From Numbers: Rational and Irrational by Ivan Niven
mathematics
A complex systems approach to biology – a review of How Leopard Changed its Spots by Brian Goodwin
How the Leopard Changed it Spots by Brian Goodwin talks about a different approach to biology. After the genetic and molecular biology revolution 1950s onwards, increasingly the organism has been shifted out of focus in biology. Instead genes and their effect, genocentrism or neo-Darwninism, have taken the central stage. Everything in biology is seen as an “action” of genes in addition to natural selection. This translates to reductionism, everything is reduced to genes which are considered as the most fundamental units of life. This is the dominant approach in biology for some decades now. The terms such as “selfish gene” basically highlight this point. Such an approach sidelines the organism as a whole and its environment and highlights the genes alone. Good draws analogy of “word” of god seen as the final one in scriptures to the code alphabet in the genes, as if their actions and results are inevitable and immutable.
Goodwin in his work argues against such an approach using a complex systems perspective. In the process he also critiques what is an acceptable “explanation” in biology vis-a-vis other sciences. The explanation in biology typically is a historical one, in which features and processes are seen in the light of its inheritance and survival value of its properties. This “explanation” does not explain why certain forms are possible. Goodwin with examples establishes how action of genes alone cannot establish the form of the organism (morphogenesis). Genes only play one of the parts in morphogenesis, but are not solely responsible for it (which is how neo-Darwinist account argue). He cites examples from complex systems such as Belousov–Zhabotinsky reaction, ant colonies to establish the fact that in any system there are different levels of organisation. And there are phenomena, emergent phenomena, which cannot be predicted on the basis of the properties constituent parts alone. Simple interactions of components at lower level can give rise to (often) surprising properties at higher level. He is very clear that natural selection is universal (Darwin’s Dangerous Idea?!)
> What this makes clear is that there is nothing particularly biological about natural selection: it is simply a term used by biologists to describe the way in which one form replaces another as a result of their different dynamic properties. This is just a way of talking about dynamic stability, a concept used for a long time in physics and chemistry. We could, if we wished, simply replace the term natural selection with dynamic stabilization, the emergence of the stable states in a dynamic system. p. 53
Goodwin uses the term morphogenetic space to convey the possible shape space that an organism can occupy. Thus seen from a complex systems perspective, the various unit of the organism interact to generate the form of the organism. Natural selection then acts as a coarse sieve on these forms with respect to the environmental landscape. The “aim” of the organism is not the climb the fitness lanscape but to achieve dynamic stability.
> The relevant notion for the analysis of evolving systems is dynamic stability: A necessary (though by no means sufficient} condition for the survival of a species is that its life cycle be dynamically stable in a particular environment. This stability refers to the dynamics of the whole cycle, involving the whole organism as an integrated system that is itself integrated into a greater system, which is its habitat. p. 179
Goodwin takes examples of biological model systems and shows how using mathematical models we can generate their forms. Structure of acetabularia (a largish ~1 inch single cell algae), the structure of eye, the Fibonacci pattern seen in many flower structures being the main examples. Also, how the three basic forms of leaf arrangement can be generated by variations on a theme in the morphospace are discussed in detail. The model shows that three major forms are the most probable ones, which is actually substantiated by observations in nature. In these examples, an holistic approach is taken in which genes, competition and natural selection only play a part are not the main characters but are interacting and cooperating with levels of organisation of the organism, environmental factors in the drama of life.
> Competition has no special status in biological dynamics, where what is important is the pattern of relationships and interactions that exist and how they contribute to the behavior of the system as an integrated whole.The problem of origins requires an understanding of how new levels of order emerge from complex patterns of interaction and what the properties of these emergent structures are in terms of their robustness to perturbation and their capacity for self-maintenance. Then all levels of order and organization are recognized as equally important in understanding the behavior of living systems, and the reductionist insistence on some basic material level of cause and explanation, such as molecules and genes, can be recognized as an unfortunate fashion or prejudice that is actually bad science. P.181
Since I am already a believer of the complex systems perspective, I was aware of some of the arguments in this book, but the particular worked examples and their interpretation for biology was a fresh experience.
The Calculus Bottleneck
What if someone told you that learners in high-school don’t actually need calculus as a compulsory subject for a career in STEM? Surely I would disagree. After all, without calculus how will they understand many of the topics in the STEM. For example basic Newtonian mechanics? Another line of thought that might be put forth is that calculus allows learners to develop an interest in mathematics and pursue it as a career. But swell, nothing could be farther from truth. From what I have experienced there are two major categories of students who take calculus in high school. The first category would be students who are just out of wits about calculus, its purpose and meaning. They just see it as another infliction upon them without any significance. They struggle with remembering the formulae and will just barely pass the course (and many times don’t). These students hate mathematics, calculus makes it worse. Integration is opposite of differentiation: but why teach it to us?
The other major category of students is the one who take on calculus but with a caveat. They are the ones who will score in the 80s and 90s in the examination, but they have cracked the exam system per se. And might not have any foundational knowledge of calculus. But someone might ask how can one score 95/100 and still not have foundational knowledge of the subject matter? This is the way to beat the system. These learners are usually drilled in solving problems of a particular type. It is no different than chug and slug. They see a particular problem – they apply a rote learned method to solve it and bingo there is a solution. I have seen students labour “problem sets” — typically hundreds of problems of a given type — to score in the 90s in the papers. This just gives them the ability to solve typical problems which are usually asked in the examinations. Since the examination does not ask for questions based on conceptual knowledge – it never gets tested. Perhaps even their teachers if asked conceptual questions will not be able to handle them — it will be treated like a radioactive waste and thrown out — since it will be out of syllabus.
There is a third minority (a real minority, and may not be real!, this might just be wishful thinking) who will actually understand the meaning and significance of the conceptual knowledge, and they might not score in the 90s. They might take a fancy for the subject due to calculus but the way syllabus is structured it is astonishing that any students have any fascination left for mathematics. Like someone had said: the fascination for mathematics cannot be taught it must be caught. And this is exactly what MAA and NCTM have said in their statement about dropping calculus from high-school.
What the members of the mathematical community—especially those in the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM)—have known for a long time is that the pump that is pushing more students into more advanced mathematics ever earlier is not just ineffective: It is counter-productive. Too many students are moving too fast through preliminary courses so that they can get calculus onto their high school transcripts. The result is that even if they are able to pass high school calculus, they have established an inadequate foundation on which to build the mathematical knowledge required for a STEM career. (emphasis added)
The problem stems from the fact that the foundational topics which are prerequisites for calculus are on shaky grounds. No wonder anything build on top of them is not solid. I remember having very rudimentary calculus in college chemistry, when it was not needed and high-flying into physical meaning of derivatives in physics which was not covered enough earlier. There is a certain mismatch between the expectations from the students and their actual knowledge of the discipline as they come to college from high-school.
Too many students are being accelerated, short-changing their preparation in and knowledge of algebra, geometry, trigonometry, and other precalculus topics. Too many students experience a secondary school calculus course that drills on the techniques and procedures that will enable them to successfully answer standard problems, but are never challenged to encounter and understand the conceptual foundations of calculus. Too many students arrive at college Calculus I and see a course that looks like a review of what they learned the year before. By the time they realize that the expectations of this course are very different from what they had previously experienced, it is often too late to get up to speed.
Though they conclude that with enough solid conceptual background in these prerequisites it might be beneficial for the students to have a calculus course in the highschool.
Experiments, Data and Analysis
There are many sad stories of students, burning to carry out an experimental project, who end up with a completely unanalysable mishmash of data. They wanted to get on with it and thought that they could leave thoughts of analysis until after the experiment. They were wrong. Statistical analysis and experimental design must be considered together…
Using statistics is no insurance against producing rubbish. Badly used, misapplied statistics simply allow one to produce quantitative rubbish rather than qualitative rubbish.
– Colin Robson (Experiment, Design and Statistics in Psychology)
The logician, the mathematician, the physicist, and the engineer
The logician, the mathematician, the physicist, and the engineer. “Look at this mathematician,” said the logician. “He observes that the first ninety-nine numbers are less than hundred and infers hence, by what he calls induction, that all numbers are less than a hundred.”
“A physicist believes,” said the mathematician, “that 60 is divisible by all numbers. He observes that 60 is divisible by 1, 2, 3, 4, 5, and 6. He examines a few more cases, as 10, 20, and 30, taken at random as he says. Since 60 is divisible also by these, he considers the experimental evidence sufficient.”
“Yes, but look at the engineers,” said the physicist. “An engineer suspected that all odd numbers are prime numbers. At any rate, 1 can be considered as a prime number, he argued. Then there come 3, 5, and 7, all indubitably primes. Then there comes 9; an awkward case, it does not seem to be a prime number. Yet 11 and 13 are certainly primes. ‘Coming back to 9’ he said, ‘I conclude that 9 must be an experimental error.'”
– George Polya (Induction and Analogy – Mathematics of Plausible Reasoning – Vol. 1, 1954)
On mathematics
Mathematics is regarded as a demonstrative science. Yet this is only one of its aspects. Finished mathematics presented in a finished form appears as purely demonstrative, consisting of proofs only. Yet mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies; you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.
– George Polya (Induction and Analogy – Mathematics of Plausible Reasoning – Vol. 1, 1954)
Unreal and Useless Problems
We had previously talked about problem with contexts given in mathematics problems. This is not new, Thorndike in 1926 made similar observations.
Unreal and Useless Problems
In a previous chapter it was shown that about half of the verbal problems given in standard courses were not genuine, since in real life the answer would not be needed. Obviously we should not, except for reasons of weight, thus connect algebraic work with futility. Similarly we should not teach the pupil to solve by algebra problems which in reality are better solved otherwise, for example, by actual counting or measuring. Similarly we should not set him to solve problems which are silly or trivial, connecting algebra in his mind with pettiness and folly, unless there is some clear, counterbalancing gain.
This may seem beside the point to some teachers, ”A problem is just a problem to the children,” they will say,
“The children don’t know or care whether it is about men or fairies, ball games or consecutive numbers.” This may be largely true in some classes, but it strengthens our criticism. For, if pupils^do not know what the problem is about, they are forming the extremely bad habit of solving problems by considering only the numbers, conjunctions, etc., regardless of the situation described. If they do not care what it is about, it is probably because the problems encountered have not on the average been worth caring about save as corpora vilia for practice in thinking.
Another objection to our criticism may be that great mathematicians have been interested in problems which are admittedly silly or trivial. So Bhaskara addresses a young woman as follows: ”The square root of half the number of a swarm of bees is gone to a shrub of jasmine; and so are eight-ninths of the swarm: a female is buzzing to one remaining male that is humming within a lotus, in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees.” Euclid is the reputed author of: ”A mule and a donkey were going to market laden with wheat. The mule said,’If you gave me one measure I should carry twice as much as you, but if I gave you one we should bear equal burdens.’ Tell me, learned geometrician, what were their burdens.” Diophantus is said to have included in his preparations for death the composition of this for his epitaph : ” Diophantus passed one-sixth of his life in childhood one-twelfth in youth, and one-seventh more as a bachelor. Five years after his marriage was born a son, who died four years before his father at half his father’s age.”
My answer to this is that pupils of great mathematical interest and ability to whom the mathematical aspects of these problems outweigh all else about them will also be interested in such problems, but the rank and file of pupils will react primarily to the silliness and triviality. If all they experience of algebra is that it solves such problems they will think it a folly; if all they know of Euclid or Diophantus is that he put such problems, they will think him a fool. Such enjoyment of these problems as they do have is indeed compounded in part of a feeling of superiority.
On not learning or con in the context
We will, we will, fail you by testing what you do not know…
We live in a rather strange world. Or is it that we assume the world
to be non-strange in a normative way, but the descriptive world has
always been strange? Anyways, why I say this is to start a rant to
about some obviously missed points in the area of my work. Namely,
educational research, particularly science and mathematics education
research.
In many cases the zeal to show that the students have
‘misunderstandings’ or are simply wrong, and then do a hair-splitting
(micro-genetic) exercise on the test the students were inflicted
with. Using terse jargon and unconsequential statistics, making the
study reports as impossible to read as possible, seem to be the norm.
But I have seen another pattern in many of the studies, particularly
in mathematics education. The so-called researchers spent countless
nights in order to dream up situations as abstract as possible (the
further far away from real-life scenarios the better), then devise
problems around them. Now, these problems are put in research studies,
which aim to reveal (almost in evangelical sense) the problems that
plague our education. Unsuspecting students are rounded, with
appropriate backgrounds. As a general rule, the weaker socio-economic
background your students come from, the more exotic is your study. So
choose wisely. Then these problems are inflicted upon these poor,
mathematically challenged students. The problems will be in situations
that the students were never in or never will be. The unreal nature of
these problems (for example, 6 packets of milk in a cup of coffee! I
mean who in real life does that? The milk will just spill over, the
problem isn’t there. This is just a pseudo-problem created for satisfying the research question of the researcher. There is no context, but only con.
Or finding out a real-life example for some weird fractions) puts many off. The fewer students perform correctly happier the researcher is. It just adds to the data statistic that so many % students cannot perform even this elementary task well. Elementary for
that age group, so to speak. The situation is hopeless. We need a
remedy, they say. And remedy they have. Using some revised strategy,
which they will now inflict on students. Then either they will observe
a few students as if they are some exotic specimens from an
uncontacted tribe as they go on explaining what they are doing or why
they are doing it. Or the researcher will inflict a test (or is it
taste) in wholesale on the lot. This gives another data
statistic. This is then analysed within a ‘framework’, (of course it
needs support) of theoretical constructs!
Then the researcher armed with this data will do a hair-splitting
analysis on why, why on Earth student did what they did (or didn’t
do). In this analysis, they will use the work of other researchers before
them who did almost the same thing. Unwieldy, exotic and esoteric
jargons will be used profusely, to persuade any untrained person to
giveup on reading it immediately. (The mundane, exoteric and
understandable and humane is out of the box if you write in that
style it is not considered ‘academic’.) Of course writing this way,
supported by the statistics that are there will get it published in
the leading journals in the field. Getting a statistically significant
result is like getting a license to assert truthfulness of the
result. What is not clear in these mostly concocted and highly
artificial studies is that what does one make of this significance
outside of the experimental setup? As anyone in education research
would agree two setups cannot be the same, then what is t
Testing students in this way is akin to learners who are learning a
new language being subjected to and exotic and terse vocabulary
test. Of course, we are going to perform badly on such a test. The
point of a test should be to know what students know, not what they
don’t know. And if at all, they don’t know something, it is treated as
if is the fault of the individual student. After all, there would be
/some/ students in each study (with a sufficiently large sample) that
would perform as expected. In case the student does not perform as
expected we can have many possible causes. It might be the case that
the student is not able to cognitively process and solve the problem,
that is inspite of having sufficient background knowledge to solve the
problem at hand the student is unable to perform as expected. It might
be the case that the student is capable, but was never told about the
ways in which to solve the given problem (ZPD anyone?). In this case, it might be that the curricular materials that the student has access
to are simply not dealing with concepts in an amenable way. Or it
might be that the test itself is missing out on some crucial aspects
and is flawed, as we have seen in the example above. The problem is
systemic, yet we tend to focus on the individual. This is perhaps
because we have a normative structure to follow an ideal student at
that age group. This normative, ideal student is given by the so-called /standards of learning/. These standards decide, that at xx age
a student should be able to do multiplication of three digit
numbers. The entire curricula are based on these standards. Who and
what decides this? Most of the times, the standards are wayyy above
the actual level of the students. This apparent chasm between the
descriptive and the normative could not be more. We set unreal
expectations from the students, in the most de-contextualised and
uninteresting manner, and when they do not fulfil we lament the lack
of educational practices, resources and infrastructure.
What is a mathematical proof?
A dialogue in The Mathematical Experience by Davis and Hersh on what is mathematical proof and who decides what a proof is?
Let’s see how our ideal mathematician (IM) made out with a student who came to him with a strange question.
Student: Sir, what is a mathematical proof?
I.M.: You don’t know that? What year are you in?
Student: Third-year graduate.
I.M.: Incredible! A proof is what you’ve been watching me do at the board three times a week for three years! That’s what a proof is.
Student: Sorry, sir, I should have explained. I’m in philosophy, not math. I’ve never taken your course.
I.M.: Oh! Well, in that case – you have taken some math, haven’t you? You know the proof of the fundamental theorem of calculus – or the fundamental theorem of algebra?
Student: I’ve seen arguments in geometry and algebra and calculus that were called proofs. What I’m asking you for isn’t examples of proof, it’s a definition of proof. Otherwise, how can I tell what examples are correct?
I.M.: Well, this whole thing was cleared up by the logician Tarski, I guess, and some others, maybe Russell or Peano. Anyhow, what you do is, you write down the axioms of your theory in a formal language with a given list of symbols or alphabet. Then you write down the hypothesis of your theorem in the same symbolism. Then you show that you can transform the hypothesis step by step, using the rules of logic, till you get the conclusion. That’s a proof.
Student: Really? That’s amazing! I’ve taken elementary and advanced calculus, basic algebra, and topology, and I’ve never seen that done.
I.M.: Oh, of course, no one ever really does it. It would take forever! You just show that you could do
it, that’s sufficient.
Student: But even that doesn’t sound like what was done in my courses and textbooks. So mathematicians don’t really do proofs, after all.
I.M.: Of course we do! If a theorem isn’t proved, it’s nothing.
Student: Then what is a proof? If it’s this thing with a formal language and transforming formulas, nobody ever proves anything. Do you have to know all about formal languages and formal logic before you can do a mathematical proof?
I.M.: Of course not! The less you know, the better. That stuff is all abstract nonsense anyway.
Student: Then really what is a proof?
I.M.: Well, it’s an argument that convinces someone who knows the subject.
Student: Someone who knows the subject? Then the definition of proof is subjective; it depends on particular persons.Before I can decide if something is a proof, I have to decide who the experts are. What does that have to do with proving things?
I.M.: No, no. There’s nothing subjective about it! Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you’ll catch on.
Student: Are you sure?
I.M.: Well – it is possible that you won’t, if you don’t have any aptitude for it. That can happen, too.
Student: Then you decide what a proof is, and if I don’t learn to decide in the same way, you decide I don’t have any aptitude.
I.M.: If not me, then who?
Mathematical Literacy Goals for Students
National Council of Teachers for Mathematics NCTM proposed these five goals to cover the idea of mathematical literacy for students:
- Learning to value mathematics: Understanding its evolution and its role in society and the sciences.
- Becoming confident of one’s own ability: Coming to trust one’s own mathematical thinking, and having the ability to make sense of situations and solve problems.
- Becoming a mathematical problem solver: Essential to becoming a productive citizen, which requires experience in a variety of extended and non-routine problems.
- Learning to communicate mathematically: Learning the signs, symbols, and terms of mathematics.
- Learning to reason mathematically: Making conjectures, gathering evidence, and building mathematical arguments.
National Council of Teachers of Mathematics. Commission on Standards for School Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Natl Council of Teachers of.
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
= 13How 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.
Topological Art
ILLUSTRATIONS FOR TOPOLOGY
From the book Introduction to Topology by Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko. The book was published by Mir Publishers in 1985.
ILLUSTRATION TO CHAPTER I
The central part of the picture presents the standard embedding chain of crystalline groups of the three dimensions of Euclidean space: their standard groups embedded into each other are depicted as fundamental domains (Platonic bodies: a cube, a tetrahedron, a dodecahedron). The platonic bodies are depicted classically, i.e., their canonical form is given, they are supported by two-dimensional surfaces (leaves), among which we discern the projective plane (cross-cap), and spheres with handles. The fantastic shapes and interlacings (as compared with the canonical objects) symbolizes the topological equivalence.
At the top, branch points of the Riemann surfaces of various multiplicities are depicted: on the right, those of the Riemann surfaces of the functions w=5z√ and w=z√; on the left below, that of the same function w=z√, and over it, a manifold with boundary realizing a bordism mod 3.
ILLUSTRATION TO CHAPTER II
The figure occupying most of the picture illustrates the construction of a topological space widely used in topology, i.e., a 2-adic solenoid possessing many interesting extremal properties. The following figures are depicted there: the first solid torus is shaded, the second is white, the third is shaded in dotted lines and the fourth is shaded doubly. To obtain the 2-adic solenoid , it is necessary to take an infinite sequence of nested solid tori, each of which encompasses previous twist along its parallel, and to form their intersection.
Inside the opening, a torus and a sphere with two handles are shown. The artist’s skill and his profound knowledge of geometry made it possible to represent complex interlacing of the four nested solid tori accurately.
ILLUSTRATION TO CHAPTER III
The canonical embedding of a surface of genus g into the three-dimensional Euclidean space is represented 0n the right . A homeomorphic embedding of the same surface is shown on the left . The two objects are homeomorphic, homotopic and even isotopic . The artist is a mathematician and he has chosen these two, very much different in their appearance, from an infinite set of homeomorphic images.
ILLUSTRATION TO CHAPTER IV
Here an infinite total space of covering over a two-dimensional surface, viz., a sphere with two handles, is depicted. The artist imparted the figure the shape of a python and made the base space of the covering look very intricate. Packing spheres into the three-dimensional Euclidean space and a figure homeomorphic to the torus are depicted outside the central object. The mathematical objects are placed so as to create a fantastic landscape.
ILLUSTRATION TO CHAPTER V
A regular immersion of the projective plane RP2 in R3 is represented in the centre on the black background. The largest figure is the Klein bottle (studied in topology as a non-orientable surface) cut in two (Moebius strips) along a generator by a plane depicted farther right along with the line intersection; the lower part is plunging downwards; the upper part is being deformed (by lifting the curve of intersection and building the surface up) into a surface with boundary S1; a disc is being glued to the last, which yields the surface RP2. The indicated immersion process can be also used for turning S2 `inside out’ into R3.
On the outskirts of the picture, a triangulation of a part of the Klein bottle surface is represented.
A detailed explanation of this picture may serve as a material for as much as a lecture in visual topology.
On Division of Education
‘Consider how we design our educational programs. We take the major subjects apart and reduce them to a number of main sub-sections. Then we subdivide the sub-sections. We continue until we have a large collection of little pieces that we believe that children can understand.
‘As a result we present our students with disembodied fragments of subject matter … fragments that they can’t possibly make sense of … fragments that they can’t use for anything. Most of them never sense the full power of the subjects that they encounter.’
via |Turtle Speaks Mathematics
Rousseau on education…
There are, indeed, professors. . . for whom I have the greatest love and esteem, and think them very capable of instructing youth were they not tied down by established customs. . . . Perhaps an attempt may be made some time or other to remove the evil, when it is seen to be not without remedy.
Jean-Jacques Rousseau
A parable on…
A Parable
Once upon a time, in a far away country, there was a community that had a wonderful machine. The machine had been built by most inventive of their people … generation after generation of men and women toiling to construct its parts… experimenting with individual components until each was perfected… fitting them together until the whole mechanism ran smoothly. They had built its outer casing of burnished metal and on one side, they had attached a complex control panel. The name of the machine, KNOWLEDGE, was engraved on a plaque set in the centre of the control panel.
The community used the machine in their efforts to understand the world and to solve all kinds of problems. But the leaders of the community were not satisfied. It was a competitive world… they wanted more problems solved and they wanted them solved faster.
The main limitation for the use of machine was the rate at which data could be prepared for input. Specialist machine operators called ‘predictors’, carried out this exacting and time consuming task… naturally the number of problems solved each year depended directly on the number and skill of the predictors.
The community leaders focussed on the problem of training predictors. The traditional method, whereby promising girls and boys were taken into long-term apprenticeship, was deemed too slow and too expensive. Surely, they reasoned, we can find more efficient approach. So saying, they called the elders together and asked them to think about the matter.
After a few months, the elders reported that they had devised an approach that showed promise. In summary, they suggested that the machine be disassembled. Then each component could be studied and understood with ease… the operation of machine would become an open book to all who cared to look.
Their plan was greeted with enthusiasm. So, the burnished covers were pulled off, and the major mechanisms of the machine fell out… they had plaques with labels like HISTORY and GEOGRAPHY and PHYSICS and MATHEMATICS. These mechanisms were pulled apart in their turn… of course, care was taken to keep all the pieces in separate piles. Eventually, the technicians had reduced the machine to little heaps of metal plates and rods and nuts and bolts and springs and gear wheels. Each heap was put in a box, carefully labelled with the name of the mechanism whose part it contained, and the boxes were lined up for the community to inspect.
The members of the community were delighted. Their leaders were ecstatic. They ‘oohed’ and ‘aahed’ over the quality of components, the obvious skill that had gone in their construction, the beauty of designs. Here, displayed for all, were the inner workings of KNOWLEDGE.
In his exuberance, one man plunged his hand into a box and scooped up a handful of tiny, jewel-like gear wheels and springs. He held them out to his daughter and glancing, at the label on the box, said:
“Look, my child! Look! Mathematics! ”
From: Turtle Speaks Mathematics by Barry Newell
You can get the book (and another nice little book Turtle Confusion) here.
Gel’fand’s Quote
This is taken from The Method of Coordinates by I. M. Gel’fand
E.G. Glagoleva A.A. Kirillov
Of course, it was not our intention that aIl these
students who studied from these books or even
completed the School should choose mathematics as
their future career. Nevertheless, no matter what they
would later choose, the results of this training re
mained with them. For many, this had been their first
experience in being able to do something on their own
— completely independently.
1 would like to make one comment here. Sorne of my
American colleagues have explained to me that
American students are not really accustomed to think
ing and working hard, and for this reason we must
make the material as attractive as possible. Permit me
to not completely agree with this opinion. From my
long experience with young students aU over the
world 1 know that they are curious and inquisitive and
1 beIieve that if they have sorne clear mate rial pre
sented in a simple form, they will prefer this to aIl
artificial means of attracting their attention — much as
one ,buys books for their content and not for their
dazzling jacket designs that engage only for the
moment.
The most important thing a student can get from the
study of mathematics is the attainment of a higher
intellectualleveL In this light 1would like to point out
as an example the famous American physicist and
teacher Richard Feynman who succeeded in writing
both his popular books and scientific works in a
simple and attractive manner.
I. M. Gel’fand
Heaven and Hell
Circle Limit IV
Heaven and Hell
by M C Escher
Yesterday I have put up Escher’s Circle Limit IV – Heaven and Hell on my new desk. The Circle Limit series of drawings was drawn by Escher are essentially what are known as his hyperbolic tesselations. The new computer table that I have got has an odd shape. On one end the side is circular and it smoothly metamorphises into rectangle on the other side. Though it is not at all comparable to what Escher has accomplished, I feel bad even when I use the word metamorphosis for this, but I have not found anything better. The table is designed for use with a desktop. So it has sections for different parts of the desktop like the monitor, CPU keyboard etc.
Anyways the main point that I want to tell is that the table at one end is circular. Since I had put Escher’s Three World on another table, I thought it would be a good idea to use a ciruclar print of Escher for this part of the table. Of all the prints I had, which I had taken when I had at my disposal A3 sized printers, the one which fitted the purpose seemed to be Circle Limit IV – Heaven and Hell.
Let us see what Escher himself has to say about this series of works viz. The Circle Limits:
So far four examples have been shown with points as limits of infinite smallness. A diminution in the size of the figures progressing in the opposite direction, i.e. from within outwards, leads to more satisfying results. The limit is no longer a point, but a line which border’s the whole complex and gives it a logical boundary. In this way one creates, as it were, a universe, a geometrical enclosure. If the progressive reduction in size radiates in all directions at an equal rate, then the limit becomes a circle. [1]
And he says this about Heaven and Hell:
CIRCLE LIMIT IV, (Heaven and Hell)
[Woodcut printed from2 blocks, 1960, diameter 42 cm]
Here also we have the components diminishing in size as they move outwards. The six largest (three white angels and three black devils) are arranged about the centre and radiate from it. The disc is divided into six sections in which, turn and turn about, the angels on a black background and then the devils on a white one, gain the upper hand. in this way, heaven and hell change place six times. In the intermediate, “earthly” stages, they are equivalent. [1]
Like most of Escher’s drawings this one also takes you to a different world. A world which is far away from the reality. A world of mathematics. A world of abstraction. But then as always we can make connections between this abstract world and the real world. The connections that we can make are dependent on the world view that we have. Some people fail to make the connection. They cannot `see’.
The Circle Limit series is what brought Escher to the eyes of the mathematicians. H. S. M. Coxeter used Circle Limit II as an illustration in his article on hyperbolic tesselations. Since then the other works of Escher have been examined by the mathematicians, and we find that very deep and fundamental ideaso of mathematics are embedded in them. As to how Escher did it is amazing. The kind of clear insight that Escher exhibits in his artwork is astounding. He could visualize the mathematical transformations in his head and then transform them onto the artwork he was working with. Escher has said
I have brought to light only one percent of what I have seen in the darkness. [2]
This must be certainly true, as most of his artwork is nowhere close to what we see in the light. I rate the artwork of Escher as greater than that of the renessaince artist’s as they had just beautifully drawn what one could “see.” But with Escher we go a step beyond, imagination takes the control. What interests me in Escher is that he can make you imagine the unimaginable. What you know is not possible is demonstrated just in front of your eyes. Logic is discarded. Rather it is kept in the basement which is upstairs for Escher.
Yesterday you start to believe what you thought was impossible tommorow.
The way different things merge for Escher is just unparalled in the work of other artists. What has now become known as “Escheresque” is just the typical of his style. Lot of later artists are influenced by the works of Escher, I have found one Istvaan Orosz particulary good. There are others who are equally good but I don’t remember their names now….
Coming back to Heaven and Hell. The main artwork is in a woodcut format in black and white. For me this is a kind of dyad which represents the world. The idea of two opposing forces one termed to be evil and the other good are all permeating in the Universe. Here also the bat-devils and the angels are the representative of the same. There is no part of the Universe where these two are not present. It might seem that somewhere far out there there is nothing, but it is not so. Even there, the design is the same, it is just too far for us to see. This is what harmony in the universe is about. It is the same everywhere, when you have a broad enough world-view. The cosmologists say that the Universe is homogenous and isotropic, if you choose to “see” it at the right scale. The cosmologists often use Heaven and Hell to illustrate this point. For me introduction to Escher came in a talk by a cosmologist who used The Waterfall to illustrate the idea of a perpetual motion machine. Since then I have become addicted to Escher, as has everybody else who has some sense of imagination. For those who cannot appreciate Escher, I can just pity at their miserable imagination.
References:
[1] The Graphic Work of M C Escher by M C Escher
Ballantine 1975, ISBN 345246780595
[2] M. C. Escher (Icons) by Julius Wiedemann (Editor)
Taschen 2006, ISBN 3822838691
Zero
For a proper understanding of the evolution and the need for the concept of zero we need to understand how our current number system has evolved from its ancestors. The very need for the concept of zero did not arise till the number systems themselves were well developed. The advancement in the number system necessitated the need for the concept of zero as we now know it. We can identify two distinct manifestations of zero; one is zero as a placeholder and the other is zero as a number, the former has much earlier origin than the later.
Humans probably before having the concept of numbers or counting then, would have begun with enumeration. By enumeration it is meant that we simply keep a track of objects in a collection or a set by matching the objects with other objects used as counters. A shepherd can keep the track of sheeps in the flock, by keeping pebbles which are equal in number to the number of sheep s in the flock or equivalently [if possible] by counting body parts. Then just by matching each sheep with each pebble the record of number of sheep s can be maintained. When the number of sheep s is increased or decreased the same number of pebbles or other counters can be increased or decreased correspondingly. The other counters that one can have for this type of counting can include the human body itself. In fact many primitive societies do indeed have a counting system based on the body parts. This is the most basic system of counting that we can have. No language is needed for such one-to-one counting.
When the languages developed, particular words were created for various body parts, so these words were used instead of the body parts themselves. This is a transition from enumeration to numeration. Thus one has to remember only the word names in order for counting. But this does not imply the idea of cardinality of number being present in this numeration. For the notion of cardinality of a number to be used in the idea of numeration it required some time. When the questions were asked in the form How many…? in the ancient texts, the answers to these type of questions are given best in terms of the cardinal number. From this further growth would be, the concept of ordinality i.e. the order of things is not important when counting objects. It relates to the fact that the last number enounced in a set not only assigns a certain name to the last object in the set to be matched but also tells us how many objects are there in that set altogether.
The further development of this numeration is the formation of numeration systems. The need for the number systems typically arose from the following question:
What is to be done when the finite ordered sequence of counters is exhausted, yet more objects remain to be matched?
This particular question was answered in different ways only one of which led us to the current number system we have. One of the most simple solutions to this is to extend the ordered sequence of counters. So that we invent new symbols or names to accommodate the excess objects that are to be matched. But this approach makes no sense when we have large number of objects that are to be matched.
A simpler way which lends itself well to the written representation, was extension by repetition. The extension by repetition implies a number system which is based on the additive principle. Most of the primitive number systems are based on the additive principle. Here the figures are entirely free. Their juxtaposition entails adding together their values. In a number system based on the additive principle it makes no difference where you place the symbols corresponding to the numbers. Some of the numbers systems based on the additive principle are; Egyptian, Cretan, Hittite, Greek, Aztec, Roman, Sumerian etc. As an example of the additive principle we consider the Egyptian system. In this system if we want to represent the number 5247 it can be represented in following ways:
When we break down the representation based on the additive principle we get the following:
Thus we see that in the representation of a number in the number systems based on the additive principle. Since addition is both commutative and associative, irrespective of where we place the base numbers the final number that is represented by the various combinations of these numbers remains the same.
This system though seems simple puts a lot of cognitive load on the user. First of all there are different symbols for different numbers and in many of these number systems the symbols have some intuitive association [at least in the lower range] to the number that they represent. So to represent large numbers a large number of different symbols were to be used. In our example of representing the number 5247 in the Egyptian hieroglyphic notation we have used a total of 18 symbols. Many times for representing large numbers new symbols had to be introduced. The arithmetic operations with these systems presented another difficulty. The number systems based on the additive principle are not well suited for arithmetic operations. For example consider the following sum in the Roman notation:
The above sum gives us no clue to what is supposed to be done. Though there are methods to perform this operations, but the procedures involved are very complicated. The above sum in the current notation would be:
In the number systems based on the additive principle the number signs are static in nature, which have no operational significance. The number signs in this case are more like abbreviations which can be used to write down the results of the calculations performed by some other means. To do arithmetical calculations, the ancients generally used auxiliary aids such as abacus or a table with counters.
The enumeration, numeration as we have seen do not have any requirement for the concept of zero as a number or a placeholder. The same is true with the number systems based on the principle of addition, in these systems there is no requirement of the concept of zero.
The next step in the evolution of the number systems was the hybrid system, called so because it involves use of both addition and multiplication. In the hybrid system when the symbols for lets say symbols for 1000 and 5 are presented together, they meant 5 x 1000 = 5000, whereas in the additive system they will mean 1000+5=1005. In the hybrid system there were basic symbols for the numbers, and symbols for various powers of the base, for example in a base 10, system the symbols for 100, 1000 etc. These number systems used the additive principle for representing numbers below 100.
In case of complete hybrid systems there were special symbols for the numbers 1 – 9, and all numbers including the tens were represented as a product of these base numbers and the powers of 10. This increased the range of numbers that can be represented. The notable hybrid systems are Assyro-Babylonian, Phoenician, Singhalese, Mari, Chinese, Ethiopian, Tamil, Malayalam, and the Mayan. We consider an example from the complete hybrid systems to represent the number 5247 from \cite{uni1}.
When we break down the representation based on the multiplicative principle we get the following:
The hybrid systems thus need a specification of the powers of the base which, determine the value of the number in a given position. This brings us a step closer to the positional number systems based on the multiplicative principle. The hybrid system are not all forgotten and are still in use today. When we verbally read a number it is more of a hybrid number system that we use that a positional number system. That is to say when we read the number 5247, we spell it out as five-thousand two-hundred and forty-seven. Here when we verbally read a number we also explicitly give its corresponding powers just like in case of the hybrid number system. Even in this case the need for zero is not there, the hybrid systems can work without the use of the concept of zero.
So to conclude the hybrid systems are “Systems based [at least after a certain order] on a mixed principle [both additive and multiplicative] that invokes multiplication rule to represent consecutive order of units.”
We now move to the positional systems or multiplication based systems. These systems have a more abstract representation. The value of a figure in these positional systems varies according to the position in which it occurs in the representation of the number. Due to this the coefficients of the power of the base, into which the number has been decomposed appear. For example in a particular representation the actual value of a number, lets say 5 will depend on which position 5 is present in. If 5 is present in the units place then it represents 5, when it is present in the tens place it represents 50, and so on. If in the hybrid system if we remove the symbols used and just have the numbers only we have a positional number system. In this case the powers of the base for our case take base as 10, are implicitly figured out from the position of the numerals in the representation of the number. We know that in the positional representation of the number 5247, 5 is in the thousands place, 2 is in the hundreds place etc. Once this order is fixed then can we represent a number without any ambiguity? If we just consider the coefficients of the number 5247, the the answer to this probably seems to be true. But is it always so? For answer to this consider another example. Suppose we want to represent a number 1043 in the positional number system. In case of hybrid number system the representation would be like this:
so if we now drop the powers of the base, and just take the coefficients we are left with:
But this is not correct, since 143 is another number and not 1043.Similarly if we take just the coefficients of the number 10403, they are again 143. In case of the non-positional system this was not a problem, since every power and the corresponding coefficient was made explicit. But here if we just consider the coefficients of the number in a particular base, we cannot be sure that the number that we are representing is correct, unless we know for sure that a particular coefficient corresponding to a particular power is not present. In case of 1043 we have the coefficient of 100 absent. Some of the earliest positional systems that were developed suffered from the same problem. In case of the Babylonian system, we are not sure of how to read a particular number in many clay tablets, and the number has to be guessed from the context of the problem. Since the Babylonians used a base of 60, so a number [lets take 5247] was represented as:
In this case there was no ambiguity in base 60 number would be written as [1;27;27]. But even in this case there was no guarantee that the number represented is the number that we want. Suppose if we want to represent 3627 in this notation, then it would be represented as:
which is very easy to confuse with
Thus we see that in case of the positional number system we required a notion that tell us whether a particular coefficient is absent. This requirement initiated the need for the concept of zero. So the discovery of zero was therefore a necessity for the strict and regular use of the rule of the position, and it was therefore a decisive stage in the development of mathematics. So how do we make sure that something is not present in a particular position in a given positional representation of a number. It becomes essential then to have a special sign whose purpose is to indicate the absence of anything in particular position. This thing which signifies nothing, or the empty space, is in fact the \textsl{zero}. As \cite{uni1} pg. 668 puts it: “To arrive at the realisation that empty space may and must be replaced by a sign whose purpose is precisely to indicate that it is empty space: this is the ultimate abstraction, which required much time, much imagination, and beyond doubt great maturity of mind.”
The concept of zero has been discovered three times in the history independently. It was discovered first by the Babylonians, the Mayans and the Indians. All these three civilizations used the positional number system for which the concept of zero is needed. The Babylonians tried to get away with this difficulty by leaving empty space where the missing coefficients of particular order were to be found. Hence they would write a number such as [1; 6] for lets say 3606. But this did not solve the problem completely. In copy or reading these spaces could be overlooked, and particularly when two or more space were to be given it could be confused with one space. But since the Babylonians has the base as 6o the need for writing numbers with zero in between arises on a very few occasions than it does in the number system with base 10. In case of the sexagesimal numeration only in 59 integers below 3600 this arises; as compared to 917 cease in the base 10 system \cite{boyer}. The Babylonian zero is the first zero to arrive on the scene. To denote absence of a coefficient of a particular order in their representation of the number, the Babylonians used a special sign [after fourth century BCE], which is the a cuneiform sign looking like a double oblique chevron. The Mayans developed their positional system with base 20, but they were not consistent with the use of the powers of the base after the third position \cite{uni2} pg 670. The Mayans understood the concept of zero sign, but they did not have its operational usability due to their inconsistent positional system. In case of the Babylonians it was never understood as a number synonymous with empty and never corresponded to the meaning of null quantity. So we see that in spite of having the notion of zero the Mayans and teh Babylonians did not get much further in this. The Mayan and the Babylonian zeros are as given in the figure.
If we work out the number represented in these notation the numbers are:
In the Babylonian notation.
In the Mayan notation.
The credit of having a well conceived positional system, which is operationally useful goes to the Indians. This step was taken by simplifying the hybrid notation, by suppressing the signs indicating the powers of the base. This required a much higher level of abstraction: the zero. This can be regarded as “… the supreme discovery of mathematicians who soon would come to extent it, form its first role of representing empty space, to embrace truly numeric meaning of a null quantity.” The Indian civilization was the only one to achieve this great feat. This system came up as a result of conjunction of three great ideas :
1.The idea of attaching each basic figure with signs removed from intuitive associations.2. The idea of a positional number system, in which the value of a number depends on its position in the representation.3. The idea of a full operational zero, filling the empty spaces of missing units and at the same time having the meaning of a null number.
In the system thus developed it does not matter what signs or base we use for the system, if it rests strictly and rigorously of the principle of position and incorporates the full concept of the symbol for zero. The discovery of zero in India and the place value were inventions unique to the Indian civilization. The roots of the development of the positional number system in India can be traced to the use of spoken sanskrit [संस्कुत] numeral system [Treatment of the development of Indian positional system follows from \cite{uni1}, \cite{uni2}]. The sanskrit spoken language has for each power of ten an individual name, “… so that to express a given number, one only had to place the name indicating the order of units between the name of the order of units that was immediately below it and immediately above it.” In fact there are names to the powers of 10 till 10^140 \cite{uni2} pg. 134. This is what is required in a positional number system. From the sanskrit spoken numeral system the Indian system of numerical symbols was formed. As soon as place value system was rigorously applied to the nine simple units, the use of a special terminology was indispensable to indicate the absence of units of a particular order. The sanskrit language already possessed the word shunya [शुन्य] to express void or absence, which also an element of mystical and religious philosophy. So to express the new mathematical notion of zero the term shunya could be used. This is how the word came to perform the function of zero as a part of the counting system.
Indian mathematicians before discovering the place value system, used their fingers or concrete mathematical devices. The most common was the abacus; from left to right, the columns representing the various powers of ten. The first nine numerals were traced in sand or dust, inside the column of a particular decimal order. Thus the number 5247 would have been represented in the following manner :
If a particular order of units was missing, one only needed to leave that particular column empty. Thus for representing 5047 we would write:
So with all this the necessary ‘ingredients’ for the creation of the written place value system had been amassed by the Indians:
- Distinct representation of one to nine numbers, which had forms unrelated to the number they represented.
- Discovery of the place value system.
- Invention of the concept of zero.
Still some things were still absent for the perfection of the number system:
- The nine numerals were only used in accordance to addition principle for analytical combinations using numerals higher than or equal to ten, the notation was very basic and limited to numbers below 100,000.
- Place value system was only used with sanskrit names for numbers.
- Zero was only used orally.
The only thing that remained was to combine these ideas. By using the nine bramhi [ब्राम्ही] numerals on the dust abacus this stage already had been reached.
The two methods of expressing the numbers bramhi numerals and sanskrit names of numbers were known to the Indian mathematicians. In the dust abacus the numbers were drawn in contemporary style. The numbers in sanskrit were expressed in orders of ascending powers of ten; from the smallest to the highest. So that 4769 is written as:
And it is read in sanskrit as:
नव शष्टि सप्तशत् च चतुरसहस्त्र
Meaning: nine sixty seven hundred and four thousand.
In the written numerals however the opposite order was used. The evidence for these methods goes back to third century BCE. IF we look at these two opposite ways of representing the number, indicates an inconsistency. This is what the Indian mathematicians expressed as :
अंकानाम वामतो गति:
Meaning: principle of the movement of numerals from the right to the left.
Since the brahmi had a limited numeral base [highest number expressed was 90,000], so any calculation larger than this was to be expressed in the sanskrit names for the numbers. In the dust abacus extremely large computations could be performed, and the successive columns in the abacus always rigorously corresponded to the consecutive powers of ten. The same mathematical structure was present in the sanskrit counting system. Thus each system was a mirror image of the other. Though the numbers are read from the right to the left from the smallest to the largest. The structure of the abacus is such that the mathematician has no other choice but to follow the principle
अंकानाम वामतो गति: principle of the movement of numerals from the right to the left.
The solution to write a number in this way was to start with the column for the simple units. This led to the abandonment of the old system. By beginning with highest power of ten, one immediately knows the size of number we are dealing with, but this did not facilitate drawing. Hence the opposite system was adopted; no matter how high a number, there could be no mistake as to which column to write it in. This was conserved when the positional notation was invented using numerical symbols.
All this lead to the following notation, “the numbers reading from left to right in descending powers of ten, constituting a faithful reproduction, minus the columns, of its representations on the abacus, as well as reflection of the abridged form of the corresponding sanskrit expression. Thus came the decimal position values which were given to the first nine numerals of the old notation. This was the birth of the modern numerals.
Now to convey the absence of units in a particular decimal order a new symbol was necessary. This was not required in the case of the abacus, but in the new positional system it became a necessity. The language already had the word symbol that expressed the concept zero, the shunya, it also conveyed the concepts such as sky, space etc. The circle has been considered as the representation of the sky, hence through a simple transposition of ideas it came to represent the concept of zero. Another sanskrit term representing zero was bindu [बिंदु], which literally means “point”. The point is the most insignificant geometrical figure, but for Indians the point represents the universe in non-manifest form. The point is the elementary of all geometrical figures, with potential for creating all the shapes, and hence was associated with zero. Zero is the most negligible quantities, but most fundamental of all abstract mathematics. The point also thus came to represent the zero. The two forms of the Indian zero are as shown in the figure below.The most likely time that the positional value system and zero were discovered is in the middle reign of the Gupta dynasty which ruled the Gangetic plains from about 240 to about 535 CE.
Along with the loaded philosophical connotations that were associated with the word shunya it served to mark the absence of units within a given decimal order in any position; the point or the little circle were used in the same way. This zero was also a mathematical operator; if placed after a number, it meant the number was multiplied by ten. Thus the three significant ideas that we have mentioned earlier were combined to give us the modern positional number system. Soon after this the concept of zero was perfected. Zero was given the status of a number, i.e. to say its cardinality was recognised. After this various arithmetic operations on and with zero were defined, which led to foundation of modern algebra .
The Arabs got this positional number systems from the Indians. The Europeans in turn got this system from the Arabs. The origin of the word zero or cipher can be traced back to this transfer of the positional number system to the Europeans from the Arabs. The Indian word for zero is shunya, from this the Arabic name sifr meaning vacant was given. When this was transferred to the Europeans the sound was kept but not the sense; Fibonacci called it zephirum. This was then passed over as zeuro, ceuero, and zepiro, which finally led to the current day synonyms which are the zero and the cipher.
References
Boyer C. B. :
Zero: The Symbol, the Concept, the Number
National Mathematics Magazine, Vol. 18, No. 8 , May 1944
Irfah G. :
The Universal History of Numbers I
Penguin, 2005
Irfah G. :
The Universal History of Numbers II
Penguin, 2005
Ore O. :
Number Theory and Its History
Dover, 1948
Temet Nosce 