Exploring the World of Mathematics: An Eight Week Curriculum

This semester my capstone math class started off with the question of, "Is mathematics discovered or invented?". As a class, we didn't spend a whole lot of time discussing this, but it stayed a common theme throughout the entire semester, and this is what sparked the idea for my final project.

Working together with Lauren Grimes, a friend of mine, we decided that this question was one that should be addressed in mathematics classrooms. This question could open up discussions and explorations into the world of mathematics and could even help students figure out how mathematics is applied to things in their personal lives. Therefore, with this in mind, we set out to create a curriculum that would form around this question.

The final curriculum we created resulted in an eight week unit of one day a week, for a time ranging from 30-60 minutes each day. This curriculum is made up of six different lessons that are focused on a variety of people, objects, ideas, and passions and create a way for students to see the ways that mathematics is used outside of the classroom. The curriculum is centered around these questions,

"Did we, as humans, create mathematical concepts to help us understand the universe around us? Or
is math the natural language of the universe itself, existing whether we find it or not?",

and the main goal is that students would form their own opinions and beliefs about mathematics based on mathematical history, facts, and discoveries that pertain to their interests and passions. After the six weeks of lessons, which include personal research and exploration for each student, the seventh and eighth weeks implement time to bring together all the things discussed, to determine an argument for whether mathematics was discovered or invented. To further show what I mean by this, here is what the general layout for a given lesson might be:

Welcome/Overview

Warm-Up

Introduction

Individual Activity

Group Activity

Discussion

Journal

 

The final two weeks, as mentioned previously, would be structured in a different way, allowing time for students to formulate solid opinions and beliefs during the seventh week, and allowing time for a full class debate during the eighth week.

After reading this description, it might seem as though this curriculum is tackling some big questions that involve a lot of work. However, while the theme of this curriculum is definitely a big topic, Lauren and I wrote this curriculum in such a way that we feel helps students to be excited about learning and discover meaning within mathematics. There are many different ways that this curriculum could be implemented, and there are several different topics that could be discussed within this curriculum. Our hope is simply that, if used in a classroom, this curriculum would produce an outcome that positively affects students in their mathematical knowledge, understanding, and appreciation, no matter where they fall on the 'discovered or invented' debate.

I know this post only provides a brief overview of what our complete curriculum entails, so I know there may be many unanswered questions. With that being said, I am more than willing to share a copy of our curriculum with anyone who desires to read and look through it, and we of course are open to comments and suggestions about how to make it better. I believe that if this curriculum eventually can be used alongside regular mathematics courses in schools, our students will develop a deeper understanding and appreciation for how mathematics is used in our world.

A Year In Review

Almost exactly a year ago I wrote a blog post titled "Confessions of a Soon-to-be Teacher (Maybe)". I reflected on where I was in my life at that moment and talked about the struggles of determining whether or not I wanted to continue with a degree in education. Today I'm looking back, reflecting on that post and the past year.

At the end of last year's fall semester, I was ready to be done with school. I had hit a wall and was feeling broken down. I was completely unsure of what was going on in my life regarding a future career, and I had come to the point where I thought the best thing for me was to just be done. I was burnt out. Over Christmas break I had time to unwind, clear my mind, and think seriously about what I felt God was leading me to do.
In the end, I started the winter semester having changed nothing, but feeling at least slightly rejuvenated from the long break. My mindset didn't completely change of course, but I was able to adapt my mindset to that of being confident that this was where God needed me for the time being, meaning in school to pursue a degree in math education. After a wonderful summer, I entered this current school year with that same idea in mind; that God wasn't calling me to be anywhere else right now, that He wasn't directing me away from this journey, and that I was going to graduate with a degree in education.

Although this year has proven to be difficult on multiple levels (senioritis is a real thing friends), I still have been holding on to this reality. I've hit some bumps over the past few weeks and had a few very real moments in which I struggled to understand why certain doors were being closed, but every time, I've been redirected to focus on God's plan. As teacher assisting continues to get closer, I am reminded that God has me in this place for a reason. This season of life is one of challenge and doubt, but it's also one of faithfulness and change.

I recently received my placement for teacher assisting next semester, and although I still feel nervous and unsure, I also have been given a feeling of peace knowing that everything is in God's hands. I have been put here, placed in one specific school, because it's a part of God's plan for my life. God has given me the gift, ability, and desire to work with children, to teach, and to use my knowledge of mathematics to help these children grow.

It's been hard to keep an open mindset about the direction my life might take, and it's been hard to accept that following this path means giving up other things. However, I am confident that if I embrace wholeheartedly this path set before me, God will open up new doors and help me recognize everything there is to celebrate in that. If God wants me to teach math at the end of graduation, doors will be opened to make that happen, and I will be there ready to embrace it if it does.




If you want a look at what I wrote last year, here's a link to that post.
https://missyorkinthemaking.blogspot.com/2016/11/confessions-of-soon-to-be-teacher-maybe.html

A Look Into the Purpose of Teaching Mathematics

I am in my seventh semester of college, and I have hit a wall. I'd like to say that I've been doing well this entire semester up until last week, but the truth is I've been feeling 'done' with college before this school year even started.
Proven by many recent conversations, it seems to me that many people are feeling the same way right now, probably due to the obnoxious weather changes and the fact that we don't get a fall break (hint hint). Something that has come up during many of these talks is the frustration with certain classes that are required, mainly, as you may have guessed, mathematics courses. Most, although not all, of these conversations have been had with individuals who are not mathematics majors, which made these frustrations even more interesting to explore and has made me begin to think further about my own opinions on the subject. Overall the questions were posed: why are mathematics classes required for (fill in the blank) major? When will I ever need to use this in my field? Why can't I focus on classes that are specific to what I want to pursue?

Let's rewind back to middle school and high school days. In those years it was expected to hear these questions asked in a math class probably multiple times a day. But the answers given by teachers then were always in some regard to a future career, a 'life' reason, or a vague reason about connections to later math courses that essentially avoided the question all together. Now, in college, we're pursuing those careers, we're dealing with real life, and we're taking 'later' math courses, and people are just as confused as ever. Here are a few comments I've heard from some friends recently:
"I'm a dance major; why do I need to take any math!?"
"Tell me why I would ever need to know more than simple math as a nurse! Shouldn't I be more focused on things that will directly apply?"
"I'm an elementary math major. I'm never going to teach anything remotely close to this!"

I don't mean to pick on mathematics of course, this could be applied to any other subject as well. A friend of mine looking to be a nurse, who was studying for a biology exam recently stated, "I don't understand what good these classes are for me either. Until I get into classes more specific to the nursing program, all that's happening is studying like crazy for an exam and then forgetting everything I supposedly learned." For a future doctor or nurse, an individual is not going to look back during an emergency situation and attempt to use the knowledge gained from a 100 level math class they took. Nor will they even look back to try and remember the facts read from a textbook for a 200 level biology class. It's the hands-on, action based experiences that are going to make an impact.

In this same way, it seems a little over the top to me to have a student wishing to be a future middle school mathematics educator to take a class like Calculus 3 or Complex Variables. When describing a degree in education, it's always said that education is the degree and the content area is the emphasis. Shouldn't this mean that college classes should be focused more on teaching than on content? These difficult classes in college highly exceed any level of mathematics that an elementary or secondary education teacher would need to know, and introduction to actual teaching isn't really a focus until the final year. This means that while these students work to grasp somewhat insane concepts, the knowledge and memory of middle or high school concepts that will need to be taught, is decreasing, forcing college of education students to scramble up lost knowledge when thrust into the busy life of teacher assisting and student teaching.

Thinking about teaching mathematics with this in mind makes me wonder what might change if the way math was taught or the concepts required to pass for a given math class were edited to place an emphasis on careers. In high school very few people know what they want to pursue in college, and people usually don't start thinking about it in depth until their junior or senior years. However, what if high school became a place where mathematics courses could be a way to help these people explore potential careers? What if mathematics courses were only a requirement for two of the four years of high school, and the empty block the following two years was able to be filled with classes more specific to what each student might want to pursue? What if the general education requirements in college weren't given as much emphasis and students were able to begin exploring their future career more quickly?

I think students in all grades would succeed and achieve more if their mathematics courses were implemented based on interests rather than complexity and grade level. All students learn and grasp concepts at different levels, so why not allow all students to determine what kind of mathematics they enjoy and use that to further their education?
Again, I don't want to lay all the blame on mathematics, but math is the subject that more often than not is recognized as the one that students dislike. There may not necessarily be a successful way to make changes as I mentioned above, but as a potential future teacher, and for anyone interested in students' learning, it's definitely something to think about.

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.

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.

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