Through these blog posts, I have written about a variety of topics revolving in some way, shape, or form around math. Whether it be a book review, a personal story, or an in-class program, I have shared some insight on what mathematics means to me. Today, I want to discuss what mathematics meant to another woman, one considered a "revolutionary mathematician."
pursued her mathematical studies.
Before we dive into those studies, lets take a look into Germain's upbringing. Sophie Germain was a home-body. She was known as being with-drawn as a child, and while her siblings went off and got married in time, Sophie Germain remained single and lived at home all her life. Because of her shy characteristics, Germain spent countless hours growing up reading, one of her favorite authors being Archimedes. At the age of thirteen, she was fascinated by his mathematical findings and the fact that he lived a life "untouched by the confusion of reality." Unfortunately, Germain's parents were not encouraged by the with-drawn behavior, especially the fact that most of her time was spent studying...
She studied mathematics on her own, and Libri relates that her parents were so opposed to her behavior that she took to studying at night. They responded by leaving her fire unlit and taking her candles. Sophie studied anyway, swaddled in blankets, by the light of smuggled candles.
Clearly the devotion started at a young age and continued on as she continued to explore a variety of "current" mathematicians during that time. She spent time exploring number theory for a while corresponding with the well-known mathematician, Carl Friedrich Gauss. She created an open way of communication sending him her ideas surrounding number theory. "He was thrilled to find that his "pen pal" was a very gifted woman," however after Gauss moved on from number theory, Germain was left in need of another mentor. From number theory, Sophie Germain went on to study applied mathematics. Germain spent years working on a project that offered some aspect of competition. Germain was not granted success right away, but rather had to fail many times before finally taking the win. Once again, her commitment and devotion to the realm of mathematics is admirable.
Sadly, the life of Sophie Germain was cut short at the age of 55 after a long, hard battle of breast cancer. Unfortunately, a lot of Germain's work was not recognized when she was still in the land of the living. However, her legacy continues to live for her development in the world of mathematics and her work ethic.
how can we learn from sophie Germain?
Sophie Germain had passion, had commitment. She wasn't going to let anything stop her from pursuing what she wanted. While my passion may lie in the realm of mathematics, my other passion lies within a classroom of students. I can learn from Germain's commitment. My commitment will be to my students. I will not let anything stop me from accomplishing my goals as their teacher. Just like I admire the strength, commitment, and passion of Sophie Germain fuel my own desires, I hope that my students will look and see the same in me, their teacher, and strive for success in their own dreams, whatever they may be.
Have you ever heard of Daily Math Routine? In my current Teacher Assisting placement, we conduct the Daily Math Routine on a (as the name implies) daily basis. Are you familiar with this routine? If not...stop and watch the video below to watch a second grade class conduct this routine.
Math Expressions, the program used within my Teacher Assisting placement, has created this routine. As you saw, the Daily Math Routine is broken down into five different elements all of which give students insight to number sense.
At the beginning of the year, twenty-two second grade students arrived. With a class of six, seven, and eight year olds, the beginning of the year was spent setting expectations, building routines, and creating a classroom community. Daily Math Routine is one of these routines. Now, over three months into the semester, the students still engage in this ten to fifteen minute routine on a daily basis. The question to consider: Is it worth it?
The answer this, let's break this routine down to the five different "segments:" 120 chart, finger flashes, money chart, number path cards, and secret code cards. As students conduct these routines, they are expected to grow more and more familiar with the way in which they visualize numbers.
a card. As you can see, the three number options a student can draw is a 5, 6, or 7. After the card is chosen, Sue asks the class, "What is my new number?" The class responds, in which case, she asks the follow-up question, "Will I make a new ten?" Circles are made, equations are written, and a new number is deemed.
A new student then leads the class through the money flip chart. Ben, a pseudonym, will ask similar questions like, "What is my number?," "Will I make a new ten?" and "What is my new number?" As you can see in the far left image above, the visuals shown are represented in money form, exposing students to a new way of thinking about numbers (in the form of cents and dollars).
Javier, another pseudonym, then leads the student through the secret code cards (pictured in the center). He simply will de-construct and re-construct the new number. For example, if the new number is "124," Javier would say "124 = 100 + 20 + 4 = 124" using the code cards as a visual.
Hannah (you guess it...a pseudonym) runs the class through the number path cards, each student has their own number path card at their desk. As you can see pictured on the far right, an equation is written and the new number is shown using base ten blocks. Again, students are exposed to a variety of number senses through this Daily Math Routine.
Finally (and not pictured), David, a pseudonym, leads the class in the final finger flashes. If you watched the video at the beginning of this blog, you saw students counting by hundreds, tens, and ones to get to the new number for the day.
Above, I broke down the Daily Math Routine. It sounds pretty awesome because it is student led and exposes students to so many different ways to represent numbers. The Daily Math Routine is very differentiated in its instruction and connects with many types of learners. However, after seeing this routine performed day after day in a second grade classroom, I am wondering if it is really worth it.
In my own placement, the students seem to struggle to remember what their job is at the different station. They are often looking for my help or my cooperating teacher's help to guide them through their specific job. Also, because this is a "routine," it is hard to see if the students are really understanding the concepts within the routine or just saying based on the rote questions and answers.
What do you think? Do you think this routine is helpful in a second grade classroom? Do you think it forces students to think about numbers differently? There clearly are many pros, but also several cons. If you were the teacher, would you continue to cut fifteen minutes out of your day to complete this routine?
College. Four and a half years of my life (up to this point). Hundreds of credit hours. A variety of classes. Thousands of assignments. A burden of dollars. Lots of tears (out of every emotion: sadness, anger, joy ect.).
Have I ever wondered why? Absolutely! Have I ever questioned why I return ever semester? Of Course? But, yet, I still do. I still work hard in my classes to earn respectable grades. I still sacrifice time with friends and family so that I can finish up a big project. I still stay up later than I care to admit, loosing precious hours of sleep. I still push through, even when my "to-do" list is never-ending and I am fried in every dimension (mentally, physically, emotionally). I still work exhausting hours so that I can fund my education. Have I wanted to quite? Throw in the towel? Give up? You bet...but I don't. I can't. I need to be a teacher.
be fearless in the pursuit that sets your soul on fire
Let's take a moment to remind back to when I was a little girl growing up. Being the youngest of four big brothers (yes, I am the only girl), I grew up playing up playing "war," digging trenches, building rafts, and participating in backyard football and ice hockey. Barbies were replaced with G.I. Joes. Disney movie were exchanged for Star Wars and Lord of the Rings.
fully understood at that young age, but this was the spark that ignited a passion that has me where I am today.
Fast forward a couple years, several grade levels and we have sixth grade Abby and the amazing (I didn't realize it) subject of math. My education was in a little different situation...something called homeschool, normally not a problem, except I was a horrible (and I mean horrible with a capital "H") students. Don't believe me? Well, what if I told you I didn't do my math homework for a solid month. Yup! My mom taught me a lesson every day, assigned homework, but I didn't do it. Well, you know as good as the next reader that math builds on itself. I will never forget the day when my mom went to teach me a new lesson. For some reason (maybe because I haven't done my math homework) I was just not understanding the concept. In response, my mom asked me to grab my homework notebook. You can imagine my shame as she flipped through the empty pages of homework. I got it such big trouble and as a consequence, had to complete every skipped assignment. As I worked day after day to get caught up, I found myself actually enjoying the math problems. Yes, I was having fun...the content made sense so there was satisfaction in completing my homework (imagine that). This was when I caught hold of another passion. One that resulted in me bringing my math
Which brings me back to where I am today...a "Super" senior by status but none-the-less pursuing my great passion. As this semester has progressed, the "why" has only grown as I have experienced real world classroom life in my Teacher Assisting placement. There is nothing that warms my heart more than teaching these second grade students, especially when they have that all-too-familiar lightbulb moment in math when they understand, the isle when they see math really isn't such a horrific subject.
This is why I keep coming back.
This is why I push through.
This is why I am fearless...
because my heart is set on fire when I am in a classroom teacher math.
In her book, How to Bake Pi, Eugenia Cheng takes the reader on a two-part journey, exploring the wonder behind the concept called mathematics. If "pi" wasn't enough of a hint, then the reader shortly realizes upon reading the Prologue that this text does not actually tell the reader how to bake the delicious desert that has a crust on the top and bottom, with some, usually fruit flavored, filling on the inside. Rather, the title comes from the series of recipes the author supplies at the opening of each chapter, which she then uses to relate to mathematics in some way, shape, or form.
What is math?
Like stated earlier, Eugenia Cheng breaks apart her book into two sections. In the first half of the book, she spends 156 pages (at least in my copy) answering the most simple question: What is math? Or maybe not so simple as she introduces mathy words like abstraction, generalization, axiomatization and more that make any non-math-geek cringe. Do not worry, for Cheng uniquely defines each by breaking down the process to the simplest form. As a reader, I was impressed with how capable I was reading her text and understanding the point she was trying to get across. That's because Cheng uses real-life, everyday encounters to explain the most complex of mathematic processes. For example, each chapter begins with a recipe. Sometimes the recipe was very vague in the ingredients it offered, or extremely detailed in the method of making the treat on hand. In either scenario, Cheng used the recipe to explain the process of axiomatization or some other ten-plus character word. As a result, the reader (no matter what math background they may have) is more likely to continue reading as oppose to quitting after fifty pages due to complex material.
As Cheng comes to a close in her writing of this section, she defines math as the "study of anything that obeys the rules of logic, using the rules of logic" (143). Seem redundant? She has lots of textual evidence that supports this definition, it is now up to you to read her text to fully understand why she defines "math" in such a way.
What is category theory?
Eugenia Cheng, after defining math in the first half of the book, then moves on to define category theory, a term that I was unfamiliar with prior to picking up this book and reading it. If the definition of math did not make you want to read the text, the way in which Cheng defines category theory will make you drop all tasks on hand and find yourself a copy. In seven words, Cheng defines category theory as the "mathematics of mathematics" (162). Do better help you wrap your mind around, let me present a quote within the text. It is a little bit of a spoiler of the contents of the first half revolving math, but it will better engage your brain at what Cheng means by category theory.
"In the first part [of the book] we saw that mathematics works by abstraction, that it seeks to study the principles and processes behind things, and that it seeks to axiomatize and generalize those things. We will now see that category theory does the same thing, but inside the mathematical world. It works by abstraction of mathematical things, it seeks to study the principles and processes behind mathematics, and it seeks to axiomatize and generalize those things" (162).
Some deeper mathematical thinking is introduced in this half of the book, bringing up lots of ideas I have previously explored in my math past, whether that be in a high school class, or college course. Once again, the way in which Cheng breaks down category theory, exploring it in relation to structure, sameness, universal properties and more, is done in such a way that draws any reader, not matter how math-smart. For example, for as long as I can remember, I have struggled with modular math. I have never fully wrapped my mind around the process and have consistently struggled through many course (both in high school and in college). In the text, Cheng uses modular math in relation to a clock, a physical object every human is aware of. I can now say, that I understand the meaning behind modular math, THANK YOU CHENG! This isn't the only example, throughout the text, I found myself often saying, "Man, I wish I was taught it this way in (insert math class name here)" or "If only (insert math concept) had been explain to me this way, I would have understood so much more." It really is amazing how Cheng masterminds her ideas into simple language. As a result of such efforts, I walked away from this section of the text with lots of new information regarding category theory.
Four out of five starts
After reflecting on the Eugenia Cheng's book, How to Bake Pi, I really cannot complain. I recommend this book to my math colleagues, to my friends, to any looking to explore the crazy subject we call math. The way in which Cheng pens the text is extremely engaging and easy-to-read. While many of her chapter contained lots of subheadings that often did not relate with one another, Cheng did her very best to break down the aspects of math into the most distinct parts and succeeded in writing in an understandable way. Your job now is to drop everything and go find a copy now and start reading. No matter how your personally feel about the text, at least you have several recipes on hand to try. A win-win situation for all!
I decided to spend this blog today digging deeper into the ideas of mathematics from the past. Based on the title of my blog, you can conclude that we will be exploring Mayan Mathematics. While I could have spent my time and words exploring many different histories revolving around mathematics, I decided to explore Mayan Mathematics as a follow-up to a class I took a year and a half ago. In MATH - 222, Math for Elementary Teachers, my professor, Dr. Nancy Mack, introduced Mayan Math, specifically the ideas surrounding their number system, when we were exploring our own number system with base ten blocks. She introduced, but did not give too much insight into the ideas of this ancient civilization, therefore, my interest has been peaked ever sense and here I am today writing about what I found.
beauty. They found flat foreheads to be attractive, therefore, when babies were born, it was normal to press a board to the babies head to create this culturally ideal flattened surface. The same goes for having cross eyes, therefore, the Mayan's would dangle objects in front of newborns eyes until they became permanently cross eyed. I think we can all agree that Americans have set far different standards about what they consider to be "beauty." Do you think the Mayan's think we are crazy, just as we may think about their qualifications for physical attractiveness? Aside from physical appearance, the Mayan culture took lots of pride in providing excellent medical practices. "Health and medicine among the ancient Maya was a complex blend of mind, body, religion, ritual, and science" (Top 10 Fascinating Facts About The Mayans). Therefore, only a few select would be thoroughly educated to carry the title of "shamans." There are many other fascinating things about this society, but its now time to take a look at the mathematical system they developed. Due to the importance of astronomy and calendar calculations to the Mayan civilization, the development of a mathematical system was not only wanted by absolutely required. The Mayan civilization was very resourceful of what they had. Therefore, they developed a number system using everyday materials that
based vigesimal number system off the number of fingers and toes on the individual. As the image depicts, they used a shell to represent the value "0," a dot/stone to represent the value "1," and a stick/horizontal bar to represent the value "5." For larger sequences of numbers, the Mayans used a vertical stick/bar to represent any power of 20. Based on this number system, the Mayans were able to develop calendars as well as make impressive astronomical observations. For example, they were able to use their mathematical system to measure the length of a solar year. The calculation that they found ended up being more accurate than what Europe had calculated. The same results exist for the lunar month, which I find to be quite astounding.
Why does this matter?
So, why do we care? Well, I don't know about you, but I find learning a new (and historical mind-you) counting system to be quite valuable as an up and coming teacher. I could see myself introducing this alternative counting system as a way to integrate history and mathematics in the classroom. I found an amazing resource that would not only educate students on the Mayan past, but also navigate them towards perfecting this number system, using the Mayan materials to create the number. Check out a screen shot below...
I can already see my future students eating up a program like this. Learning about a new counting system can only help solidify the ideas revolving around the counting system we use in the United States (base ten). When thinking about my own K-12 education, I rarely recall learning about the "history" behind the subject. I believe that teaching students where the math from today has evolved from is not only informative, but valuable.
After exploring the Mayan number system, I can finally return to my Final Exam Extra Credit for MATH - 222 and answer the question. Too bad it's a little late to actually get extra credit.
Mastin, Luke. “Mayan Mathematics.” The Story of Mathematics, www.storyofmathematics.com/mayan.html. Accessed 19 Sept. 2017.
“Top 10 Fascinating Facts About The Mayans.” Listverse, 16 June 2014, listverse.com/2009/09/21/top-10-fascinating-facts-about-the-mayans/. Accessed 19 Sept. 2017.
Here is the link to the online "Maya Math Game"
Cheng sways the readers own personal beliefs about what mathematics is by proposing her own term, category theory, also known as, "the mathematics of mathematics." Confusing right? I thought so too, but in a sense, math can't be defined by one thing. For example, math is not just about numbers, symbols, or even measurements. Rather, Cheng uses category theory to define math as being all about "relationships, contexts, processes, principles, structure, cakes, custards." If that isn't a more enticing Prologue to a book, I don't know what else is. So, naturally, I kept reading...
A simple question to some, but a vague question to others. To me, there are so many elements that comprise what we call "math" that pages and pages of pages of text couldn't begin to cover all of the topics. However, when we consider this term in the realm of its presence, it is fair to say that math is everywhere. Take a moment to examine things around you: from the chair you may be sitting in, to the open window to the left in your office, to the water bottle on your desk. All around us we see shapes and numbers and if not those in particular, we see things that were made using equations, formulas, and much much more! Math is beyond anything we can every fully wrap our minds around...therefore, math is everywhere. Without math...well, that thought is a little scary, so instead, let's just be thankful we have the beautiful and amazing concept of math.
Mathematics is the most beautiful and most powerful creation of the human spirit
With that in mind...what have been the greatest "math" moments to date. Through the years, there have been many discoveries that have advanced the realm of math. Examine the list below of what I believe are some of the TOP 5 discoveries in the history of math (not listed in order of importance).
(1) Fermat's Last Theorem...having almost every math professor mention this
theorem must make it somewhat important.
(2) The Creation of Pi...let's be honest we are thankful for the number, as well as
the baked good.
(3) The Pythagorean Theorem
(4) Geometry...referring to the findings around area, volume, and surface area of
(5) Addition, Subtraction, Multiplication, and Division...if they weren't important,
why did we start learning them in 1st grade?
Thank goodness there is no limit on discovery because mathematics is a growing and changing idea that will continue to revolutionize, just as it has in the last 3,000 years.
My name is Abby Niemiec and I am in the midst of my final year as a undergraduate student at Grand Valley State University. I am a double major in Mathematics and Education, with my minor focusing on Elementary Education. Within this blog, I will be sharing mathematical ideas, perspectives, thoughts and much more! Stay tuned...and enjoy the read!