on 13-Aug-2018 (Mon)

Incremental Video Test

The difference of two squares identity is a squared number subtracted from another squared number to get factorized in the form of

\[a^2-b^2=(a+b)(a-b).\]

Rewrite \(5^2-2^2\) as a product.

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We have

\[5^2-2^2 = (5-2) \times (5+2) = 3\times 7.\]

Calculate \(299\times 301\).

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You can brute force the answer to this problem by using a calculator, but we have a sweeter way. We can apply the difference of two squares identity.

At first we may think about using the long multiplication method, but it wastes time and is, of course, boring. Notice that \(299=300-1\) and \(301=300+1\), so

\(299\times 301=(300-1)(300+1)=300^2-1^2=89999\).

Show that any odd number can be written as the difference of two squares.

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Let the odd number be \( n = 2b + 1 \), where \(b\) is a non-negative integer. Then we have

\[ n = 2b+1 = [ (b+1) + b ] [ (b+1) - b ] = (b+1)^2 - b^2. \ _\square\]

What is

\[234567^2-234557\times 234577\ ?\]

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Using the same method as the example above,

\[\begin{align*} 234567^2-234557\times 234577&=234567^2-\big(234567^2-10^2\big)\\ &=234567^2-234567^2+10^2\\ &=100.\ _\square \end{align*}\]

What is \(99^2 - 98^2 \, ?\)

Note: Try it without using a calculator.

Which of the following equals \(\dfrac { { a }^{ 2 }-{ b }^{ 2 } }{ a-b } \) for \(a\neq b\)?
\[a+b\]

\[\large \color{blue}{2014}\color{blue}{2014} \times \color{blue}{2014}\color{blue}{2014} - \color{blue}{2014}\color{red}{2013} \times \color{blue}{2014}\color{fuchsia}{2015} = ? \]

Don't use a calculator!

Japan has become seen as a much-admired and emulated exemplar of these active, “understanding-centered” teaching methods. But what’s often missing from the discussion is the rest of the story: Japan is also home of the Kumon method of teaching mathematics, which emphasizes memorization, repetition, and rote learning hand-in-hand with developing the child’s mastery over the material. This intense afterschool program, and others like it, is embraced by millions of parents in Japan and around the world who supplement their child’s participatory education with plenty of practice, repetition, and yes, intelligently designed rote learning, to allow them to gain hard-won fluency with the material.

The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. Worse, students often believe they understand something when, in fact, they don’t. By championing the importance of understanding, teachers can inadvertently set their students up for failure as those students blunder in illusions of competence. As one (failing) engineering student recently told me: “I just don’t see how I could have done so poorly. I understood it when you taught it in class.” My student may have thought he’d understood it at the time, and perhaps he did, but he’d never practiced using the concept to truly internalize it. He had not developed any kind of procedural fluency or ability to apply what he thought he understood.

There is an interesting connection between learning math and science, and learning a sport. When you learn how to swing a golf club, you perfect that swing from lots of repetition over a period of years. Your body knows what to do from a single thought—one chunk—instead of having to recall all the complex steps involved in hitting a ball.

In the same way, once you understand why you do something in math and science, you don’t have to keep re-explaining the how to yourself every time you do it. It’s not necessary to go around with 25 marbles in your pocket and lay out 5 rows of 5 marbles again and again so that you get that 5 x 5 = 25. At some point, you just know it fluently from memory. You memorize the idea that you simply add exponents—those little superscript numbers—when multiplying numbers that have the same base (10⁴ x 10⁵ = 10⁹). If you use the procedure a lot, by doing many different types of problems, you will find that you understand both the why and the how behind the procedure very well indeed. The greater understanding results from the fact that your mind constructed the patterns of meaning. Continually focusing on understanding itself actually gets in the way.

What I had done in learning Russian was to emphasize not just understanding of the language, but fluency. Fluency of something whole like a language requires a kind of familiarity that only repeated and varied interaction with the parts can develop. Where my language classmates had often been content to concentrate on simply understanding Russian they heard or read, I instead tried to gain an internalized, deep-rooted fluency with the words and language structure. I wouldn’t just be satisfied to know that понимать meant “to understand.” I’d practice with the verb—putting it through its paces by conjugating it repeatedly with all sorts of tenses, and then moving on to putting it into sentences, and then finally to understanding not only when to use this form of the verb, but also when not to use it. I practiced recalling all these aspects and variations quickly. After all, through practice, you can understand and translate dozens—even thousands— of words in another language. But if you aren’t fluent, when someone throws a bunch of words at you quickly, as with normal speaking (which always sounds horrifically fast when you’re learning a new language), you have no idea what they’re actually saying, even though technically you understand all the component words and structure. And you certainly can’t speak quickly enough yourself for native speakers to find it enjoyable to listen to you.

Chunking was originally conceptualized in the groundbreaking work of Herbert Simon in his analysis of chess—chunks were envisioned as the varying neural counterparts of different chess patterns. Gradually, neuroscientists came to realize that experts such as chess grand masters are experts because they have stored thousands of chunks of knowledge about their area of expertise in their long-term memory. Chess masters, for example, can recall tens of thousands of different chess patterns. Whatever the discipline, experts can call up to consciousness one or several of these well-knit-together, chunked neural subroutines to analyze and react to a new learning situation. This level of true understanding, and ability to use that understanding in new situations, comes only with the kind of rigor and familiarity that repetition, memorization, and practice can foster.

As studies of chess masters, emergency room physicians, and fighter pilots have shown, in times of critical stress, conscious analysis of a situation is replaced by quick, subconscious processing as these experts rapidly draw on their deeply ingrained repertoire of neural subroutines—chunks. At some point, self-consciously “understanding” why you do what you do just slows you down and interrupts flow, resulting in worse decisions.

When learning math and engineering as an adult, I began by using the same strategy I’d used to learn language. I’d look at an equation, to take a very simple example, Newton’s second law of f = ma. I practiced feeling what each of the letters meant—f for force was a push, m for mass was a kind of weighty resistance to my push, and a was the exhilarating feeling of acceleration. (The equivalent in Russian was learning to physically sound out the letters of the Cyrillic alphabet.) I memorized the equation so I could carry it around with me in my head and play with it. If m and a were big numbers, what did that do to f when I pushed it through the equation? If f was big and a was small, what did that do to m? How did the units match on each side? Playing with the equation was like conjugating a verb. I was beginning to intuit that the sparse outlines of the equation were like a metaphorical poem, with all sorts of beautiful symbolic representations embedded within it. Although I wouldn’t have put it that way at the time, the truth was that to learn math and science well, I had to slowly, day by day, build solid neural “chunked” subroutines—such as surrounding the simple equation f = ma—that I could easily call to mind from long term memory, much as I’d done with Russian.

Time after time, professors in mathematics and the sciences have told me that building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their success. Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

Advantage of interruption in learning:

improving attention: whenever attention declines, change of the subject is the simplest remedy other than taking a definite break from learning

you will quickly discover that multiple cloze deletions on a single paragraph are not a good idea

The term Information Fatigue Syndrome has been coined recently to refer to stress coming from problems with managing overwhelming information. Some consequences of IFS listed by Dr. David Lewis, a British psychologist, include: anxiety, tension, procrastination, time-wasting, loss of job satisfaction, self-doubt, psychosomatic stress, breakdown of relationships, reduced analytical capacity, etc. The information era tends to overwhelm us with the amount of information we feel compelled to process. Incremental reading does not require all-or-nothing choices on articles to read. All-or-nothing choices are stressful!

The term Information Fatigue Syndrome has been coined recently to refer to stress coming from problems with managing overwhelming information.

Incremental reading is best suited for articles written in hypertext or in an encyclopedic manner. Ideally, each sentence you read has a contribution to your knowledge and is not useless without the sentences that follow.

The hardest texts may not be suitable to reading in increments. For example, a piece of software code may need to be analyzed in its entirety before it reveals any useful meaning. In such cases, when the text (here the code) comes up in the incremental reading process, analyze it and verbalize your conclusions. The conclusions can then be processed incrementally. You will generate individual questions depending on which pieces of knowledge you consider important and which become volatile.

Japan is also home of the Kumon method of teaching mathematics, which emphasizes memorization, repetition, and rote learning hand-in-hand with developing the child’s mastery over the material.

American consumers have done fabulously well in recent decades

According to the skilled strategist Sun Tzu, strategy is about winning before the battle begins, while tactics are about striking at weakness.

“Strategy without tactics is the slowest route to victory. Tactics without strategy are the noise before defeat.”