“The Monty Hall Problem”: My/Our Persistent Statistical Error

I haven’t posted here in a while because I’ve been busy with school, but I decided to upload a paper I wrote for Intro to Statistics. Even if you’ve never taken statistics, you might enjoy this. 

But first, go to https://www.youtube.com/watch?v=mhlc7peGlGg and watch the short video. Then go to http://betterexplained.com/articles/understanding-the-monty-hall-problem/ and play the game itself.

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                                               “The Monty Hall Problem”:
                                         My/Our Persistent Statistical Error

                                                           Karen Leh

                                                             10-9-14




Trying to understand “The Monty Hall Problem” presented quite a challenge, but evidently, I have good company (e.g., most of the human race). When I watched this video, I chose—incorrectly—not to switch doors, figuring that my chances were fifty-fifty. Even the explanation that was given did not fully convince me. My mind kept saying, “But, but.…”
 
What really helped was visiting a site that simulates the game show’s choices and keeps track of how well the player does. With this simulation, I did fifty trials where I opened the same door I had picked initially, then fifty trials where I switched doors. Here is how I fared: when I did not switch, I lost 72% of the time and won 28% of the time; when I switched doors, I won 50% of the time and lost the remaining 50%. I did a repeat trial of the door-switching to see if I’d do better or worse and came out farther ahead with the wins (56% in my favor, 44% losses). This activity, more than any fancy explanation, helped me to accept the realities of the game.
 
On the video, the narrator explains the game’s rules and how, when you stay with your first choice, you are really limiting yourself to that initial 33% chance of winning. (Since there are three doors and the contestant gets to choose one, the probability of winning, at the beginning, is one-third or 33%). Altering your choice, however, means a 66% chance of success. Although you might win with your initial choice, the narrator tells us, you actually increase your odds by changing position (the graphics emphasize how this is working). A simpler way of saying this is that the door you picked first has a 33% chance, and the 66% chance is still “out there.” That is why when you switch you increase the likelihood of winning. But as the video tells us, this realization is “counterintuitive” (Clarke, 2007).
 
It took a while, but I began to understand and accept how statistics was operating within the game. The author of the simulation I played notes that sticking with your first choice occasionally wins you the car, especially in a small number of trials, which is a demonstration of The Law of Large Numbers (Sullivan, 2013). The more games you play, the more the probability settles into its usual outcome. I witnessed this law in action when I repeated the switched-doors strategy and got a number closer to the predicted 66%. I’m sure that if I repeated the same-door strategy, that too would tend toward its usual 33%-66% divide. But a contestant isn’t likely to see the long view; he/she is in the present moment and not likely to see anything beyond a possible short-run windfall.
 
The other laws I could see influencing what was happening were The Conditional Probability Rule and The Complement Rule. I got a little lost in the math, but I liked how one particular writer set this up: “…let us call the curtain picked by the contestant curtain A, the curtain opened by Monty Hall curtain B, and the third curtain, C. We will define the following events: A, B, and C are the events that the prize is behind curtains A, B, and C respectively. O is the event that Monty Hall opens curtain B. The Monty Hall Problem can be restated as follows: is Pr {A|O} = Pr {C|O}?” (Zomorodian, 1998). This person went on to discuss Bayes’s Theorem, which I didn’t completely understand, but once he got me to that last understandable equation, I realized I could make use of The Complement Rule. If Pr {B} = 0, and Pr {A} = one-third, then Pr {C} = 1- Pr {A}, which leaves two-thirds. Monty has actually assisted because he’s already opened one of the doors behind which a goat awaits. As one writer put it, “[b]ecause Monty’s choice was not random...the remaining probability of two-thirds gets squeezed, as it were, into the third box” (The Economist, 1999). Another writer explains it this way: “…when Monty opens a door, he is reducing the probability that it contains a car to zero….When Monty shows…a goat behind one of those two [remaining] doors, the two-thirds chance is only for the one unopened door because the probability must be zero for the one that the host opened” (Math Images, 2010). I found these explanations especially helpful, and I then began wondering how I’d been fooled.
 
Because I was definitely fooled. And then surprised, which was followed by disbelief and confusion. I had to read a number of articles for the facts to start making sense to me. The simulation was incredibly helpful, but it took many diagrams and explanations for the situation to make sense.
 
Part of what happens is that Monty’s opening of the one door draws attention away from what’s really happening. It’s an illusion that we have a fifty-fifty chance, and the host actually “helps” us to make that assumption (Better Explained, n.d.). He’s also limited the number of doors to a total of three; if we were offered more doors, we would probably be more likely to switch (Better Explained, n.d.). But we humans usually fall for Monty’s tricks if we’re not in on the game.
 
I read one article about an experiment where simple pigeons were actually better at adjusting their behavior in these circumstances—that put in a bird-friendly experiment with feed as a reward, they would switch doors when it made sense to do so. And when the researchers changed the rules of the game so that the pigeons were rewarded for staying with the same door, the birds modified their behavior. “Replication of the procedure with human participants showed that humans failed to adopt optimal strategies, even with extensive training” (Herbranson, 2010).
 
I find that fascinating. In a way, it’s reassuring to know I’m similar to most everyone else. But now I also know better.




 
Sources:
 
Better Explained. (n.d.). Understanding the Monty Hall problem. Retrieved from betterexplained.com/articles/understanding-the-monty-hall-problem/
 
Clarke, Ron. (Jan 21, 2007). The Monty Hall problem. Video retrieved from https://www.youtube.com/watch?v=mhlc7peGlGg
 
The Economist. (Feb. 18, 1999). Getting the goat. The Economist 350. Retrieved from www.economist.com/node/187166
 
Herbranson, Walter T., and Schroeder, Julia. (2010). Are birds smarter than mathematicians? Pigeons (Columba livia) perform optimally on a version of the Monty Hall Dilemma. Journal of Comparative Psychology 124(1): 1–13. Retrieved from www.ncbi.nlm.nih.gov/pmc/articles/PMC3086893. doi:10.1037/a0017703
 
Math Images. (2010, July 19). The Monty Hall problem. Retrieved from mathforum.org/mathimages/index.php/The_Monty_Hall_Problem
 
Sullivan, Michael, III. (2013). Statistics: Informed Decisions Using Data (4th ed.). Upper Saddle River, NJ: Pearson Education, Inc.
 
Williams, Richard (2004). Appendix d: the Monty Hall controversy. Course Notes for Sociology Graduate Statistics I. Retrieved from www3.nd.edu/~rwilliam/stats1/appendices/xappxd.pdf
 
Zomorodian, Afra. (1998, Jan. 20). The Monty Hall problem. Retrieved from http://www.cs.dartmouth.edu/~afra/goodies/monty.pdf

Little House on the Internet Highway

This last week I finally picked up the first book in Laura Ingalls Wilder’s autobiographical series to read to my twelve-year-old son Luca. I had given him my dusty set for Christmas, but then I second-guessed myself: Maybe only girls enjoy this sort of thing, I thought. And when my inner feminist rejected that notion, I thought, This book might be too realistic, as Luca’s tastes tend to run toward fantasy and science fiction.

After reading the first few chapters to him, I asked, “Does this just seem weird to you?”

Luca has autism and words do not easily form on his tongue. But he gave me this look that said, “I can’t believe you’re traumatizing me with stories of people having to carry guns because they fear bears. Not to mention the descriptions of slaughtering hogs.”

Several days went by, though, and when I presented him with a new book as well as this old gem, he clearly chose the latter.

“Really?” I asked.

“Yes,” he answered.

So, on we went, and the farther we got, the more I remembered why I had adored these books as a child. They are, in fact, as strange as any science fiction.

In one early chapter, Ma, Laura, and her sister Mary are churning butter (churning butter???), and Laura talks about how Ma has assigned to each day a different chore: the clothes washing, the housecleaning, the bathing of the children. Come Sundays, no one does anything, which must have been a blessed relief to mothers everywhere but was nothing but anathema to fidgety kids. The family’s food is grown or hunted, then cooked, smoked, or otherwise preserved, and then tucked away. Maple syrup and honey represent special delicacies, and store candy is “too pretty to eat.” Snow banks against their log cabin, then the “sugar snow” arrives, and later the snow with dents in it because what’s on the trees falls in great, melting dollops. Every time a friend or a neighbor or a family member shows up at the gate, a kind of holiday ensues.

More than a few times as I reread the book aloud, I wondered to myself, How is this working? Why am I still engaged? There is really nothing happening here.

But I think that’s exactly what kept me going: the quietness of it all, the little miracles, Laura and Mary holding hands at the edge of their field and looking on with astonishment at the new-fangled but to us primitive threshing machine doing the work of three men in one day. Here, in one image, is the advancing industrial revolution, pushing ever westward.

When I was a child, in the late sixties and seventies, Laura’s world was already three generations removed. For my son Luca, the distance is probably four or five generations, but it may as well be a thousand. And it’s becoming more distant at an ever-increasing speed.

The reason I write about this is not to promote the book series, though I have enjoyed rereading the first volume. I am definitely not some Luddite who shuns technological progress (I’m writing this on a laptop in my air-conditioned kitchen). A part of me is certainly remembering why I wrote and self-published a novel last fall (Dream of an Inland Sea) wherein the young heroine embodies some of this over-the-shoulder longing for the unspoiled American West.

But mostly, I find myself thinking about our speed—about what has been gained, what lost.

My sense is that, collectively, we have a pretty good idea about what’s been gained. Most of us do a little jig every time Apple puts out a new iPhone, and no one—not even the most backward among us—wishes his or her laptop would get larger or slower.

What’s harder to grasp is what’s been lost. But Little House in the Woods brought me face up to it. When I read Laura’s descriptions of foods, I can really taste them. Some of that may be my own pure brand of nostalgia, I admit, but a part of this sensation is also my understanding of the toil the family invested in growing and cooking everything they ate, brought forward by the writer’s evocations. A tree stump—which ordinarily seems obnoxious—looks different after Laura talks about it. Suddenly, it is less pesky, more tolerable, even desirable—let’s all have multiple tree stumps in our yards, I think, so we can jump from one to another. Perhaps—I think, after reading the chapters entitled “Sugar Snow” and “Dance at Grandpa’s”—I have failed to see my cousins in the magical light with which I should view them (and all other people, for that matter); maybe I’m not looking at what’s around me and should travel more often by wagon or sleigh, or in the very least, take more walks; and have I ever really learned to cook, shouldn’t I start my own garden and make more meals from scratch?

Somehow—maybe by virtue of how slowly Ms. Wilder’s life moved—I slowed down. And this simple act made me start to question the ferocious speed at which everything in my own life was and is traveling.

I suspect we all have sudden premonitions that the human mind is incapable of keeping up with the rate of change occurring around us. And if you question that premise,  consider the fact that when I sat down to write this essay, my computer—in the course of about three minutes—fixed itself of some internal systems issue. Most of us cannot fathom how it did that, nor do we care, but this occurrence is the sort of thing I’m talking about. Once upon a time (say, a mere ten years ago?), I might have had to take it to a computer guru, but not anymore. The computer “knew” it was sick, diagnosed itself, and then, a physician unto itself, performed its own successful operation. Thirty years ago, who could have imagined that? It’s uncanny, fantastical.

And at the same time, confounding. If I can’t grasp these advances, understand the processes involved, it’s as if Time itself is tricked. Of course, it isn’t really: someone somewhere understands what’s occurring inside my computer; they coded it to do just what it did. But I, the lowly un-tech-savvy consumer, don’t entirely get it. If enough of these kinds of advances occur simultaneously, I begin to feel as if I am being pulled willy-nilly over the clockwork. Even if I let go, allow myself to advance at the rate of progress, I start to feel somewhat “played,” confused. Time isn’t obviously what I thought it to be, and that’s fine. But my body also cannot seem to catch up. I experience a kind of technological hangover. It’s low-level, but it’s real.

Luca has not told me what he liked about Little House in the Woods, but I think Laura’s descriptions of various processes might be key—even if those very processes are no longer central to how we live. How does Pa maintain his firearm and make bullets? How does Ma create butter pats? How would they smoke venison or tap a maple tree and process its syrup? The slow unwinding of “how” holds intrigue for us humans, and it counteracts the effects of technology’s onrush. Those descriptions are also a legacy of sorts; they tell us we matter, that the world is complicated, but there are those who will take the time to explain in a way our minds can grasp. All of us can appreciate that, but children, especially.  

So the question becomes: How do I honor in myself both slowness and speed? Because to live without slowness is to forget my hunger for nature, for the minuscule awakenings in my midst, for the full satisfaction of my senses, all of which seem essential to my humanity. And to reject speed is to banish myself from ordinary company, to be dumb in the face of ingenuity, to spurn the amazing apparitions of the creative mind. To somehow incorporate within ourselves these different speeds is to become acquainted with Beauty in the broadest sense, isn’t it?

I haven’t been able to answer that question about how to honor both slowness and speed, but I can feel myself pivoting around an answer. If I can reread a children’s book, and suddenly become aware of the plasticity of modern life, how I’m not altogether registering what’s going on, if I naturally ride the wave of the “new” and gather up speed because that is what most fully surrounds me, then I clearly have some work to do.

I will think on this. But first, I will stop, go outside, smell the air, and survey the sky.

 

 

The Pluses of On-Demand Publishing

As I said in my previous blog, most writers (myself included) would prefer to have a contract with a publishing house, as that confers a certain degree of respectability and comes with an advance and access to the resources of the company (such as publicity). Having said that, I have come up with a list of positive attributes for the on-demand publishing side of things.

 

1)      Good for the environment. Your book is not mass-produced, so there’s no pre-publishing guess about how many copies will be needed by the public. Each book represents a desired commodity, and no books get remaindered. This is good for our forests. And as Kindle use grows, even fewer trees will meet the axe.

 

2)      The book never goes out of print. Once computer files of the book are uploaded, your book will always be available unless you decide to take it down.

3)      No timetable for publicity. If the book is always available, you can do publicity bursts whenever and however you choose.

 

4)      Corrections can be made quickly and easily.  This point might not seem very important, but if you’ve ever worked in book publishing (as I have), you know the process authors go through to correct typos, etc. By the time changes are incorporated, many thousands of books may have already been produced and purchased.

5)       No second-novel pressure. As a fiction writer, I’m not supposed to talk about this, but many published writers feel some panic about producing a second book. But in this case, if your second title doesn’t sell like the first, you’re the only one to know. In fact, commercial success is moot, though plenty of on-demand publishing authors sell very well (a friend of mine mentioned to me that one of her students had sold more than 10,000 copies).

 

6)      Less ego involvement. We all like our ego stroked, but there’s a delicious sweetness in sending your book out into the world without fanfare. Like a parent, you trust this creation (which is yours and not yours) to go forth and make its way.

7)      No publishing house author hatred. If you’re a writer whose book has traveled the ordinary publishing house route, this might surprise you, but oftentimes there’s some copy editor or production person gnashing their teeth at the mere mention of your name. I remember this well…such comments as, “Will he ever stop making corrections?” “She’s in my office making kissy sounds over the phone with her husband, and I can’t get her to leave.” “He only thinks he knows Spanish.” “She thinks she lives here, and she keeps staring at my breasts.” “Can you believe that advance? Good god, how I hate her.” (By the way, if you are a publishing house author, follow Cristina Garcia’s--Dreaming in Cuban—lead and send gracious notes to everyone who worked on your book. You’d be surprised how few do this, and a simple thank you goes a long way. Your agent and editor are not the only significant people in the development of your book.)

 

 

Self-Publish or Perish

Roughly two years ago, I finished writing a novel called Dream of an Inland Sea. Unfortunately, I have terrible timing, as Lehman Brothers had just gone into bankruptcy, and most of us were simultaneously learning about real estate derivatives and sensing that our economic security had been horribly undermined.

Many years before, I worked in book publishing in New York City (HarperCollins, Alfred A. Knopf), and I knew that this industry was not unlike others; when people become more conservative regarding financial risks, every business is affected. In book publishing, this means agents and editors view all incoming manuscripts with more reserve than they did before—and this was in addition to the reserve they’d already developed due to Americans’ changing reading habits, the growth of the internet, and the availability of e-books.

But, ever hopeful, I began sending my book to agents. After three or four rejections, I almost had one, but someone at the agency with veto power prevailed. At this point, I went through some soul-searching. Is the book flawed? I wondered. Do I need revisions, or am I satisfied with its vision and execution?

I decided I was happy with the book but that I didn’t have the energy to pursue finding an agent. This decision was, no doubt, affected by the journey I’d been on with my twins, both of whom have special needs. Publishing the book was no longer my most important focus, and I was at peace that I had managed to finish the manuscript. I decided to put it away and bide my time. I had children to raise, and maybe the book didn’t need to be in the marketplace.

Then my high school band mate Dean Fetzer and his wife, Debra, came for a visit. They live in England, and Dean is also a writer. He, too, had not found a place within traditional publishing, but with energy and can-do, he’d begun successfully to self-publish his mystery novels online. During his visit, he said, “Why don’t I help you? Can I read the book?”

At first, I was dubious. When I went through the graduate program at the Iowa Writers’ Workshop, “vanity publishing” (which was how self-publishing was once described) was  considered just that: evidence of the author’s feeling of self-importance, a purple signpost announcing the person’s inability to write or, at the very least, to successfully edit his- or her own work.

But the publishing industry has changed, and I decided that I should, too. I have friends who have been successful with the typical publishing route, and I applaud them, and of course, I still wish to join them, as it would be ever so lovely to have an advance and a publicity machine. But because I was at peace about the book, I thought, Why not? The novel might not sell a lot of copies, but maybe someone will enjoy it. That was an electrifying thought, as no writer honestly writes entirely for the self.

So…my book is now available in paper and ebook form. This novel means a great deal to me, and I would be so honored should any of you decide to read it. Here are the links:

PAPERBACK:

US
http://www.amazon.com/gp/product/0956158188/ref=as_li_ss_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0956158188&linkCode=as2&tag=deafet-20
UK
http://www.amazon.co.uk/gp/product/0956158188/ref=as_li_ss_tl?ie=UTF8&tag=deafet-21&linkCode=as2&camp=1634&creative=19450&creativeASIN=0956158188

KINDLE:

http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Ddigital-text&field-keywords=dream%20of%20an%20inland%20sea

 

 

 

Steal This Business Idea (Please!)

It’s Mother’s Day, 2012, 10 a.m., and I’ve

1)      Performed part of a gastronomy tube feed on one 11-year-old child,

2)      Bathed and changed the clothes (and beds) of both this same child and his twin (they both have autism),
3)      Cleaned the kitchen after a lovely breakfast made by my partner, Lisa,
4)      Set up one child in the master bedroom with a video about the moon,
5)      Set the other up in the living room with a Signing Time video, on a loop, the way he likes it.

A pretty good morning, actually. (Don’t ruin it by telling me I’m a bad mother for “parking” my children in front of the telly; I already know this.)

So if it’s a good morning, why is it I feel that someone should create the Karen Leh Home for the Perpetually Perplexed?

No, this is not the business idea (but come to think of it, you might want to stay tuned on that one). The business idea would actually counteract my perpetual confusion and make such a home unneeded. What I’m thinking of is a nonprofit called Rent-a-Mom. Not a mom for my kids—no. For me!

Before anyone thinks I’m putting down my own mother, I should say this: I adore my mother. We’ve come through a great deal together, but she’s 75 years old, and her health is not that great. Plus, she lives two hours away. And our town home has (count ‘em!)  29 steps—which isn’t even counting the three to our front door or the seven into the lowest part of the basement. In short, it’s a septuagenarian’s nightmare. (And to punctuate this household issue, Lisa has just called me on the land line from our basement. No joke.)

So why would Rent-a-Mom be necessary and what would it entail?

Unfortunately, I have to back way up. Picture yourself with twin babies, then imagine flying halfway across the country, both three-month-old infants in tow, so that one can get open-heart surgery that will then go very wrong. Five months, seven corrected heart defects, one heart valve, one tracheotomy, one gastronomy tube, and a brain bleed later, you are back at home. One child lies on the floor. He has a tube that has been lassoed around his neck and is blowing oxygen and moisture at his trach, the fog of which billows up around him, prompting an overseas friend who sees a picture of this get-up to remark that he “looks like an astronaut.” The other baby seems to be doing okay, is hitting his milestones, but eventually takes to screaming day and night, forgets how to use his eating utensils, loses his vocabulary, stops giving eye contact, and begins incessantly lining up Matchbox cars.

I could go on, but you get the idea. When my children were small, I used to threaten to put a flagpole on my roof where I’d attach a white surrender flag so that someone would come and rescue me—or at the very least, feel like crap for NOT coming to rescue me. I also used to fantasize about getting in the car and driving.

And driving. And driving. Away, anywhere. One of my friends, who also had a critically ill child, told me, “Oh man, if you do that, call me. I want to come.”

Somehow, we all survived, though with what mental defects I do not know. I remember one particularly horrible instance where one of our twins, at about age six, had diarrhea, took off his pants, danced through it, then oh-so-artfully fingerpainted his walls, the curtains, the windows, and everything else. This had happened on many other occasions but never this badly.

“Lisa,” I said, “we’ve had a poop disaster, and I think we’re going to have to tear up the carpet.”

“Oh, it can’t be that bad,” she insisted.

I went back upstairs, and after several minutes, she came to survey the damage. “Holy Mary, Mother of God,” she muttered. “We’re going to have to tear up the carpet.”

“Yes,” I said, jaw set.

She left the house to purchase a truckload of cleaning supplies and tools we’d need. I will not describe her mental state or the particular swear words she chose for the occasion, but suffice it to say, she was not happy.

I was, if I remember correctly, shaky. But I was clear-headed enough to call one of the boys’ paraprofessionals from school. “Can you come?” I asked after explaining the situation.

Fortunately, she did. I’d never had anyone help before, but I probably had never sounded like that before either. She was, no doubt, somewhat alarmed. Sherry (the parapro), shepherded the kids, and Lisa and I proceeded to take the room down to the subfloor. I opened the windows, cleaned the walls, threw away bedding, curtains, stuffed animals, books, you name it, moved the one boy into a different room with his brother, then locked the door to the ruined room and left it closed off for something like nine months. Sherry, being the sweetheart that she was and is, entreated Lisa and me to go out for dinner afterward, just the two of us. We sat in a Chile’s in numb, stunned silence for quite a while, as I recall, like people just returned from some unknown circle of hell.

I’ve heard others tell similar tales. One mom I met several years ago told me that after her autistic son pulled a brand-new $2,000 plasma television set down on himself, she promptly packed the whole family off to the ER. Not because anyone was hurt but simply because they “needed monitoring.” But who can afford to do that every time a TV is destroyed or a room is redecorated with feces?

Many of you might think, But Rent-a-Mom already exists—it’s called Therapy. But again, I tell you, too expensive, especially for those of us who are parents of special needs children (for those of you who don’t know, most of these parents 1) have filed for bankruptcy, 2) totter somewhere near the brink of it, and/or 3) have avoided it only by virtue of loving family). Plus, if you’ve come through a room-destroying scenario like the one I described, a 45-minute “therapeutic hour” will never suffice. And, honestly, complaining only goes so far. I’d be happier if someone came and did my laundry or just sat and brushed my hair. The way I picture Rent-a-Mom, it’s a funded service, gratis for the person being served.

Sherry was my hero that day. But what if there was always someone like that that you could call? Even mothers of neurotypical children could use such a service. Hell, even single, childless people could use it once in a while (read my post entitled “Karen’s ‘Helpful Hints’ on Coming Out” if you don’t believe me). Most of us have dear friends and family, but I have to tell you, it’s incredibly hard to ask for help, or to keep asking for help. And sometimes the help you need is just someone to make sure your kids are safe while you do a hard thing. Someone to make you a cup of tea and remind you to breathe. Someone to say, “Go out and eat, go grab a nap, go for a walk—I’ve got it.”

In short, a mom. But a younger, able-bodied, no-emotions-invested-in-this-situation kind of mom. (My own mother knows better and would never do this, but the last thing you want someone to say in a situation like the one I described is, “Why did you leave him alone in his room for twenty minutes?”)

This nonprofit I’m thinking of would offer short-term but essential help. I would start it if only I didn’t still need it so badly (argh!), but maybe this is your calling? Go ahead, I dare you—steal this idea.

To the Madonna of the Healthy Children

Yes, you—the one who came into the coffee shop lugging a car seat and a beautiful baby girl who is still fast asleep and enfolded in gauzy pink blankets while you sip a hot drink on the couch with your friend. I know I’m not supposed to be listening, but my table is inches away, and I can’t help hearing you any more than I can help peeking at your child.

“Typical” children astound me—I look at them as if they’re aliens because neither of my children has full health (they have been given six diagnoses between them, and both are considered autistic). I know in my head that fully functioning children are still the norm, that women’s bodies (by and large) know instinctively what to do to create life and deliver it in perfect packages. (That happens every day, I remind myself, astonished.) I am currently studying to become a nurse, and one thing that surprises me daily is that despite the complexity of the body, most of us are extremely healthy. We also possess an incredible ability to heal when our health goes awry.

I know that my pregnant body, too, did a good job. I have gorgeous children who possess all their fingers and toes and functioning muscles. They can produce words, are capable of affection and emotion as well as humor, and they do learn. But one’s heart folded incorrectly, probably in those first couple weeks in the womb, which set off a whole series of post-birth operations, surgical accidents, and developmental fall-out. His twin mysteriously regressed at about two and lost all the milestones he’d achieved.

Needless to say, I’ve been living in a whole different world. And it’s all right—I really mean that. I think I’m privy to some insight that I wouldn’t have otherwise, which is extremely valuable to me though it’s been costly. I’m not saying these things to call forth your pity or excessive admiration. But my life is very different from yours. So different, in fact, that I sometimes can’t relate to “ordinary” people. They mystify and sometimes downright irritate me. In a week in which the CDC announced another rise in the number of autistic children (1 in 88), I’m no doubt a little more cranky than usual.

But I’m off the point: Listen, you are complaining about the loss of sleep, the two o’clock feedings, the soreness of your breasts, about managing your other two children and your husband’s work schedule, and I want to yell, “Stop! Do you not see the bliss on your newborn’s face, the blush in her cheeks? Could she be any more perfect? Do you realize what a privilege it is to have these so-called problems? So seven-year-old Michael’s not doing as well as his peers with reading, and you have to nag Natasha to practice the piano—but Michael’s only seven, and he’s reading, for godsake, and Natasha—don’t even get me started. She can decipher music and manipulate her fingers across a keyboard well enough to play? That’s simply amazing to me. And here you are at a coffee shop, looking for all the world like someone who’s showered and is wearing clean clothes.”

This may sound a little like “when I was young, I had to walk three miles through the snow to get to school,” but bear with me. Basic daily tasks were tough to manage, if not absolutely impossible, when my children were infants. How, for example, do you take a shower and ensure that your baby is still breathing through his tracheotomy? How do you help your significant other get enough rest to function at work when you both are awakened by a tantrumming toddler who screams from two until five in the morning many nights of the week for years? When does the laundry get washed—okay, maybe it can be wedged in, between occupational therapy at one and speech therapy at three, but who knows?

And now that my kids are eleven, I still have trouble with these tasks. I certainly could never bring them into a coffee shop for more than maybe ten minutes. One child would bother patrons who are studying their computer screens (because he loves computers) or would go behind the counter so as to investigate the cash register or push buttons on the microwave (he blew up the one in our home). The other child will be overwhelmed by the smell of coffee, the large number of strange people, and heaven-only-knows-what-else because his sensory system just can’t cope with the intricacies and—what is to him—the onslaught of our world.

I don’t mean to be persnickety although I’m sure I’m coming across that way. And I don’t mean to imply that I do not ever enjoy my children because I do. I realize, too, that neurotypical children can also be challenging. Besides, difficulties are relative, right? If you’ve no experience other than the reality you’re living, your life can seem quite difficult. I, too, am reminded of this fact when I see people in desperate poverty, in our country or elsewhere. Or when I think of friends who have an autistic child (or children) who are divorced and parenting solo, or those whose children are more severely impacted, having to spend weeks or months in hospitals or lock-down units because they’re so dysregulated that they’re violent or self-abusive. I have a friend whose life was forever changed by her son with special needs, only to have him die at eight and a half, which was beyond unfair. Believe me, I know I have a great deal for which to be thankful. Still, I don’t know if my children will ever be fully independent, or even independent enough to take themselves regularly to a bathroom. This is a whole different level of worry.

So go ahead and complain—sometimes we just need to. I get that. But don’t fooled. Your troubles are not that bad.

I was recently reading a piece about Angelina Jolie because…well, I’m a bit obsessed—she's beautiful, her husband’s beautiful, they have this exotic, beautiful menagerie of children. She and Brad were attending some banquet (I forget which one) where Brad was nominated for an award. The interviewer asked Angelina something to the effect of “Does he get nervous about these things?” And her reply was something like “Oh, a little. But really, if your children are healthy, what is there to be nervous about?” So now she’s my personal hero. How can that come out of the mouth of a starlet when so many day-to-day people don’t even realize such a thing? Talk about a true madonna.

Some people make so much of their children’s “issues” that it feels to me like they’re making things worse. There was a woman in here last week who was hectoring her son relentlessly about his math, really going at him, which made me feel embarrassed and sorry for him (irony of ironies, he had the same name as one of my sons). And a friend recently told me about a hockey practice she went to where a father yelled out at his child, “Try not to be a dumb-ass.” Say what?!

I don’t think you will do that (I certainly hope not). But please recognize that half the things you are worrying about—okay, I’ll just say it—are stupid. Honestly, if no one’s dying, there’s not a problem. (I sat at the bedside of my heart-baby for two weeks and didn’t have a place I could even touch him, he had so many tubes coming out of his jaundiced, swollen body, so you can take my word on this one.)

Breathe. Look around (it’s a stunning day). Listen to your friend—really listen. Enjoy that baby’s sleep. For heaven’s sake, enjoy that baby’s sleep.

Obscure Lessons from Math & Science

Maybe these lessons are not so obscure, but as I’ve been taking prerequisites for a nursing program, they represent another dimension of what I’ve been learning/relearning. (I apologize ahead of time for the uneven tone of what follows: some of these “lessons” engender my humorous side, while other inspire my inner philosopher or social psychologist.)

What’s the status on your permeable membrane?  Not a typical question, I know, and I’m not talking about your skin or the envelope around your cells, from which I drew the example. Most people, I assume, fall into the psychological category of semi-permeable. (If you’re in that category, bully for you. Go on to the next lesson.) The rest of us possess the unfortunate quality of taking in too much or too little.

I’m afraid that since I fall into the first category, I won’t be much help on the second. If you don’t take enough in, all I can really say is this: Be assured that you are this kind of person if you find yourself on an episode of “I Married a Mobster.” (I mean, come on. There were signs.) One should also be alarmed if 1) a significant other carries on a year-plus affair, 2) an accountant or investor steals all of one’s life savings (though those folks can be tricky, it’s true), 3) everyone at the conference table is howling with laughter, but you “just don’t get it.”

Unfortunately, as I intimated, I don’t know how to help. Ask more questions? Develop some healthy paranoia? Ask the heavens to unstick the cobwebby goo from around your senses? (I’m at a loss, so please forgive me. Because I’m at the other end of the spectrum, trying to come up with solutions is like an uber-wealthy politician trying to relate to your joblessness or end-of-the-month inability to pay bills).

For those of us in the overly permeable category, there’s a panoply of options. A very incomplete list would include 1) unremediated schizophrenia or other mental illness, 2) pharmaceuticals!, 3) heavy use of alcohol and/or street drugs!, 4) permanent hermit status, 5) endless years of therapy, 6) endless journaling, or my personal favorites, 7) yoga and meditation, 8) relentless internal “housekeeping,” whereby you toss on the ash heap various useless assumptions, obsessions, feelings, beliefs, and relationships, or  9) some combination of the above. Believe me, I do not place judgment on anyone’s use of any of the above strategies, although homelessness, jail time, and psychosis present obvious drawbacks.

Oversensitivity sucks. However, seeing the underside of nearly everything has its upside. (Did I just say that?) Maybe it’s even a gift—at least, if one manages not to drink oneself into oblivion, adopt an unwavering victim status, or redirect one’s hurt or sickness in an outward direction (in other words, please avoid becoming the next Unibomber or Timothy McVeigh).

There is usually more than one way to solve a problem. This is true even in addition; it’s called the commutative property (2 +1 can also be written 1+2).

The online algebra classes I’ve taken the last two semesters have solidified this point for me. Pearson Learning Solutions produces these fabulous textbooks with an online interface that I cannot stop raving about. After reading a section of text, I then go into something online called My Math Lab to do the homework. Each question may be answered three times incorrectly before I get dinged. But even so, I always have the option to click a button that pulls up a similar problem so I can try again. Another great feature allows me to pull up an application that will walk me through the steps to the solution. The end result: no math anxiety. I can try a problem, then try it again and again. I can go step by step with the online tutorial—multiple times, if need be.

The second advantageous aspect of this system concerns experimentation. The problem appears on the screen, and say I don’t know quite how to approach it. No problem! Sometimes I’ll just give it a go and see what happens. And sometimes I’ll end up solving the problem in a less than typical fashion. Woo-hoo, I’m a genius, I’ll think. I’m not really, but the mind has amazing ways of working toward solutions.

I’ve found this idea of “more than one way to solve a problem” a useful tenet, and it’s essential to call to mind when someone tells you that you simply must do things a certain way. While it may be true that theirs is more expedient, sane, healthier, safer, or more long-lasting, believe me—it doesn’t represent the only way. And if you follow your own path, you might discover a startlingly wonderful new method of arriving at an answer.

We are all one. Yeah, yeah, I know—this is a spiritual truth and not something one is supposed to derive from math and science. Poppycock. In biology, one thing I’ve been learning about involves systems and their interrelationships. An individual organism is part of a larger population of similar organisms, which then participate in a community, which takes part in an ecosystem, which takes its place in the overall biosphere. This trend moves the other direction as well: from individual to organ systems and organs to tissues to cells to organelles and then molecules and atoms. If someone ingests food with an awful microbe (God help you), that person is likely to feel systemic effects. In the same way, if you toss chemicals into a water body, there will be ever-enlarging effects in the greater world. We understand this in a fundamental way, even if we don’t always acknowledge it (it’s certainly not politically expedient to acknowledge it, that’s for sure).

Here’s another example that should make us pause and look at one another. Human beings—according to what has been learned through the Human Genome Project—are 99.99%  genetically similar. What we might observe about our differences can seem extraordinary, and we put a premium on their value, but statistically speaking, those differences are small.

An experience I’ve had since I stopped working for a local hospice speaks to this point as well. Sometimes I’ll be driving around, and I’ll pass the home of someone I took care of during their final days or months, and then I’ll pass by another and one more. I’ll suddenly be struck by the web of death across Boulder County, by how we’re all connected through that experience—not one of us gets out of it. And I’ll think, Well, if we’re all connected by death, why not by life? Why are we not more aware of our interconnectedness? I don’t have an answer for that one, but I figure there’s some kind of payoff in not acknowledging it. After all, if I’m not connected to you, what do I care if you lost your job, or if the nearby plant or pipeline poisoned your water, if my kid called your kid “retarded,” if the state’s roads look like hell and the neighborhood school stinks and you suffer from some disease that requires my tax dollars but I believe the government shouldn’t take any of my money because I need it for a vacation to the Bahamas?    

Just because you’re solving for X and Y, don’t assume you understand the whole alphabet. By this, I’m referring to what I’m learning regarding graphs (and of course, extrapolating from that). I’ll sometimes graph an equation in algebra, and the line represented by it will shoot up indefinitely—or down, depending. But I’m usually solving for one or two variables, not an infinite number of them. Who knows what that would look like—I’m certainly not there mathematically speaking.

I’ve just started working with a graphing calculator, which is kind of fun. Basically, you have to input the minimum and maximum values for X along the horizontal axis; then you input the minimum and maximum values for Y along the vertical axis. If you don’t set up the window correctly, it may appear as if the graph of the line is shooting up or down at an uninterrupted angle; but widen the window, and whoa! Sometimes the line actually hits its full height a little higher up and then changes direction.

And here, because I’m odd, I think of Whitney Houston (may she rest in peace). What a meteoric rise, right? What a fabulous talent. But it’s hard to know what else was going on—why the sharp downfall, the recovery, her eventual death? None of us can account for all the variables that had an impact on her life. None of us could have predicted the dips and recoveries. Sometimes our “window” isn’t wide enough. And people are full of mystery; you can’t put them on a graph because sometimes their influences are unknown to us (possibly even to them), and influences don’t all possess equal pull. The lesson to me: show some humility, girlfriend.

The body possesses an amazing capacity to heal. You cut yourself with a serrated knife, and five days later, the injury appears to be much better. Experiments with stem cells return function to the malfunctioning body part of a lab rat. Former Congresswoman Gabrielle Giffords is shot, and months later, though still struggling with her recovery, she is able to speak and understand, to carry out many tasks of her pre-injured life.

This last example highlights a difficult aspect of this truth: unfortunately, we often learn more about the body’s healing capacity from those events where it fails to do so, or fails in part. I think, too, of my twin sons (who, for those of you who don’t know, share about six medical diagnoses between them). I think of that scar down one son’s chest from heart surgeries, which reminds me that eventually he’ll require another operation; of that inch-long scar on the side of his neck, where the heart-lung machine's tubing entered his body the second time he was hooked up, which reminds me of the brain hemorrhage that occurred during that period because of the blood-thinners he had to be on; of those times my other son finds a phrase and then cannot stop repeating it, the synapses of his brain firing in some fashion I cannot fathom; of the wide-eyed, anxious look that sometimes appears on his face because stimuli from the world is impinging upon his nervous system.

But even in offering these personal examples, I still look at my boys and realize that in so many ways, their little bodies developed according to the usual plan. Muscles attached to bones, their hearts beat, their eyes track an object, they try with amazing courage to communicate, they run and jump and laugh. And from the time of their various diagnoses, they have traveled a long, long way toward recovery.

Also, after reading about meiosis—that very intricate process whereby our parents’ DNA is copied, crosses over, divides, and then divides again, with individual gametes later joining to become each of us—I cannot help being awestruck by my children’s individuality and perfection. Mutations, my textbook assures me, happen with great rarity. But people don’t often find a chunk of meteor either, so if my sons’ genes mutated in some way, I can’t help thinking that it’s as if I had not one but two chunks of star-stuff fall into my backyard. The very fact of their being seems so specific to me, so personal, that I simply assume we are bound up in a grand universal plan of some sort. Hokey-sounding? Maybe, but not from my vantage point.