The following article is heavily influenced by Paul Lockhart’s brilliant article, ‘A mathematician’s lament’. I only hope to add my experiences as a Pakistani student to back his stance in the debate over Mathematics Education.
Throughout my life I have hated mathematics with a passion. I hated its rules and notations. I hated the fact that I had absolutely no say in whatever was going on in the class. I just had to sit there and listen to my math teacher go on and on about formulas, notations needed to write these formulas, practice questions which would help us memorize these formulas and eventually “practical problems” which were supposed to exhibit the relevance of these formulas in everyday life although even the eight year-old me could tell that these were merely the same practice questions loosely disguised in the most unlikely of social situations known to man. And frankly, I didn’t care. I didn’t care where x was, or how much older Mary was than her brother Mark or when train A would reach London. As far as I was concerned math was an obsolete science to which I didn’t want to contribute to and which, for the most part, didn’t really want me to contribute to it anyway.
Therefore it comes as a surprise to many people that I am currently a Computer Science major focusing on theoretical computer science, which is basically a branch of mathematics. I, who had once famously given a speech to my seventh-grade math class about the pointlessness of mathematics, am now the one trying to explain to other people the beauty of Erdos’ brilliant proofs. And it all started with the following beautiful proof of the infinity of prime numbers:
For any finite set {p1,p2…pr} of primes consider the number n= p1..p2..p3…pr +1. This n has a prime divisor p but this is not one of the {p1,p2…pr}, otherwise p would be a divisor of n and the product p1..p2..p3…pr , and thus also of the difference n-( p1..p2..p3…pr) =1, which is impossible. So a finite set {p1,p2…pr} cannot be the collection of all prime numbers.
I first heard of this proof in the first lecture of a discrete mathematics course I took during my sophomore year at university. The instructor didn’t even write the proof down, with all its messy set notation. He just told us about the idea of putting the prime numbers together in a group and showed us what goes wrong if we assume the group to be finite. At first I thought this was one of those introductory shenanigans professors deploy in the first class to get students interested. How could something so simple be counted as math? Where were the fancy symbols and the list of variables with their definitions? Where was the list of steps used to reach the conclusion? Where were the ten similar questions I needed to solve at home for practice? This was simply a clever idea used to solve a problem. Surely, this couldn’t be math! But, as I have learnt in the past year, this is basically what math is: a set of simple ideas used to solve problems. Sometimes the problems can be simplified to older problems for which people have already come up with solutions. Sometimes ideas which have been used to solve a certain problem can be used to solve an unrelated problem. But the simplicity of the process remains intact. It is the ‘idea’ which is at the heart of all mathematics, and to come up with ideas you just need creativity (and maybe a pencil and a notebook).
If a course can change the path of a person’s life, then this discrete math course changed mine. In the course of nine weeks, I was introduced to the kind of math I hadn’t even known existed. For the first time in my life I didn’t feel like a robot while doing math. I actually had to think about the problems and figure out strategies for solving them. While I was introduced to techniques like induction and graph theory, for the most part my assignments and exams required me to come up with my own strategies based on these techniques and my own logical arguments and common sense. Math was like an elaborate game and finally I felt like it actually wanted me to take part.
So, this brings us to the central question: why did I, and countless other students, hate elementary and high school math? What needs to be done to make mathematics more interesting to students? Although I do not have any experience teaching mathematics, I do remember the reasons why I hated it so much and know exactly what eventually made me realize that I wanted to study a branch of mathematics as my major. For the sake of this article, I am going to ignore factors which affect all subjects alike and focus on why math has become such a hated subject.
Looking back at my years of struggling with high school math the first word that comes to mind is boredom. And this was not caused by a lack of interest in school because I was generally a very enthusiastic kid. I loved studying languages, history, and science. It was just math that I dreaded. And looking back at the way math is taught it comes as no surprise. While all other subjects are taught as an amalgamation of the history, foundations, rules and applications of the subject, math is mainly limited to the rules of the subject. Take a typical sixth grade science class. I remember learning about the effect of different factors on the rate of evaporation by placing different shaped beakers filled with water all over the school campus. What followed was a memorable class in which we all had mock “evaporation races” as we timed the beakers to see which one would lose its water first.It was only once we had made our own conclusions about which factors affected evaporation, that our teacher explained Brownian motion to us. She also mentioned factors such as surface area and wind-speed, which most of us had been able to conclude for ourselves based on the observations we had made.
Now compare this to a typical sixth grade math class. Looking back, sixth grade was when some of the most wonderful mathematical concepts were introduced to us. It was in the sixth grade that we first encountered the idea of a variable and started to really analyze shapes. Statistics was introduced, and we started manipulating probabilities to get results which even now give me the feeling of being able to predict the future. But in the midst of all these amazing ideas, this is how a typical math class would go:
Teacher: An isosceles triangle is a triangle which has two sides of equal length. Okay?
Students: YES!
Teacher: So what is an isosceles triangle?
Students: A TRIANGLE WHICH HAS TWO SIDES OF EQUAL LENGTH !
And you can bet one of the questions on the progress test would be: “What is an isosceles triangle?”. In such a situation who would be interested in math? And these are not just two extreme examples I have mentioned to prove my point. Science that year continued to keep us hooked: we grew plants in inky water, caught insects in jars, experimented with mirrors and discovered the material we were supposed to learn, while in math we moved on to triangles which had no sides of equal length (I honestly don’t remember what they were called, though I think it begins with an s) and other lexical atrocities.
You may argue that science is an extreme example and that math just doesn’t have the exciting material needed to keep students hooked. While science teachers can use models, take their students outside or perform simple experiments to demonstrate their material, math teachers have nothing to interest a group of thirty kids. Not only do I disagree with this, I actually claim that it is the other way round and that it is the math teachers that have it good. While science teachers need extensive (and often non-available) funding to buy lab equipment and take their students out on field trips, all a math teacher needs are thirty pencils and notebooks. And how does he keep them interested? Well, he actually asks them to do some math. Do you remember the puzzle we probably all tried as kids in which we had to draw a house without lifting our pencils. That is just a simple example of a Eulerian path. And those complicated strategies for winning card games that our older siblings tried to explain to us were mostly simple applications of probability. The tower of rings of increasingly small diameters which we had to shift to another peg is the most common example given for recursive algorithms. The list of interesting mathematical problems which we solved willingly as kids is endless. Nim, Hex, magic tricks, and riddles in which we had to find loopholes in logical arguments are all example of the math we enjoyed as children and it is these problems which should be bought to the classroom to make math classes more interesting.
Another issue which I find with the way mathematics is taught, which is closely related to the first, is the extreme and almost exclusive emphasis on the utterly mundane aspects of mathematics. Take the isosceles triangle example above. Would it really have mattered if we had called the triangles, “triangles with two equal sides”? Maybe shortened to TWTES (pronounced tevtes). What’s important are the properties of these triangles. Instead of asking a child to spend time trying to memorize the pronunciation and spelling of this weird word, she should be asked to think about how they are made, and how the angles inside this triangle are related to each other. I am pretty sure if a child made a dozen different TWTES’ she would figure out most of their properties for herself and she would actually enjoy the mental excursion of discovering these properties instead of hastily be given a list of them in the last fifteen minutes of class.
Admittedly, there are some terms and jargon that a student of mathematics must learn in order for the classes to be held smoothly and for the students to eventually take part in the wider mathematical discourse. But no other subject puts even half of the emphasis that math places on its lexicon. Take the example of chemistry. If a subject has the right to focus on terminology it is chemistry, with it’s multitude of symbols, chemical formulas and specific reactions. But not once do I remember a chemistry teacher reciting the names of the elements along with their atomic symbols. Instead, we focused on the elements and their reactions and any time we needed help deciphering a symbol we could simply look it up on the huge periodic table taped to the classroom wall. Maybe that is what mathematics needs: a periodic table of shapes and functions which would be taped to the wall of every classroom. Then, children all over the world could forget about mathematical terminology and actually do some math.
And by ‘doing math’ I don’t mean the mindless repetition, or solving exercise problems at the end of every chapter. As a result of school mathematics, most people end up believing math is the application of known rules to problems that we know the rules can solve. That is the job of an accountant or a cashier or an insurance planner. A mathematicians job is much simpler. He must come up with the rules that other people are to use. When faced with a problem, he is not told that it can be solved using the second trigonometric identity; that is what he must figure out. And while this is harder than simply applying a set of rules, the result of coming up with a solution is infinitely more rewarding. You can compare the two as the difference between the joy a child feels in having an adult place him on a bike and push him along, and the joy he feels when he races through the park himself. It is hard to teach him how to ride and it might take him ages to learn but all parents understand that the end result is worth it. Math teachers should definitely do the same with their students.
And if difficulty was such a major barrier, why doesn’t it stop teachers of other subjects from trying to get their students to appreciate the beauty of their fields? By the end of high school most of us have faced the toughest aspects of most of the other subjects. We have read Iqbal’s poetry and critiqued it with our peers. We have a deep understanding of how the major systems of the body work. We have built electrical devices and have made original pieces of art in a range of different mediums. Then, why is it that most of us only experience the joy of coming up with a true mathematical proof well into our undergraduate programs? Surely there is something wrong going on here.
When it comes to maths, Bart Simpson had it right: ‘I never thought it was possible, but it both sucks and blows.’
Take heart, Mariyam … Greater minds have felt the ‘suck’ of mathematics, until it blew them away. Einstein famously didn’t care much for math until he got stuck trying to generalize his Special Theory and was rescued by his buddy Marcel Grossmann & tensor calculus. So, ‘boredom’ probably is in the DNA of mathematics …
But, what do I know. I’m a math atheist! :)
People (like Einstein) say various things at various points in their lives, under very many different circumstances. Einstein also said things positive about Mathematics…. “poetry of logical ideas” , [mathematical equation] “stands for ever” etc. But anyway, that is not what I want to say. What I want to highlight is ….
That one of the things that drove Einstein towards the theory of relativity, was his conviction about the correctness of Maxwell’s equations. Maxwell (in the 1800s) joined Electricity and Magnetism under one banner. How did Maxwell do it? He stared at the bunch of equations explaining electro-magnetic phenomena. He saw that things are not tallying — and he wondered how would one make the equations consistent. It was a mathematical exercise … and in the end it explained wonderfully well, what nature is doing.
So here is another of those Feynman quotes:
” From a long view of the history of mankind — seen from, say, ten thousand years from now, there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade. ” — Feynman Lectures on Physics, Vol 2.
A good read. Though I feel stupid now, because I really enjoyed math in secondary school! Algebra, probability, kinematics.. it was all so interesting to me. The fun was in how math provided ‘tools’ for practical situations, this would be kinematics. And algebra, algebra was amazing – put down a scenario in symbols and variables, manipulate the equation to get what you want. It was so fantastic and amazing for me. And even more stupidly enough, its exactly the aspect of mathematical proofs that I abhor now in university… its so tedious and abstract.
Hmm, or maybe what I was really liking back then was actually physics…
Lockhart’s article specifically refers to people who enjoyed math the way it is currently taught. According to him (and I agree), the subject itself is so beautiful that we can even find beauty in the mangled mess of mathematics that is taught to us. And as you said, it gets tedious after a while and there is no joy in it. I hope you get a chance to take a nicely taught math course focusing on proofs and strategy so your childhood love of mathematics is rekindled.
I think there are usually more factors involved than simply liking or not liking the subject. Maybe someone reaally does enjoy the subject but some external factors are leading to their success.
My child hood experiences with Math and Science were ditto.And now, I am a software Engineer how studies math and physics as hobby subjects.
It has been an amazing turn around for me. I can attribute this to two factors. Incessent rote learning during my school years in Pakistan turned me off from Science and Math.
My western undergraduate study experiences allowed me to reason/question/inquire and not just memorize for sake of exams.
As HEC pushes forward with it’s agenda of Higher education reform it must remember that education is a path to learning/enquiry and not an exercise in achieving a degree.
True, you can’t have fun with math like with science, but it was always my favorite subject. I loved trying to mentally figure out Mary and Mark’s age, as well as what time the train would arrive at London.
I guess it just differs from person to person :) Good article though!
A fun read, i took Discrete Math too, did the same proof and i agree with you totally, high school math does have a pronounced saddening effect on students. What is more strange is how little even undergraduates know about mathematics and mathematicians, by which i mean, what is math all about and what is it that mathematicians do. Nice effort in covering that up :D
Enjoyed…
Sure our math teachers in Pakistan talked to chalkboard plus they had very little exposure to applied maths. Besides, the more complex they could make maths, the better they felt about themselves..
During graduate studies I enrolled in engineering maths class and one student asked the professor units of the final answer of a triple integral to which our professor responded that he was a mathematician and not an engineer and cared less for the units… Such humble ignorance is universal.
On a different note, I was helping my son with high school algebra and was impressed by the quality of contents however as widely as this book is claimed to be in use in high schools, it is not reflected in the math skills of engineering students I come across regularly (with few exceptions.)
I think, math skills are a blessing and regardless of good or poor teaching, these qualities finally emerge.
I can assure you (having recently graduated from high school 3 years ago) that as beautiful and complex as the algebra your son is studying is to you, he and most of his classmates will have little appreciation for it. It will be merely formulas they have learnt and will have forgotten by the time they enter university.
As you say math skills are important. It would be nice if the education system encouraged and honed mathematical skill and didn’t just expect it to “emerge” in a random few.
Interesting that a very similar argument (maths is fun) was made in a series of NYT articles by Steven Strogatz (with several fun inducing examples!)
http://opinionator.blogs.nytimes.com/2010/01/31/from-fish-to-infinity/
upto
http://opinionator.blogs.nytimes.com/2010/03/14/square-dancing/
(You can link to the other articles in the series from the last link).
Good job.
problem is that, we dont have experienced teachers, you cannot expect from a high school or middle school teacher to explain you the ideas given by the great mathematicians although those are simple sometimes. as mathematics models the physical phenomenon in its own language, means x,y and other notations, so it looks boring. when i was student i performed very well in mathematics but understood nothing, but now when im in my thirties and have master degree in engineering, i can say mathematics is a divine knowledge, its absolutely marvelous and i love mathematics. this world is nothing but mathematics.
I agree with you that it hard to teach mathematics as the article says it should be taught. But why is it that we don’t expect the same level of expertise from our math teachers as we do from those who teach other subjects. Parents and educational administrators (principals etc) wouldn’t accept an Urdu teacher who couldn’t read Iqbals poetry or an English teacher who couldn’t recognize flaws in sentence structure. But we are okay with math instructors who can’t understand fundamental mathematical works.
And no matter how hard teaching math is, pretending the way it is being taught is alright can never be the best solution to the problem
That was an interesting read.
What is Math? Or more importantly, its purpose?
“Math is but a tool to do Physics”, Feynman, a Physics Nobel Laureate.
This quote seems to by and large summarize the popular view people hold of Math.
Math itself can be divided (albeit wrongly so) into Pure and Applied. Applied Math may serve the purpose of providing tools, to say Science etc. Basically, Science etc. take a result from Pure Math. They may put a constraint on it (this is where the so-called real world/universe comes in) and apply it to solve the problem at hand. Lemme give an example:
In Physics, relationship between two quantities, x and y, may be seen this way: x changes as y varies. So here comes differentiation (or differences if its discrete?).
x/y is constant (or conserved) and hence integration takes over.
Now both these concepts are from Calculus. Differentiation and integration are just say operators there with many distinct properties. However, the Math of it doesn’t take into consideration any physical quantity existing in the world out there. Yeah this is the ‘true’ Math, the Pure Math.
The problem with Math education (worldwide, not just in Pakistan) is that its viewed and taught as just a problem solving tool for Science, Engineering etc. How many people out there don’t know that 2 + 2 = 4? Hardly anyone? Now how many can actually prove that this is indeed the case?
And this is the problem with Math education. Math is not taught in schools (grade 12th, say) at all. Students coming to college may be exposed to the real thing. However by then, its a bit too late for many. Most of them can’t come to terms with the fact that there is any such thing as a proof at all. That truths are NOT discovered, rather they are INVENTED or CREATED!
Now this is probably what makes Math genuinely hard. There is NO sensory input associated with it to appreciate it. This was the view of one of my instructors in college. Ok given that there is no sensory input, but do you know that there is supposed to be none? That Math has absolutely NOTHING to do with the ‘real world’?
So why then the real world ‘feels’ to be so Mathematical? That’s because the universe is nothing more than Math restricted to a certain set which is “isomorphic to the real world”. Yes sir, that’s all there is to the real world.
Now how can a teacher, if teaching real Math and not just computations/numerics, interest her pupils into Math? That is the question? Will it be by applications of Math to the world? But then it wouldn’t be Math at all! How then? I guess I don’t know.
“Don’t confuse Math with Science. Science tries to explain/understand the universe. Restricting Math to the universe would be grossly limiting its scope”.
“Where do you draw the line between Pure and Applied Math? It gets to Applied where the rigor disappears, and so all the wrong things begin”.
Both of these quotes are by Fields Medalists!
I don’t disagree with what you wrote in general, I think Feynman might have been taken out of context there. Maybe you already know all of this, but this is just for the benefit of others.
I will not take the time to confirm, but I guess Feynman was not talking about Mathematics, but what Mathematics should mean to a student of Physics. And that is completely different. For example, for a ‘businessman’ a computer may just be a ‘tool to store tables in Excel”. But that is not all what a computer is about.
Feynman also said that it is highly improbable that any one would now do something significant in Physics without strong grounding in Mathematics. Btw there is also an ‘integral’ known as Feynman Integral. For example if one browses the book: “Garden of Integrals”, one finds amongst a couple others: “Riemann Integral” , “Lebesgue Integral” , “Hanstock Kurzweil Integral” , “Feynman Integral”. The first one is the integration we learn in high school, and the one developed by Newton etc. The second one is getting very important nowadays, and much more powerful.
When a physicist like Feynman says that “mathematics is a tool” … it probably doesn’t mean the same when an engineer says that “mathematics is a tool”. When Physicists are learning ‘tools’, it could look pretty abstract pure mathematics to many others.
Results from Graph Theory may be applied to certain situations arising in Computer Networks. But Graph Theory is a subject in its own regard.
Wow, Nice piece of information,Sure yes yes very well said, our math teachers in Pakistan talked to chalkboard plus they had very little exposure to applied maths.During graduate studies I enrolled in engineering maths class and one student asked the professor units of the final answer of a triple integral to which our professor responded that he was a mathematician and not an engineer and cared less for the units… Such humble ignorance is universal. In my opinion, math skills are a blessing and regardless of good or poor teaching, these qualities finally emerge. nice piece of discussion and well maintained site, keep it up. nice one.
QUOTE: “By the end of high school most of us have faced the toughest aspects of most of the other subjects. We have read Iqbal’s poetry and critiqued it with our peers. We have a deep understanding of how the major systems of the body work. We have built electrical devices and have made original pieces of art in a range of different mediums. Then, why is it that most of us only experience the joy of coming up with a true mathematical proof well into our undergraduate programs?”
Do you think writing mindless “tashreehs” to pass the course is a worthy of being called a critique of Iqbal’s work! Now given that most students understand Urdu it’s possible some small but significant number of students actually think about it as a part of their course study but then you say our high school graduates have deep a understanding of body’s working and also that they build electrical devices! Now keeping exceptions aside, in general, most Pakistani have at most a cursory understanding of the subject matter of their courses, let alone have an in-depth knowledge or understanding of it. An average FSc student, whose coursework includes stuff that is usually only taught at Bachelors or some university level, will read a sentence in his book and if the tiny amount of English language that his matriculation has given him allows him to understand that sentence, well and good, otherwise the general trend would be to “just remember it the way its written”. Their way of doing experiments in the lab: Put this wire in this slot, this one in that slot and there you go, its works! Practical marks in exams are very often allotted on the basis of personal relations with the examiner or based on parents’ little “discussion” with the examiner. In a system where students are never given a chance to appreciate the beauty of even the applied sciences, which are easier to comprehend given the right direction, there is definitely no chance for them to appreciate the abstract art of pure mathematics. Hell, I have met university math graduates (not famous unis, but still, unis with charters in Pakistani Law!) oblivious to the greatness of their subjects! The failure does not lie in not being able to instill an interest in math, but, in failing to provoke curiosity and develop scientific thought overall. Otherwise, we would still have a few names to take proudly in other fields at least! (lets not make the SINGLE obvious name an excuse :D ).
Let me carry this ‘lament’ forward to the university level. Unfortunately, I don’t remember where was the link, but there was a link to past papers in Punjab and Sargodha Univ on STEP. I saw 2-3 math papers there (MSC, BSC)… definitely not all. But still, the questions that were being asked in those papers…. from the top of my head i recall questions like:
1) “Prove that [definition of a Normal Subgroup], satisfies the properties for being a subgroup” …. well that would encourage rote learning! There are entire sections (or even chapters) devoted to “Normal Subgroups” in the usual textbooks on Algebra. The proof that is being asked for, should come as one of the most major and first theorems in those chapters. When we ask to repeat the proof of something that big; that well known … we are only asking for memory.
I am not cherry picking. Most questions looked like that to me.
2) The paper on “Complex Analysis and Differential Geometry” … I think a high-school student should be able to attempt those. The only thing he needs is a bit of vocabulary and definitions. A good proportion of questions were like:
“Define meromorphic functions” —- again, a very important (and well known) concept in Complex Analysis. It’s like asking: “did you at least listen in the class?”
Another one was something like: “Distinguish between Curvature and Torsion” —- again basically just asking for definitions. And it went on.
I have read some mathematicians say, that there is a difference between an “exercise” and a “problem”. An exercise is a direct application of what you know. The methodology is obvious as soon as you read the question. You just need to do it. A ‘problem’ on the other hand, looks confusing and it is not immediately obvious how to solve it. You have to go ahead, think and try. The usual math student is more of an ‘exercise solver’ rather than a ‘problem solver’. And taken to extreme, the ‘problem solving’ questions can crush ones confidence (the median score in some Math Olympiads is 0 ).
Generally, we do deal with exercises and not problems. But I thought that even ‘exercises’ should be a bit more thought provoking than what i saw. Ideally, maybe around 10% of the questions should be ‘problems’.
Finally got around to reading Mariyam’s article (as well as Lockhart’s). Lockhart in his 25 pages has expressed my feelings better than I could have — probably should recommend it to people in the future rather than arguing about it with them. A few points that came to mind:
—————————————————
a) The same analogy as one of Lockhart’s (the Music society), is there in the preface of the book “Visual Complex Analysis” by Needham. The first page of the preface is at: http://books.google.com.sg/books?id=ogz5FjmiqlQC&pg=PR7
He doesn’t dwell on this analogy for too long (it’s a textbook). But the preface is still nice, and the first few pages of chapter 1 of this book (and many other books on Complex Analysis) is worth a read.
————————————————————–
b) “Numbers and Geometry” by Stillwell, is a fairly nice book. It has a historical tinge to it. It is about the subjects of Numbers, Geometry, and their intimate connection with each other, from a much better perspective than that in high school. Some more advanced theorems are also there. As prerequisites one doesn’t need more than high school math; I don’t remember calculus being used. The level of ‘math maturity’ required is probably a bit more than that in high school…. but it is very readable.
—————————————————
c) As one keeps on reading mathematics, things do invariably get a bit tough. Getting through to the *idea* might not be as easy, or even possible to do within a few pages of text. But probably one should heed the advice of George F Simmons in this regard: “A mathematical proof should be read again and again, until it becomes a single idea”. At least one should try to achieve this ideal as much as possible.
Btw, Sometimes we need to read ahead a bit and come back, in order to manage to reduce a given proof to a coherent whole. It’s not always possible on the given spot to do so, because we sometimes simply don’t know the bigger picture.
————————————————–
d) One could similarly argue about Science education too. Although these subjects generate greater excitement amongst the young students, but the perspective given in ‘science’ classes, I believe can be made better (closer to what the scientists generally have) and more exciting.
Plugging in F = ma to solve inclined-planes problems (or other engineering problems), doesn’t tickle one’s spine with the grandeur and awesomeness of Mother Nature, which I am confident is a feeling shared by most good/famous scientists.
For example: http://www.hal.rcast.u-tokyo.ac.jp/~drebes/value.html .. which is a public address in 1955 given at the meeting of National Academy of Sciences. It gives a peek on the way a scientist can feel towards Nature. This talk is one of my favorites.
But I agree, Math probably does suffer a bit more than the sciences in early years of education. Science can still generate more excitement for those students, as its results are more readily available to our senses.
I do not think the disinterest or passion for maths at school has much to do with teachers, I mean normal teachers. It is more a matter of interest combined with some gray cells. Some persons are just “spaetzuender” (late igniters)
- Show quoted text -
Hi
Challenge Websites
Modern Scientist, Philosopher, Scholar, College & University Teacher, Math Dr & Mathematicians, do not know ” The Science & Technology of Zero”, Discovery of Atom & Difference between Number & Numerals. ETC
See proof in my site.
http:0angles.20m.com/
http://0science.20m.com/
Numerical Science Researcher
Munawar Butt