A Mathmas Carol

by Cristina Cidnay

This little tale may scare some or deceive others. Not by the ghosts, spirits or apparitions that may exist in the world (about which I can’t pronounce myself, because I’ve never seen them, and science hasn’t yet proved their existence, so that I could believe even without seeing them), nor by the strange but funny appearance of one of its main characters. Yes because of the terrible subject in question, and yes because of the name of this same character, who seems to inspire a festive and warm time, but who, after all, is related to another one, normal and laborious for the majority of us.

I don’t know if the reader will be part of the majority or the minority. In any case, I want the character I spoke of to vibrate that investigative rope, sometimes loose or broken, to the point where, as if by magic, it returns to its natural state in tune with the world, with nature, with the Self.

Note: Any resemblance to “A Christmas Carol” of a CD author is purely coincidental. Neither I have the art of this author nor he my ingenuity, since we live in different times and our subjects are diverse. Relevance is deserved to the first and acclaimed author.

I

Charles, the tormented tormentor

Charles was a teacher, a math teacher. Yes, that discipline that torments many and only some brings happiness. A happiness that, unlike any other happiness in life, doesn’t attract jealousy or great disputes. A happiness predestined for some, which, in the eyes of others, looks more like a curse. A dull, musty, archaic, and painful happiness moving under a cloak of axioms, mottos, theorems, corollaries, deductions, inductions, symbols, operations, and even contradictions, imagine it yourself… Are we talking about happiness? Even I feel my heart clench, my brain becoming nervous, and a knife tearing at my belly. Hello, Professor Charles! Goodbye, Professor Charles!

‘Fred! Where are you going? The exercise is not over yet!’ shouted the angry teacher.

‘Professor, it’s time. You know, my mother is waiting downstairs.’ said the student, already standing, trying to wriggle out with his best excuse.

‘But let’s not leave the exercise in half,’ insisted Charles, ‘It’s still necessary to determine the intersection point of the two lines.’

‘The system is impossible for me, Professor,’ said Fred, ‘Impossible to continue. I don’t understand anything.’

Don’t understand anything? What is he saying?, thought the teacher. I spent an hour explaining how a system is algebraically and geometrically solved, and he ends it with ‘I don’t understand anything’? There can only be a problem … And the problem is not me, nor the math, he concluded. Another one… I could devote a whole week to him that, in the end, he wouldn’t know much more than at the beginning. Math is truly out of reach for some…

‘But you’ll understand, Fred,’ he sought to reassure himself more than the student. ‘A few more explanations and you’ll get it. Just don’t give up, okay? ‘

Fred nodded and ran for the door, glad to be back in the real, everyday world, carefree, imperfect, and with few rules, only understandable and useful rules, he thought. A colorful, fun world with adventures and emotions. The living room where Professor Charles gave explanations was at the antipodes of this world.

The clock struck five o’clock in the afternoon. Time for one more student.

Ah! Now someone who understands me!, Charles thought, rubbing his hands together happily.

‘Professor Charles, I’ve found the solution to the challenge you posed me,’ said Alex, shining his teacher’s eyes.

It didn’t take much to excite Alex. He had always liked numbers, complicated calculations, and problems. His mind didn’t bother when he plunged into the sea of fractions, powers, or functions that his dear professor Charles presented him. It would be impossible to sink into this sea. His logical depths, with geometric corals and algebraic algae, made him feel like the authentic Captain Nemo, one of his favorite characters in literature.

Too bad not all students are like that … , Charles thought as he watched Alex struggling to solve an equation. Why can’t everyone see what this student sees? Why is it so hard to understand math?

This was the real problem of Professor Charles’ mathematics, yet to be proven, and that often plagued his well-deserved rest.

II

The ghost

That night, a night seemingly like so many, Charles dreamed. It wasn’t a pleasant dream, it wasn’t a nightmare either. Few things scared him, in fact. He considered himself a brave man and never evaded his duties.

In this dream, his student Fred was arguing with his other student Alex. Fred felt slighted by Alex, who insisted that his solution for a math problem was the best. Fred showed him a drawing, without any calculation, that proved the solution of the problem. Alex waved a sheet with various calculations, saying that only that way was possible to prove the result and that Fred’s visual process was inadequate to solve the issue.

Faced with the confusion, Charles intervened and analyzed both resolutions. Clearly, he preferred Alex’s resolution, formal, mathematically correct, and could even simplify it a little and make it, with greater mathematical aesthetics, more concise.

Fred’s resolution shunned the usual kind of resolution, was visual only, and therefore seemed incomplete. Although it did show, to those who carefully analyzed it, the mathematical relationship illustrated in the picture of the problem.

The teacher’s face must have spoken, as Fred turned away from them, looking offended by their facial expressions, shouting ‘Math is stupid, the problem is stupid and you are stupid! Stupidity to the cube!’

To the cube… to the cube… Is this cube yours, Charles? You can’t see me yet… here! Is this cube yours?

Cube … what cube? … What time is it? Charles wondered, still with his eyes closed.

When he finally managed to separate the upper eyelashes from the lower ones, he thought he was seeing in 3D some sheet of an exercise book from one of his students. A cube with a strange character on it. This character resembled the mathematical Pi — the one that is approximately 3.14 — but adorned with a green Santa barret, with a wide white band and a white pompom on the end, and with many yellow stars around it.

Well, something sounded very strange here. This Pi had two eyes, one on each leg of this Greek letter (eyes on the legs is really new!). And that was the only resemblance to a human being. Nonsense!Pi was and always will be a Greek letter and a mathematical symbol. Only that. But it looked human, with two eyes…

This Pi was staring the teacher of this tale, more used to telling than to believing in tales. Pi’s gaze was defiant, though his rosy cheeks might indicate shyness. Faces? What faces? If there is no face, only legs, how can it have faces? thought Charles now. Faces has the cube where it’s sitting.

Pi didn’t speak now. Yeah, there’s no mouth! Just legs. — concluded the teacher. But I thought I heard a voice…

‘Yes, it was me. Santa Pi at your service’ heard Charles.

The teacher saw no movement beyond the gleaming twinkle of Pi’s eyes. How did it speak? Is it any mathematical sound column?

‘I’m not a sound column, okay?’ snapped Santa Pi. ‘I’m a mathematical wizard. You hear me in your head. The same head where you do calculations and solve problems, where your thoughts are born and live. I live there too.’

‘You’re an idea then…’ Charles answered aloud, feeling embarrassed talking to himself. He had no such habit. He kept his ideas well to himself.

‘You can talk to me out loud, Charles. No problem. It’s a great way to reflect on your ideas. And right now, about me’ Santa Pi soothed him.

‘Right’, thought Charles, speaking at last to this imaginary being. ‘You’re an idea. In my head. I even find it a funny idea. And what brings you here, Pi?’ He continued, now amused and in a good mood.

‘SANTA Pi, that’s my name. I’m a mathematical wizard, more than a number. Although the Pi number is already quite important in itself’, corrected the wizard, and went on, in a gravelly thoughtful tone.’ You must take me as a ghost of your past life.’

‘Ah, Santa Pi… as Santa… Santa Claus,’ concluded Charles. “But he’s not a ghost, as far as I know.” He doesn’t go around scaring young children, but giving them presents.’

‘Three truths and a note: you’re not a little child. And now three more truths: I, Santa Pi, the mathematical wizard, will have to frighten you like a ghost and lead you to review your past so that you can understand the present and prepare the future’, the wizard solemnly revealed. And he added: ‘The spirit of the present and the apparition of the future will also come to visit you. Listen to me and listen to them in your thoughts, for we will give you the opportunity and hope you need to be better.’

‘A ghost… a spirit… an apparition… to be better?’ asked Charles incredulously. ‘Better in what way?’

At that very moment Charles’s room began to spin around him, seeming to become a downward spiral paper scenario. Only he and the wizard sitting on the cube remained motionless. Charles began to hear Santa Pi recite his own decimal places, 3… point… one… four… one… five… nine… two… six… three… five… eight… nine… seven… while, in increasing speed, both began to twirl following the downward spiral.

The landing was theatrical. Charles had fallen on top of Santa Pi, crushing the deformed cube as well.

‘Charles, get off me!’ shouted the wizard. ‘You’re going to ruin my pompom.’

The teacher, still seeing the yellow stars of Santa Pi swirling around in his head, could only say, ‘Where are we?’

Santa Pi, already free of the teacher, whistled and the yellow stars returned to their usual place around his green and white barret.

Charles heard, then, the sound of chalk writing on a blackboard.

‘Oh! That’s me’, he exclaimed, ‘And these students… they don’t look like the ones I have now…’

‘Of course not, Charles. Today you no longer write with chalk on a blackboard. Even though you miss that time, I know,’ observed Santa Pi,’ And indeed, there is little difference between this classroom and yours at present.’

‘No, I find it a lot different. These students are much quieter and more attentive. They want to learn. Not like today’’ added the professor.

‘Yes, quieter, I just don’t know if they are more attentive and interested, as you say. At least not all’, continued Santa Pi. ‘Watch that girl playing with the pencil. Is she listening to you? Or that boy, who seems to agree with you, with his head? ‘

‘This one is for sure. There’s no doubt’, said Charles.

At that very moment, Professor Charles of those past times questioned precisely this boy. Soon to be very angry. ‘How is it possible, David, that you don’t know how to answer correctly? Haven’t you been listening to me? ‘ Yes, the boy was listening to him, or at least was watching him speak, if he could give an honest answer. But he was thinking about the characters in a tale he’d like to write, and he’d already been lost in the teacher’s explanation a good ten minutes ago. He thought it was a quality, to look alert when he was really daydreaming. It didn’t cause him any trouble, but neither did it increase his wisdom, a truth he could not deny.

These ideas popped into Professor Charles’ mind, so suddenly, upon seeing his embarrassed former student, more for the teacher’s criticism than for not being able to answer. Charles looked at Santa Pi, who winked at him. It was really Santa Pi that made him think and see what he couldn’t understand then.

‘But here are really good students,’ said the teacher.

Groups of children are playing in the snow and throwing snowballs at one another.

Process print after E. Girardet

Wellcome Collection

CC BY

Process print after E. Girardet

Wellcome Collection

CC BY

‘Of course, Charles. You have always been a good teacher’, concluded Santa Pi,’ But not all your students of that time gained a taste for mathematics. Just some. Few enjoyed the discipline and many decided their own future by trying to escape from it.’

‘Was it the case of David?’ asked Charles, afraid of the answer.

‘By chance David even followed the math course. But only after learning philosophy. Because he was very fond of letters and logical ideas, he began to make sense of mathematics’, said Santa Pi. He began to understand that mathematics is about ideas rather than numbers, and that sparked his interest. But many in this room could have been excellent engineers…’

‘Am I to blame?’ asked the sad teacher.

‘Yes and no,’ answered Santa Pi, ‘Yes, because you couldn’t understand that your message didn’t reach all the students. No, because you were so taught, and that was how most teachers taught’, and continued, ‘Teachers taught and were the authority. No contestation or conflict. For these children, little world existed beyond the classroom walls. The windows showed the exterior of their school, where other children from their village or town passed, more or less well-known adults, perhaps a plane that would go somewhere far and only known by television, which, too, showed little more than that the reality of their country. The books really showed fantastic and unknown things. Images that didn’t exist in books were created in the imagination.’ and it ended jokingly, ‘Of course, in the case of mathematics the imagination really had to be of the highest quality.’

‘Imagination?’, Charles was surprised, ‘More than imagination, the capacity for abstraction that arises from hard work.’

‘Some have it more developed, some have it less. In any case, it has to be stimulated with creative activities, and it’s at this point that the capacity for abstraction begins to construct increasingly complex ideas and concepts.’

‘Is imagination the most important then?’ asked Charles.

‘It’s the beginning of big ideas. And that’s why it’s so important to stimulate imagination from an early age’, said Santa Pi. ‘Without it, we wouldn’t be talking to each other. I only exist in your imagination, remember?’

Santa Pi was right. This strange character was the product of his imagination. He was dreaming, in deeper and deeper sleep… deeper… more deeper sleep…

III

The spirit

‘You again?’ wondered Professor Charles, suddenly waking up from his deep sleep, ‘Aren’t you supposed to be a fat spirit, transmitting abundance, as in Dickens’ tale?’

‘No’, answered the voice humourously, ‘this isn’t Dickens’ tale and I don’t have to convey abundance, just wisdom. Notice, however, that my costume is similar to that of the Spirit of Christmas Present.’

Charles was in front of Santa Pi again. Just as he recalled it before he went into deep sleep. With the green and white barret — indeed! — and with its yellow stars, brighter than before.

‘I’ll make you aware of your present’, recalled Santa Pi, and warned him: ‘You better pay attention to everything you see, reflect on and start making a resolution.’

‘Why? Is my future that bad? ‘ asked Charles, startled.

‘No future is completely bad, no future is completely good. However, your actions have an impact on many people’s lives, so you should always reflect and seek to choose the best path, especially the best path to the common good.’

‘So lead me to see with other eyes what I couldn’t see so far.’

‘Grab my pompom’, said Santa Pi, ‘let’s go travel.’

They left through the half-open window of Charles’ room. It seemed incredible he could fly with a simple pompom, a pompom of a number — something unheard of — and beyond that an irrational number. Only thanks to his imagination. Come on, he thought, if something went wrong on this unusual flight, he could cling to one of the digits of Pi’s infinite decimal expansion. That is, it was impossible to die in the fall. That thought made him laugh.

‘Stop laughing’, said Santa Pi, ‘you’re rocking yourself and unbalancing me. I already see strands of my pompom in the air. Don’t you dare to spoil it. It was a gift from Santa Claus.’

This latest information didn’t help Charles stop laughing, quite the contrary. Which led Santa Pi to make an emergency stop in mid-air, and thereby to give Charles a great fright, who found himself literally hanging by a thread.

‘Stop it, ok?’, shouted Santa Pi, even without mouth but furious. ‘You don’t want to know how dangerous math can be!’

‘Sorry, sorry, Pi! I don’t laugh anymore’, said Charles, his laughter choked with fear of falling. After all, the digits of Pi were not visible. It was in its exact form.

More pleased, Santa Pi continued his flight, remembering that its name was SANTA Pi, a mathematical wizard.

They flew over villages, towns and cities. The main difference between them, seen from above, lay in the amount of green nature. In all these places they could see people, young or old, talking to each other, in the streets, in the cafes, near the houses, and when Charles peered through the windows, in their living rooms or kitchens.

The human being is primarily gregarious. Charles saw individualism in any of these places, but the instinct and willingness to be with others, to share, prevailed.

These thoughts were interrupted by Santa Pi’s voice. ‘Do you like what you see? People feel happy when they share.’ Charles smiled.

They began to descend and Charles’ feet landed softly on a school playground and Santa Pi in his right hand, the one that had held this nice character by the pompom until that moment. It wasn’t his school, which made him a little disappointed for that he thought he would see himself teaching in the present. It would be interesting to see himself once again, as an outside observer. Isn’t that the best way to reflect on yourself?

‘Look inside,’ said Santa Pi. ‘A math class is going on.’

‘Where are we?’ asked Charles.

‘It doesn’t matter the place, it matters the class. Watch carefully.’

Inside the spacious classroom, with walls lined with papers, cardboards, maps, manuscripts and a computer, were many students sitting in small groups at circular tables. There were conversations between them. With no indiscipline or jest. They even seemed to be engaged in solving some mathematical task. There were, however, five students sitting at individual desks, facing a whiteboard. They would be doing individual work, Charles thought, it wouldn’t be a punishment, for not only were they calm and focused on their task, but they occasionally questioned a fellow colleague about some aspect of their work. The teacher circled the room, responding to students’ requests and also throwing more questions and ideas among the groups and with the students at the individual tables.

At one point, the teacher began to take pictures of the students’ work, and when the rounds were over, she went to the interactive whiteboard. She projected the photographs of the work she had taken and a spokesperson from each group explained the group’s work and conclusions to all students. And even students who had done individual work were part of this larger group and participate in the conversation. Some students explained, others questioned, the teacher moderated the conversation. And they all seemed motivated and committed to explaining the main problem, which was related to the area of forest land in the region in which they lived.

‘Interesting…’, sighed Charles, ‘but hard to achieve at my school.’

‘This isn’t difficult, you just have to want it,’ countered Santa Pi.

‘These students know how to work in groups. Mine, no.’

‘You’ll have to teach them, don’t you think?’

‘They always end up in small talk,’ justified Charles.

‘Group work has to have well-defined rules and roles, like any organization of people. And explained and achievable goals, nothing can be left to chance’, explained Santa Pi. ‘And everything goes from habit’, he added.

‘Difficult. I don’t think I could have the support of my fellow teachers. And besides, we have a lot of subjects to teach, we can’t waste time’, complained Charles.

‘Losing or winning…’ ventured Santa Pi, ‘Especially winning students for our cause, isn’t it?’

Charles looked thoughtful. Meanwhile, Santa Pi waved his green-and-white barret, and the many yellow stars around it flew toward the students in this classroom, entering through a small wicket. They seemed to be subdivided into more stars, these of very bright gold, falling over the students’ heads. Their mathematical conversation got even livelier and several raised their fingers as they found a new solution to the problem being discussed. As the main stars returned to Santa Pi’s barret, it shook them again and they flew quickly to the teacher in the classroom, describing circular movements that enveloped her. She, like the students before, didn’t realize the presence of the stars, but said she had ideas for a new project. Charles noticed that Santa Pi’s eyes also glowed.

The stars then flew towards them, but didn’t stop above Santa Pi. They had gone to an old house, which was at odds with all the surrounding scenery. The stars could illuminate the ajar door. Staring at this entrance to the house, Charles jumped. Two children, a boy and a girl, very poor, poorly dressed, dirty, with broken and bumpy teeth, hair in disarray, miserable looking, even heinous. The air smelled rotten now.

‘Do you know who they are?’, asked Santa Pi.

‘Ignorance and Misery’, Charles recalled from Dickens’ Christmas tale. Of all the books he’d read, all the movies he’d seen, this was the scariest passage of all the passages ever written. He remembered that Dickens had advised, in the voice of the spirit, special care for Ignorance, for it brings condemnation.

‘Yes, condemnation,’ Santa Pi continued the thoughts of Charles. ‘Ignorance perpetuates differences, injustices, produces conflicts and wars, causes discrimination and prejudice, instigates revolt. It condemns the future and succeeding generations are more likely to remain ignorant. And so often in misery.’

‘That’s why my work is so important!’, concluded Charles. ‘I have some power and some strength to change the lives of children like these.’

‘Even if you don’t see children like these again, you can always educate others so that they can change your world for the better. And so you will be changing it for the better too.’

Charles looked once more at these wretched children and once again at the school where happy children learned from each other. The world should be like this classroom, free, creative, sensitive to the individual and the group.

Absorbed in these thoughts, he didn’t even notice the fog that had formed in the meantime. Santa Pi was gone. The house was gone. And even school had ceased to exist. There were no children. A figure was approaching. Charles was afraid.

IV

A apparition

The figure was approaching. Not silently, like the fog that preceded it. There was a rhythmic sound that became more and more frightening. The figure didn’t walk, it marched. Two excessively long legs, with each stride opened in an inverted V. A stride that was not human but mechanical. Each time was closer. The head… the head… What a horror! It has no head. It’s a hammer! Charles jumped.

A song began to grow from the fog.

We don’t need no education

We don’t need no thought control

No dark sarcasm in the classroom

Teachers leave them kids alone

Hey, teachers, leave them kids alone

All in all it’s just another brick in the wall

All in all you’re just another brick in the wall

Charles remembered this song… Pink Floyd! The hammer is the hammer of the video. I won’t end at the giant meat cruncher, will I?, thought the teacher.

‘No, Professor Charles. This shredder was just for students turned into clones without individuality’, said the giant-legged hammer that, in a split second, turned into Santa Pi.

‘You again?’ marveled Charles.

‘Would you prefer the hammer?’ asked Santa Pi, adjusting the green and white barret, which unleashed the yellow stars previously glued to it.

‘You’ve picked a suggestive song for the moment,’ said Charles, still nervous. ‘What do you mean with it? Reveals something of the future?’

‘I’m an apparition, but I can’t predict the future, I’m sorry. You build your future. Do you think the lyrics are indicating what will happen?’

‘I really hope not. Without education and without teachers society wouldn’t survive. My class has no sarcasm, nor do I try to control students’ ideas. They’re all free to think for themselves.’

‘Is it really so?’ asked Santa Pi. ‘Is there in your class the creative and idea-sharing freedom that makes students think for themselves?’

Charles was now looking very seriously at Santa Pi. The yellow stars around his cheerful and festive barret took on their true meaning. Lights that represented the creative ideas that may arise in mathematics.

‘…you’re just another brick in the wall’, repeated Santa Pi. ‘Are you just another brick in the wall, Charles?’

‘No!’ Exclaimed Charles emphatically. ‘I don’t build walls, but bridges.’

‘But sometimes the school creates more walls than bridges. It creates few opportunities’, reflected Santa Pi.

‘I don’t think so. The school always creates opportunities for everyone. Everyone always learns something at school’, said Charles.

‘Something may not be enough to overcome Ignorance and combat Misery,’ countered Santa Pi. ‘ A student doesn’t go to school to learn something, but to develop its potential. Shouldn’t this be the premise?’

At that moment, Charles saw one of Santa Pi’s legs grow and become one of the music hammer’s legs again. With a dry blow, this leg tore down a high wall, not of stone but of thousands of stacked books. These books now flew like birds with wings spread in the wind. And there was a classroom, which had no cement walls but glass. Everything that happened there could be seen from the outside. It was a magnificent classroom. Santa Pi waved his barret and its stars flew to the open door of the room, whirling, as if inviting Charles inside.

He accepted the invitation. The room had several work areas, in an open, large and cozy space. Each zone had specific equipment. The teacher there began by launching a project idea, the renovation of the school library space, and asked students to investigate on this subject. In their area were computers, microscopes and small robots. The students devoted themselves in small groups to looking for information on how to organize a library and how to build a 3D model of their school library using mathematical scale and picture similarity. They could do it in cardboard and in the computer.

Next door was a spacious nook, which was after all a recording studio. Students then went through the research phase to create a short film about the original organization of the library and the new organization they proposed, following the rules and script that one of the groups had prepared. They often went to a larger space with an interactive whiteboard, a desk with diverse materials, a computer and projector, and on a large wall a huge whiteboard where various ideas and schemes were drawn. Students collaborated, exchanged ideas and notes on the whiteboard, did computer simulations and records. They also communicated online with colleagues from other schools, in video and in writing. However, in another more sheltered area of the classroom, students would explained math to each other and listen to the explanations of a teacher who pointed and wrote on an interactive whiteboard. They used mobile phones and tablets in search of information and in carrying out activities.

Students and teachers later met in a small amphitheater, where each group presented the work they had done, projecting on screen or explaining the model. The teacher indicated that they could deepen what they had studied in the area opposite, informally equipped with circular tables and beanbags, with board games, laptops and headphones.

(en) Camel (in Rhythmic Landscape with Trees). Paul Klee. Kunstsammlung Nordrhein-Westfalen, Düsseldorf . Public Domain Mark

Charles was amazed by the dynamics and diversity of this classroom, quite different from his own. And especially with the kind of work developed there. The motivators were the students. The teachers guided the work. There was more noise than in a traditional classroom, but it was a productive noise and everyone seemed very involved in the task. Did they learn better?

‘And you, what did you learn best and do you remember most as a student?’ Santa Pi asked immediately, reading Charles’ thoughts.

‘I learned better what I did for myself and I remember my friends and the conversations and games we had together,’ replied Charles.

‘Exactly. That’s what I wanted to show you’, concluded Santa Pi.

Charles’s eyes returned to this classroom, full of life and science. The key would be to balance individual work with the collective work of each student. What I do for myself and what I do with others.

‘Sounds interesting, Santa Pi… Santa Pi?’

The mathematical wizard was gone. Charles was back to his bedroom. The light of a new day came in through the window.

Gezicht op het eiland Malang met een kerk onder palmen en een opkomende zon. Rijksmuseu.Public Domain Mark

V

The end… or the beginning

Yes! I’m back. A new day begins. It happens to be Saturday. There’s no school. But I have individual explanations in the afternoon. And why not join Fred and Alex today?

Charles phoned Fred’s mother and Alex’s father. Yes, there would be no problem having two hours together instead of one, so they could share ideas and some new tasks.

The teacher hastened his bath and breakfast to devote the morning to math research to propose later some nice exercises to the two young men. He found on the Internet the math task he had dreamed of. He might have seen it before, but he had given it no value back then.

When they arrived, Professor Charles proposed the task of his dream, as an individual activity. And just as in the dream, each student solved it differently, one by an arithmetic process, the other by a visual one.

1x(1×1) square with smaller area, one time

2x(2×2) area of the square on side 2 with the areas of the two adjacent rectangles, which added together have the same area of that square

3x(3×3) area of the square on side 3 that appears 3 times

… from there on through to case 6:

6x(6×6) area of the square on side 6, which appears 5 times, with the areas of the two largest rectangles, which together have the same area of that square, i.e. 6 squares on the side 6

(1+2+3+4+5+6)x(1+2+3+4+5+6) is the measure of the area of the total figure which is a square

Therefore,

1x(1×1) + 2x(2×2) + 3x(3×3) + … + 6x(6×6) = (1+2+3+4+5+6) x (1+2+3+4+5+6)

Charles smiled at the two resolutions, which pleased both students. He asked each one to explain his idea, and found that both had understood the problem and found a solution. He asked Fred to write the calculations involved in his reasoning, which consisted of dividing the figure into unit squares and counting squares with relation to the problem statement. The teacher sought to bring the two ideas together in a more detailed resolution. He proposed that they look for the general case, valid for natural exponents. And from here he managed to get to the study of sequences, then establishing a bridge to the study of functions, which was the topic they were working on at school. In the second hour, each student returned to a more individual work, performing the exercises that each of his school teachers had requested. At one point or another, the two students came back to share ideas when the exercises were similar. It was going well, the professor thought. The three were happy. And there was still time to return to systems of equations, relating them to functions, a subject Fred hadn’t understood in the last explanation.

Charles no longer believed in the existence of a “mathematician’s brain.” It seemed possible that both Alex and Fred could learn math, even at a higher level, if they dedicated themselves to it. It would all depend on their degree of dedication, persistence and work. It was important that they believe in their potential and develop it to the best of their ability. To make a mistake was more important than to make it right, because it led, understood the error, to clearer ideas and varied solutions, which pointed new and creative ways. Speed was less important than depth of study. Seeing an idea represented in different ways, especially visually and creatively, seemed to help Fred a lot. Repeating the same idea over and over had not helped him until that day, Charles thought. Now he saw Fred gain confidence. Even Alex was amazed at his colleague’s new abilities, repeating ‘You’re a genius!’ in a nice and genuine way.

‘Professor’, Alex said at the end of the second hour, ‘next week my school celebrates Pi Day.’

‘Pi Day?’ asked Charles, confused. He was convinced Christmas was approaching. What nonsense! It could only have been from the dream, which was reminiscent of Charles Dickens’ “Christmas Tale.”

‘What’s Pi Day?’ Fred was interested.

‘Think: Pi, 3.14. What day is it?’ threw Alex to his colleague.

‘3 point 14… March 14?’ suggested Fred.

‘You’re a genius!’ joked Alex.

For a few moments it seemed to Charles to see yellow stars above Fred and Alex’s heads. Stars formed geometric shapes as triangles, squares, rhombuses, pentagons, hexagons, circles. Behind them, perched on the bookshelf, next to a book entitled ‘The Rhythm of Forms’ danced Santa Pi, very lively and with eyes brighter than its stars. Charles also began to shake his head, following Santa Pi’s movements.

Fred and Alex looked at each other with a common thought: He freaked out! And they laughed a lot, all three.

References

The NRICH Web site publishes free math resources designed to challenge, attract, and develop mathematical thinking among students ages 5–19. NRICH also provides support for teachers through the publication of Teacher Resources for use in the classroom.

EUROPEANA is the European digital platform for cultural heritage. This platform provides free, online access to more than 50 million digital resources from over 3 700 libraries, museums, archives and galleries across Europe.