My goal as a teacher is for my students to develop qualitatively new ways of thinking about the real and imagined worlds they live in. I believe that they are best able to do this when they are active participants in the creation of knowledge. Accordingly, my curriculum is structured around guided discovery, design challenges, problem-solving, and dialogue. To help my students set and achieve goals, I provide them with a constant stream of information about their progress through standards-based formative assessment. I work hard to create a classroom environment free of discrimination where all students feel safe enough to take the intellectual risks I expect them to.
Harbord Collegiate Homework Club, November 2012
During my first practicum, I volunteered at Harbord Collegiate Institutes’s after-school homework club. This “club” is a room open Tuesdays and Thursdays from 15:15 to 16:45, and generally attracts about 20 students per day. Of these, half are there to socialize, do homework, and use the computer facilities, and the other half are there for the free tutoring. There are usually two “regular” teachers at the program each day, but, as many of the students were there for math help and there was only one math teacher there for one of the days each week, I found that I was able to be very useful. (Besides tutoring math, I also tutored English, Chemistry, and Physics, whenever there was nobody more qualified available). I spent a total of approximately 12 hours in this volunteer position.
In the past, I have found that working individually with struggling students has been an excellent way to build a reputation among all students as a caring and competent teacher. This was certainly the case with my work at Harbord. I believe that my visibility at the homework club was a significant factor in the positive relationship I was able to quickly build with my students.
Because the students I worked with at the homework club have such different backgrounds (they were mainly from Asian-Canadian or Middle-Eastern families)) from the students I am accustomed to working with in the BC and the Yukon (who are mainly Euro-Canadian or First Nations), helping them one-on-one broadened my perspective on small-scale teaching. I also gained a sense of how powerful positive social interactions between students and teachers can be, even if they do not involve any actual “teaching” or even much extended discussion. Simply by being around the students for large amounts of time, and treating them with casual friendliness and respect, I felt that their willingness to learn from me during class time, or to ask me for help at the homework club, increased greatly.
Another thing I gained from my praxis experience was a better sense of how the school functioned as a whole, rather than being only exposed to a few different classes, all in the same subject. This impression was even stronger because the homework club was located in the GLC classroom. For many students, the homework club was a safe and comfortable place to hang out after school more than it was a tutoring centre, and my conversations with those students were enlightening.
In summary, the volunteer work I did during my praxis assignment both improved my relationship with the school at which I did my practicum, and helped me to develop as a teacher. The specific standard that this refers to is “support for student learning.”
The Mathtwitterblogosphere, 2009-Present Ever since I became interested in teaching mathematics, I have sought ideas and inspiration from a variety of sources. Some of the best resources I have found have been from a group of loosely interrelated math teacher bloggers that their unofficial Godfather, Sameer Shah, has described as the “mathtwitterblogosphere.” For a full list of those that I have found the most useful, click here.
The ideas that these teachers discuss constantly challenge me to improve my approach to teaching. For example, Dan Meyer‘s blog has transformed my educational philosophy in ways I could not have imagined five years ago; in fact, in ways that are in direct opposition to some of the reasons I first became interested in teaching math. When I began I was excited about how, the higher you went in math, the more different things became manifestations of a shrinking number of increasingly abstract structures. I wanted to share this excitement with my students by teaching from the general to the specific: give them just enough theory to deduce what they need to know through logical argument. I’m still excited about it (and I still plan to teach deductive proofs), but Meyer has convinced me that top-down abstraction is a pedagogical dead-end: to really care about mathematics, my students need to experience abstraction from the bottom-up.
So now, rather than beginning with “here are the field axioms,” I get my students to work with structures that display mathematical patterns – whether that means literally embodying polygons undergoing plane transformations or folding origami proofs of the Pythagorean theorem – with the philosophy that they should contribute to performing the resulting abstractions, rather than passively benefiting from my concise understanding of mathematics. This has been a radical change in my approach to teaching, and relates to the specific standard “knowledge of teaching practice.”
Pelly Road Math Trail, January 2013
As a review activity for a group of Calculus students I volunteered with in 2012-2013, I designed this winter-themed math trail in the woods behind Riverdale, a subdivision in Whitehorse. There were three goals I wanted to accomplish with the trail: first, for the students to deepen their understanding of the techniques they had learned so far in the course by applying them to solve open-ended problems; second, to provide a fun team-building event as they began the difficult final month of the course; and third, to informally assess their proficiency in the key skills of the course. The trail consists of six (and a half) stations and is designed to be usable for differentiated instruction in the future: each station contains one page of questions that are answerable by a junior high-school student, and a second page of calculus questions.
On the day of the math trail the weather was on our side: the temperature was an unseasonably warm -5° C, which made writing notes and answers easier for the students and myself than I had anticipated. The students used measuring tape, string, and a clinometer they built by modifying a protractor to answer in situ questions that required the use of optimization, integration, and infinite series. Everyone enjoyed themselves, but the event felt rushed, and the students solved only a few of the problems to the depth I was hoping for.
This event directly relates to the following standard of practice: “[Members] use appropriate pedagogy, assessment and evaluation, resources, and technology in planning for and responding to the needs of individual students and learning communities.” In this case, my students’ learning need was to broaden their understanding of Calculus, my form of assessment was formative, and the resource was the natural environment.
The next time I run this math trail, or an activity similar to it, I will reduce the number of stations (or, even better, work with another teacher, divide the students and stations, and then have each group present their findings to each other back at the school). This will allow for a better discussion about each question, and thus help achieve my goal of having students achieve improved solutions through dialogue with each other and with me.
Logarithms Lesson, November 2012
During my first practicum, I delivered the first few lessons for the logarithms unit in Grade 12-U Math. Since logarithms are one of the most notorious topics for students learning through rule-memorization rather than understanding, I took a historical approach to the subject in an effort to help my students see why logarithms were invented. (I stole the basic idea of what follows from Shawn Cornally). I describe this lesson here to demonstrate how I “integrate a variety of teaching and learning strategies, activities, and resources.”
I began the lesson, as I always do, by posting a question aimed at connecting the students’ prior knowledge to the new material I want them to learn. In this case, I asked the students to explain why the following “laws of exponents” are true: and . (Since logarithms are the inverses of the exponential functions with the same base, this theorem is equivalent to the “logarithm laws” and that my lesson was focused on).
Once the students had reacquainted themselves with the mechanics and proof the the exponent laws, I began the part of the lesson where I try to make my students feel an intellectual need for what I want them to learn. So I asked them which of the following two questions is easier to do by hand: or
Of course, the answer is that addition is easier. Then I asked them to imagine how hard it would be to try to do serious science – like figuring out how and why the planets and stars move – without access to a calculator, and pointed out that this is exactly what some people were trying to do in Renaissance Europe. Finally, I posted the following arithmetic problem on the board, and promised them they would be able to solve it in minutes by the end of the class: (Lesson slides here). (Aside: this problem isn’t flashy, or even that innately interesting, but I was banking on my students’ natural love of challenges. It worked).
The students spent most of the remainder of the lesson working on the worksheet I made for them that was intended to help them come up with the formulae and and to see how they could be applied, together with a table of logarithms, to solve difficult multiplication and division problems. Since my goal was for the students to be able to do something with their knowledge as quickly as possible, I had them focus exclusively on base-10 logarithms for this first lesson. At the end, I re-posted the monster arithmetic problem from earlier, and about a third of the class was able to solve it.
I felt good about this lesson, but of course there are things I wish I had done better. For one thing, the more I teach math, the more I want to make review – but not review drills – a bigger part of my teaching. I think the main difference between those students who really benefited from this lesson and those who did not was in how comfortable they were with the reasoning behind the exponent laws, and if I could do this unit again I would probably spend an entire day on some kind of activity designed to re-active their knowledge of exponents. I also think it would have been worthwhile to have the students create their own logarithm tables (using the exponent key on their calculator and binary search) to get a better sense of how they work and how changes as increases, before I asked them to apply the tables. (Of course I would never ask any one student to calculate for more than a few values of this way, so we would crowdsource it). Both of these changes would help the students move slowly but steadily up the ladder of abstraction, rather than being pulled up it by their bootstraps.
Three-Act Math Translation Project, Upcoming
As I mentioned above, one of my most important influences as a math teacher is Dan Meyer. Among Meyer’s main contributions has been the popularization of “three-acts” style lessons, in which students are presented with a compelling situation and must determine what the most appropriate mathematical question is, and what information is needed to answer that question. Meyer maintains a database of the three-act lessons he and others have put together.
The problem, for me, is that all of this material is in English, and sometimes I teach math in French. To solve this problem for myself and for other Canadian math teachers, I have volunteered to lead a translation project at my school in Whitehorse. Meyer has stated that he intends to place our translations alongside his original material on his website.
The standard that this addresses is to “act both as team members and as team leaders.”
FH Collins Secondary Problem-Solving Club, October 2011 – June 2012
While working as a substitute teacher in Whitehorse, I noticed that I was getting better-than-expected reactions from students whenever I put an enrichment math problem upon the board. In an effort to take this energy and go somewhere with it, I began a problem-solving club that quickly reached over a dozen members – far more than I was expecting.
The problem-solving club studied a variety of different mathematical topics with a focus on the number theory, abstract algebra, and calculus. Some of the students also learned how to write computer programs using Python. Eventually, some members joined who were in younger grades, so I began to write problems like this one that were approachable by students at all levels.
The specific standard of practice that this relates to is to “motivate and inspire through sharing [my] vision.” In a voluntary, unevaluated setting, I encouraged students to achieve excellence in mathematics.
Through the problem-solving club I was able to genuinely inspire some students to pursue mathematics more seriously. Still, there are things I would do differently next time. For example, the material the club covered reflected the opinion I held at that time that math instruction should follow the axiomatic method – an opinion I no longer hold in most circumstances. I think that a better way to teach enriched math would be to balance strict logical approaches to mathematics with real-world challenges and the development of students’ mathematical intuition.
MOOCS (Massive Open Online Courses), 2011-2012
I graduated from university with a weaker background in science and mathematics than, in hindsight, I wished I had. To fix this problem, during and after my MA program, I took courses from both the University of British Columbia and the University of Athabasca. But the courses I took that have had the largest effect on what I believe is possible in education are two massive open online courses I took through the computer science department of Stanford University and its private offshoot, Udacity. The specific standard that this relates to is to “engage in a variety of learning opportunities both individual and collaborative that are integrated into practice for the benefit of student learning.”
While there had been MOOCs prior to Sebastian Thrun’s 2011 Stanford course on artificial intelligence, none had ever achieved the same level of media buzz or enrolment. I signed up for the course as much to see what all the buzz was about, and what contributions such courses could make to my own teaching practice, as I did to learn about the content. Although I certainly have some criticisms of the way it was taught (see below) the experience really opened my eyes to the possibilities the Internet will create for students, especially for students with special interests or who learn at a faster rate than their peers. Giving students access to world-class researchers and a huge variety of possible subjects to study, no matter where they live, definitely increases my ability to meet the needs of all of my students. This is especially relevant in the Yukon, which lacks many of the higher-end academic facilities found in large Canadian centres.
There was one thing about the Artificial Intelligence course I took through Stanford that I found very strange: it was billed as being the same as the course being taught to actual Stanford students, but that was false advertising. While both courses covered the same content, the levels of richness were drastically different. In the online course, I, along with the other 160,000 students across the world were responsible for demonstrating our knowledge of the basic algorithms used in the field of artificial intelligence through weekly quizzes and a final exam. By contrast, the students who took the course in person at Stanford were involved in applying their knowledge to writing increasingly complicated (and fun) programs based on the classic video game Pac Man. As a result, they no doubt learned the material on a much deeper level. Directly experiencing this disparity of approaches to the same topic by the same professors helped me articulate one of the cornerstones of my philosophy of teaching: that I want to help my students move creatively back and forth between theory and applications, in order to reinforce both. However, I still believe that MOOCs do offer me something as a teacher: I would like to try using a MOOC as the lecture part of a “flipped” classroom in which the in-class sessions are dedicated to rich problem solving and projects.
So, my participation in the two MOOCs both helped develop my content knowledge in the areas I teach and gave me new ideas about teaching itself.
Keith Devlin famously described mathematics as “the science of patterns.” Since art is composed largely of patterns – of colour, shape, sound, and texture, among other things – it is not surprising that many artists incorporate mathematics into their work, and that many mathematicians produce works of art based on their field.
Recently, I have begun consuming, exploring and producing different forms of mathematical art. So far, my own work has been un-self-consciously derivative, as I have been practising the basic techniques that define different mathematical art forms. For example, inspired by Margaret Wertheim, I have experimented with exponential crochet in order to create a hyperbolic annulus, one of the basic shapes she used to create her famous crochet coral reef. I have also been learning a variety of advanced origami techniques using designs developed by experts, with the ultimate goal of learning how to create my own designs using Robert Lang’s textbook.
I look forward to bringing art into my math classes for multiple reasons. First, art is a powerful and often overlooked context for some topics in the high school math curriculum: vector graphics are based on function transformations, origami provides a number of opportunities for problem solving in a variant of Euclidean geometry, and families of curves and polar co-ordinates can make beautiful designs. Second, art has the potential to engage some students who are less excited by the traditional science-based approach to math (while I do not wish to be essentialist, I believe there is a gender component to this). And third, using art in the math class produces artistic products, which can then be posted in hallways and serve to improve the reputation of mathematics as a whole within the school.
The specific standard that this relates to is that I “understand that teaching practice is enhanced by many forms of knowledge, ways of knowing, and ways to access that knowledge.”