Math Art Project Individual Reflection

My group collaborated on each part of the assignment but we each focused more on one part than the others. The part I focused the most on was the creation of a new piece that extended the concepts from the original. A part of this process was figuring out how to make a tile that would look nice when connected to other tiles regardless of the rotation. How we ended up doing this was ensuring that the lines touched the sides of the tile in the same spot on each side. Then I started experimenting with different patterns to connect the spots on the edge of the tile. Every pattern I tried was interesting or cool looking in some way. This was interesting to me because I think there is a lot of potential for this to be used to teach about rotations. Each kid could make their own design and then would have to look at different ways that design rotated. Because this type of design seems to always create an interesting pattern, this could be a good way to allow kids to create something themselves but to give them an outline for it that would help make sure the end result was something cool they would hopefully like. I think being able to create something that is visually interesting with rotations would add interest and motivation to the topic of for lots of kids. 

This insight made me think that when I am teaching it may be good to look for and think about art made with the type of math I want to teach. If I hadn’t seen the original piece and worked with it, I would not have realized that these types of patterns could be made using rotations. I think that knowledge could easily be used to make a more interesting lesson than I would otherwise be able to make.

I also think this project served as a good reminder that different ways of doing things are interconnected. What I mean by this is that math can be done through art. This seems to me to be particularly important because it is likely that some students may not be excited about math but may be verry passionate about art. So, incorporating non-traditional ways of doing math makes it easier for students who don’t normally enjoy the subject to have a good experience of it in your classroom.

Another, thing this reminded me of is the effort required to create something. For me this came in the form of gluing 100 tiles to cardboard, cutting each one out, lining them up in a grid, and then attaching Velcro to each one. The reason this is a good reminder is that this project inspired me to incorporate similar projects in my own class. However, remembering that these types of creative projects take time is important when I consider assigning them to my own students. If students feel stressed and rushed to finish a project, they likely will not enjoy it, nor will it lead them to have a positive experience of math. So, the reminder that stuff like this can be time consuming is good, so I don’t accidently overburden my own students

Truchet Tiles Math Art Project

Art Piece: Cable-Knit Truchet Tiles

Original Artists: Lisa Marks and Owen Rowm 

Group Members: Emily Scott, Eleiah Hengeveld, Adam Barlev

Lisa Marks and Owen Rowm collaborated to produce cable-knit truchet tiles. Since the original piece was fully knitted from yarn, we decided to incorporate fibers into the recreation. The fabric and yarn were purchased from a community fabric store called 'Our Social Fabric' which saves ends of rolls from being dumped in the landfill. 

  

Polyurethane spray adhesive was used to attach the white fabric, reminiscent of the cable-knit pattern, to 25mm wooden squares. Then attached magnets to the back. These magnets allow the tiles to be easily repositioned and rotated to express variations of the tiling pattern. 

  

This construction required us to create 100 identical tiles, a daunting task. What we discovered when making art of this nature is that the first few are very slow and awkward to make, but as the done pile increases, efficient ways to work with the material become apparent, and a form of mastery and meditative state engage. Hours passed and the sun moved across the sky as more and more squares came together. When we finally looked down at our finished handiwork, we recognized that the look was unique. At that moment, we knew all that effort was worth it.

  

To expand this piece and make it our own we started with the idea of making a new 10x10 grid but with a different tile. The tile used in the original piece had two rotations, so we challenged ourselves to create a tile for our piece that had 4 rotations. We also wanted to use colour more than it had been used in the original piece (the original piece used colour more simply, it had a white background with yellow lines). How we practically went about achieving these goals was we started by trying to find a tile that, when put into a grid, had lines that connected between tiles. To make sure this happened we made the points where our pattern touched the outside of the tile the same on all 4 sides of the square. Then we sketched a design that connected the lines and that had 4 distinct rotations. Using a digital drawing software, we were able to shrink, duplicate, and rotate the tile to see how the pattern we had chosen would look in the final piece. Next, we chose how to incorporate colour. We experimented with a few different ways of doing this, we wanted to insure that however we chose to use colour it didn’t interrupt the continuous flow we were trying to create between our tiles. Once we had chosen how we wanted to incorporate colour we printed 100 copies of our tile. We then glued them to cardboard backing, cut out each tile, arranged them in a 10x10 grid, and then placed Velcro dots on each tile to connect it to our board. 

 

Some considerations that came up for us were what was practical and achievable and how we could ensure the pattern made by our tiles was atheistically interesting. We knew early on that we wanted both of our pieces to have tiles that were able to rotate so we could demonstrate how different patterns could be made using the different rotations of the tiles. However, we considered multiple different ways of doing this including wooden tiles on pegs, magnets or Velcro on wooden tiles and cardboard tiles with Velcro which we ended up deciding on. A big part of the reason we ended up deciding to build it how we did was because, given that we had to crate 100 identical tiles, we felt that it would be wise to be pragmatic and realistic about which method would be the most achievable. Another thing that came up was how to choose a pattern that would be astatically interesting. What we focused on to achieve this was ensuring we had a pattern that felt like it flowed between the tiles. This meant ensuring our lines connected across tiles. We made sure to carry this through our art not only creating the pattern in a way that ensured connection but also choosing our method for the use of colour in a way we knew wouldn’t interrupt the flow of our piece.

 

We weren’t entirely sure how to relate our art project to math at first. It was just a beautifully interesting image that looked like something we could understand and recreate with our own touches. 

 

When we first stared at the image it reminded us of a maze. Could we determine the probability of any given pattern having a maze that’s solvable, what would be the conditions that dictate if the maze could be solvable? We were looking for a test for mazeability, like the vertical line test to see if a graph represents a function. While we were running through ideas on probability we were struck with the realization that none of the lines cross, and therefore the grid could very well represent the non-crossing partition problem in combinatorics. 

 

The grid very much represented the non-crossing partition problem, the lines had to follow the same rules for Catalan number problems. Each line on the edge has a starting and ending point. The lines don’t cross and therefore if the start of a line is +1, and the end of a line is -1, the sum must always be greater or equal to zero, i.e. the number of starting points is never less than the number of closing points. I started with an example of every line that could be finished with the first point. The results were initially promising, so I continued with a full example.

  

With a 2x2 there are 8 points for lines to start and end, thus 4 lines. The Catalan number for 4 is 14, but we have two rotations per square and four squares, thus 2^4=16. I was wondering which answers were repeated, the answer was 5 of them, giving only 12 unique answers. Meaning 2 were missing. I went about drawing all the answers to both problems for a 2x2, and I realized which answer our grid couldn’t produce, and it was limited by our geometry. It’s at this point I decided to switch to a lesson about geometry and orientations/permutations. 

We will begin by giving the class a single tile and asking them to determine how many unique orientations it can have through rotation. We will then introduce tiles with different symmetries and ask whether rotating, reflecting, or inverting them produces a genuinely different image. From there, we will scale the problem up to a grid of tiles. If each tile has a certain number of possible orientations, how many possible arrangements can the entire grid produce? Finally, we will return to the lines themselves and ask what restrictions the geometry of the tiles places on the patterns that can be created, particularly when the lines cannot cross. 

 

Battleground Schools Response

The first time I stopped while reading this article was when the reasons behind a conservative baseline existing in regard to mathematics education in America were presented. This made me stop for two reasons. The first is the idea of a negative view of math being something that is a
societal truth. Immediately upon reading it, it seemed obvious to me but for some reason I had never thought of it like that. I had always sort of thought “a lot of people just don’t like math” and never thought about the societal view of math that is behind this. The second thing that caused me to stop when reading about the reasons behind the conservative baseline was the idea that people who thrive in traditional (lecture based and with a focus on the correct answer over process and understanding) math settings support conservative math education. This made me stop because it made me wonder if in some cases by creating more progressive math classrooms are we causing difficulties for kids who might thrive in a more traditional environment. Given that many of the subjects kids learn in school (social studies, art, English, and often science) are almost always taught in a way that involves a lot of group work, discussion of how personal experience relates to the subject and, the idea that there is not one right answer, it seems likely that students who struggle with group projects and subjective grading might find comfort and reprieve in a more traditional math class. A more traditional class might let them get a break from the expectation that they work in groups and participate in class discussions and instead let them have the chance to strive for a more straightforward goal of finding the correct answer. This isn’t to say that I would not want to implement many of the ideas correlated to progressive math education into my own classroom. However, I think also bringing an awareness that the structured unemotional aspects of math could be important to some students into my teaching seems like it would be wise.

I also as I was reading found myself comparing “New Math” and the NCTM standards. I think logically what the NCTM standards did is closest to what I think would be best for the math classes I may teach in the future. Despite this I can’t help but think that I personally would do best in a math classroom that taught something closer to New Math. New Math seemed to take the aspects of progressive teaching that I found the most important to me as a student while also teaching math in an abstract version of math that I think I would have enjoyed when I was younger. This makes me wonder how best to teach, when I know that not every student will see math the same way I do or will enjoy learning in the same ways I like to learn.

Three curricula all schools teach Response

Something that made me stop while I was reading this article was the examples of implicit curriculum the author used. Some of these examples were things I could relate to from my own experience in school but a few of them weren’t. This made me think about what implicit curriculum I was taught in school without even realizing it. The main differences between how I was taught and examples the article uses came when competitiveness was discussed. I believe I was taught competitiveness implicitly during school however there were some key differences between how I was taught and the examples the article uses. These differences make me think that I was taught a version of competitiveness that I find to be more beneficial. The article spoke about a grading system where the top 10% of students got a A. This is different from how I was taught where our grades were not attached to how others did. We were definitely still competitive, but it was not a type of competitiveness where we were implicitly taught that our success was predicated on someone else’s failure. The article also talked about students who had high enough academic achievement being allowed into “higher” stream classes. My experience in high school was that we were allowed to choose our own streams. This implicitly taught us to self-evaluate and to determine what was needed for the goals we wanted to achieve. I was glad that there were differences between how I was taught and the examples used in the article because it really exemplified to me how differences in how schools are set up can change what we are implicitly taught.

I also stopped when the concept of Null curriculum was introduced. It made me ask myself is it better for curriculum to be designed in a way where students have a general understanding of a lot of things or, in a way where students know a few things really well. Many of the examples of fields that aren’t taught in school are things that are briefly touched on in other classes (ex. Econ is touched on in social, communication in English law in social, etc.). So, I think that most educational systems currently prioritize general knowledge of a lot of things. I think this is the right way to do it especially in grade school where you want to expose students to lots of different fields, concepts and ideas. This is good because it lets them find things they may be interested in by trying lots of different things. How ever I do wonder if there might also be value in allowing students to have the feeling of being a “subject matter expert” in something. I think allowing students to know the difference between having general knowledge and specific knowledge is important because being able to know what you don’t know is a really important life skill. In some ways I think that that feeling of being a “subject matter expert” in something is part of the null curriculum unless teachers give their students specific chances to do this.

Locker Problem

 

I found that if n has an even number of factors than the nth locker would be open at the end and if not, it would be closed at the end. I did this by trying trial and error for the first 10 lockers and 10 people. Then, by looking at that I realized that the lockers changed states when the locker number mod the person number was 0 so it changed when the locker number is divisible by the person number. When it changed an even number of times it would end up open. When it changed an odd number of times it would end up even. So, it is dependent on whether the locker has an even or odd number of factors. I added a picture of what I wrote as I was thinking through this.

Favourite and Least Favourite Math Teacher

My favourite math teacher was a teacher I had for many of my high school math classes. The reason this teacher was my favourite was because he explained concepts and why they worked instead of just presenting us with rules to memorize and use. I also really enjoyed his class because he was really enthusiastic and passionate about math. It made class seem exciting, he always made it seem like a big deal when we reached a spot where we were able to connect topics or when topics started to “click” for us. High school was when I started to enjoy math and I loved having a teacher who matched my enthusiasm for the topic.

My least favourite math teacher was also a high school teacher. I was lucky to never have a math teacher I really disliked. But this math teacher was very monotonous. This teacher also didn’t do much to ensure the students played attention in his class, so most people just talked to their friends the whole class. It made it hard to learn. It was hard to learn form a teacher who was kind of boring to listen to and who also didn’t do anything to stop students from having other (sometimes more interesting) conversations while he thought. It was hard to be motivated to learn math instead of listening to the fun things my friends were saying.

Skemp Article Response

While I was reading this article I instinctively agreed with Skemp that teaching relationally is superior. As the article went on, I did not change my mind about this, but I did have some thoughts about when instrumental mathematics might be useful. I don’t think that instrumental mathematics is what should be exclusively taught in schools. However, I think instead of discarding it completely it might be better to instead use it as a tool to help teach relational math. The example that comes to mind for how this could be a useful tool is something that we see more in university math than in high school. In university textbooks and lectures we often see theorems where the proof has been omitted. This is often done because the theorem is needed to understand (relationally) the topics being taught but the proof is too difficult for the level of the class. This is essentially instrumental math since we are given a theorem and told to believe it without being told why the theorem holds. However, in many cases this is done to give us a stronger relational understanding of other theorems. This is similar to how Skemp says teachers may use instrumental math because “a skill is needed for use in another subject (e.g. science) before it can be understood relationally with the schemas presently available to the pupil.” (and him saying this is likely what made me think about this). Except sometimes it is not within another subject that a skill is needed but instead within math itself. Sometimes a topic or mathematical process is mathematically useful within the context of teaching relational math. But the explanation of why that process works is still too complicated for the current level that the students are at. So, I think instrumental understanding can be a good tool if it is used for the broader goal of relational understanding.

Hello World

Hello!

Here is a picture of my dog :)