During the past few semesters, I have done a lot of work in design thinking. For my final project for MTH 495, I decided to follow the design thinking process.
Before I began the project, I had to consider the ‘users.’ For this project, the users were my fellow MTH 495 classmates who are all math majors in a class about the nature and history of math; and myself, a fellow math major and an art enthusiast.
After considering my audience, I decided that I wanted to create a painting of a timeline collage of the history of math. I wanted to work on a project that I knew I would enjoy working on, and that I could connect to the content of the course.
For my project, it made the most sense for me to combine the ‘Ideate’ and ‘Prototype’ stages. I reflected on what concepts that we discussed in class I could include in my painting that would create visual appeal. Then, I created a sketch of each of these concepts. I cut out each sketch and stuck tape on the back so that the pieces would be easy to manipulate into possible prototypes. Here are some of the prototypes I came up with:
I spent the majority my time working on my project at this stage. To begin, I drew the outlines of each of the shapes on canvas, using a ruler as a straightedge. Then, I consulted with an artist that Professor Golden connected me with on Twitter (Paula Beardell Krieg) on how I might create a sense of direction in my painting. She suggested that I create a sense of movement through time by altering both line and color. I decided to create the illusion of direction in my visual timeline by gradually increasing the weight of the lines from left to right, and also by gradually darkening the color of the shapes from left to right. My goal in varying line and color was to help guide the viewer’s eye through the timeline.
Once I filled in the line and color of each object on the canvas, what was left to do was to create the grid in the background. At first, the grid occupied only half of the canvas, stopping abruptly in the center of the canvas, which created a feeling of disconnect between the left and right sides of the painting and ultimately detracted from my goal of creating direction. After brainstorming with my roommates, who are graphic design majors, we came up with the idea of adding a “fall-off” effect to the grid on the left side of the painting. Not only does the ‘fall-off’ effect connect both sides of my painting, but it also reflects the idea that math has evolved over time.
Here are some pictures I took over time as I worked on the painting:
The Final Product:
The concept of infinity has not always been as widely or easily accepted as it is today.
The Pythagoreans believed strongly in Number, and thought of ‘finite’ quantities as good and ‘infinite’ as evil.Other Greek mathematicians considered infinity as a possibility, as it described the endlessness and unboundedness of time and space. Since you could add one to a really big number and get a bigger number, they considered ‘potential’ infinity, which was allowed only in dealing with finite numbers. They hesitated to consider a ‘continuous’ infinity because they could only really quantify known. So at this time, the modern concept of infinity was not accepted because it describes the unknown.
A few centuries later, early European mathematicians reflected on the concept of infinity, only in the context of religion, to describe the influence and ‘unlimitedness’ of God.
The modern concept of infinity was not established until the emergence of Calculus. Georg Cantor, an esteemed mathematician who contributed to the fields of Number Theory and Calculus, is deemed as the first mathematician to really understand what infinity means and gave it a more precise definition. Cantor sad that if you can add finite numbers like 1 and 1, and 50 and 50, then surely you could add infinity and infinity. Instead of thinking of infinity as a really, really big number, he realized that there are infinities even larger, more infinities even larger, and so on. This idea forms the basis of the argument for why infinity is a number. We know that for every number there exists a number that is even bigger and even smaller. Cantor had a similar idea about the concept of infinity- there can be bigger infinities and smaller infinities. Cantor’s concept of infinity pushed the boundaries of mathematics, and made it possible for future mathematicians to study more abstract fields of mathematics involving the concept of infinity.
What does 'Infinity' Mean to Us Today?
Today, we think of infinity as a way to describe the abstract notion of “something without any bound or larger than any number.” Infinity is viewed as something ‘numberlike’ in the sense that it can be used to count or measure things “in infinite terms”, however it is not a number like the real, imaginary, complex, or irrational numbers.
My impression of infinity is that I understand the argument for why it is not a number. I think of ‘infinity’ as a word we use to describe something, like time and space, which go beyond the concept of Number. When we think of really, really big numbers that we cannot possibly quantify or plot on a number line, we call it ‘infinity.’ Infinity is a boundless quantity without a beginning or end. I also think that the way we interpret infinity changes based on the given mathematical situation.
After further reflection, I also understand the argument for my ‘infinity’ is a number. My understanding of a number is that it is a measure of quantity, or a way of describing “how many” or “how big” a given set of mathematical objects is. To gain a true understanding of number, we must consider the three aspects of a number: quantitative, verbal, and symbolic. In other words, a number is “a mathematical object used to count, measure, and label.”
Considering this definition of a number, I can see why infinity must be a number. In theory, we can count the number of objects in an infinite set of objects. We know that the size of that set is ‘infinity”. We also know that the measure of a line of infinite length is ‘infinity.’ Furthermore, since we can measure and count an infinite value, we can also label mathematical objects with ∞. Therefore, because infinity has all the qualities of a number, it must be a number.
In class, we talked about how women in mathematics often face bias, and their contributions in mathematics are often unrecognized. This discussion prompted me to reflect on my own experiences with facing bias in math class and also made me want to do some more research on gender bias in the classroom.
The following is some research I collected about gender bias in the classroom, from http://www.education.com/reference/article/gender-bias-in-teaching/:
Our bias against girls in school is often unintentional, as teachers often do not even realize they are projecting bias toward girls. A study on gender bias in the classroom done in 1987 (Tobin & Gallagher) found that the students who dominated the teacher’s time by asking and answering questions were primarily white and male. In the sciences, the study found that teachers also tend to ask boys harder and more complicated questions, as opposed to girls. Though such behaviors are unintentional, when teachers do this they are subliminally sending a false message that boys are smarter and more capable than girls when it comes to math and science.
The stigma that boys are better than girls in math is a stigma that I have faced personally during my journey as a math student. Many people that I meet appear surprised when they inquire about my major and I tell them that I am studying math. They just assume that I am studying Language Arts or Social Studies, because I am a woman and there are not very many female elementary math teachers.
As an elementary student, I really struggled in Math, but I told myself that it was okay because I am a girl and girls are not inherently good at math. If I continued to carry this mindset with me, I would not be the scholar and lover-of-math that I am today.
Gender Bias in History
There were many female mathematicians in the past who faced discrimination in the field of mathematics, just as I have as a student. One such mathematician was Mary Fairfax Greig Somerville, who lived from 1780-1832. Somerville grew up during a time when women weren’t allowed to be educated. She had two brothers who went to school, but her parents saw no need for her to be educated since she was a girl. As a young girl, the only education Somerville received was from her Mother, who taught her how to read, but did not teach her how to write. Later in her childhood, Somerville attended a boarding school for girls, but eventually left because she was unhappy and didn’t feel that she was getting a good education. Soon, she began to educate herself by reading any book she could get her hands on, but she was often ridiculed by her family members, who considered the act “unladylike” and felt that it was silly for a girl to be reading. Somerville developed a passion for mathematics through study of “Euclid’s Elements” and later published several books and papers about Math and Science. She became a supporter of Women’s Education and Women’s Suffrage, and is one of the greatest female Mathematicians in history.
Sommerville’s story is one I identify with both as a fellow woman and lover of math, as it shows the struggles many women had to overcome in the pursuit of their passion. Her story shows how strong women can be in the face of bias and adversity, and also exemplifies how important they really were in the history and development of Math.
A Message for Teachers
As teachers, we need to be more aware of these biases against women in Math and Science, and be mindful that we aren’t passing on the stigma that girls are not as smart as boys in these subjects. We need to be more mindful of what we say and do, of our body language, of how we group students, who we call on, the tasks we assign, and of the hundreds of other choices we must make in the classroom every day. It is our responsibility to empower all of our students to be successful no matter their gender, heritage, past experiences, or other backgrounds.
Overall Review & Reflection
Love and Math is a really good story about the struggles of being a mathematician, and it is also about the author’s own struggle with facing antisemitism in Russia during the Cold War. Frenkel was originally interested in studying only all things Physics, but a Professor at the University took Frenkel under his wing and opened his eyes to the beauty of mathematics. In his book Love and Math, Edward Frenkel aims to show his readers the true beauty of mathematics and how seemingly different concepts of math are actually connected in some way.
My favorite part of this book was Frenkel’s effort to make the book relatable for readers from any background. What I really liked about this book was that whenever possible, Frenkel used an analogy to make the idea or definition relatable for readers. For example, one of the analogies from the book that I really liked was Frenkel’s analogy that illustrated the concept of one-to-one correspondence. He said that if you have 5 pencils and 5 pens, then there is a pencil for every pen. This analogy used familiar objects and a visual of placing a pen and a pencil together, instead of just including the technical definition of the concept, which is what made the concept more accessible to readers who do not have a math background. At the beginning of the book, Frenkel also includes a warning about the chapters that are really math-heavy and cover the more abstract ideas. He told readers to feel free to skip those chapters, which I found funny but also useful advice.
I selected this book because I too have a love of math, and I was hoping that this book would show me more things that I could love about math. What I found out after reading this book is that the math that I love is much different than the math that Frenkel loves, and that is okay. There were parts in this book that I found very challenging, even as a math major. Reading this book made me realize that the math that I know seems very limited, and there is so much more out there to explore.
Connections to Teaching
In the book, I noticed many themes that relate to my future career as a teacher. For example, at one point in the book, Frenkel discusses about his newfound appreciation for teachers after acting as a mentor. He says, “It’s hard work being a teacher! ...You have to sacrifice a lot, not asking for anything in return,… [but] the rewards can be tremendous” (129).
Another theme that I noticed in the book was collaboration. Collaboration is a really important idea because so many people believe that Mathematics is an independent art. Frenkel attributes his success to collaborating with peers, mentors, and other leaders in the Math community. He admits that a lot of what he learned was learned by communicating and sharing ideas with others. This is a sentiment that I want present in my classroom. I want my students to feel welcome to share their ideas and learn from their peers. I can teach my students about all of the math that I know, but I think that they will benefit most by sharing and understanding new ideas from their peers.
Another idea from the book that I found really useful for teaching was the idea that different people have different interests. Frenkel mentiones that each of his mentors and colleagues were mathematicians, but they were each interested in different areas of math. As teachers, we need to recognize that our students are different, and they each bring in their own experiences and interests into the classroom. I think that striving to really get to know our students and trying to attend to each of their different interests can make learning more engaging and enjoyable for all of my students.
Overall, this was a really great read, though a bit difficult to follow at times.
What is a dimension?
In geometry, a dimension describes measurement and direction of an object. We can measure an object using coordinates, standard units, or nonstandard measurements. When we talk about the direction of a dimension, we refer to length, width, height, and the fourth dimension. The number of the dimension (i.e. the 2 in 2-D) refers to the number of coordinates needed to identify each point on the object. For example, a point on a 3-dimensional object would be named by three coordinates to indicate the location of the point in each direction (height, length, and width, etc.). (http://mathworld.wolfram.com/Dimension.html)
Each dimension can be extended into the next higher dimension by “stacking.” For example, to expand a 1-Dimensional space to achieve a 2-dimensional space, you can “stack” 1-dimensional spaces, then to expand a 2-D space into a 3-D space, you “stack” 2-D spaces, and so on. (http://www.maa.org/meetings/calendar-events/the-fourth-dimension)
What are the different types of dimensions?
Geometry in the fourth dimension still follows the rules of Euclidean Geometry in the third dimension, but is extended to one additional dimension. When we describe objects in the fourth dimension, we often add the prefix ‘hyper-’ to the name of the object (ex: hypercube, hyper sphere, etc.) (http://mathworld.wolfram.com/Four-DimensionalGeometry.html)
An example of a Tesseract:
Why do we study dimensions?
The study of dimensions in math is relevant to our daily lives. For example, we apply knowledge of the 0-Dimension when we use coordinates to find an address, city, or place on a map. When we describe how to get from place to place when we are driving, we use 1-dimensional geometry. We use 2-dimensional geometry when we calculate the area of a living room to determine how much carpet we need to buy. We study the 3rd dimension because every object around us has three dimensions. In regards to geometry in higher dimensions, it is much harder to come up with examples of everyday applications, except for use in physics (http://www.math.brown.edu/~banchoff/STG/ma8/papers/anogelo/hist4dim.html).
We study dimensions in mathematics because the concept of ‘dimension’ helps us better visualize or conceptualize “the complexity of any geometric object.”
Studying the dimension of any given geometrical object, either concrete or abstract, gives an idea of the 'size' of the object or the amount of space it occupies. Once we know how big an object is, we can begin to compare it to other geometric objects. When we study dimensions in math, we gain a better understanding of the qualitative properties of geometric objects. In studying dimensions, we gain insight to the topological properties of geometric objects, but we also gain new perspectives that help us better understand the world around us.
As teachers, we introduce students to the concept of dimension when we teach students about shapes; how to plot points, lines, and segments on the coordinate plane; how to calculate volume and area, etc. Here is a link to some of the specific standards of geometry that allude to dimension:
Here are some cool manipulatives you can use in the classroom to help teach students about dimensions:
Some examples of manipulatives on this website are geoboards (or XY coordinate pegboards),
geosolids, folding geometric shapes, etc.
When we worked on the Greek timeline in class, one of the developments in math that really caught my interest was that the Sumerians ‘invented’ zero as a placeholder around 300 B.C., but they did not consider it as a number (http://www.ccsdk12.org/mclemens/Projects/mathhist.htm). I found this fact very interesting because it made me wonder why the Sumerians did not regard zero as a number, and also what practical use for zero they had. Zero has always been such a mysterious number to me, so I decided to do a little digging to find the answers to some of my many questions about the number Zero. Here is what I found:
Zero has many functions in mathematics and number sense. To name a few, the digit “0” is used as a placeholder to distinguish between 1, 10, 100, etc., and it is also used to measure the length of a point and serves as a reference point for distance on a number line. Zero makes it possible for us to subtract a number from itself (ex: 4-4), is the additive identity, and also plays an important role in set theory as it is the cardinality of the empty set (http://www.ams.org/samplings/feature-column/fcarc-india-zero).
Zero was not always considered to be a number as it is today. In the past, Zero was just seen as a placeholder and was not regarded as a true number. There is some debate in the math community about who created zero and when, and the digit appears in writings in many different cultures, namely in the fertile crescent, India, during the time of the ancient Mayans, and also during the time of the Sumerians. It is difficult to really tell when the first zero appeared, because there is not much of a surviving written record, and a lot of math was part of oral tradition (http://yaleglobal.yale.edu/about/zero.jsp). It wasn’t until around 1200 during the time of Fibonacci that Zero became the number we regard it as today.
In class, when we debated what the definition of a number is, many of us considered a number to be a symbol that represents a finite distance from zero. I think that the reason that zero was not so widely accepted as a number in the past because it represented the absence of a number. Numbers were commonly used for measurement or for counting money or possessions (http://www.scientificamerican.com/article/history-of-zero/). How can we measure something or count something that is not there? Zero is the answer. We can count the number of coins we have in our pockets, but we can’t count them if we don’t have any. Since zero is not concrete like the other finite numbers, I think it is the number’s abstract quality that makes us question whether or not it is a true number. I also think that the definition of a number as a measure of distance also extends to zero because zero is also a measure of a distance of zero from itself, and it also describes the distance between every number and itself (ex: 0-0=1-1=2-2=3-3=4-4=…=0), therefore zero is a number.
Part of being a Mathematician is being able to solve, represent, or understand a problem in different ways. After solving the Diophantus riddle algebraically in class, I wondered if there was a way that I could represent the problem visually so that I could better understand it. From a teacher viewpoint, I know that each student learns and understands differently, so some of my future students might better understand a problem like this if it is represented visually. In some of my teaching classes, when dealing with fractions, we have worked with visual models of fractions like arrays. I worked through the Diophantus riddle again and found a way to represent each part of the riddle visually in an array.
'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'
The first thing that I noticed when I worked through the riddle again was that I could establish a common denominator of 84 for the fractions 1/6, 1/7, and 1/12, since the least common multiple of 6, 7, and 12 is 84:
Once I established a common denominator of 84, I knew that I could represent each of these fractions visually in a 7 x 12 array, which makes 84 equal-sized units. In the model below, I represented each value in the problem in a different color:
Since Diophantus was 84 years old when he died, based on this model, we know that:
When I was in Elementary, Middle, and High School, I thought that math was all about solving dozens of addition, subtraction, multiplication, long division, fraction, or trig calculations per night, using the same method my teacher showed me in class. I was not the best math student, and I remember feeling frustrated every night when I looked back at the notes I took in class and still couldn’t understand my homework. I thought that math was about finding an answer, and the answer could either be right or wrong, and there was no in-between.
When I came to Grand Valley, I chose to study math because all my life, it had been some big scary monster that I could never really understand. I wanted to overcome my fear and hatred of math and finally get some answers to those things about math that were always left unexplained. During my studies here at Grand Valley, my hatred and confusion of math turned into passion and curiosity. I realized that math is not all about numbers and finding the right answer-- for me, math is all about the process, not the answer. In the past, I believed that there was only one way to find the right answer to a problem, and that was the way that my teacher showed me how. All of my professors here at Grand Valley challenged me to find my own solutions to problems by approaching a problem in different ways, and they encouraged me to collaborate with my peers to find many different solutions to the same problem. Now, I think that math is more about solving problems creatively than about crunching numbers. Real math goes beyond pencil-and-paper and extends out into the real world. It is what we use to analyze and describe patterns and problems we observe around us. Math solves problems. Math challenges. Math creates. Math inspires.
Although I’m a mathematician, I admit I don’t know much about math history. What I do know about the history of mathematics is that math has been around for a very, very long time, and it is responsible for some of humankind’s greatest accomplishments. Without math, the Egyptians would not have been able to design the pyramids, and ancient astronomers wouldn’t have learned very much about the Universe. Mathematics has always been a tool that people have used to help them make sense of the world around them, and it is a tool we all use in our daily lives today. I think that the most important development in the history of mathematics was the modern number system, because it made math universal. Some of the famous mathematicians whose work I am familiar with include Euler, Euclid, Hippocrates, Fibonacci, and Pythagoras. I am really looking forward to learning more about the history of mathematics in this course.