What IS math?

For this blog entry, we were asked to answer the question: "What is math?"

To help with my exploration of this question, I asked some friends and acquaintances for their answers. Here are some of the answers people gave me:

"Math helps to make the world make sense in formulas."

"Math is the study of calculation and the language of science."

"Math is hard."

"Math is using derived equations and functions to solve numeric problems." 

"Math is working with numbers."

"Math is the abstract study of quantity, using letters, numbers and structures to solve equations." 

To me, math is necessary. The world and everything in it has to do with math. I do not have a great background in math but I do know that it is all around me. Even the computer I am typing on right now is creating binary equations to allow me to do everything I need to do for my homework.

In order to further explore this question, I did some research. I found this article (http://www.livescience.com/38936-mathematics.html) by Elaine J. Hom, who says : "Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building blocks for everything in our daily lives, including mobile devices, architecture, art, money, engineering, and even sports." Hom seems to have considered this question a fair bit and I learned a lot from reading her article. Although I had known that math has been around since the beginning of time, I never considered the fact that society demands mathematics - and the more complex the society, the more complex the math needs to be. I also had not considered how old math is. You tend to think of people in ancient times as not intellectually advanced because of their lack of world knowledge and technology. But algebra, that branch of mathematics that had me tearing out my hair in high school, was invented in the 9th century. The NINTH century! So when the Vikings invaded Ireland and Scotland, people were doing algebra. That alone amazes me. The type of number manipulations and equations that those ancient people where able to do, 12 centuries ago, is the same thing that people are learning today in high schools. Hom has all sorts of information in this short, very readable article that I had never known about math until now.

To expand upon the subject in question, I then asked "what does it mean to do mathematics?" Some people gave me these answers:

"It means to work with numbers to solve an equation."

"It means writing down something on an exam to try and make it look like you know what you're doing."

"It means using equations and formula to get from a question to an answer."

"It means measuring, counting and calculating just about everything in your everyday life."

Then I decided to see what Google had to say about this. Keith Devlin of Stanford University (http://www.maa.org/external_archive/devlin/devlin_04_05.html) writes on his blog "'Doing math' involves all kinds of mental capacities: numerical reasoning, quantitative reasoning, linguistic reasoning, symbolic reasoning, spatial reasoning, logical reasoning, diagrammatic reasoning, reasoning about causality, the ability to handle abstractions, and maybe some others I have overlooked. And for success, all those need to be topped off with a dose of raw creativity and a desire - for some of us an inner need - to pursue the subject and do well at it." This was the first time I ever considered math as being an outlet for creativity. I suppose if you had a mathematician's level of knowledge, you could look at a problem and get creative, coming up with lots and lots of different ways to answer the one question. Then Dr. Stordy read us a book in class called The Math Curse and I saw very clearly, in the form of a children's book, another way to get creative with math. The book used math in unconventional ways and talked about all the different ways people can use math in the run of a day but generally don't, or do and don't even realize they're doing it, such as when they are buying something or deciding who gets how many cupcakes.

The third thing I asked my subjects was "If you are thinking mathematically, what exactly are you doing?" I received these answers:

"You're calculating in your head."

"You are doing equations for little things without even realizing it."

"You think in mathematical terms."

"You come up with the most logical answers to questions by having done the math in your head."

"You can solve problems really well and really quickly because your mind is good at it and used to it from all the time you've spent doing math."

I actually think there might be something to that last answer, although at first glance it doesn't seem to make a whole lot of sense. I think thinking mathematically really means that your mind works in such a way that it does small equations and calculations without it being obvious and you can understand why things are the way they are without having to write everything down. For example, a child without a mathematical mind might see a tall skinny glass of water, and a short fat empty glass, and decide that there is no way that the water is going to fit into the short glass because the tall glass is 'so much bigger'. A child with a mathematical mind will probably have no issue knowing that it's going to fit just fine because the volume of water is the same, just the shape of the container is changing. To think mathematically means that we probably wouldn't even think twice about this because it just makes sense to us, mathematically. 

Thinking mathematically, on one extreme, I think is like thinking in another language. This would apply to all those mathematicians out there - the Sheldon Cooper's of the world. He is a physicist but also a mathematician and some of his sentences are so convoluted, you have to really think about what he's saying to decipher it. I understand he is a fictional character, but I actually know some people that are near his level of genius and that is how they speak. They inadvertently make everyone around them feel inferior because their language is so complex. Keith Devlin of Stanford University is seen here in a video in which he is introducing his online course that he began offering in 2012. Ignoring the information about the course, the way he explains and defines mathematical thinking is very straightforward and sums up just about everything I've read on the topic - thinking mathematically is about thinking outside the box. http://www.youtube.com/watch?v=YFs06zgBfMI 

Stay tuned for thoughts from our class discussion!! 







This is a link to an e-book called Thinking Mathematically which I found online, after reading an article about how amazing this book is. Here is the link if anyone is interested:

http://f3.tiera.ru/2/M_Mathematics/MPop_Popular-level/Mason%20J.,%20Burton%20L.,%20Stacey%20K.%20Thinking%20Mathematically%20(2ed.,%20AW,%202010)(ISBN%209780273728917)(O)(265s)_MPop_.pdf

This is a link to another e-book called Learning to Think Mathematically. This is a textbook written by professors of the graduate education program in California University. It does a fantastic job of helping readers to understand mathematical thinking, what it is, how it came to be, and how to use it.

http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf 

Response to Sir Ken Robinson's talk - Do schools kill creativity?

My thoughts on this video featuring Sir Ken Robinson...


Sir Ken Robinson is a very engaging and convincing speaker. I was interested in what he was speaking about throughout the whole video, but more importantly I was immersed in thought, considering that of which he was speaking. Robinson put the idea that schools, or more accurately, today's education system, does not sufficiently encourage education, in a very clear light by using ideas that are easily accessible by everyone. By this, I mean that he did not throw philosophical phrase after philosophical phrase at the audience. He was very straightforward with his point: in order to educate students for a future for which we may not be alive, we need to allow and nurture the emergence of their creativity, not just their academic knowledge, and not "educate them out of it". 

Robinson touched on a few things that I found rather troubling. There is a theory that children do not grow into creativity, they grow out of it. This statement was not overly surprising, but the thought it provoked within me caught me off guard. I can personally vouch for the fact that we grow out of our creativity, and we are even educated out of it. And on a less severe but also truthful note, I can understand what he meant by many talented and creative people are being steered away from the things they love because there are no job prospects. When I was in elementary school, I was constantly being praised by my teachers for my writing. I wrote stories and poems all the time. I remember writing them for in-class work and would often have my writing read aloud by the teacher to the class as an example of good work. I would write on my own time...if I could recover the hard drive from our old Windows 95 computer, I would probably find about 50 finished and unfinished stories and poems that I wrote. I even won the Staples Canada short story-writing competition for Newfoundland when I was in grade 6. I truly loved it, and as young as I was, I was very good at it. When I got to junior high, however, writing became about essays and arguing points. Aside from a small number of creative writing activities I can remember doing, I stopped writing the way I used to. Any time I was asked as a child what I wanted to be when I grew up, after I grew out of the "I want to be a ballerina and a movie star" stage, I would say I wanted to be an author. More specifically, a children's author. As I got older, I was told how difficult it is to write something that could actually be published. Sadly, I cannot remember the last time I wrote a poem or story for my own enjoyment. To be honest, although I am really close to finishing my arts degree, I actually moved to education because of the ridiculous number of people that told me an arts degree was useless. I know I'm a good writer, but I've been made to feel that that is a useless commodity to have in today's society and job market. Robinson is truly disheartened by the number of young people who convince themselves that they are not as good at something as they think they are because they are told it is not a practical or useful skill to have. You go to school to get a degree to get a job...no one ever really emphasizes the fact that you should enjoy what you do or be able to put your own creative twist on what it is that you do. Education is certainly a way that I can be creative and now that I've learned so much about children and teaching them, I can see myself maybe attempting to be a children's author one day.

I found a couple of other things really resonated with me in Robinson's speech. I found the story about Gillian Lynne very thought-provoking. How many children have been in similar situations, and with the out-of-control diagnosis of ADHD today, have had their creativity and talents suppressed because they do not fit near the top of the hierarchy of education. I also found what he said about degrees these days (that students today go home after having obtained their degree and continue to play video games) all too familiar. I know a lot of people who have done just that, or are still working at their retail or food service jobs. Sadly, a lot of those people are teachers. But it's so true...my parents are both educated professionals, and neither had a difficult time breaking into their careers, and neither have had to switch due to lack of jobs available. Not even my mom, who has been teaching for almost thirty years. But now, there are hundreds and hundreds of unemployed teachers, arts majors and science majors floating around the province without work. I think that in some cases, not all but some, if these people knew how to bring their own creative flare to their profession, they would stand out from the crowd and stand a better chance at landing a position and gaining respect from colleagues and superiors. Unfortunately, the education system has suppressed creativity in a lot of ways.

A couple of points that I will take with me from this video are:
- We need to be careful that we use the gift of the human imagination wisely.
- Intelligence is diverse, dynamic and distinct.
- The education system tends to mine our minds for the same reason we mine the earth: for a particular commodity. This can be, in no way, beneficial for the future.
- We need to rethink the fundamental principles upon which we are educating our children.
- We need to educate a child's entire being because, even though we may not be around to see changes in the future, they will be, and they need to be prepared.
- We need to see our children for the hope that they are.

I would highly recommend this video to people who are in the education profession but even to young people who are concerned about what to do with their lives. The awful, anxiety-ridden first years of university when you are trying to figure out what to do and what degree will get you what kind of job, is still a very fresh wound for me. Ken Robinson is an extremely smart man and I learned a lot from him, even in this short 20-minute video.

"We can't prepare the future for our youth, but we can prepare our youth for the future." 
                                                                       - Theodore Roosevelt

My Math Autobiography

Faced with trying to remember what my experience with mathematics was like in primary/elementary, I am hit with the realization of how long ago that really was! I started kindergarten in 1996 - 18 years ago. Although that seems like a really long time, I have a few memories of learning mathematics and how I was taught.

In my elementary school, I remember that some classrooms were more "decorated" with math paraphernalia than other classrooms. I suppose it depended on the preferences of the teacher. In my grade four classroom, my teacher, Mrs. Howse, had a great amount of enthusiasm for math and there were math posters throughout the room - ones that had to do with counting money, 100s charts, multiplication patterns and so on. I remember the textbooks we had for math in the higher grades, and in the primary grades, we had these booklets with pages of math problems stapled together, and the only thing I can really remember about them was that the pages were really soft and if you made more than 1 mistake, you'd erase a hole into your page! My memories of actually learning math are very vague, unfortunately.

My worst memory about mathematics was in grade 3. Mrs. Barfitt would call on each of us individually and ask us one of the multiplication tables. This used to cause me a lot of anxiety because I had a really hard time memorizing the tables, especially the 6, 7 and 8 tables. To help me, my mom wrote out the times tables that I was struggling with and taped them around my room on the walls, so that I could examine them as I was going to sleep at night. It definitely helped! But I know that this affected my confidence in math. Even now, I often second guess myself when it comes to mental math. I often have to double-check things on a calculator when giving change to my tables at work, even though I know I was right the first time. I'm very paranoid about making a silly mistake and not giving the correct change!

In elementary school, although I struggled with the multiplication tables, I never considered myself "not good" at math. Well, at least not in elementary school. I had to have my cousin tutor me in grade 9 and even though I was in advanced math in high school, my boyfriend at the time had to help me with it fairly regularly. In my mind I was a "smart" kid and I never wanted to sell myself short by doing the regular math when I knew I could get through the advanced math with a bit of work. But from kindergarten to grade 6, I aced every report card, and classmates often looked to me for help with many things, including math.

As I mentioned, my experience with mathematics in high school was not quite as pleasant. I was lucky enough to have John O'Reilley as my advanced math teacher in grade 12, but within a month of school starting he left the school to dedicate all his time to his MathReviews business. The replacement teacher was not as effective, but I know that I would have struggled no matter what. My physics teacher, whom despite his sarcastic and "suck it up, buttercup" attitude is one of the best and most influential teachers I have ever had, once called me a "math weenie" because I lost points on my quizzes due to several simple math mistakes, several times.

With John O'Reilley's MathReviews help, I managed to do really well on my math placement test for university. But as I said, I was not overly confident in my math abilities, so I decided to start with Math 1090. My prof taught the very basics of the concepts which I did fine with, but the final was made up by the other professors who taught more complicated concepts....needless to say going into that exam with a 92% was the only thing that saved me from failing the course altogether. The following semester I took Math 1000 and learned it by memorizing the types of problems I was expected to answer, with the help of my ex-boyfriend and John O'Reilley. I didn't really understand what I was doing first nor last, but I finished that course with a 91% which, for me, was something of which I could be proud.

There are some out-of-classroom occasions on which I notice using mathematics. I grew up learning music, and I remember noticing how much math has to do with music theory. It has a lot to do with counting. Figuring out intervals and time signatures has a lot to do with math. I notice that I use math when I'm baking, in terms of using measurements. When I work out at the gym, I calculate how many calories I can burn in a minute and set a goal for myself, therefore I know how long I need to spend on the treadmill. This brings me to telling time, a type of math which we all do every single day and don't even realize it. The most obvious, though, would be when shopping or purchasing something, or changing up money.

Today, I take comfort in the fact that the most complicated math I will ever have to do anymore in my life is that at the grade 6 level! It was never my favourite subject and it still causes me a bit of anxiety if I ever have to really use it. Having said that, I do hope that my attitude towards math will improve throughout this course as I realize how many ways in which it can be used and applied, and how important it is for children to have a good mathematics experience. I don't think I had the best experience with math growing up and looking back, I think it left me with a natural aversion to it and I do not want that to be obvious to my students. If I can eliminate that altogether, that would be the most ideal thing for me! I know that I will have many math resources around my classroom, and I will most certainly not be making my students answer multiplication tables on the spot! Math should not create anxiety in students like it did for me, and I hope to be the type of math teacher that produces confident mathematicians in my classroom! 

Welcome to Maria's Blog for Education 3940: Mathematics in the Primary/Elementary Grades

Welcome! 

This blog's purpose is to give myself and my classmates the best possible learning experience as we explore mathematics and the many techniques and approaches there are to teaching it to students in the primary and elementary grades.