Art Project Reflection

 Our art project expanded on Katelyn Owen’s “Her,” a piece centred around the Fano Plane (or PG(2, 2), where “PG” stands for “projective geometry”).  Owen's work features paintings on each edge of the Fano Plane, with each edge representing a theme from her life.  Each of the seven art pieces corresponds to the intersection of three themes, encouraging introspection and the exploration of connections that might not have been considered without the geometric structure of the Fano Plane.  We took this concept further, both artistically and mathematically, by working with PG(3, 2), the three-dimensional extension of the Fano Plane.  This allowed us to explore 35 themes through 15 art pieces, each representing the intersection of seven themes.  We also categorized these themes by colour to simplify brainstorming and connection-forming, though this symmetrical and aesthetically pleasing approach isn’t necessary for constructing PG(3, 2).

My experience with this project reminded me of being an undergraduate, researching mathematics that initially felt out of reach. That being said, once I was able to re-familiarize myself with the basics of projective geometry, the research I did surrounding the Fano Plan was exciting. I found myself excited to convey the beautiful symmetries in this structure, though I I’m unsure if I was successful. The longer I spent time with it, the more my appreciation grew. I saw tremendous potential in leveraging this structure to explore many areas of life – political structures, math curriculum, introspection, and everything in between. Despite my excitement, I’m not sure how well I was able to present the deep symmetry of this structure to the class. I personally take a long time to process definitions in mathematics, and I’m unsure as to how well I explained the axioms in our short presentation. I hope to find better techniques in the future.

As a teacher, I fully expect to execute some variation of this project. However, I plan to place much more importance on ensure the art itself is meaningful to those who are working with/creating it. It is clear that there is value when we project mathematics through a creative lens, but unless you have a feeling for the mathematics, the art feels abstract and out of context. For example, when I initially saw Owen’s ‘Her’, although the symmetry of the Fano Plane was beautiful, the depth of the art was lost on me. It was only after understanding the interconnectedness of the Fano Plane that I was able to appreciate the completeness of the art.

I think given the abstract nature of math, in combination with the vast array of unique creative abilities and hobbies I hope to find in my students, any assigned art project will be highly open-ended. I hope to offer students an opportunity to leverage both art and math so as to better understand something they are already familiar with. I speculate this is most clearly possible after completing the ‘Combinatorics’ unit in Foundations 12.  

Battleground Schools - Article Response

 I wasn’t expecting to have as strong a reaction to this article as I did. This is the first time I have heard of the New Math, and it is the first time that I’ve read any account of the explicit style of mathematics taught to my parent’s generation. Frankly, this article made me angry. 

First, I was very impressed by the type of mathematical education suggested by Dewey. Given the quality of public mathematic education in North America for the past 60-ish years, I was surprised to learn that in the early 20^th century, Dewey advocated for unpredictable classrooms driven by student autonomy. Mathematics is returning to this style, demonstrating how modern Dewey was in his thinking. As suggested in this article, I can only speculate that the unbearably intense global politics of the mid-20^th century scared the public away from Dewey’s slower, more authentic approach to education. It makes me wonder what qualities of Dewey's education were infused in this generation of students. It is perhaps noteworthy that many of the young hippies of the 60s would have had parents educated under Dewey's system. Perhaps parents raised to see the value of student autonomy were more inclined to let their children pursue authentic ways of living. This is deeply speculative. 

My mom, born in 1969, was traumatized by her math class. This article gives me the understanding that the education she received was some abomination of ‘New Math’ and neo-liberal politics, completely absent of authentic problem solving, geometry, and critical thinking. For her, mathematics is instrumental calculation. This did not align with her strengths as a student – consequently, she has actively avoided any semblance of mathematics for nearly 50 years. It is terrifying to speculate on the number of elementary teachers who share this disposition.

Perhaps the most sinister quality of New Math mentioned in this article was its attempt to create a global, ‘teacher-proof’ curriculum. As has been discussed, teachers in the modern age do NOT exist as sources of content. It is only by cultivating authentic intellectual relationships with their students that are teachers able to facilitate the transmission of knowledge. A ‘teacher-proof’ curriculum blatantly goes against this pedagogy. If a 'teacher-proof' curriculum was possible, there would be no (human) teachers in the 21st century. 

Finally, I have a much greater appreciation for the role the NCTM has played in the current state of mathematics. The NCTM’s prophetic attitude towards standards enabled teachers to articulate their ideals, as well as to preserve those ideals against the static, fundamentalist world view of the religious right. Upon concluding this article, I feel a much greater motivation to improve my understanding of the NCTM standards. In particular, I plan to understand how they align with the BC Curriculum.

Lockhart's Lament - Response

 I am aligned with Lockhart. His lament has helped articulate the ‘wrongness’ that we’ve all felt in high school classrooms for so long. It is tragic that our schools continue to operate in the same way, despite us having writing like this in the world.

I am very drawn to the style of teaching he proposes. I believe that enabling autonomy in students will deeply enrich their learning, relative to what it is now. That we can provide resources and guidance, but ultimately, they will pursue the paths that appear to teach of them. In a separate class, I recently wrote about what qualities I believe describe ‘authentic assessment’. I noted that generally, when authentic art is produced, it is done so without expectation – the artist is not attempting to mimic anything. I speculated that authentic assessment should be absent of as many expectations as possible. I feel this would align very well with Lockhart’s conceptions of math class, since students would be pursuing their own problems without some concrete destination. This feels Authentic.

Lockhart speaks about giving students the opportunity to discover math. This resonates hard – as I’ve mentioned earlier, a book on Greek Mathematics provided me with this experience first-hand. If we could cultivate these experiences in students, math class would become a favorite for many.

Lastly, I want to note when Lockhart states: “Teaching is not about information. It is about having an honest intellectual relationship with your students”. This sentence gives me confidence that Lockhart’s ideas are grounded in humility. He understands that a teacher’s role is to engage with students, listening and guiding them to math’s natural wonders. What’s more is he offers (general) methods of accomplishing this - rich questions, student autonomy, and minimal expectations. These are qualities I hope to instill in my classroom.

The Locker Problem

 Teacher bird here. 

I'm not sure why, but when I first approached this problem I was expecting primes to play a role. Not sure why this was, but I initially thought both primes and squares would be open, until I saw that 5 was closed, and realized I had no justification for primes being open...

Initially I wrote out a sequence of 10 to get a feel for things. I only wrote out the changes in position, and then a final line which showed the position of the first 10 lockers after 10 students. I didn't notice that it was only the squares at this point, but had a feeling it was related to the factors of numbers. I wrote out the factors for 60 and was reminded that they come in pairs. This helped me realize that every locker would be closed except for those with an odd number of factors. Since only squares have an odd number of factors, the problem was solved. 

I love this question, though I'm not sure if the ceiling is as high as I originally thought. Once someone understands the algorithm, all higher cases are trivial. This might be good for a Math 9 class. If was to present this to a Math 9 class, I would ask questions in this order:

Given the problem, how many lockers are open if there are:

a) 10 Lockers

b) 50 Lockers

c) 100 Lockers

d) N Lockers

I would make sure to only show them one part of the question at a time so that they are forced to realize brute force will not work because of their own experience with the question, NOT because they are anticipating a harder case later on. 

Favorite and Least Favorite Math Teachers

 In truth, this exercise had me believing that I didn’t have a favorite math teacher. That being said, there are several moments in time which I know cultivated my joy for mathematics.

1.       Solving countless 2D and 3D spatial puzzles in video games as a child.

2.       My Grade 9 math teacher, Mrs. W, who regularly offered us geometry puzzles to solve (with and without the use of trig. Functions).

3.       As an engineering undergraduate, reading a book about the history of Greek Geometry. This book presented the history in a mostly chronological way and offered the reader many opportunities to prove Lemma’s for themselves using tools derived in the book.

Of these three sources, I may point to the book of Greek Geometry as my greatest teacher.

As for my worst math teacher, Mr. R from Pre-Calculus 12 comes to mind. He read directly from the textbook and frequently was shouting at students who repeatedly got wrong answers (generally after repeatedly receiving the same explanation). There was a silent tension in the class, though it often broke with laughter from the absurdity of it all. The knowledgeable students were frequently pointing out mistakes during lectures, and I do not recall feeling as though he could help me learn anything. Fortunately, his tests we’re predictable, meaning my final grade was not harmed (something I was deeply concerned about at the time). I later learned that Mr. R was an ex-NFL player. Hopefully helmets are better now.  

Reflecting further on my greatest teachers, I see that they are actually activities which cultivated a sense joy in mathematics and problem solving. In particular with the book of Greek Geometry, I felt that the incorporation with history offered authentic motivation, which in turn gave me the feeling that I was personally discovering the math. I consider this to be my richest learning experience and is one I hope to recreate for future students.

Thoughts on the Relationship between Instrumental and Relational Understanding - September 9th Group Discussion

 In my initial response to Skemp's article, I speculated that Instrumental understanding should be taught before relational understanding. I justified by stating:

1. Instrumental understanding can act as a frame of reference / landmark for Relational understanding

2. Instrumental understanding can be used to inspire / motivate Relational understanding. 

During our class discussion today, I came to realize that there is (likely?) significant personal bias in this opinion. As it relates to point (1), there is nothing to suggest that relational understanding cannot act as a frame of reference for instrumental understanding. I am confident that for me personally, instrumental understanding helps me with process relational understanding. However, I now see that there is likely value for others, namely those who learn differently from me, to approach understanding in the opposite way. 

As it relates to point (2) - I now speculate that given any subject, either relational OR instrumental understanding may offer content which serves as motivation for the novice of the subject. In this way, one should consider each subject individually so as to determine how best to inspire future interest. 

The Three Curricula That All Schools Teach - Discussion Post

Eisner’s explanation of what could be (in terms of implicit curriculum) were far more exciting that his evidence of the existing curriculum. Specifically, his explanation of schools cultivating initiative made me stop. The ability to guide my own learning, the awareness to plan rich and meaningful days, and the capacity to define intrinsically motivated goals are essential skills that were absent during my high school experience. I speculate that schools designed to cultivate initiative would also cultivate these essential skills, giving rise to generations with a far stronger sense of identity and purpose.

Generally, his description of the implicit curriculum of most schools seems to align with the true, intended curriculum that most public schools aim to teach, namely preparing students for a monotonous workforce.

The idea of a Null curriculum also resonated. So many of the most meaningful books and subjects I’ve engaged with as an adult were absent in high school. Any formal introduction to psychology or spirituality was absent; two subjects that I feel could have an extraordinary effect on a teenager’s capacity to appreciate alternate perspectives. Why don’t high school’s teach Jung?

As teachers in BC, we have tremendous autonomy in our classroom. Although we are confined to a specific explicit curriculum, we can choose (almost any) implicit curriculum. It is in this way that teachers can dramatically affect the next generation. Personally, I believe we need to leverage this opportunity to give students the capacity for critical, independent thought, ideally inspiring aspirations beyond the monotonous workforce.  

Relational and Instrumental Understanding - Discission Post

Skemp’s elucidation of relational and instrumental understanding resonates with me deeply. As an engineering student, I often felt a ‘shallowness’ in the mathematics we were being taught – I realize now that this is because it was taught in such a way as to achieve instrumental understanding. I can understand the motivation – engineers are expected to interact with the real world and being able to get a correct answer - quickly - is essential. I could write pages on this point, but I’ll leave it at that.

Early in the article, Skemp notes that instrumental understanding is often used even by those with relational understanding. He later provides a very relatable analogy regarding the navigation of a city. One first develops routes through the city that function well, but do not provide an understanding of the city. It is only after one has some well defined routes that one is inspired to develop relational knowledge of the city. As such, I speculate two roles for instrumental understanding in the development of relational understanding:

1.       Instrumental understanding acts as a frame of reference for relational understanding. To walk blindly through a city is pleasant, but to without any routes/buildings/landmarks from which to relate these new places, they are lost in a cloud of information that only becomes clear after a long exploration. Having instrumental understanding allows you to more quickly process new relational information.

2.       Instrumental understanding inspires relational understanding. As a musician, I recall learning the guitar. At first, I was not interested in scales or theory – I just wanted to make music. As such, learning how to play chords and produce basics songs was extremely inspiring. This provided me with a glimpse of what was possible and served as essential building blocks later on in my musical life.

Lastly, although I mostly agree with Skemp, I disagree with his assessment that the general ‘negative attitude’ towards mathematics can be attributed to our failure to teach it with the goal of relational understanding. Both instrumental and relational understanding are necessary and can be taught in either a good or bad way. To consider instrumental mathematics as a ‘major cause’ for negative attitudes seems overly simplify the situation. I speculate that instrumental mathematics is the only type of mathematics that can be taught when teachers employ methods which are the true cause for math dysphoria.

Hello World

 Welcome to my blog! I hear I should be attempting to use concise language, but as this sentence exhibits, it is not my forte. Below is a picture of a talking drum. 

To play these drums, one holds them between one's torso and arm. By compressing the drum with your arm at different strengths, you vary the pitch of the drum. This creates opportunities for very dynamic drumming!