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A non-violent anarchist Mathematics teacher and Christian who has views about everything under the sun.

  • The Tyranny of Numbers (And How to Break Free)

    Launching Pad

    José Vilson is a middle school mathematics educator based in New York City. (Source: X)

    I launched this blog on Friday, 1 March 2024. I knew that I wanted to have a post specifically for International Women’s Day on Friday, 8 March 2024, which became Primordial Prime Ordinals. So I had planned something else as my third post for this blog. However, after the first post, The Eye of the Beholder, I received many responses indicating that people were somewhat apprehensive about reading a blog on mathematics. A few confessed to having some kind of phobia about anything related to the subject. Others said that they just could not understand anything related to it. I am thankful to those who, despite their apprehensions or expected non-comprehension, went ahead and read the posts.

    One of the purposes for this blog is to enable readers to appreciate – dare I say, grow to love? – mathematics. But if so many wrote back expressing apprehensions, I’m sure there were many more who remained silent about them. Having enjoyed and loved mathematics for as long as I can remember, I find it impossible to place myself in the shoes of someone who is wary of, or even hates, the subject.

    However, I have encountered this before, most often from students who have been forced to take the subject even though it was an elective. For example, in one school I worked at, students had to choose between Art, Environmental Management (EVM) and Mathematics. Now Art is a skill based subject and caters to a small niche of students. And EVM, unfortunately, is not known as being a rigorous subject. Hence, many students who were not skilled enough to take Art and who were reluctant to take a not-so-rigorous subject like EVM ended up taking Mathematics! Given that there are constraints that schools face, such unlikely options are inevitable. Nevertheless, the bottom line remains – many, if not most, of the students just did not want to be in my class, primarily because they have developed an indifference to or an aversion for or hatred of the subject in their earlier years at school.

    So I have often wondered, “Why is it that so many people actively dislike this subject that I love so much?” Granted that it is foolish to expect everyone to love what I love, is it too much to hope that there wouldn’t be as much indifference, aversion, and hatred going around for it?

    An Important Insight

    The first thing that we need to grasp before we can address the indifference, aversion, and hatred is that mathematics is not really about numbers. If this thought seems strange to you and you have done at least a little mathematics beyond 8th grade, then I would request you to pause here and write to or call your mathematics teacher and ask her/him why she/he stiffed you when it came to teaching you. I’m serious. This post will still be on your screen when you return after writing or calling!

    William Paul Thurston was an American mathematician who specialized in low-dimensional topology. (Source: Comap)

    So why do I say that, if you do not know that mathematics is not really about numbers, you should confront your mathematics teacher? Well, first, as a teacher, I would have hoped your teacher knew more than you did! More not just about the procedural aspects of mathematics, about which I have no doubts she/he far outstripped your knowledge. What I’m more concerned about is communicating to the student the idea that, important as what we teach them may be, what they learn in school is not just barely scratching the surface of the subject but also, and more importantly, what they learn in school is largely misrepresentative of it! If you are surprised, prepare for more.

    In his excellent online course Introduction to Mathematical Thinking, Keith Devlin asserts that mathematics is the study of patterns. The patterns could present themselves as patterns of chance or change or quantity or relationships or shape, etc. In other words, mathematics is concerned about the patterns that emerge when we focus on any aspect of the real, or even imagined, world. It is so much bigger than what we are introduced to in school.

    Impediments to Learning

    In school, what we are introduced to are procedures. We learn how to add, subtract, multiply or divide. We learn how to find the Least Common Multiple (LCM) or the Greatest Common Divisor (GCD). We are taught how to perform long division. In other words, we are given ‘recipes’ to carefully follow lest we flounder at the ‘cooked’ up problems we are given to solve.

    Now, it is true that some students may be given some patterns to observe. For example, they may be told the rules for divisibility by 2, 3, 4, etc. But do we have the patience and do we allow the space for a journey of mathematical discovery that would lead the students to have their own eureka moments?

    John Wesley Young was an American mathematician who produced the axioms of projective geometry. (Source: Comap)

    Unfortunately, we have crammed each year of mathematical ‘learning’ with endless ‘recipes’ and convince ourselves that we are giving the students a robust education. Due to this we do not have the ‘luxury’ of allowing the students to embark on a journey of discovery. More damaging, however, is the conclusion that students, and unfortunately many teachers, reach that mathematics is about mastering these ‘recipes’.

    Now, there was a time when mathematics was about mastering these ‘recipes’. But that is simply because mathematics had not advanced as much then as it has now. In the 17th century the pinnacle of mathematics was Newton and Leibniz‘s development of calculus. However, even those pioneers did not think their discoveries could be put to use in the study of economics, epidemics, and politics, as they are today. Mathematics as a subject has developed as much as any other field in the intervening three centuries. However, mathematics that we teach in school remains stuck in a time warp while other areas of knowledge at least include newer ideas.

    Other areas of study also include the history of their subject. For example, chemistry students are taught the original Periodic Table developed by Dmitri Mendeleev based on atomic mass and how it differs from the modern Periodic Table based on atomic number. Similarly, physics students are taught Newton’s Law of Gravitation and are at least given an introduction to Einstein’s Theory of Gravitation with some discussion on the superiority of the latter, yet sufficiency of the former to land humans on the moon.

    Despite this, mathematics is presented as ahistorical, with one formula after another, one ‘recipe’ after another thrown at the students. However, I believe that students have – and are right to have – a suspicion that an ahistorical body of knowledge is actually inhuman! Please read the previous sentence again. If all we have in mathematics are final results rather than at least some indication that there were human struggles behind the development of those results, students will not be motivated to learn because they know that all genuinely human knowledge is developed in and through time and space.

    But somehow we have convinced ourselves that mastering the ‘recipes’ implies mastering the subject. And we teachers tell our students that. But not everyone is interested in numerical ‘recipes’, just as not everyone is actually interested in cooking!

    In addition, we mathematics teachers have developed a strange version of the idea that ‘practice makes perfect’. However, it is only in the classroom that we are presented with 25 identical problems with only the numbers changed! Indeed, if the law of diminishing returns has anything to tell us it is that we can well overdo this ‘practice’ and make the students repeat things mindlessly just to say they have finished an assignment. In that case, with the students having the attitude of an automaton, it is quite likely that very little learning is actually happening.

    It is my hypothesis that, because we tell students that learning mathematics involves the mastery of ‘recipes’, those students who find it easy to master these ‘recipes’ will float along under the illusion that they excel at mathematics, an illusion often shattered when they enter either Grade 11 or college, when the mathematics involved is more abstract and not based on ‘recipes’. And those students who do not care about numerical ‘recipes’ will disengage from the subject and become at best indifferent to it and at worst develop a hatred for it.

    The Encroachment of Numbers

    This is compounded by the fact that the ‘bread and butter’ of these ‘recipes’, that is, numbers, seem to be exceptionally aggressive entities that transcend the well defined silos our education systems hold sacrosanct. What do I mean?

    Well, you have probably recited the alphabet and realized that ‘m’ is the 13th letter of the English alphabet. But have you ever heard anyone say that ’13’ is the mth counting number? Even the superscripted ‘th’ in the previous sentence refers back to ordinal numbers! What gives these ‘numbers’ the right to intrude where they don’t belong?

    Richard Dawkins is a British evolutionary biologist. (Source: Comap)

    But that’s not even the worst. Every student has her performance in every subject indicated by a number. Of course, some schools or grading systems may try the deceptive path of assigning letter grades. But if you push them and ask, “Why did I get a B and not an A?” your teacher will probably come back with, “Well, you scored 87 and the grade boundary was 88” or something of the sort. So even though some of us pretend that we aren’t using numbers to evaluate the student’s performance, lurking behind our ‘recipe’ or algorithm for assigning even letter grades are those very numbers we pretended we weren’t using.

    In other words, we have (ab)used numbers and enslaved them to do a task that makes assessment convenient for us rather than meaningful for the student. And as with all (ab)use, there is enough trauma to go around.

    You see, it is my hypothesis that many students slowly develop a fear of and hatred for numbers precisely because, on the one hand, the algorithmic approach to mathematics leaves nothing to the imagination, thereby asking inherently inquisitive and curious young children to shut down those faculties, and because, on the other hand, these entities of cold rationalism, i.e. numbers, are nonetheless used to measure everything they do. It is as though a child was told that the playground bully would be writing up the report about every aspect of their lives!

    This ‘tyranny of numbers’ is ubiquitous and is, in my view, what causes many students to develop an aversion toward numbers, just as any people group under the rule of a tyrant would develop a hatred for the tyrant.

    Does that mean, then, that we are stuck with mathematics being really enjoyable for a few students while being irrelevant to or despised by most others? Can we not find a better way to teach mathematics, one that will enable more young students to appreciate it, rather than live in a world in which high school mathematics teachers constantly bemoan the fact that our classes are full of students who just don’t want to be there? I believe there is a better way. Of course, I do! Otherwise, I would not be writing this post. So what could a better way be? I will delineate this in five parts.

    Rejecting the Tyranny of Procedures

    Georg Cantor was a Russian mathematician who made pivotal contributions to set theory. (Source: Comap)

    First, we need to move our teaching away from the procedural. Does it really matter how a student adds or subtracts, multiplies or divides? I mean, suppose she was told to add 14 and 29. Does it matter how she arrives at the answer 43? She could stack them one on top of the other and add using ‘carried’ numbers. Or she could count up from either of the numbers. Or she could split them into units and tens, add them and then combine. But does it really matter how the addition was done?

    While I think that the above five pronged approach will lead to a situation in which more students will enjoy mathematics and less students will hate it or be apprehensive of it, I am not hopeful that any of them will be adopted. From schools to assessment boards, from individuals to nations, from teachers to students, we have become accustomed to what we are currently doing. And we can barely see any alternative to the tried and true. However, I put this five pronged approach forward with the slimmest of hope that perhaps somewhere down the line we will humanize mathematics and break free from the current cycle of tyranny.

    Quite obviously some methods are more efficient than others. But can we leave the discovery of the pros and cons to the students? When one student realizes she is slower than her classmates, then, if she is interested, she will try to learn a new, more efficient method. Note the words ‘if she is interested’. It could be that the student is content with a much slower method. Does that mean she does not know how to add? Absolutely not! She is just content with an inefficient method.

    But trust me, at a later date, when she realizes that the inefficient method is holding her back, she may want to learn a more efficient method. But what we are doing is linking the understanding of the operation, which is the crucial element, to the speed with which the student is able to execute the operation, which is really not all that important.

    Don’t misunderstand me. Procedures are important. And efficiency is desirable. However, when we elevate the procedural above the understanding of how the various operations work, we make our students begin to believe that memorizing the procedures is crucial for their mathematical understanding. Then, when they find some procedure difficult they erroneously conclude that they do not understand the underlying concept that the procedure was designed to serve.

    Avoiding the Tyranny of Time

    Second, we need to move away from timed assessments. I claim that speed is not important because it is we who have artificially made speed important by assessing students in contrived timed situations the like of which they will never face when they hold down jobs of their own. We have done this because it is convenient for us to have a closed form, tightly structured environment for assessing students, rather than because it is actually beneficial to the students. We may say that this allows for uniform testing conditions. But who said that uniform testing conditions are the holy grail of student assessments?

    Maryam Mirzakhani was an Iranian mathematician who made ground-breaking contributions to hyperbolic geometry. (Source: X)

    Suppose one student solves a problem in 5 minutes while another takes 15 minutes. Does the difference in time actually reflect a difference in understanding? Apart from a life or death situation, in which I am confident one’s ability to use the quadratic formula will never play a role, I cannot fathom why the first student is considered better. All we are doing is privileging speed for, if the exam were an hour long, the first student would solve 12 questions while the second would solve only 4. In fact, in a timed context, we are specifically sacrificing our ability to assess the student’s understanding of the subject involved because we are placing an additional constraint of time that has nothing to do with the subject. This is true of most subjects, of course, except perhaps a course in bomb diffusing! Hence, it would require complete overhaul of the way we conduct assessments in school. The tyranny of time is rendered all the more acute if all we are testing is procedural proficiency, for then we are assessing the speed of execution of algorithms rather than conceptual understanding.

    Now don’t get me wrong. This is not a case of sour grapes for I am an excellent test taker. I am able to function really well in high stress time situations. However, I really do not think timed assessments should have any role to play in assessing students. Just because the number of students is inordinately large does not justify making assessments easier for the teacher or assessing body. If we are truly concerned for assessments to reflect student learning then we need to find ways of assessing them that does not artificially introduce a constraint that has absolutely nothing to do with what the student is learning.

    Project based learning holds a lot of promise. However, that would actually render large assessment bodies like the CISCE, CBSE, CAIE, and IB obsolete since they would not be able to administer such assessments. It would increase the burden for fair assessments on local schools, which could prove to be a problem. Also, given the prestige that association with some of these bodies gives schools, thereby also allowing them to artificially inflate fees, this will prove to be one of the biggest roadblocks should what I am saying hold any water.

    Shunning the Tyranny of Quantity – Aspect 1

    Third, we need to streamline our curriculum. Since it is relatively easy for motivated students to learn how to replicate algorithms and since we have convinced ourselves that memorizing algorithms is what mathematics is about, we are unable to think of reasons for which another algorithm should not be added to the curriculum.

    However, many things we teach are just different ways of combining previously memorized algorithms rather than being cases of new knowledge for the students. For example, once a student knows that the angles around a point add up to 360° and understands how to apply the sine and cosine rules to a triangle, does she need to be burdened with learning how to ‘apply’ this to a situation with bearings? Unless she is planning a career in aviation or maritime navigation it is unlikely she will ever come across the idea of a bearing. Given that there are only about 350,000 commercial airline pilots in the world, this would seem to be quite a niche requirement. Yet, we ask most of our students to learn how to use bearings.

    Another example is the early inclusion of statistics. In many schools we introduce students to ideas of mean and median from Grade 1, perhaps earlier. However, mathematics teachers know that, at that age, we cannot really explain the nuanced differences between the mean and the median nor when one would be more appropriate to use than the other. Yet, all we are doing is bringing in another context within which we ask the students to repeat previously memorized algorithms – addition and division in the case of the mean and ordering in case of the median. But these algorithms are almost trivial to learn. Why can we not introduce the ideas of mean and median at a stage when we can also explain at least some of the nuances?

    What we do when we artificially bloat our curriculum with inconsequential, unimportant, or niche concepts is communicate the idea that the subject is beyond the grasp of most students. This is exacerbated with our textbooks, which enable our students to give professional weightlifters a run for their money! When we have to complete more than 20 chapters in a year, we only create a high stress atmosphere for the teacher – stress that then permeates into the subconsciousness of the students. The students have no time to stop and appreciate the mathematics they have learned because we are dragging them immediately to the next item on our ever burgeoning checklist.

    Refusing the Tyranny of Calculators

    Fourth, we should move away from the use of calculators. If we are honest, any motivated student will be able to learn how to use any calculator in a matter of days or, at worst, weeks. Just look at how adept at using personal computers and smartphones the young people are. They can master a calculator, if needed, quite quickly.

    However, once you expect students to have a calculator, you need to justify this requirement. This is evident most clearly in the area of probability and statistics. For instance, if a student understands what a discrete probability distribution is, for example, the binomial distribution with small numbers, how much additional learning is needed before she  can apply what she learnt to the context of the Poisson distribution or the geometric distribution or the binomial distribution itself but with larger numbers? The inclusion of other distributions only changes which keys on the calculator the student has to press. Is that really how pathetic our idea of learning has become?

    Another area where this shows up is the use of exponential and logarithmic functions in questions that require calculation of derivatives or integrals. For example, it is well known that a closed form anti-derivative of ex2 does not exist. What then is gained by a student being asked to find the area under this curve when it requires the use of a calculator? What is lost if instead the question used the function ex for which a clear anti-derivative exists and is known? Conceptually, nothing. But the latter can be done without a calculator while the former requires it. With no new conceptual understanding being granted to the student, the inclusion of the former can only be to justify the requirement of a calculator.

    It is my hypothesis that there is a nexus between assessment boards and calculator manufacturers that ensures the unnecessary requirement of calculators from too early an age. And I am confident that the use of calculators actually hinders rather than furthers student understanding.

    Shunning the Tyranny of Quantity – Aspect 2

    Fifth, we need to move away from counting how many problems we solve in a given class. At one institute, my boss expected me to complete 25 questions in a 90 minute class. Could I do it? Sure! I knew all the problems and the solutions. I could certainly have written all the solutions on the board. And the students could have been faithful simian stenographers, mindlessly copying the scribbles on the board to their notebooks!

    Edsger Dijkstra was a Dutch mathematician and computer scientist. (Source: X)

    But this does not mean that any learning has happened. And this was a bone of contention with my former boss all through my tenure at that institute.

    To the contrary, I remember one year when we unpacked a particularly gnarly probability problem over the course of many classes! With each suggestion the students made, we looked at the implications and whether it actually solved the problem. When we finally solved the problem, it was the students who had done it, not me. And though they were exhausted after the marathon, they were immensely satisfied. And I knew they had learned not just what the correct answer was but, more importantly, why the approaches we discarded didn’t work and why the one we finally settled for was more efficient than the discarded ones and other, more tedious, but viable approaches.

    Some may say I wasted precious time by devoting so much to one problem. However, if Keith Devlin is right that mathematics is the study of patterns, then those students learned fruitful and unfruitful patterns of thought during those classes. I still think it was worth the time investment. But probably more to the point those students themselves remembered this years later and told me that they not only learned how to think but developed perseverance.

    However, if we insist that quantity rules quality, we will never be able to ensure that any valuable and long-lasting learning happens. This ties back to the previous point about calculators. A calculator can help a student reach the answer with a few button presses in a much shorter time than if the student did not have a calculator. And a mathematically adept student will certainly know why she pressed those particular buttons. So nothing would seem to be lost by allowing a calculator and increasing the number of problems solved by the mathematically adept student.

    However, we do not cater only to mathematically adept students! There are students who struggle with mathematical concepts. To them a calculator is just a magic box that churns out answers. Such a student may memorize which buttons to press for certain kinds of problems but she would not be learning anything because she is recognising not the mathematical patterns, but the types of problems. She may be able to even keep up with a mathematically adept student because she has memorized the problem types and the button presses for each type. But she is learning nothing because the trusted genie inside this lamp gives her the answers on demand.

    In other words, increasing the number of problems solved based on the premise that calculators speed up the work hinders the learning precisely of those students for whom the subject is proving to be particularly difficult. This is a travesty!

    However, if we expect students to think through their approach, we will, as teachers, have the opportunity to address any misconceptions or misunderstandings that would otherwise have been hidden inside the wirings of the magic box!

    Breaking the Cycle

    In conclusion, I propose a five pronged approach to addressing the malaise most students experience at the thought of mathematics. First, we need to embark on the much needed process of stripping down the school mathematics curriculum. The aim should be to include only topics that introduce new mathematical insights rather than combinations of old ones, since mathematical concepts can be combined in endless ways. Rather than teach students specific combinations of concepts, we should teach them how concepts can be combined.

    Second, the focus should be on furthering mathematical thinking rather than memorization of procedures. Students should be asked and engaged to ask questions like, “Why does this procedure actually work?” and “How could this procedure be made more efficient?” This kind of reflection on the mathematics they have employed will enable them to recognize future dead ends before they proceed down those unfruitful roads.

    Third, we should move away from giving students problem sets or worksheets that involve mindless repetition of memorized procedures. Rather, the problems we give them should be carefully curated with follow up ‘thought provoking’ questions asked that would further the development of mathematical thinking. We should also give them underspecified problems in which, by trying to solve the questions, they realize they need more information, which they can then obtain either from the teacher or through research.

    Fourth, we should not fall prey to the lure of calculators. While I personally find that no significant mathematical insight is gained by allowing students up to high school access to calculators, others may certainly differ. However, at the very least we should limit access to calculators to the final two years of high school than allow it too early, thereby actually giving students a crutch on which they grow ever more dependent.

    Fifth, as mentioned in the previous post and here, we should introduce our students to the historical development of mathematics. This will allow them to see that it was humans like us who struggled with problems and who used their imaginations and intuitions to break new mathematical ground. Students will appreciate that mathematics is not some esoteric body of knowledge that just dropped from the sky, but rather the fruit of human imagination and intuition.

  • Primordial Prime Ordinals

    Personal Experiences

    This post is of a more personal nature and really does not deal with mathematics per se. Rather, it uses my experiences as a student and teacher of mathematics as a springboard to talk about women in the field of mathematics. For as long as I can remember, I have been reasonably adept at mathematics. And as mentioned in the previous post, it would not be misleading to say I ‘loved’ mathematics all along.

    Now, I am quite a competitive person by nature. And I become better when I face stiff competition. Hence, I am glad my parents enrolled me in a co-educational school because some of my fiercest competitors, at least for the honor of scoring the highest in mathematics, were some girls in my class. You/they know who you/they are. Today, I see a trend away from competition in most schools. However, competition itself is not damaging as there is a healthy kind in which you attempt to improve yourself, not pull the other person down. And I had this sort of competition with the girls in my class. I doff my hat to them since I believe my understanding and appreciation of mathematics wouldn’t be half as good as it is now were it not for their friendship and competition.

    Despite this, I still notice a reluctance among parents especially to see that their daughters could very well have a keen mathematical mind. I have had parents deliberately choose for their daughters college programs in the humanities or the biological sciences specifically because these programs involve less mathematics. Don’t misunderstand me. There is nothing wrong with a girl wanting to study the humanities or the biological sciences. All I am bemoaning is the closing of certain doors to girls based on some a priori notion of what girls are able to do.

    In my over three decade long career I have had hundreds of students. Without painting too rosy a picture, I have found the girl students to be on average more diligent than the boy students. This is probably because they face more obstacles and, therefore, have more to prove. I don’t think that there is any statistically significant difference in the inherent mathematical abilities of girls and boys. However, the very fact that they have to face greater challenges just to be accepted in fields that are heavy with mathematics develops in them the tenacity that is essential in the acquisition of mathematical understanding and the development of mathematical intuition.

    This, of course, should come as no surprise to most mathematics teachers. We know that, while the number of male mathematicians is significantly more than the number of female mathematicians, there have been very significant contributions to the subject made by female mathematicians. Unfortunately, while we know about them, our curriculum barely has the space to include the historical study of the development of the subject, so packed it is with inane procedures and inconsequential concepts! Hence, we are barely able to mention a handful of mathematicians over the course of an entire 12 year program of study in mathematics. This invariably means that students would be fortunate if they were told the name of even one woman mathematician, let alone more.

    I may be mistaken, but I think mathematics might be the only area of study in schools where we barely learn about the history of the discipline. This is because we have been sold a lie – namely, that, since mathematical concepts seem to transcend space and time, there is no reason to study their development in space and time. In order to redress this situation, I have consistently given historical information in my classes. One memorable time was when I was in Albania. I had asked all my students to give a 5 minute presentation on one mathematician. I am glad to say that over a quarter of the students, including boys, chose to study women mathematicians. Yes, there have been that many women in mathematics! But unfortunately, this is not common knowledge, even among mathematics teachers!

    This is a travesty because it not only means that students are unaware of the contributions of women to mathematical knowledge but also deprives girls today of real life role models who demonstrate that women can be as much at home in the mathematical world as are men. It is time, then, to address this lacuna. I cannot, of course, cover even one of these women to sufficient depth in a blog post, let alone even mention them all. So on the occasion of International Women’s Day, I will briefly consider three women mathematicians in this post. For further reading, I direct the reader here, here, here, here, and here. The lists are not exhaustive. There is also significant overlap in some cases, with a few names appearing in most lists.

    So how have I chosen the three mathematicians for this post? One bridges two of the worlds in which I live and breathe and have my being – mathematics and Christian faith. Another bridges another set of two worlds – mathematics and literature. The third was a contemporary, whose all too short life hits home at a very personal level. I will refer to them by their first names, except when drawing comparisons with other mathematicians, when I will revert to their last names. I do not mean this as a sign of disrespect. I believe after many years of learning about these women I have somehow gotten to know them as friends. Of course, they far outstrip my mathematical ability. But I think that one can befriend a superior!

    The Philosopher

    Hypatia, by Julia Margaret Cameron, 1867. The J. Paul Getty Museum. Digital image courtesy of the Getty’s Open Content Program.

    One of the earliest woman mathematicians and one who makes almost every list is Hypatia of Alexandria. It is not just that she is the earliest known woman mathematician. Rather, her accomplishments speak for themselves. In order to understand the importance of her accomplishments, however, we need to look at the context within which she worked.

    By all accounts Hypatia was born somewhere between 350 CE and 370 CE and died in 415 CE. The latter half of the fourth century to the start of the fifth were tumultuous decades in the history of Northern Africa and Europe. Just about a century before her death in 415 CE, the Roman Emperor Constantine had issued the Edict of Milan, in collaboration with Emperor Licinius, in which the official policy of the Empire towards the Christian Church was changed. The edict officially recognized Christianity as a religion and gave everyone in the Empire the freedom to worship according to their conscience.

    Quite obviously such a drastic change was not received by everyone with open arms and it wasn’t until 380 CE and the Edict of Thessalonica by Emperor Theodosius I that Christianity forcibly became the official state religion of the Roman Empire.

    Suddenly being thrust into a role with power, I am anguished to admit that the Church did not respond with the grace it purportedly preached. Rather, the Christians, now backed by the military might of the Empire, began to crackdown on those who were not Orthodox Christians. This included those with Heterodox beliefs and of course those who held pagan views.

    Hypatia was born between the issuance of the two edicts. In other words, she was born into a world in which Christianity was a recognized religion. However, when she was in her early teens to late twenties, suddenly she would have found herself in a strange situation.

    You see, her father, Theon, was the head of an exclusive school called Mouseion, which endorsed conservative Neoplatonism. In other words, the school was a bastion of pagan doctrine. But suddenly the philosophy which had ruled the roost in the Empire for centuries had been relegated to a supporting role at best and a shunned and despised one at worst. Suddenly, the philosophy with which she was familiar and in which she had been educated was looked on with suspicion, especially by some of the more radical elements within the Church.

    Hypatia, nevertheless, was tolerant of Christians and even welcomed them as students, becoming a mentor of a future bishop, Synesius. In fact, most of our personal information about her comes from extant letters from Synesius to her. Since she bridged both the worlds of academia and religion, the Roman prefect, Orestes, relied heavily on her for advice.

    While Hypatia is not known to have made any innovations in mathematics, she was reputedly a more accomplished mathematician than her father. She was known to hold impromptu lectures when people asked her questions in public.

    However, Hypatia is known to have written commentaries on many works spanning astronomical calculations, arithmetic algorithms, algebra, and conic sections. Unfortunately, most of her writings were destroyed in the furore that ensued following her death. I would have really liked to know what she wrote about the conic sections.

    Her death, of course, is tragic. Hypatia had maintained friendly relations with the Christian leaders of Alexandria, even being considered as an ally by bishop Theophilus. This was despite Theophilus’ opposition towards Neoplatonism, which itself indicates the irenic spirit with which Hypatia conducted herself. When Theophilus died in 412 CE, he had not named a successor. The power vacuum resulting from his death only caused more conflict within the Church at Alexandria between Theophilus’ nephew, Cyril, and his rival, Timothy. The struggle between Cyril and Timothy was violent and made Hypatia distance herself from the Church leaders.

    Despite this, she remained in close relationship with the prefect Orestes, a new convert to Christianity. In one of his letters, Synesius urges her to use her good graces with Orestes to enable Cyril and Timothy to reach a peaceful resolution to their conflict. However, some radical elements within the Church used her advisory relationship with Orestes as a pretext to accuse her of inciting anti-Christian sentiments with the prefect.

    On one fateful evening in Lent of 415 CE, while Hypatia was returning home, a mob of Christians led by a lector named Peter, stopped her carriage, pulled her out, stripped her naked, dragged her through the streets, and then flayed her to death with roof tiles.

    The murder of Hypatia at the instigation of the lector Peter. (Source: Smithsonian Magazine)

    For me Hypatia stands as a symbol of inspiration and a totem of warning. Though living in a violent world, she maintained her commitment to nonviolence. Though living in a world dominated by men, she proved herself worthy of being considered to have overshadowed most of her contemporaries. While she may not have made any innovations, the writing of commentaries to explain difficult ideas is a crucial contribution to the abstract area of mathematics and its often esoteric writings. Her murder at the hands of people who held to the same faith as I do is a warning to me that religious zeal can quite often be destructive. But I write about her to honor a teacher of mathematics like me and hope that she will inspire many of my students, especially girl students, to wade into these deep and refreshing waters of mathematical discovery.

    The Poet

    Portrait of Sofia Kovalevskaya. (Source: MacTutor)

    From North Africa in the fifth century CE, we wend our way to Russia in the nineteenth century. Not that there weren’t any other woman mathematicians in the intervening fourteen centuries! Of course there were! But Sofia Kovalevskaya has a special place in my heart because of the lyrical way in which she communicated mathematics. But that is to get ahead of myself.

    Sofia was born in 1850 CE. Her father belonged to a house of minor nobility while her mother’s family had immigrated to Russia from Germany. When she was 11 years old, a miscalculation of wallpaper needed to cover the rooms of her home led to there being one roll of wallpaper less than what was required. Her parents decided to paper the walls of her room with some old sheets of paper lying in the attic. These turned out to be her father’s notes from Mikhail Vasilevich Ostrogradski‘s lectures on differential and integral analysis. Perhaps this was a foreshadowing of the future? With an introduction to calculus at such an advanced level, it is hardly surprising that she would develop a keen mind for mathematical analysis.

    To encourage her interest in mathematics, her father employed a tutor, under whose guidance Sofia’s love for the subject grew. In her words, “I began to feel an attraction for my mathematics so intense that I started to neglect my other studies.” At this, her father chose to put an end to the tutoring.

    But the fire had been kindled. She smuggled a copy of Bourdon‘s Elements of Algebra, which she read in bed after the rest of the family had gone to sleep. Yet, she knew that her knowledge of mathematics would be limited to what was in that book if something did not happen to change her father’s outlook.

    Soon after, a neighbor presented her with a physics book on optics, which required the use of trigonometry. However, since her study of mathematics had been limited to algebra, she was ignorant of trigonometry. Yet, she took it on herself to derive the Trigonometric identities by herself and did so in almost the same way as had been done historically. The neighbor began to persuade her father to allow her to study mathematics, which he succeeded at after two years! Sofia’s study of mathematics included analytic geometry and calculus and she excelled at these.

    Shortly after this, the family made the acquaintance of the great Russian novelist Fyodor Dostoevsky. During one of their visits to his house Sofia’s sister, Anya, showed some of Sofia’s poetry to the author, who was impressed with her way with words.

    When she was eighteen, she entered into a marriage of convenience with Vladimir Kovalevsky. The marriage gave Sofia the freedom to move away from her birth family and follow Vladimir. But soon her aspirations were dashed when they moved to Heidelberg and discovered that women could not graduate from the university there. Somehow she persuaded the university to allow her to at least attend the classes. They reluctantly agreed subject to the condition that she obtain prior approval from all the professors whose classes she wanted to attend. Nothing could stop her now and she met with the professors and obtained their approval.

    Two years later, on the advice of Leo Königsberger, Sofia moved to Berlin hoping to study under Karl Weierstrass. Since the university at Berlin did not allow her to attend classes since she was a woman, Weierstrass agreed to tutor her in private. Under the guidance of Weierstrass, in 1874 she produced three ground breaking papers on Partial differential equations, Abelian integrals and Saturn’s Rings. The first of these was published in Crelle’s journal in 1875.

    Her work on partial differential equations (PDE) is particularly significant. PDEs are notoriously difficult to solve. This is so even when we know the initial conditions. Sofia extended a special case framed by Augustin-Louis Cauchy to a general case now known as the Cauchy–Kovalevskaya theorem, which specifies the kinds of initial conditions under which the PDEs can be solved. (For the stout of heart, the proof of the theorem can be found here.)

    Despite all the obstacles she faced, Sofia became the first woman to get a doctorate in mathematics, the first woman professor in the world, the first woman editor of a mathematics journal. Shortly after being appointed as professor at the University of Stockholm, she published a seminal paper on the mathematics of crystals and was subsequently also appointed as the chair of Mechanics at the university.

    Cover of a collection of Sofia Kovalevskaya’s poems and plays. (Source: Goodreads, Available on Amazon.)

    Though being so accomplished in mathematics and physics, two fields that are often considered to be rigid and rule based, Sofia claimed, “It is impossible to be a mathematician without having the soul of a poet.” In an attempt to dispel an unfortunately common perception of mathematics, she wrote, “Many who have had an opportunity of knowing any more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination.” In asserting that there is a poetic element and essential role of the imagination in mathematics she wrote, “It seems to me that the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing.”

    In keeping with her openness to the imagination, Sofia was also a poet, author and playwright. With her friend, Anna Leffler, she co-authored The Struggle for Happiness, a play that drew heavily from her relationship with Vladimir. She also wrote a memoir of her younger days titled Recollections of Childhood about which one critic wrote that she “should without doubt occupy one of the most prominent places among Russian authoresses.” Her novel The Nihilist Girl, containing a lot of her political views, was repeatedly banned by the Russian authorities.

    With so many ‘firsts’ under her belt in the face of so many policies that discriminated against women, Sofia showed herself to be perseverant and tenacious. Her contributions in areas of literature and mathematics gives the lie to the commonly held view that those who are good at mathematics have poor language skills and vice versa. She is a powerful example to girl students today, not just in the area of mathematics, but as an inspiration to fight against all kinds of policies that discriminate against women.

    The Artist

    From Moscow and Stockholm we move our gaze to Tehran, Iran, where our next woman mathematician was born. Maryam Mirzakhani was born in 1977, two years before the Iranian Revolution.

    She completed her schooling and undergraduate studies in Tehran itself and demonstrated an exceptional mathematical mind when she won the Iranian Mathematical Olympiad in her junior (11th) and senior (12th) years at high school. While in school she also won the International Mathematical Olympiad in 1994, becoming the first Iranian woman to do so, and in 1995, becoming the first Iranian to achieve a perfect score. After she won the Iranian Mathematical Olympiad in 1994, one of her teachers challenged her to find what are known as complete tripartite graphs, promising her a dollar for each one she found. As a response to this challenge, she derived a method for forming complete tripartite graphs of any size, meaning that her teacher technically owes her an infinite amount of money!

    Standard picture of Maryam Mirzakhani used by Iranian media prior to her receiving the Field Medal. (Source: Muslim Mirror)

    In March 1998 she had a brush with death when a bus she was riding in fell off a cliff. Her success at the Iranian Mathematical Olympiad allowed her direct admission to the Sharif Institute of Technology1 where she earned a BSc in 1999.

    After her undergraduate studies Maryam enrolled at Harvard University where she worked under the supervision of Curtis T. McMullen, who observed that she was “distinguished by determination and relentless questioning.” This shouldn’t come as a surprise since she once declared, “The beauty of mathematics only shows itself to more patient followers.” Treating mathematics as a field that reveals its beauty only to those willing to invest in the search she said, “You have to spend some energy and effort to see the beauty of math.” These twin statements resonate with me, as seen from the previous post. Maryam received her Ph.D. from Harvard University in 2004. From 2004 to 2009, she was a research fellow at the Clay Mathematics Institute and a professor at Princeton University.

    Maryam had incredible mathematical insight, managing to produce a simple proof2 of Schur’s Theorem, cleverly employing the principle of mathematical induction to produce an elegant proof. However, as can be seen here, there is a silencing of women’s voices. The reader will easily recognize that Mirzakhani’s proof is much more elegant than the one by William Gustafson presented in the post. Despite this the author of the blog says Gustafson’s proof is the most elegant. While beauty is indeed in the eye of the beholder, no one would confuse the Antilla-like patchwork of Gustafson’s proof with the Lotus Temple-like simplicity of Mirzakhani’s.

    Maryam’s areas of interest included Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. If the previous sentence seemed like it was in another language, you are not alone. Apart from hyperbolic geometry, I had not heard of any of the other areas before I started learning about Maryam. And I don’t claim to understand any of those esoteric areas.

    In 2013 Maryam was diagnosed with breast cancer. By 2016, the cancer had metastasized to her bones and liver. She died in July 2017 aged only 40. She is survived by her husband, Jan Vondrák, and their daughter, Anahita. Following her death, the Iranian leaders broke protocol and posted pictures of her not wearing a head covering. This made Maryam the first Iranian woman to be officially portrayed without head covering and that too with the short hair she chose as her style following the cancer diagnosis!

    Maryam’s contributions to the field of mathematics are enormous considering that she was a professor for less than 15 years. The international mathematics community recognized her contributions by awarding her the prestigious Fields Medal in 2014, making her the first and only woman, the first Iranian, and the first Muslim to receive the medal.

    Maryam Mirzakhani receiving the Fields Medal at Seoul in 2014. (Source: ABC News)

    Maryam’s work on Riemann surfaces meant that she had to hone her skills of visualizing the complex mathematical structures she was imagining as none of them could accurately be displayed on a 2-dimensional surface, or for that matter even a 3-dimensional space. I think it is her superior ability to visualize these mathematical structures that made her so adept at what she did.

    Despite being diagnosed with cancer, Maryam continued to work and contribute to the field. Despite the debilitating effects of the illness and the treatment, she continued undaunted. There are very few lectures of her on the internet. I have placed a two part series of lectures on the Dynamics of Moduli Spaces of Curves in a playlist with an honest confession that I understood only about 10% of the lectures! These lectures were given after she was diagnosed with cancer. And yet we see her thoroughly animated and full of life when talking about what she loved.

    While Maryam’s focus was not on the application of her discoveries, her advancements nonetheless have found applications from such blasé areas as vehicle painting and roadway design to some really intriguing areas as M-theory and 3D animation. Since her work involved a lot of drawing, she would doodle as she worked, leading her daughter to think that she was actually an artist! Some mathematicians would admit that Maryam’s daughter was not far off in her appraisal!

    Conclusion

    Hypatia, Sofia, and Maryam are just three examples women who have made exceptional contributions to the field of mathematics. In Hypatia we saw a woman who also extended herself in an advisory role to the local government leaders and who strove to form bridges between warring factions. In Sofia we saw a woman who allowed both sides of her brain to be used, not restricting herself only to mathematics, but expressing herself also in poetry, novels, and plays. In Maryam we saw a woman whose tenacity and playfulness revealed mathematical insights that the world is still coming to grips with. All three of them died very young. Hypatia may have just about reached 50. Sofia was 41 when she died and Maryam just 40.

    All three women faced obstacles in life. Hypatia’s struggles were of a religio-political nature. Sofia’s obstacles were institutional in nature. And Maryam had to contend with the rebellion of her own body. Yet all three overcame these obstacles and left lasting legacies that should inspire all of us and surely young girls, especially those who might want to embark on a lifelong study of mathematics.

    1. I tried to get a link to an Iranian site, but none of those links work seem to be working in India. Most of the links that did work were blatantly anti-Iranian. The Wikipedia article was the least polemical of the lot, but even this is clearly and heavily biased. ↩︎
    2. The reader who has not studied number theory beyond high school may find this ‘simple proof’ far from simple. However, given how convoluted most mathematical proofs at this level are, this is actually quite beautiful in its elegance. ↩︎

  • The Eye of the Beholder

    I was in the 12th grade when I came across what I, along with many others, have come to recognize as the most beautiful equation in Mathematics. This was way back toward the end of 1986. Unfortunately, none of my Mathematics teachers that year ever bothered to show me and my fellow students the beauty behind this equation. Perhaps beauty is indeed in the eye of the beholder and my teachers had just not become beholden to the beauty that would eventually hold me.

    It was while I was studying the topic of Complex Numbers that I came face to face with this mathematical beauty – elegant in its austere frugality and sensuous in its visual appeal. Unfortunately, it was many years later, when I was a Teaching Assistant at the University of Texas at Austin that my eyes were opened to this grand vision. As mentioned in my post on my motivations for starting this blog, I had to keep regular office hours. During the office hours many students used to visit me. But their questions pertained mainly to High School Mathematics rather than the Control Systems or Adaptive Control, for which I was TAing. Since both these courses required a heavy application of Complex Numbers, I had to really delve into the subject matter to be well prepared to address the questions the students might have had.

    So I went back to the basics. I knew the rudiments of Complex Numbers of course. I could perform the four operations – addition, subtraction, multiplication, and division – on Complex Numbers. What more could there be? But when I went to self-study this topic, I realized that I wanted a hefty refund from the educational institutions I attended in the 11th and 12th! I had been thoroughly short-changed! It was as though someone had described the Sistine Chapel by telling me how Michelangelo had mixed his paints rather than tell me about the beauty of the actual frescos that he had painted on the ceiling!

    Fresco of the Creation of Adam from the Sistine Chapel by Michelangelo, c. 1508-1512. (Source: Britannica)

    Of course, Michelangelo could not have painted the frescos without knowing how to mix the paints. But mixing the paints wasn’t the goal! The purpose of mixing the paints was the beauty that be behold on the ceiling! Similarly, the purpose of studying Mathematics cannot be relegated to the success of its procedures. If it does not extend to the beauty of the subject students will not develop the skill of recognizing that austere and frugal beauty.

    And so, not being satisfied with knowing how to mix ‘mathematical paints’ I allowed my ‘mathematical breath’ to be taken away by the beauty that these numbers held but that most students of Mathematics are just not introduced to at the time when the beauty can have the most impact on them – that is, when they are first learning about them.

    Of course, the problem arises right when we first hear about these numbers. Almost without exception students will hear, if not learn, about these numbers in the context of learning how to solve quadratic equations. So the teacher will tell the students to consider the equation:

    Here a couple of points about the notation above are needed. First,

    tells us that a, b, and c are real numbers. While later study in complex numbers will get rid of this condition, it is needed at this stage to show the students that, even if we remain fully committed to using only real numbers, our solutions could end up being non-real. Second,

    tells us that, in addition, a cannot be equal to 0. This ensures that we are actually dealing with a quadratic equation rather than a linear equation.

    The teacher will then (hopefully) show the students that the solutions to the quadratic equations are:

    Now the teacher will focus on the expression under the square root sign. It is called the ‘discriminant‘ and I was recently quite shocked to learn that many teachers have stopped telling the students this technical term. This is a travesty in my book! All areas of knowledge use technical terms that help communication between people working in those areas. Hence, to deprive students of the terminology and the reasons behind the terminology1 is to deprive them of a robust education!

    Anyway, the teacher will tell them that the ‘discriminant’ tells us something about the nature of the solutions. If the discriminant is positive, we get two real and distinct solutions; if it is zero, we get one real and repeated solution; and if it is negative, we get complex solutions.

    Now quadratic equations are normally introduced in the 8th or 9th grade. Prior to this, the students would have learned some basic patterns for multiplication. Most relevant, in this context, is the fact that any non-zero number when squared yields a positive number. Inquisitive students will then probably ask the teacher how it can be that we can have a negative number under the square root because that would means that there are number, which when squared yield a negative number.

    And here we encounter the first roadblock to the study of Complex Numbers. Remember, this question has arisen in the context of trying to solve an equations involving (presumably) only real numbers. We have ensured that a, b, and c are real. But the formula for the solutions raises the possibility that x could violate a consistent property of real numbers, namely that their square cannot be negative. So what kind of number when squared would yield a negative number? Unfortunately, the history of Mathematics itself proves to be a hindrance because these numbers were called ‘imaginary numbers‘. This unfortunate terminology has led many teachers of Mathematics to conclude that, since these numbers are ‘imaginary’ they must not exist, much like those voices in my head that I keep trying to silence!

    The irony is that Mathematics gets increasingly abstract as it is. Still Mathematicians have compounded the problems students have with the subject by actually calling some numbers ‘imaginary’. If ever there was need for some prescient wisdom to reject the unfounded whims of René Descartes, it was certainly with respect to his derogatory use of this term! Unfortunately, the name has stuck and we are stuck with it.

    The second roadblock to the student of Complex Numbers is the term ‘complex numbers’ itself! When most students already think that the subject is complicated enough, intentionally labelling an important class of numbers ‘complex’ is like shooting yourself in the foot!

    Anyway, students are introduced to these ‘complex’ and ‘imaginary’ entities and have their phobias duly stoked. This is especially true if the teacher does not bother to communicate the beauty inherent in the concepts and focuses only on the procedures behind the operations on these numbers.

    However, when I came across the equation I opened this post with and spent time with it, I realized that these numbers, far from being a figment of our imagination, which might be conveyed with the word ‘imaginary’, have a very tangible existence in the real world. Moreover, far from being complicated, as might be suggest by the word ‘complex’, these numbers actually enable the linking of parts of Mathematics that we might otherwise be led to think are unconnected.

    I have hyped this equation a lot. And you may be thinking there is no way I can deliver on my promise. Let us see.

    The equation that I have been referring to, of course, is:

    This is referred to as Euler’s Identity, another case of unfortunate terminology since identities normally contain variables and this equation has none!

    So what’s so great about this equation? First, it contains three mathematical operations – addition (indicated by the ‘+’ sign), multiplication (hidden between the ‘i‘ and the ‘π‘), and exponentiation (the power of ‘e‘). Given that subtraction is just addition in reverse and division is multiplication in reverse, this equation contains all the basic mathematical operations. Second, it links the additive identity (i.e. ‘0’), the multiplicative identity and unit of real numbers (i.e. ‘1’), the base of the natural logarithms (i.e. ‘e‘), the number that describes the shape of a circle and the angle measure for a semicircle (i.e. ‘π‘) and the unit of imaginary numbers (i.e. ‘i‘). Third, the equation links arithmetic (indicated by the presence of ‘0’ and ‘1’), geometry (indicated by ‘π‘), growth (indicated by ‘e‘) and complex analysis (indicated by ‘i‘). The equation, of course, is a special case of a more general equation, known as Euler’s Formula, in which the links with trigonometry and calculus are also explicit. Fourth, the equation also links integers (‘0’ and ‘1’) with irrational numbers (‘e‘ and ‘π‘) and imaginary numbers (‘i‘). So many connections across so many concepts and branches of Mathematics, all contains in one single, simple equation with 7 mathematical symbols!

    Unfortunately though, the High School Mathematics syllabuses in most countries divide the course content into almost rigid and impervious categories, isolated from each other. Since most of these syllabuses are artificially bloated to communicate the false notion of rigor, teachers scarcely have enough time to bulldoze through the syllabus. Stopping to smell the ‘mathematical roses’ is all but impossible. Hence, an equation that, in its simplicity and efficiency, demonstrates the inter-relatedness of all parts of Mathematics, thereby shattering the illusion that the various parts of Mathematics can – or God forbid are expected to – function in isolation from each other, should be one with which I believe teachers of High School Mathematics should become intimately familiar.

    1. The expression is called the ‘discriminant’ because it allows us to ‘discriminate‘ concerning the nature of the solutions without having to solve the equation. ↩︎

  • Motivations

    As I start this blog, I need to ask myself what I hope to accomplish through it. And why! If there is no purpose in my mind, this will soon devolve into a collection of inane posts, probably about mathematics issues, but will nothing more. So let me describe what I hope to accomplish through this blog.

    I have enjoyed – perhaps ‘love’ might not be too strong a word – Mathematics since I can recall. I have memories from when I was just past the toddling phase of attempting to find patterns that were revealed in various mathematical operations.

    My mother was a teacher of high school Mathematics for a considerable portion of her career. And I remember sitting with her while she graded papers and helping her total the marks. Patterns for addition and subtraction revealed themselves to me during this phase. For me, these patterns were a source of joy and pleasure. And I could not, for the life of me, understand how so many of my classmates were also not enamored with the subject.

    When I started as a Teaching Assistant at the University of Texas, Austin, I had to keep regular office hours. During those periods quite a few students came to me for help. But they needed help, not in Engineering concepts. Rather, most of them needed help in high school Mathematics. They too, like many of my classmates, were either disinterested in the subject or actively detested it. And as a consequence their ability to grasp concepts was hindered. It is difficult to learn when you hate something! Or, for that matter, if you fear it.

    Over the course of my reasonably long career teaching Mathematics, I have come to the conclusion that any hatred that most students have for the subject stems mostly from a fear of it. Mathematical concepts and operations seem like some arcane body of knowledge that only a special and very small group of initiates can hope to fathom. I do not fault the students for this.

    I teach students in the High School, mostly in the 11th and 12th grades. These are the final two years of their schooling and, unfortunately, by the time they reach me, most of them already have a gripping fear for the subject and, for some, an utter disdain for it. The damage has been done. And there is very little a teacher can do at that stage to mitigate the fear.

    So as an experiment, I decided to teach in the Middle School one year. I took on the task of teaching students in Grade 6. And what I discovered distressed me. They had no clue about how to use the order of operations (BODMAS or PEMDAS). They had been given some inane phrases to remember while performing operations rather than the rationale behind the order. I spent almost the whole of the first term trying to get them to unlearn the horror that had been foisted on them – to limited success, I must confess.

    I realized something very important. Children are curious. And if we teach in a way that suppresses their curiosity, they will think that memorization is learning. And there are many students who memorize tricks related to the operations and believe they are the cat’s whiskers when it comes to Mathematics only to flounder later when ideas become more abstract and the illusion of concreteness that numbers conjure recedes into the background with the study of algebra, trigonometry and calculus.

    However, if we capitalize on the innate curiosity of children while teaching them, allowing their questions to jerk us from our complacent stupor, not only will they learn well and learn to love the subject, but also we teachers will reach a better understanding of our subject.

    So in this blog I hope to deal with all sorts of things related to Mathematics. I will address mathematical ideas that have piqued my interest. I will also tackle issues of how to teach some concepts to students. While I may not fully be able to make things ‘not obtuse’ (i.e. not difficult to comprehend), I hope that the posts will be ‘acute’ (i.e. characterized by keen discernment and intellectual perception).

  • What’s with the name?

    When I decide to start this blog, I was faced with the task of deciding on a name for it. I wanted something enigmatic and intriguing. I didn’t want the word ‘Mathematics’ to be in the title. At the same time, I didn’t want the title so distanced from Mathematics that it would seem unrelated to what I hope to accomplish with the blog.

    I asked some people whom I trust to suggest names and a few of them responded. A good friend and former student suggested ‘So Obtuse’. That got me thinking. ‘Obtuse’ goes with ‘acute’. Could I find a way to include both? And what would it mean?

    Of course, there are the geometry related meanings that most people who read Mathematics blogs will have readily recognized. An acute angle is one whose measure is between 0° and 90°. And an obtuse angle has a measure between 90° and 180°. So far so good. But that would have been too plain for my purposes. I wanted something deeper.

    So I checked the meanings of both words. Among the various meanings listed, the Merriam-Webster Dictionary states that ‘acute‘ meant ‘marked by keen discernment or intellectual perception especially of subtle distinctions’ (meaning 4a). That was promising. What about ‘obtuse’? Well, the Merriam-Webster Dictionary says that ‘obtuse‘ could mean ‘difficult to comprehend’ (meaning 2b). I was getting closer to what I wanted to do with this blog.

    So in the end I have chosen to go with ‘Acutely Obtuse’, with the proposal that I will be attempting to provide insightful, discerning and perceptive commentary on mathematical ideas that may be difficult to comprehend but that will, hopefully, become more lucid through my commentary.