Math/Art Assignment Individual Reflection

This project was really eye opening in a few ways. Firstly, the presentation itself highlighted how important it is to try and anticipate any challenges/misinterpretations of instructions that students could have with an activity. When we were describing the first part of our interactive activity (the sudoku/latin square building), some groups had instead jumped to trying to create orthogonal latin squares (which was intended to be the second part of our activity) instead of just latin squares. In hindsight, I can see how this was an easy misstep to make given the instructions I gave for the activity, and while it wasn’t a big deal for this presentation, it really cemented just how easy that is when running a lesson. Especially with a group of high schoolers who don’t have as much of a background in math and education as my peers in this class do, I can imagine that it is even easier to have confusion around instructions. It taught me that it is really valuable to try and get ahead of any of this by either trying to anticipate areas for confusion yourself, or running it by someone not involved in the project first to see where they get lost (especially since it can be hard to see when you are so deeply involved in the planning). Another lesson I took away from this whole process was that art takes a lot longer than you think! We had initially thought the construction part of this process would take a few hours and it instead took 6; measuring, cutting, and gluing should not be underestimated! This will be valuable in my own teaching, as it is really important to have an accurate understanding of how long seemingly menial tasks can take, whether that’s for planning an in class activity, or the amount of time you give students for a project. 

This project had its challenges. The amount of time it took to construct (as previously mentioned) was one of them, but I also found it difficult to initially wrap my head around the math used in the original artpiece, since translating what is essentially four different matrices onto a 2D artwork was difficult to parse. The first step we took – when we did a digital mockup – was really helpful here, as it was easier to see how each layer was pairwise orthogonal with all the other layers when we could deconstruct it. It also helped to spend some time on the more numerical side of the math (when we were looking through papers on the subject), and build our 7x7 mutually orthogonal latin squares as a sequenced list rather than attempting to do so with all the art pieces. Once we had built it, though, it was really gratifying to see it all come together in a really visual way. It helped me understand how so much of art really is math: symmetry, patterns. Definitely worth the time put in. 

Ultimately, the project really illustrated how intertwined math and art really are, and how having art based projects can just as easily demonstrate understanding of concepts as a traditional test can. This is really important (having varied forms of assessment within the class), especially since there are many students that really struggle with traditional test-taking methods. Not only does it give students more agency and choice, it also helps them build creative and critical thinking skills (it’s not always easy seeing how math is present in art!) which are incredibly important in a multitude of aspects of life. I believe that this project has given me a better appreciation for this, and helped me take a closer step to being a more well-rounded mathematics teacher that is comfortable stretching the limits of what I had initially considered ‘math’/teaching math!

Math/Art Assignment Group Reflection

Group Members: Sarah Dicastri, Tiffany Gong, Kia Prezeau 

Original Artwork: Sudoku Without Numbers by Dru Horne and Shannon McKillip

We decided to remake and extend the original piece using cardstock and markers instead of fabric, as none of us had experience with quilting. This proved challenging in that it was quite time-consuming to cut out every individual piece and glue them all together. Additionally, we needed to glue all background squares together in a way that was structurally sound, taking us 6 hours to make just the extended art piece. Coming up with a unique concept for each layer was also difficult to decide on, as we wanted to select distinct features that layer well on one another and still show the other elements beneath. We ultimately settled on adding features like coloured borders, coloured squares, and hand-drawn icons that could fit around our larger icons without blocking the background.

We first focused on extending the piece mathematically by scaling the canvas to a 7x7 matrix and layering 6 different mutually orthogonal latin squares. We arrived at this number of layers by using theorem 7 as described by Ballif (2008), which produces 6 as the maximal number of mutually orthogonal latin squares that can be determined from this matrix, given that 7 is a power of a prime. We also used this theorem to construct a full matrix of six layers in a numbered sequence, by assigning a number of 0 through 6 to each element in each latin square to construct the full picture that ensured all entries were distinct. This part of the process required some care and could be a point of challenge for others interested in recreating, as any mistakes could result in duplicate entries.

We then extended the art by focusing on how we could tie it into learning from place, specifically by having one of the Latin squares be icons from Coast Salish symbols for different local animals (sources were cited in our presentation). This was beneficial both for ourselves to explore more local Indigenous art and artists, as well as to tie in the BC curriculum’s goal of incorporating Indigenous ways of knowing.

We designed our interactive activity to be a smaller/simpler version of the mutually orthogonal latin squares that we constructed. Specifically, we created a 4x4 matrix with two fixed elements and a movable third that students can use to layer their own latin square. We hope this can cement an understanding of orthogonality between latin squares, as well as spark discussion on how many possible arrangements exist when they compare with other students. We decided on a simpler matrix to have confidence it can be completed during the short allotted time during lecture, while still giving everyone a chance to have hands-on experience with these concepts. It also brought the content difficulty closer to the high school level – as combinatorics is only introduced in Foundations of Math 12 – instead of the university-level math the original piece employs.



Original digital mockup.


Progress Pictures


Final Product


Bibliography:
Ballif, S. (2008). Mutually orthogonal Latin squares. [Lecture notes]. Department of Mathematics and Statistics, Dalhousie University.  

https://www.mscs.dal.ca/~janssen/4370/Orthogonal_Latin_Squares_text.pdf

Reading Response Sep. 23: Battleground Schools

 The first thing that made me stop when reading this article was the fact that the word “elite” was used in the table on page 392 when describing the group of students who would receive a higher level of instruction for mathematics (in the conservative column). This was especially interesting as it ties into some conversations we’ve been having in other classes, and particularly the idea that school was initially formed and structured around the idea of socialising students and preparing them to continue in their social class, instead of purely for educational purposes. It’s interesting to see that reflected here as well, as that further cements the idea that school was, and still is, a social institution with ulterior goals or impacts on students lives rather than just giving them a well rounded education. 

Another thing that stopped me was the idea that there is a general fear around mathematics that the public holds, and especially that it is something that most people aren’t good at and should indeed avoid. This is of particular poignance to me as I think one of my main goals as a mathematics teacher is to destigmatize the subject and the idea of ‘failing’ at math, as I know first hand just how demoralising it can be. I hope that my own pedagogies will help address some of the complications that this article outlines, as this is something that will likely need time to disseminate throughout generations in order to overcome the fearful ideas parents have “from their own schooling”. Although, with the increasing emphasis on more exploratory and inquiry based methods of teaching math, I would think that we are already making headway in that department. 

The third thing that made me stop was the sheer scope of the role that politics and outside factors played on the shape that mathematical education took over the last century, and in particular the impacts of the cold war on mathematical reform as outlined in The New Math section. While conversations throughout the last two weeks have made it clear to me that education does not exist in a vacuum and is often serving a larger purpose, this revelation was really fascinating, as it was really not something I had considered. It goes back to my first point about how schools can act as socialisation industries to prime children for particular roles in life – in this case, scientists to beat the USSR in the space race. It’s intriguing to think about how much outside influences impact the way education is formulated, even in arguably recent times, and it makes me wonder what possible events could happen in the future that might have a similar impact.

Sep 21 Reading Response: What is meant by curriculum?

 

Something that made me stop in this article was when Eisner talked about some examples of implicit curriculum, and in particular how something as seemingly trivial as school architecture conveys meaning to students. While I had given some thought to other ways that we implicitly teach students things (such as the diversity of representation we imply with our in-class examples, or the inherent value we place in certain subjects given the time allocated there), this was something I had never considered, but makes total sense. It seems strange to think that we can hope to inspire a level of confidence and belonging from our students when the environment they are expected to foster such qualities in can be so isolating and cold. Particularly with how important it is to me in my desire to create a safe and comfortable space in my classroom, this is definitely something to consider and hopefully address within the scope of the school I work at. Another thing that made me stop was that there were two separate excerpts on individuals from the 30s (Mumford) and the 70s (Illich) who were expressing concern over what the extensive use of machines and tools respectively would mean for our relationship with them, and specifically that it would eventually sour into humans being governed by them – that it would become “master” to us. I think this is really interesting, as it is arguably even more relevant today with how widespread technology and in particular AI use is. I find it fascinating that we can have the same anxieties almost a hundred years ago, especially considering how new and unprecedented the issue of AI seems to us. It is slightly comforting to know that these were worries that have existed for a while, and have yet to come to true fruition (arguably), and yet possibly slightly disheartening as well that these are things that have plagued us for so long. 

This has definitely expanded my own idea of curriculum, as before I had considered the word at face value: a set of content and competencies that teachers are required to teach to their students. A way of meeting standards. After this article, I have a greater appreciation for how all encompassing curriculum actually is. It is not just the standards we hope to teach, but the ideas we implicitly impart both through what we do as well as what we don’t. I think that the new BC curriculum is hopefully well positioned to address this, especially with its emphasis on core competencies and foundational skills instead of focusing solely on subject content, as well as its emphasis on expanding and diversifying the perspectives we focus on. Although I do not have extensive knowledge of the past BC curriculum, this new one does seem to have a good theoretical goal of helping teachers guide students to become productive, critically thinking, well rounded individuals, focusing a lot more on the implicit curriculum Eisner discusses through the types of activities and methods chosen, and how we structure our classroom. In terms of the null curriculum, although I am sure the new curriculum has expanded upon content that was not covered before, there does still seem to be an emphasis on the typical subject areas and content that have been the main focus for years (math, science). I would be curious to delve more into the art and elective curriculums to see what has been added there, and how BC is addressing what Eisner was speaking on in terms of a lack of emphasis on certain subjects and life skills (such as law, economics, art etc.). Overall, it seems like the new curriculum is addressing a lot of the implicit curriculum more explicitly, with still having room to grow, as I’m sure there will always be. 

Math Puzzle: The locker problem (september 16th)

(Read: top left, top right, bottom left, bottom right)

Here is my work for the locker problem. I started with trying the problem with the first 20 lockers, and trying to see what patterns arose. From that, I saw that any number that had an odd number of factors would be closed (since the first person closed all the lockers, the second person opened every second (any number with 2 as a factor), the third then changed the state of every third (any number with 3 as a factor) and so on, thus any locker that was 'handled' by a student an even number of times went from open to open, and an odd number of times went from open to closed). After that, I investigated how you would find out how many numbers below 1000 would have an even/odd number of factors, initially trying to look at primes, before backtracking when I realised that would be incredibly complicated to try to brute force, and usually puzzles like these have clever and neat ways of solving them. I then looked for more patterns by actually breaking down the factors of the first 16 ish numbers, and discovered that any square has an odd number of factors, whereas all the others have an even number (from my investigation). After that, I just needed to figure out how many squares there are below 1000, which I did by brute force before realising I could have figured out the first square to surpass 1000 and going from there. In the end, I discovered that there are 31 squares below 1000, and so there would be 31 numbers with an odd number of factors. Thus, there would be 31 lockers left closed at the end of the puzzle (all of which would be squares), with the rest remaining open. 

Favourite and Least Favourite Math Teachers Response (september 16th)

One of my favourite teachers was my 10th grade mathematics teacher. She was a really kind woman who had a clear passion for the subject which really encouraged us to see math through her eyes and not as something boring/to fear. She encouraged group work throughout the term, with both small group problem-solving, and whole-class discussions that helped make math feel like a team sport. I especially appreciated that we would spend a class before a test making a collective mind-map of everything we thought was important for that unit as a sort of study guide. It not only helped us actively recall what we had just spent the last few weeks learning, but I imagine it was also beneficial for her to see ahead of time where we might have deficits. She also made a point of taking as much stress out of assessments as possible, giving students colouring sheets to decompress before and after tests if they wanted, and having a generally approachable persona that helped with coming to her with concerns. 

On the other hand, some of my least favourite math teachers have been in my university career, the reasons for which I think are probably largely attributed to the sheer scale of university classes compared to high school, but which nonetheless I found did not work for me as a student. In particular, I disliked that some of my professors didn’t seem to make an effort to scale down/explain the math they were employing to solve certain problems, possibly because to them (who've been in the field for years), it seemed too basic to bother to explain. As a result, it often felt like math was being ‘made up’ on the way to the answer, and left me feeling incredibly discouraged and having to spend my own time finding outside sources to help me. This, coupled with the fact that in many cases I didn’t get the sense that they cared all that much if we actually did well, just that they conveyed the information they needed to, made it hard to approach them with questions for fear of seeming “stupid”.  Again, I acknowledge that in class sizes that are often in the hundreds, it’s much harder to achieve a level of attention and care for your students than it is in a room of below 30, which probably played a large part in these factors. Nevertheless, it was definitely a more challenging experience as a student. 

Both of these experiences have greatly informed the kind of math teacher I want to be. I know how isolating it can be when it feels like your teacher doesn’t care about you, and conversely how empowering it is when you feel like you are a part of a team. I want to ensure that I leave my students with the impression that I am here for them and any questions or concerns they may have. I also want to be sure that I create an environment where there are no “stupid” questions, and students feel comfortable enough to ask questions, ideally by anticipating some big ones by appropriately scaling the definitions/content to their age level, but also by encouraging discourse through group-work and regular check-ins. On a more menial note, my 10th grade teachers' methods of creating mind maps and offering means of destressing pre/post assessment has really stuck with me, and something I would like to try implementing in my own classroom. I think in these ways, I can hopefully be a better math teacher.

September 14th Reading (Skemp and instrumental versus relational learning) Response:

    The first thing that made me stop was at the beginning of the article when Skemp described the difference between relational and instrumental understanding/mathematics, as this was a concept I implicitly knew, but had never heard put to paper before. It was gratifying to have those definitions spelled out as I believe it helped me organise my own thoughts on the subject. The second thing was when he talks about how building relational understanding is difficult when students are already used to largely instrumental ways of learning, and so the value of implementing relational ways at an early age. It makes me wonder what sort of values versus obstacles there would be to enacting this at a secondary level if the students are coming from years worth of instrumental learning backgrounds – or if they even are, as things might have changed since this article was published. And finally, I was also really struck by Skemp’s mental map analogy at the end, as this really helped make things concrete in my brain on the difference between relational versus instrumental, especially since I am not very music minded and the previous music analogy was thus not as influential. I also feel that this analogy would even be appropriate to share with our kids if we feel that there is value in describing these ways of doing mathematics in order to help guide them towards attempting more relational learning than instrumental. Definitely something I’ll keep in mind.

    I think there is real value in what Skemp argues for relational mathematics, as it has incredible implications not only for student’s enjoyment and success within their math careers, but its transferable skills to other subjects and daily life. I do want to acknowledge, however, that while the ideal of teaching exclusively relational mathematics is a noble one, the feasibility of that in schools today could be in question. Especially since Skemp made it clear that relearning the way to learn (i.e. from instrumental to relational) can be incredibly hard and demoralising for students (if they even attempt to at all), and particularly coming into the secondary program, where students have had years of mathematics behind them - most likely instrumental - it might be near impossible to effectively shift all their mindsets to relational within the school year. So, in theory I am in support of his argument, but I also want to leave room for acknowledging the faults it could have in complete implementation at the high-school level, and I am eager to get first hand experience on what type of instruction is currently going on during our practicum (especially since this was written so long ago).