Monday, October 9, 2017

Genius at Play: A Book Review

Genius at Play by Siobhan Roberts is a biography about "the curious mind of John Horton Conway", a raging mathematician and absolute genius. This book is written in it's own sort of curious way as it is written in the form of a kind of interview in some parts, but as a general story in others. Roberts includes numerous accounts of conversation with Conway in the text, incorporating direct quotes and allowing the reader to hear Conway's voice. The main focus of this book, although the title seems to focus on Conway's mathematical intelligence, is in my opinion the character of Conway as a person rather than as a mathematician specifically. While mathematics is definitely involved in Conway's life, I felt that math was the subtopic behind Conway himself in this book, which is something I was not expecting. Overall, the author uses Conway's life to explore certain mathematical concepts, as Conway did impact the world of mathematics immensely.

Genius at Play definitely wouldn't be a book that I would recommend to anyone who was not in some form interested in math. Even as an individual who is a math major, I personally felt that this book was difficult to read and I struggled to get through it. The mathematical content that is addressed in this book is often breezed by, so any form of proof or explanation for a given problem is hard to find. Therefore, this book would be a good read for anyone who enjoys exploring and forming proofs and discovering those kinds of connections. There are also several parts in this book that mention a theorem or game of some sort that Conway proved or invented, so it would be easy for an individual interested in that sort of thing to find lots of material as well.

Although I felt that it was difficult to read, there were still things included that I liked. As mentioned earlier, the author included specific quotes from Conway from interviews and conversations with others in the writing, which added another perspective and gave the book more personality (partly because Conway has quite the personality). The author also included various drawings and graphics that Conway presented while forming a new game or deciphering certain theorems. This is a nice change of pace as well because it allows the reader to explore as well in an attempt to understand and follow along with Conway's thinking.

All in all, given an individual who desires to deepen their mathematical knowledge and challenge themselves with the mind of John Conway, this book could be a really strong, beneficial read.

Monday, September 18, 2017

The Creation of Mathematics

Anyone who knows me knows that I enjoy writing as long as I can write about something that I have passion for. Despite being a mathematics education major, the history of mathematics isn't actually something that I care too much about (sorry), so coming up with a topic for this post has posed to be a little difficult. However, anyone who knows me also knows that I am crazy passionate about Jesus and diving deep into the Word of God. So, to tag along with my previous blogpost about being able to find and use mathematics in all situations, we're going to attempt to tie mathematics and the Bible together in what may be a feeble attempt to create a solid blog post.

In class we have been following the progression of mathematics and discussing many great mathematicians and philosophers who created varying theories regarding mathematics. These are things that while I've never really experienced learning about them in other classes, I've also never really wondered about them. Mathematics in my mind is one of those things that just seems to have always existed. But today (literally today) I started thinking about the other side of that assumption. Where did mathematics begin really? Is there one person who first explored and discovered mathematics? How was mathematics actually created?

The way I see it is like this:
The Bible begins with the story of creation. In fact, the very first two verses (Genesis 1:1-2) say, "In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters." This means that before God spoke the earth into creation, there was literally nothing. Just God. Then, at the voice of the Lord, over a period of six days, everything in and of the earth was formed. In the book of Colossians, it is stated, "For in him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities: all things have been created through him and for him" (Colossians 1:16). Although these passages in the Bible don't come out and directly state, "and then God created mathematics", the concept of mathematics clearly had to come from somewhere. The Bible tells us that God created all things, so does that mean that God created math as well?

Again, here's how I see it:
Mathematics isn't an object; it isn't a tangible thing like a person or an animal. But, God still created everything. God is and always has been, ever-present. He knows the details of everything of this earth; everything in it, everything on it, and everything that happens within it. He knows the specifics of all things before they happen, and He knows each and every new earthly discovery before it's made. God designed this earth according to how He saw fit so that He may be glorified. We, as humans, have not actually created anything, but have rather been given the gift of discovering the vastness of the creation that God has already so carefully constructed.

I think in this way, mathematics is something that has actually been present seemingly for forever. God created the shapes of the land, the movement of the waters, and the properties of everything in between, and while it may be interesting to learn about the great mathematical discoveries of our time, I personally think, God is the ultimate creator and arguably therefore, the ultimate mathematician. 

Tuesday, August 29, 2017

So What is Math?

As a math major, it's somewhat surprising to realize that no one has really ever asked me to answer the question, "What is math?". Whenever mathematics, or rather the idea of my majoring in math, is brought up in conversation, it seems to me that there exists a common theme involving statements such as, "I hate math", or "I understood math until letters got involved", or "Wow, I would never be able to do that", and that's the end of it. There is no effort from either side to pursue exploring the depths of what math actually consists of. So, in the spirit of tackling this newly proposed, seemingly impossible question, here's my personal take on what I believe is math.

The obvious answer of course, is that math involves numbers. But it doesn't just end there; math also includes letters, units, patterns, rules, and amazingly, things that don't even make sense to exist upon first glance, such as imaginary numbers. Mathematics also involves methods. In fact, mathematics itself can be explained as a method. It's a method of critical, deep thinking; one that involves problem solving in an effort to make a discovery. These discoveries can range from seemingly simple things, to things that completely blow your mind in such a way that not even you, the discoverer, can understand. Mathematics can also be defined by a set of calculations, the construction of a graph, or the identification of a shape. It can be used to help construct a patio, to give correct change at a grocery store, or to appropriately interpret how strongly a group of people represents a given issue. In short, mathematics is not just one thing, there is not just one definition, and there is definitely no lack of mathematics usage in the world. Mathematics is an all encompassing practice, that whether we like it or not, has been and always will be a part of our every day life.

With that being said, here are what I believe to be the five most important milestones, or discoveries, in the history of math thus far:

1. The Pythagorean Theorem
2. The Defining of Pi
3. The Identification of Patterns
4. The Formulation of Area and Volume Equations
5. The Knowledge and Use of Addition, Subtraction, Multiplication, and Division

All five of these milestones have been instrumental in determining other factors and aspects of life. Like it or not, mathematics is used everyday, and it isn't going away anytime soon.



For a fun way to see and hear more about mathematical discoveries and how mathematics is used in everyday life, the link below is a good choice!

Donald Duck in MathMagicland

Monday, December 5, 2016

My Math Teacher's Kitchen

After reading my previous blog post, you might be wondering what I could possibly be writing about for my last one. As it turns out, I've been planning my last post throughout this entire semester and even though I may not be certain that this career is right for me anymore, I've still been impacted quite a bit by a certain teacher during this time.

This year, to save some money, I decided to live with a friend that lives only five minutes away from the Grand Valley Allendale campus. This friend just so happens to be a teacher, and it also just so happens that eight years ago, I was a student in her math class. Our student to teacher relationship slowly became a coach to coach relationship, slowly transforming us into kinda sorta friends. From here our relationship became that of babysitter to parent and it was only a matter of time before that relationship turned into a strong friend to friend one. I promise it isn't as weird as it sounds.

In any case, now that I'm living in said location, it has become routine for us to get home and swap stories regarding my classes for the day and her teaching experiences for the day. Generally, for some unknown reason, this takes place in the kitchen. Several weeks ago, she (let's just call her Susan) shared a particular experience that I now want to share with all of you.

Susan teaches three different seventh grade math classes during the school day. On one Friday, after she had looked at the total number of missing assignments for each class for the week, she decided a talk needed to be had with one of those three classes. This class, just in one week, had had 39 missing assignments. The other two classes only had two and five missing assignments for the week. Here's how Susan decided to approach the situation: How many of you have ever failed at something the first time you tried it? Almost every hand in the class went up. Susan then told them a story of the first time she tried to jump-rope. The first time she tried, she tripped over the rope and bashed her face on the cement floor of her garage, splitting her chin open bad enough to need stitches. Talk about an epic fail right? (Sorry Susan). Now that she had the attention of the class, Susan kept going. She moved the focus to sports and asked the class, "What if you never practiced for your sports team? What if you just showed up to the games to play, but never did anything to practice or prepare yourself?" This question was met with a lot of "Why would you do that?" and "That would be stupid!" comments from the class. Susan then went on to relate this back to the classroom explaining that this class is the 'game' and homework and other outside of class assignments are the 'practices'. If you don't do the assignments, you're skipping all the practices and expecting to still do just as well in the game as the ones who are doing the assignments. It wasn't because the students couldn't do it, it was because they simply weren't. After Susan played this scenario out and told her students the number of missing assignments they had, it was silent. She ended it like this, "What if I had shown these numbers to the other classes? Would you have been embarrassed?" The whole class nodded yes.

When Susan got home that day, she was so proud of these students. After their conversation, the students had gotten down to business and worked hard for the entire rest of the class time. They understood now that they had the ability to do just as well as the other two classes, they had just needed some encouragement.

Sometimes you just have to show students that they can do it. Sometimes it might take a more personal story to help them see it, but when they do see it, it changes them. I think some of the most important aspects of being a teacher involve this encouragement. In this case, it involved a personal story; an opportunity to be open, to be vulnerable, and to provide a connection.

I know I'm not a teacher yet, and I know that being a teacher probably isn't where I'm going to end up anymore either, but I am still confident that these three things are something we should strive to do in any conversation. I felt the pull to do share what I'm struggling with in my previous blog post even though I didn't necessarily feel super comfortable doing it, and since then I have already connected with others who, wouldn't you know, are dealing with the same thing I am.

In the eight years that I have known Susan so far, she has taught me so much, whether she knows it completely or not. She is the one that originally inspired me to become a teacher, and despite my confusion regarding that now, she continues to inspire me in other aspects of my life. It's amazing what can happen when someone goes out of their way to make a connection, and even if I don't become the next 'Susan', I know that her lessons will continue to influence every part of my life. I only hope that no matter where I end up, I can do the same for others.

So, for the last time, I am a math teacher in the making (maybe), a fellow math nerd, and these are just some of my thoughts. Thanks for reading.

Tuesday, November 29, 2016

Confessions of a Soon-to-be Teacher (Maybe)

Being a teacher, in my opinion, is one of the most underappreciated jobs in our world. It's not that this career path isn't appreciated by anyone, it's just that the life of a teacher, everything a teaching job actually entails, is unknown by the general public, and therefore disregarded. We (as the general public) have a very basic understanding of what goes on in a teacher's life; it doesn't seem like many people understand what a teacher really does behind the scenes, unless they themselves are already a teacher or are in the process of becoming one, and this is why a teacher's job seems underappreciated.

I decided that I wanted to be a teacher when I was in seventh grade. I know it isn't typical that people already know what they want to do that early on, but for me, there wasn't any hesitation in knowing that teaching is what I wanted to pursue. When I entered my junior and senior years of high school, when others begin to ask you what your plans are for college, the responses I received about my career choice were somewhat shocking and frustrating to hear. While I did hear the occasional, "You're going to make a great teacher!", so many other times I heard, "But you're so smart, why would you do that?", or "Wait really? You could do something so much better!", or my personal favorite, "Why would you want to do that? You aren't going to make any money.".

Hearing these things started a fire in me. Why did so many people think negatively of my choice? Didn't anyone realize that teaching was so much more than those surface level things? As a middle school student, I already had the pull, the passion, the understanding that accompanies the desire to become a teacher. I had experienced these things in some of my own middle school teachers, and I had already formed the desire to give back and pour into future students, the way they had done for me. I wanted to show people that I could be a teacher, and that I could be a good one. I wanted to prove that I could be the teacher that students went home and talked about because of something exciting that had happened in class. I wanted to be the teacher that students felt they had a relationship with, not just as teacher-student, but as friend-friend or mentor-mentee.

That was me in seventh grade; passionate, confident, and determined to prove to everyone that this was where my life was supposed to go. Here's the thing though; that was eight years ago, and even though I had never thought twice about changing that path, this year has posed to be a difficult one in that regards. After working towards this career for 2.5 years now in college, I still find myself thinking about the questions above, almost more than I did when they were originally brought up in my life. For the past several months, I have had some sort of inkling that maybe I'm not pursuing the right career path anymore. It's not to say that I can't still picture myself having my own classroom, teaching math to middle school students; that picture is still pretty visible in my mind. But, there also exists a picture in my mind that doesn't include teaching, a picture that God has decided to put in my life, even though I have no idea why. As someone who had never doubted or second-guessed her degree choice, I struggle with what to do now. This new picture is blurry; it doesn't show me what else I might be doing in the future, it just doesn't show teaching to be something I pursue. So here I am, a junior in my college career, not sure that teaching is what I'm being called to do anymore, despite the fact that I felt that calling for eight years.

I'm not trying to make this post about the inner struggles of Kelsey York's life, but I do feel like these thoughts have made me more observant to the things that teachers don't generally get recognized for. Being in the education program, you get all the background information, all the stuff no one thinks about until it's staring them in the face; things like the need to care for students as if they're your children, the importance of teaching students that it's okay to fail, showing students that they aren't just a number in the grade book, the need to connect with students and form relationships, to get to know students on a personal level rather than looking down on them, the importance of meeting students where they're at, both as students and as people/children outside of the classroom; things like how to talk about politics when everyone's on edge, or how to address religious or cultural issues and how much personal input to include. These things are what make teachers who they are. These things are powerful; they show the true grit and passion it takes for a person to decide to be a teacher and these should be the things that teachers are recognized for. These are the things that I picked out as a seventh grade student, because I saw them in many of my teachers, and although I don't know for sure that teaching is where I'll end up anymore, I am still confident that these things, these connections, are what I will strive to fulfill until I figure that out.

Now I know I've jumped around a bit, but I promise I'm about to tie it all together, so keep reading. Please. Going through this struggle of not knowing what to do with my career path these past few months has been, and still is, a pretty stressful situation for me. There has always been a pull from society to know what you're doing with your life the minute you step into the college world; what's your major, your minor, what are doing with that degree when you graduate, where are you going to live, where are you going to work, and so many more. We even encourage this thought process in students who are only in middle school and high school. Shouldn't the focus simply be to learn for the sake of learning; to grow for the sake of growing; to form relationships with others for the sake of learning to maintain those relationships? Believe me, going through the 'life choices' issue as a junior in college is no fun, but that doesn't mean we should force it upon students who barely know what they're passionate about yet. In my opinion, in order for teachers to fully encompass the ideals and connections mentioned above, encouraging students to take life one step at a time is the only way to go. When college comes and it's time to decide the path you want to take from there, it's okay to not have any idea yet. The ideas will come, and until then it is simply a teacher's job to support them wherever they end up. In this way, and only in this way, can we encourage students to truly find what they are passionate about, what they're interested in, and what they want to work towards someday, hoping that those things will be made clear to them by our actions, not our teachings.

Sorry for the jumbled thoughts, but life is jumbled sometimes anyway. In any case, I am a math teacher in the making (maybe), a fellow math nerd, and these are just some of my thoughts. Thanks for reading.

Sunday, October 30, 2016

Redefining Failure & Remodeling Homework

"The biggest mistake you can make is being afraid to make one." I pulled this quote from a book called Mathematical Mindsets, a book that is designed to help teachers understand how they can assist students in finding their potential [in math]. One of the things that this book talks about is the importance of making mistakes for all students. We live in a society where we look down on mistakes; we want to do everything perfectly the first time, and if we mess up, we are automatically embarrassed. In regards to school, I think this is definitely most prevalent in math classes. When a student volunteers their answer in class and is wrong, it is more common for a teacher to simply tell them they are wrong and dismiss them without explanation or praise for trying. Teachers check homework for correctness and deduct points for wrong answers, but generally don't tell the students what they did wrong, not wanting to repeat lessons, hoping that their students will eventually figure it out on their own. Doing these things definitely doesn't encourage mistakes, and because of this, students remain nervous and afraid to volunteer responses in class.

One of the biggest outcomes of teaching in this way is that of stress. Obviously there are many things in life that cause stress, but I think in regards to school, our expectations of straight-A, mistake-free students is a huge factor as to why we as a society are so stressed now. Like I said above, we are a high-achieving society, drawn to be the best of the best in all that we encounter. While it isn't necessarily a bad thing to have the desire to do well, it is unhealthy to instill the mindset of not wanting to make mistakes on students immediately upon starting school. Mistakes should be something that students are encouraged to do, in every aspect of their lives, to ensure learning and the ability to do better the next time. Striving to do our best will only come from first making those mistakes, followed by making the connections to discover what went wrong. In my mind, the stress formed at school never goes away; it continues to grow as we continue on into college, then into the real world and the work force. The stress of not wanting to make mistakes in school carries into the rest of our lives, throwing us into an already fast-paced society, that is now one where you're required to be free of mistakes as well. There's always the fear of disappointing someone with your mistakes, of embarrassing yourself in front of other people, and on top of that, the stress of time. All because of the stress that is piled upon us as students.

So what does all of this have to do specifically with homework? Well, I think another idea we have stuck in our heads is that repetitive homework problems is the best way to make sure our students don't fail, especially in a math class. Generally, students in math classes get assigned an insane amount of problems to do from their textbook for homework, every single night. There has been a lot of debate about whether or not this is an effective way to give homework, or whether homework is even necessary at all. In my opinion, the process of learning requires variation more than repetition. In one of my previous posts, I mentioned the idea of memorizing vs remembering. By using repetitive structures such as homework problems assigned from a textbook, we are encouraging the idea of memorizing. This method of teaching leaves nothing for the students to grasp onto, it simply implements the stress of needing to pound information into your head before you can finally forget everything after the end of year. Remembering is encouraged by incorporating more assignments or activities that fall along the lines of 'out of the box' thinking. It's important to try new things, to allow our students to experiment with their hands before being given the full tools to solve a problem. In using this approach, students find a more comfortable, less stressful environment, and can leave with a memory rather than a fact. I can't help but think that if we, as teachers, started including activities that encouraged experimentation, there would be less stress on students to feel the need to be right, and therefore, less of an emphasis on incorrect work. In this way, I think we can slowly start to encourage and show students that failing or being incorrect is definitely not a bad thing, but in turn, it actually helps guide the way to understanding.

To close I want to go back to Mathematical Mindsets. Carol Dweck, a psychologist and fellow writer says, "Every time a student makes a mistake in math, they grow a synapse." So am I saying that students should just decide to not try? That they should just purposely fail at everything they do in an attempt to grow? Of course not. Believing in yourself is still incredibly important in the development of each student's mindset. In my opinion, the key to tying failure and believing in yourself together, is finding that balance. It's not specifically one or the other that helps the brain grow, but the idea that when you believe in yourself, you can fail as many times as you need to, because you also know that at some point, you'll succeed. Jo Boaler, author of Mathematical Mindsets, claimed that the people in our world who are the most successful, have made the most mistakes on their way to achieving where they are know. Instilling this knowledge in our students and showing them that the value of correct work is much less important than the value of mistakes is a great place to start. I can only imagine how much more our teachers and our students could change the world if we work together to achieve this new mindset.

Note: I am definitely not trying to generalize and say all teachers are guilty of making their students feel this way. I am saying that this might not always be noticeable and that it might not be a bad idea to think about how current teaching methods are affecting students. I am simply calling out a problem that I have seen and experienced in an effort to help change the stereotypes about failure.

So, as always, I am a math teacher in the making, a fellow math nerd, and these are just some of my thoughts. Thanks for reading.

Sunday, October 9, 2016

Observations from the Outside

Of all the different 'in-school' experiences I've had since deciding I was going to pursue a teaching career, I had never gone to a school for student/teacher observing up until a couple of weeks ago. As a requirement for a class right now, I needed to find a teacher who was willing to let me come observe his/her classroom for several days throughout this semester. As someone who is from the general area, I decided it would be easy to go back to my high school and observe a teacher there. I ended up asking a math teacher who I had never had while I attended high school, although I took the class that he is teaching. Mr. K is a math teacher at Grandville High School and teaches Pre-Calculus. Because I had never had the experience of having him as a teacher, I thought this would be a good route to go, as it would give me a new perspective that I had never observed before. My younger brother also happens to be in one of Mr. K’s classes right now and really likes the way he teaches; another reason I thought he (Mr. K) would be a good choice. 

One of the first things I noticed about Mr. K, was that he was a very genuinely nice person. Although we had never met, he was more than willing to help me out, and I didn’t feel uncomfortable at all while sitting through the class. Throughout the class time, you could tell that he wanted his students to have more of a personal connection to him than just that of “I am the teacher, you are the student”. He wanted them to enjoy his class, not dread it. This was also evident in the way the students acted with him. They were plenty comfortable with asking (and answering) questions, but they were also plenty comfortable with joking around with him and him joking back. I think the relationship that a teacher has with his/her students is one of the most important things for a teacher to establish. In some cases, students see their teachers more often than they see their parents, and because of this, it is important for there to be a positive connection, or at least the opportunity for a positive connection, between teacher and student, and I thought that was great to see in Mr. K’s classroom.

Mr. K’s general schedule for the day included time for questions about the homework, time for his lecture, time for questions about the lecture, time for example homework/practice problems, and time to work on the homework in class. I thought this was a great way to organize the time, in an effort to ensure that all his students understood what went on in the previous class period, as well as what is going on in the current one. During this class period, Mr. K just so happened to be teaching “Polynomial Long Division”, which we reviewed not too long before in our 229 class. It was interesting to be able to compare things we talked about in class to what Mr. K was implementing into his classroom. One of the things he did to begin this lesson was do a warm-up problem with synthetic division. I found this especially interesting, because this was not something I ever learned. The first time I ever saw synthetic division played out was in our class, this year. So it was interesting to see not only how he used concepts we talked about in our class, but also to see what has changed slightly since I had attended this high school. When Mr. K actually started the process of explaining polynomial long division, he followed the generic process that I have always been taught as seen below (we also did this in class). In any case, I thought he did a great job of explaining the method and process of how he obtained an answer. After explaining the first one, he took questions and then moved on to another example, explaining again very thoroughly. He repeated this until there were no more questions.


Another thing I noticed during this observation was that Mr. K didn't plan his lecture to last the entire class period. I think sometimes as teachers, we feel the need to fill the entire hour with the lecture, scrambling to assign the homework one minute before the bell rings, and therefore not allowing students to ask questions about the homework before they bring it home. Each of these classes at Grandville are only an hour long, and somehow Mr. K still had plenty of time to review questions from the previous night's homework, as well as to allow his students to work on their homework problems in class. Seeing this was, in my opinion, a really impressive way to see how Mr. K organizes his class time to ensure that his students were going home confidently, rather than confused.
Overall, after just one hour of observing so far, I am very impressed with the way that Mr. K approaches his classroom, and I am excited to see what I learn from him throughout the rest of this observation time. As someone who has always been on the inside, experiencing these things while they're happening, it was great to now have the opportunity to observe on the outside, looking in.

With that, I am a math teacher in the making, a fellow math nerd, and these are just some of my thoughts. Thanks for reading.