Mandelbrot's Life & Career

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Math is about Creativity

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"Dance" was never meant to be a repetitive act of the same moves over & over. Wasn't the person who originally invented a move improvising, after all? Same goes for music, math, or any subject really. The main issue I think with the education system around the world, is in fact that it doesn't let kids shine and thereby strengthen their inherent creativity, but rather it is founded on the false premise of rote repetition. Math has been about exploration and creativity from the start.

There is one particular teaching method, called the Moore method, which encourages just that. How would you like to sit in a college class, where you get to discover & invent the subject together with your fellow classmates? The future of education is not online, it's inside of us, our very own inner light.

I began doing research in like the second month of college, which eventually became my masters thesis. That's in no way special. You too can pull such a move, if you lose your pride, and approach mathematics with stupefying egoless humility, where you can stumble, fall, get up, and grow.

Intellectual Growth and Personal Life

In striving for intellectual growth, one must be careful when considering ideas, or systems of ideas. Their aesthetic appeal should not sway one in their critical objective consideration. Even a single flaw can have a devastating effect on the validity of the idea / system. The consideration must be multidimensional, transgressing one's personality and location in space and time. The potential effect of an idea is its harshest measure, and ideas without effect are worthless.
 
On the other hand, if an idea system takes away from one's personal life rather than make you something more, then self-reflection is required. It's better to approach ideas critically, with humility and measure, and to reflect this to your environment. The point of this life is not just intellectual but also personal growth, and if you isolate yourself that won't happen. In seeking intellectual growth, one's introversion can strengthen with potential negative effects on personal life. If one picks up arrogant attitudes due to intellectual growth, rather than trying to see another's point of view, that doesn't help either. Besides, what is intellectually cool, is not necessarily valid or worthwhile.
 
I think it is a grave mistake to allow perceived intellectual growth to overtake one's very real personal life. The only valid evaluation of philosophies is based on their end products in practice, for oneself and for society. Choose what you indulge in carefully.
 
"Bethink you, then, of old age which cometh all too soon, and not an instant will you lose. While yet you may, and while you yet enjoy the spring-time of your years, taste of the sweets of life. The years flow on like to the waters of a river. The stream that fleeteth by, never returns to the source whence it sprang. The hour that hath sped returns again no more. Make the most of your youth; youth that flies apace. Each new day that dawns is less sweet than those which went before."  —  Ovid

Code of Ethics for Researchers

1. Act with skill and care in all scientific work. Maintain up to date skills and assist their development in others.
2. Take steps to prevent corrupt practices and professional misconduct. Declare conflicts of interest.
3. Be alert to the ways in which research derives from and affects the work of other people, and respect the rights and reputations of others.
4. Ensure that your work is lawful and justified.
5. Minimize and justify any adverse effect your work may have on people, animals and the natural environment.
6. Seek to discuss the issues that science raises for society. Listen to the aspirations and concerns of others.
7. Do not knowingly mislead, or allow others to be misled, about scientific matters. Present and review scientific evidence, theory or interpretation honestly and accurately.

Devised by Sir David King.

A Vision of the Future of Research

The following was posted as a comment on Tim Gowers's blog post "How might we get to a new model of mathematical publishing?"

Interesting post. I have proposed a very similar idea several months ago to a colleague, namely a cross between arXiv and a social networking site. I also suggested to get the idea rolling initially via a blog/journal publishing arXiv articles, peer-reviewed in the blog comments - for the sake of technical simplicity. So essentially what you've described.

Having experienced other social networks with a creativity streak - mostly in art like Deviantart - the danger of "gamification" or "score-hunting" does arise as many here have realized in advance. Then there is the issue of getting lost in the crowd unless the ranking algorithm is phenomenal - hardly ever. Any ranking system would reduce a person to a score, but how could the scoreless / no-reputation newbie researcher be noticed for something ingenious? People would just follow the big fish, because there would be way too many little fish tiring their eyes. (Just consider, we all follow Terence Tao's blog, but who follows ours?)

So I have got to the point where I discarded the above idea altogether (though I still am in a dilemma whether to create such a blog with a limited group of researchers inviting each other). Instead I asked myself, why are we doing research in the first place? Is it not to contribute to something greater than ourselves... the development of mathematics / science? Do we really need to be credited for our contributions? Does it have to be an ego game? How about a world of research where no-one cares about who did what? See where I'm getting at...? Wikipedia! Sure, there are wiki sites on math, but what if that's all research would become?

The above idea is a little naive in its brevity and humility. The actual website would have to be a little more sophisticated than a wiki, to account for measuring contributions and the relevance of those contributions (one will still need to put something on their resume to land a job). Possibly a thumbs up/down system per each wiki edit per user (stars don't work btw). If an edit gets too many downs, it would get removed. Let's just keep it simple. (No wiki discussion pages = battlefields either, they are soul-killing.) Then on one's personal profile, the contribution stats could be shown, exportable for resumes. So quantity vs. quality would be displayed analytically.

As far as the edits themselves, a user would have total freedom as on a wiki. Brief to longer edits ok. An edit would be highlighted as new for let's say 3 days, at which point the votes would decide if it remains included in the body of text. But if a person gets too many thumbs down on edits, they could be identified as potential spammers by the admins. The site should require a scan of one's master's degree and edits would be only permitted in that field of science, as verified by admins upon registration. (Thus no need for a weighted voting system, since users would be professionals by default, and a newbie's vote is potentially as accurate as others'.) The site could be peer-invitation only in its infancy, to ensure coherence of material through existing collaborative relationships. The first researchers invited should be prime representatives and authors of a field, so their writing would serve as the foundations of the website. We would always see where research is at by just scrolling to the end of an article. There would be no need to write intros to topics as we did in papers, since the info would all be there.

So to sum up the idea, research would become a coherent evolving online encyclopedia striving for absolute complete knowledge, just like it should have always been in real life in my opinion. Let's admit, the current journal system has always been just a substitute for the real thing, that we couldn't yet make happen without the technology... but now we can! What do you think? Worth a try?

Initiatives

• The Polymath Project was first suggested by Tim Gowers in a Jan. 2009 blog post, with a locked wiki as of Jul. 2015.
• The WorkingWiki by Lee Worden is a MediaWiki extension, first online in Jun. 2009, presented as a paper and a demo video, with installation instructions.

Self-Reflection for Identifying my Ideal Learning Experience

Response paper for the session "Understanding the Learner"
Certificate in University Teaching Program - University of Waterloo

A commonly voiced belief is that the learning environment greatly influences the learning experience of students. In fact, mostly the teacher is the one to blame, when students' expectations are not met in regards to their imagined ideal of classroom instruction. In this essay, I will reason that learning is a bipolar experience, greatly dependent on both the student and the teacher. Via an acute self-reflective analysis, I will identify my own learning characteristics in the context of two very different learning experiences - one negative the other positive. Meanwhile, I shall keep in consideration the potential experiences of other types of students, based on my retrospective observations. I will begin with a brief description of myself as a learner, to provide a working foundation for the subsequent analysis of my experiences.

Currently I am a graduate student working on my PhD in Applied Mathematics, primarily occupied with doing research, which is quite different from my undergraduate mindset, when studying was my primary focus. Therefore much of this discussion will be a travel back in time, though my main characteristics as a learner have not changed, only become less elaborate and refined. When I was an undergraduate in Hungary at the Eötvös Loránd University, I was literally like a sportsman of the mind - constantly monitoring myself for what shape I was in. This was of utmost necessity so that I could allocate my time and efforts to the most appropriate times for a particularly or less demanding subject. I monitored my diet, breathing, everything - it was a bit insane, but the tasks at hand required it. I was also very independent as a learner. I did not value having to sit through lectures, where I was superimposed with a "sequential" learning experience in time, in which I was unable to flip back and forth as I would do by myself with a book, to connect formerly learned concepts. I am what I can now properly categorize as an absolutely "global" learner. Even in doing research, I feel more like a bird flying over the savannah than an antelope galloping through it. Whenever I notice something interesting, I zoom in somewhat, then zoom out, or zoom in more if I find it to be of value, guided by my intuitive inner compass. Just as a bird, I am a great lover of freedom, and being restrained in any way, whether by a non-liberating lecture or else, can be very disheartening to me. I feel lucky to have been endowed with a considerately adaptive supervisor who understands me by instinct, and is willing to act as a companion, versus a guide.

Considering the above, the ideal learning condition for me is most likely an environment in which I am given a chance to contemplate the material with freedom, allowing me to internalize it in a gradual process of digestion, in which I reassemble the buffered information packets into a new didactic whole within myself, via a highly independent process. Is this possible through any kind of lecture or tutorial format? Does not my own sense of independence make the reception of any external learning projection impossible? Sometimes it does feel like I have this impediment. In rare occasions however, I encounter a learning environment in which sequential spoonfeeding is not the primary mode, and which I find highly enlightening and motivational, activizing my neurons on all levels.

A specific negative experience was my Partial Differential Equations (PDE) course as an undergraduate. It was undoubtedly a large bulk of information we had to internalize, or perhaps only seemed like it? The professor acted like a sage on the stage throughout the lectures, speaking his mind like a radio, and scribbling away on the board, without any consideration for our understanding. Certainly contrary to a multilevel learning experience, that would have sent off sparks in the minds of various types of learners. I personally felt being given no chance to reflect on the material, the pace was too fast and uninterruptibly monotonic, which made me feel disconnected, from both the professor and the material. I was given no opportunity to make a connection either, as it felt like the instructor had no idea there was even an audience. I was not motivated in any way to learn the material, and suffered greatly when I "had to" in the exam period.

I believe a good teacher not only connects with the audience, but monitors their level of understanding throughout a lecture, and sets a fluctuating pace accordingly. Before the process of information flow is even initiated however, the professor must set a base tone for the rest of the lecture in some way, much like a painter might paint a base color for a painting to be laid out upon. The base tone might involve some motivational words of wisdom on the nature and relevance of the material, its historical importance, or their own enthusiasm for the topic. These may serve as an extension of a hand to help students on board for a journey of sailing the waters of knowledge. Most of all, teachers must primarily know themselves and their own motivations, so that they may accompany students on their own journey of learning, whether as captain or companion. In a classroom, a teacher must above all create a multilevel learning experience, in which learners of all kinds shall thrive.

A specific positive experience was my Functional Analysis course as an undergraduate, which I remember as my favorite one of all time. Unlike the PDE course in which I never got past learning the toolset, here I internalized it from the notes quite naturally and largely independently, since I was eager to apply it in assignments given at the tutorial. The tutorial was also held by a professor who I have had for years in Analysis, and found his approach reliable and respectable. His opinion and marking represented to me the unquestionable law carved into stone. In other words I truly trusted him, regardless of him being a difficult personality, who did not think much of "applied" mathematician students. So clearly, I had an incentive to prove myself in attempting to solve the rather difficult problems given by him. Indeed week after week I handed in each assignment in full completion, even though a vast majority of students sabotaged the tutorial by missing it and not doing the homework - the general consensus was that this was a teacher to be disliked. Yet, I saw him as a representative of mathematical rigour from the past, and his very character projected reliability and wisdom. So even though I was a largely independent learner, his character still had an influence on my motivation, captivating my interest as if by undetectable magic. I was not alone though among those who were motivated - some were hard-working by nature, some saw me as competition, and a few were just trying to get by on a mediocre level. I however saw this class, which presented the pinnacle of modern Analysis (Calculus), as a chance to mature into an able mathematician. When the course was over, it felt like I have gone through a training period harsher than ever. Yet by overcoming independently the challenges I have faced, I have reached a new level of mathematical awareness and understanding, inaccessible via any other learning process. Then and there, I became a mathematician.

How was any professor able to evoke all these feelings alone? Analyzing my memories, I believe the answer is that he did not do it alone, I was an equally necessary ingredient. In fact, being the experienced educator he was, I believe he tailored his classes to specifically the kind of student that I was. This was quite irresponsible, since the rest of the class was left behind. I have observed however, that many professors grow tired over the years of having to deal with other - perhaps less motivated - students, and hold their classes for the most worthy. I must however voice my criticism, being a great believer in student potential. While I may have felt good in this particular classroom experience, it was unfair to the other students. Therefore I once again arrive at the conclusion, that a good educator creates classes which run on multiple levels, as an orchestral concert involving winds, strings, and percussion, for various musical sounds which may appeal in varying degrees to individuals in the audience. For instance, there are lovers of rhythm (percussion) and those who are more receptive to tunes (winds or strings), whether brief or overarching. A great symphony like a good lecture, weaves elements in a magical fashion, sparking the minds of as many listeners as possible, ie. optimizing the learning experience for the entire class.

In conclusion, I deduce that the ideal learning experience for me is one where I am granted freedom to explore, and the material is shown to me to be a toolset for practical application in the gradual exploratory process. Even though I have always thought that the personality of the instructor has no effect on me as a learner, I have now realized that it is indeed a major factor towards connecting with the material. A demandingly harsh yet nurturing environment, can be better than the nowadays common easy-going attitude of many professors, on the lookout for high student ratings. A professor can still remain accessible for a personal connection, in many ways. In regards to presentation of material, I have reasoned for a multilevel orchestral manner of information propagation, for optimal exploitation of student potential. Though certain manners of instruction I may see selfishly as ideal to me personally, as a teacher I feel I must become a more complete individual, growing beyond my own natural ways of learning.

Benefits and Risks of the Moore Method in Mathematical Education

Response paper for the session "Interactive Teaching Activities"
Certificate in University Teaching Program - University of Waterloo

A commonly observed phenomenon in undergraduate mathematical education, is the lack of motivating conceptual instruction. All too often, students are lectured on techniques of problem-solving, without a parallel development of their conceptual understanding. This lack of depth often continues on to graduate-level classes, for the sake of rational efficiency. The method developed by Robert Lee Moore, is a form of mathematical activity spanning a single class event to potentially entire terms. In a sense, it may be viewed as ideal for the conceptual development of students in science disciplines, primarily in mathematics. This paper will consider its hypothetical application in a first-year Calculus setting onwards of Pure and Applied Mathematics students.

First-year Calculus for a student can be a time of great transitions with unusual concepts, formulas, and methods, that many instructors seek to hard-wire in students, to establish a foundational toolset for later courses. Most instructors achieve just that, storming by underlying concepts and philosophies that may only occur to the most contemplative students, in such a fast-paced setting. Likely most professors would cite lack of time, size of classes, and efficiency of information propagation. On the other hand, elementary Calculus could easily become an ideal platform for activizing students, and involving them in the creative development and learning of the material. Bringing to the surface and upholding students' inherent mathematical acumen, must be the goal of any program which honestly wishes to raise able mathematicians and future researchers.

Indeed, most students accepted to university, possess the mathematical background and potential to develop the Calculus of Newton, Leibniz, and Cauchy - however outrageous this claim may sound. Encouraging them to do so in a classroom setting, may potentially turn some into quite capable and confident mathematicians once they become researchers themselves. The suggested alternative activity would hard-wire deep research and collaboration behaviours they stumble upon themselves, instead of the shallow toolsets of technique they are mostly provided. Thus my argument that follows for using the Moore Method in first-year Calculus classrooms, is primarily intended for students who may become researchers one day, in the fields of Pure or Applied Mathematics.

I will hereby outline the overall class as well as the classroom activity itself, which the entire longspan method would reduce to. The Moore Method has numerous existing interpretations, branching off from the original central idea behind it, and this is reflected in the way the actual classroom activity may be carried out. Often the interpretation reflects the course it is applied to, so what follows is my own interpretation, as I would potentially apply it.

The central idea of the Moore Method is to involve a class of students in the development of historical theories of mathematics and science, from the ground up. Students are generally given a set of basic knowledge, often in the form of definitions and axioms, providing a foundation for the relevant initial questions and directions. In regards to elementary first-year calculus, the initial concepts provided to the students might entail a vague idea of limits, continuity, and tangent / area determination, which they would be first tasked with refining. The refinement process itself for these basic definitions may last 1-2 classes - to be determined experimentally. Then as a syllabus, the task of putting the definitions in practice, such as for the evaluation of specific limits, then later determining continuity and differentiation of functions could follow. While solving problems they invent for themselves together, the students may encounter cases that will imply the invention of the Squeeze Theorem, the Intermediate Value Theorem, or even L'Hopital's Rule - not missing out on relevant ideas. The same implications via applications would follow for differentiation and integration, as well as their inverse relationship, which the students would be likely to discover. Specific formulas and techniques of integration, would be developed by the students, for the sake of creating a toolset for themselves, so that they can crack the nuts found in problems they encounter in their exploration. Instructor involvement would amount strictly to asking occasional motivational questions that may blow the students' sail in the right direction. Otherwise, students would be expected to create and collaborate independently, both during and after class, representing their "assignments". Soon, such a Calculus class may have the effect of occupying the students' minds 24/7, preparing them for the analogous experience they may encounter as researchers in their future careers.

The classroom activity itself would possess a loose yet consistently coherent structure, with explicit rules of conduct such as free speech balanced with mutual respect, stated in advance. These explicit rules may serve to instill and induce principles of integrity, and serving as foundations for engagement in future research. Furthermore, the instructor would provide the initial definitions, motivational ideas and questions, to ignite both group discussion and individual efforts, in a sentiment of "collaborative competition". The incentives being inherent mathematical interest, the thrill of building something together, or even peer pressure to perform.

In addition to the aforementioned benefits of a Moore-based activity which shape its very format, we may recognize an even more fundamental characteristic in the method. By placing the learning process in a creative social setting, the intructor immerses the students in a primordial soup which they are used to functioning in, and from which they may naturally evolve as maturing mathematicians. Thus each student would find their natural function in the micro-social environment of the classroom, paralleling their social preferences in real life. Thus collaborators would collaborate, thinkers would contemplate, leaders would emerge, tasks would be divided, global minds would present the vista, and sequential ones the process to explore it. All would find their own role in such Stone Age conditions, in which we are genetically programmed to rely on one another and contribute as we can.

Much of the above is hypothetical to me, and based on pure rationale, versus any prior implementation. To me, the above is an idealization of a mathematical learning activity I would have always wished to be a part of. I have been observing the inefficiency of the classical lecture-tutorial settings, ever since I became a university student, and I find this to be a solution. Though I realize that the Moore method may not be for all students. The initial selection of the students must be based on whether they find the challenge appealing, and not based on academic standing. The worst possible outcome for an aspiring mathematician, would be realizing that they are not fit for doing research, which is still better than a realization in graduate school or real life. On the other hand, many who are unsure initially, may gain self-confidence for the long run. I personally had doubts about my own research abilities as an undergraduate, projecting an unpredictable dark cloud into my future. I believe that all students should be given an opportunity to put themselves to the test, in classroom activites of this kind. Furthermore, the professor may serve as a mentor in this setting - accompanying, guiding, and nurturing potential talent.

The benefits of such an adventure, even if experimental, undoubtedly outweigh the risks. As the Ancient Greek aphorism of virtue recommends "Know thyself!", so should mathematicians learn as much about themselves as possible early on. What better way to do so, than a setting that mimics what they are apt to experience later on in real life. This is akin to teaching children to swim in shallow water so that they would be ready for the deep, while throwing them in deep water at once may prove fatal. Is it not the responsibility of experienced researchers and educators to prepare students for the deep water, besides striving to enhance their overall conceptual understanding?