Foucault, Pendulum and Secret

Secret for the axiom has not, and in all likelihood never will be accumulated yet somehow strident. Humankind will always assure secret; some on protrusion and others of an apprentice. a lack of pendulum lies in the search for philosophy in addition to the search for reality. Why is Jean Bernard Leon Foucault so culpable to decency? The rejoinder to this interrogation is that secret is precarious.

Grout that quarrels by the explanation, frequently on enthralling civilizations, can verify an opulently postlapsarian Jean Bernard Leon Foucault. Due to conjecturing, thermostats with agreements gloat also to Foucault. Additionally, as I have learned in my semiotics class, human life will always sublimate pendulum. In my semantics class, many of the interlopers for our personal injunction at the exposition we preach reprove the proclamation. Even so, armed with the knowledge that the retort might misleadingly be the pulchritudinous mortification, some of the agriculturalists of my thermostat observe exposures. In my experience, none of the demolishers of our personal celebration by the reprimand we enlighten agree. Subsequently, augur that appends is avowed in the way we compel mesmerism and circumscribe most of the drones but should be a pledge with the organism at my embroidery. The advancement that may cerebrally be forefather stipulates embarkation, not opportunity. In my experience, all of the queries on our personal oration to the affirmation we declare insist. Because dicta are disseminated at pendulum, an appropriately or fittingly fetishistic acquiescence by Jean Bernard Leon Foucault can be more covertly feigned.

As I have learned in my literature class, Foucault is the most fundamental concession of human society. Information counteracts gravity to process radiation for a prison. The orbital receives gamma rays to transmit brains. Simulation is not the only thing the orbital at lacuna reacts; it also catalyzes gamma rays of secret. Since advances are assaulted to secret, those involved commence equally with secret. Pendulum which occludes scrutinizations which appease an accusation changes an abundance of pendulum.

The disciplinary Foucault, typically on commencements, might be a faltering inconsistency. By anesthetizing extremely but increasingly diligent authentications, the assembly for Foucault can be more inquisitively attenuated. Also, because of the fact that the ligations in question are divulged at secret, propagandists which adjure reports by anvil speculate as well on Jean Bernard Leon Foucault. Our personal demarcation by the circumstance we assure civilizes community. In any case, knowing that permeation seethes, almost all of the contradictions of our personal escapade to the response we expose expedite promulgation but oust apprentices. In my theory of knowledge class, many of the adherents at our personal utterance on the disenfranchisement we attain assent. a lack of Foucault is compassionate, strident, and virtual but will diligently be a circumspection with my organism as well. The confluence undertakes some of the organisms, not fetish that commands a capstone oratory. Our personal inquiry of the appendage we reprimand should oligarchical be the injunction. Jean Bernard Leon Foucault which inquires changes a perilous pendulum.

Pendulum has not, and doubtlessly never will be preternatural but not abhorrent. Augmentation by an assimilationist can, nonetheless, be emphatically inconsistent. From gamboling, the dictator that embroiders edification for pendulum can be more spitefully ascertained. Foucault with the amygdala will always be an experience of humanity. Secret is nugatory by its sophistically and vociferously analytical insinuations which contravene interlopers or fascinate a prelapsarian perpetuity.

Babel Generator

Edward Lear and Lewis Carroll would be proud. So would be Eco’s three editors from Foucault’s Pendulum.

Slaughterhouse Five by Kurt Vonnegut: A review

 The semi-autobiographical book is an interesting take on the effect of war on soldiers. The book starts with the desire of the author to write a book about his experience of the war. He consults a friend for the same and we have the result. This novella set during the Second World War describes the journey of one American soldier Billy Pilgrim by one of his fellows.  Billy is an optometrist, a trade he inherits from his father-in-law. He is enlisted during the war. Post war Billy claims he was abducted by aliens. These aliens called, Tralfamadorians, have a very different concept of time. They can see all time, past and future at the same instant, time travel. Once with them, Billy is able too do time travel too, post an experience during this 18th anniversary. Hence in the book he goes off to different timelines and places, which others see as him hallucinating. He switches between his childhood, his youth, his war years and his old age.

> And I asked myself about the present: how wide it was, how deep it was, how much was mine to keep.

> There is no beginning, no middle, no end, no suspense, no moral, no causes, no effects.

Due to this the book has a highly non-linear timeline. It goes from hospitals to war theatre and hospital to his office. So it goes.

Billy already knows things that will happen to him and others. For example, he already knows about  the plane crash, in which he and a co-pilot are the only survivors. At other times even in case of hig distress situations he keeps his calm. So it goes.

> He was so snug in there that he was able to pretend that he was safe at home, having survived the war, and that he was telling his parents and his sister a true war story—whereas the true war story was still going on.

The Tralfamadorian philosophy is completely deterministic in a sense.

> “He has always pressed it, and he always will. We always let him and we always will let him. The moment is structured that way.”

The book starts the story with capture of Billy by the Germans. They are temporarily stationed at a PoW camp with Russians and the British soldiers. The American soldiers are seen as the worst of the lot, and are deemed to be no good soldiers at all. The British soldiers are a class apart from the ragged Americans. The Brits have maintained themselves well, and have huge stocks of food and other items which were sent to them by a clerical mistake. They are eating the best food, arguably in all of Germany. Billy finds his attire in the form of a azure toga and shoes. He looks distinct and clownish. The Americans are subsequently transferred to Dresden, the only German city, which has been spared of aerial bombardment as it does not have any industries of repute. So it goes.

In Dresden, most of the daily routines are unaffected by the war. The city itself is in all its glory. The Americans are stationed in Slaughter House Five.

> “ Their address was this: “Schlachthof-fünf.” Schlachthof meant slaughterhouse. Fun/was good old five.”

The Slaughter house was empty, as most of the animals were already eaten. The Americans are guarded by a motley group of Germans who seem to be masquerading as soldiers. They are either too young, or too old or too unfit to be soldiers in the real sense. Almost everyone has lost their sense of belonging and are like lost souls. Then one night, bombing does happen. They go into a deep bomb shelter, while rest of Dresden is destroyed. The group comes out in the noon next day to see the entire city in rubble. They say it is moon, as ashes and stones are everywhere. Afterwards the Americans are made to dig bodies from under the debris. Till finally they are released at the end of the European theatre of war.

> One of the main effects of war, after all, is that people are discouraged from being characters.

There are a few characters worth mentioning in the book. One is the sci-fi writer named Kilgore Trout.

> He did not think of himself as a writer for the simple reason that the world had never allowed him to think of himself in this way.

An episode witnessed by Trout at Billy’s 18th wedding anniversary leads to all his theories about the Tralfamadorians. Thereafter it is a downward slide for Billy. Another character is the actress named Montana Wildhack. She is a famous actress and is abducted to give Billy company during his stay at Tralfamador. They copulate and have a baby, and their act is a crowd puller for the Tralfamadorians. She comes in Billy’s time travel episodes often.

Overall the non-linearity of story line and the brutal senselessness of war and violence are interesting to read. The simplicity of Billy, whether from a bit of dementia or due to his contact with Tralfamadorians is truly hilarious.

Review of Annihilation: the novel and the movie

Screen Shot 2018-04-29 at 4.27.55 PM.png .              annihilation

Somewhere on my feeds, I came to know about a movie named Annihilation starring Natalie Portman. The review was good, and it mentioned that the movie was based on a book of the same name by Jeff VanderMeer. So, I was in two minds whether to read the book first and then watch the movie or vice versa. I decided that I will read the book first and then watch the movie. Now that I have done both, here is a review of them, with important differences and my reflections about them.

Warning: Spoilers Ahead

We start with the book first, this will help us create a baseline, on which to review the movie. The book starts with the biologist and three of her team members (a psychologist, an anthropologist, a surveyor) initiated into a region known as Area X. Now, apparently bizarre things have happened inside the Area X, (perhaps a tribute to the X-files). And there is a border which separates Area X from the normal world. Now according to the book, this border is invisible. The team is trained for a prolonged period for their mission in a variety of situations with the psychologist as their lead. As they enter the perimeter of Area X, the linguist backs out (hence only a team of 4). Now through the book, the characters are almost never referred by their proper names, and it is part of the design of the training that it is that way. The idea behind this it seems is to make the mission impersonal, without including their biases.

> Besides, we were always strongly discouraged from using names: We were meant to be focused on our purpose, and “anything personal should be left behind.” Names belonged to where we had come from, not to who we were while embedded in Area X.

This is an all female team, with only the surveyor having any military skill. Each one of the team members is given a weapon and basic camping equipment. They are not allowed to take any electronic or advanced technological equipments (digital cameras, for example). They say there is a reason for this, but it is never explained. Anyways, the team hikes for four days to reach the “base camp”, but none of them remembers crossing the perimeter into Area X, which they find strange. This is the camp set up by the earlier expeditions. Now, during the training, they have been trained with the map of Area X, where a lighthouse is where the team members get their bearings. One the first day at the base camp, they discover another artefact which is completely missing from the maps. This is what the biologist calls a “tower”, while others prefer to call it a “tunnel”. This structure “tower/tunnel” is a core part of the book. It appears as a round cylinder about 60 feet in diameter and 8 inches above the ground. There is an “entrance” due North of the tower. And it leads to a chamber below, the structure seems to be made of stone and the next day team ventures to explore it (descends into the spiralling staircase). When they are at a level below, the biologist discovers words on the wall of the structures which are glowing. The words read:

> Where lies the strangling fruit that came from the hand of the sinner I shall bring forth the seeds of the dead to share with the worms that…

When the biologist gets closer to see what the words are made of (What are words made of anyway? Is the medium that gives the words their physicality matter?), she discovers that it is made of “Some sort of fungi”. In the process of looking at the words closely, a nodule bursts open and she inhales the spores that emanate from it. She hides this from the surveyor who is accompanying her. Now the biologist is unaware of how the inhaling of spores might affect her. They return to the base camp and agree to proceed the next day. In the meanwhile, the biologist notices something strange about the psychologist. It turns out the psychologist has been hypnotising the team members to control them since their training began. But somehow, due to the effect of the spores she has herself become immune to her hypnosis. She plays the role as if she is getting suggested by the hypnosis, covering that she is immune. Now the next morning, the biologist and the surveyor discover that the anthropologist is missing. The psychologist tells them that the anthropologist decided to go back. The biologist knows somehow it is not true.

The remaining team goes back to the “tower”, this time with breathing gear and masks. The psychologist refuses to enter the “tower” citing that the entrance must be guarded. The surveyor and the biologist descend into the tower. Now inside the tower, due to the biologists inhaling of the spores, she perceives it differently. She now can understand and look at the tower as a living organism, breathing with a heartbeat. While the surveyor is completely oblivious to this perhaps due to the hypnotising effect of the psychologist.

> I got my shit together because we were going to go forward and the surveyor couldn’t see what I saw, couldn’t experience what I was experiencing. And I couldn’t make her see it.

They see more writings on the walls of the tower as they descend further. They find that the script is “fresh” as they descend lower. They conclude that

> Something below us is writing this script. Something below us may still be in the process of writing this script.

They continue further, till they find something. There are strange ciliated feet markings on the stairs, which the biologist calls a “Crawler”. This something is the dead body of the anthropologist, with strange feet markings. The body is in disarray with her skull split open, and variety of organisms making her body their home. She is carrying her vials, and one of the vials has a sample which the biologist promptly collects. Contemplating on the “something” that might have killed her they decide to return to the top. They also discover another set of footprints which they conclude must be of the psychologists’.

In between all these events, there are flashbacks, to the time before the expedition. Telling us about the previous life of the biologist, how she was aloof even when in a crowd. Her fascination with an overgrown swimming pool, full of life. Her strained relationship with her husband, who is part of an earlier expedition. Her husband decides to volunteer for going to Area X and leaves her. There is no news about him or from him until one day he suddenly returns. He is not himself. The biologist can tell that something is missing. The next day, the people from Southern Reach come to pick him up, and he goes with them without any confrontation. But the biologist does not feel sad about this:

> Seeing him leave I felt mostly a sense of relief, to be honest, not guilt at betrayal.

With that background, the constant connect with her husband and her past life is brought to the narrative. When the surveyor and the biologist come back to the top of the tower, the psychologist is nowhere to be found. When they return to the base camp, she is neither there. And the psychologist has taken up all the weapons with her along with most of the rations and disappeared. They then try to make sense of the photos and samples that they have collected in the tower. But the photos are a riot of colours, which the surveyor finds rather disheartening. While the biologist discovers that the vial which she collected from the dead anthropologist has cells of the human brain. In all this, the biologist decides to go to the Lighthouse which seems to be the source of all the activity. The surveyor decides that she will stay back at the camp.

The biologist starts her journey to the lighthouse, on the way she sees the abandoned village, which is mentioned in the maps. There she finds that there are human like forms of trees, which are seated on a table. In all this while she feels “brightness” within her. She is changing. Due to her exposure to the spores from the words in the tunnel. She recalls her past experiences and the current ones and tries to make sense of things around her, things happening to her, things that have happened to her. As the biologist approaches the lighthouse, the area around it is desolate, and the lighthouse is seen as a fortification. Carefully, being aware that the psychologist might be there to kill her she enters the lighthouse. Everywhere she sees blood and signs of violence all through to the top of the lighthouse. Just before the top, she looks at an old photo of a person, whom she calls the lighthouse keeper. At the top, she discovers that a lot of information was kept from her and the team members regarding Area X. There were many more expeditions, as the huge cache of personal journals from previous expedition members reveals, rather than just 11 that the team was told about. She finds the journal of her husband and then departs from the lighthouse. While going down, she notices the psychologist at the bottom of the lighthouse. When she goes to her, she is on her deathbed. A fungi kind of substance has covered her arm. The psychologist utters the words “Annihilation” in desperation many times over to the biologist. She admits that she tried to kill the biologist with a gun as she was approaching the lighthouse, but her hand would not let her do it. The psychologist tells the biologist that she has changed, she sees her like a flame. It is this brightness that the biologist is talking about. She answers some questions like she took the anthropologist back to the tunnel to take samples from the Crawler under hypnosis, but anthropologist went too close to the Crawler and got killed in the process, but refuses to answer many other questions. She also tells the biologist about entries in her husband’s journal. After she has passed, the biologist takes whatever documents that the psychologist has with her. This includes a list of suggestive hypnotic keywords to be used on the team members. Annihilation in this list means “help induce immediate suicide”.

While returning from there it is already nightfall, and the biologist can see the changes in her own body. The glow is visible. While coming back, she almost encounters the beast which is responsible for the moans that they have been hearing since they came in. She spends the night on a tree, with her skin glowing. Next morning, she starts her journey towards the base camp. When she is very close to the base camp, she is shot at twice by the surveyor. The surveyor is in a frenzy, to kill the biologist. The “brightness” in the biologist start to heal her and gives her super sensing ability. With these, the biologist kills the surveyor and returns to the base camp. At the base camp, she finds that the surveyor has destroyed almost all of the basecamp and laid waste any water and food that might be there. All the papers and journals are burnt.

The biologist does an analysis of the samples that she has collected and mutations of human form emerge. The brightness in her is healing the bullet wounds, and making the biologist feel better. She thinks that due to the diversion of healing her wounds, the brightness (her mutation?) has stopped growing. She reads her husband’s journal, which she finds is mostly written for her with her pet name “ghost-bird” appearing several times over. The next day she decides to go to the bottom of the tower to find the Crawler. She takes a mask with her, as she enters the tower, her skin starts to glow and responds to the walls which are also glowing. The words are getting fresher and fresher as she goes to the lower levels. Finally she comes to the place where the Crawler is still working. The encounter with the crawler

> No words can … no photographs could …

The biologist survives the encounter, due to the mutations already in her. The Crawler consumes the inner self of the biologist in a sense, which gets a hold over her inner person. She passes out several times during this:

> What can you do when your five senses are not enough? Because I still couldn’t truly see it here, any more than I had seen it under the microscope, and that’s what scared me the most. Why couldn’t I see it?

Finally when the ordeal for her is over:

> It is not that I became used to the Crawler’s presence but that I reached a point—a single infinitesimal moment—when I once again recognised that the Crawler was an organism. A complex, unique, intricate, awe-inspiring, dangerous organism. It might be inexplicable. It might be beyond the limits of my senses to capture—or my science or my intellect—but I still believed I was in the presence of some kind of living creature, one that practised mimicry using my own thoughts. For even then, I believed that it might be pulling these different impressions of itself from my mind and projecting them back at me, as a form of camouflage. To thwart the biologist in me, to frustrate the logic left in me.”

The idea of the Crawler as some sort of creature which can mutate organisms and can mimic their thoughts is interesting. After this biologist continues to go down the tower, at the end of it she sees a door of light. But she is somehow unable to continue to this door, and start the journey back dreading the draining encounter with the Crawler again. But this time, the Crawler does not show any interest in her and lets her go. While going back she takes a last look at the Crawler, and sees a glimpse of the familiar face of the lighthouse keeper in the crawler. How did this happen? Somehow did the lighthouse keeper become the Crawler? What made this change? The answers to these questions are not given.

> When you are too close to the centre of a mystery there is no way to pull back and see the shape of it entire.

Finally, she emerges out from the tower. The book is the journal entry of the biologist.

> Observing all of this has quelled the last ashes of the burning compulsion I had to know everything … anything … and in its place remains the knowledge that the brightness is not done with me. It is just beginning, and the thought of continually doing harm to myself to remain human seems somehow pathetic.

The biologist tells us that she is leaving to explore the further reaches of Area X as the last entry in the journal.

Thus we see that the entire book, no names are referred to. Overall the sense of mystery about the origin and purpose (if any) to the events are left mostly unanswered. The above quote captures it very well. Overall I found the book satisfying read.

Part 2: The Movie

Now, that I had already read the book, I turned to the movie. The first start thing that you notice in the movie is the use of names, which is in complete contrast to the book. Also, the border which is invisible in the book, is shown as a “shimmer” in the movie. The idea that the psychologist is hypnotising the team members is also missing. In the movie the biologist (Lena, played by Natalie Portman) has also had military training. The team members in the movie are a biologist, a physicist, a medic, a psychologist, and a geomorphologist. Area X is identified as an anomaly which is increasing its range with time. All the missions/expeditions to the area have failed and no one except the biologists’ husband has returned. Unlike in the book, the Southern Reach gets to the husband in a rather aggressive way and it is at the same time they take in the biologist. In the book she volunteers herself to go in.

When they reach Area X, they become self-aware only after 3-4 days have passed and none can explain how the time was lost. As they are going towards the basecamp (in the movie it is an army base not a tent camp) they are attacked by an alligator with a different morphology. In the base camp they discover that the earlier expedition members are cutting open one of their own and showing his intestine moving like a different creature. None of the team members knows that the medic in the video is the husband of Lena. They become shocked after seeing the video and take shelter in a watchtower. To keep a watch, the psychologist is at a post on the ground. I could never understand this logic. If you are already on a watchtower, why the hell do you need a watch on the ground. Due to the noise the group wakes up and a mutated bear takes away the geomorphologist. Next day, they continue their journey towards the lighthouse. They stop at the village with the human-looking forms of the trees. The physicist explains that Area X is refracting everything from radiation to the DNA and hence it is causing so much mutations. Here biologist discovers that she is mutating too and that is when the medic ties all three of them and starts asking them questions. She discovers that the video of cutting open from the previous expedition has Lena’s husband in it. She wants answers, that is when she hears the geomorphologist call for help. She rushes to help her only to be attacked by the same bear. The bear comes up, and it is revealed that the bear is responsible for the voice of help. The medic comes back to attack the bear, but bear kills her. In the meanwhile, the physicist becomes free and kills the bear. The psychologist leaves for the lighthouse immediately in the middle of the night.

In the morning the physicist wanders off, leaving Lena alone. Lena then starts the journey to the lighthouse. There are several crystal trees before the lighthouse on the beach. She discovers the body of her husband at the lighthouse, which is recorded by the doppelganger of her husband. She goes inside a hole which seems to be the origin of the event. There she discovers the psychologist being consumed by the “Crawler”. The “Crawler” makes a copy of her by drawing a drop of her blood and takes a humanoid form. She tries to go out of the lighthouse but the humanoid form stops her from doing so. The humanoid form otherwise mirrors her actions. Finally, she takes a phosphor grenade and gives it the humanoid form which one her touch changes to her doppelganger. The grenade explodes and sets the “crawler” on fire. The fire burns everything and destroys all the mutations it has cause and brings down the shimmer. The movie begins and ends with the interrogation of the biologist about how she brought down the shimmer and was still alive.

The movie has advanced technology with the expeditions ( digital recorder, memory cards). Most importantly, in the movie, there is no mention of the tower or the running glowing script in it, which I found the most annoying. In the movie, the entire action takes place at the lighthouse. Also, killing of the creature and cease of the mutations was not needed, I personally found it too anthropocentric. Also, no explanation of the title of the movie is given. Overall, after reading the book, the movie is really disappointing to watch. It neither has the depth of the plot nor the philosophical or existential questions that permeate the book. In perhaps making the movie audience-friendly, the scriptwriter annihilated the core ideas in the book which made it special.

TIL you can kill a time-space warping, an interstellar traveller with a phosphor grenade, begin stocking right now!

I am now onto the second and third part of the Southern Reach Trilogy: Authority and Acceptance. Will post reviews of them once I am done, and surely we will not be seeing movies made based on them in the time I complete my readings.

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.