Maths+Reflections

//By George Dimitriadis// //Lakeside Secondary College//
 * What is mathematics?**

"Numbers", someone will say, others will reply with, "Doing things with numbers and letters and other symbols," whilst pragmatists will contend that "Mathematics is a necessary tool for day to day living." The philosophers among us will avow that mathematics is replete with drama and emotion and an end in itself. Any functional definition must acknowledge that mathematics is a language comprising an alphabet (numbers, symbols and mathematical operators such as + and x), and rules for sentence construction (mathematical properties such as 1 x 0 = 0). To the non-mathematician, some things seem magic, but to the mathematician, it is simply a case of following the rules. To demonstrate this point, consider the following set of instructions. Think of any number, double it, add fourteen to your result and take half of your answer. Finally, subtract the number you started with. I predict that your answer will be the same as the number of days in one week. It is not a coincidence that the language of mathematics was first known as natural philosophy and practised by philosophers to explain a myriad of observations. Their approach was more akin to describing events rather than adopting symbolic structure. Some extremists among them had even sought to prove the existence of God! Today, their legacy is evident in a branch of mathematics known as mathematical logic, except that verbal descriptions have been replaced by a vast array of symbols anyone skilled in short-hand will drool over! Once we accept that mathematics is a language, the rest easily follows. Just as an infant learns to talk by first mastering basic sounds and phrases, someone studying mathematics will first need to master the building blocks on which mathematics is built. These of course are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the basic operations of addition, subtraction, multiplication and division. However, numbers in themselves are meaningless until we give them meaning, until we give them a purpose and make them work for us to unlock doors to a world of adventure and discovery. Mathematicians use the language of mathematics to produce works of elegance and beauty to rival the great works of Shakespeare. Think of Einstein's formula E=mc^2. This shows the equivalence between the mass (m) of an object and its energy (E), with the square of the speed of light (c^2) holding it all together. Who could have envisaged -certainly not Einstein- that this formula has major applications in nuclear development? Notwithstanding the level of inflexibility a study of mathematics entails, much can be still be done using the tools we already have. When Peter is asked what he did on Sunday morning, he may say, "I read the newspaper." When John, Emeritus Professor of English Literature, is asked the same question, he may respond with, "I perused the daily chronicle of contemporary events." Both Peter and John did the same thing on Sunday morning, but Peter is obviously the clearer of the two thinkers. Mathematics requires us to structure our information clearly, concisely and unambiguously so that another mathematician knows exactly what is happening. For instance, as a child, Karl Friedrich Gauss, a famous mathematician of the nineteenth century, was instructed by his teacher to add the whole numbers from 1 to 100. Dismissing the laborious and time consuming computational process that his class mates sheepishly adopted, Gauss paired numbers from either end of the sequence; 1+100, 2+99, 3+98, ,50+51 to obtain 50 lots of 101, that is, 5050. Gauss avoided considerable computational drudgery and sowed the seeds for what was to become the theory underpinning the sum of an arithmetic series. To learn mathematics is to learn a second language. It may not be easy at first, but the effort is well worth it if you wish to be rewarded with the poetic cadence afforded by proving that the left hand side of the equation is equal to the right hand side.

//By George Dimitriadis// //Lakeside Secondary College//
 * It is not easy to teach mathematics**

Students instinctively recoil in fear, disgust or apathy at the mere mention of the subject. Indeed, a student foolhardy enough to mention to their friends that maths is not that bad -that in fact, they actually like doing Algebra- will quickly find themselves asked to return to the alien planet which spawned them! The following episode remains one of my happier memories as a maths teacher, even eclipsing the usual concerns of being overworked and underpaid. The lesson began with the simple instruction, "I want you to think carefully before replying to the question I will ask you in a moment." I assumed the silence that greeted this introduction was acquiescence. "If I said to you I am a liar-" I continued. "We know you are, sir," Daniel interrupted with a grin. "My maths report proves it." "Daniel, please go to the arena. The lions are waiting for you," I was goaded to reply. This brought appreciative laughter and cries of "Go, sir!" I tried again."If I said to you I am a liar, am I telling the truth?" "Yes, er, no, I mean yes, I think, maybe," Francesca vacillated, furrowing her brows in concentration. "You've certainly made up your mind, haven't you, Francesca?" I asked wryly. "Anyone else?" I asked as I scanned the room, eyebrows raised. Helda offered a critical response. "If you are a liar, then your statement ‘I am liar’ is not true. This means 'I am a liar' is really ‘I am not a liar'. Therefore you must be telling the truth." A round of applause acknowledged the alleged success of Helda's Socratic Method. Wendy, however, was pensive. She stood up and proposed a corollary to Helda's analysis. "But if Sir is now telling the truth, then it was true when he said ‘I am a liar’ and we're then back where we started." Not the clearest of thinkers at the best of times, Jimmy decided to forfeit his membership of the society of natural philosophers by claiming battle fatigue. "I have a headache, sir. Helda and Wendy are claiming truths are lies and lies are truths. Something's wrong somewhere." He rested his head resignedly on the desk. "Exactly, Jimmy," I agreed consolingly. "We are chasing our proverbial tail." I tried to offer clarification. "What we have is a paradox, a statement that contradicts itself. Here's one you may already know. The irresistible force meets the immovable object." Enlightened, Stavros called out, "I know that one. It's like your big brother threatening you to mow the lawn just when you are getting ready to watch the footy Grand Final on TV." "I'm not scared of my brother," Chris boasted gratuitously. He was rebuked by calls of ‘liar’ by his friends. "Anyhow," I continued, "this leads us to a series of paradoxes first proposed by the Greek philosopher Zeno, born in Italy about two and a half thousand years ago." "Hey!" Jimmy protested emphatically as he raised his head from the desk, "How could he be Greek if he was born in Italy?" I tried unsuccessfully to clarify the situation. "He was born in Southern Italy, which at the time was a Greek colony." "Now you say Italy is Greece and Greece is Italy," Jimmy wailed before lowering his head again onto the desk. "That is a paradox in itself," Janice remarked proudly, believing her comment was an epiphany of sorts. "Okay, now I need an assistant," I stated. Steve's hand was up first. "Good. Steve, go to the corner. Now, I'm going to prove to all of you that Steve is incapable of walking to the other side of the room." "Are you going to break his legs, sir?" Alex called out hopefully. "The room is about eight metres in length, right?" I asked, soliciting agreement. I waited for nodding heads before continuing. "Before Steve can reach the other side, he must first reach the four metre half way point, correct?" More head shaking followed. "But before he reaches half way, Steve has to make it a quarter of the way across, which is two metres from the start. Are you still with me?" I wanted to know. All heads were motionless but I knew by the dazed expression on every face that brains were thinking. "And to make it a quarter of the way through, Steve must first walk one metre to reach one eighth of the required distance, but before that he must get to one sixteenth of the way, and so on." I paused to catch my breath and to allow my minions to assimilate the significance of the vulgar fractions assailing them. "We are halving an infinite number of times," I stated. Now the punch line. "And since infinity never finishes, Steve will never be able to take that first step." A defiant Steve now pretended that an invisible force was preventing him from taking a step forward. Encouraged by class laughter, he continued his portrayal of a victim attempting to break free from leg shackles. Finally, Yao encapsulated everyone's puzzlement by exclaiming, "That's weird!" To allay her concerns I asked, "There is a flaw in Zeno's method of bisection. Can someone suggest what it is?" "What does bisection mean?" Janice asked. "Maybe they didn't have fractions in ancient times," Lino offered simplistically. "It depends on the units they used," was another naive hypothesis. In exasperation and to obtain absolution for my incompetent teaching I pleaded, "Helda, what do you think?" "If you halve something an infinite number of times, sooner or later you will get to the end," she again argued convincingly. Her sophistry was persuasive but incorrect. Jimmy raised his head and completed the puzzle. "When you add fractions together, you always get an answer." "Precisely," I responded. "Zeno assumed that the sum of an infinite sequence, in our case 4, 2, 1, and so on, is limitless, but in fact, it has a sum of 8, which is the length of our room." To visualise the idea of a limit, I got each student to measure a length from one wall and to stand on the chalk line marking their endpoint. Steve measured 4 metres, Helda was assigned 2 metres, and so on. The seventh person, Jimmy, took his position about six centimetres from his neighbour, and the eighth person stood no more than two centimetres from Jimmy. In fact, most of the students were required to be so close together that they believed that they could physically coalesce if they pressed against each other hard enough, which they tried to do!. Some even pretended they were playing a game of Twister! At one point the jostling created a snapshot moment when the group resembled Joe Rosenthal's famous World War 2 photo, Raising the Flag On Iwo Jima. Laughing, the contorted bodies collapsed to the ground. Just as the lesson had begun with my simple instruction, it now ended with my simple instruction, "Thank you." Only this time it was delivered emotionally.

//By George Dimitriadis// //Lakeside Secondary College//
 * Student Mathematical Engagement Via Chopsticks!**

We were drinking coffee in the staffroom amidst a cornucopia of gossip, rumour and regrets. We were also immune to the cacophony of sounds authored by a phalanx of teachers dispensing their professional duties. Tom was passionately explaining some aspect of psychology. "Ultimately, our behaviour as teachers is a self-fulfilling prophecy. A classic study of Maslow's Hierarchy Of Needs." "Huh?" I teased. Tom teaches psychology and, when excited, he is prone to erudition. The bell sounded, reminding me of the unusual lesson I was about to give. "Good luck," Tom called out as I walked towards the door. Twenty pairs of eyes stared intently at me the moment I entered the classroom. I was dressed as a chef, complete with a chef's hat and neckerchief. A Fu Manchu moustache and long fake nails completed my persona. Without comment I unfolded a tablecloth that I was holding, placed it on the floor and with a oriental accent I began chanting, Ad Infinitum plus one,Ummm, Ad Infinitum plus one, Ummm.... Without warning I thrust both hands in my pockets, grabbed their contents and tossed a myriad of bamboo chopsticks high in the air. Mesmerised, all eyes followed the paths of the chopsticks as they performed aerial gymnastics. "Me much need now two volunteers," I requested with my assumed accent, at the same time gesturing with both of my index fingers. All hands shot up. Everyone wanted a role in my histrionics. "Vely vely good. Me choose Steve and Rosa. Steve, you count all chopsticks and tell me number. Rosa, please tell me chopstick numbers how much land on the tablecloth yellow stripes." Steve and Rosa arranged themselves around the tablecloth as if they were having a picnic. Steve opted for the brute force approach. He thrust a finger at every stick and grunted 1, 2, 3,. Rosa displayed finesse. Only the barest movement of her eyes betrayed her purpose. "Thirty-five, honourable one," Steve said, keeping up the spirit of the occasion. "I counted twenty-three, sir," Rosa added. Both returned to their seats. "You both top students," I praised them and I traced a tick symbol in the air with my long fingernails. "Now I have secret," I revealed. "Me go to Chinese restaurant last night. Made me think too much, eh?" "Do you often do that?" Jimmy asked. "Me go to restaurant many times, Jimmyboy" I informed him. "No, sir, I mean do you often think?" "It is old Chinese saying, he who mocks teacher fails subject," I said, maintaining the accent. "How did eating in a restaurant last night get you thinking?" prompted Helda, the most incisive thinker of the group. She was assisted in her inquisition by Teng who remarked, "What's that got to do with maths, anyway?" "Confucius say, he who is patient will inherit egg," I replied. "Restaurant tablecloth similar to this one. Length of chopstick same as width between stripes. Now you understand, eh? Just as magpie fly free when libellated, I libellate chopsticks to use for you now." I quelled several accusative cries of "thief" and begged to be allowed to continue my discourse uninterrupted. "Nearly 300 years, maybe little more, French mathematician, very, very clever man, his name Comte de Buffon, found very, very clever way to estimate pi. You know pi, the number 3.14159. Go forever, this funny pi number, right?" I said, teasing my moustache between my fingers. Jimmy raised his hand. "Sir, do you mean that a buffoon became famous by playing with chopsticks in a Chinese restaurant in France 300 years ago?" I knew the aftershock was imminent. And then it came as Jimmy added pityingly, "Sir, don't you know that stealing chopsticks is wrong?" "You please, Jimminy cricket, say of clever man his name is pronounced Bew-fon," I said, ignoring his allegations of criminality, insanity and the historical anachronism. "Now, please everyone you listen. Monsieur Buffon say that if you divide twice the total number of chopsticks by chopsticks that fall on stripe, it will give you goodly approximation to pi." I grabbed an abacus that I also had in the bag. "Me have 35 chopsticks and 23 of them fall on stripes. So ratio is 70/23 or 3.04. This is good start. If use more sticks, must get better answer, not so?" "It's a coincidence", Francesca decided. " You want maybe try again?" I challenged. "Do another go?" More trials were conducted with similar results. There were accusations of skulduggery. Helda finally conceded. "What's the trick?" she asked. " Honourable ancestors tell me mantra you heard me say before!" I replied. "Ad Infinitum plus one, Ummm. It is much magic." Helda placed a hand over her mouth to suppress laughter, but the action was an invitation to the rest of the class. Order was eventually restored. "Father of Confucius say he who no eat has empty stomach. Now we order Chinese?" I took of my chef's hat, held in upside down and walked between the tables calling out like a town crier... "Show me money. You no pay, you no eat! Chinese yum yum, ..."

//By George Dimitriadis// //Lakeside Secondary College//
 * Can You Teach Maths By Going On An Excursion?**

"How can I teach mathematical probability?" I reflected as I entered the classroom. I moved my chair from behind the desk so that it faced the class, and I sat down without talking. My students soon stopped their conversations and looked on with interest. "I'm driving to school," I said as I flailed my arms and legs in fair imitation of how Marcel Marceau might behave if he was high on caffeine. At the same time I released childish car sounds that should only be used at a kindergarten playgroup. "Ahead of me are traffic lights that stay green for thirty seconds and show red for sixty seconds." I avoided an imaginary collision by thrusting my leg forward, imitating a driver braking and the car screeching to a halt. "Did you crash, sir?" Janice enquired in an effort to appease me. "Missed it by that much!" I replied, mimicking Maxwell Smart's catchphrase. Janice turned to Julia and remarked, "There but for the grace of God go I." Ignoring Janice's idiom, I asked, "What is the probability that I have to stop?" "Zero, because you never stop," Stavros advised, giving the thumbs down to my histrionics. "The probability is two thirds," Lino proposed, "because the lights are red for sixty of the ninety seconds, and sixty over ninety is two thirds." Yao raised her hand to disagree. "I think it's one half because at any given time the light is going to be either red or green," she correctly concluded. "Can we go on an excursion?" Jimmy asked excitedly. "We can learn more about probability that way," he lied. The truism that excursions are synonymous with self flagellation is a tenet in the teaching profession, but I agreed to Jimmy's request. "When is it?" the Principal wanted to know. "The Ides Of March," I replied, "I'm taking my group to the research and development section of ProtoSoft. And I'll try to stay out of trouble," I added quickly. " Your career is ephemeral. Even if something goes wrong, it won't matter," he retorted with a trace of a smile. "Et tu, Brute? You're a real Job's comforter," I complained. The bus trip on the morning of March 15 was uneventful except for the singing that inexplicably irked the bus driver. Exasperated, he finally turned his head whilst driving and not too kindly said, "Sir, please stop singing! Can't you behave like your students?" At ProtoSoft we were welcomed by Bill Stranks, Chief Executive Officer, who ushered us through tight security along an air-conditioned passageway. On either side were glass panelled rooms inside which Bill Gates clones gazed at whiteboards smeared with indecipherable mathematical calculations, vulgar fractions, circuit diagrams and other questionable symbols that are best hidden from polite society. "We are exploring Combinatorics, Network Decision Making and Conditional Probability," Mr Stranks informed us as we walked. "We have developed a level of data security not thought possible, therefore we take security precautions very seriously here." "Do you mean you've found better encryption methods?" I asked. "Yes," he replied as we entered a large area filled with technological paraphernalia. "Our algorithm will be a hacker's worst nightmare. Up to a googolplex of operations will be required from them to be successful; an impossible hurdle to overcome." "These are working prototypes," he continued, waving his hands expansively, "please feel free to test them." Like children opening presents on Christmas morning, everyone except for Jimmy began to play with mechanical robots and with machines sporting strange protuberances and appendages. In fact, Jimmy was uncharacteristically keeping to himself. I was attracted by a music synthesiser that creates sounds using random numbers but I abandoned the device after my students commented that my compositions were nothing short of caterwaul. At one stage, Jimmy edged close to a door emblazoned with the warning, STRICTLY NO ADMITTANCE. "You can't go in there," Mr Stranks warned with authority, but after reflecting for a moment he unlocked the door, entered the room and reappeared pushing a trolley on which rested a monitor, a keyboard and apparatus resembling the innards of a computer. "This is a working model of ProtoSoft's algorithm," he revealed proudly. "It simulates the efforts of a hacker who must enter sixty-four characters in the correct order to break the code." We were all listening attentively. "It's been running non-stop for 5 years, trying random combinations and decryption methods. So far it has not found the correct sequence, and it won't." "What is the correct code?" Helga asked. "We don't know and we don't want to know. It will compromise the validity of the experiment. A computer generated a random sequence which is kept in the safe." "Can we try to break the code?" a few students eagerly asked. "Why not?" was the smug reply. "It's impossible to crack." And it appeared to be so. As each student confidently entered a combination of characters, the computer announced failure by the message, "No way, mate!" and a snippet from the tune, “Waltzing Matilda". At first, Mr Stranks sardonically called out, "Sorry" or "Bad luck", but eventually he tired of the proceedings, ennui set in and he finally lapsed into complacency, staring at nothing. He did not notice when it was Jimmy's turn and he did not witness Jimmy's fingers move at lightning speed across the keyboard. When Jimmy finished there was deafening silence. "What happened?" I finally asked, dumbfounded. As if awakened from a trance, Mr Stranks falteringly replied with, "It must be a computer malfunction." He checked connections, reset the system and randomly entered sixty-four characters. His reward? "No way, mate!" and "Once a jolly swagman...." "Please," he motioned to Jimmy, "can you try again?" Jimmy was eager to oblige. His fingers again expertly worked the keyboard and for the second time the result was a blank screen and no hint of a swagman. We left soon afterward, leaving a bewildered CEO in the middle of his soliloquy, repeating the dirge, "Not possible, oh, not possible, oh." Did Jimmy change the course of History? Or at least, did he slow it to a walking pace? I think not. In fact, I am certain of it. The following day I found a note on my desk. Dear Sir, I didnt do nothing wrong yesterday. When we were supposed to be playing with the electronic stuff I saw a room with its door open and I noticed there was an open safe near the door. After all the things Mr Stranks was saying about security I went to close it but I took a quick look inside, all there was inside was a peace of paper with A3&e@#0GeT8*U+%N h12O$OPU 9^#G5WQ2 65M )Y(10975Ha6349Dg63/T on it. Then I shut the safe and came back and then when I was asked to try the machine I remembered. Honest sir I didn't touch or see anything. Please tell him I'm really really sorry but he should not leave doors and safes unlocked because its not safe. Thank you.

I laughed and was on the verge of calling Bill "Security" Stranks to let him know all is well. But wait! First I must ask Jimmy to tell me how he memorised so much in so short a time.