Adventures in Education: Problems with Visualization

In a previous post, I remarked on the beauty of abstract algebra. This post regarding the subject is not so positive.

Since then, the material has gotten quite sophisticated. Simultaneously, significant events in my personal life have taken my focus away from the subject. Moments of clarity have been few and far between. I am using strange rules to justify arguments that I don't believe. I am blind in a world where visualization has always been my most valuable tool. Now what do I do?

Earlier in the course I could see the structures I was learning about in my mind. I could watch the elements interact, and manipulate their behavior to simulate a result. It was not easy to do, but I was operating on some level of intuition. That intuition is now completely gone, and I'm finding this type of mathematics a most frustrating endeavor. I'm memorizing rules and trying to remember when I am supposed to use them. Algebra is not supposed to be like this. The only things that keep me going are the hope that I will finally understand the material (better late than never), and the fact that I'm not going to let this subject best me.

I wonder if there are few math-o-phobes who have tried learning the subject blindly by memorizing rules and techniques. I recall a few high school math lessons where teachers would recite, "Well, if you see a question like this then you do this. See? It works." How dreadfully boring! If I had learned all math that way, I would have sworn off the subject a long time ago. It's an awful way to go. There is no fun in it! It's easy to see how frustrating this method of learning is when I'm actively doing it.

So, to end on a positive note, here is a graph theory question that everyone can have a crack at. It's intuitive, it requires logic, and most of all, its FUN! It was on my last assignment. Don't worry; it's a relatively easy question and it doesn't require any formal training in graph theory.

Suppose you wanted to colour the lines on this diagram. (This is called the Petersen graph, in case you were interested.) Here are the rules: each line gets exactly one colour. You may use the same colour for two different lines if you wish, but only if those lines do not share a dot. Can you convince yourself that it is impossible to do it with only three colours? (I.e. you need at least four colours.)

Planning our Wedding

Spoiler alert: this post is a rant, and it's probably poorly written. If you are looking for something lighthearted and funny, check out this video instead.

My fiance and I are planning our wedding, and I've been reading magazines and watching wedding-related reality TV shows in the name of research. Have you seen Four Weddings? It's a show where people get married and guests complain that the free signature cocktails aren't to their taste, and that the wedding dress doesn't fit around the hips, and how the five course dinner had steak and chicken. (I'm not sure why that was a problem, but trust me it was.)

Not too long from now, Mr. Right and I will stand in front of our friends and family and promise to be a team for the rest of our lives. We will work together to make our lives amazing, and take care of each other even when everything sucks. This is a significant decision and a huge promise, and although it has taken us eight years of consideration before agreeing to take the plunge, it is a decision we make with confidence.

Frankly, chair covers, specialty linens, and centerpieces seem insignificant. During this process, I have not once thought, "Gee, I'm about to marry my match. Therefore, all chairs must wear matching outfits. Yes, chair covers are essential; I wouldn't want my guests sitting on an uncovered chair like a savage. Only $500? Well worth every penny. And while I'm at it, why not spend the PhD/house fund on pipe-and-drape and twinkle lights, so our decor can be competitive with the weddings I've seen in magazines. I must have decor. It's so essential for a wedding. That's what guests should remember."

There are a few traditional elements I'm genuinely excited about. My dress is fantastic and I can't wait to wear it. I might even purchase a little flower to wear in my hair. My playlists have been perused to find that perfect "first dance" song. I'm especially excited to get some professional photos taken. Finally, photos in which neither of us is making a funny face! I never thought the day would come.

Most of all, I'm excited to celebrate with the family and friends who have supported us in our relationship over the past eight years. We'll host a good party for our loved ones. We're gonna feed 'em! There will certainly be dancing and drinking and laughing and carrying on. It will probably be loud, and at times inappropriate (hopefully not too inappropriate). What I want most of all (other than a loving commitment from Mr. Right, of course) are a great set of stories to begin our new family repertoire. In ten years, I hope we can sit around the dinner table and reminisce about the goings-on at our little wedding.

There are few times in our lives when we can count on our families to take a day or two off so that everyone is together to celebrate. Other times, we find excuses for absence in mundane tasks and empty commitments. We ask "When was the last time we were all together?" A wedding is such a grand opportunity to take enjoyment in our family and friends. We need this time together. We need to take these memories with us on our journeys. I hope our modest little wedding is rich with humor, love, joy, and family. If we can pull that off, we don't really need chair covers.

Adventures in Education: An instant of revelation

In my last post, I discussed how strongly I wanted to understand abstract algebra. Today, I am getting closer.

Mathematics is a play. In this play, there is a character (appropriately named the characteristic) that seemed alien to me. Its definition was quite simple, but the character was mysterious. I knew what it did, but not how it worked or what it was capable of. But then, in an instant, I learned a side of the character previously hidden. As if in soliloquy, it divulged just enough information to give me a taste of it's ambitions and agendas. And then, the rest of the story began to make sense. Those mysterious theorems, those awkward propositions, and those questionable conclusions began to fit together in a cascade of connections. A grand mystery has been resolved, and I can still feel the afterglow of the euphoria in my rosy cheeks and my tingling fingers. It felt like the tipping point in a Sudoku solution, when the identification of a single entry immediately reveals all of the others.

This character reached out and handed me a key. It is not a key that opens the entire subject of abstract algebra, but it is a key to its front foyer. Finally, I could step inside and marvel at the intricate workings of the subject! I expected it to be complicated and detailed and expansive, much like a grand hall with gold leaf designs and high ceilings and fine Victorian architecture. Much to my surprise, it was simple, compact, and clean like a photo in an Ikea catalogue. Where I expected a grand cabinet of tools I found a small cupboard with a few multipurpose theorems. It is incredible that such a great set of perfect ideas can collapse easily into such a small yet robust framework of elegantly simple rules.

Today, I went to class and understood everything the prof said. I didn't find the minor error in his notes. I didn't ask the clever question about a strange definition. I couldn't repeat the lecture back to you. But I understood what he was talking about the entire time. Hallelujah!

Some weeks are good weeks and some weeks are bad weeks. This week is a good week.    

Adventures in Education: At the back of the pack

Abstract algebra has become both a favorite hobby and the bane of my existence. The nuances and caveats are so grand, they swallow me up like sink holes. It's difficult to appreciate the great landscape of the subject from deep inside the chasm of a single theorem. The detailed elegance of every  piece of the algebraic machine takes my breath away. But putting those pieces together... well that's a different story.

So far, I have struggled to tie the subject together. I puzzle over problems for days, only to find the solution to be, of course, simple and perfect and easy. Slowly, very slowly, these beautiful theorems and quaint definitions are weaving together. The other students in the class ask intelligent questions and correct small errors in the teacher's definitions. They nod and smile and assimilate the ideas quickly and gracefully. I write every word the prof says and pray that later on it will make sense. This course is the biggest mental challenge I have undertaken, and I am sitting at the back of the pack.

My primary goal is not to achieve a high score or out-compete my classmates (although an A would be nice.) This subject is so beautiful and every detail is so precise, that it would be immensely satisfying to open my eyes wide and behold the magnificence of the theory. That's what I want. Not the grade, but that satisfaction. The solution to the grand puzzle. The insight. Thankfully, this stuff is much more important than my big ego.

Adventures in Education: New Beginnings at UVic

A degree wasn't enough. My week now consists of commuting back and forth to Victoria to take undergraduate math and computer science courses. If all goes well, next September I'll be in a Masters program. If not, I get to spend eight months learning undergraduate mathematics from geniuses. Win-win.

I learned very quickly that VIU (my home institution) and UVic (my new school) have slightly different approaches to teaching math. At VIU, in-class solutions are often quite detailed. By contrast, yesterday a prof wrote a solution as follows: "Proof: Induction." This is analogous a recipe for chocolate chip cookies that reads "Method: Bake." The student is still expected to understand all of the details of the proof. A benefit to this method: I must keep up with the reading. The drawback: I must keep up with the reading. Independent study skills? Yes, I think I'll need those. This semester presents an opportunity for substantial personal growth.



When I arrived in my first Abstract Algebra II* class, the front row was full and the desks had been pushed forward as far as they could possibly go. Strange, I thought, since there were only seven students in the class. The keeners in the front knew something I didn't. The prof came in with a few notes scrawled on the back of a receipt and a used letter envelope, and then plunged into an amazing lecture. At least, the parts I could hear were amazing. He was quiet. Mental note: get a seat at the front for every lecture from this point forward. Anyways, this guy knew every detail about the theory he was describing, as if it was his first language. Amazing.

As I adjust to the expectations of a new department, my emotions transition between excitement, intimidation, fear, and hope. I remind myself of  these words of wisdom from a VIU math instructor (very, very paraphrased):

In math, it's not about smart you are. Success is the result of being able to keep your head in the game. Get immersed in a problem. Do your best. Sometimes you don't get results quickly. It happens to everyone. It's not your intelligence that will define you; it is far more important to be able to find your way back into a groove when you are lost. We do this because we love it. You have to love it to be good at it, because you have to have the patience to stick with a tricky problem, even if you aren't getting anywhere quickly. If you really want to, and if you are persistent, you will do well.

I'll try my best.



*Abstract algebra is a subject about algebra of objects that may or may not be numbers. These objects could be all of the numbers, certain types of numbers, remainders, symmetries, systems of equations, the arrangements of sets of objects, or something else entirely. It's easier than it sounds.

The best things about September

Cool evenings.

Cool mornings.

Hot coffee.

Cinnamon in coffee.

Cinnamon in hot apple juice.

Cinnamon in everything.

Scarves.

New jeans.

Sunny afternoons.

Walks in the forest.

Walks by the beach.

Camping at the beach.

Homework at the beach. (Yes, homework!)

Fresh school supplies.

New courses.

High expectations.

High work ethic.

Classmate reunions.

Potluck parties.

Curried zucchini and apple soup.

Comfort food.

Warm hearts.

Cool evenings.

Counting to infinity

     Riddle me this: if you have the collection of all the positive integers (1, 2, 3, 4, 5, and so on) and you remove all of the odd numbers, is the remaining set of even numbers (2, 4, 6, 8, and so on) smaller than your old set?

     I've spent the last three weeks or so thinking hard about infinity. It's a required portion of one of my second year classes, and although we have covered the material and moved on I am still haunted by paradoxes and strange dreams. Heads up: this article may get a little technical and thought provoking.

     In the 19th century, a German mathematician named Georg Cantor founded a field of math known as set theory. His new ideas were met with criticism, and he was repeatedly institutionalized for mental illness. (After studying just a few of his ideas, I don't blame him. This stuff would drive anyone mad.) However, David Hilbert welcomed Cantor's revolutionary ideas, and stated that "No one will drive us from the paradise that Cantor has created." The logic used in set theory has been used to prove remarkable results in modern mathematics.

      I'd like to share some really interesting ideas with you. Let's suppose that we have an infinite set of objects and some way to count them. For example, if our collection was the set of positive integers (1, 2, 3, 4, 5, and so on), we could agree that there are infinitely many and that we could count them. We would say that this set was countably infinite. If we included 0 and the negative integers as well, we could still count the set. (0, -1, 1, -2, 2, -3, 3, -4, 4, and so on). As long as the set can be put in some sort of list, all elements are guaranteed to appear in the list, and the list goes on forever, then we consider the set to be countably infinite. (Note: some sets of numbers, such as the real numbers, cannot be ordered in such a list. We call these sets uncountable, but we will not deal with these sets here.)

     We can find the size of a finite set by counting the number of objects. For example, the set {a, b, c, d, e} has five elements in it, so it has size 5. But what size does an infinite set have? Can there be different sizes of infinities? What does "infinite size" even mean?

     Cantor suggested that we could consider two sets to have the same size if we could pair up each element from the first set with a unique element from the second. If each element had a partner and no elements were left over, then the sets are the same size. So if you have a crowd of people boarding a train and each person takes a seat and all seats are taken, then the number of seats is equal to the number of people. Now imagine that the train has infinitely many seats and the line of people is infinitely long. Cantor supposed that we could use the same logic on infinite sets; if there is exactly one seat per person then there are the same number of seats as there are people. (Author's note: I have not yet accepted the use of this logic on infinite sets, but I'll share some results with you anyways.)

     Let's look back at our original question. Consider the set of positive integers {1, 2, 3, ...}. Suppose we removed all of the odd numbers from this list. We would be left with the evens {2, 4, 6, ...}.  If you take any number from the first list and multiply it by 2, you will get a number from the second list. Also, each integer when multiplied by two will produce a unique even integer. Furthermore, for every even number, you can divide it by two to get a number from the first list. We are pairing up our lists like so:

{1, 2, 3, 4, 5, ...}
{2, 4, 6, 8, 10, ...}  => (1,2) (2,4) (3,6) (4,8) (5,10),...

We pair each element from the first set with a unique element from the second set, and no elements are left over. By Cantor's logic, these sets are the same size.The number of positive integers is equal to the number of positive even integers. Oh dear.

     This means that you can take a countably infinite set, remove infinitely many objects, and the set will be the same size. In fact, you could add a countably infinite number of objects and the set would be the same size. You could add a countably infinite number of objects a countably infinite number of times, and the set would still be the same size. Holy paradox, Batman!

      Some people are not bothered by this result. After all, infinity is infinity, right? Doesn't it make perfect sense that infinity as the same size as infinity? Well, not quite. It gets weirder. I'll leave you with one more strange thought.

     It turns out (for reasons that I will not address in this post) that there are different sizes of infinity. Some infinities are bigger than others. For example, there are more numbers between 0 and 1 (including decimals like 0.5, 0.111234, and 0.101010...) than there are integers (0, -1, 1, -2, 2, etc.)

     Stay tuned for more insomnia fueled posts about infinity. I'd like to address my issues with Cantor's logic, sets that should be the same size but aren't, and Hilbert's infinite hotel. Check your intuition at the door. It will serve no purpose when discussing infinity.

Rain on a Plastic Skylight

It's early. I've left my fellow adventurers to snooze, and gone outside to blog in an early morning shower. The soundtrack this morning features melodious bird song against the rhythm of raindrops on tin roofs and plastic skylights: an orchestra without a key or time signature.

Last night, my coworker and I walked along a beach on the south side of Malcolm Island at dusk. The surface of the bay was roughened by the weather, and had been etched with lines from boats and winds and physics. The atmosphere moved past us excitedly, but the mountains and the skyline seemed to hold their breath in anticipation of nightfall. A family of ducks, oblivious to the impending darkness, swam and dove in synchrony. Meanwhile, birds and insects softened  the harsh silence of paradise after sunset.

We didn't speak much, but in my mind I placed boundaries around the universe of my memories. I realized (as I often realize but choose to forget) that life, although composed of an infinity of instants, is a short thing with a definite beginning and a firm end. There are only so many evenings in which we can feel the sharp bite of a cold ocean around our ankles or the playful hands of a western breeze across our faces. Furthermore, personal philosophy and preference are dynamic; there may only be so many evenings in which we desire these experiences.

How many of those who would risk our coastlines for their bottom lines have stuck their toes in the ocean at dusk? How many? Are they unaware of the majesty of the waterline, or is it simply less impressive than a large number in a spreadsheet?

As I sip the last of my coffee, I listen to the sound of the rain. When I leave, I'll remember the sound of this cold, foggy morning: bird song and rain on a plastic skylight.

Pho on the Go

I live across the street from a Vietnamese restaurant. Every month or so, my man and I will arrive home from work too drained to cook. We put our jackets on and dodge the traffic Frogger-style until we arrive at the doors gates of the small restaurant heaven. Our order is always the same: number 21. Pho tai sach. Beef noodle soup. Pho at its finest.

I sometimes joke about opening up a Vietnamese take-out restaurant and calling it "Pho on the Go." The fatal flaw with this plan is that I don't cook any Vietnamese food.

Last week, I saw a recipe for beef pho in Canadian Living magazine. I decided that I will attempt the dish tonight (if I fail, I will run across the street, order some take out, bring it home, pour it into bowls, ditch the styrofoam containers, and take credit when the man arrives home.) However, I have some reservations about using a Vietnamese recipe from a "Canadian" magazine. What if it is not authentic?

I could use your help. Have you ever made Vietnamese beef noodle soup at home? Do you have any tips or suggestions to enhance the recipe?

Thanks! In the words of Joey, "Mmm, noodle soup."

 

Letters about a Pipeline

Last week, Joe Oliver, Canadian Minister of Natural Resources, published an open letter to Canadians regarding the public proceedings over the Enbridge Northern Gateway pipeline project. Many media outlets picked up on the letter and made it the topic of their news programs. Read the letter for yourself here. Not long afterward, Elizabeth May published a response, which you can read here.

Hold on to your hats, folks. This debate is going to be one for the history books, and it's just heating up.

One Morning in the North

The recent news coverage of the proposed Northern Gateway pipeline project has reminded me of how much I love the BC north coast. Last year, some friends and I took a phenomenal cruise through the inside passage to Prince Rupert, and then carried on to Haida Gwaii. We disembarked from the ferry in Skidegate (at 6:00 am!) and drove the short distance to Queen Charlotte City before parking at the beach to wait for the town to awaken. I sat on a damp wooden picnic table and counted a dozen eagles while they enjoyed their breakfast in the soft morning sunlight. The air was brisk and invigorating, while by contrast my coffee was hot and soothing. The breeze carried an essence that was, well, different from my urban home. Finally, I understood the phrase "fresh air." It was in that moment, a moment that I will never forget, that I re-prioritized and re-defined "the best things in life."

I wish that I had words that could do justice to the experience that I had that morning. Fortunately, I have some pictures.

Some of these were taken by myself, some by Ginny Casey, and the rest by Jeremy Krogh. I owe many thanks to VIU's ACER project for the opportunity to take this trip.  

Joe Oliver vs Evan Solomon

There has been recent political debate regarding the proposed Enbridge Northern Gateway pipeline. This pipeline would carry oil from the Alberta tar sands to Kitimat, BC. This oil could then be very easily traded with countries in Asia Pacific. Canada could stand to gain strong economic growth and trade diversification from the project, which could lead to improved social services such as education and healthcare. However, the risks include hazards to the environmental health of the BC coast, as well as changes to the way-of-life of the Aboriginal people who live there. Additionally, by extracting fossil fuels, Canada is contributing to climate change which has a large portfolio of economic and environmental risks such as increasing global surface temperatures, increased severity of storms, melting of alpine and polar ice, and decreased biodiversity.

In the video below, CBC's Evan Solomon interviews Joe Oliver, the Canadian Minister of Natural Resources. Solomon asks Oliver some very tough but fair questions regarding the public relations tactics used by the government who strongly support the pipeline project. Take a look at the video here.


Although I have formed an opinion regarding this debate, I am not going to state it on here. The point of this article is not to persuade you to agree with my political views. I just liked the types of questions that Evan Solomon asked of Joe Oliver in this interview. They were tough, but they were also clear, fair, and extremely relevant. It is essential to ask these types of questions of those on both sides of this debate in order to hold them accountable to their statements. This helps to preserve the integrity of the information that we, the public, receive from the media.