It's Long Division

Arithmetic seems to terrorize a significant number us, students both young and adult. And nothing is more emblematic of that terror as long division.

The alternative rock band Fugazi captures the sense:

It's a long time coming,

It's a long way down,

It's long division,

Crack and divide.

There is nothing more daunting to students we have tutored than long division. It seems a special torture invented in Hell for their personal torment.  But it need not be so.

Arithmetic was invented largely to keep track of human trade and commerce. How else would one keep things straight? Purchased 200 wheels of cheese, sold 192, lost 8 to vermin, made a modest profit. Without arithmetic, it would be impossible to track.

These early traders weren’t schooled mathematicians, but they weren’t stupid. They learned how to mark sticks or make imprints on clay tablets to record numbers. They added and subtracted to determine sales and inventories and profits and losses. Arithmetic was not an abstraction but the means to a very important end.

At some point they realized that when a customer ordered five crates of clay pots, each crate containing 10 pots, there might be a quicker way to determine the total of pots needed than to add 10 together 5 times. Hence was eventually born multiplication, with a variety of  creative solutions: the Ancient Egyptian method, the Russian Peasant method, the Gelosia method. Human ingenuity knows no bounds when trying to avoid tedious labor. These methods required a few brain cells but were better than adding a column of the same number, over and over and over.

But, then, technology forced us to take many steps back.

Imagine in the 1940s, the slowest, most unwitting arithmetician among us. That would be an electromechanical computer. The computing machines of Alan Turing’s day (of “Imitation Game” fame) were notably obtuse. Huge, hot, electric sparking and fumes, they could add a column of numbers not that much faster than you could. They had but two fantastic advantages: absolute accuracy and immunity to fatigue.

Oddly enough, though, these electrical behemoths didn’t know how to subtract. Or multiply. Or divide. All they knew how to do was add.

So how did the machines of Turing’s era perform subtraction and multiplication and division when all they knew was addition?

Addition is a basic, primitive operation. If you are given two apples, then three more, you will have a total of five apples.

                2 + 3 = 5

Even the Common Core methods will agree with that.

But say you are given five apples, then two are taken away. We know the answer is three – we did the subtraction in our heads. But Turing’s machine can’t subtract. So, brilliantly, we instruct it to add five and the negative of two:

                5 + (-2)= 3

Still using only the basic operation of addition, the machine has succeeded in subtracting by simply adding the negative of the subtrahend (the number being subtracted). We can now say that the machine knows how to subtract:

                5 – 2 = 3

…but in fact, we know that it is adding the negative to do so.

How about multiplication? That is simple for the sparking behemoth as well. After all, it is tireless, and simply replicates the feat of early humans  before the many multiplication methods were invented. It merely adds, repeatedly. Five cases of ten clay pots is calculated thusly:

                10 + 10 +10 + 10 + 10 = 50

Multiplication is just repetitive addition. (With some fussing with decimal points, but we can ignore that for the time being).

Now for the promised key to long division. If multiplication is repeated addition, might you suppose that division is repeated subtraction? Yes, you might, and you would be absolutely correct.

Let’s try to divide 49 by 12 given this method. Keep track of how many times we can subtract 12 and stop only when the balance is not greater than 12.

                49 / 12

                49 – 12 = 37         (1^st subtraction)

                37 – 12 = 25         (2^nd subtraction)

                25 – 12 = 13         (3^rd subtraction)

                13 – 12 = 1           (4^th subtraction)

                1 left over

The answer is 4 with a remainder of 1. That’s division, using only subtraction. And remember that subtraction is just the addition of negatives. This is how Turing’s useful idiot, the sparking monster that could only add, was able to subtract and multiply and divide as well.

The moral of this story is that arithmetic is ancient. It is rooted in the real world, not an abstract implement of student torture.  And, hopefully, by realizing that a quite stupid computer can do this, you will have the confidence that you can, too.

Now, to those of you who found this all very simplistic, it’s time to volunteer as a tutor. Some yearning student needs your insights. This is how you give back.

Bucket list - Italy!

Aunt Grace on her first morning in Sicily

It is rare, the occasion that one gets to check off  an item from one's bucket list. And to complete two in under four months is nearly unheard of. And that both involved a trip to Italy is fantastic luck.

The first, of greatest import.

Our Aunt Grazia (Grace) was born to newly emigrated Sicilian parents in the early 1920s. Remaining in Sicily were three first cousins and their families, corresponded with, but never seen nor touched for a veritable lifetime. The bucket list trip occurred to us on the eve of her 90^th birthday. When, if not now, would she be able to embrace her family?

A furious round of planning commenced, involving your intrepid writer, his wife, and two sisters. We studied the logistics of an international trip with an advanced octogenarian. The AAA travel folks in the South Attleboro office were fabulously effective. Domestic air travel to gather in Boston, then international to eventually land in Sicily, and a large rented van to accommodate us on our way to the final destination.

Finally, one day in October, the plan was put into effect. One sister flew directly to Providence. The other, to Texas to join Aunt Grace and to bring her to Boston. Finally, all gathered in Attleboro, we had a family dinner at the Heritage Tap in Pawtucket, a family-friendly place if ever there was one.  Next day, how to better enjoy the local cuisine than lunch at Tex Barry’s. You locals know what that means.

Then, finally, the limo ride to Terminal E at Logan Airport. Aunt Grace with her crisp, brand new passport, having never before exited the country of her birth, was ready to go.

The flight to Rome was long, but the wine and dinner service were decent and we managed to sleep a bit. At Rome’s Fiumicino Airport, we awaited our transfer to Palermo, Sicily. The cappuccinos (cappuccini) from the airport coffee bar were excellent, a prognostication of the wonderful gustatory delights to come.

After a relatively short flight, we arrived at Palermo and secured our Hertz rental van, an eleven passenger behemoth adequate for the five of us and our considerable luggage.

The first night was spent in Palermo, on the bay, in a lovely hotel with beautiful grounds and a wonderful restaurant.  To close the circle, this was the same hotel to which we had brought our parents in 1997, both since passed. It was very sentimental, but the focus was on tomorrow – the Sicilian relatives were in Agrigento, some 80 miles to the south.

Morning dawned and we checked out, only to spend nearly an hour in Palermo rush hour traffic.

But then the traffic thinned and we finally headed south, into the dry, dusty hills and mountains of central Sicily. After nearly two hours, we emerged on the southwest coast, with the Mediterranean aglow below us. This was the land of our grandparents. Arid, poor, but with a wealth of olive groves, lemon trees, irrigated vegetables, and the endless bounty of the sea.

Finally checked into our hotel, a small family-friendly, former estate, we began the phone calls and arranged meetings. Three first cousins, all octogenarians, and we met them all, with multitudinous nieces and nephews and grandkids. To see Aunt Grace embrace her kin for the first time was beyond touching. Hugs abounded, tears flowed, and the tables were never empty (yes, Italians do insist that you eat – mangia mangia). She was welcomed into the family as if her parents had never left. Their  mountain village, almost unchanged for over 500 years, celebrated her presence.

A bucket list item is often thought of as being a personal thing. ”I want to see Mount Everest. I would like to visit Tokyo.”

But this bucket list trip was different. The five of us were delighted to meet the first cousins and numerous nieces, nephews, and grandkids. There were perhaps fifty people touched by this one event. That’s the way to do a bucket list trip.

The other trip, some three months later, to ski the Italian Alps for the first time, was personal and selfish, not worth recounting here. But that it was to Italy, with the scenery and the food (the food!), made it fabulous.

No, the adventures of Aunt Grace, on the eve of her 90th birthday, on her first trip ever out of the country, to see close relatives she had never met… that’s what made this  bucket list trip magical. Most probably, never to be matched.

But always remembered.

The Winter that Would Never End

Sidewalk buried under "Mount Bronson," County Street

It has been a long, cold, dark, snowy, ruthless winter in New England. It is usually said, given such conditions, that only the skiers are happy. But this winter, even ardent skiers are yearning for spring. And the plow drivers – bless them for having not yet slipped a gear.

In Attleboro, the potholes on Thacher street below the railroad bridge are of a particularly vicious nature. Multitudinous, cavernous, sharp-edged, a recipe for bone-jarring, heart-stopping whacks which test the vehicle’s suspension and the owner’s wallet.  More experienced drivers are aware of the danger, slowing to avoid these chasms . The younger ones, perhaps, with some life lessons to learn.

Persistent winds across the Locust Valley, or in opposite direction sweeping Leach’s meadow, have squeezed Locust Street to barely a lane and a half. Witness to the ineffectiveness of lightweight trucks in achieving curb to curb clearance, and perhaps calling for that Vermont Yankee solution, the snow fence. (Snow fences are temporary structures erected after golfers have abandoned the links and dismantled before the robins return and daffodils bloom). A simple solution, perhaps too old fashioned to appeal.

Repeated plowings from both the street and parking lot sides have raised up a veritable “Mount Bronson” of snow at the Bronson building on Country Street. The “Do Not Enter” sign seems balefully directed at pedestrians, those mothers with strollers and elderly trekkers who are forced to trudge in the busy street. While the city was able to quickly clear the site of the Winter Festival, Mount Bronson has remained impervious for over four weeks. We are apparently following the North Carolina model of snow removal here – “Just wait long enough and it will melt.”

Animals, domestic and wild, are having to persevere. The deep snow pack and tall, plowed banks are channeling dogs and their walkers into hoary canyons. This has the effect of concentrating the inevitable detritus of dogs, only some of which is picked up by diligent human attendants. Ah, well, a problem to be dealt with only if the thaw ever arrives.

Squirrels have been notably absent for weeks, in semi-hibernation as food sources shrivel. Only recently have they slowly reemerged to do battle with the birds twittering about the feeders, scattered black oil sunflower seeds, their common pursuit.   

Who knows where the deer and turkeys have gone? Huddled, perhaps in a deep piney  copse or thick swamp, desperate by now for a buried acorn or tuft of dormant grass or a low-hanging cedar branch on which to nosh. The next few weeks will be tough for them, as the snow pack slowly melts enough to reveal hidden nourishment and, eventually, sustain new growth.

All in all, we should pat ourselves on our collective back. There are millions of people in our broad land who have never had to deal with such a winter. But we have done it, with some aplomb. Residents with snow blowers and plow drivers honchoing big machines have done a credible job of keeping an enormous amount of snow at bay. Roofs have been cleaned, fireplugs unburied. Exercise enthusiasts are seen jogging (mostly in the streets), and many have resorted to cross country skiing or snow shoeing as hearty alternatives.

The sun is much stronger now, we can feel it. Higher in the sky, brighter, hotter, closer it seems. Even on frigid days, the ice is beginning to melt from long-encased driveways. We will survive, with stories extolling our strength and virtue to mock our snowbird friends as they make their eventual, fainthearted return. (Although, perhaps not so secretly, we envy them).

Yes, we have been tested and have risen to the challenge. There is only one imperative thing we could have done better.

Let’s clear the sidewalks. Pedestrian lives matter.