Rendered on a Commodore 64, then assembled on a PC.
Showing posts with label math. Show all posts
Showing posts with label math. Show all posts
From Fish to Infinity
Steven Strogatz, a professor at Cornell University, just concluded a great series of articles in the NYT articulating math concepts (from simple to complex) in accessible language. I've really enjoyed reading these over the past several months...here's the first article, and a link to the entire series.
Yesterday, while building wooden train track #10,768 with Max on our living room rug, the little guy surprised me by counting to 33 in odd numbers: 1, 3, 5, 7, 9, etc. I didn't even know he could do that! By around 3-1/2, and in the months approaching 4, Max was able to count to 100 and beyond, to 1,000 by hundreds, do addition and subtraction in his head up to around 12, hand-write several words (including his name of course), type even more words, phonetically read many short and medium length words, and sight read a larger list of words. He can finish the candy level of We Love Katamari by himself, he's a force to be reckoned with on the tricycle, he loves to romp, loves silly dance contests, and has been known to break into an impromptu chorus of 'I've Been Working On The Railroad' at any given moment. I look at the kid and constantly think: It must be a lot of fun to be Max.
Esmé is blossoming. In the past two months as she approaches one year old, her personality has really started to come out. She loves to be pushed around the neighborhood in her little plastic car (clapping enthusiastically as she goes), and she gives Max a run for his money in silly dance contests. She is so beautiful it's distracting...big eyes, curly hair (lots of it), enormous smile (it seems to have grown in the past month). She waves at nearly everything and everybody, with her delicate little fingertip-model-wave. Loves her big brother, loves to chase him and be chased by him; she crawls so quickly she looks like a little beetle scurrying. She's so sweet we call her Zen baby--she's not the intense micro-manager Max that is. She's sweet, that is, until you take away that toy she was playing with...
Esmé is blossoming. In the past two months as she approaches one year old, her personality has really started to come out. She loves to be pushed around the neighborhood in her little plastic car (clapping enthusiastically as she goes), and she gives Max a run for his money in silly dance contests. She is so beautiful it's distracting...big eyes, curly hair (lots of it), enormous smile (it seems to have grown in the past month). She waves at nearly everything and everybody, with her delicate little fingertip-model-wave. Loves her big brother, loves to chase him and be chased by him; she crawls so quickly she looks like a little beetle scurrying. She's so sweet we call her Zen baby--she's not the intense micro-manager Max that is. She's sweet, that is, until you take away that toy she was playing with...
Last night on the way home from Rose’s birthday dinner, and from visiting my new nephew (quite a full day), Esmé was sleeping, Max was looking out the window at the lights along the highway, and Rose and I were holding hands enjoying the peaceful drive.
About halfway home, we heard Max in the back seat talking to himself: “Eighty-one, eighty-two, eighty-three…eighty-eight, eighty-nine…Beedy.”
Eh?
He continued: “Beedy-one, beedy-two, beedy-three…beedy-eight, beedy-nine…Seedy.”
Huh?
It took us a few minutes, but we figured it out. He was counting by tens using the alphabet as a base: A0 (“A”-dee, not “eighty”), A1, A2, A3, A4, A5, A6, A7, A8, A9, B0 (pronounced “BEE-dee”), B1 (“Beedee-one”), B2, B3… C0, C1, C2, C3, etc. He counted all the way from A0 to Z9 (“Zeedee-nine”), and then proclaimed, “I counted to alphabet-hundred!”
My inventive young man had actually counted to 261, in his own special way. Rose and I were absolutely cracking up the whole time he was doing this (once we realized what was going on, that is). Ah, the mind of a three-year old.
About halfway home, we heard Max in the back seat talking to himself: “Eighty-one, eighty-two, eighty-three…eighty-eight, eighty-nine…Beedy.”
Eh?
He continued: “Beedy-one, beedy-two, beedy-three…beedy-eight, beedy-nine…Seedy.”
Huh?
It took us a few minutes, but we figured it out. He was counting by tens using the alphabet as a base: A0 (“A”-dee, not “eighty”), A1, A2, A3, A4, A5, A6, A7, A8, A9, B0 (pronounced “BEE-dee”), B1 (“Beedee-one”), B2, B3… C0, C1, C2, C3, etc. He counted all the way from A0 to Z9 (“Zeedee-nine”), and then proclaimed, “I counted to alphabet-hundred!”
My inventive young man had actually counted to 261, in his own special way. Rose and I were absolutely cracking up the whole time he was doing this (once we realized what was going on, that is). Ah, the mind of a three-year old.
“99 Out of 100” or “The Last 1%”
While raking leaves the other day, I was thinking about “the last one percent.”
Raking leaves, like much physical labor, is all about the most effective use of movement. In a sense, you could say it's about minimizing movement--getting the work done as quickly as possible with the least amount of effort. Engaged in this task, you think about things like the optimal pattern in which to rake the yard, how many times you bend to fill each wheelbarrow, etc.
So, in the process of minimizing my movements, I began thinking about “the last one percent,” or more informally, “just how clean does the yard need to be?” Do I pick up every single leaf? Do I pick up 90% of the leaves? 95%? 99%?
Of course, getting every leaf would only be possible through extraordinary (and foolish) effort … just as removing every drug dealer from the streets would be possible only at astronomical cost of money and human life ... just as guaranteeing that no person would ever again die in a car accident would make automobiles impossibly expensive and impractical to drive … choose any example you'd like. One hundred percent is the limit--the conceptual goal never to be realized but only to be approached. How close one gets depends on the cost one is willing to pay. This seems to be the way that the natural world operates.
However, I can think of situations when this “principle” is turned on its head. For example, in my business, housing, it seems that the last 1% of construction cost is where you really add the most perceived customer value. You can build a great building, with strong architectural integrity, intelligent planning, etc., but if the surface finishes in the kitchens and bathrooms aren't appealing, no one will buy. The other 99% of the building has to be built … there's no way around it. But without that last 1% of quality, the other 99% becomes much less valuable.
Another counter-example … something I've written about before, “the bottom of the bowl.” When you eat a salad, or a bowl of pasta, or whatever, the stuff that collects in the bottom is the best tasting stuff.
You hear the aphorisms, “god is in the details” or “the devil is in the details,” imploring us to be detail-oriented. You taste the stuff at the bottom of the bowl and realize how good it is. And yet nature and human social structures seem to operate in a way that makes attaining the last 1% impractical, even foolish.
I must think about this more … I can't yet reconcile these two interpretations.
While raking leaves the other day, I was thinking about “the last one percent.”
Raking leaves, like much physical labor, is all about the most effective use of movement. In a sense, you could say it's about minimizing movement--getting the work done as quickly as possible with the least amount of effort. Engaged in this task, you think about things like the optimal pattern in which to rake the yard, how many times you bend to fill each wheelbarrow, etc.
So, in the process of minimizing my movements, I began thinking about “the last one percent,” or more informally, “just how clean does the yard need to be?” Do I pick up every single leaf? Do I pick up 90% of the leaves? 95%? 99%?
Of course, getting every leaf would only be possible through extraordinary (and foolish) effort … just as removing every drug dealer from the streets would be possible only at astronomical cost of money and human life ... just as guaranteeing that no person would ever again die in a car accident would make automobiles impossibly expensive and impractical to drive … choose any example you'd like. One hundred percent is the limit--the conceptual goal never to be realized but only to be approached. How close one gets depends on the cost one is willing to pay. This seems to be the way that the natural world operates.
However, I can think of situations when this “principle” is turned on its head. For example, in my business, housing, it seems that the last 1% of construction cost is where you really add the most perceived customer value. You can build a great building, with strong architectural integrity, intelligent planning, etc., but if the surface finishes in the kitchens and bathrooms aren't appealing, no one will buy. The other 99% of the building has to be built … there's no way around it. But without that last 1% of quality, the other 99% becomes much less valuable.
Another counter-example … something I've written about before, “the bottom of the bowl.” When you eat a salad, or a bowl of pasta, or whatever, the stuff that collects in the bottom is the best tasting stuff.
You hear the aphorisms, “god is in the details” or “the devil is in the details,” imploring us to be detail-oriented. You taste the stuff at the bottom of the bowl and realize how good it is. And yet nature and human social structures seem to operate in a way that makes attaining the last 1% impractical, even foolish.
I must think about this more … I can't yet reconcile these two interpretations.
Conservatism: Failure to change (or changing slowly) one’s own mind in light of new information/evidence.
Recency: The most recent events dominate those in the less recent past, which are downgraded or ignored.
Anchoring: Predictions of the future are unduly influenced by initial information, which is given more weight than it deserves.
Illusory correlations: Belief that patterns are evident and/or that two variables are causally related when they are not.
Optimism, wishful thinking: People’s preferences for future outcomes affect their predictions of such outcomes.
Underestimating uncertainty: Excessive optimism, illusory correlation, and the need to reduce anxiety result in underestimating the uncertainty of future events.
Adapted from Spyros G. Makridakis, “Forecasting, Planning, and Strategy for the 21st Century,” 1990.
I haven't had a whole lot to say lately because most of what's been going through my head looks something like:
Ah, yes, academia. Waking the brain after several years of... I won't say "rest" because that's not really true... several years of... let's say a comfortable routine... has not been all that easy in certain areas. Some of the "areas" to which I'm referring include math.
Now, don't get me wrong, I was pretty good at math in high school, and college too for that matter. In eighth grade I was even in "MathCounts," one of those geeky competitions for kids who can't think of a good enough excuse to convince their parents that they shouldn't have to participate. But we all know what happens to the brain as it falls into a routine. Atrophy. When I graduated from college, taking a derivative was like breathing. Now I have to stuggle just to remember the order of operations. (Remember that from seventh grade? Exponents first, division and multiplication second, addition and subtraction third... oh, boy.) And it's only been four years. If ya don't use it, you lose it, I suppose.
It seems that no matter where I am in life, I always seem to want to put myself in a situation where I'm not comfortable anymore. And we ask ourselves: "Why?"
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