Mr. Trusz Proves that 0 = 1

When I was a youngster in the MSAD #64 school system, I dreamt of the day when I would at long last be challenged in my classes.  I was the kind of kid who listened to his classmates and their struggles with the material being presented and wondered what it would be like to need actually to do the homework assigned to learn the material—I usually did mine just because it gave me something to occupy my time.  I admit readily that I was one of those smart kids who ruined the curve for the others, the kid for whom most everything came very easily.  I was not “the norm” in that regard.  There was no “gifted and talented” program at that level (and the one that had existed at the middle school level was not very well developed), only a tiny number of AP classes, and the “college prep” classes weren’t all that taxing either.

I can tell you though, from experience as both a student and a teacher, that when you raise the bar in educational settings, the students rise to meet it; if you lower it, they do the same.

It would be wrong of me to brush with a broad and negative brush stroke the entire school system.  There were a few bright spots in my formative education.  Charlene Farnham and Lucille Koncinsky, about whom I have previously written, remain in my mind two very positive role models for my own teaching.  They exemplified caring, and love of their craft; and, that translated directly into excellent opportunities for learning for the students in their care.

Another of my teachers who merits my eternal gratitude is Mr. Trusz, a masterful mathematician and general unrelenting techie of the first order.  When I first met his son, Daniel, as a classmate, I was on occasion invited to his house to play from time to time.  Daniel’s dad, Richard “Dick” Trusz had “everything”.  He had a computer that accepted not only floppy disks of varying sizes, but could also read an actual cassette tape and store information on it.  The computer functioned in “basic language” and you could write code for it so that it would draw a three-dimensional box on the screen, or even make a string of text fly up and across the amber-colored screen, repeating itself endlessly.  It had a printer that took paper that was really one long sheet with perforations every eleven inches.  It was magical—or would have seemed so if one considers that those early Tandy machines weren’t even as sophisticated as the little Apple Macintosh that I would buy myself when I got to college a half dozen years later.

Mr. Trusz had joined the CHS teaching faculty in 1969, only a couple of years after fire had ravaged and destroyed the East Corinth Academy.  (Richard Grant was the school superintendent then, and the new school replacing the old Academy had been officially named “Central High School” two years earlier in March.)  Whoever accepted his application and gave him the job did the school and all those who attended it a tremendous favor.

Mr. Trusz was in charge of a small group of us for an algebra class.  We met each morning in a tiny room off the library with no windows.  He was the kind of teacher who would not accept work completed on a calculator—you had to show the steps that you took to get from the problem to the solution.  He felt strongly that we needed to learn to reason mathematically as much as we needed to arrive at a correct answer.  He also expressed a desire to see textbooks where we weren’t always given all of the information needed to solve the problem, to see if we could find the solution on our own using reasoning and logic.  (I am not sure such a book exists even today.  Books that encourage synthetic reason, such as Mr. Trusz desired, are certainly not available in the foreign languages that I teach.)

Mr. Trusz had a very dry sense of humor.  The first Gulf War was raging in Kuwait.  One day, when a loud noise came from outside the classroom, he turned to us and asked, “What was that?  A SCUD Missile?”  Mr. Trusz was also fond of proving to us that he had in fact discovered the way to become very wealthy.  He could get something from nothing, he told us, and went on to prove that in fact zero was the equivalent of one.  His mathematical proof was unassailable, but all these years later, I was unable to recall it.  I knew I had to contact my old friend Daniel.  “Okay…”, says son Daniel, “My dad gave me the proof. Here it is:”

Given: ‘a’ and ‘b’ are integers such that ‘a = b+1′

Prove: ’0 = 1′

————————————————–

a = b+1 | (given)

(if a= b+1 then a-b =1) | (implication)

a(a-b) = (b+1)(a-b) | (multiplication prop)

{multiply both sides of equation by (a-b) }

a^2 – ab = (b+1)a + (b+1)(-b) | (distributive prop)

a^2 – ab = ab + a – b^2 – b | (distributive prop)

-a = -a | (reflexive prop)

a^2 – ab – a = ab – b^2 – b | (subtraction prop)

a( a-b-1) = b(a-b-1) | (reverse distributive prop)

a = b | (division prop)

{divide both sides of equation by (a-b-1)

a = b+1 | ( given)

a = a +1 | (substitution prop)

a = a | (reflexive prop)

0 = 1 | (subtraction prop)

Thus 0=1.

————————————————–

Now, I didn’t share this proof with you so that you could take it with you the next time you were at the bank disputing an overdraft fee… “But how could I have nothing in the account?  I have at least one dollar left!”  I shared this proof with you to help you see what I saw as a student in Mr. Trusz’s class—there is joy to be found in all things.  Even algebra problems!

All said, Mr. Trusz (whose wife was our school nurse) was also a pretty wonderful guy.  He served as the faculty mentor for our school math team.  He taught us leadership—encouraging some of the older boys to take charge of the club’s meetings and practices, imploring more of the female students in his classes to join the group.  He was convinced that math wasn’t just for boys, which was pretty radical thinking at the time.

Most importantly for me, though, was the willingness Dick Trusz displayed to go the extra mile to help those of us in need.  While I was not in need of “remedial” lessons of any kind, I was at one point enrolled in a math class with Mr. Johnson.  I was doing marvelously well, but was so bored with the work and was likely contemplating mischief, as I am sure Mr. Johnson could tell.  Mr. Trusz offered to step in and create a “class for one”, just for me.  He saved me, he truly did.  Finally a challenge.

While Mr. Johnson’s class met across the hall, Mr. Trusz and I met in what I am sure was but a broom closet only hours before our first meeting.  As others sat there, coming to terms with the fact that no matter what you do, a-squared plus b-squared will always equal c-squared, Mr. Trusz and I hammered out curves and lines, the areas beneath them, and also the likelihood that statistically-speaking we were right in our estimations to between 3 and five points.  I alone consumed an entire year’s worth of free periods for Mr. Trusz and I am not certain that I ever said thank you properly.  (I do know that he knows of my admiration for him since I have shared that with Daniel several times over the years.)

What I am not advocating here is that “good teachers” always sacrifice their planning periods so that they can take on additional unpaid duties on behalf of the school.  I teach, and I loathe the “good dooby-syndrome” in some of my colleagues.  No, I believe in fair pay and fantastic administrative support as a way to encourage excellence in teaching.  If Mr. Trusz wasn’t compensated for his time with me and I learn of it, I have a list of names of those people in the school’s administration who should be very ashamed of themselves!

What was extraordinary in this case, in my eyes, is that Mr. Trusz saw a student who was languishing in a system that was not set up to handle more than the average student, and he took the steps necessary to find a solution.  He was a problem-solver.  He could have very easily have left me in that other class.  No one would have blamed him.  He could have also have set me up in that other room to work at my own pace, on the home-schooled model, checking in on me only on occasion.  I would have been fine with that solution too.  Rather, Mr. Trusz took me under his wing and guided me through an entirely different curriculum.

Mr. Johnson was no dummy.  He had also taught my Mom and later my brother, and knew a bit about me before I got there.  I think he sensed that since I was not only smart, but also creative—which is not always a nice combination, mind you, if you should be on the receiving end—that I might in the end be a morale killer in the room more than an aide.  Moreover, he had likely already spoken to Mrs. Campbell, the study hall monitor, who had gained a certain level of notoriety for telling students, “If you can’t behave, I am going to send you to the office so FAST!”  One day, as she shouted her mantra for the umpteenth time, I asked her right out right, in a smart-assed tone, “Just how long would it take to get to that office exactly?”  Damned if she didn’t give me the chance to find out!  Of course when I arrived at the office and told Mrs. Wiggins the reason for my visit, she thought I was clearly joking.  When Mrs. Campbell confirmed over the inter-com that indeed she had sent me there, I was quickly ushered in to the guidance office where Mrs. K. found me another class to fill my free time.  Yes, bless Mr. Johnson for having spoken with Mr. Trusz about my discontentment in his class.  Being challenged in that one 50-minute period of the day was just what I needed.

What made Mr. Trusz an excellent teacher was the fact that he knew his craft very well, and also cared about the pedagogy and didactics behind it.  He told me once that it did him no good to have all this knowledge about mathematics if he couldn’t share it with others, just like me.  He also believed as I do that you can bring all the tools to the workshop, give all the instruction on the method of performance, but you can’t do the learning for the student.  They have to be willing to do that themselves, and if you find one who is willing to do more, for heaven’s sake DON’T DISCOURAGE THEM!  In my own teaching, when I see a student who is even remotely bored or who seems ahead of his/her classmates, I always take that student aside and offer to provide materials which go above and beyond what the others are working on—I never let students languish in boredom.  If I had a small room the size of a broom closet and free periods to sacrifice for a gifted learner, I too would find a small chalk board and set up a curriculum for one.  Honoring those who love to learn, that is what I learned most from Mr. Trusz.  There is joy to be found in all things, and great teachers help us to see that.

Salads for my friend, Debra

I spent a leisurely afternoon in the Adirondak chair on my front lawn, reading a new recipe book that my friend Kristin had recommended to me this spring.  I was contemplating getting in to the kitchen to whip something yummy up, but the lure of the sunshine was too great and I remained seated.

As I pondered some new dishes, I turned to thinking about a couple of friends of mine, now living in Guatemala.  Debra and I had previously worked together on her study of Spanish and became good friends over the years.  She is the one who inspired my version of “Hilachas Guatemaltecas”, her favorite dish of her adopted land.

Debra is a self-admitted novice in the kitchen, and after a pleasant gathering among friends last summer, her husband Earl asked if I would share a couple of my recipes with him.  Without hesitation, I wrote out the recipe to a pasta salad he had enjoyed as well as a recipe for a salad dressing which is just delicious and not at all over powering—both of these recipes would be great this time of year now that the summer heats are on the way, and farmer’s markets are back in town.

Gregory and I had been to a restaurant a few months prior to this party that I mention and they were serving a chicken salad sandwich “for adults”.  I thought to myself, “This would be great as a pasta salad too!”  So here is how I created the salad for the party.

Waldorf Pasta Salad

 

Chicken, chopped, about 1 cup

cherry or grape tomatoes cut in half (about 1 cup)

red grapes cut in half (about 1 cup)

apple chunks (1 large apple or two medium apples, peeled and chopped)

1 stick of celery, chopped

Pasta (about 1 cup)–any small pasta will do.  We like Bowtie pasta, so I use that.  Just cook up the pasta, run it under cold water to cool it off, drain, and stir in to the rest of the mix.

Add Ranch dressing to taste  (I like mine a bit more moist)

For the sandwich version, eliminate the pasta and use a little less dressing or substitute the Ranch for mayo.  Serve on good sour dough or other bread of your liking.

I also love a good salad dressing for greens, and our friend Lyn had asked me to make a dressing for the party.  Sometimes, I think that vinegar is too over powering for a salad, so I make a vinaigrette (which is at its base an acid and an oil) which uses lemon juice instead

Lemon Vinaigrette Dressing

 

1 teaspoon of honey (I often make mine without the honey but I had some I was trying to use up)

1 teaspoon of Dijon mustard (don’t use a traditional American yellow mustard like French’s.  I tried that once to disastrous results)

juice from one lemon (I often use a bottled lemon juice instead of being bothered with a fresh one)

roughly the same amount of olive oil

1/4 to 1/2 teaspoon of dried lavender (you could substitute the same amount of an Italian seasoning mix, or if you go to a spice store, pick up some Herbes de Provence which has lavender in it)

Shake well and serve

Shadow Race

Shadow Race

Every time I’ve raced my shadow

When the sun was at my back,

It always ran ahead of me,

Always got the best of me.

But every time I’ve raced my shadow

When my face was toward the sun,

I won.

–Shel Silverstein

A Light in the Attic

A few years ago, I decided to digitize my family’s old photos.  Photos are precious links to the past and to a host of memories, almost always good since people rarely wasted film (or the money to develop it) on the family’s hard times.  My hope for the project was that I would be able to share these instants in time more easily with my siblings if anything ever happened to my parents, and also create a record of them lest something terrible ever happen to my parent’s home.  My mother was fortunate enough, when her two beloved aunts passed away each in their own time, to have inherited their family albums, as well as play the role of safe-keeper of our own family’s memory books.  Our recent past, otherwise said, is well documented, or at least better than some.

Scanning photos takes a fair amount of time, but is an activity which bears its own reward.  For one, it is a history lesson.  My brother was born in the black and white era of the brownie camera, the nifty little brown box which captured snipets of time for so many American families just like mine.  My sister and I came along in the next generation of photography, growing up with the 126 cameras upon which four-sided “flash bulbs” could be attached, greatly enhancing indoor shots.  As we neared the end of our primary school days, we as a family transitioned to the polaroid where the photos came out of the front of the apparatus and developed themselves right before your eyes, and later graduated to the 33-mm camera that had taken control of the photography market, becoming the standard for most digital cameras too.

Among all of these images, I stumbled across snapshots of our family trip to the Niagara Falls, ones when we visited an aunt in Connecticut and others in Florida.  There were pictures of the birthday cakes that Grammy used to decorate for us each year.  (My mother insisted that just because my sister and I were twins didn’t mean that we had to “share” a cake—Grammy made one for each of us, in separate themes).  As my father added on to the garage, or we bought a new truck, Mom was there to snap a picture of it, her own chronological paper keepsake.

After my mother’s passing, I dug a bit deeper in to the older albums she had tucked away at the bottom of the pile.  In the older black and white albums I discovered, ones which I don’t recall ever having seen, my mother’s later childhood is reborn.  In some of the neatly arrayed images, displayed on black paper with little paper holders at each corner, my then teenaged mother sits astride her white horse Snowball.  She rode snowball in the Olde Home Day parade that year before she graduated high school.  Amid crowds of cheering people, she and friend Wendy, also a lover of horses, proudly displayed their equine friends to the rest of the town.  Since there is photographic evidence of the event, it can be assumed that Grammy and Grampy were there on the sidelines cheering the girls on in their saddles.

I have long held a special place in my heart for black and white photography.  There are a range of emotions which display themselves richly when you take away some of the “color” noise of today’s clichés.  Moreover, when I was in the seventh grade, I learned about photography and the art of darkroom printing and more from my science teacher, Derwin Emerson.  He was a wonderful teacher who allowed me for the first time to see that my personal gifts and talents were not only valuable and exploitable, but worthy of dreams—dreams which I should be following.  Mr. Emerson repeated often that in reality, the images caught on film were just shadows etched in to silver in that instant when the light hit it.  Just fancy little shadows.

I, along with one of the girls in the class ahead of mine, worked many afternoons after classes had let out to take pictures of the school’s events for the yearbooks we eventually produced.  When it was a sporting event, I would focus the camera and set the aperture settings on a spot ahead of the athletes in anticipation of their arrival in the frame.  If the shot I was setting up were a portrait, I would try to get the light to conform to my needs, backlighting the staged area so that the person’s face and features were clear and bright.  I was, in essence, always trying to get under a shadow if the sun were too bright, or run ahead of one if the room were already too dark.  In most cases, in this shadow race, I was victorious.  On those occasions when I would lose, I could always get a rematch when I made my way in to the dark room, lightening or darkening an image as necessary with the chemical processes Mr. Emerson had taught me.

I celebrated my thirty-ninth birthday this spring.  Among the comments from friends and family, pointing out that there is a lot of white in my beard now, I was reminded that quite a lot of time had passed since Mr. Emerson introduced me to the darkroom laboratory at the newly constructed Central Middle School; that even more time had elapsed since Mom was caught on film carrying her large round Christmas box, containing the new jacket her Father had so proudly bought for her.

I was informed that I am almost officially a part of what is known as “Middle Age”—that period in your life when your memories seem to become more precious to you than the dreams you once chased.  On my birthday, I looked at a few more of these old snapshots sitting on my desk waiting to be scanned, and I saw family gatherings and other happy occasions when those people I cherished the most were still with us.  I closed my eyes to recollect those moments before and after the photographer intervened.  As I pondered, one by one the people started dimming.  They grew pale and then transparent, and finally they disappeared.  My Mom’s father disappeared first and then his sisters, next to their luxurious new cars, clad in long cloaks still wearing gloves and kerchiefs over their hair.  Slowly, the images themselves seem to have become less and less populated, until the only one left was that little boy I used to be, standing there all alone.  Not really knowing what has become of my cousins with whom we gathered on Christmas each year, or their children now on the cusp of adulthood, I realized that as I have gotten older, I have said goodbye to more of those closest to me than I have perhaps said hello to new faces and personalities.  That link, my connection, with a past was receding beyond memory as I contemplated the first half of my life.  I am grateful for those photos which help me remain in touch with those people who were there for me in my formative years.

I resolved at that moment to finish my photo project, and encourage you to do the same.  But don’t just scan those pictures to your computer’s hard drive where they will sit precariously, fearing some snafu which might wipe them out entirely.  Make a cd-rom of the images and share them with your local historical society so that a record of them can be kept there as well—help to cement further your own town’s past by preserving your personal and family story for the future.  Be the kind of person who runs toward the sun and wins the shadow race that is the family album.

*If you are from Corinth, Maine, as I once was, then contact the Corinth Historical Society, P.O. Box 541, Corinth, Maine  04427, or write to me and I will help facilitate your gift to them.