On respect in the classroom

If you are a teacher (of any sort) and teach young people, don’t be disheartened if the students in your class don’t respect you or listen to you or maintain discipline. Even great philosophers like Socrates and Aristotle has a tough time dealing with their students

Socrates grumbled that he don’t get no respect: his pupils “fail to rise when their elders enter the room. They chatter before company, gobble up dainties at the table, and tyrannize over their teachers.” Aristotle was similarly pissed off by his stu­dents’ attitude: “They regard themselves as omniscient and are positive in their assertions; this is, in fact, the reason for their carrying everything too far.”Their jokes left the philosopher unamused: “They are fond of laughter and conse­quently facetious, facetiousness being disciplined insolence.”

– Judith Harris The Nurture Assumption

That being said, the students are also very perceptive about the knowledge of the teachers, and know who is trying to be a cosmetic intellectual.

Cosmetic Intellectuals (+ IYI)

In the last few years, the very connotation of the term intellectual has seen a downward slope. Such are the times that we are living in that calling someone an “intellectual” has become more like an insult rather than a compliment: it means an idiot who doesn’t understand or see things clearly. Now as the title of the post suggests it is this meaning, not the other meaning intellectuals who know about cosmetics. Almost two decades back Alan Sokal wrote a book titled Intellectual Impostures, which described quite a few of them. In this book, Sokal exposed the posturing done by people of certain academic disciplines who were attacking science from a radical postmodernist perspective. What Sokal showed convincingly through his famous hoax, is that many of these disciplines are peddling out bullshit with no control over the meaning contained. Only the form was important not the meaning. And in the book, he takes it a step forward, showing that this was not an isolated case. He exposes the misuse of the technical terms (which often have precise and operational meanings) as loose metaphors or even worse completely neglecting the accepted meaning of those terms. The examples given are typical, and you cannot make sense of what is being written. You can read, but cannot understand. It makes no sensible meaning. At this point, you start to doubt your own intelligence and intellectual competence, perhaps you have not read enough to understand this complex piece of knowledge. It was after all written by an intellectual. Perhaps you are not aware of the meaning of the jargon or their context, hence you are not able to understand it. After all there are university departments and journals dedicated to such topics. Does it not legitimise such disciplines as academic and its proponents/followers as intellectuals? Sokal answered it empirically by testing if presented with nonsense whether it makes any difference to the discipline. You are not able to make sense of these texts because they are indeed nonsensical. To expect any semblance of logic and rationality in them is to expect too much.

Nassim Taleb has devised the term Intellectual Yet Idiots (the IYI in the title) in his Incerto series. He minces no words and takes no bullshit. Sokal appears very charitable in comparison. Taleb sets the bar even higher. Sokal made a point to attack mostly the postmodernists, but Taleb bells the cats who by some are even considered proper academics, for example, Richard Dawkins and Steven Pinker. He considers entire disciplines as shams, which are otherwise considered academic, like economics, but has equal if not more disdain to several others also, for example, psychology and gender studies. Taleb has at times extreme views on several issues and he is not afraid to speak of his mind on matters that matter to him. His writings are arrogant, but his content is rigorous and mathematically sound.

they aren’t intelligent enough to define intelligence, hence fall into circularities—their main skill is a capacity to pass exams written by people like them, or to write papers read by people like them.

But there are people who are like IYIs, but don’t even have the depth of the content or knowledge of IYIs. They are wannabe IYIs, all form no conent. They are a level below IYIs. I term such people as cosmetic intellectuals (cosint). We have met them before: they are the envious mediocre and the ones who excel in meetings. The term cosmetic is used in two senses both as adjectives. The first sense is the Loreal/Lakme/Revlon fashion sense as given from the dictionary entry below:

cosmetic

  • relating to treatment intended to restore or improve a person’s appearance
  • affecting only the appearance of something rather than its substance

It is the second sense that I mean in this post. It is rather the substance of these individuals that is only present in the appearance. And as we know appearance can be deceiving. Cosints appear intellectuals, but only in appearance, hence the term cosmetic. So how does one become a Cosint? Here is a non-exhaustive list that can be an indicator (learn here is not used in the deeper sense of the word, but more like as in rote-learn):

  1. Learn the buzzwords: Basically they rote learn the buzzwords or the jargon of the field that they are in. One doesn’t need to understand the deeper significance or meaning of such words, in many cases just knowing the words works. In the case of education, some of these are (non-comprehensive): constructivism, teaching-learning process, milieu, constructivist approaches, behaviorism, classroom setting, 21st-century skills, discovery method, inquiry method, student-centered, blended learning, assessments, holistic, organic, ethnography, pedagogy, curriculum, TLMs. ZPD, TPD, NCF, RTE, (the more complicated the acronyms, the better). More complicated it sounds the better. They learn by association that certain buzzwords have a positive value (for example, constructivism) and other a negative one (for example, behaviorism) in the social spaces where they usually operate in, for example, in education departments of universities and colleges. Not that the Cosints are aware of the deeper meaning of there concepts, still they make a point of using them whenever possible. They make a buzz using the buzzwords. If you ask them about Piaget, they know the very rudimentary stuff, anything deeper and they are like rabbits in front of flashlight. They may talk about p-values, 𝛘2 tests, 98.5 % statistical significances, but when asked will not be able to distinguish between dependent and independent variables.
  2. Learn the people: The CosInts are also aware of the names of the people in their trade. And they associate the name to a concept or of a classic work. They are good associating. For example, (bad) behaviorism with Burrhus F. Skinner or Watson, hence Skinner bad. Or Jean Piaget with constructivism and stages (good). Vygotsky: social constructivism, ZPD. Or John Dewey and his work. So they have a list of people and concepts. Gandhi: Nayi Taleem.  Macauley: brought the English academic slavery on India (bad).
  3. Learn the classics: They will know by heart all the titles of the relevant classics and some modern ones (you have to appear well-read after all). Here just remembering the names is enough. No one is going to ask you what was said in section 1.2 of Kothari Commission. Similarly, they will rote learn the names of all the books that you are supposed to have read, better still carry a copy of these books and show off in a class. Rote learn a few sentences, and spew it out like a magic trick in front of awestruck students. Items #1 through #3 don’t work very well when they have real intellectual in front of them. A person with a good understanding of basics will immediately discover the fishiness of the facade they put up. But that doesn’t matter most of the time, as we see in the next point.
  4. Know the (local) powerful and the famous: This is an absolute must to thrive with these limitations. Elaborated earlier.
  5. Learn the language aka Appear academic (literally not metaphorically): There is a stereotype of academic individuals. They will dress in a particular manner (FabIndia?, pyor cotton wonly, put a big Bindi, wear a Bongali kurta etc, carry ethnic items, conference bags (especially the international ones), even conference stationery), carry themselves in a particular manner, talk in a particular manner (academese). This is also true of wannabe CosInt who are still students, they learn to imitate as soon as they enter The Matrix. Somehow they will find ways of using names and concepts from #1 #2 #3 in their talk, even if they are not needed. Show off in front of the students, especially in front of the students. With little practice one can make an entire classroom full of students believe that you are indeed learned, very learned. Any untoward questions should be shooed off, or given so tangential an answer that students are more confused than they were earlier.
  6. Attend conferences, seminars and lectures: The primary purpose is network building and making sure that others register you as an academic. Also, make sure that you ask a question or better make a tangential comment after the seminar so that everyone notices you. Ask the question for the sake of asking the question (even especially if you don’t have any real questions). Sometimes the questions devolve into verbal diarrhea and don’t remain questions and don’t also have any meaning that can be derived from them (I don’t have a proper word to describe this state of affairs, but it is like those things which you know when you see it). But you have to open your mouth at these events, especially when you have nothing substantial/meaningful to say. This is how you get recognition. Over a decade of attending various conferences on education in India, I have come to realise that it is akin to a cartel. You go to any conference, you will see a fixed set of people who are common to these conferences. Many of these participants are the cosints (both the established and the wannabes). After spending some time in the system they become organisers of such conferences, seminars and lectures definitely get other CosInts to these conferences. These are physical citation rings, I call you to my conference you call me to yours. Year after year, I see the same patterns, so much so I can predict, like while watching a badly written and cliche movie, what is going to happen when they are around. That person has to ask a question and must use a particular buzzword. (I myself don’t ask or comment, unless I think I have something substantial to add. Perhaps they think in same manner, just that their definition of substantial is different than mine.) Also, see #5, use the terms in #1, #2 and #3. Make sure to make a personal connection with all the powerful and famous you find there, also see #4.
  7. Pedigree matters: Over the years, I have seen the same type of cosints coming from particular institutions. Just like you can predict certain traits of a dog when you know its breed, similarly one can predict certain traits of individuals coming from certain institutions. Almost without exception, one can do this, but certain institutions have a greater frequency of cosints. Perhaps because the teachers who are in those places are themselves IYI+cosints. Teaching strictly from a  prescribed curriculum and rote-learning the jargon: most students just repeat what they see and the cycle continues. Sometimes I think these are the very institutions that are responsible for the sorry state of affairs in the country. They are filled to the brim with IYIs, who do not have any skin in the game and hence it doesn’t matter what they do. Also, being stamped as a product of certain institution gives you some credibility automatically, “She must be talking some sense, after all he is from DU/IIT/IIM/JNU/”
  8. Quantity not quality: Most of us are not going to create work which will be recognised the world over (Claude Shannon published very infrequently, but when he did it changed the world). Yet were are in publish or perish world. CosInts know this, so they publish a lot. It doesn’t matter what is the quality is (also #4 and #5 help a lot). They truly are environmentalists. They will recycle/reuse the same material with slight changes for different papers and conferences, and surprisingly they also get it there (also #4 and #5 help a lot). So, at times, you will find a publication list which even a toilet paper roll may not be able to contain. Pages after pages of publications! Taleb’s thoughts regarding this are somewhat reassuring, so is the Sokal’s hoax, that just when someone has publications (a lot of them) it is not automatic that they are meaningful.
  9. Empathisers and hypocrites: Cosints are excellent pseudo-emphatisers. They will find something to emphathise with. Maybe a class of people, a class of gender (dog only knows how many). Top of the list are marginalised, poor low socio-economic status, underprivileged, rural schools, government students, school teachers, etc. You get the picture.  They will use the buzz words in the context of these entities they emphathise with. Perhaps, once in their lifetimes, they might have visited those whom they want to give their empathy, but otherwise, it is just an abstract entity/concept.(I somehow can’t shake image of Arshad Warsi in MunnaBhai MBBS “Poor hungry people” while writing about this.) It is easier to work with abstract entities than with real ones, you don’t have to get your hands (or other body parts) dirty. The abstract teacher will do this, will behave in this way: they will write a 2000 word assignment on a terse subject. This is all good when designing things because abstract concepts don’t react in unwanted ways. But when things don’t go as planned in real world, teachers don’t react at all! The blame is on everyone else except the cosints. Perhaps they are too dumb to understand that it is they are at fault. Also, since they don’t have skin in the game, they will tell and advise whatever they have heard or think to be good, when it is implemented on others. For example, if you talk to people especially from villages, they will want to learn English as it is seen as the language which will give them upward mobility. But cosints, typically in IYI style, some researchers found that it is indeed the mother tongue which is better for students to learn, it should be implemented everywhere. The desires and hopes of those who will be learning be damned, they are too “uneducated” to understand what they need. It is the tyranny of fake experts at work here.

    He thinks people should act according to their best interests and he knows their interests… When plebeians do something that makes sense to themselves, but not to him, the IYI uses the term “uneducated.” (SITG Taleb)

    Now one would naturally want to know under what conditions that research was done? was there any ideological bias of the researchers? whether it is applicable in as diverse a country as India? What do we do of local “dialects”? But they don’t do any of this. Instead, they will attack anyone who raises these doubts, especially in #6. They want to work only with the government schools: poor kids, poor teachers no infrastructure. But ask them where their own children study: they do in private schools! But their medium must be their mother tongue right? No way, it is completely English medium, they even learn Hindi in English. But at least the state board? No CBSE, or still better ICSE. Thus we see the hypocrisy of the cosint, when they have the skin in the game. But do they see it themselves? Perhaps not, hence they don’t feel any conflict in what they do.

So we see that IYI /cosint are not what they seem or consider themselves. Over the last decade or so, with the rise of the right across the world is indicating to everyone that something is wrong when cosints tell us what to do. The tyranny of pseudo-experts has to go.  But why it has come to that the “intellectuals” who are supposed to be the cream of the human civilisation, the thinkers, the ideators, so why the downfall? Let us first look at the meaning of the term, so as to be not wrong about that:

 The intellectual person is one who applies critical thinking and reason in either a professional or a personal capacity, and so has authority in the public sphere of their society; the term intellectual identifies three types of person, one who:

  1. is erudite, and develops abstract ideas and theories;
  2. a professional who produces cultural capital, as in philosophy, literary criticism, sociology, law, medicine, science; and
  3. an artist who writes, composes, paints and so on.

Intellectual (emphasis mine)

Now, see in the light of the above definition, it indeed seems that it must be requiring someone to be intelligent and/or well-cultured individual. So why the change in the tones now? The reasons are that the actual intellectual class has degraded and cosints have replaced them, also too much theory and no connect with the real world has made them live in a simulacrum which is inhabited and endorsed by other cosints. And as we have seen above it is a perpetuating cycle, running especially in the universities (remember Taleb’s qualification). They theorize and jargonise (remember the buzzwords) simple concepts so much that no one who has got that special glossary will understand it). And cosints think it is how things should be. They write papers in education, supposedly for the betterment of the classroom teaching by the teachers, in such a manner that if you give it to a teacher, they will not be able to make any sense of it, leave alone finding something useful. Why? Because other cosints/IYI demand it! If you don’t write a paper in a prescribed format it is rejected, if it doesnt have enough statistics it is rejected, if it doesn’t give enough jargon in the form of theoretical review, and back scratching in the form of citations it is rejected. So what good are such papers which don’t lead to practice? And why should the teachers listen to you if you don’t have anything meaningful to tell them or something they don’t know already?

The noun to describe them:

sciolist – (noun) – One who engages in pretentious display of superficial knowledge.

School as a manufacturing process

Over most of this century, school has been conceived as a manufacturing process in which raw materials (youngsters) are operated upon by the educational process (machinery), some for a longer period than others, and turned into finished products. Youngsters learn in lockstep or not at all (frequently not at all) in an assembly line of workers (teachers) who run the instructional machinery. A curriculum of mostly factual knowledge is poured into the products to the degree they can absorb it, using mostly expository teaching methods. The bosses (school administrators) tell the workers how to make the products under rigid work rules that give them little or no stake in the process.
– (Rubba, et al. Science Education in the United States: Editors Reflections. 1991)

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.

Examinations: Students, Teachers and the System

We think of exams as simple troublesome exchanges with students:

Glance at some of the uses of examinations:

  • Measure students' knowledge of facts, principles, definitions, 
    experimental methods, etc
  • Measure students' understanding of the field studied
  • Show students what they have learnt
  • Show teacher what students have learnt
  • Provide students with landmarks in their studies
  • Provide students with landmarks in their studies and check 
    their progress
  • Make comparisons among students, or among teachers, 
    or among schools
  • Act as prognostic test to direct students to careers
  • Act as diagnostic test for placing students in fast 
    or slow programs
  • Act as an incentive to encourage study
  • Encourage study by promoting competition among students
  • Certify necessary level for later jobs
  • Certify a general educational background for later jobs
  • Act as test of general intelligence for jobs
  • Award's, scholarships, prizes etc.

There is no need to read all that list; I post it only as a warning against trying to do too many different things at once. These many uses are the variables in examining business, and unless we separate the variables, or at least think about separating them, our business will continue to suffer from confusion and damage.

There are two more aspects of great importance well known but seldom mentioned. First the effect of examination on teachers and their teaching –

coercive if imposed from the outside; guiding if adopted sensibly. That is how to change a whole teaching program to new aims and methods – institute new examinations. It can affect a teacher strongly.

It can also be the way to wreck a new program – keep the old exams, or try to correlate students’ progress with success in old exams.

Second: tremendous effect on students.

Examinations tell them our real aims, at least so they believe. If we stress clear understanding and aim at growing knowledge of physics, we may completely sabotage our teaching by a final examination that asks for numbers to be put in memorized formulas. However loud our sermons, however intriguing the experiments, students will be judged by that exam – and so will next years students who hear about it.

From:

Examinations: Powerful Agents for Good or Ill in Teaching | Eric M. Rogers | Am. J. Phys. 37, 954 (1969)

Though here the real power players the bureaucrats and (highly) qualified PhDs in education or otherwise who decide what is to be taught and how it is evaluated in the classroom. They are “coercive” as Rogers points out and teachers, the meek dictators (after Krishna Kumar), are the point of contact with the students and have to face the heat from all the sides. They are more like foot soldiers most of whom have no idea of what they are doing, why they are doing; while generals in their cozy rooms, are planning how to strike the enemy (is the enemy the students or their lack of (interest in ) education, I still wonder).  In other words most of them don’t have an birds-eye-view of system that they are a focal part of.

Or as Morris Kline puts it:

A couple of years of desperate but fruitless efforts caused Peter to sit back and think. He had projected himself and his own values and he had failed. He was not reaching his students. The liberal arts students saw no value in mathematics. The mathematics majors pursued mathematics because, like Peter, they were pleased to get correct answers to problems. But there was no genuine interest in the subject. Those students who would use mathematics in some profession or career insisted on being shown immediately how the material could be useful to them. A mere assurance that they would need it did not suffice. And so Peter began to wonder whether the subject matter prescribed in the syllabi was really suitable. Perhaps, unintentionally, he was wasting his students’ time.

Peter decided to investigate the value of the material he had been asked to teach. His first recourse was to check with his colleagues, who had taught from five to twenty-five or more years. But they knew no more than Peter about what physical scientists, social scientists, engineers, and high school and elementary school teachers really ought to learn. Like himself, they merely followed syllabi – and no one knew who had written the syllabi.

Peter’s next recourse was to examine the textbooks in the field. Surely professors in other institutions had overcome the problems he faced. His first glance through publishers’ catalogues cheered him. He saw titles such as Mathematics for Liberal Arts, Mathematics for Biologists, Calculus for Social Scientists, and Applied Mathematics for Engineers. He eagerly secured copies. But the texts proved to be a crushing disappointment. Only the authors’ and publishers names seemed to differentiate them. The contents were about the same, whether the authors in their prefaces or the publishers in their advertising literature professed to address liberal arts students, prospective engineers, students of business, or prospective teachers. Motivation and use of the mathematics were entirely ignored. It was evident that these authors had no idea of what anyone did with mathematics.

From: A Critique Of Undergrduate Education. (Commonly Known As: Why The Professor Can’t Teach?) | Morris Kline

Both of the works are about 50 years old, but they still reflect the educational system as of now.