Nature of Math: Alternative Methods and Anxiety

The nature of mathematics is a tricky subject to write on. I avoided it (outside of my first blogpost) because of this. Since I am going to be a teacher, I wanted to write about the nature of math in today’s culture. It won’t so much delve into the nature of higher math, but will deal with the anxiety that exists in our culture today. Societies view on math is just as relevant to the nature of mathematics as a professor’s papers on the subject. All kinds of buzz populate our media and social networks; buzz filled with judgments on the common core and the plethora of standardized tests. The weird difference about this topic is that everyone seems to think that they are an expert in it. Most everyone agrees that they didn’t like the way they learned math, yet they all seem to want to keep it the same.

I was very gifted in math at an early age and was always told that I was smart in math. (Obviously this isn’t exactly a great practice for teachers.) Early on in my school career, I began getting pulled a few days a week to work on more advanced math with a small group. My teachers used Everyday Math, that is more of a reformed set of math books, but the teachers were not trained in how to use them. Instead of having rich mathematical discussion in which we could share our reasoning, we answered questions that always always always always always ended in “Explain your reasoning;” one could hear the groan each time a student got to this part of a problem. The ideas behind Everyday Math were strong, but the implementation was week. This continued till I rejoined the class in algebra in eighth grade. Algebra and on was a more traditional method that lacked the explanations that Everyday Math had asked for. It wasn’t until I reached a level beyond calculus that I got to a mathematical problem that I couldn’t answer quickly. The mathematical practices in the common core that ask for perseverance in solving mathematics did not exist within me. It wasn’t until I went back and studied math pedagogy that I understood my own strengths and weaknesses, and those in the program I had gone through.

Teaching alternative methods are is part of the common core. Everyday Math is a big one for teaching multiple ways to solve a problem. Looking back I can understand the pedagogical decisions behind this. Many of today’s curriculums ask for an environment in which many of these methods are discovered or taught to clarify a concept. But they don’t demand that the quickest, rote, way of solving a problem be perceived as the best. Instead it asks for methods that are tied to the concepts. I love this idea. Everyday Math attempted to present this same idea by teaching multiple ways to solve problems that have a traditional method i.e. division. In the end it didn’t demand a method be chosen above the others and many teachers either supplemented the book with traditional methods or lacked the understanding to demand that students find, and stick, to one method. This led to a lot of confusion in my childhood development of mathematical methods and to this day I can’t perform division or multiplication with ease. Thus, I understand and agree with the ideas behind today’s more reformed curriculum, but I can see how the ideas alone don’t make a good math curriculum; it is the teacher’s determination to create a rich mathematical environment in the classroom that makes math come alive.

Alternative methods didn’t always provide a natural progression to a more efficient way of solving problems for me. There were more concept based methods that lent themselves to understanding better; I even remember learning methods like these. But these methods weren’t always the most efficient and the most efficient can be useful when variables are added into the mix. When I was in a calculus course we needed to know how to do long division with variables. Without a strong understanding of the traditional method this became much more difficult. Instead of leaving a child with a method that is less efficient, I prefer to move the child to a more efficient method once the concepts are thoroughly understood. I do understand that without a strong foundation these methods can prove difficult but the methods are useful and deserve a place in the curriculum. An understanding in the concepts and most efficient methods seems like a happy median. I was blessed with an ability to understand numbers and I have always tested well. I tend to find myself dismissing alternative methods since I never struggled in these areas. But honest reflection demands that I learn how all types of students work, so that I can empathize with their unique situations. This will be a system of trial and error for me. I hope that I have the ability to create a rich mathematical environment that is accommodating to all students, despite my personal hurdles.

Text anxiety is a huge issue. In order to deal with this many schools have elaborate systems in place to support students. The weeks of preparation, the letters home, the relaxation methods, all seems over the top. The success of the students on this test matters, I’m not denying that these can be helpful, but not to the extent that our society suggests. It distracts students and takes away from valuable teaching time. But many teachers give up at this point and say “if you can’t beat ‘em, join ‘em” and proceed to teach one how to operate in the world we live in. The weeks preceding the test are given a significant amount of weight though. Why? This creates stress by giving the test weight. Being the best teacher I can be is important to me. I will not waste several weeks preparing my students for a test that I don’t give a damn about. Instead, I will explain to them the situation like they are adults, and expect them to perform well. Teaching to a test is obviously something that society would discourage, but the mathematical environment they create is still bound by the shackles of the test looming in the spring. A rich mathematical environment will allow for the creativity of the students to come out and go in directions that support all students’ mathematical creativity. I argue for creating a rich environment and to make this an exception day where there is a test taken.

So the nature of mathematics in today’s society is a grim one. It is laden with debates on reformed vs. traditional methods and test anxiety. I only briefly talked about these two issues, but there is hope. Classrooms can create rich mathematical environments that are not bound by standardized tests. I will do my best to never back down from a belief in a culture that can support children in their mathematical journeys.

Ramanujan

I have now gone through all the different types of blog posts that were assigned in our course. I am looping back to looking at the history of mathematics in this blog since I feel that my last post on the history of mathematics was a tad recent.

Ramanujan interested me right away when we talked about him in class. I didn’t know exactly what drew me to him at first, though by the end of my research I was a little clearer. I watched a BBC documentary on him for one of my class daily’s and decided that writing a brief history on him would be an enlightening experience for why I was interested. The first part of this blog will be a brief history of his life, followed by a section on his work, and finishing with a critical view of his life and legacy.

Ramanujan is one of those mathematicians that come around once in a hundred years. He had an innate ability to understand math in ways that were far from traditional. Growing up in a small town in India, his ability in math could not carry him forward in traditional academics since his other subjects were far from proficient. He taught himself mathematics from limited sources and began communicating with GH Hardy in his early twenties. In the early 1900’s Ramanujan left India and came to Cambridge to work with Hardy. His life was far from happy here. He was a strict vegetarian because of his Hinduism. But because of the intensity of his vegetarianism (nothing could be cooked in a pan that had once held meat) he often didn’t eat. He wrote home to his wife to send food “the cheapest route possible.” Because of the weather and his lack of nutrition Ramanujan was frequently sick. He left after six years in England, but was so sick that he lived barely a year in India.

According to his widow, Ramanujan was always doing “sums.” His mathematics was focused on infinite series, continual fractions, and other areas of mathematics. His development of a theorem that can predict the number of divisions for any given number is revolutionary. It combines areas of mathematics that many wouldn’t equate to each other. His way of doing math was simple and not like that of traditional math that requires rigorous proofs. Because of this he wasn’t well received in an academic environment that was more traditional. It is only because of Hardy that his work was known at all. Even though Hardy acknowledges the brilliance of Ramanujan, much of Ramanujan’s work is written with Hardy instead of alone.

The last part of this blog will be a more critical view of his life and legacy. Ramanujan is viewed as this brilliant mind that left us with so much that we don’t understand. Rightfully so, there is a critique of how he was treated in England. But more than anything else the story is just sad. In the entirety of my research there seems to be a prevalence of sadness. The unforgiving school system in India didn’t support him, the racist undertones in England didn’t respect his work unless he conformed to their norms, the unsupported restrictions of his religion and diet, and more led to a life that was sad in England. But all of this sadness can be blamed on others. My thoughts are that he partly chose this sadness as well.

There is a story from his widow that talks about how he would be so focused on his sums that he wouldn’t stop to eat. His wife was forced to put rice in his hand so that he wouldn’t forget to eat. This makes me think the guy was a jerk. What kind of person doesn’t care enough about his wife to stop and eat dinner with her? I understand that I can’t judge a culture by my standards, but I don’t like this. Ramanujan also leaves his wife alone for years while he pursues his math in England. I don’t care how brilliant one is and how important one’s work is, there has to be a level of humanity in oneself.

What I would argue is that there needs to be a broader scope when looking at the life and legacy of Ramanujan. There needs to be a continuation of understanding that England’s academic system was not as receiving as it should have been. But there needs to be an affirmation of the support he did get from Hardy. There needs to be a disdain for a culture that was less welcoming than it should have been for someone who came with a different way of doing things, but through the proper hermeneutical and forgiving lens. There needs to be honesty when looking a Ramanujan’s personal life that allows one to be critical while again looking through the correct lens. Ramanujan is a man who will not be forgotten but a story that is laden with sadness. 

No Controversy Post on The Eight Queens Problem

This blog is going to take the place of our third blog post. The one topic that I haven’t covered yet is communicating mathematics. I doubt this will be as controversial of a piece as the previous post was since it will be more informative and less opinion based. So if one is reading this looking for something similar, I apologize.

In looking back through our course page on different thinkers around the time of the third blog post I was drawn to Thabit ibn Qurra since I knew so little about him. I wanted to delve into his mathematics for my blog. Looking at his Wikipedia page, I noticed a link to a chess solution. From that page I found a link to the eight queens puzzle. Following my interest and Wikipedia links I decided to settle on this as my topic for this blog.

The problem is as follows. Is there a way to place eight queens on a standard chessboard so that none of them could attack each other either diagonally, horizontally, or vertically? The history of this problem is part of my interest.

Max Bezzel first introduced the puzzle in 1848 in a German magazine. Over the next six years there were a variety of solutions published in various magazines. With a reposing of the question the problem shifted to a question of how many different solutions to the problem there were. Franz Nauck was the one responsible for this reposing and asserts that there were 60 solutions to the problem. This sparked a debate between two friends Gauss and Schumacher. They communicated with letters for around a month. During that time Nauck corrected himself and affirmed that there were actually 92 solutions, but offered no proof of this assertion. Gauss referred to Nauck’s correction and laid out how to go about proving it, though never actually proves it. It seems that this task wasn’t of extreme importance to Gauss since he didn’t spend the time to solve it. These letters were published about fifteen years later.

The solution that Gauss proposed was clever. I’ll try to communicate as best I can the solution that was proposed. Gauss reformulates this problem into an arithmetic one. Gauss begins by assigning a position to each permutation of the numbers one through eight. Thus 12345678 would have the first queen at a1, the second queen at b2, the third queen at c3, and so on. The permutation 45632178 would have the first queen at a4, the second queen at b5, the third queen at c6, and so on. Using this idea of position Gauss was able to work with the solutions more easily. He found that if each permutation was summed with 12345678 and separately with 87654321, and if these sums were all different in each number that this was a solution to the problem. This is difficult to understand without an example so I will share the example that Gauss sent Schumacher.

All these sums are different in this permutation of 15863724. This means that it is a solution to the eight queens problem. In the next permutation of 13425867 we can see that 10, 9, and 8 all appear twice. This means that it is not a solution to the problem.

  

By setting the problem up this way Gauss was able to more quickly sift through solutions. The setup itself prevents queens from lying on the same row or column. The addition of the permutation 12345678 prevents queens from lining up diagonally to the right; the addition of the permutation 87654321 prevents the queens lining up diagonally to the left.

The solution isn’t flushed out arithmetically that I could find, but was explained to have 12 fundamentally different solutions that, through rotations, are flushed into all 92 solutions. This geometric solution is nice, but I wanted the arithmetic solution.  

The problem continued to evolve into boards of differing sizes and got into complications that I won’t flush out. As stated earlier, Gauss’ passion didn’t seem to be evident in this problem since he didn’t bother to create every permutation, or offer an argument for why there were only 92 solutions. I wish that his argument had been clearer and that Gauss had had more passion in the subject. This was a fun problem to post on because it combines my passion for chess and math!

David Kung

In my capstone class one of our “dailys” (or homework that we could do for that day) was to watch a lecture by David Kung. I watched this video over a week ago but it is still on my mind. Deciding what to cover for this blog post I decided that it should be something that was relevant to me. Since this lecture was so fascinating, I decided to focus on that. I would label this as studying the history of mathematics since it is looking at the contributions of this mathematician to his field. I will give a summary of the lecture, my reactions to it, and end on how I think Kung could change his lecture.

To start the post I will simply give a brief summary of the lecture. The lecture was on diversifying the mathematical community. He began by presenting the audience with a series of statistics on the lack of diversity in all STEM education. He would start with other subjects represented in a line graph that would show something like, the number of women with a doctorate in that subject over time. He would then ask the audience to draw their best guess as to where mathematics lay. Every type of diversity had a dismal report in the field of mathematics. Women, Blacks, Hispanics, etc, were all underrepresented in every part of the mathematics field. From here, Kung laid a framework of how he would change the type of mathematics education to be more conducive to fostering those minorities who aren’t typically part of math education. He finally ended with a story that told the story of one student who was able to succeed in a mathematics degree despite the social and institutional hurdles in her way.  

The suggested method of closing this achievement gap was through pedagogy, though there is an acknowledgment that pedagogy alone will not solve the problem. Passive teaching in STEM subjects, especially at the colligate level, is keeping students away from these degrees. Passive lecturing is becoming available in more and more forms and has the added benefit of being “pauseable.” Kung calls passive lecturing “professional misconduct.” There is a shift to more interactive pedagogies in earlier grades, but the colligate level is not following suite. Kung sites studies that find that interactive pedagogies, which focus on a growth-mindset opposed to a fixed-mindset, are able to help close the achievement gap by being more accessible to a wide variety of students.

Another method for closing this achievement gap is through recognizing the inherent bias that exists in classrooms already. In order to recognize this Kung wants to look at questioning patterns in the classroom. It has been shown that more time is spent with males than females. A teacher is more likely to call on a male student and is more likely to ask deeper, more thought-provoking- questions to said students. This is also true for minority students as well. By recognizing our bias and understanding our prejudices, teachers will better be able to approach all students fairly.

The reason that this video has stuck with me so long is the inherent and unrecognized bias that David Kung brings to the table. Taking a step back, I need to be careful about how I phrase this since I will be publishing this online. I want to affirm that my opinion is not backed up by research and that it is only my opinion based on my experiences as a white male. That being said, I feel that Kung’s attempt at being diverse and accepting, purposely ignores the views of an entire group based on their skin and sex--i.e. white males. If his goal is to be tolerant and accepting of others in education he can’t ignore this entire group. The most telling part of this is during a story about how women will do poorer on a test if there is a question about their sex before the test begins. After his point is made he tells how this research is backed up because they tested male’s golf skills with a similar question about if it was testing athletic ability or just for fun. His first story gets a bunch of quiet and sad nods from the crowd but his second story draws a laugh. I’m not doing the story justice, but this story is marginalizing a group. And everyone laughs. If one wants equality, there cannot be jokes on others behalf either.

There is so much talk about prejudice and how we need to draw out our inherent bias, but bias is bias. One can’t have it both ways. There seems to be a culture of blame in this lecture and one that I have seen in many other situations. Blaming the majority is an easy thing to do, but it doesn’t establish the culture of respect and authority that one is so desperately seeking. The argument that the majority can take it since they are “ahead” is not accurate though; it is simply using a system that one wants to unravel. At one point Kung talks about a picture of a selection of excellent professors at his university. He jokes that he is the diversity factor in the picture of fifty or so white men. But isn’t this marginalizing their accomplishments? Why is it their fault that they succeeded? I struggle to understand his lack of equality while operating under its guise.

I am all for diversity in every facet of life. I understand the lack of diversity that is in higher academia. But whose “fault” it is that diversity isn’t as prevalent as it should be is up in the air.  According to Kung, Berkley professors in the 1970’s blamed students not working hard enough for the lack of diversity. Today that blame has shifted to society. But it seems that society has become synonymous with white males. It won’t be until this mentality of blame has shifted to encompass all that actual change can happen.

I don’t want to affirm the Berkley professors of the 1970’s but I want the burden to go back to the students. I want to create a society in academia that is supportive of all groups so that success is based on the student. If this culture is created, and all black females are the ones who succeed, I don’t want there to be a culture of blaming them for marginalizing others. Those who try the hardest and perform the best are the ones who should succeed in a true democratic society. There needs to be equal opportunity and no judgment of those who succeed. Kung needs to follow his own practices and recognize his bias. By removing that and his inability to not blame others for society's problems he could accurately address the problems. 

"How Not to be Wrong: The Power of Mathematical Thinking" Review

Blog Post Review

 I recently read How Not to be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg. To say that I enjoyed this book would be a stretch.

            

The biggest problem with this book was that it had no idea who its audience is. There was a disconnect between much of the information being given. On one had the author would provide us with an overly simplistic explanation of math that a sixth grader could understand, and then go into overly complex math that is explained in an incredibly illogical and confusing way. His jumping from topic to topic without building on previous issues further muddies the waters on who he is trying to reach. His introduction needed to define his audience and explain his thesis for said audience. This clarification would have better organized his thoughts for how the book progressed.

This disconnect made for a very long and tough read. But the authors attempt to tie these concepts together is perhaps the one guiding light in the book. His attempt to connect concepts was by using an overarching example that he could keep referring to. This was done almost every chapter and gave relevance to the concept that her was trying to get across in said chapter. And these stories were fantastic and relevant. His use of the lottery story in Massachusetts allowed him to constantly give context to a series of mathematics that wasn’t incredibly interesting. If he had been able to tie the chapters together like he tied individual chapters together, it would have been a fantastic book. Instead the book continued to feel disconnected.

Unfortunately, this is where he fell apart. Though the stories were good and relevant, the author failed to come back to his overall thesis. The thesis of the power of mathematical thinking wasn’t addressed in a relevant enough way. The book read as a series of small vignettes and didn’t come to a conclusion. I wanted to know how and why the book was going in its direction. Even the movement of the mathematics wasn’t connected in a relevant enough way to even hint at a thesis. Individual chapters hold weight. I could see splitting this book into different articles if better concluded in each chapter. In this way, at least, the author could have better argued individual points about mathematical thinking.

I guess, in conclusion, that the real problem with this book is that the majority of the mathematics lies in an area I just don’t love. Statistics is imprecise and just isn’t up my alley. I understand that it is used in ways that get at the heart of mathematics, a way that describes the world around us, “it is the extension of common sense.” And statistics embodies this. I just like to live in the abstract world when it comes to mathematics. The author attempts to describe a mathematical world that we live in but falls short. A few of his insights give credence to his views, but he fails to share them in a cohesive way that argues for any type of point. Mathematics describes the world we live in and mathematical thinking allows us to navigate within it, but complex, unconnected stories does not, a thesis make. 

Chautisa Yantra

This blog had to be screenshotted from word since I couldn't figure out how to transfer it over once I wrote the equations. Regardless...here ya go!

Blog Post One: Greek Influence

My first post since talking about the nature of mathematics is based on the influence of Greek mathematics. I enjoy history and especially the history of how ideas came into being. During my time as a philosophy major I was typically more interested in the history of how ideas developed; how different thinkers countered each other using their best logic. Mathematics interests me in a different way than philosophy does. I enjoy the computational logic of watching a long problem come together with perfect clarity. I enjoy the satisfaction of having struggled through a problem and achieved the singular correct answer.

Thus my tendencies lie in two different camps when comparing how I approach philosophy and math. I love being able to argue with no answer in sight as a philosopher, and I love the ability to come up with the “correct” answer in math. My hope throughout this course is to unify these two opposing forces and enjoy the history and development of how math arrived at my doorstep as a young mathematician.

The first part of this journey had me delve into the history of Greek mathematicians on modern mathematics that I have seen in my life. This requires me to look back on my own mathematical journey as a mathematician. To be brief…I grew up as a “gifted” child in mathematics. I never struggled in math and excelled in every class I ever took. I was frequently told how smart I was and never spent much time on math until I got to the math beyond calculus. Suddenly I was expected to struggle with mathematics, the only problem being that I had never struggled before. I didn’t know how to act and it took copious amounts of work to succeed. But the math content I saw was the typical American curriculum. It was everyday math, or Chicago math. It focuses on procedure and explaining ones reasoning. My teachers didn’t fully utilize this though and followed a more traditional method. I did the algebra, geometry, advanced algebra (trig), stats, pre-calc, calc route which most students who like math go through.

So when looking at the Greek mathematicians I have two things to look at when thinking how they influenced my mathematical journey. The first thing to look at is the culture of math that I was raised in and the second being the progression of the content I went through.

In terms of content I think that Euclid was a huge influence on how geometry is taught. I don’t think there is anyone who would disagree with me on that. His Elements has profound implications on how math was/is done for generations. Geometry in high school was a series of propositions that led to theorems and finally proofs. This logical structure for proof writing was instilled in us throughout the whole year. I loved the class. It relied on logic and used logical steps to move forward. This Euclidean influence could not be more obvious.

Strangely though, this is not how geometry was first introduced to me. From an early age we studied shapes and their properties. This type of spoon-fed information that I used to regurgitate volume, area, length, etcetera would have driven Euclid crazy. The logical structures for my understanding were not based in logical deductions; they were based in the authority of my teachers and textbooks. This is a culture that is against how Euclid would have looked at math; in fact this is against how many early mathematicians would have looked at math.

            I’ll speak more to this in the next paragraph, but math was taught in an abstract way to me as a child. Symbols and their meanings were taught to me with the express purpose of solving math problems. Using these new symbols and skills I could solve bigger and harder problems and discover new symbols. In essence, math was taught to me so that I could solve math problems. The Greeks would have scorned this and expected an application for the math being done.

In reading the blog by David Mumford I look at one of his key take-aways being that math was based in applied math. Math was accomplished in order to solve real-world problems. The Greeks would have had this in the back of their minds as they solved math problems. Thus, much of their math was geometry at its core. Math today separates much of geometry from other subjects, where the Greeks used it as a basis for math. Mathematics was a culture of discovery in the Greek day, mathematics in our schools today is a culture of solving problems. So the Greek influence is not strong on this front. Math is frequently not seen as a culture of discovery. Our schools need to teach what is at the core of mathematics again and follow our Greek brethren.

In summation, I can see the influence of Greek mathematicians in how mathematics was taught to me, but I can also see a blatant dismissal of much of the underlying principles that determined why mathematics was done.