Mascot Mantis

mantis-01.JPG

A mascot is any person, animal, or object thought to bring luck, or anything used to represent a group with a common public identity, such as a school, professional sports team, society, military unit, or brand name. [Source]

A few months back, while I was riding my Motorcycle to work, I noticed that a small green praying mantis was stationed on the speedometer. I thought it would move away once the bike starts. So didn’t give it much thought and started the drive. I have to cover three traffic signals to go to work. Now when I stopped at the first signal I noted that the little mantis was still there, holding on to the rim and the glas of the speedometer. As the signal went green, we started the journey again, this time we had appreciable speed as we  were on the highway. I was keeping a tab on the mantis, thinking that it would be flicked away by the air flow. It crouched and held on, no matter how much I accelerated. Its antennae went back with the wind and at times it really struggled to keep on the position. It looked as if it was determined to come with me (or lead me) to the workplace. A personal mascot for me! Leading me through my journey of life. When I finally came to my office, I placed it over the nearby shrub hoping the best for it. (Unfortunately, I did not have my camera at that time, so could not take any photos. )

Having a mantis as a mascot is not a bad thing at all. In the manga Baki Son of Ogre (Vol 2), the hero shadow fights with a mantis for practice, as by weight they are perhaps the strongest of animals. It is claimed that if the mantis was ~100 kg they could hunt a full grown African elephant singly (Note that manga are read from right-to-left). That is the ratio of their weights and size to that of their preys. Since a mascot is supposed to represent you and your qualities, these are no bad at all.

So I am all for a mantis as a mascot, after all, it choose me!

The photo at the top has another story to it. One evening I was just strolling on a terrace when I found this impressive specimen. I placed my camera on the ground to get closer and better photos. The mantis became aggressive
(too friendly??) and came on the camera itself, just after this photo was taken.

 

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Multiple citations, biblatex and APA

Issues with multiple citations of an author and biblatex-apa

Recently I had to refer an author multiple times in a document. I had recently shifted to biber and biblatex. The American Psychological Association (APA) style was the recommended style.

And in the final references

The author name was, let us, was Wilbur-Markus Rowland, and he had about 10-11 articles. The author had many co-authors, in some cases this author was the first author in other cases he was the second or even third one.
Now the issue that I was facing was this: The same author name was getting cited differently at different places.
For example, in some cases it appeared like
W.-M. Rowland, in other cases Wilbur-Markus Rowland, and in some cases even as Rowland. This was very confusing. And on top of all that the final references had entries like

Rowland W.-M. (W. M.)
Rowland W.-M. (Wilbur-Markus)
Rowland W. M. (Wilbur-Markus)
Rowland W. M. (W.-M.)

I checked and rechecked the bibtex entries in hope of finding some error but it was not to be found. I must have done a Clean All a dozen times, after editing the bib file.. I tried adding same name entries for Rowland as first author to be the same, but it didn’t work. This was really frustrating.

Similarly a couple of references with two different authors with a common surname were giving intials in the main text. For example, D. R. Cook and M. P. Cook. Now all other refernces (except the Rowland one) were coming as per the APA requirement, which is (Author, Year), so ideally they should have been (Cook & Weston, 1999) and (Cook, 2004). But rather they were being displayed as (D. R. Cook & Weston, 1999) and (M. P. Cook, 2004). Now no change in the bib file was changing any of this. I thought there was some issue with the bib entries. I deleted the entries and entered them again manually, but no avail.

Then I chanced upon this link

 even though the year of publication differs in the two Campbell (Cook in our case) references, the lead author’s initials should be included in all text citations, regardless of how often they appear.

So the Cook mystery was solved. biblatex was compiling correctly as per the APA guidelines, the initials for the two different Cook entries must be there. This is because

 Including the initials helps the reader avoid confusion within the text and locate the entry in the reference list.

Now for the Rowland entry, the next part of the blog gave me a hint towards the possible problem:

Although this rule seems straightforward, one thing that trips up some writers is how to proceed when different lead authors with the same surname are also listed in other references in which they are not the lead author.

After this I checked the entries where Rowland was second or third author. These entries differed from the entries where he was the lead author. And this was causing the problem. For each different entry style of his name, biblatex was considering him as a separate author. Hence the initials and the different references entries with names in brackets. Once all the entries for Rowland were made consistent the problem disappeared. Phew!

This issue bugged me for almost a couple of working days, to find the cause and subsequent addressing was most rewarding.

Note that the bibtex references that I had were taken from google scholar, hence differing styles in author name. Please make them consistent. This is a warning for future me and others who are reading this post.

 

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Hymn of Creation from Rig Veda

This wonderful Hymn of Creation one of the oldest surviving records of philosophic doubt in the history of the world, marks the development of a high stage of abstract thinking, and it is the work of a very great poet, whose vision of the mysterious chaos before creation, and of mighty ineffable forces working in the depths of the primeval void, is portrayed with impressive economy of language.

“Then even nothingness was not, nor existence.
There was no air then, nor the heavens beyond it
What covered it? Where was it? In whose keeping?
Was there then cosmic water, in depths unfathomed?

“Then there were neither death nor immortality,
nor was there then the torch of night and day.

The One breathed windlessly and self-sustaining.
There was that One then, and there was no other.
“At first there was only darkness wrapped in darkness.
All this was only unillumined water.

That One which came to be, enclosed in nothing,
arose at last, bom of the power of heat.
“In the beginning desire descended on it
that was the primal seed, bom of the mind.

The sages who have searched their hearts with wisdom
know that which is is kin to that which is not.
“And they have stretched their cord across the void,
and know what was above, and what below.

Seminal powers made fertile mighty forces.
Below was strength, and over it was impulse,
“But, after all, who knows, and who can say
whence it all came, and how creation happened?

The gods themselves are later than creation,
so who knows truly whence it has arisen?
“Whence all creation had its origin,
he, whether he fashioned it or whether he did not,

he, who surveys it all from highest heaven,
he knows— or maybe even he does not know.

From – The Wonder That Was India – A. L. Basham

 

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Review of I Am A Strange Loop by Douglas Hofstadter – Part 2

Part 1

The toilet flush is one of the simplest and common feedback mechanisms that we find. There is a float which rises with the water level which controls the inflow of water. After a certain height is reached the water inflow is stopped. Do we attribute intentionality to the water flush? We usually do not. And this is the theme that Hofstadter explores in Chapter 4 Loops, Goals and Loopholes.

But what kinds of systems have feedback, have goals, have desires? Does a soccer ball rolling down a grassy hill “want” to get to the bottom? 52

We anthropomorphize objects and impart them our human attributes. Adding a “purpose” or a “goals” to any system is considering it from a teleological perspective. Teleology is the explanation of phenomena by the purpose they serve rather than by postulated causes. Considering examples of variations on this theme, we can say that answer to the above question is not clear cut. There are no black-and-white answers but are judgment calls. We tend to move towards the idea of teleology and intention for a system when the feedback mechanisms are not directly perceptible.

Among other examples, Hofstadter considers plants which in normal time will appear to be static and without any “goals”. But a time-lapse of the same would show that they have “goals” and “intentions” and use strategies to achieve them.

The question is whether such systems, despite their lack of brains, are nonetheless imbued with goals and desires. Do they have hopes and aspirations? Do they have dreads and dreams? Beliefs and griefs? 53

The claim is made that presence of a feedback loop in a system, triggers in us a response which shifts the description from a goalless level of mechanics to a goal-oriented level of some cognitive mechanism. Things have the desire to move!

So far we have considered basic feedback loops. Now we move onto a more complex idea of a positive feedback loop. In a positive feedback loop, a part of the output of the system goes into increasing the output by a certain factor. With each iteration the output increases, which causes the next output to increase even more. A small change in input can cascade into a very large change (exponential) in output.

Perhaps the most common example of a positive feedback loop is the unpleasant, high pitch sound one hears in an auditorium or a meeting. This happens when a microphone gains some of its output as an input and produces an ever increasing pitch and volume of the input sound. An example is given below:

Now one can imagine that due to the exponential nature of growth, any little disturbance in such a system might lead to a sound which will eventually destroy everything.

In theory, then, the softest whisper would soon grow to a roar, which would continue growing without limit, first rendering everyone in the auditorium deaf, shortly thereafter violently shaking the building’s rafters till it collapsed upon the now-deaf audience, and then, only a few loops later, vibrating the planet apart and finishing up by annihilating the entire universe. What is specious about this apocalyptic scenario?

But this is a fallacious argument. The first fallacy is the physical nature of the setup and the amplifier in our scheme of things. If the roof falls, it will destroy the amplifier too! The second case is the nature of the amplifier, it doesn’t amplify in an unlimited way. After a certain gain, due to the physical design, the amplification becomes equal to unity and the system stabilizes at its natural frequency. It so happens that the natural high frequency of an audio amplifier is close to a high pitch scream. This is achieved by the system tends to go towards that pitch in series of rapid iterations. These are the screeching high pitch oscillations that we hear. It seems the systems “wanted” to go there, the stable point of its existence. Thus we see that

Similarly, we can also “see” visual feedback loops, when the output of a camera is given back to the camera. This can be most easily setup by pointing the camera towards a screen which is showing a live output of the camera. The cover image of the book is one such image, captured during Hofstadter’s “experiments” with the visual feedback system. One of the difference, in this case, is that the camera is not an amplifying device, it just transmits. Yet the pictures it produces are bizarre and beautiful. Seeing images of video feedback gives one a sense of mystery and wonder. There is some inherent beauty in it, yet it seems un-natural to watch.

Feedback — making a system turn back or twist back on itself, thus forming some kind of mystically taboo loop — seems to be dangerous, seems to be tempting fate, perhaps even to be intrinsically wrong, whatever that might mean. 57

Shifting gears, we get a Hofstadter’s introduction to Gödel when he was fourteen. What intrigued him was the thought that one could have an entire book about a single book. The book was Nagel and Newman’s Gödel’s Proof, published in 1958. Hofstadter wrote the introduction to the new imprint in 2001. He was fascinated by footnote on formal use of quotation marks.

So here was a book talking about how language can talk about itself talking about itself (etc.), and about how reasoning can reason about itself (etc.). I was hooked! I still didn’t have a clue what Gödel’s theorem was, but I knew I had to read this book. 58

This is something that happens to me too. Some time back (almost a decade now) I had posted about books attracting me. Perhaps it happens to many people.

We next look at famous Russel’s Paradox. One of the examples derived from it is Barber’s Paradox

The barber is the “one who shaves all those, and those only, who do not shave themselves.” The question is, does the barber shave himself? [.]

There is also a loop here and there is contradiction too.

This loophole (the word fits perfectly here) was based on the notion of “the set of all sets that don’t contain themselves”, a notion that was legitimate in set theory, but that turned out to be deeply self-contradictory. 60

Russell tried to overcome this by formally re-defining the concepts of sets to save this, but it didn’t work out well. Rather it became too complex, though built on solid, atomic (in the mathematical sense) ideas.

In Principia Mathematica, there was to be no twisting-back of sets on themselves, no turning-back of language upon itself.  61

But why is self-reference considered problematic? Here Hofstadter quotes from his column Metamagical Themas (an anagram of Martin Gardner’s Mathematical Games) in Scientific American on Self-Referential sentences. But all were not receptive to the idea, some of the readers were sceptical about the utility of self-reference and denied any meaningful output of such activities.

In the next chapter On Video Feedback we explore the theme of video of video feedback and Hofstadter’s experiments with it. He explores and explains many of the images which were made by adding slight things in the image, fox example, truncated corridor, endless corridor, helical corridor etc. The common element in all these video feedback is the repeating of the primary image in scaled down fashion till the resolution of the screen can support (theoretically infinite). During one the experiments, he covers the lens and then removes his hands. During this, the movement of his hand is captured and forms an endless image which is moving, even when the hand is removed. This action has formed a loop and is feeding itself in a cyclic setup.

A faithful image of something changing will itself necessarily keep changing! 67

A similar phenomenon is that of dogs barking in sync. Some dog somewhere, starts to bark for something that is passing near it. Now, other dogs pick up and start barking too. And the chain goes on. Once setup, it doesn’t matter what was the reason for the first dog to bar, it may have gone away. But the chain of barking sustains itself. During one the flights, I have seen this happen with small babies. There were about 5-6 babies on the flight. It so happened that one of them started to cry for some reason. Then the rest joined in one-by-one. Perhaps the others were crying because the heard another one cry. And the event became self-sustaining. This went on for quite some time.

This is one of the core idea of an emergent phenomenon, once

In general, an emergent phenomenons omehow emerges quite naturally and automatically from rigid rules operating at a lower, more basic level, but exactly how that emergence happens is not at all clear to the observer. 68

The video explorations led to some fantastic images, many of which are reproduced in color in the central pages of the book. In the last part of the chapter, Hofstadter drives towards one of the central themes which we will explore in the remaining book. The idea is that strange and robust (self-sustaining) structures can emerge from the process of looping.

Once a pattern is onthe screen, then all that is needed to justify its staying up there is George Mallory’s classic quip about why he felt compelled to scale Mount Everest: “Because it’s there!” When loops are involved, circular justifications are the name of the game. 70

Some of the images I myself have collected are shown below:

The locking-in gives rise to abstract phenomena at higher levels.

In short, there are surprising new structures that looping gives rise to that constitute a new level of reality that could in principle be deduced from the basic loop and its detailed properties, but that in practice have a different kind of “life of their own” and that demand — at least when it comes to extremely finite, simplicity-seeking, new level of description that transcend the basic level out of which they emerge. 71

Whether we will be able to actually do it, or want to do it is another question. This reminds me of the saying: In theory, there is no different in theory and practice, in practice there is.

Here are a few more:

 

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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.
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Review of I Am A Strange Loop by Douglas Hofstadter – Part 1

I recently finished I Am A Strange Loop by Douglas Hofstadter. The book is an introduction to the core ideas about self, self-reference, feedback loops and consciousness as  an emergent phenomena. The core question that is considered is

What do we mean when we say I?

Hofstadter in the preface indicates his angst at many people missing out on the core ideas of Gödel, Escher, Bach: An Eternal Golden Braid. No doubt GEB is hard to read, and each one makes their own meaning of it.

Years went by, and I came out with other books that alluded to and added to that core message, but still there didn’t seem to be much understanding out there of what I had really been trying to say in GEB. xiii

I Am A Strange Loop is sort of a prequel to GEB, which came afterwards. In the book the focus is on developing an idea of emergent self, in which our consciousness is seen to emerge from feedback that we have by interacting with the world. Hofstadter uses a variety of examples to drive home the point of recursive feedback loops, giving rise to strange phenomena. The central claim is that we, our sense of self, our idea of consciousness derives from recursive interactions and feedback that we get via our senses.

He starts with a dialogue he wrote as a teenager between Plato and Socrates about what is it to be alive and being conscious, this in a way sets the stage for things to come. In the first chapter On the Souls and Their Sizes we are made to think about presence of souls in different foods that we eat (he himself doesn’t partake mammalian meat). We non-chalantly eat a tomato, irritatingly squish a mosquito, but what happens when we eat higher life forms, like chicken, pigs and sheep? Do they have souls? Do all living beings have souls? If so, then does the soul of a human is greater than that of a cow (now here I must be careful, there are people in my country who judge the soul of a cow much much greater than that of a human being), of a pig, of a chicken, of a mosquito of a tomato?

Does a baby lamb have a soul that matters, or is the taste of lamb chops just too delicious to worry one’s head over that? 18

The suggestive answer is  given in a conciousness cone, in which we normal adult humans are at the top and atoms are the start of the cone. But then granted that we have a soul, are we born with a fully developed one? Here Hofstadter takes a developmental approach to the concept of the soul. The idea is that we are born with some essence of what appears to be soul, then gradually over the years it develops. The concept of soul here is used interchageably with “I”. The main take home point in this chapter is whatever this is, we do not get the fully developed version of it from birth. Rather it is a developmental process which takes place in the real world, shaped by experiences. The said developmental changes are in degree, rather than a black/white switch.

In the second chapter This Teethering Bulb of Dread and Dream we look at possible ways of studying the mechanisms of the brain which might potentially shed some light on the puzzle that we are after. In general the idea of studying the hardware of the brain seems to be set in agenda of many neurologists. But Hofstadter argues against this way of studying thinking.

Saying that studying the brain is limited to the study of physical entities such as these would be like saying that literary criticism must focus on paper and bookbinding, ink and its chemistry, page sizes and margin widths, typefaces and paragraph lengths, and so forth. 26

Another analogy given is that of the heart. Just like heart is a pumping machine, brain is a thinking machine. If we only think heart as an aggregate of cells, we miss out on the bigger picture of what the cells do. The heart surgeons don’t think about heart cells but look at the larger structure. Similarly to study thinking the lower level of components may not be the correct level to study highly abstract phenomena such as concepts, analogies, consciousness, empathy etc. This is pointing towards thinking as an emergent phenomena, emerging from the interactions at lower levels which are composed of objects/entities which are not capable of thinking.

Hofstadter then takes philosopher John Searle to task for his views regarding impossibility of thinking arising from non-thinking entities. The analogy of a beer can to a neuron is taken apart. What is suggested by Searle in his thought experiments is equivalent to memory residing in a single neuron. But this certainly is not the case. We have to think of the brain as a multi-level system. But going too deep in these levels we would not get a comprehensible understanding of our thinking.

Was it some molecules inside my brain that made me reshelve it? Or was it some ideas in my brain? 31

Rather it is ideas that make more ideas.

Ideas cause ideas and help evolve new ideas. They interact with each other and with other mental forces in the same brain, in neighboring brains, and, thanks to global communication, in far distant, foreign brains. And they also interact with the external surroundings to producein toto a burstwise advance in evolution that is far beyond anything to hit the evolutionary scene yet, including the emergence of the living cell. Sperry as quoted on 31-32

Another analogy that is given is that of Thermodynamics and Statistical Mehcanics. Just as atoms interact in a gas at a micro-level to create gas laws which can be observed at a macro-level. The macro-level laws also makes it comprehensible to us, because of the sheer amount of information at mirco level that one would have to analyse to make sense. (Provided that we can in theory solve such a massive set of equations, not considering the quantum mechanical laws.) Similarly the point is made that for understanding a complex organ such as the brain, which contains billions of interacting neurons, we should not look at the hardware at the lowest level, but rather look for macro-level patterns.

Statistical mentalics can be bypassed by talking at the level of thinkodynamics. 34

The perception of the world that we get is from sensory inputs, language and culture. And it is at that level we operate, we do not seek atomic level explanations for the dropping of the atomic bomb. This simplification is part of our everyday explanation, and we choose the levels of description depending on the answers that we are seeking.

Drastic simplification is what allows us to reduce situations to their bare bones, to discover abstract essences, to put our fingers on what matters, to understand phenomena at amazingly high levels, to survive reliably in this world, and to formulate literature, art, music, and science. 35

The third chapter The Causal Potency of Patterns provides us with concrete metaphors to think about emergent phenomena and thinking at levels. The first of such metaphors is a chain of dominoes, which can be thought of as a computer program for carrying out a given computation. In this case finding checking if a number is prime: 641. Now a person watching the domino fall right upto 641 can presumably give two answers, the first one is that the domino before 641 did not fall, while other is 641 is a prime number. These two answers are many levels apart. The second example is of Hofstadter sitting a traffic jam, The reason why you are stuck in traffic, is because the car in front of you is not moving. On the other hand this does not tell you anything about  why the jam arose in the first place, which may be due to a large number of cars going home after a game or a natural disaster of some kind. The main idea is that we can have two (many?) levels of explanation each one looking at the system from a different level of detail, for example, the car ahead of you local,  the reasons for the jam global. As far as the causal analysis goes we can look at answers at different levels.

Deep understanding of causality sometimes requires the understanding of very large patterns and their abstract relationships and interactions, not just the understanding of microscopic objects interacting in microscopic time intervals. 41

Similar example is that of a combustion engine. The designers of the engine do not think about molecular level of interactions, the level that is relevant for them is the thermodynamic level of pressure, temeperature and volume. The properties of individual molecules like their locations, velocities is irrelevant in such a description, though the properties of the ensemble is.

This idea — that the bottom level, though 100 percentresponsible for what is happening, is nonetheless irrelevant to what happens — sounds almost paradoxical, and yet it is an everyday truism. 42

Another example that is given is of listening to music. Lets say you hear a piece of music, and you experience some emotions due to it. Now, consider there was a slight delay before playing started, the actual molecules which vibrated to get you the music, would be different than in the first case. Yet, you would experience the music in the same way even though the molecules that brought you that music were completely different.

The lower-level laws of their collisions played a role only in that they gave rise to predictable high-level events. But the positions, speeds, directions, even the chemical identity of the molecules – all of this was changeable, and the high-level events would have been the same. 42

Thus we can say that a lower level might be responsible for a higher level event and at the same time is irrelevant to the higher level.

 

The next metaphor we consider is that of careenium and simmbalism. (No points for guessing what the intended puns are here!) There are many witty puns throughout the book, and Hofstadter uses them very effectively to make his points. This Gedankenexperiment is referred to many times in the book. Simms (small interacting marbles) are very small marbles, which can crash into each other and bounce off the walls in a frictionless world. They are also magnetic so that if they hit each other with low velocity they can “stick” to each other and form clusters called simmballs. A simmball can be composed of millions of simms, and may loose or gain simms at its boundary. Thus we have tiny and agile simms, and huge and nearly immobile simmballs. All this bashing and boucing happens at frictionless pooltable, the careenium.

After setting this metaphorical system we add another complexiety that external events can affect the simmballs, thus we can have a record of history by reading the configurations of simmballs. Now a reductionist approach to this system would be that we really need to know only about nature of interaction of the simms, rest are just epi-phenomena, which can be explained by behavior of the simms. But such a view isnot helpful in many ways. One of the issues that is raised is that of enormous complexity raised by such approach will render it meaningless. But, whether we can even describe a phenomena in a truly fundamental way, just by using basic laws is itself questionable.

A interesting reading in similar line of though is by Anderson (Anderson, P. W. (1972). More is different. Science, 177(4047), 393-396). He gives examples from physical science which seemingly defy solutions or explanations on basis of the fundamental laws. He strongly argues against the reductionist hypothesis

The main fallacy in this kind of thinking is that the reductionist hypothesis does not by .any means imply a “constructionist” one: The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe, In fact, the more the elementary particle physicists tell us about the nature of the fundamental laws, theless relevance they seem to have to the
very real problems of the rest of science, much less to those of society.

Anderson draws three inferences from this 1) Symmetry is of great importance to physics; symmetry the existence of different viewpoints from which the system appears the same. 2) the internal structure of a piece of matter need not be symmetrical even if the total state of it is.

I would challenge you to start from the fundamental laws of quantum mechanics and predict the ammonia inversion and its easily observable properties without going through the stage of using the unsymmetrical pyramidal structure, even though no “state” ever has that structure.

3) the state of a really big system does not at all have to have the symmetry of the laws which govern it; in fact, it usually has less symmetry.

Starting with the fundamental laws and a computer, we would have to do two impossible things – solve a problem with infinitely many bodies, and then apply the result to a finite system-before we synthesized this behavior

Finally Anderson notes:

Synthesis is expected to be all but impossible analysis, on the other hand, may be not only possible but fruitful in all kinds of ways: Without an understanding
of the broken symmetry in superconductivity, for instance, Josephson would probably not have discovered his effect.

Going back to Hofstadter, he considers a higher level view of the Gedankenexperiment with simms, simmballs and careenium. To get a birds eye view of our  have to zoom out both space and time. The view that we will get is that of simmballs, simms would be to small and too fast for us to view at this level. In fast forward of time, the simmballs are no longer stationary, but rather are dynamic entities which change their shapes and positions due to interactions of simms (now invisible) at lower level. But this is not evident at this level, though the simms are responsible for changing the shape and position of simmballs, they are irrelevant as far as description of simmballs.

And so we finally have come to the crux of the matter: Which of these two views of the careenium is the truth? Or, to echo the key question posed by Roger Sperry, Who shoves whom around in the population of causal forces that occupy the careenium? 49

The answer is that it all depends on which level you choose to focus on. The analogy can be made clear by thinking of how billions of interacting nuerons form patterns of thought, analogy, interacting ideas. Thus while trying to think about thinking we should let go of observing a single neuron, or the hardware of the brain itself, it will not lead us to any comprehensible description or explanation of how we think. Nuerons are though responsible for thinking they are irrelevant in the higher order of thinking.

 

 

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The True Purpose Of Graphic Display – J. W. Tukey

John Wilder Tukey, one of the greatest Statistician of the last century points to what the purpose of a graphic display should be:

  1.  Graphics are for the qualitative/descriptive – conceivably the semi quantitative – never for the carefully quantitative (tables do that better).
  2. Graphics are for comparison – comparison of one kind or another – not for access to individual amounts.
  3. Graphics are for impact – interocular impact if possible, swinging-finger impact if that is the best one can do, or impact for the unexpected as a minimum – but almost never for something that has to be worked at hard to be perceived.
  4. Finally, graphics should report the results of careful data analysis – rather than be an attempt to replace it. (Exploration-to guide data analysis – can make essential interim use of graphics, but unless we are describing the exploration process rather than its results, the final graphic should build on the data analysis rather than the reverse.)

From:

Tukey, J. W. (1993). Graphic comparisons of several linked aspects: Alternatives and suggested principles. Journal of Computational and Graphical Statistics, 2(1), 1-33.
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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.

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Erudition without…

Erudition without bullshit, intellect without cowardice, courage without imprudence, mathematics without nerdiness, scholarship without academia, intelligence without shrewdness, religiosity without intolerance, elegance without softness, sociality without dependence, enjoyment without addiction, and, above all, nothing without skin in the game.
(A letter of advice to a younger person) source

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Automation is set to hit workers in developing countries hard | The Outline

In the absence of a heavy tax on robots, the report notes, “developing countries should embrace the digital revolution” by “redesigning education systems to create the managerial and labor skills needed to operate new technologies.”

Source

Wonder where we actually stand on this in India?

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