Author: The Mitr

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

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.
 
 

The Mitr

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.
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.

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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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Reductionism in Science

Many scientists look on chemistry and physics as ideal models of what psychology should be like. After all, the atoms in the brain are subject to the same all – inclusive physical laws that govern every other form of matter. Then can we also explain what our brains actually do entirely in terms of those same basic principles? The answer is no, simply because even if we  understood how each of our billions of brain cells work separately, this would not tell us how the brain works as an agency. The “laws of thought” depend not only upon the properties of those brain cells,but also on how they are connected. And these connections are established not by the basic, “general” laws of physics, but by the particular arrangements of the millions of bits of information in our inherited genes. To be sure, “general” laws apply to everything. But, for that very reason, they can rarely explain anything in particular.
– Marvin Minsky in The Society of Mind pp. 26

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They Thought They Were Free

In this post we will look at some experiences that people in Germany had during the rise of Nazi Party. Overall the trend is that you make it almost impossible for anyone opposed to your thought as an outcast, and others just follow the herd. Many measures of the present incumbent have parallels to this. And especially the current drama of demonetization of high denomination currency notes.

“This separation of government from people, this widening of the gap, took place so gradually and so insensibly, each step disguised (perhaps not even intentionally) as a temporary emergency measure or associated with true patriotic allegiance or with real social purposes. And all the crises and reforms (real reforms, too) so occupied the people that they did not see the slow motion underneath, of the whole process of government growing remoter and remoter.”

They say that it was essential that it should have been kept secret, otherwise the point of exercise would have been futile. People of the country are asked to make sacrifices for the betterment of the country. Otherwise the country was in crisis. So we had to take emergency steps. What is happening in all this introduced chaos is the issues which need to go in public imagination are removed. These are issues which the government doesn’t want people to discuss, debate. Like a magician they are directing the public attention with gimmicks and shenanigans when their slight of hand remains invisible from public scrutiny.
In all these perception managing exercise the ever changing breaking news in our main-stream-media plays an ubiquitous role. They are supposed to be a pillar in the democratic process, but instead we find that they are malleable and play hand-maiden’s role for diverting and capturing public imagination. Most of the time this is in sync with what the incumbent government wants.

“The dictatorship, and the whole process of its coming into being, was above all diverting. It provided an excuse not to think for people who did not want to think anyway. I do not speak of your ‘little men,’ your baker and so on; I speak of my colleagues and myself, learned men, mind you. Most of us did not want to think about fundamental things and never had. There was no need to. Nazism gave us some dreadful, fundamental things to think about—we were decent people—and kept us so busy with continuous changes and ‘crises’ and so fascinated, yes, fascinated, by the machinations of the ‘national enemies,’ without and within, that we had no time to think about these dreadful things that were growing, little by little, all around us. Unconsciously, I suppose, we were grateful. Who wants to think?”

Thinking is also outsourced. Experts are called in, to provide excerpts from ideas too complex and too common for common citizens to comprehend. Each idea is digested in studios, what is generated is a pre-digested version of the ideas so that you don’t need to do it. You perhaps do not have time to do it. There are more relevant things than ruminating about rationalisations regarding political policies. And if at all you do question or think about these, one of the basic logical fallacy of ad hominem is employed. Shoot the messenger, we already have the message (or massage after McLuhan). Messenger is the mess-maker. Here in public imagination the questioner becomes the questioned. The questions are irrelevant, motive, history and ideological stance of the person asking the question is more important. Questioning policies and performance metamorphose from act of trying to understand to act of treason to undermine.
The perpetrator becomes predated. Overnight they are condemned to become public and hence national enemies. Any one who does not support becomes anti-national by default. To live here you have to live by our rules, otherwise you should go away. Who gave this authority to them? This is again questioned back, you must have something to hide, hence you are not supporting this. Then it captures public imagination, those questioning are enemies within. Dissent is treason.
And we have in form of Pakistan the “Other”. The national enemy without. When there is a dullness in the public imagination, raise the ante in form of the bogey man for all our troubles. Again here the pattern is well laid out.

“Uncertainty is a very important factor, and, instead of decreasing as time goes on, it grows. Outside, in the streets, in the general community, ‘everyone’ is happy. One hears no protest, and certainly sees none.

What might happen next, what event will break the news, tomorrow is unknown. Through surveys through debates it is brought to fore that “All is well.” If it is not well for you, the trouble is with you. All the problems are only for people who are enemies within. Those supporting, are the ones who are honest, happy and hardworking. Rest of you need to prove you allegiance, we already have by token sloganeering, literally and figuratively both.

“And you are an alarmist. You are saying that this must lead to this, and you can’t prove it. These are the beginnings, yes; but how do you know for sure when you don’t know the end, and how do you know, or even surmise, the end? On the one hand, your enemies, the law, the regime, the Party, intimidate you. On the other, your colleagues pooh-pooh you as pessimistic or even neurotic. You are left with your close friends, who are, naturally, people who have always thought as you have.”

If you think too much you become the pseudo-leftist-communist-atheist-liberal. Taking a stance against the establishment is an act of defiance against the country. The content of the stance is not important, questioning is. Your thought is against the wisdom of the entire society. The entire system acts against you. The system forces you to choose. AADHAR is a case in this category. All the arguments against are drowned in a sea of arguments which do not address the concerns raised. Look at the benefits for the poor they say, those who fall in line, are normal. It is only people like you living in ivory towers feel bad about it. Outside everyone is using it, and they are happy about it. The dubious and shady way in which it was legalised itself should have sent shudders, but rather it has evoked a lukewarm response. Those in power are so intoxicated that even supreme court ruling that it should not be mandatory is ignored left right and center. Each day incrementally small changes and notifications are provided. Each day it is becoming near impossible to live without it. This is already under the premise that it is needed. No one can question that. And if you do, they ask what is your problem? Why can’t you fall in line? Just accept it will you. People must have bank account and must have ID cards, who cares if it is not constitutionally mandated?

Now you live in a world of hate and fear, and the people who hate and fear do not even know it themselves; when everyone is transformed, no one is transformed.

But till we find out and till we realise the water is already boiling and we and our coming generation is already cooked.
They Thought They Were Free  – Milton Mayer

The Mitr

Millions of Computers for Millions of Children

Yesterday ( it should be now read “a couple of years back”)while giving a talk, I was asked this rhetorical question (not verbatim, but close):

“What did you say was the sample size of your study?”
“Two. This was a case study.”
“So, considering that the activity that you have designed requires a computer and expeyes (a hardware for collecting data). How can you scale it up to schools which have millions of children?”

It seems that the person who was asking the question, for lack of any other question asked this. In seminars and academic institutes, there are always people like this, who will ask the question for sake of it. Just to make their presence felt. Anyways, it was good for me. I was expecting that this question would be asked. And I was very happy that it was asked.
The short answer that I gave was:

“You give a million computers to a million children!”

one-computer-per-child
Some people thought, this was a rhetoric answer to a rhetoric question, which incidentally was also humorous, as it also generated a lot of laughter, but this was not the case. In this post, I would like to elaborate on the short answer that I gave.
Of course, most of these ideas have come from reading and hearing Seymour Papert (who has recently demised, the article was started before that, but due to my lethargy never seen completion). The memes have been transferred, and now I am trying to make sense and adapt them to my own experience. And I would like to assert again that reading Papert has been an immensely rewarding and enriching experience for me. His are perhaps few books which I do not mind reading again and again. I like his writing style of giving parables to explain points in his arguments because the points he wants to make do not need a backbone of statistics to survive. Here also I will give a hypothetical example (derived from Papert) to explain what I meant.
The technological tools that children are using now mainly in the traditional school system are the pencil and the book. In this case, almost all educationalists would agree that every child would require to have one pencil to write and book for study. Even then there are some children who do use computers, some because their parents have them, some because the school has them, some have both. Now we consider a time 50 years back. Computers were almost non-existent, as we know them now. Computers were one of the most complicated and expensive technological artefacts that humans produced. But the enormous amount of money and efforts were put in the miniaturization of computers. So finally now we have computers that have become devices that we now know. In the last 50 years, the computer technology has grown exponentially, while the prices for the memory and computing power that one gets are falling, their usage.
Consider a classroom of 50 years back. Though there were computers they were something to be wondered about, something like very very expensive toys. The computers were not mature enough that children could handle them. In the classroom, the only available technological artefacts were used. The technology in the classroom was the pencil
and the printed book and a notebook to write with the pencil and of course, there was the blackboard.
Wait, you might be thinking we are in a digital age technology by default means computers, be it in your smart-phone, laptop or a desktop or at least a projector for god’s sake. But here I would like you to think about somethings which are very deeply embedded in our cultural psyche. The very fact that many things which we take for granted are
all technologies. For example, the writing instruments that you have to be it a pencil or a chalk are all technologies. But most of us don’t think of them as such because they are so common and most of us have had our experience with them. The mystery is lost. As the Arthur C. Clarke once said about technology and magic as his Third Law:

Any sufficiently advanced technology is indistinguishable from magic.

So deeply embedded this image is that we define it as the normal for our learners to be able to use this technology. Rather the entire edifice of our educational system rests on it. For example, your educational achievement is more or less based on the fact how much you can “write” in a limited time, from memory. And this we call assessment, examination and the like. Also the written text, from the time of Gutenberg, has more
or less complete hold over our intellectual activities. The text formed the basis of our discourse and analysis of the world. Why do children use to write with a pencil on piece of paper in order to learn. The drill typically starts with the children trying to
recreate elegant fonts in some shape or form which is decipherable for the teacher. You have to write “A” 500 times to get it right, ok? How would you write words when you cannot write alphabets? How would write sentences when you cannot write words? How will you write examinations if you cannot write sentences?
Is it the only way in which we can learn language? If we observe this in detail we see that only reason we ask them to write “a” 500 times in a notebook is because it comes from an era when there was no other technology to write. And this is the same learner who can converse well and answer questions, but yet we need them to write it down with their hands. It was the only possible solution. And generations of humans were trained using this method. So much so that most of us still think this is the only method for education. Any deviation from hand-written text is seen as a abomination. But typing on a computer provides us, and especially, young learners with cognitive offloading of immense task of holding a writing instrument and shaping an alphabet, a word, a sentence out of it. Children learn to type much much faster than they learn to write with a pen. And what is even more important is that the written text is in electronic form, which can be revised and shared with their peers and teachers. In hand written text there is no question of revision, the original takes too much effort to produce so there is no question of revising it.
one-pencil-per-child
Considering the amount of cognitive load the child has to undergo to produce decipherable alphabets, words and sentences in order to “write”, a thing which he can perfectly do orally, are the results worth the effort? Are there any studies which show that this is an efficient method? Yet is used everywhere without exceptions and we accept it meekly without challenge because this is how it was done in the past and someone in the past must have had good reason to use this hence, we should also use this. Papert calls this as “QWERTY Phenomena”. Somethings just get culturally embedded because the are
suited for an bygone era, the are like relics in the current era. And writing with pencil and paper is just one of them.
Now consider the question that was asked at the beginning of the post. Replace the computer with a pencil. The question then becomes,

“So, considering that the activity that you have designed requires a
pencil and a notebook. How can you scale it up to schools which have
millions of children?”

one-pencil-per-1000-child-cyan
Suddenly question seems rather bizzare and at the same time sotrivial. Of course you might say but the pencil and notebook is so much cheaper than the computer. Yes. It. Is. But if you consider that a well designed laptop like OLPC, can serve a learners for 5-6 years and can remain with them through the schooling years. Then calculations show the investment that we seek is rather modest. In general when something becomes more
common, it also becomes cheaper. Mobile phones provide an excellent proof for this argument. And it is not happening in some first world country but in our own. What has promoted a rapid growth in the number of mobile users? How do tariff plans compare
from 15 years back to now? How come something which was something exclusive for the rich and the famous, just a few years back, is now so common? It is hard to find a person without a phone these days. Even people who do not have access to electricity have a
phone, they get it charged from some place else. Now if some sociologist would have done some study regarding usefulness of mobile phones for communication, perhaps 20 years earlier, they might have had some statistics to show, but critics would have said,

“but the cost is too prohibitive; this is perhaps ok for a case study you seriously
think all (or most) of the people can have this; and people who cannot
read and write will be able to use this; people do not have
electricity and food to eat and you want to give them mobile phone?”

But look at where we are, because people found contextual and personal value in using a mobile, it became their personal assistant in communicating with others, an inherent human trait, they got it. With proliferation of the mobiles, the cost of hardware came down, the cost of tariffs came down, almost everyone could afford one now.
It is sensationalist to compare a pencil and laptop in terms of cost, but when you consider the kinds of learning that can happen over a computer there is simple no match. They are not different in degree but in kind. Note that I have used “can happen” instead of will happen. This is for a reason, a laptop can be used in a variety of ways in learning. Some of the ways can be subversive, disruptive of the traditional education system, and redefine radically the ways our children learn. But in most cases its subversion is tamed and is made submissive to the existing educational system. And computers are made to do what a teacher or a textbook will do in a traditional context. So it is blunted and made part of a system which the computer has the potential to alter radically.
Some people then cite “research studies” done with “computers”. These studies will typically groups “with” computers and “without” computers. Some tasks are given and then there are pre and post tests. They are looking at the submissive action set in a highly conservative educational system. Even if such studies show the use of computers in a positive light, all these studies are missing the point. They are just flogging a dead horse. The point that computers when used in the right way, the constructionist way, can change the way we learn in a fundamental way. There are many studies which “prove” the counter-point. That computers don’t improve “learning”. Typically children will have limited access both in terms of time and sharing it with more people. One computer shared by three people, one hour in a week. Even then children learn, with computers if
used correctly. Continuing with out example of the pencil, consider this: one pencil shared among three children, once a week! Seems absurd isn’t it? But this is what typically happens in the schools, children are not allowed to develop a personal relationship with one of the most powerful learning ideas that they can have access to. Access is limited and in most cases uninformed involving trivialisation of the learning ideas that can redefine learning.
one-computer-per-1000-child

The Mitr

On who controls who

PUNCH AND JUDY, TO THEIR AUDIENCE
Our puppet strings are hard to see,
So we perceive ourselves as free,
Convinced that no mere objects could
Behave in terms of bad and good.
To you, we mannikins seem less
than live, because our consciousness
is that of dummies, made to sit
on laps of gods and mouth their wit;
Are you, our transcendental gods,
likewise dangled from your rods,
and need, to show spontaneous charm,
some higher god’s inserted arm?
We seem to form a nested set,
with each the next one’s marionette,
who, if you asked him, would insist
that he’s the last ventriloquist.
-Theaodore Melnechuk