Blog: Articles filed under mathematics

Once again Margaret Wente, my favourite Globe and Mail columnist, has delved into the gritty underworld of math education to expose the truth. This time she is concerned that we’re not teaching basic arithmetic in schools any more. She takes issue with recent trends in math education, which emphasize discovery-based learning over drill or rote-based learning. As a consequence of this shift, the standard algorithms for addition, subtraction, multiplication, and division are no longer a core part of the curriculum. Wente, as well as some parents and teachers, thinks this is a bad idea. And while I agree with her on one point—it’s essential for students to know basic arithmetic as they go on to high school—once again I have to protest how she has chosen to argue that point.

Before I discuss Wente’s arguments, I think it’s important to mention one thing that Wente does not make explicit. Education falls under the mandate of the provincial governments. Hence, every province and territory in Canada has different math curricula. There are similarities, but we still have to be careful when we are talking about math education across the entire country as if it were some uniform curriculum.

Canada is “Behind the Times”

One of Wente’s more absurd reasons for rejecting the current curriculum is that “this approach to math education has been repudiated” in the United States, and this apparently makes us “behind the times”. Heaven forfend that we don’t copy the United States in every respect! You would think we’re a sovereign country or something crazy like that. We are allowed to structure our curriculum differently from our neighbour to the south. And without being too indelicate, let’s just say that the American education system in all its forms does not instil much confidence, at least for me personally. I’m not sure it’s something we should be striving to emulate.

This Is a Plot by Private Tutoring Firms (And Thus Puts the Poor at a Disadvantage)

Since kids no longer learn arithmetic in school, parents are forced to turn to private tutoring companies—Wente names Kumon as one example—for these skills. At the conclusion of her article, Wente decides to deploy the heavy Scare Tactic weapon:

The biggest losers aren’t your kids, of course. The biggest losers are the kids of parents who can’t afford tutoring, or don’t have the time to teach them times tables, or don’t even know their kids need help. It’s called two-tier education. And it’s here.

I love the way Wente phrases this: the biggest losers aren’t your kids; they’re the kids of those poor people. Those two sentences tell you all you need to know about who Wente assumes is reading The Globe and Mail.

I’m not sure how to refute this argument simply because it’s a conspiracy theory, and Wente doesn’t even try to disguise that fact. I suspect that if our curriculum magically amended itself to reflect Wente’s visions, then Kumon and its ilk would find other ways to get clients. They are a business and target their marketing accordingly. It just so happens they’ve found a niche here.

But two-tiered education has always been around. That second tier is called private school.

The Standard Algorithms are Better Because They are Efficient

Wente calls the standard algorithms “efficient and foolproof”. And they are. She blasts the alternative methods, “such as breaking numbers into units of thousands, hundreds, tens and ones” for not being efficient. I would disagree, but first we probably need to decide what we mean by efficient.

If one’s goal is to add two numbers efficiently, then my suggestion would be to use a calculator. Savants aside, computers are just better than humans at adding and subtracting. And now that calculators are commonplace, not just in schools but on our computers and even in our phones, there is no reason not to encourage their use. Should you be able to add basic numbers without a calculator? Absolutely! There will be instances where a calculator isn’t in reach, for whatever bizarre reason, and you will be glad you can do arithmetic. But those instances are becoming increasingly rarer. And so when they come, are we really worried about efficiency?

An algorithm is a series of steps that one repeats until one reaches a pre-determined stopping point. The first algorithm that most of us learned (and one that is apparently no longer taught, to Wente’s chagrin), is the long division algorithm. In this case, you repeat the same step over (dividing a digit of the dividend) until the remainder is less than the divisor (your pre-determined stopping point). As Wente points out, the nice thing about algorithms is that they are foolproof. Whether you are dividing 30 by 12 or 3000 by 1250, assuming you recall the steps correctly and don’t make a mistake while implementing them, you will always come up with the correct answer. This is comforting.

But algorithms are also cumbersome for humans. Unlike computers, which thrive on algorithms because that’s the way we built them, our brains do not always think linearly. We make intuitive leaps, and we often think spatially. Thus, training ourselves to use algorithms to do arithmetic might be a waste of our brains’ potential. The method that Wente disparages, which we can call the “place-value method” is ingenious: it short-circuits arithmetic by allowing us to take advantage of our base 10 number system.

How would you divide 110 by 5? Margaret Wente would like you to use long division, in which case you would follow these steps:

1. Recognize that 5 goes into 11 evenly 2 times.
2. Multiply 2 by 5 to get 10.
3. Subtract 10 from 11 to get 1.
4. Bring down the 0 to get a new divisor of 10.
5. Recognize that 5 goes into 10 evenly 2 times.
6. Multiply 2 by 5 to get 10.
7. Subtract 10 from 10 to get a remainder of 0.

Or, you could do it this way:

1. Recognize that 110 = 100 + 10.
2. Divide 100 by 5 to get 20.
3. Divide 10 by 5 to get 2.
4. Add 20 and 2 to get 22.

I suspect that the second method is probably closer to what most people do in their heads, whether they were taught that way or not. You can do this for multiplication too. It’s all thanks to the nifty distributive property. And hey, look, suddenly instead of memorizing two algorithms, you only need to know one strategy. Furthermore, the long division algorithm is exclusive to, well, long division; it’s very difficult to use it as a template for solving different types of problems. In contrast, knowing how to exploit the place values of our number system will leave you in good shape for a variety of problems. Generalized knowledge!

I’m sure some people prefer the long division algorithm instead, and that’s fine. In fact, that brings me to Wente’s next argument.

Discovery-Based Learning Sucks Because Students Have to Start from Scratch on Every Problem

Wente dislikes the alternatives to the standard algorithms because, instead of just giving other methods to students, teachers instead encourage students to find those methods themselves. In addition to her lament that this is not efficient enough for her tastes, it also means

every time a student sees a new problem, he has to start from scratch—and pick his “strategy”. It’s like playing the piano without ever learning scales, or hockey without basic drills.

Those are quite evocative analogies; it’s a shame they’re false. Solving math problems bears little resemblance to playing hockey and even less to playing the piano. When playing piano, the goal is to reproduce a series of sounds by triggering the correct sequence of keys. For a given composition, that sequence is always the same—and you know the sequence beforehand (unless you’re playing some kind of weird piano game where you reproduce a sonata by ear). A new math problem, by definition, is one a student has not seen before.

Let me tell you from personal experience on my practicum: if you put a problem on a test that is identical, except for the numbers themselves, to one on the review, the majority of students will not recognize this fact and will instead approach it as a novel problem. Encouraging students to make connections is one of the most difficult tasks a math teacher faces. Those moments when a student goes to ask a question and then says instead, “Wait, it’s like what we did yesterday, right?” are golden—and far too few.

So let us suppose students do have trouble making such connections, that they do approach each problem from scratch even if it is of a type they have seen before. What can we do to help them solve the problem anyway? Wente would have the student, like a good computer, apply one of the standard algorithms and arrive at the solution. No need to find a new strategy! Yet this assumes the student recognizes which operations are necessary to find the solution. And therein lies the crux of the problem. Incidentally, this is also why computers suck at solving word problems.

Discovery-based learning actually works better in this case. By encouraging students to look at what they know and what they need to find out, then develop their own strategy to get there, we are building general-purpose skills that will work whether they recognize the type of problem or not. This is the beauty of mathematics: there is one correct solution but not one correct method. Wente would rob students of this beauty.

Failing to Learn the Standard Algorithms Makes It More Difficult to Learn “Higher” Math

I left this argument for last because it’s the one with the most validity. One of the reasons I am so passionate about teaching high school mathematics is because I have seen why my peers are struggling with their university math, and it’s usually not because the university concepts are too hard. No, most university students just suck at fractions. And I saw this while on my practicum too: fractions and basic algebra are concepts that students fail to master in grades 7, 8, and 9, and it haunts them for the rest of their schooling.

So Wente has a point here: students do need basic skills in order to go on to higher-order thinking. And it’s not clear-cut, despite what either side might have you believe, whether drill-based or discovery-based learning is superior in teaching these skills. I can’t really evaluate them properly, because despite my passion for this subject, I’m a fledgling teacher with very little experience in the field. I can tell you what I have reasoned a priori, but experienced teachers are expressing frustration, so there must be something else going on.

While on my practicum, I had the opportunity to sit in on a meeting between Grade 9 math teachers at my school and Grade 7 and 8 teachers from the “feeder” schools. This very dilemma came up during our discussion: the push from the Ministry of Education and curriculum experts is to have students discover their own strategies and take ownership of their learning. At the same time, however, these teachers feel a responsibility to ensure that students are prepared for high school and for their EQAO tests, and sometimes discovery-based learning makes this difficult—for one thing, it can take more time. So I can see why there is frustration among teachers who are trying to work with this new curriculum but seeing less-than-stellar results.

Of course, the curriculum will always need refinement. Continual revision and renewal of the curriculum at regular intervals is a hallmark, at least in Ontario, of the high quality of our education system. It’s never going to be perfect, and as our society and our needs change, so too will the curriculum. Right now, I think a lot of what we are seeing is simply growing pains—teachers who are used to the previous curriculum are still finding the their way with this new curriculum. Moreover, this is clearly a complex issue, one with a plurality of perspectives that should be considered.

And that’s why I take issue with Wente’s column: I agree that arithmetic is important, but once again I wonder why she feels the need to create a dichotomy where none need exist. She would have us return to the methods that turn kids off math and lend credence to their cries that “math is boring”. Picture me going to my knees and pleading as I say this: it doesn’t have to be that way. Math can be fun and full of wonder. Please, parents, don’t make math boring. Computers do their math in binary, but there is no reason our math education has to be an either/or scenario. And I wish The Globe and Mail would talk about that instead of choosing to be sensational and blame it all on the corporate interests of Big Tutoring.

I woke up on Friday to see a page from Thursday’s Globe and Mail on the living room table. My dad had flagged an article by Margaret Wente as something that I might find relevant. You can find it online under the title “Too many teachers can’t do math, let alone teach it”, but in the paper itself it was published with the headline, “Go figure, because teachers can’t.” I encourage you to read the article, but the gist goes like this: elementary teachers, according to Wente, are failing to teach students the basics of math, because faculties of education don’t take their responsibility to prepare those teachers seriously enough.

By way of disclaimer, I am preparing to teach at the Intermediate/Senior level (I/S), or grades 7–12. As an I/S teacher, and as a formally-trained mathematician, I have to admit to a bias when it comes to this subject: I do worry about how well-prepared elementary teachers are to teach math. I’ve marked for a course that teaches elementary concepts to prospective teachers, and some of the answers to the assignments are … creative. However, my concern isn’t so much with their knowledge of content; I worry more about their attitude toward learning and using mathematics.

When I tell—more like confess, it sometimes feels—fellow teacher candidates that my teachable is math, I’m usually met by some type of cringe, as if the very concept brings up bad memories of a grade 10 test review. As I said in my previous post, I feel like there is a perception of math as something you can either do or you can’t, and if you can’t, then there’s no reason to bother wasting time learning anything beyond what you need to punch into a calculator. Of course, this might be the result of our education system and how we teach math. Whatever the cause, I worry less that teachers won’t be able to teach the content and more that teachers will transmit their anxiety about mathematics to their students. I’m not saying all elementary teachers must love mathematics, but how can one foster an appreciation for mathematics if one does not share that appreciation and is merely teaching it as part of the curriculum?

But I digress.

Wente might be on to something when she points out that elementary teachers need more thorough preparation in math. I don’t know; I am not familiar with the research and can’t step to that claim. (Here’s a York professor’s rebuttal with actual data analysis.) I find it interesting that Wente does not mention any of the current methods that faculties of education use to prepare elementary teachers: here at Lakehead University, Primary/Junior teachers must complete a content test to demonstrate their understanding of elementary concepts in mathematics. The way Wente presents faculties of education makes it sounds like they are resting on their laurels:

Today’s faculties of education have much loftier goals in mind. According to them, their main job is to sensitize our future teachers to issues of social justice and global inequality.

Gasp! Teaching our teachers to respect diversity and, shock!, be aware of factors affecting equality among our students? Those naughty faculties of education! Who do they think they are?

What I find really bizarre is how Wente goes on to devote the rest of her article to criticizing this one aspect of education—but at no point does she give any evidence for a causal relationship between the teaching of social justice and a decline in the quality of math education! Dripping disdain, Wente writes:

No wonder little Emma doesn’t know her times tables. She’s way too busy learning how her Western position of privilege entrenches gender relations. Or something like that.

(Wente does not, in general, have a very high opinion of social justice and related fields of study. Earlier this year she wrote a controversial piece about how the “war for women’s rights is over”; the original post is behind a paywall, but there is a good rebuttal on Shameless.)

I hope I’m not making a straw man here, but Wente seems to be saying that teaching social justice, either to teacher candidates or to students themselves, is a waste of time. Apparently it’s a move worthy of “the wacky wing of the NDP”. Yet not once does Wente bother to link this emphasis on social justice with elementary teachers’ abilities to teach mathematics. I guess she’s implying that we spend too much time teaching teacher candidates about social justice instead of teaching them math?

As part of the Differentiated Instruction in Math and Science (mouthful, I know) course I’m taking this year, we are learning how to teach math through social justice issues. Talk about two birds, one stone. This probably wouldn’t placate the Wente, however, for in her concluding paragraph she chooses to take a cheap shot at discovery-based learning, claiming we need to focus more on “practice and problem-solving”. This is a false dichotomy, and presenting these teaching strategies as such is irresponsible and even harmful: discovery-based learning is problem solving. In order to engage students, we provide them with problems they haven’t encountered—problems that are relevant to issues in their lives—and ask them to apply skills and discover new (to them) methods to solve the problems.

Wente concludes by reiterating that teachers need to know math in order to teach it. I agree with this statement; it’s just too bad that the rest of the article is somewhat incoherent. Wente does faculties of education a disservice even as she frames a legitimate concern—preparation of elementary school teachers to teach math—in a way that is confusing and unhelpful. The public, and especially parents, have every right to observe and critique the preparation of teacher candidates, for teachers have an awesome responsibility in our society. I just hope that when they do so, they refer to better sources than this piece, which is far more sensational than sensible.

Last Friday marked the end of my summer research term. For reasons I don’t entirely understand and don’t need to understand, Jessica made a pie to celebrate the milestone. It was raspberry (my favourite fruit) and, more importantly, it was delicious. This summer feels like it has gone by extremely quickly, and I’m not yet eagerly anticipating school. I have two weeks off now, returning early on August 29 to begin the intense final year of my concurrent education degree. My schedule does not seem all that bad, as far as classes go, but I‘m not sure what the workload will be like—I hear it’s heavy but not difficult.

As far as my research goes, I can’t help but compare this summer to last summer. Overall, I was not as interested in my project this time around—it’s the same project, so it is no longer fresh. Working on it on a full-time basis for 16 weeks was intense. This year was also quieter around the office; Jessica was not around as much, and Rachael had a research project, but it only lasted eight weeks. Aaron came in pretty consistently several days a week, despite not being on any kind of schedule. Aaron and I have been working our way through the Portal 2 co-op levels. It’s hilarious. A typical level involves me dying, followed by Aaron going, “Oops, wrong portal”, and then Aaron dying because I have terrible timing. The levels themselves are really well-designed, though, and I love some of the solutions to the puzzles. The beauty behind the physics engine is a lot more obvious when you can watch your partner go flying through the air (to his death).

This summer my supervisor and I collaborated to write a paper based on my research! Aside from my honours thesis (which is now available in the math section), I have very little experience writing about mathematics for an audience. This was the most interesting part of my research, and I really enjoyed the opportunity. We are polishing up the paper now, and then my supervisor plans to submit it to a few journals. I have no idea if it will be accepted for publication, but it was fun to write anyway.

Using SHARCNET was once again very cool, although the clusters I used this year gave me more headaches. I have begun using Git for managing my own coding projects (including this site), and SHARCNET’s clusters have Git installed. So I’ve put all the code into a Git repo, and you can view it on GitHub.

I‘m experiencing no small amount of trepidation regarding student-teaching this year. However, this summer has reaffirmed my desire to become a high school teacher. I don’t want to spend the rest of my life doing research! As a thank-you to my supervisor, I gave him a signed print of this xkcd comic.

With two weeks off, what do I plan to do? Read, of course! And get my life in a semblance of order before school starts—that is, I want to tidy up my room and get some other projects finished, started, or just kicked into gear. I have several blog posts I want to write, and some improvements I want to make to my site. Although I intend to drive myself in these next two weeks, my priority goals are relaxation and reading, in that order. All too soon, university will loom again. I know because they took my money last week.

Sunday was mostly an odds-and-ends day. I cleaned my room, organized things, and finished some books. Although the threat of rain hovered constantly in the air, I even managed to do some reading outside. So I had a pretty good weekend.

I managed to finish both Persuasion and the Iliad. My to-read shelf was finally empty, which meant I could restock it with books from the rather oppressive overflow stack. I have forty more books on the shelf now, and the overflow now fits comfortably inside that blue milk crate! My goal is to empty the shelf again by the end of July—this is ambitious, I‘m aware, and made even more so by the fact that I also have to get through the Hugo Voters Packet by the end of July.

I’m voting in the Hugo Awards again this year. I first voted last year, when John Scalzi alerted his readers to the fact that the Worldcon organizers distribute a packet containing electronic copies of most of the nominated works. This year, the attending membership at Renovation is only $50. That is a small price to pay for access to all these wonderful works, not to mention the privilege of voting in the Hugos themselves. I’ll blog more about the awards once I have read more of the nominees.

My weekend was rather relaxing, and certainly not as active as my brother‘s. He spent almost the entire weekend outside in our driveway, doing body work on his truck. Brad’s dedication and work ethic never fail to amaze me. I’ll come home from my seven hours of math research, which includes high speed Internet and tea, collapse into a chair, and declare myself exhausted. Brad, on the other hand, leaves earlier than I do, comes home later from a physically-demanding job, and goes straight to work on his truck. He’s always working on his truck—and when he’s not, he’s helping his friends with their trucks, or going mudding. None of these activities particularly appeal to me, but I am glad I have someone around who knows how to fix my car when it breaks. Especially when he’s the one who broke it!

And now I’ll talk about my research for two paragraphs, which means some fairly intense math jargon. You have been warned!

This is the sixth week of my summer research. So far, it has been very similar to last year, which doesn’t surprise me. I have mostly been trying new approaches to computing the spreading number by looking at the symmetry of the graph. We can perform rotations and reflections on sets of vertices using the symmetric group, and Dr. Van Tuyl and I hoped this would lead to better algorithms for finding the spreading number (which, you may recall, is the cardinality of the maximum independent set on the graphs we are studying). Alas, although we have made many valiant attempts, a feasible solution remains beyond our grasp. We have several interesting algorithms I’ve been testing, but they either use too much memory or do not produce tight enough lower bounds.

This week I think I am going to finish up my look at the spreading number, regroup, and redirect my efforts. I will turn again to the covering number; last year I had a fair amount of success with a greedy algorithm to find an upper bound (specifically, a minimal clique covering). Despite our lack of success in computing new bounds for the spreading number, the time I’ve spent so far this summer has furnished me with some new tricks that I hope to put to good use in improving this upper bound algorithm. Also, I would really like to understand why the covering number in four variables corresponds to this integer sequence.

And so my summer continues: lots of reading, plenty of math. As we now ease into June and hopefully receive more sun, I want to get more writing and more programming (mostly for this site) done as well. I’ll try to keep the blog posts coming.

So I‘ve finished my first week of summer research, which I began on Tuesday (as Easter Monday was a holiday). I am re-revisiting the spreading and covering numbers, those devilish little fiends from combinatorial commutative algebra that plagued me last summer. You can read about last summer’s research here. I shall try to blog often about this summer’s efforts as well.

This week was very much about settling in and trying to get my mind into a math research mode. I am starting earlier than I did last year, because I will be returning to school in late August. We professional year students have to start early to finish classes in time for student-teaching in November! Unfortunately, this means that I didn’t get much of a break between the end of classes and starting research. I have tried to seize as much downtime as I could. Still, as far as summer jobs go, I won’t complain about doing research. It’s pretty choice.

I’m in the same office as I used last year, and you can see photographs in this blog post. I‘m in the desk in which Aaron is sitting in those photos, as my former desk is still occupied by a sessional professor’s things. Aaron, who is not in concurrent education, has also finished his math degree and is going on to graduate school here at Lakehead. He promises he will be getting a key to the office, however. Also, Rachael is returning to academic research starting next week, so it will soon feel like old times!

There is a new face in the office this year: Tim also has an NSERC grant to do some math research; he’s a third year student who has switched to math from biology, and so far his project involves recurrence relations.

There is one material difference in the office this year: books. The math resource room is getting new carpet, so Tim and I spent some time this week removing all of the books from the metal bookcases, packing them in what boxes were available, and stacking those boxes in our office. We cannot reach the chalkboard any more; moreover, we ran out of boxes, so there are a lot of freestanding stacks right now. It’s messy and musty and dusty and kind of cool, though I want my chalkboard back. I didn’t take any photos; I shall do that on Monday and update this post. Updated with photos! And again, now that Aaron has joined us. One more time, because Rachael insisted I have a photo of my workspace here.

I shall not say too much about how my research is going right now, because there’s not much to say. I’ll be using Macaulay2 and SHARCNET again. So far we’re looking at the orbits of vertices of my graphs; these are the equivalence classes created by the action of the symmetric group on the graph’s vertex set. We hope to find some relationship between the orbits and the size of the maximum independent set and thereby find a way to compute the spreading number. Next week I’ll be investigating this further.

Last updated Tuesday, May 3, 2011 at 8:22 PM

I‘m almost finished my fourth year of university, and with it, my HBA in Mathematics. It doesn’t feel like four years! It feels like barely yesterday I was a nervous first-year student trying to figure out how to get around our campus (which I now realize is tiny compared to other campuses).

I won’t be graduating at the end of the year, because I’m actually in a five-year concurrent education program. For those of you unfamiliar with it and with teaching certification in Ontario, let me give you a brief run down. Instead of completing my mathematics degree and then doing a one-year education program (“consecutive education” or colloquially known as “teacher’s college” around these parts), I have for the past four years been enrolled in concurrent education. As the name implies, I‘m taking education courses concurrently with the courses I need for my math degree. So at the end of the five years, assuming I complete the program, I’ll have an HBA in Mathematics and a BEd. In Ontario, teachers are certified to teach in a specialization defined by grade level. Mine is “Intermediate/Senior,” or I/S, which means grades 7-12. I really want to teach high school, but of course, those with seniority will get to choose what they teach first, and I’ll get what’s left over.

In the I/S specialization, I need two “teachables.” Mine are math, naturally, and English. This is usually where I get odd looks from people and something along the lines of “that’s an unusual combination.” Maybe it is in practice, but I don’t think it is in fact. Mathematics and English share in common a need for clear, precise communication. I love rigorous proofs; I love grammatical constructions. Mathematics is about exploring the beauty of abstract thought; English is about exploring the beauty of our minds, bodies, and hearts. They are complementary.

So anyway, next year is my “professional year,” the big culmination of my education degree. I’ll take courses related to my two teachables, as well as a few more general courses like “classroom management” and some electives. Yesterday I attended an information session run by the Faculty of Education’s Department of Undergraduate Studies (mouthful, that) where they told me about what I could expect prior to and during my professional year. I was sceptical about how useful this session would be, but I actually found it very enlightening. The speakers provided precise information, both written and oral, about what I could expect; I no longer feel like my understanding of professional year is vague at best.

Oh, when I got to the session, the woman at the door asked me what my teachables are. When I said “Math and English,” she looked at the combined teachable schedules she had printed off, and then said, “Email me for your schedule.”

I‘m looking forward to professional year in the sense that I still don’t really feel like I’ve learned much about teaching. My education classes thus far have run the spectrum from “absurdly unhelpful” to “academically interesting, with some practical applications.” Educational Psychology is an example of the former and Educational Law the latter. For the most part, however, I don’t feel like I’ve learned much about the more practical parts of teaching, like preparing lessons and lesson plans, ensuring I meet curriculum expectations, etc. So I’m hoping those tantalizingly-practical titles like “Classroom Management” will indeed contain the golden nuggets of truth that will set me on my way. By which I mean, I know there’s a lot I’m going to have to figure out for myself, but at least this should give me some idea of what my options are.

Of course, the counterpart to these classes is my student teaching, or placement. My year will consist of two blocks of 9 weeks of classes followed by 5 weeks of placement, with Christmas holidays sandwiched between them. I’m very nervous about placement, and the information session went a long way to quelling that trepidation by assuring me that the department offers as much support as it possibly can to teacher candidates, especially in the field. Hopefully, after those 9 weeks of classes, I will feel slightly more prepared to re-enter a classroom, this time as a teacher.

Between now and then, I will once again be researching for the summer. I received another NSERC USRA, and I will be re-revisiting the spreading and covering numbers. Once again, I will miss working with my coworkers at the Art Gallery over the summer, but I’m also excited to spend another summer thinking about math. I start my research on April 26. Until then, I have to finish this year, of course, and try to find time to relax before my summer begins.

This Friday, Saturday, and Sunday I attended the eighth annual Combinatorial Algebra meets Algebraic Combinatorics Conference. No, I didn’t record awesome video diaries as I did when I attended the 2010 Canadian Undergraduate Mathematics Conference. I did meet many experts in these fields, listened to interesting talks that I didn’t really understand, and gave a talk of my own!

Combinatorial algebra and algebraic combinatorics are, as the conference’s title and purpose expresses, two sides of the same mathematical coin. They are areas of mathematics that combine techniques from combinatorics and abstract algebra (notably, commutative algebra) to solve a variety of problems in algebra, combinatorics, and even algebraic geometry. Now, these fields are specialized. I got the impression that even among the thirty or so graduate students, postdocs, and professors in attendance, many of them were struggling to keep up with some of the talks, because the topics in this area, as with any specialized field, can get pretty esoteric. One fellow gave a talk on cluster algebras, and the room was rather silent when it came time for questions.

Still, it was exciting to attend the conference even though I, as an undergraduate student with only two courses of basic abstract algebra under my belt, understood very little of any of the talks. I was invited to speak at the conference by Adam Van Tuyl, chair of our mathematics department and one of the conference organizers. He supervised my summer NSERC USRA. I previously gave a talk about that research in the fall, and he felt it would be a good fit for the conference. I was a little sceptical, not to mention a little intimidated by the notion of talking in front of all these learned academics. Nevertheless, I acquiesced—I mean, that opportunity might not come again. I‘m getting a lot of mileage out of this talk.

If you are interested, I’ve set up a page explaining my research on the spreading and covering numbers. Unless you are familiar with abstract algebra or graph theory, most of it will sound like gibberish, but check it out any way. You can also download a copy of the talk I gave, as well as the Macaulay2 code I wrote.

Giving my talk, which was well-received, was one of the high points of the conference, of course. For one thing, I‘m pretty sure everyone there followed what I was talking about, since I was presenting it on a more elementary level than a postdoc or professor would. And that’s fine. More importantly, a few of the attendees had some interesting ideas that might help me in the future. I am currently applying for another NSERC grant to continue working on this project this summer; hopefully I’ll get the grant and be able to put some of those ideas into practice. If anything, going to the conference has made me more excited about working on this problem again.

Another high point was meeting Tony Geramita. He co-authored the paper that introduces the spreading and covering numbers, essentially making him the originator of what I studied. And he knows his stuff; he seemed to switch gears effortlessly between each talk and ask intelligent questions (or at least, from my limited understanding of the topics, they seemed intelligent) whenever he needed clarification. So meeting him, and giving a talk about these spreading and covering numbers in front of him, was kind of a big deal. Plus, my natural tendency toward introversion means it takes me a while to warm up to new people, especially ones whom I meet in an artificial, arranged way like this.

So imagine my surprise and amusement when, at lunch, I brought out my copy of Forest Mage, and he said, “Ah, you’re reading Robin Hobb.” From there we conversed about our mutual love of science fiction and fantasy. Later, we started talking about eBooks, and he spontaneously asked if I had a thumb drive on me so he could give me a 1 GB library of eBooks he has on his computer. I was somewhat taken aback by this random and generous windfall. (I used my phone, since it had 11 GB free on its internal SD card. I should probably get an external one too.) This unforeseen icebreaker made it easier for me to think of him as a person, not just a Smart Math Individual, and much easier to give my talk.

Saturday night, after the conference, we went to the Masala Grille for dinner. Although my dad and I have ordered takeout from this Indian restaurant in the past, I had never actually been there to eat, so that was an interesting new experience. We had the upper room to ourselves, and the food was good (although I made the mistake of putting too much sauce on my plate). I had some interesting conversation with the people at my table about a variety of things, mathematics and non-mathematics alike, including an opportunity to talk to an Iranian fellow who is at Dalhousie for the summer. This was his first trip outside of Iran, and it was cool to hear about the situation in that country from someone who has grown up and lived there.

All in all, I have to admit the conference was a great experience, even though it did have people at it and did not in fact consist of me sitting in a chair reading a book all weekend. Sacrifices had to be made, and they were worth it! But don’t think this means I’m going to grad school just yet, despite the fact that more-than-hints have started to drop! But that is another topic for another blog post. Now I have to concentrate on finishing the rough draft of my honours thesis, for it is due on Thursday.

Hello September. I have missed you. You might be my favourite among all months, but don’t tell the others. And no, it’s not because my birthday is in September (although that helps). Nor is it because September signals the start of fall television, with new episodes of Castle, Chuck, House, Stargate Universe, etc. More than any other month, even that notorious January, September is a month of changes and new beginnings. For those of us biased in our perceptions by our position in the northern hemisphere, summer will soon be a memory; the leaves will change colour; and I’ll be back in school, where I belong.

I spent this summer doing research and quite enjoyed it. We didn’t make as much progress toward a solution as I had hoped, but I learned a lot, both about mathematics and research in general. I’m comfortable using LaTeX (which is sexy) and have had some experience with Macaulay2 (also pretty hot). I even went to a conference, something that surprised me.

With my research finished, I have these two weeks off before school begins on September 13. Next week I return to work at the art gallery. I don’t look forward to returning to the job that much; my relative solitude of this summer has left me even less eager to interact with people in a customer-service-based position. But I do miss my coworkers, my fellow front desk attendants, so I look forward to returning to them.

I anticipate another great year of school as well. This is my honours year for my math degree, and the Honours Seminar will consist of a sort of research-based project supervised by a prof. We’ll have to write a math paper and give a talk. This is a nice departure from lecture-based courses (I don’t much care for lectures); also, having done research, read papers, and written up results for the past four months, I feel somewhat prepared.

And with summer endings and fall beginnings come changes. My site last had a major redesign over two years ago. I’m still happy with the design in general; however, there have always been certain rough edges I wanted to correct. Now I‘ve done so. A few weeks ago, I rolled out tweaks to the design and significant changes to the backend.

I’ve reorganized the content on the home page. It’s my portal on the Web, something that lets people access my content whether it’s on this site or elsewhere. I‘ve tried to lay it out so that everything is on offer.

You’ll also notice that I have a new background image. Now that is definitely tea. The other image was tea, but ambiguously so, and the berries were an odd addition—it was a very Christmas-like cup of tea. It was the best photo I could find at the time. This new photo is exactly what I envisioned when I originally decided to use a cup of tea as my background image, and I‘m very happy with it.

For a long time, the only real content on this site has been my blog and the About section. Everything else consists of links and a little aggregated content. I have plans to change that soon and add more pages dedicated to original content (or specific aggregated content). For example, you’ll notice that my home page no longer displays my most recent book review from Goodreads. I want to keep my home page compact, and you can easily access my 15 most recent reviews from the books on the sidebar. Instead, I intend to create a new section of the site devoted to my reading habits—not just reviews, but top 10 lists, statistics, etc.

This sort of flexibility is thanks to the new backend. I’ve finally gone over to the dark side and started using a CMS—but not just any CMS. It’s Symphony, an XSLT-based CMS that is both minimalist and developer-friendly. The custom-coded backend I was using was rubbish, and I don’t need anything as powerful as an entire framework. Symphony is exactly what I need, and I highly recommend it.

It is Saturday, but it doesn’t feel like Saturday, mostly because I’m … at school. This is the last day of the CUMC. I’m in the last talk of the day, having chosen to attend “Perfect Matchings and Shuffling.” Afterward, there is the final keynote, which Ram Murty will deliver on the Riemann hypothesis.

Yesterday I went to a talk on fractal image compression. The talk itself was not stellar, but there were some good questions on the applications of this type of lossy compression, and the speaker addressed those well.

In the afternoon Aaron, Rachael, and I took a bus—yes, a bus—down to King St. This was my first time riding public transit, and it wasn’t in my own city! Aaron wanted to visit a small record store, Orange Monkey Records, and then i checked out a used bookstore known as Old Goat Books. I bought more books than I should have, considering they need to fit in my sparse luggage—but I couldn’t resist.

The final keynote of the day was delivered by Greg Brill, of Infusion. Although titled “The Evolution of Technology,” Brill’s talk was not what I expected. He has a Masters in computer science (after coming from a liberal arts background!) but talks like a showman rather than an academic or a businessman. He discussed how mathematics—and hence, mathematicians—are essential to the development of technology, particular business products. For example, he mentioned how his company had been working with motion-sensing technology similar to Kinect, and that the main problem was not a lack of technology but a lack of the mathematics necessary to achieve what we want in that area. Brill is very keen on the idea that we are moving from an idiomatic society to an idiom-less one and is convinced that mathematicians will help make that happen.

Dinner came in banquet form, and while the food was OK, the dancing was better. That’s right: dancing. I love to dance, and I had a great deal of fun on the dance floor for about an hour or so before calling it a night. I’m not as young as I used to be.

Tonight we fly home, and tomorrow I go to my nephew’s first birthday party (no weekend recovery for me). Then it is back to math research: reading papers, re-reading papers, writing algorithms, and making tea. CUMC has been fun, but I will be glad to be home.

It is Thursday, July 8.

After the first talk this morning—on set theory, particularly ZFC—I spent time caressing the lovely wireless network by way of uploading some photos to Flickr. When attempting to geotag them, however, I ran into the slight problem, in that typing “University of Waterloo” into the Flickr map’s location finder produced no results.

So, Yahoo!, in case you are wondering why people drool over Google and its products, here is a hint: we are lazy. When I type in the name of a major university, your map should be able to find it for me. I should not have to go find a postal code on my own, enter that, and wind up in the general vicinity of the campus. (I used Google Maps to find the postal code too, which just seems wrong). It is not that I am a Google fanboy, Yahoo!—they just do it so much better.

At lunch, I did something completely out of character and chose to be adventurous, purchasing bubble tea for the first time. My less adventurous self was soon vindicated. We went to a fast food place called “The Grill” for food. I attempted to poke my straw through the seal placed over my cup—urged on by Rachael’s encouragements of, “Just do it!”—and after one mighty stab, the straw went through … and the bubble tea exploded. A plume escaped from the top, but the cup also developed a leak in the bottom somehow, and it spilt all over the table and down onto the floor. We don’t cry over spilt milk, but what about spilt bubble tea?

I also decided to be adventurous when it came to food. The menu had a “lamb burger” on it. I have had lamb before, but never in burger form, so I ordered one of those. Its taste was similar to a regular hamburger, which disappointed me.

For some reason, I was lethargic after lunch and greatly desired a nap. I blame the heat. I struggled to stay awake and pay attention to the afternoon’s talks—first one on computability theory, and then another on universal algebra. After that, we had a little break before going for dinner. Rachael and I ordered some chicken fried rice from a Chinese place, while Aaron opted for shrimp wonton soup. The price was right and the portions huge—I could not finish mine, although I came close, while Rachael ate a lot and left even more.

The morning keynote speaker was Michele Mosca, from the University of Waterloo. He talked to us about quantum computing, with a particular focus on quantum cryptography. The talk was more about mathematics than of mathematics, with only a little actual math involved. I quite enjoyed the subject. Quantum computing is a concept that sounds like science fiction, but it is real; we have quantum computers—albeit primitive ones—right now! The future is here.

It is Wednesday, July 7. The CUMC talks began today.

I went to four talks today. Rather than summarize them all—I enjoyed them all—I’ll mention some highlights. The first talk of the afternoon was both my least favourite and most favourite talk. Entitled “The Ontology of Mathematics: Do Numbers Exist?,” the presenter read from dense slides, which did not make for the most riveting experience. There was some lively discussion among the audience, however, and I enjoy talks like that.

Comparing CUMC to the Combinatorics & Optimization workshop that preceded it, I prefer the student talks of the former. The topics are so varied—there is so much choice within each time slot, that it is difficult to decide which talks to attend. The atmosphere is less intimidating, because it’s undergraduates talking to undergraduates. I almost regret not giving a talk myself—almost, for it would involve public speaking, and long gone are the days when classes made that mandatory.

There were two keynote speakers, one at lunch and one at the end of the day. First, Frank Morgan, from Williams College, gave a talk on densities and the Poincaré conjecture. As I have never studied differential geometry, most of the mathematics went over my head. The audience in general got into it, however, asked great questions, and we all tried answering the questions Morgan asked of us. In the end, I learned from the talk, which is all one can ask, right? The second talk was easier for me to understand, because it involved matrices and metric spaces. I love metric spaces! Carsten Thomassen, visiting from the Technical University of Denmark, was the speaker; he also gave two talks at the Combinatorics & Optimization workshop.

After the last keynote, Aaron, Rachael, and I walked down to the campus plaza, which has a cornucopia of restaurants. We elected to share a pizza, placed an order, and then took it back to the air-conditioned environment of another building. The Waterloo campus is beautiful, but the heat makes any sort of lengthy walk unattractive. Waterloo campus is also big—compared to Lakehead’s, at least—so every walk is lengthy.

The pizza proved a good choice, as it was tasty and filling. We walked back to the residence where Aaron and Rachael stayed, and then Rachael and I listened to Aaron’s talk, which he is presenting tomorrow afternoon (it concerns the classical Cantor set). Tomorrow I plan to attend talks on set theory, computability theory, universal algebra, and perhaps one on range-sum queries.

I’ve uploaded some photos from my trip so far. They are all accessible in this Flickr set, and new ones will be added there as well.

It is Tuesday, July 6.

Today’s four talks began with electrical networks and random walks. That is, suppose you have a graph that describes a network through which electricity flows. Starting at a vertex x, what is the probability that, when walking at random along the graph, we will arrive at a vertex s instead of a vertex t? This talk was very easy to follow (for which I am thankful), even though I don’t have any engineering or physics background with which to understand the electrical current aspects (like voltage law).

Unfortunately, the second talk involved probability. Probability is great, but I find it very difficult, so this talk was hard to follow. The third talk was about embedding locally-compact metric spaces on surfaces (it is not as scary as it sounds). Finally, the fourth talk was about matching polynomials. The speaker went rather briskly, so it was difficult to take detailed notes, but I enjoyed the subject. Before this summer, I had no idea that polynomials and graphs went so well together. Now it seems like they’re inseparable.

And that concludes the Combinatorial and Optimization workshop. There was a banquet for CUMC at the Huether Hotel, and it was not what I was expecting—very crowded, although the food was good.

Prior to the banquet, Phelim P. Boyle delivered the first keynote speech for CUMC. Boyle is a mathematician of finance, he is interested in the recent financial crisis. He discussed option pricing and the Black-Scholes equation. As with probability, finance is an area of mathematics I avoid, because of its strong dependency on number. Nevertheless, I enjoyed the talk.

I now have access to reliable wireless on campus, although such a phenomenon continues to elude me at my grandparents’ house. Never has my dependency on the Internet been so apparent.

I wrote this last night at my grandparents’ house, which has no Internet connection I can feasibly use (dial-up does not count), so I had to wait until today to post it from the University of Waterloo campus. All references to “today” refer to Monday, July 5.

This week, Rachael, Aaron, and I have travelled to Waterloo, Ontario for two math conferences. The first is the Combinatorics & Optimization Summer School, a two-day event consisting of several talks and, yes, food! The second is the Canadian Undergraduate Math Conference, which also entails much talking and eating. I was reluctant to attend at first, because I dislike travelling. However, my grandparents live in Waterloo, so this was a convenient way to visit them for a week while still getting paid. With that incentive, I managed to convince myself that these conferences would be interesting and probably even useful to my research. This was only the first day, but so far I remain convinced in those respects.

I’ve been up since 4:30 in the morning. Let me take a moment to reflect on the fact that we flew from Thunder Bay to Toronto in an hour and a half, traversing—or rather, bypassing—the largest freshwater lake in the world. And we did it in a metal behemoth that harnesses complex physics and engineering to work miracles.

Flight is awesome.

OK, science-geeky moment over: back to math.

Today there were four talks. We arrived late to the first talk, by about fifteen minutes, but it was still very interesting. It concerned the colouring of graphs on surfaces.

Following a short break, the second talk discussed the Borsuk conjecture, which asks a question about the existence of a certain partition of any set in d dimensions. This was my favourite talk of the day, for several reasons. Firstly, I learned a lot about the diameter of sets, a topic with which I was not familiar. Topology involves a lot of geometry, something for which I lack proper intuition. Yet still it interests me, probably because of its ability to formalize that geometry. I like abstraction. Secondly, the presenter told the story of how Kahn and Kalai proved the Borsuk conjecture false. They took a problem that had been open for nearly seventy years, solved it in a week, and wrote a short, about one-paragraph proof. It’s a wonderful example of how unpredictable and exciting mathematics can be: sure, sometimes math research involves long, boring days reading papers and staring at a problem on a chalkboard. Sometimes, just sometimes, it leads to the most interesting results.

After lunch, we listened to a talk about cutting cake—specifically, how to divide a cake into sections such that no one person would complain that he or she received a worse section. It was by the far the most accessible of the three talks, and the presenter had a very engaging manner. Unfortunately, my fatigue caught up with me during this talk, and I found myself nodding off during the most interesting parts. We learned a little about hypergraphs, which, as the name implies, are like regular graphs but on crack.

The last talk was on symmetric groups and their combinatoric properties. Last week, my prof showed me how we may be able to make use of the symmetric groups to solve the problem on which I‘m working this summer. The talk was more of a review of things I had already learned in group theory two years ago, which was still useful considering the gap in time.

The day began winding down as we went to a pub-like house for dinner. Then we trekked across campus to the residence where Rachael and Aaron are staying. We got lost in the process, of course, but eventually found our way thanks to a map and, moreso, a helpful student.

More to come on Tuesday’s schenanigans tonight or tomorrow morning!

Last week, I discussed how maths is hard, but I spent plenty of time solving a Rubik’s cube anyway. At this rate, you are going to get the idea that I don’t do any work at all. Nevertheless, a desire for accuracy and lulz requires me to remain truthful regarding how I spent this week in the office.

We made a piñata.

We named him Stanley the Resurrection Pig.

I don’t recall who came up with the initial idea. As with all good, crazy plots, it starts off as an innocuous hypothetical scenario: piñatas equal fun, fun equal good, we could make a piñata! This is the last week all four of us will be in the office together—Aaron, Rachael, and I are going to Waterloo next week for a conference, and Jessica is off to Ireland, returning only after Aaron and Rachael’s contracts are finished. So if ever there was a time to set aside the math papers and construct a papier-mâché animal, then savagely beat it to a pulp, this was that time.

None of us are piñata-making experts, and that was probably for the best. Rachael had some experience with papier-mâché—also for the best—so we made her foreman and gave her a silly newspaper hat to go with the title. In remarkably little time, we gathered together the hodge-podge of materials required to manufacture a piñata. We decided on a simple shape, assembled the skeletal structure from balloons, and mixed up a batch of goo to begin the work of creating Stanley.

Over three days, Stanley emerged from a series of colour balloons. He grew stubby legs, ears, and a snout. We named him Stanley because none of us knew anyone named Stanley, and it sounded like a good name for something we would beat to death. (I apologize to all those named Stanley reading this.) Jessica, in particular, was quite bloodthirsty about the whole project. By Friday, however, as we stuffed Stanley full of candy and trussed him in string, we were all savouring the anticipation of Resurrection-Pigpocalpyse.

Stanley met his demise rather quickly. We took him outside, where it was the warmest it has been all summer so far, and suspended him upon a suitable tree branch. Jessica, as the aforementioned most eager participant in this piñata-bashing, got the first swing. I had brought a thin, metal beam that had been propped up in one corner of the hallway outside our office with other thin, metal beams, but we started with a stick to maximize Stanley’s torment. After a few swings from Jessica, however, the stick broke in two. Stanley one, us zero.

So we switched to the metal beam, and Stanley’s death came swift. Jessica pretty much decapitated him with a single, fearsome blow. Aaron, Rachael, and I quickly followed, each of us contributing to his destruction in our own way, until finally he lay on the ground, battered and broken, a shell of his former self.

Stanley was no more. But in his death, he gave us one final gift: lots and lots of candy. Oh, and math riddles. But moreso candy. Really, way too much candy. We had all brought candy, and even though much of the chocolate melted from the heat, there was more than we wanted to take home with us. There is still some of it languishing in the office despite our forthcoming week-long absence.

I could talk about what I‘ve been researching this week, how my supervising prof was in town only for the two days we were dunking our hands in flour-water to make a piñata in the office. I could mention that I’ve started running programs on SHARCNET and it’s awesome. Really, all of these things pale in comparison to spending a week making, and breaking, a piñata.

This was the eighth week of my research. I’m now halfway through my summer job, and it feels like I’ve barely begun. Wow.

Farewell, Stanley the Resurrection Pig. You served but a brief, miserable existence, but you served it well. So long, and thanks for all the fish—er, candy.

I like to joke with my friends about how easy I have it this summer. I‘m sitting in a cozy little office with a fan, proximity to a kettle, and a high-speed Internet connection. Unlike a summer research student in, say, chemistry or biology, I don’t have to manipulate lab equipment or sex fruit flies (Cassie :P). The extent of my experimentation will involve uploading programs to a high-powered computing network and asking it kindly to compute a few more numbers for me. I Google math papers relevant to my problem, try to understand what they say, and see if I can come up with my own ideas. One thing I love about math research, especially in my area of interest, is how much it’s thought. All I really need is a blackboard and chalk, or pencil and paper. (That being said, the high-powered computing network does help when I get to the computation step!)

Of course, it’s not all fun and games (even though I did learn how to solve a Rubik’s cube last week). Maths is hard! And right now, even though I’ve been in university for three years, I feel like an amateur groping around an unsolved problem. I know that research can be like that in general, and I’m still having lots of fun—and learning a lot. Nevertheless, sometimes I feel like a poser. And nothing is worse than a math poser!

I was all excited, two weeks ago, because I had almost finished an algorithm to compute the spreading number recursively. I was tackling the problem as one of finding a maximum independent set. The spreading number is, among other things, the cardinality of the maximum independent set of a certain type of graph. (The covering number is an analogous clique cover cardinality). The general problem of finding a maximum independent set is NP-hard. This means that there likely isn’t a very efficient algorithm for solving the problem (if there were, then P=NP, and that’s way above my pay grade). The best I could hope for was a good algorithm for my specific case; indeed, that was my hope for this algorithm.

After returning from the weekend, I finished the algorithm and happily set Macaulay2 to work, asking it to compute the spreading numbers and compare it with the values we already know. Alas, there were discrepancies, and I quickly understood why: I had made a fundamentally flawed assumption in constructing the algorithm. So while the algorithm did exactly what I wanted it to do, it turns out that what I wanted would not give me the graph’s maximum independent set.

Back to square one!

Frustrated but not very surprised am I. The problem is non-trivial, so I did not really expect such a simple solution. And I have plenty of summer left in which to try new ideas. Right now I am looking at Hilbert series. Most computer algebra systems, including Macaulay2, use Hilbert series to compute the dimension of rings (and this is how my professor’s orginal algorithm computes the spreading number). For larger rings, this computation takes up too much memory.

The easiest solution is, of course, to throw more memory at the problem. We had hoped my computer would be able to compute at least another two or three of the numbers, but this was not to be. Even without any refinements to the algorithm, however, SHARCNET should blow my computer out of the water. This week, I am looking at ways of breaking the computation of the Hilbert series into independent tasks so I can make use of throughput computing.

Oh, and I did learn how to solve a Rubik’s cube. I obtained one in my young adolescent days, but because I have poor spatial skills, I was never able to solve it on my own. Last week I observed Rachael manipulating her cube like a pro. I expressed my admiration and awe, and she just shrugged and mentioned that it was a matter of using certain algorithms (which makes sense). I was doubtful of my ability to learn the necessary algorithms; fortunately, I think I understand enough now to solve the cube reliably. I doubt I’ll ever be a speedcuber, but that is one puzzle down.

Now back to my shiny infinite polynomial series.

Yes, yes, I know. At this rate, my weekly recap will become bi-weekly. I didn’t do a lot the week before last, owing to Victoria Day making for a shortened week. So rather than two very short blog posts, I decided to forbear and write one short blog post instead.

The last two weeks have been more reading, more learning, and a little thinking. I hesitate to ascribe a label like “productive,” since it’s hard to quantify. I think I understand my problem now, but there remains a lot for me to learn in order to start trying solutions.

I tried running the original algorithm for computing the spreading number, which was written in CoCoA, on my computer. I had hoped that my 2 GB of RAM and 1.83 GHz processor would have enough memory to compute some additional numbers. Alas, CoCoA stubbornly crashed (after several long hours) each time I instructed it to do so.

So I ported the code to Macaulay2. It’s even slower, which makes me suspicious that I’m missing something—after all, I am learning both languages, so I‘m sure that in transliterating the code I managed to miss an obvious way to make it more efficient. Still, it looks like the original algorithm won’t produce many more useful results, at least not until I stick it on SHARCNET.

My supervising prof pointed me to a series of lectures he gave on combinatoric commutative algebra. Last week I started working through those, and I’ll continue doing so this week. He’s given me several promising “leads,” I suppose you‘d call them, but at this point, I have to start exploring avenues of interest and seeing if they produce any interesting results. I’ve already toyed with some alternative approaches in Macaulay2, familiarizing myself more with the language, but I think I need more experience with the mathematics first.

Probably the most significant news of the past two weeks would be my decision to attend the Canadian Undergraduate Mathematics Conference at the University of Waterloo and the Combinatorics & Optimization Summer School preceding it. Initially I was reluctant to go, because I don’t like to travel, but Aaron and (maybe) Rachael are going, so I won’t be alone. Plus, I’ll get to visit my grandparents. That’s July 5-10, a few more weeks away. Until then … time for more learning.

Shorter entry this week, as I didn’t do much new and exciting in week 2 of my research project. I‘m still having fun, but because it’s so early in the summer, that fun mostly takes the form of reading.

As tweeted earlier, the secret to reading (and understanding) math papers is simple. First, always read it twice. Then read it again. But to make sure you really understand, you need to take notes. Write down what’s implicit in the paper, the steps the author leaves out because “it is obvious” or “it is clear to the reader” or, even worse, “this has been left has an exercise for the reader.” Once you‘ve done that, the final step is to read the paper again.

I spent all week reading two papers, one of which expands on the findings of the other. The first investigates the spreading and covering numbers in relation to the ideal generation conjecture. Much of the paper goes over my head. Nevertheless, there were some very useful figures, and the use of graph theory in one paper and set theory in another helped improve my comprehension of what these numbers are. The second paper, in particular, was devoted to finding explicit values and bounds for the covering number using a combinatorial/set theory approach.

One of my goals is to improve, if I can, upon the bounds found in these papers. The actual values computed by my supervising prof suggest that there’s room for improvement. I’m a little daunted by this prospect. I feel like I understand the proofs present in these two papers regarding the bounds for the covering number … but I‘m not so sure I understand the procedures well enough to build upon them. Granted, I’ve only been doing this for two weeks. As the summer progresses, I’ll learn more and become more confident. For now, however, I’m just a wee bit intimidated by what I will try to accomplish.

Don’t mistake trepidation for discontent. The best is yet to come! Soon I’ll be playing with CoCoA and Macaulay2. This week, I‘m learning about resolution, which leads to a generalizatio of dimension from ordinary vector spaces to modules. Oh, and I’m having a lot of fun learning how to typeset my proofs in LaTeX. Math is totally the language of the universe, and LaTeX is its markup.

I am now into the second week of my NSERC summer research project. So far, I’m having a lot of fun. The subject of my research is interesting and exactly the type of mathematics that I want to study. The “daily grind,” such as it is, does not grind at all—it helps that there are three other undergraduate students doing research this summer, and we all share the sessional lecturer office. We can distract each other, when needed, and pick each other’s brains for help with particularly puzzling proofs.

So what exactly am I doing? Well, it’s esoteric even for those who enjoyed math up until the first years of university. I‘m going to drop some math jargon in the next few paragraphs, so don’t worry if your eyes start to glaze over. Photos and hilarious video will follow!

Since my prof was leaving town at the end of the week, we met several times so he could give me some lectures and we could discuss my project. The work I’m doing relates to ring theory, which is a course I took nearly two years ago, so I have a lot of review to do. Most of the week, like the next few weeks will, involved preliminaries. I found all of the references my prof recommended to me. I began reading the three textbooks among those references, learning about monomial ideals and simplicial complexes.

These, however, are but means to an end. After I have mastered the secrets of these wonderful algebraic concepts, I can use them toward the eventual goal of finding better algorithms for calculating the spreading and covering numbers. These relate to the maximum and minimum dimension, respectively, of a monomial subspace of a vector space over all polynomials of a given degree such that the subspace fulfils two respective properties.

On Thursday, my prof went over what’s changed since he and his colleagues wrote the paper from which my research project comes. In particular, they’ve learned about a connection between edge ideals and the Stanley-Reisner ideal. In the paper, they showed that calculating the dimension of the Stanley-Reisner ring is sufficient to find the spreading number. (A similiar result makes calculating the covering number possible.)

This connection is really cool for two reasons. Firstly, it makes the connection to graph theory stronger, which gives us another avenue for exploring the problem. Secondly, it might provide an alternative way ofcalculating spreading numbers (graph theory is also useful in this respect). The algorithm in the paper finds the Stanley-Reisner ring and then uses a computer algebra system to find the dimension of the ring. They did this on a Pentium II, so they could only find a few of the numbers before the calculations became impractical given the available computer memory. Computing power has improved considerably since then, so my first step will be to see how my little laptop compares against their Pentium II using the algorithm in the paper. Later in the summer, I’ll be creating alternative (hopefully more efficient) algorithms in Macaulay2 and running them on SHARCNET.

Of the three other students sharing the office with me this summer, Aaron is in the same year as me, and Jessica and Rachael are a year behind us. Aaron and Rachel are working on the same project, which involves fractals and Cantor sets. Jessica is also working on something related to commutative algebra (affine varieties and Gröbner bases). So not only do I get to learn about simplicial complexes and monomial ideals, but I’ll be learning about affine spaces and some more real analysis as well.

And for those of you who wonder exactly what math research looks like, I can attest that it’s pretty much like this clip from The Big Bang Theory. Aaron and I spent a good deal of Friday afternoon staring at my faulty proof regarding prime ideals on the chalk board. I did manage to figure it out eventually, but imagine if we had had a montage!

This January, I applied for a summer Undergraduate Student Research Award (USRA) from the Natural Sciences & Engineering Research Council (NSERC). Lakehead University has 20 such awards to give to applicants this year, and on Monday, I learned that I am the recipient of one!

I was (still am) a mixture of elation and trepidation. Part of me is still in a state of shock and can’t quite believe that this is real. I spend a good half hour after learning I got the grant just trying to calm down so I would not run up to everyone I encountered and yell, “I GOT A GRANT!” Another part of me is saying, “What do you think you‘re doing, Ben? You don’t even understand what it is you’re going to be researching!” As anyone who has ever looked at a higher math textbook knows, the language is just scary sometimes.

I applied for the NSERC grant for two reasons. Firstly, it’s a different summer employment opportunity than my default, which is the art gallery. Don’t get me wrong: I love working at the gallery. You can’t beat the hours, and I have an awesome boss—she took the news that I wouldn’t be working there over the summer much easier than I thought she would. Nevertheless, I’ve worked there for four consecutive summers. I‘m not averse to trying something new, particularly something related to my area of interest.

Secondly, since this is a research position, I’ll get a chance to experience exactly what “math research” is all about. Sometimes people will ask me why I’m becoming a high school teacher instead of going on to graduate school and becoming a professor; usually my answer is somewhere along the lines that I‘m not sure I’d like doing “math research” and writing “math papers.” I‘m more in it for the teaching. This grant is a perfect way to see if, in fact, I like or dislike doing research, without committing to something like graduate school first.

So I’m excited about this change, but also just a little bit anxious—it is a big change in how I’ll be spending my summer, and a different responsibility. After four years at the gallery, I’m so used to doing the same thing every summer that it’s hard imagining myself doing anything else.

The position itself is a full-time for 16 weeks. My area of interest in mathematics lies in commutative algebra, so Dr. Adam Van Tuyl has agreed to be my supervisor. He’s come up with a neat project for me, and I’ll try to explain some of it. I don’t fully understand what I’m doing yet myself; for the first few weeks I’ll need to review my ring theory from last year and then work to learn new concepts we didn’t even cover in that class.

Ultimately I’ll be continuing work that Dr. Van Tuyl did on computing spreading and covering numbers for monomial ideals. One of the issues he and his colleagues encountered when they first worked on this problem was a lack of computational power for calculating values for these numbers. Later in the project, I’m going to be writing my own algorithms for calculating these numbers, and I should be able to run them SHARCNET, a network of high performance computers maintained by several academic institutions in Ontario.

I plan to blog about the project as the summer goes on. I start working on May 10, so I probably won’t have much to say on the subject until then. For now I need to focus on finishing the school year!

Tonight Stargate Universe premiered, and I wanted to share my thoughts on it. However, I feel guilty blogging about a television show when I haven’t blogged about arguably more important matters, such as life.

With a month behind me, I feel good about the school year so far. I only have four courses this year: Introductory Analysis, Partial Differential Equations (PDEs), Introduction to Mathematical Probability, and Speculative Fiction. Three math courses and an English course. All of my math courses are interesting, and I was excited to take the English course the moment I saw it offered. I’ll discuss it first, since the rest of the post will be about math.

My Speculative Fiction course is covering only science fiction this section—which is fine. Although I love literature in general and would gladly have taken something like Victorian Literature if this course hadn’t been offered, the chance to read and discuss science fiction for credit is not something I was going to overlook! We’re reading The Time Machine and The War of the Worlds, The Left Hand of Darkness, Do Androids Dream of Electric Sheep?, Neuromancer, Dawn, and Singularity Sky. We also have to watch Blade Runner (a film based on Do Androids Dream).

Of my math courses, Introductory Analysis is my favourite because it comprises my favourite aspect of math: proofs. Specifically, I love algebraic proofs—the more abstract the better. I love math but don’t like numbers so much. PDEs are fascinating and challenging as well; the course is very much oriented toward application, however, whereas I‘m more interested in theory. Unfortunately, my ardour doesn’t quite extend to probability, but I think I’ll survive—so far it hasn’t tripped me up too much.

My involvement in math at the university extends far beyond courses! Last term I marked assignments for a first-year calculus course; this fall I‘m marking a second-year linear algebra course. Moreover, I’m tutoring in the new Lakehead Math Assistance Centre (LUMAC for short). Both of these jobs are paid positions, which is a nice income in addition to my gallery job while also providing me with relevant experience for my future career.

Having spent a few sessions tutoring, I can already say that I enjoy it. We’ll see if it stays that way once the flood of people arrives the week before midterms! For now, however, it’s fulfilling. Plus, it gives me a nice review of first-year courses, like basic calculus, that contain skills I’ll always be needing but don’t always practise as I should.

So I have a very math-filled term, it appears. I like to use the phrase “inundated by math—and I love it.”

More Reasons to Love the Guild

I‘ve already preached my love for The Guild, a webseries by talented comedians and actors, including Felicia Day. Well, even as they work on a third season, they’ve released a fantastic music video:

Who Said Math Can’t Be Fun?

Well you were wrong, whoever you were. Mathematicians from Carleton University and the University of Ottawa modelled different responses to a zombie apocalypse and concluded that the best way to survive a short-term zombie apocalypse is to impulsively eradicate all zombies. Ladies and gentlemen, load your engines and start your shotguns.

I’m an Uncle

In July, my sister, Tara, gave birth to a very little boy named Clark! So I’ve got a nephew, which makes me an uncle, and that is sublime. I got to meet Clark today for the first time, which called for the typical point-and-shoot photos that wind up on Flickr somehow.¹ If I‘m short on words about Clark, it’s only because I don’t really know him yet—he doesn’t know himself yet, since he’s only a month old and still new to the world. I will report back in four or five years!

---

• [ 1 ] I blame the gnomes, if only because they haven’t unionized yet like the orcs did.