composting Pit
In this type of composting, you need a big pit. The space in which I am going to start is too small to do this but it’s still an interesting idea. You need an area marked by stones or tiles. Fill the pit with soil and put your green waste in it. To mix this, you will need a big rake. Make sure you do it every day.
Again, after a few months, take a look at the bottom of your pit and you might just find something.

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It’s Giving Time!

When the ambience triggers a certain action in us, we say, “It’s in the air.” End of a year on the English calendar (and the Diwali time on the Indian calendar) is one such time when the ambience triggers a sense of giving in you. We are not physically up in the northern hemisphere right now, but the feeling still engulfs us.

Giving is a big topic with many connotations. In this blog-post, we are not going to go into why we want to give and how much. I am not a well-known philanthropist, but I don’t want to curb my enthusiasm of giving and then writing about how we carried out a very small session of giving.

I scheduled the “It’s Giving Time” event on the school calendar and everyone accepted the invite. I chose that we keep it simple:

  1. Go to the beautiful and simple Amaya restaurant (located at the mouth of the Gokhale Institute of Politics and Economics in Pune) for breakfast.
  2. List the charities that we are interested in donating money to. Each donor can pick as many charities as they want. Our budget this year is $150 per donor.
  3. While picking a charity, there is no need to look at how popular a charity is. Just look at how indispensable it has been for you.

We reflected upon the past year of homeschooling (and before) and came up with the following:

In the above image, D, A, R, and K stand for our name initials. So, you can see that we are a DARK family and homeschool!

Everyone knows of Wikipedia and Khan Academy. I personally love Archive.org. It really is a treasure trove of all media, especially books (the way-back machine is just awesome and the way-forward machine is awesome too). Do you know pTable? It is, along with the periodic table of videos (and Prof. Poliakoff), the best resource for studying the periodic table of elements! My favorite text editor on the computer is Vim and our hats are off to computer programmers like Bram Moolenaar and Tim Pope. 3b1b and Grant have been absolutely phenomenal. SJPL, the public library of San Jose, has been extremely generous to us over the years. The OpenStax initiative by the Rice university has been very useful especially for our high school student. Solving the Project Euler is just so much fun! The MOOC revolution was fueled by the MIT Open Course Ware and OCW had to be on the list as well.

Thanks to all these wonderful people! We have received much, much more from them than what we are giving this year.

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Gelfand’s Algebra and an Application of the Greedy Algorithm

One of the joys of working with children learning mathematics (that means all of us) is to witness the accidental discoveries that they make. Let me narrate my experience of that in this rather long post.

The only way to learn mathematics is to do it. And doing mathematics is solving problems. In a delightful book that I will come back to at the end of this article, the following problem appears (reproduced here verbatim):

Problem 1. Several digits "8" and some "+" signs are inserted to get 1000. Figure out how it is done. (For example, if we try 88+88+8+8+88, we fail because we get only 280 instead of 1000).

Try to solve it before reading further (I have inserted a space-filling image below 😊). I guarantee, it would be a lot of fun. It will also be fun if you ask it to a ten- or eleven-year old and observe what she does.

Photo by Anna Tarazevich on Pexels.com

Rujuta started to think about how to solve it. “Hmm, let’s see”, she said. After a few minutes (which we both spent patiently, as I was also trying to solve it in my head; this was not a material that I had reviewed before the class).

“Maybe I’d start with the largest number that is all 8’s but which is less than 1000”, she said somewhat speculatively. “That’s interesting, sure”, said I.

“That would be 888, which means I have 1000-888 = (after some time lag) 112 remaining”, she added.

[Here, she actually made a few calculation errors, but don’t get me wrong, she had no conceptual misunderstanding. Sometimes she ends up saying or writing 76 even if she means to say 66, or thinks of 72. Do you know if this is some kind of peculiarity that some people have especially with numbers?]

“Cool, now, we have 112 that we need to get using only 8’s”, was her next thought.

I was just witnessing her “thinking out loud process” and nodding to her efforts.

“That would be one 88 and now we have 12 + 12 = 24”.

“Ah, that is three 8’s”, she said, and then wrote, “888+88+8+8+8 = 1000“. She was giggling all along and as she finally discovered the solution, she was very happy.

Quite frankly, I was stunned. I had made (quietly) the mistake of expecting her not to do it so fast (yet another reason we should get rid of expectations of any kind).

And as I witnessed it happen, I was very happy as well. I said, “Wow! You did it, and in doing so, you accidentally discovered, or perhaps invented a technique called The Greedy Approach of problem-solving. This approach provides elegant solutions to several problems, although, at times, it may lead you astray. For example, how do you know beforehand if you would end up into 24 which is just three 8’s? Of course, you don’t, but you just press on in the hope that it may work in the end.”

She agreed with me. Indeed, the so-called greedy approach is effective is many situations.

Did you solve this problem this way? Perhaps you did. It is also possible that you solved it differently. No despair. All solutions are interesting. The problem itself is perhaps trivial for mathematically advanced students, nonetheless, it is very interesting.

I then proceeded to ask her, “What if the sum were 2000? Would you still use your greedy technique? That is, would you still look for the largest all-8’s number smaller than 2000 (which is still 888) and so on?”

She thought for a few moments and said, “Well, I need 2000. That is 1000+1000. I have already solved the problem for 1000 and I can solve it once more: 888+88+8+8+8 + 888+88+8+8+8. Isn’t it?”

I was pleased to say yes. This was indeed a useful observation. We then went on to think of what will happen were the sum 10000. She tried with 8888 first (she said that that was the largest all-8’s number smaller than 9000), arranged to get the remaining 112 (9000-8888) as 88+8+8+8 and then since she had already solved the problem for the sum = 1000, she reused that part of the solution.

After hearing both these answers I told her a mathematician joke:

A physicist and a mathematician are sitting in a faculty lounge. Suddenly, the coffee machine catches on fire. The physicist grabs a bucket, leaps toward the sink, fills the bucket with water, and puts the fire out. The next day, they are sitting in the same lounge. Again, the coffee machine catches on fire. This time, the mathematician stands up, gets a bucket, and hands the bucket over to the physicist, thus reducing the problem to a previously solved one.

Then I said to her, “That makes you a mathematician.” And she just chuckled.

The problem is taken from a book, Algebra, by two well-known mathematicians, I. M. Gelfand and A. Shen. This is a brilliant book for School Algebra (and beyond). I recommend at least referring to this book when introducing Algebra to middle school students. Here is the mathematician Richard Askey’s review of this book.

We went on to see their solution and here’s its verbatim reproduction:

Solution: Assume that
...8
....
....
...8
____
1000
We do not know how many digits are used in each number. But we do know that each number ends with "8" and that the last digit on the sum is "0". How many numbers do we need to get this zero? If we use only one number, we get 8. If we use two numbers, we get 6 (8+8 = 16), etc. To get zero we need at least five numbers:
....8
....8
....8
....8
....8
_____
1000
After we get this zero, we keep "4" in mind because 8+8+8+8+8=40.
To get the next zero in the "tens place" from this 4 we need to add at least two 8's since 4+8+8=20.
   8
   8
   8
..88
..88
____
1000
We keep "2" in mind and we need only one more "8" to get 10:
   8
   8
   8
  88
 888
____
1000
The problem is solved: 8+8+8+88+888=1000.

It is clear that the authors are subtly trying to introduce Algebra to middle schoolers. I went over this solution with her and she appreciated it, but quite naturally, she liked her own solution more!

I do think that the solution presented in the book has become (perhaps subjectively speaking) a bit more tedious than the greedy approach that Rujuta found. For example, it is not immediately clear if we need to take five 8’s or ten 8’s (or fifteen or twenty 8’s) to yield the zero in the units place of the sum since both satisfy the condition: 8×5=40, 8×10=80. Similarly, with the 4 carries, to get the zero in the “tens place” with two 8’s (as demonstrated above, 4+8+8=20) or with seven 8’s (4+8+8+8+8+8+8+8=60). We will clearly choose 4+8+8 first, but the point is that we need to examine many more cases (it appears to become like searching in a “tree”).

I let Rujuta know that I liked her solution more than the great Gelfand’s solution and she was somewhat flattered. But more importantly, she looked content. She looked happy. I thought she may enjoy such exploration more even though it is expected to be challenging. In the end, maybe, just maybe, the endeavor would turn out to be worthwhile. I had my little experience of what Francis Su calls Mathematics for Human Flourishing.

That, I thought was priceless.

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A (Not-so-) New Steacher

Yes, steacher is not an English word. I just defined it:

steach·er| ˈstēCHər noun: a person who teaches with an intention to learn, especially in an environment where freedom of expression is valued (past tense, past participle: steached) Such a person believes in a socratic style of teaching.

I have accepted the job of a full-time steacher at the Free Learner’s School. I consider it a privilege. I, along with the brilliant Deepa Joshi, will be steaching the new year 2020-21 at The Free Learner’s School. This post is rather long and it tries to deconstruct my decision to accept this job.

Perhaps traditional schools are great institutions. Students go there and make friends (sometimes lifelong), build memories, and, along the way, learn things that a curriculum dictates. The idea is that they learn something so that they can become capable to carry forward this wonderful world by developing some skills that society needs. In short, traditional schools are a somewhat natural outgrowth of humans’ need to learn and teach. We will not go into the history of traditional schooling here, however. Interested readers can refer to the excellent book, Free to Learn, by Peter Gray. Though Peter questions our blind faith in today’s traditional schooling, it is obvious that it is where a vast majority of us belong. So, traditional schooling will remain a big force in our lives for a foreseeable future, even when the 21st-century thinkers, entrepreneurs, and reformers rethink education in the era that is deeply influenced by technology.

I understand the confusion that you may have after reading the above on the blog of a homeschool! I want to stress that traditional schooling is here to stay. But that does not mean there are no alternatives. In our case, homeschooling appeared like the only choice. We moved from the United States to India in November 2018 in the middle of the school year. Our son was 13 and our daughter 9. It would have still been possible to make the children go through the drudgery of exam-oriented schooling and they might have gotten used to it just fine, but it simply did not feel like the right thing to do.

I started with this picture (Figure 1) which does not reflect the reality accurately, because, unfortunately, the data are much more skewed in the favor of traditional schools. I told them that we are embarking on the road that is (much) less traveled and that there were implications of that. One of the biggest implications is the social aspects. Although these are less acutely felt in the US, in India, homeschooling families are a very small minority and perhaps not well-accepted. It appeared, however, that the kids were comfortable to satiate their social needs via other, non-school ways of interacting with the world.

Figure 1. (With apologies to Scott M. Peck) Homeschooling: a road less traveled

I said, “When something feels right and also appears objectively sound, being in the minority gives you conviction and courage, but sometimes, being in the majority gives you strength.” We then asked them if they were willing to give it a try. I was not sure if they understood, but they were willing to give it a try nonetheless. Many acquaintances (all of whom were educated, well-meaning parents) doubted our audacity citing the common reasoning, “Do they really know [the answer to this question]?” Whereas I understand that children may not know about a subject of knowledge, how could we be so sure that they won’t know?

Someone has said, “It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.”

So, we decided to give it a try.

In the light of the Harvard Law Professor Elizabeth Bartholet’s critical and somewhat unfortunate analysis of homeschooling, I must clarify here that our family is not at all religious or a right-wing fanatic of some kind. We have always believed in science and humanity and humanists are what we are, although we are not as famous as the great Carl Sagan.

After such an introduction to homeschooling in August 2018, we practiced it for a few months before committing to it in 2018 after we moved to India. Deepa steached Geometry, Music, and Numeracy to our daughter and I was a part-timer introducing Computer Science to our son. You are more than welcome to ask the students, but we believe that they enjoyed the whole experience and their childhood is something that they will remember fondly.

It must be kept in mind, however, that homeschooling is a lot of work, but it is also a lot of fun. The mathematician Richard Guy has said, “It is an old adage that you don’t really understand something until you teach it to someone else.” As a result of this, steachers at homeschools need to first understand things and then they have the privilege to explain it to students. If you are too bothered (it is quite understandable as depicted by the student’s father in this wonderful Hindi movie “तारे जमीन पर” ) by the thoughts of schooling success, cramming children’s minds with knowledge that they don’t have much connection with, and making them tomorrow’s responsible workforce even before they have lived their childhood in freedom, then you may be better off with a traditional school. The child may suffer through the environment of often misplaced discipline, strict compliance, stereotyped thinking, and dishonesty, but it may help them survive in the harsh reality of the world. It is also quite probable, however, that the child may turn into a youth that embodies the same principles that traditional schools reinforce. Of course, I am not suggesting that this is what will happen, I am just saying that it is possible or even probable.

No, this is not bashing of traditional schooling. In general, I am not interested in comparing these two ways of schooling because that is a much broader topic any amount of discussion on which may be inconclusive (if we were to choose a winner). Any of the above (apparently negative) characteristics may well be demonstrated by homeschooled kids.

Good traditional schools and good teachers are irreplaceable, but they are also rare. All I want to say is alternatives exist in shaping a child’s future and homeschooling may create the kind of world citizens that we want with a more-or-less similar probability as traditional schools. After all, like Malcolm Forbes has said,

The purpose of education is to replace an empty mind with an open one.

An enabling factor for such an undertaking is freedom. And the essence of freedom is in giving it to others when you are in a privileged position. Such freedom is perhaps hard to accept until one groks it. It makes one have no expectations from anyone (even yourself). When you are starting a school with freedom as a (and probably the only) core principle, it is better to be absolutely clear about it. Of course, you want to impart the love of learning to children and help them discover themselves. But if doing so comes at the cost of their freedom, then it is worthless and sad.

This does not mean one should not be undertaking hard work, the motive force behind any success, and perhaps happiness. It does not mean that the moment someone is experiencing hard work they should back off, do some deceptive soul-searching, and look for an alternative like a fleeting creature. It only means that hard work should become a way of life once the personal connection with the present circumstances has been firmly established.

So, with that in mind, I have begun my adventure. It has been an adventure because I have no idea how it will turn out to be. My only hope is that the time will be spent well and looking back we will have fond memories of this time.

On this blog, we (steachers) will discuss such things as:

  1. Intended meaning of some important terms (e.g. examination, society, work ethic, discipline, compliance, respect, sincerity, etc).
  2. Administrative aspects of homeschooling (remember, homeschooling is a lot of work) and it may help others to find out what seems to work for us.
  3. Actual experiences while learning and teaching something and how the learning seems to take place. These would be quite detailed.
  4. Students’ expressions.
  5. Book reviews and personal reflections.
  6. Tools that we use for various subjects and school administration (management).

I hope I continue to post my experiences now that the adventure has begun.

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The Start

This sticky post is done in a hurry, but it tells you about the kind of posts that we will write here. We may occasionally deviate from this “plan”, but we hope to stay true to it in spirit.

  • Experiments about homeschooling our children. Don’t worry, we are not religious people.
  • Thoughts about education, formal education in particular.
  • Relevant book reviews.
  • Some practical tips which should be taken with a grain of salt because we will mostly write about things that worked for us or appeared to work for us.

When one teaches, two learn — thus spake someone. So, there are no formal teachers here because almost invariably, whenever I teach something I learn. I learn from the responses I receive, by simply being empathetic about the whole experience. I can’t even count how many things my daughter (one of my “students”) has taught me simply by speaking with me.

But for the record, this school is being run by these two teachers:

  • Deepa Joshi (email redacted)
  • Kedar Mhaswade (email redacted).

The best part is going to come from the “students”. They have agreed to write about their experience as well!

Please let us know if you’d like to get in touch by clicking on the Contact page.

Learning About Physics as a Home-schooled Student

Like the first day of many college classes, we decided to start with some review.

In physics, the law of conservation of energy is very important. It states that energy can neither be created nor not destroyed. However, one form of energy can be converted into another form. What are the different types of energy? There is light, heat, electric energy, chemical energy (one example is the electrolysis of water), potential energy, nuclear energy, and more.

The prefix “infra-” as in infrared means “below”. We can translate infrared to “below-red”.

The prefix “ultra-” means “beyond”. Ultraviolet means “beyond-violet”. (It could have been “supra-violet” or “hyper-violet” because those prefixes mean “above” in Latin.) From red to violet, lies the electromagnetic spectrum of visible light. Violet has a wavelength of about 400 nanometers and red has a wavelength of 700 nanometers. Beyond violet are ultraviolet rays, x-rays, and gamma rays. Below red are the infrared light, microwave radiation, and radio waves. The wavelengths of radio waves range from a few millimeters to hundreds of kilometers. As far as we are concerned, the difference between a radio wave and a red light wave is that we can see one type of wave but we can’t see the other. 

Waves have two properties: frequency and wavelength. Only a small portion of the electromagnetic spectrum excites our neurons.

In an optical telescope, the visible light source goes through the telescope’s objective lens, continues to the eye piece, and then enters our eyes. In this process, the wavelength of light doesn’t change. A radio telescope is hit by waves that are in the range of radio waves (way beyond visible light rays). Visible light is to an optical telescope as radio waves are to a radio telescope. 

Longer wavelength means lower frequency and shorter wavelength means higher frequency. Here’s an interesting question: What does a 400-nanometer wavelength correspond to in terms of frequency? Here is a picture of a wave:

The distance between the two red dots is called the wavelength (λ). That part of the wave is one cycle of the wave. The number of cycles of the wave that happens in one second is called the frequency (f) of the wave. If 1 cycle happens every 1 second, the frequency of that wave is 1.

The scientist Heinrich Hertz created electromagnetic waves for the first time in a laboratory.

Waves are crucial in all of physics. We can go to a pond, drop a pebble into the pond, and see the ripples it creates. When the pebble is dropped, the water molecules are shaken, but they are not willing to go anywhere, so they pretend to go somewhere and then come back to their place. When the pebble is at a height, it has a lot of potential energy. When it drops and hits the water, its potential energy decreases. For example, if there is a boulder at the top of a cliff, and you are at the bottom of that cliff when that boulder falls, it might crush you because its potential energy has reduced (now that its height, in relation to the great earth, has reduced), but since we just learned that energy can not be created or destroyed, the lost potential energy is converted into another form: the kinetic energy; in other words, the boulder gains momentum. When the pebble hits the water, it creates two kinds of waves: sound waves (a faint noise that sounds like “kerplop”) and the ripples. 

Think of a few notes in music ( “do, re, mi, fa, so, la, ti, do”) that you can sing. Why does the first note sound different than the last note? The frequency is different. What is frequency in the case of sound waves? The number of times compressions and rarefactions of that medium in 1 second. The higher note which has a higher frequency will have a greater number of compressions and rarefactions happening in 1 second than the lower note.

In the case of light, pure red light will have a higher wavelength and lower frequency.


(The purple curve is the wave of a violet light. It is inaccurate but enough to explain the point.)

The wavelength of violet light is shorter than that of red light. Let’s say that the time between the two red points is 1 second. The number of cycles that happen in one second –the frequency– of the violet light is greater than the red light (again, the picture is inaccurate, but one can get the idea). From the first red dot to the second red dot is λ, from the first to the third red dot is two λ, from the first to the fourth red dot will be three λ, and so on.  

Going back to the pebble, the amount of potential energy it will have at 1 cm, 1 meter, and 1 km will be different. If there is a leaf somewhere close in that pool, it may not move when the pebble is dropped from a 1 cm height but it is likely to move when it is dropped from a 1 km height because its energy is much greater. The same pebble at a higher height will have a higher potential. 

Now comes the ultimate question: how are wavelength and frequency related? A wave that makes 100 cycles in 1 second has a frequency of 100. That particular wave had a frequency of 100 Hz. From this alone, we will not be able to find the wavelength. There is a crucial piece of information we need to find the wavelength. We know the time it takes (1 second), and we know how many cycles were created, what else do we need to know? We need to know how far the wave went! Let’s say that that particular wave went 10 meters. That means that the wave made 100 cycles in 10 meters.
λ=(10/100)
=(1/10) meters

Let us generalize this result. The number of cycles is f and the length of each of those cycles is λ. Therefore, f*λ is the distance traversed by the wave in 1 second. The distance traversed by the wave in 1 second is its speed, or, its velocity.

Therefore, we find this equation:
f*λ=v

Using the velocity of light (300,000 km/s) and its wavelength (700 nm), we can find the frequency of red light after some calculations. The frequency of red light comes out to be approximately 430 THz (terahertz).

Galileo attempted to find the speed of light but was very far from the accurate answer that we have today. Very few sources have data about the experiments he did.

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