Ignorance on the Battlefield - (Battleground Schools - For Wed Sept 23rd)

 Battleground schools and oh the casualties, of course the students. 

 It's interesting how adults always frame their objectives for the betterment of the children, but their desires are just their own and forced upon the victims, the kids. I suppose some of them are realistic about their ambitions, the New Math's goal for the preparation of the next generation of scientists, regardless of how it negatively affected the children. Schools shouldn't be a political battleground, they should be a safe place for student learning. My strong opinions aside I think that conservative and progressive math have more in common than one might first assume. The goal is to prepare students for what each believes the purpose of math is in our society. Obviously they both have very different ideas of what is important in society, but both of them are trying to look out for the future of the next generation. I honestly don't think there is a correct answer between the two ideologies, math should be both conservative and progressive, it should give students both mathematical fluency and understanding, absorbing some facts and eliciting inquiry/sense-making, etc. How do we educate the public and gain their trust in our balanced mathematical teachings at school?

The public has always been something that has annoyed me, diverse opinions are important, democracy allows all voices to be heard, but how much should we really be equally valuing the opinions of those who are not educated on the subject? Do we listen to the public and look to their input when conducting experiments on astrophysics, nuclear physics, etc? No, we and they both understand that they don't have the knowledge to positively contribute to the research or projects being done, yet the public thinks they have enough knowledge on education to direct its course. How do we help the public understand that something as "simple" as teaching and education is much more complex than the layperson understands? How do we value parents' input on their children's learning while keeping their misconceptions and sometimes dogmatic opinions out of it?
 

I don't have the answers to these questions at the moment, and I'm not sure they'll be solved anytime soon in this hostile political climate, but I can hope that my schools and districts are able to stay out of the battlefield.  

Truchet Tiles Math Art Project - A-maze-ing Art

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. 

 

 

 

Curriculum Curriculum Curriculum Curriculum, oh that's four.

What's taught and what's naught, but taught anyway. 

Eisner says there are three curricula, but are there really only three? There's the curriculum we know, the explicit, implicit, and the null curriculum. So already we have four different curricula, but what is the curriculum we know? The curriculum we know is the one the ministry gives us. It's what we're expected to teach in class. Teachers I'm sure do their best to teach the curriculum as it's defined, but I'd wager they're teaching it through their own lens of bias, probably intentionally so. The explicit and implicit curricula are what they actually teach. The explicit curriculum, I don't see it as Eisner does, it's not what the school openly states is the content, objectives, lessons, it's what the teaching in the classroom is actually teaching. I don't think this is necessarily a bad thing however. The official curriculum of a province, district, or school could take a long time to evolve and grow to adapt to the modern world and current situations the classroom is facing. A teacher who practices inquiry and reflection with efficacy will be better prepared to respond to emerging issues and ideas that the official curriculum has yet to address.

A teacher's impact on the implicit curriculum is just as important as the explicit one. I believe a teacher has an obligation to help better and shape the expectations, routines, structure, etc., of the school they work for. Great opposition does elicit discipline, so I do not encourage teachers to radicalize their classrooms, but to take slow steps to educate and improve school culture.

Ah yes, the curriculum so delicious it's already missing, the null curriculum. It's a curriculum I've honestly never consciously thought about. I recognize that not teaching about subjects like sex-ed causes teenagers to be uninformed, leading to higher rates of teen pregnancy. So the null curriculum is something I subconsciously understood, but as teachers we can't just subconsciously understand it, we have to actively reflect on how it affects our classrooms.

Looking over our three taught curricula, the explicit, implicit, and null one, we see that each teacher has a lot of liberty with how they're taught. With such leeway how do we protect or deal with teachers who are taking their liberties to go in the opposite direction of what's best for the students and their future? Is there a best direction, is every direction just a bias of its own? I suppose there are pedagogical best practices, but that direction evolves over time as the field of research grows.

Once again, all we can do is stay curious, and give a thought about what is taught.

Opening and Closing the Doors of Mathematics

 I'M LOCKED! Well if that was so it would make the problem a lot easier.

Logic wise we know that each student visits each locker they are a factor of. Each factor for a locker number would change the state of the locker. An odd number changes the state to a new one, an even number of factors keeps the state of the locker as its original state. The closed lockers have odd factors as we start with open lockers, the even ones have open lockers. 

My approach to this problem was to start out with a brute force method of the first 17 lockers to look for any patterns. The pattern I noticed was that it seemed to be that the first locker is closed, then 2 lockers are open, then a closed one, then 4 open, then 1 closed, 6 open, etc. So there seemed to be a pattern where the closed lockers were separated by increments of the sequence 2,4,6,8,...,100,102,..,2n, where n is a natural number. 

From this I wanted to make a recurrence relation for the closed lockers. The sequence for closed lockers is 1,4,9,16,25,36,... . This sequence has the recurrence relation K_n = K_(n-1) +2n + 1, where n is a natural number. I solved this recurrance relation for a closed form solution, giving h_n = n^2+2n+1 where h_0 = 1, and n is a natural number. Using this we can find all closed lockers, for a sequence of any length. Example n = 3, h_30 = 961. The factors of 961 are 1, 31, 961, which is an odd number, so it is in fact closed. Doesn't quite prove my relation, but it's close enough of a proof for this assignment. So I could give a list of all the 31 closed lockers (as n starts at 0) in the first 1000 lockers. 31 closed, thus 969 open lockers. 

Looking through the keyhole I realize I've been quite silly and the answer is simply all the perfect squares. This makes sense as if you simply the closed form solution it gives h_n = (n+1)^2. The explanation is that when we find the factors of a number, they come in pairs. For example, for 30 our pairs are, 1x30, 2x15, 3x10, 5x6. That's 8 factors. The only way to get an odd number of factors is to get factor that pairs with itself, for 9, 1x9, 3x3, thus an odd number. 

A silly answer, and a more straightforward one. At least the doors unlocked now... 

Lessons from Hatters and Caterpillars

 

Attention Please! 

My favorite math teacher was my calculus teacher in university. He would start off each class with a music video, and then there would always be fun jokes or other things throughout his lessons. On April Fools he taught something along the lines of the Rickroll method, obviously he just linked to a video to Rick Roll the class. On top of being very funny and inviting, he taught everything he could with a relational understanding. Then during all his tests he would make up all kinds of different problems where the student had to think about different ways to apply the understanding we were taught. The questions were like nothing in the homework, and so they were challenging and fun.

It was always a joy to come to his classes. He inspires me to also make my own curriculum fun, to teach a relational understanding of methods, and to create my own problems where the application of the knowledge is done a variety of different ways. 

 

 

Least favorite is a bit of a stretch, as I still enjoyed the knowledge of the class, but my least favorite professor was a university one teaching Introduction to Complex Variables. The content itself was quite interesting, but the teacher often went deep into proofs for the sake of completeness, even when they didn't provide much relational understanding. His explanations themselves were very textbook, and he didn't bring a lot of energy or excitement to the class. I think a big problem was he didn't take the time to go through his teaching material to make it more digestible to the students. He really gave me an idea of how important it is to do a great analysis of our own teaching materials, and to iterate on them to make them more engaging and understandable to our audience.

A Relationally Instrumental Understanding

 All true friends in Wonderland are of course faux amis, or my favorite, falsche Freunde.

The point Richard R. Skemp makes is one that I've very much been buying into over my mathematical learning journey. I suppose a little history of myself, I actually failed high school. At the time, as I told my teacher when I skipped school to play a World of Warcraft expansion at launch, "I care more about video games than school." Much of my learning journey happened after high school when I had the great idea to become an engineer despite having no prerequisite knowledge or credits. So I of course had no foundation for any of my learning. I started out taking catch-up courses at Langara, and I kept learning things instrumentally. This got me quite far in engineering, but after I finished and started mathematics I kept finding holes in my knowledge that slowed me down and hampered my understanding of the mathematical problems I was presented with. 

I've come to have a great respect for relational understanding of math, and it's now the goal to have it for myself and the accompaniment of my teaching. Obviously there will be students like my former self who will reject the relational understanding in favor of the simple instrumental one to get problems done. I cannot convert every student into a "born again relationalist." As such, I wish to teach a relational understanding where I can, but bow down to the instrumentalist when all else fails. 

I hope my own journey will motivate students to appreciate and desire a relational understanding for themselves, as I hope not to have any falsche Freunde in my classroom.

Hello Wonderland

 Hello visitors of Wonderland.

Welcome to my nonsensical realm. 

I hope, or not, that you're able to make sense of things here. 

Here's a not so Cheshire cat