mathematics

The Mitr

On not learning or con in the context

We will, we will, fail you by testing what you do not know…

We live in a rather strange world. Or is it that we assume the world
to be non-strange in a normative way, but the descriptive world has
always been strange? Anyways, why I say this is to start a rant to
about some obviously missed points in the area of my work. Namely,
educational research, particularly science and mathematics education
research.
In many cases the zeal to show that the students have
‘misunderstandings’ or are simply wrong, and then do a hair-splitting
(micro-genetic) exercise on the test the students were inflicted
with. Using terse jargon and unconsequential statistics, making the
study reports as impossible to read as possible, seem to be the norm.
But I have seen another pattern in many of the studies, particularly
in mathematics education. The so-called researchers spent countless
nights in order to dream up situations as abstract as possible (the
further far away from real-life scenarios the better), then devise
problems around them. Now, these problems are put in research studies,
which aim to reveal (almost in evangelical sense) the problems that
plague our education. Unsuspecting students are rounded, with
appropriate backgrounds. As a general rule, the weaker socio-economic
background your students come from, the more exotic is your study. So
choose wisely. Then these problems are inflicted upon these poor,
mathematically challenged students. The problems will be in situations
that the students were never in or never will be. The unreal nature of
these problems (for example, 6 packets of milk in a cup of coffee! I
mean who in real life does that? The milk will just spill over, the
problem isn’t there. This is just a pseudo-problem created for satisfying the research question of the researcher. There is no context, but only con.
Or finding out a real-life example for some weird fractions) puts many off. The fewer students perform correctly happier the researcher is. It just adds to the data statistic that so many % students cannot perform even this elementary task well. Elementary for
that age group, so to speak. The situation is hopeless. We need a
remedy, they say. And remedy they have. Using some revised strategy,
which they will now inflict on students. Then either they will observe
a few students as if they are some exotic specimens from an
uncontacted tribe as they go on explaining what they are doing or why
they are doing it. Or the researcher will inflict a test (or is it
taste) in wholesale on the lot. This gives another data
statistic. This is then analysed within a ‘framework’, (of course it
needs support) of theoretical constructs!
Then the researcher armed with this data will do a hair-splitting
analysis on why, why on Earth student did what they did (or didn’t
do). In this analysis, they will use the work of other researchers before
them who did almost the same thing. Unwieldy, exotic and esoteric
jargons will be used profusely, to persuade any untrained person to
giveup on reading it immediately. (The mundane, exoteric and
understandable and humane is out of the box if you write in that
style it is not considered ‘academic’.) Of course writing this way,
supported by the statistics that are there will get it published in
the leading journals in the field. Getting a statistically significant
result is like getting a license to assert truthfulness of the
result. What is not clear in these mostly concocted and highly
artificial studies is that what does one make of this significance
outside of the experimental setup? As anyone in education research
would agree two setups cannot be the same, then what is t
Testing students in this way is akin to learners who are learning a
new language being subjected to and exotic and terse vocabulary
test. Of course, we are going to perform badly on such a test. The
point of a test should be to know what students know, not what they
don’t know. And if at all, they don’t know something, it is treated as
if is the fault of the individual student. After all, there would be
/some/ students in each study (with a sufficiently large sample) that
would perform as expected. In case the student does not perform as
expected we can have many possible causes. It might be the case that
the student is not able to cognitively process and solve the problem,
that is inspite of having sufficient background knowledge to solve the
problem at hand the student is unable to perform as expected. It might
be the case that the student is capable, but was never told about the
ways in which to solve the given problem (ZPD anyone?). In this case, it might be that the curricular materials that the student has access
to are simply not dealing with concepts in an amenable way. Or it
might be that the test itself is missing out on some crucial aspects
and is flawed, as we have seen in the example above. The problem is
systemic, yet we tend to focus on the individual. This is perhaps
because we have a normative structure to follow an ideal student at
that age group. This normative, ideal student is given by the so-called /standards of learning/. These standards decide, that at xx age
a student should be able to do multiplication of three digit
numbers. The entire curricula are based on these standards. Who and
what decides this? Most of the times, the standards are wayyy above
the actual level of the students. This apparent chasm between the
descriptive and the normative could not be more. We set unreal
expectations from the students, in the most de-contextualised and
uninteresting manner, and when they do not fulfil we lament the lack
of educational practices, resources and infrastructure.

The Mitr

What is a mathematical proof?

A dialogue in The Mathematical Experience by Davis and Hersh on what is mathematical proof and who decides what a proof is?
Let’s see how our ideal mathematician (IM) made out with a student who came to him with a strange question.
Student: Sir, what is a mathematical proof?
I.M.: You don’t know that? What year are you in?
Student: Third-year graduate.
I.M.: Incredible! A proof is what you’ve been watching me do at the board three times a week for three years! That’s what a proof is.
Student: Sorry, sir, I should have explained. I’m in philosophy, not math. I’ve never taken your course.
I.M.: Oh! Well, in that case – you have taken some math, haven’t you? You know the proof of the fundamental theorem of calculus – or the fundamental theorem of algebra?
Student: I’ve seen arguments in geometry and algebra and calculus that were called proofs. What I’m asking you for isn’t examples of proof, it’s a definition of proof. Otherwise, how can I tell what examples are correct?
I.M.: Well, this whole thing was cleared up by the logician Tarski, I guess, and some others, maybe Russell or Peano. Anyhow, what you do is, you write down the axioms of your theory in a formal language with a given list of symbols or alphabet. Then you write down the hypothesis of your theorem in the same symbolism. Then you show that you can transform the hypothesis step by step, using the rules of logic, till you get the conclusion. That’s a proof.
Student: Really? That’s amazing! I’ve taken elementary and advanced calculus, basic algebra, and topology, and I’ve never seen that done.
I.M.: Oh, of course, no one ever really does it. It would take forever! You just show that you could do
it, that’s sufficient.
Student: But even that doesn’t sound like what was done in my courses and textbooks. So mathematicians don’t really do proofs, after all.
I.M.: Of course we do! If a theorem isn’t proved, it’s nothing.
Student: Then what is a proof? If it’s this thing with a formal language and transforming formulas, nobody ever proves anything. Do you have to know all about formal languages and formal logic before you can do a mathematical proof?
I.M.: Of course not! The less you know, the better. That stuff is all abstract nonsense anyway.
Student: Then really what is a proof?
I.M.: Well, it’s an argument that convinces someone who knows the subject.
Student: Someone who knows the subject? Then the definition of proof is subjective; it depends on particular persons.Before I can decide if something is a proof, I have to decide who the experts are. What does that have to do with proving things?
I.M.: No, no. There’s nothing subjective about it! Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you’ll catch on.
Student: Are you sure?
I.M.: Well – it is possible that you won’t, if you don’t have any aptitude for it. That can happen, too.
Student: Then you decide what a proof is, and if I don’t learn to decide in the same way, you decide I don’t have any aptitude.
I.M.: If not me, then who?

The Mitr

Mathematical Literacy Goals for Students

National Council of Teachers for Mathematics NCTM proposed these five goals to cover the idea of mathematical literacy for students:

  1. Learning to value mathematics: Understanding its evolution and its role in society and the sciences.
  2. Becoming confident of one’s own ability: Coming to trust one’s own mathematical thinking, and having the ability to make sense of situations and solve problems.
  3. Becoming a mathematical problem solver: Essential to becoming a productive citizen, which requires experience in a variety of extended and non-routine problems.
  4. Learning to communicate mathematically:  Learning the signs, symbols, and terms of mathematics.
  5. Learning to reason mathematically: Making conjectures, gathering evidence, and building mathematical arguments.
National Council of Teachers of Mathematics. Commission on Standards for School Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Natl Council of Teachers of.
The Mitr

Reflections on Liping Ma’s Work

Liping Ma’s book Knowing and teaching elementary mathematics has been very influential in Mathematics Education circles. This is a short summary of the book and my reflections on it.

Introduction

Liping Ma in her work  compares the teaching of mathematics in the American and the Chinese schools. Typically it is found that the American students are out performed by their Chinese counterparts in mathematical exams. This fact would lead us to believe that the Chinese teachers are better `educated’ than the U.S. teachers and the better performance is a straight result of this fact. But when we see at the actual schooling the teachers undergo in the two countries we find a large difference. Whereas the U.S. teachers are typically graduates with 16-18 years of formal schooling, the typical Chinese maths teacher has about only 11-12 years of schooling. So how can a lower `educated’ teacher produce better results than a more educated one? This is sort of the gist of Ma’s work which has been described in the book. The book after exposing the in-competencies of the U.S. teachers also gives the remedies that can lift their performance.
In the course of her work Ma identifies the deeper mathematical and procedural understanding present, called the profound understanding of fundamental mathematics [PUFM] in the Chinese teachers, which is mostly absent in the American teachers. Also the “pedagogical content knowledge” of the Chinese teachers is different and better than that of the U.S. teachers. A teacher with PUFM “is not only aware of the conceptual structure and the basic attitudes of mathematics inherent in elementary mathematics, but is able to teach them to students.” The situation of the two teacher is that the U.S. teachers have a shallow understanding of a large number of mathematical structures including the advanced ones, but the Chinese teachers have a deeper understanding of the elementary concepts involved in mathematics. The point where the PUFM is attained in the Chinese teachers is addressed. this Also the Chinese education system so structured that it allows cooperation and interaction among the junior and senior teachers.

Methodology

The study was conducted by using the interview questions in Teacher Education and Learning to Teach Study [TELT] developed by Deborah Ball. These questions were designed to probe teacher’s knowledge of mathematics in the context of common things that teachers do in course of teaching. The four common topics that were tested for by the TELT were: subtraction, multiplication, division by fractions and the relationship between area and perimeter. Due to these diverse topics in the questionnaire the teachers subject knowledge at both conceptual and procedural levels at the elementary level could be judged quite comprehensively. The teacher’s response to a particular question could be used to judge the level of understanding the teacher has on the given subject topic.

Sample

The sample for this study was composed of two set of teachers. One from the U.S., and another from China. There were 23 U.S. teachers, who were supposed to be above average. Out of these 23, 12 had an experience of 1 year of teaching, and the rest 11 had average teaching experience of 11 years. In China 72 teachers were selected, who came from diverse nature of schools.In these 72, 40 had experience of less than 5 years of teaching, 24 had more than 5 years of teaching experience, and the remaining 8 had taught for more than 18 years average. Each teacher was interviewed for the conceptual and procedural understanding for the four topics mentioned.
We now take a look at the various problems posed to the teachers and their typical responses.

Subtraction with Regrouping

The problem posed to the teachers in this topic was:

Lets spend some time thinking about one particular topic that you may work with when you teach, subtraction and regrouping. Look at these questions:
62
– 49
= 13
How would you approach these problems if you were teaching second grade? What would you say pupils would need to understand or be able to do before they could start learning subtraction with regrouping?

Response

Although this problem appears to be simple and very elementary not all teachers were aware of the conceptual scheme behind subtraction by regrouping. Seventy seven percent of the U.S. teachers and 14% of U.S. teacher had only the procedural knowledge of the topic. The understanding of these teachers was limited to just taking and changing steps. This limitation was evident in their capacity to promote conceptual learning in the class room. Also the various levels of conceptual understanding were also displayed. Whereas the U.S. teachers explained the procedure as regrouping the minuend and told that during the teaching they would point out the “exchanging” aspect underlying the “changing” step. On the other hand the Chinese teachers used subtraction in computations as decomposing a higher value unit, and many of them also used non-standard methods of regrouping and their relations with standard methods.
Also most of the Chinese teachers mentioned that after teaching this to students they would like to have a class discussion, so as to clarify the concepts.

Multidigit Multiplication

The problem posed to the teachers in this topic was:
Some sixth-grade teachers noticed that several of their students were making the same mistake in multiplying large numbers. In trying to calculate:
123
x 645
13
the students were forgetting to “move the numbers” (i.e. the partial products) over each line.}
They were doing this Instead of this
123 123
x 64 x 64
615 615
492 492
738 738
1845 79335
While these teachers agreed that this was a problem, they did not agree on what to do about it. What would you do if you were teaching the sixth grade and you noticed that several of your students were doing this?}

Response

Most of the teachers agreed that this was a genuine problem in students understanding than just careless shifting of digits, meant for addition. But different teachers had different views about the error made by the student. The problem in the students understanding as seen by the teachers were reflections of their own knowledge of the subject matter. For most of the U.S. teachers the knowledge was procedural, so they reflected on them on similar lines when they were asked to. On the other hand the Chinese teachers displayed a conceptual understanding of the multidigit multiplication. The explanation and the algorithm used by the Chinese teachers were thorough and many times novel.

Division by Fractions

The problem posed to the teachers in this topic was:

People seem to have different approaches to solving problems involving division with fractions. How do you solve a problem like this one?
1/(3/4) / 1/2 = ??
Imagine that you are teaching division with fractions. To make this meaningful for kids, sometimes many teachers try to do is relate mathematics to other things. Sometimes they try to come up with real-world situations or story-problems to show the application of some particular piece of content. What would you say would be good story or model for 1/(3/4) / 1/2 ?

Response

As in the previous two cases the U.S. teachers had a very weak knowledge of the subject matter. Only 43% of the U.S. teachers were able to calculate the fraction correctly and none of them showed the understanding of the rationale underlying their calculations. Only one teacher was successful in generating an illustration for the correct representation of the given problem. On the other hand all the Chinese teachers did the computational part correctly, and a few teachers were also able to explain the rationale behind the calculations. Also in addition to this most of the Chinese teachers were able to generate at least one correct representation of the problem. In addition to this the Chinese teachers were able to generate representational problems with a variety of subjects and ideas, which in turn were based on their through understanding of the subject matter.

Division by Fractions

The problem posed to the teachers in this topic was:

Imagine that one of your students comes to the class very excited. She tells you that she has figured out a theory that you never told to the class. She explains that she has discovered the perimeter of a closed figure increases, the area also increases. She shows you a picture to prove what she is doing:
Example of the student:
How would you respond to this student?

Response

In this problem task there were two aspects of the subject matter knowledge which contributed substantially to successful approach; knowledge of topics related to the idea and mathematical attitudes. The absence or presence of attitudes was a major factor in success
The problems given to the teachers are of the elementary, but to understand them and explain them [what Ma is asking] one needs a profound understanding of basic principles that underly these elementary mathematical operations. This very fact is reflected in the response of the Chinese and the U.S. teachers. The same pattern of Chinese teachers outperforming U.S. teachers is repeated in all four topics. The reason for the better performance of the Chinese teachers is their profound understanding of fundamental mathematics or PUFM. We now turn to the topic of PUFM and explore what is meant by it and when it is attained.

PUFM

According to Ma PUFM is “more than a sound conceptual understanding of elementary mathematics — it is the awareness of the conceptual structure and the basic attitudes of mathematics inherent in elementary mathematics and the ability to provide a foundation for that conceptual structure and instill those basic attitudes in students. A profound understanding of mathematics has breadth, depth, and thoroughness. Breadth of understanding is the capacity to connect topic with topics of similar or less conceptual power. Depth of the understanding is the capacity to connect a topic with those of greater conceptual power. Thoroughness is the capacity to connect all these topics.”
The teacher who possesses PUFM has connectedness, knows multiple ways of expressing same thing, revisits and reinforces same ideas and has a longitudinal coherence. We will elaborate on these key ideas of PUFM in brief.
Connectedness: By connectedness being present in a teacher it is meant that there is an intention in the teacher to connect mathematical procedures and concepts. When this is used in teaching it will enable students to learn a unified body of knowledge, instead of learning isolated topics.
Multiple Perspectives: In order to have a flexible understanding of the concepts involved, one must be able to analyze and solve problems in multiple ways, and to provide explanations of various approaches to a problem. A teacher with PUFM will provide multiple ways to solve and understand a given problem, so that the understanding in the students is deeper.
Basic Ideas: The teachers having PUFM display mathematical attitudes and are particularly aware of the powerful and simple concepts of mathematics. By revisiting these ideas again and again they are reinforced. But focusing on this students are not merely encouraged to approach the problems, but are guided to conduct real mathematical activity.
Longitudinal Coherence: By longitudinal coherence in the teachers having PUFM it is meant that the teacher has a complete markup of the syllabus and the content for the various grades of the elementary mathematics. If one does have an idea of what the students have already learnt in the earlier grades, then that knowledge of the students can be used effectively. On the other hand if it is known what the students will be learning in the higher grades, the treatment in the lower grades can be such that it is suitable and effective later.

PUFM – Attainment

Since the presence of PUFM in the Chinese teachers makes them different from their U.S. counterparts, it is essential to have a knowledge of how the PUFM is developed and attained in the Chinese teachers. For this Ma did survey of two additional groups. One was ninth grade students, and the other was that of pre-service teachers. Both groups has conceptual understanding of the four problems. The preservice teachers also showed a concern for teaching and learning, but both groups did not show PUFM. Ma also interviewed the Chinese teachers who had PUFM, and explored their acquisition of mathematical knowledge. The teachers with PUFM mentioned several factors for their acquisition of mathematical knowledge. These factors include:

  • Learning from colleagues
  • Learning mathematics from students.
  • Learning mathematics by doing problems.
  • Teaching
  • Teaching round by round.
  • Studying teaching materials extensively.

The Chinese teachers during the summers and at the beginning of the school terms , studied the Teaching and Learning Framework document thoroughly. The text book to be followed is the most studied by the teachers. The text book is also studied and discussed during the school year. Comparatively little time is devoted to studying teacher’s manuals. So the conclusion of the study is that the Chinese teachers have a base for PUFM from their school education itself. But the PUFM matures and develops during their actual teaching driven by a concern of what to teach and how to teach it. This development of PUFM is well supported by their colleagues and the study materials that they have. Thus the cultural difference in the Chinese and U.S. educational systems also plays a part in this.

Conclusions

One of the most obvious outcomes of this study is the fact that the Chinese elementary teachers are much better equipped conceptually than their U.S. counterparts to teach mathematics at that level. The Chinese teachers show a deeper understanding of the subject matter and have a flexible understanding of the subject. But Ma has attempted to give the plausible explanations for this difference in terms of the PUFM, which is developed and matured in the Chinese teachers, but almost absent in the U.S. teachers. This difference in the respective teachers of the two countries is reflected in the performance of students at any given level. So that if one really wants to improve the mathematics learning for the students, the teachers also need to be well equipped with the knowledge of fundamental and elementary mathematics. The problems of teacher’s knowledge development and that of student learning are thus related.
In China when the perspective teachers are still students, they achieve the mathematical competence. When they attain the teacher learning programs, this mathematical competence is connected to primary concern about teaching and learning school mathematics. The final phase in this is when the teachers actually teach, it is here where they develop teacher’s subject knowledge.  Thus we see that good elementary education of the perspective teachers themselves heralds their growth as teachers with PUFM. Thus in China good teachers at the elementary level, make good students, who in turn can become good teachers themselves, and a cycle is formed. In case of U.S. it seems the opposite is true, poor elementary mathematics education, provided by low-quality teachers hinders likely development of mathematical competence in students at the elementary level. Also most of the teacher education programs in the U.S. focus on How to teach mathematics? rather than on the mathematics itself. After the training the teachers are expected to know how to teach and what to teach, they are also not expected to study anymore. All this leads to formation of a teacher who is bound in the given framework, not being able to develop PUFM as required.
Also the fact that is commonly believed that elementary mathematics is basic, superficial and commonly understood is denied by this study. The study definitively shows that elementary mathematics is not superficial at all, and anyone who teaches it has to study it in a comprehensive way. So for the attainment of PUFM in the U.S. teachers and to improve the mathematics education their Ma has given some suggestions which need to be implemented.
Ma suggests that the two problems of improving the teacher knowledge and student learning are interdependent, so that they both should be addressed simultaneously. This is a way to enter the cyclic process of development of mathematical competencies in the teachers. In the U.S. there is a lack of interaction between study of mathematics taught and study of how to teach it. The text books should be also read, studied and discussed by the teachers themselves as they will be using it in teaching in the class room. This will enable the U.S. teachers to have clear idea of what to teach and how to teach it thoughtfully. The perspective teachers can develop PUFM at the college level, and this can be used as the entry point in the cycle of developing the mathematical competency in them. Teachers should use text books and teachers manuals in an effective way. For this the teacher should recognize its significance and have time and energy for the careful study of manuals. The class room practice of the Chinese teachers is text book based, but not confined to text books. Again here the emphasis is laid on the teacher’s understanding of the subject matter. A teacher with PUFM will be able to choose materials from a text book and present them in intelligible ways in the class room. To put the conclusions in a compact form we can say that the content knowledge of the teachers makes the difference.

Reflections

The study done by Ma and its results have created a huge following in the U.S. Mathematics Education circles and has been termed as `enlightening’. The study diagnoses the problems in the U.S. treatment of elementary mathematics vis-a-vis Chinese one. In the work Ma glorifies the Chinese teachers and educational system as against `low quality’ American teachers and educational system. As said in the foreword of the book by Shulman the work is cited by the people on both sides of the math wars. This book has done the same thing to the U.S. Mathematics Education circles what the Sputnik in the late 1950’s to the U.S. policies on science education. During that time the Russians who were supposed to be technically inferior to the U.S. suddenly launched the Sputnik, there by creating a wave of disgust in the U.S. This was peaked in the Kennedy’s announcement of sending an American on moon before the 1970’s. The aftermath of this was to create `Scientific Americans’, with efforts directed at creating a scientific base in the U.S. right from the school. Similarly the case of Ma’s study is another expos\’e, this time in terms of elementary mathematics. It might not have mattered so much if the study was performed entirely with U.S. teachers [Have not studies of this kind ever done before?]. But the very fact that the Americans are apparently behind the Chinese is a matter of worry. This is a situation that needs to be rectified. This fame of this book is more about politics and funding about education than about math. So no wonder that all the people involved in Mathematics Education in the U.S. [and others elsewhere following them] are citing Ma’s work for changing the situation. Citing work of which shows the Americans on lower grounds may also be able to get you you funds which otherwise probably you would not have got. Now the guess is that the aim is to create `Mathematical Americans’ this time so as to overcome the Chinese challenge.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

The Mitr

Topological Art

ILLUSTRATIONS FOR TOPOLOGY
From the book Introduction to Topology by Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko. The book was published by Mir Publishers in 1985.

ILLUSTRATION TO CHAPTER I
The central part of the picture presents the standard embedding chain of crystalline groups of the three dimensions of Euclidean space: their standard groups embedded into each other are depicted as fundamental domains (Platonic bodies: a cube, a tetrahedron, a dodecahedron). The platonic bodies are depicted classically, i.e., their canonical form is given, they are supported by two-dimensional surfaces (leaves), among which we discern the projective plane (cross-cap), and spheres with handles. The fantastic shapes and interlacings (as compared with the canonical objects) symbolizes the topological equivalence.
At the top, branch points of the Riemann surfaces of various multiplicities are depicted: on the right, those of the Riemann surfaces of the functions w=5z√ and w=z√; on the left below, that of the same function w=z√, and over it, a manifold with boundary realizing a bordism mod 3.

ILLUSTRATION TO CHAPTER II
The figure occupying most of the picture illustrates the construction of a topological space widely used in topology, i.e., a 2-adic solenoid possessing many interesting extremal properties. The following figures are depicted there: the first solid torus is shaded, the second is white, the third is shaded in dotted lines and the fourth is shaded doubly. To obtain the 2-adic solenoid , it is necessary to take an infinite sequence of nested solid tori, each of which encompasses previous twist along its parallel, and to form their intersection.
Inside the opening, a torus and a sphere with two handles are shown. The artist’s skill and his profound knowledge of geometry made it possible to represent complex interlacing of the four nested solid tori accurately.

ILLUSTRATION TO CHAPTER III
The canonical embedding of a surface of genus g into the three-dimensional Euclidean space is represented 0n the right . A homeomorphic embedding of the same surface is shown on the left . The two objects are homeomorphic, homotopic and even isotopic . The artist is a mathematician and he has chosen these two, very much different in their appearance, from an infinite set of homeomorphic images.

ILLUSTRATION TO CHAPTER IV
Here an infinite total space of covering over a two-dimensional surface, viz., a sphere with two handles, is depicted. The artist imparted the figure the shape of a python and made the base space of the covering look very intricate. Packing spheres into the three-dimensional Euclidean space and a figure homeomorphic to the torus are depicted outside the central object. The mathematical objects are placed so as to create a fantastic landscape.

ILLUSTRATION TO CHAPTER V
A regular immersion of the projective plane RP2 in R3 is represented in the centre on the black background. The largest figure is the Klein bottle (studied in topology as a non-orientable surface) cut in two (Moebius strips) along a generator by a plane depicted farther right along with the line intersection; the lower part is plunging downwards; the upper part is being deformed (by lifting the curve of intersection and building the surface up) into a surface with boundary S1; a disc is being glued to the last, which yields the surface RP2. The indicated immersion process can be also used for turning S2 `inside out’ into R3.
On the outskirts of the picture, a triangulation of a part of the Klein bottle surface is represented.
A detailed explanation of this picture may serve as a material for as much as a lecture in visual topology.

The Mitr

A parable on…

A Parable

Once upon a time, in a far away country, there was a community that had a wonderful machine. The machine had been built by most inventive of their people … generation after generation of men and women toiling to construct its parts… experimenting with individual components until each was perfected… fitting them together until the whole mechanism ran smoothly. They had built its outer casing of burnished metal and on one side, they had attached a complex control panel. The name of the machine, KNOWLEDGE, was engraved on a plaque  set in the centre of the control panel.
The community used the machine in their efforts to understand the world and to solve all kinds of problems. But the leaders of the community were not satisfied. It was a competitive world… they wanted more problems solved and they wanted them solved faster.
The main limitation for the use of machine was the rate at which data could be prepared for input. Specialist machine operators called ‘predictors’, carried out this exacting and time consuming task… naturally the number of problems solved each year depended directly on the number and skill of the predictors.
The community leaders focussed on the problem of training predictors. The traditional method, whereby promising girls and boys were taken into long-term apprenticeship, was deemed too slow and too expensive. Surely, they reasoned, we can find more efficient approach. So saying,  they called the elders together and asked them to think about the matter.
After a few months, the elders reported that they had devised an approach that showed promise. In summary, they suggested that the machine be disassembled. Then each component could be studied and understood with ease… the operation of machine would become an open book to all who cared to look.
Their plan was greeted with enthusiasm. So, the burnished covers were pulled off, and the major mechanisms of the machine fell out… they had plaques with labels like HISTORY and GEOGRAPHY and PHYSICS and MATHEMATICS. These mechanisms were pulled apart in their turn… of course, care was taken to keep all the pieces in separate piles. Eventually, the technicians had reduced the machine to little heaps of metal plates and rods and nuts and bolts and springs and gear wheels. Each heap was put in a box, carefully labelled with the name of the mechanism whose part it contained, and the boxes were lined up for the community to inspect.
The members of the community were delighted. Their leaders were ecstatic. They ‘oohed’ and ‘aahed’ over the quality of components, the obvious skill that had gone in their construction, the beauty of designs. Here, displayed for all, were the inner workings of KNOWLEDGE.
In his exuberance, one man plunged his hand into a box and scooped up a handful of tiny, jewel-like  gear wheels and springs. He held them out to his daughter and glancing, at the label on the box, said:
“Look, my child! Look! Mathematics! ”
From: Turtle Speaks Mathematics by Barry Newell
You can get the book (and another nice little book Turtle Confusion) here.
 

The Mitr

Gel’fand’s Quote

This is taken from The Method of Coordinates by I. M. Gel’fand
E.G. Glagoleva A.A. Kirillov

Of course, it was not our intention that aIl these
students who studied from these books or even
completed the School should choose mathematics as
their future career. Nevertheless, no matter what they
would later choose, the results of this training re­
mained with them. For many, this had been their first
experience in being able to do something on their own
— completely independently.

1 would like to make one comment here. Sorne of my
American colleagues have explained to me that
American students are not really accustomed to think­
ing and working hard, and for this reason we must
make the material as attractive as possible. Permit me
to not completely agree with this opinion. From my
long experience with young students aU over the
world 1 know that they are curious and inquisitive and
1 beIieve that if they have sorne clear mate rial pre­
sented in a simple form, they will prefer this to aIl
artificial means of attracting their attention — much as
one ,buys books for their content and not for their
dazzling jacket designs that engage only for the
moment.

The most important thing a student can get from the
study of mathematics is the attainment of a higher
intellectualleveL In this light 1would like to point out
as an example the famous American physicist and
teacher Richard Feynman who succeeded in writing
both his popular books and scientific works in a
simple and attractive manner.

I. M. Gel’fand

The Mitr

Heaven and Hell

Circle Limit IV
Heaven and Hell
by M C Escher
Yesterday I have put up Escher’s Circle Limit IV – Heaven and Hell on my new desk. The Circle Limit series of drawings was drawn by Escher are essentially what are known as his hyperbolic tesselations. The new computer table that I have got has an odd shape. On one end the side is circular and it smoothly metamorphises into rectangle on the other side. Though it is not at all comparable to what Escher has accomplished, I feel bad even when I use the word metamorphosis for this, but I have not found anything better. The table is designed for use with a desktop. So it has sections for different parts of the desktop like the monitor, CPU keyboard etc.
Anyways the main point that I want to tell is that the table at one end is circular. Since I had put Escher’s Three World on another table, I thought it would be a good idea to use a ciruclar print of Escher for this part of the table. Of all the prints I had, which I had taken when I had at my disposal A3 sized printers, the one which fitted the purpose seemed to be Circle Limit IV – Heaven and Hell.
Let us see what Escher himself has to say about this series of works viz. The Circle Limits:
So far four examples have been shown with points as limits of infinite smallness. A diminution in the size of the figures progressing in the opposite direction, i.e. from within outwards, leads to more satisfying results. The limit is no longer a point, but a line which border’s the whole complex and gives it a logical boundary. In this way one creates, as it were, a universe, a geometrical enclosure. If the progressive reduction in size radiates in all directions at an equal rate, then the limit becomes a circle. [1]
And he says this about Heaven and Hell:
CIRCLE LIMIT IV, (Heaven and Hell)
[Woodcut printed from2 blocks, 1960, diameter 42 cm]
Here also we have the components diminishing in size as they move outwards. The six largest (three white angels and three black devils) are arranged about the centre and radiate from it. The disc is divided into six sections in which, turn and turn about, the angels on a black background and then the devils on a white one, gain the upper hand. in this way, heaven and hell change place six times. In the intermediate, “earthly” stages, they are equivalent. [1]
Like most of Escher’s drawings this one also takes you to a different world. A world which is far away from the reality. A world of mathematics. A world of abstraction. But then as always we can make connections between this abstract world and the real world. The connections that we can make are dependent on the world view that we have. Some people fail to make the connection. They cannot `see’.
The Circle Limit series is what brought Escher to the eyes of the mathematicians. H. S. M. Coxeter used Circle Limit II as an illustration in his article on hyperbolic tesselations. Since then the other works of Escher have been examined by the mathematicians, and we find that very deep and fundamental ideaso of mathematics are embedded in them. As to how Escher did it is amazing. The kind of clear insight that Escher exhibits in his artwork is astounding. He could visualize the mathematical transformations in his head and then transform them onto the artwork he was working with. Escher has said
I have brought to light only one percent of what I have seen in the darkness. [2]
This must be certainly true, as most of his artwork is nowhere close to what we see in the light. I rate the artwork of Escher as greater than that of the renessaince artist’s as they had just beautifully drawn what one could “see.” But with Escher we go a step beyond, imagination takes the control. What interests me in Escher is that he can make you imagine the unimaginable. What you know is not possible is demonstrated just in front of your eyes. Logic is discarded. Rather it is kept in the basement which is upstairs for Escher.
Yesterday you start to believe what you thought was impossible tommorow.
The way different things merge for Escher is just unparalled in the work of other artists. What has now become known as “Escheresque” is just the typical of his style. Lot of later artists are influenced by the works of Escher, I have found one Istvaan Orosz particulary good. There are others who are equally good but I don’t remember their names now….
Coming back to Heaven and Hell. The main artwork is in a woodcut format in black and white. For me this is a kind of dyad which represents the world. The idea of two opposing forces one termed to be evil and the other good are all permeating in the Universe. Here also the bat-devils and the angels are the representative of the same. There is no part of the Universe where these two are not present. It might seem that somewhere far out there there is nothing, but it is not so. Even there, the design is the same, it is just too far for us to see. This is what harmony in the universe is about. It is the same everywhere, when you have a broad enough world-view. The cosmologists say that the Universe is homogenous and isotropic, if you choose to “see” it at the right scale. The cosmologists often use Heaven and Hell to illustrate this point. For me introduction to Escher came in a talk by a cosmologist who used The Waterfall to illustrate the idea of a perpetual motion machine. Since then I have become addicted to Escher, as has everybody else who has some sense of imagination. For those who cannot appreciate Escher, I can just pity at their miserable imagination.
References:
[1] The Graphic Work of M C Escher by M C Escher
Ballantine 1975, ISBN 345246780595
[2] M. C. Escher (Icons) by Julius Wiedemann (Editor)
Taschen 2006, ISBN 3822838691