A.P. Keaveney taught mathematics at Castleknock College for several years. He was a regular contributor to "Dublin Opinion".
It is about thirty years to the day since I first laid eyes on Noodle.
He was sitting in the third desk, a bright-eyed, intelligent looking boy in the Third Year Pass Class. I noticed nothing unusual about him; it was not till the end of September that he came fully under my notice.
We were working at algebraic facctors and Noodle gave the square of (a + b) as a2 + b2. I told him this was wrong, explained why it was wrong, and told him it should be a2 + 2ab + b2. The thing didn't seem very reasonnable to Noodle, but he was a nice boy and he took my word for it. Besides, in Noodle's eyes, the difference was not worth all the talk about it. I had only begun teaching and I knew that a boy, however slow, had only to have a point fully explained to him by an intelligent teacher, like myself , and the information was his for all time.
Accordingly, I was surprised to find a few weeks later that the square of (a + b) was again a2 + b2. I explained the whole position ab initio, with the help of a geometrical diagram and by giving simple numerical examples. This time he seemed to understand me fully, but two days later he reverted to type, this time with (x + y).
This made me very interested in Noodle. I regarded him as a' challenge to my powers as a teacher, and I set to work on him. I soon found that my powers as a teacher were by no means what they should be; and when Nooodle left my class for the summer vacaation in June he was hardened in his errors, and his views on all branches of mathematics were the same views that he held when he entered it in September .. He was a nice boy; but I was glad to see him go as he was the symbol of my failure.
Imagine my surprise and disapppointment, on entering the new Third Year Pass the following September, to see before me Noodle again. Not the same Noodle but another, completely differing in physical appearance, but with identical views on arithmetic, algebra and geometry. I was four or five years teaching before the knowwledge slowly seeped in on me (I reefused to accept it for as long as I could) that there would always be a Noodle. And events proved me right. I changed schools a few times but always Noodle was there before me in the new school. Noodle has pursued me for over a quarter of a century, as fresh to-day as the day we started; on me the years have left their mark but Noodle has eternal youth. Every September I look out for him and there he is. I forewarn him of the errors he will make but, with Noodle, forewarned is not forearmed.
Figures mean nothing to Noodle.
He will pay a man over £5,000 as the simple interest on £500 for a couple of years. He is not in the least surrprised when he gets the father's age as seven years and his son's age as eighty-three, or the speed of an exxpress train as three miles per hour. He will make a man walk 8,395 miles per hour, sound barrier or no sound barrier. He can never get to grips with a sum until he has brought the tons to ounces (I speak of pre-metric days) and the £'s to farthings and the miles to inches. Then he stops. Time up.
Let it not be thought that I dislike Noodle or resent him in my class. After my thirty-odd years life would be dark without Noodle. In an Honnours Class a boy will ask a question that I can answer, but Noodle asks such questions that no man-can answer. In this he is a standing challenge to me. For example, when I have finished a long problem in algebra, beginning with, 'Let x equal the father's age', I ask him if he understands how I got the result. Noodle is my testing ground.
"Yes, sir, I understand the end part of it."
"Well, Tom", (or Jim, or Joe, or Bill.
For Noodle's name is legion) "What part of it do you not understand?"
"I don't understand where you got the x. Where did the x come from, sir? "
Now, this is a question that I could never answer, since the answer, if there is one, woufd involve the very same x, a shocking example of arguing in a circle; and the whole thing becomes an endless, cursed regression, like the Problem of Time. In order to get away from this ever-recurring x I tell Noodle, "You can begin by letting x, or y or z equal the father's age, or any letter you like."
"Then all the letters in algebra have the same value, sir?"
"Not necessarily, Tom. You select any letter and let it suffer the same changes as the number you are looking for."
This Shakespearean use of the word 'suffer' was unfortunate as it brought up the whole Problem of Pain.
In geometry, Noodle is out on his own. He can learn by heart the proopositions of Euclid (Now-a-day, gone with the wind). He can do them on the blackboard provided the diagram is lettered exactly as in his book; but change the lettering or a triangle from ABC to ACB and Noodle is floored. But deductions or riders on the proopositions he solves in a flash; everyything is equal to everything else, the lines always bisect the angles and bisect each other. The idea of two -lines intersecting without bisecting each other is repugnant to Noodle. He can't see any difficulty.
I ask the class: "Here is a given straight line and a given point not on the line. Show how to construct a circle of two inches radius passing through the point and touching the line".
Noodle is insulted. He was expectting a problem. His hand goes up like a flash of lightning, his clicking finngers demanding attention.
"Well, Tom, how is it done?"
"You open your compass two inches, sir, and describe the circle touching the line and passing through the point."
"But how do you find the centre of the circle?"
"This is the centre here, wher,e you put the point of the compass."
Noodle cannot see the least diffiiculty in the question; there's nothing to it but to draw the circle and if it does not get the point the first time, just move it over a bit till it does.
The problem of getting a boy to solve a problem where he himself sees no problem, is, for me, a problem that will remain, forever unsolved. In geoometry I cannot grapple with him In geometry I cannot grapple with him because he is intangible. Reasoning with him is as futile as trying to overrtake a person in front of you on a merry-go-round: no matter how fast your hobby-horse goes he is always the same distance ahead of you. In algebra Noodle may ask me imposssible questions, but in geometry he knows all the answers, Noodle loves geometry.
Noodle always reminds me of the great French mathematician, Laplace, because there is such a vast difference between them. Laplace was the author of Mechanique Celeste, a tremendous work of original matheematics which extended and improved on the work of Isaac Newton on the attraction between the heavenly boddies, and who was the discoverer of the immortal equation of the potential which will endure as long as the heavvenly bodies keep to their paths, namely:
d2y d2v d2y
- + - + - = 0
dx2 dy2 dz2
Laplace had social ambitions and requested Napoleon to make him Minister of the Interior. Napoleon, ever desirous of tilling his Court with men of learning, consented to his request. But after six weeks Napoleon had to dismiss him for incompetence. It is Napoleon's verdict on his exxminister that links Noodle with Lapplace. Here is that verdict: "Laplace, a geometer of the first rank, was not slow to show himself an administer worse than mediocre. Laplace did not see any problem in its true perspective, he saw difficulties where none existed, and he carried the idea of the 'infinitely small' into his administration."
Laplace saw difficulties everywhere; Noodle, as a geometer, sees difficulties nowhere. Laplace looked everywhere for subtleties, Noodle for simplicities.
The more I ponder on Noodle, the more I am convinced that Sigmund Freud wrote a lot of piffle on commplexes, especially inferiority complexxes. Noodle has no inferiority complex. He understands mathematics all right, sir, maths is OK with him but he just can't understand what I'm talking about. He thinks that I complicate things too much and that he is the boy who simplifies them. Indeed, he is quite interested in mathematics. Reecently he asked me what was the hardest sum in the whole world and if I could do it. In order to bolster up his morale and to show that the best of us has his limitations, I told him that I couldn't (which is only too true) and that the hardest sums could only be done by machines. Noodle was not surprised.
Noodle is only biding his time till he leaves school and makes plenty of money. One of my greatest Noodles, who left school after failing all examminations has half-a-dozen University graduates working for him. In fact, he would have given me a job in his place some years back only that my degrees did not quite please him. Yes, Noodle is a complete success when he leaves school. He has the lowest handicap in the golf club and can sing a song or do a tum with the best of them; he is hail-fellow-well-met with everybody; he knows barmaids by their Christian name and hotel porters by their nick-names. He always has plenty of money and often stands me a real good dinner, telling me of his latest trip abroad. He gives me tips for the races, advising me to put on a couple of bob as he is putting on a pony. I don't know how much money a pony is, but Noodle does and what's more he has plenty of them. He talks in thousands, which always reminds me of the time when he did not know whether there were four or seven noughts in a thousand.
Recently I had a chat with a downnand-out who lamented bitterly that he did not make better use of his time in school. Thinking of all my Noodles, I said to myself: "Me, too."
When I see how successful Noodle is when he leaves school, I often wonnder if brilliance in mathematics has a stultifying effect on the human mind. I rather think it has. One of the most illustrious mathematicians of modem times, a man of world-wide reputation, submitted to an intelligence test and was rated an imbecile. I myself am not one of the greatest mathematicians, but I am one of the greatest Noodles that ever sat behind thirteen cards at the Bridge table; I look for subtleties everywhere, I see difficulties where none exist. I would make one rotten Minister of the Interior, worse than Laplace.
It has been truly said that each of us is ignorant, only in different subbjects, and when I reflect on all the Noodles of the past as well as, with God's help, the Noodles yet to be, I am convinced that one of the very finest subjects to be ignorant in, is mathematics. A.P.K.