Axiomatic Method : A little explanation

I) Take an English-into-English dictionary (any other language will also do). Start with any word and note down any word occurring in its definition, as given in the dictionary. Take this new word and note down any word appearing in it until a vicious circle results. Prove that a vicious circle is unavoidable no matter which word one starts with , (Caution: the vicious circle may not always involve the original word).

For example, in geometry the word “point” is undefined. For example, in set theory, when we write or say : $a \in A$ ; the element “a” ‘belongs to’ “set A” —- the word “belong to” is not defined.

So, in all branches of math or physics especially, there are such “atomic” or “undefined” terms that one starts with.

After such terms come the “axioms” — statements which are assumed to be true; that is, statements whose proof is not sought.

The following are the axioms based on which equations are solved in algebra:

1. If to equals we add equals, we get equals.
2. If from equals we take equals, the remainders are equal.
3. If equals are multiplied by equals, the products are equal.
4. If equals are divided by equals (not zero), the quotients are equal.

More later,

Nalin Pithwa.

Discuss the following “proof” of the (false) theorem:

If n is any positive integer and S is a set containing exactly n real numbers, then all the numbers in S are equal:

PROOF BY INDUCTION:

Step 1:

If $n=1$, the result is evident.

Step 2: By the induction hypothesis the result is true when $n=k$; we must prove that it is correct when $n=k+1$. Let S be any set containing exactly $k+1$ real numbers and denote these real numbers by $a_{1}, a_{2}, a_{3}, \ldots, a_{k}, a_{k+1}$. If we omit $a_{k+1}$ from this list, we obtain exactly k numbers $a_{1}, a_{2}, \ldots, a_{k}$; by induction hypothesis these numbers are all equal:

$a_{1}=a_{2}= \ldots = a_{k}$.

If we omit $a_{1}$ from the list of numbers in S, we again obtain exactly k numbers $a_{2}, \ldots, a_{k}, a_{k+1}$; by the induction hypothesis these numbers are all equal:

$a_{2}=a_{3}=\ldots = a_{k}=a_{k+1}$.

It follows easily that all $k+1$ numbers in S are equal.

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Regards,

Nalin Pithwa

Patterns in the primes: Clay Math, James Maynard, 2015

(shared from Clay Math website and shared for my readers. Many thanks to James Maynard and Clay Math :-))

Patterns in primes

Check your talent: are you ready for math or mathematical sciences or engineering

At the outset, let me put a little sweetener also: All I want to do is draw attention to the importance of symbolic manipulation. If you can solve this tutorial easily or with only a little bit of help, I would strongly feel that you can make a good career in math or applied math or mathematical sciences or engineering.

On the other hand, this tutorial can be useful as a “miscellaneous or logical type of problems” for the ensuing RMO 2019.

I) Let S be a set having an operation * which assigns an element a*b of S for any $a,b \in S$. Let us assume that the following two rules hold:

i) If a, b are any objects in S, then $a*b=a$

ii) If a, b are any objects in S, then $a*b=b*a$

Show that S can have at most one object.

II) Let S be the set of all integers. For a, b in S define * by a*b=a-b. Verify the following:

a) $a*b \neq b*a$ unless $a=b$.

b) $(a*b)*c \neq a*{b*c}$ in general. Under what conditions on a, b, c is $a*(b*c)=(a*b)*c$?

c) The integer 0 has the property that $a*0=a$ for every a in S.

d) For a in S, $a*a=0$

III) Let S consist of two objects $\square$ and $\triangle$. We define the operation * on S by subjecting $\square$ and $\triangle$ to the following condittions:

i) $\square * \triangle=\triangle = \triangle * \square$

ii) $\square * \square = \square$

iii) $\triangle * \triangle = \square$

Verify by explicit calculation that if a, b, c are any elements of S (that is, a, b and c can be any of $\square$ or $\triangle$) then:

i) $a*b \in S$

ii) $(a*b)*c = a*(b*c)$

iii) $a*b=b*a$

iv) There is a particular a in S such that $a*b=b*a=b$ for all b in S

v) Given $b \in S$, then $b*b=a$, where a is the particular element in (iv) above.

This will be your own self-appraisal !!

Regards,

Nalin Pithwa

Teaching (or learning on one’s own) and the pursuit of clarity: Prof Samir Mathur, Ohio State University

Further reference: Chapter 11, “Teaching and the pursuit of clarity”, in the book “Secrets of Good Teaching”, Edited by Viney Kirpal, The ICFAI University Press, ISBN: 81-314-0323-8;

By the way, the actual article, reproduced below is meant for teachers of physics, especially. But, some of the suggestions of the author, who is a famous physicist and a good teacher of physics, can be modified and applied towards one’s own self-study program in IITJEE Mains Math or Physics, or even Regional Mathematics Olympiad or INMO (Indian National Mathematics Olympiad) or even the prestigious IMO (Internat ional Math Olympiad).

I have found the article extremely useful especially for teaching/coaching math to my talented students, and for my own self as a I am self-learner in Mathematics 🙂 I would request my students, past and present to go through it at least once…If you like, drop a word of thanks to Prof. Mathur at OSU !!

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Suppose you are walking down a corridor in a University, in the Department of Physics. You peep into two different classes, and observe the way the students are studying. Which of these two modes of teaching would you say is “good”?

A: In room A, you see that the teacher has written a problem on the board. The solution is long, and the students have been given an hour to obtain it. The students are looking through their textbooks, trying to grasp the needed tools, while the teacher sits in a chair at the head of the room.

B: In room B, you find that students are clustered around a set of “hands on” demonstrations, where they can explore the principles of physics by working with gadgets. The students are divided into small groups, and can talk among themselves. The teacher walks around the class, answering questions.

My impression is that in today’s world of education almost all people would argue that the teacher in room B is doing the right thing, that educational resources must be geared towards ensuring that all students should one day be taught in this way and that teaching in the manner of room A should be made a thing of the past. I will probably surprise you by asserting that neither A nor B is a useful path to excellence; further, if you force me to choose between just these two options, I would argue that the students of room A have a higher choice of success later in life, at least if we assume that working with physics is their goal.

So, what is the missing ingredient? Here is my firm belief:

“The only time we learn is when we answer a question that we have posed on our own.”

How does this work? Isn’t this the teacher who has to ask questions, and we discover the answers? Not quite, I would argue. The educational system — the textbook, the teacher, the exam — can bring some broad questions to our attention. These questions force us to acknowledge that we are confused, we don’t understand many tiings. At this point, we must undergo a process of introspection, sitting by ourselves to make sense of things. The book can help us, and on occasion we may ask the teacher a question. In the process of solving any one question, we must ask ourselves hundreds more, and evolve answers to these to make a complete picture. At the end, we would not only have answered the questions posed by the teacher, but would have mastered the area as well.

Let us assume for a moment that this is indeed the way we should learn. Then, we see that a lot of the effort of modern education is simply taking us away from our goal. First, I find that the students in high school and college are heavily loaded with coursework. They stagger from day to day, trying hard to balance class attendance, a crushing load of homework, extracurricular activities, and possibly a need to work some hours a week to support their education. The last thing they have time for is to puzzle out on their own an understanding of the subject. Most things are learnt partially.If I ask my undergraduate class “Do you know Fourier transforms?” I get a response — “Well, sort of …” The material was “seen”, some homework done, some partial understanding obtained, and life moved on to the next course. But in mathematics or physics, concepts are like building blocks; if they are hard and well-made you can stack them on top of each other to reach great heights, while if they are soft and fuzzy they don’t stack well at all. So it seems the more we try to “stuff knowledge” into the student, the weaker they might end up becoming. I will summarize this in my second belief:

What we learn we should learn thoroughly, an “exposure” to ideas is not very useful.

Clearly, one must make a distinction here: it is the basic ideas that students must learn thoroughly, and it is certainly useful to have an “exposure” to a wide variety of other facts. But, in our attempt to stuff knowledge into students at an ever increasing pace where do we get to make this distinction? I think we have got ourselves into this mess by thinking that we deliver good education when the students “know” more things, or acquire more “skills” like learning to use the Fourier transforms. I would argue differently.

Students need to develop the ability to puzzle out results by sustained thought on a problem; acquiring “knowledge” or “skills” is not a worthwhile goal of education.

This exposes another weakness in modern education. The increased reliance on “short multiple choice” questions, where one either knows the answer as a “fact” or obtains it by the “skill” of plugging into a formula. By contrast, when faced with a well-designed “long” problem the student must understand the setting, evolve a method of attack, explain to the grader (examiner) his steps (in the process understanding them much better himself), gather together many different concepts, and arrive at a final answer which he may cross-check using other physics reasoning. The ability to think and reason with concepts is thereby developed, and I would argue that only this represents true learning.

If developing “sustained thought ability” is a goal, then we see why the tools of modern education are a distraction. When students work in groups, they do no think by themselves; the individual thought processes are continuously interrupted by conversation and when the group arrives at a feeling that between them they understand the issue then the students move on. It can of course be useful for just two or three people to get into an argument on an issue which each one has thought about previously and resolve differences in their understandings; in fact, this is an extremely useful step in learning. But I find that when the initial learning itself is done as a part of a group, then a very hazy picture of the material is generated in the students’ minds.

Much is made of the value of “demonstrations or “hands on” learning where the student explores concepts by seeing them in “real life”. I recall an episode from my early years at MIT, teaching introductory mechanics. The main instructor would give a lecture and do demonstrations, and the next day the “recitation instructors” (I was one of them) would divide the class into smaller groups and iron out ideas or do problems. The lesson of the day was “Impulse”,  which is the idea that if we deliver a sharp blow to an object then a useful measure the impact is the product of the (large) force and the (small) duration of the force, rather than the force itself. The main instructor was a very dedicated teacher, but in addition, he was a Karate expert, as was the rest of his family. The class first watched a video on “Impulse” where cars smashed into walls and other “fun” stuff happened, and some formulae made their appearance. Then, the instructor, his two sons, and little daughter, all broke a variety of boards, to the  cheers of the class.

The next day I met the class for the recitation, and asked them to explain “impulse”. There was no response for a while and then somebody offered the word “force”. Somebody else said “sudden force”. Another said “impact”. Moving on from there, over the hour we managed to get together the basic ideas and formulae for “impulse”. The hour long “movie and demo” session had by itself not left any clear ideas, though everyone agreed that it was “cool” and made physics “interesting”.

But after the class my mind flew back 15 years, when as a high school student I had to learn the concept myself. The teacher had just written it on the board, and it was up to us to make sense of the notions before we would be faced with an actual examination. At home, with the book on the table, I made up examples of my own, checked out units of various quantities, tried to understand when the concept was useful and learn about the topic. By contrast at MIT, we had taken the students through several pathways to the notion, and they had faithfully followed us on this journey, but at no stage did we leave them the time or energy to think in peace over the idea themselves. I have the highest regard for the professor who had the energy to arrange such a wonderful “demo” session, and the students I had were outstanding as well. But, I would say that at the end, I was left dissatisfied with the level of clarity that we had managed to impart to them. I would summarize this feeling in the belief:

We don’t need “hands on” learning, we need “minds on”:

Let me be more explicit about this. It is of course possible to devise, with enough effort, an apparatus that will demonstrate a physical principle. But in most cases it is more helpful to imagine what the physical entities are doing, letting the mind make a “virtual” lab where objects move, attract and repel. Consider a simple electromagnetic device: the capacitor. This consists of two plates, separated by a thin layer of dielectric material. Wires are connected to each plate, and by connecting a battery to these we can store charge — positive on one plate and negative on the other.

But if you buy a real capacitor you find that the plates are rolled up into a tight cylinder, which is then encasted in plastic and sealed, so there is not much that you can actually see about the capacitor. One may connect a battery across the capacitor and observe it charge and discharge. But I would contend that what can be learned from this “demo” is limited, much more can be learned by letting the mind visualize the charging process: the battery pushes the electrons onto a plate, the electrons accumulate there until enough pile up to repel further ones from joining there. The positive charges on the other plate attract the negative charge on the first plate, holding it there and thus increasing the charge that can be stored. An evening spent in such a “minds on” exercise leads to deep understanding of the capacitor. What is more, this understanding can be transferred to understanding a spherical capacitor, which has two spherical plates, one inside the other, or to the idea of stray capacitance, which arises in circuits whenever one wire passes near any other metallic object. It can also help us understand time dependent voltages where electrons surge to and fro from plates. Obviously, we cannot keep making physical “demos” to illustrate these ever more complex principles. But the modern student mind seems to be getting increasingly attached to “learning by doing” in lab setting, while the power to do “minds on learning” seems to be fading away, to the detriment of the overall education process.

The reader must be wondering: when do we get to the important part, the role of the teacher? The notions I have advanced above will help us define the role. Should the teacher explain the physics very clearly, making sure that the students understand every little concept? No, for we have argued that the important thing is for the student to ask his own questions and clarify his thoughts. Should the teacher do a lot of nice “demos”? No, for these have limited success in advancing student learning. Before suggesting what the teacher should do, I want to talk about two other ways in which the influence of a “good” teacher can be negative rather than positive.

I have said that the student mind must be in a state where he is continually asking questions and refining his understanding. The most important tool that he needs is confidence. What kind of a mind will be able to ask its own questions and try to make its own answers? A mind that has some faith that its questions are good ones, relevant ones, not stupid ones whose answers are obvious. In any class setting, the students who are ahead of the pack are the ones with the courage to ask bold questions, the others feel that the answers are probably obvious to everyone else so they will pick up the facts “later”. Obviously, we must strive to make all students feel that they are as valuable as others in the class. But, a more important, though somewhat subtle, effect is played by the competence of the teacher. Suppose the teacher in a college course is a leading member of the scientific profession, with many famous results to his name. He approaches the class with an atmosphere with confidence, and thus conveys a subtle feeling that everything in the subject is known and clear, its just a matter of time before he will be able to convey it to the students and then the students will know what they needed to be taught.

if such is the case, what effect does it have on the students? Even the good ones retreat into a passive mode, where the teacher leads and they follow. They do not originate questions themselves, because a complete presentation of all issues will be given to them in due course. If something is not addressed then they imagine that it cannot be of much importance, since the teacher is a leader in the field and will completely cover all that is really needed.

Conversely, I find that the truly animated students are to be found in a course where the students believe that they know about as much as the teacher, or at any rate that they can obtain a clearer alternative understanding of some t opics if they strive hard enough. There is joy in leaping over the authority represented by the teacher, but even more exhilarating is the feeling that any ideas they explore from this point on are “new territory”, at least as far as their local knowledge sources are concerned.

I remember that in my course of introductory physics in college we had lectures (ably given by a principal instructor) followed by recitation sections carried out by other teachers (the education at IIT Kanpur was modeled after the system at MIT, Boston.) Our recitation instructor would put a problem on the board, and say quite simply, that he did not have a good idea of how it was to be done. I recall a feeling of eagerness to step up to the board and show the teacher and the class a path to the solution, and many other students were equally eager to chip in. We did not have a high opinion of the teacher (we just thought of him as a ‘nice guy’) but we learnt quite a bit of physics. In later years, when I look back at those classes I realize that perhaps that teacher fooled us all; he probably knew the answers perfectly well, but in an ultimate act of self-sacrifice made us believe that we students controlled the learning in the class.

A second, related problem is that in the modern method of teaching the teacher decides what he will teach, chooses what he will ask in the exam, and assigns the grades; he is in complete control of learning in that course for that semester. While it is often said that this allows the teacher to be “creative” about what he will teach, and to tailor the course to the class needs, I find that it also has the effect of ‘enslaving’ the students to the teacher. They learn only what he teaches; they don’t read the book. Often they don’t buy the book, or they sell it when the course is done. The exam questions are similar to what the teacher said in class, so they just try to remember what the teacher said, rather than learn in depth the basic principles.

I recall the graduate qualifier examination at Ohio State. The students take graduate courses on mechanics, electrodynamics, quantum mechanics etc., and then must pass the exam to start their Ph.D. Sometimes the exam in a subject was set by the same instructor who taught the corresponding course; in this event, the students all did quite well. But when the questions were made by a different professor, the  scores were much lower! If the teacher sets the exam himself then the students get a false sense of confidence; the questions may look hard but they can be answered by closely repeating what was done in class.

With all these issues in my mind, let me try to outline how I would plan and teach a course. I will take as an example an undergraduate course in electrodynamics, a course that I have had occasion to teach over the years. Rather than detail my experiences from any one quarter of teaching let me make an imaginary vignette out of the experiences that I had over the different times that I taught the course. I will also add suggestions that I have not had occasion to implement but would like to, the next time I teach the course.

At a typical University this course could have mostly juniors (third year of the undergraduate curriculum), though there will also likely be some sophomores and seniors. The first thing is to assign a good text for the course; in this case, the book by Griffiths has served me well. The important thing is that I will then follow the text closely, so that the students do so too and learn to use that book as a resource for all time.

I also state at the start of the course what chapters will be covered. The students can thus read ahead if they wish, and the good ones will. I would like to ask a different professor to make the exams. The students should know that the exam will not be made by me, so it is up to them to absorb with clarity what is in the book. The exam questions should be long ones where they have to develop a whole train of thought and computation, not multiple choice type questions. It is good to ask for at least one “derivation” of a formula. This is rather uncommon these days: Students are usually taught how to use formulas/formulae, derivation is considered a waste of time. But by working through derivation they learn the process of scientific thought by which results are arrived at, and if they are to obtain any new results of their own later in life it is invaluable that they imbibe the process of systematic and rigorous reasoning needed to obtain a result.

The next important issue is to identify the basic principles of the subject, and insist on 100% clarity on these topics. Just ‘exposing’ students to a wide swath of material is not particularly useful. II would keep the syllabus comparatively light, and would avoid excessive use of demos, movies, etc. so that student time is freed up for thorough, introspective learning. For the course under discussion this means for instance that we learn 100% clearly the force law between charges, both stationary and moving, and we work through a detailed derivation of the Gauss law, which is an important insight obtained from the force law. I have not found it very helpful to conduct demos showing giant sparks jump between electrodes, such demos do have entertainment value, but it is questionable what physics stays in the mind at the end.

The first time I was teaching this course I had a strange experience. The students would do very well on quite difficult questions, and then get stuck on seemingly simple one. We had come to a point where the students had to find the electric field produced by a charge distributed uniformly through a cone. Only a few could do this; the rest asked for ‘similar solved examples’ which they could then use to do the problem at hand. After some discussion with them I realized that this had become the way they always did problems. Look at solved examples, and then try to fit the given problem as closely as possible into the pattern of one of these examples. But this makes a mockery of the very purpose of assigning problems for homework! I would think that the idea of homework problems is that students try to absorb the BASIC principles by struggling to apply them to specific situations.

To remedy this I tried the following strategy, which worked out surprisingly well. I invited all those students who were unsure of how to attack this and similar problems to show up for an evening session, 7.00 pm to 9.00 pm. About half the class, 15 students, came. I put the question on the board and sat down at the back of the class, asking the students to figure out a solution. After some initial hesitation one girl got up and made a first attempt. I made it clear that small steps and potentially wrong directions were all fully welcome since they help to channel thinking. When the first student got stuck, someone piped in with a suggestion and took up the chalk. Over half an hour of intense discussion the essential idea emerged — one had to slice up the cone into thin discs; the result for a disc was known, and these contributions could be then added up. But this is essential idea underlying the whole of calculus! So with a bit of thought the class had uncovered the philosophy of calculus for themselves. My only contribution was to point out this fact to them, which they found quite thrilling.

We held more such sessions, and I could feel that as time passed the students were gradually moving away from “plug and play” as the method of solving problems to a more introspective approach where puzzling a few hours (or even days) over a confusion became an acceptable way to spend time.

A last point about the course. When preparing a lecture one develops a ‘story’ for each topic. What is the issue, why it is important, how it is to be understood, how it relates to other things we have learnt. For example, one comes across the force law on moving charges. If I have to prepare a lecture, I have to start with the electric force on static charges, how relativity tells us that there must be a magnetic force on moving charges, and finally go through a derivation to arrive at the force. It is only the complete story that can hold interest, the final answer by itself is only a dry formula just like so many others. It turns out to be useful to have the student develop understandings of little topics, and have them teach it to the rest of the class. Rather than learn isolated facts they learn to see the complete picture around a formula. For example, the students can give mini-lectures on the solenoid and the field it creates, the Biot-Savart law for magnetostatics and its derivation etc. The students pay much more attention when one of them is struggling to get the ideas across, and they learn with more active participation.

In conclusion, let me return to the theme I have tried to develop here. It may seem that this essay has been rather negative; many modern approaches to teaching have been criticized, and it may seem that there is a reactionary desire to return to the ‘good old days’ familiar from my own youth. But, I would say instead that the message here is one of hope. Suppose great education needed teachers who are great pedagogues and masters of their field, and expensive ‘hands on’ equipment. Then, only a privileged few will be able to receive this great education, perhaps the students at expensive Universities like MIT; this is certainly true of my field in physics, and I see a similar pattern among those that work in engineering or management related professions. Software engineers from a variety of Indian colleges challenge the best in the world today, and multinational biotech companies are setting up shop in India to avail of our educated talent pool.

So, in the end, we see that learning does not have that much to do with ‘great teaching’. It is the student’s mind that has to reach out and gather knowledge from books and introspection, with the teacher being at best a facilitator for the process. The student does not have to despair that he cannot find that magical personality who will inspire and make it all clear. The inspiration to learn is in all of us, though all too often the burden placed upon the mind by educators dims our desire to ask and understand. And, a good teacher is one who understands this, understands that learning is the students’ own journey in which the teacher is an occasional helper, not the master.

Samir Mathur obtained his Masters in Physics from Indian Institute of Technology, Kanpur in 1981 and a Ph.D. at TIFR, Bombay in 1987. This was followed by postdoctoral work at TIFR and Harvard. He was on the faculty of MIT 1991-1999 and is currently a Professor at Ohio State. He received the Distinguished Teaching Award in 2003-04.

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Many thanks to Prof Samir Mathus from Nalin Pithwa and his readers/students.

Some random problems in algebra (part b) for RMO and INMO training

1) Solve in real numbers the system of equations:

$y^{2}+u^{2}+v^{2}+w^{2}=4x-1$

$x^{2}+u^{2}+v^{2}+w^{2}=4y-1$

$x^{2}+y^{2}+v^{2}+w^{2}=4u-1$

$x^{2}+y^{2}+u^{2}+w^{2}=4v-1$

$x^{2}+y^{2}+u^{2}+v^{2}=4w-1$

Hints: do you see some quadratics ? Can we reduce the number of variables? …Try such thinking on your own…

2) Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be real numbers such that $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=0$ and $\max_{1 \leq i . Prove that $a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2} \leq 10$.

3) Let a, b, c be positive real numbers. Prove that

$\frac{1}{2a} + \frac{1}{2b} + \frac{1}{2d} \geq \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a}$

More later

Nalin Pithwa.

PROMYS: Boston; Program in Mathematics for Young Scientists

https://promys.org

For promising Indian students, the Mehta Scholarships are also available.

— Shared by Nalin Pithwa for spreading awareness of math in India also…