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.