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Mr. Trusz Proves that 0 = 1

May 23, 2012

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.

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