Mathwars or MathTrek?

I must again turn to my early childhood experiences as an example of how relational and instrumental learning are synergistic as opposed to in conflict. I distinctly remember having the realization, in 5th grade or so, that long division was somehow different from multiplication, despite being it's opposite. The feeling that doing a problem in one direction was relatively easy, yet doing the reverse was very difficult, sometimes impossible (in the case of a remainder) seemed to appear in my mind without any prompting. It was never explicitly stated by any teacher or any text, and if it had, I doubt I could have appreciated it in abstract. Now in my roll as a tutor, I often use multiplication and division as an archetype to explain how PREDICTING and NMR spectrum from a compound is straightforward, and yet SOLVING the compound from the spectrum is often much more challenging. Even if my pupils don't remember how to practice the mechanics of long division, the experience of learning it and it's challenges are a powerful teaching tool which is not quickly forgotten.  

I find it interesting that the TIMSS study was collecting video evidence of different forms of math teaching and comparing it to the statistics of the math scores. In the 1990's this was a very technically challenging feat, both collecting the video evidence and analysis and comparison. I'm curious to read the materials and methods of the original report. The conclusion that the relational model of math is correlated with higher scores is heartening but I want to see the details.

I was aware of some of the history of 'new math' but I hadn't ever heard it characterized on the political spectrum as inherently conservative. The relationship with the military industrial complex and the cold war are interesting as well. I sought out 'abstract' math like set theory and logic as a tool to help me understand real-world situations in Chemistry. No one ever forced me to do it... If they had, I might have rebelled and resisted. I see education as a long game... A particular battle might be won, but has the war? I hope to be able to plant the seeds so that my students might develop their own interests and seek out their own answers, and the confidence to believe that such learning is possible.

On Skemp

    In math 11, before I fell in love with the subject, I had intense frustration with 2*2 matrix multiplication. I struggled to remember which index was which, and the long formulas for each element were not intuitive. The 3*3 case was even worse. No amount of practice made it click. Then it all changed when the TA, a grad student, told me the secret... Just dot the row with the column! Why had no one ever said that? All of a sudden, any matrix size had a simple procedure for determining its products. The formulas which had vexed me so needed not even to be written. My love affair with matrix algebra began on that day. 

    What elements led to my breakthrough? The eureka moment was perhaps an example of what Skemp calls relational understanding. In a flash, I got a glimpse of what matrix multiplication really was. But in another sense, it was clearly instrumental understanding. Indeed, it was a process I could use to get the correct answer to a given problem. It relied upon another instrumental understanding, the fact that I knew what the dot product was. My thesis is that these two ways of understanding are not mutually exclusive. A successful math teacher must be mindful of both and how they support eachother. 

   The example of the error made when computing an area with incorrect units highlights a new goal in relational understanding. Let us treat each mathematical instrument, such as a procedure of multiplying length and width, as a heuristic. Then once a heuristic is learned, then the next important task that we must take on is to map the limits of validity of that heuristic. So many mistakes in reasoning come down to inappropriate applications of heuristics. 

    As with many aspects of education, there is a practical aspect, that Skemp notes that relational understanding is more difficult to quantify than instrumental. Perhaps this is itself a misapplication of the heuristic that all understanding must be quantified and used to sort and rank students. Another way of describing this mistake is 'putting the cart before the horse.' 

introspective writing

In 3rd grade, my class had daily math worksheets called 'minute math'. I never finished them in time and I was very frustrated to not be able to keep up with the class. My teacher decided to run an experiment... They read me the answers out loud and I couldn't even write them down in 60 seconds. I appreciate the effort to figure out what was holding me back. 

In 4th grade, I had to do my math homework on graph paper, with precise numbers of squares between my answers. I could never get it right... Lots of late nights crying, feeling stupid. Now I appreciate that having the answers in a grid is much easier to grade. As a student it was torture. I loved the problem solving aspect, but the way it was graded destroyed my confidence. 

I had an art teacher who was also a family friend, who gave me a book of Origami. Tomoko Fuse's classic, 'Multidimensional Transformations.' I've spent thousands of hours in a flow state executing her designs and exploring different types of symmetry. I will forever be in her debt. 

Math art project: Knitting Truchet tiles

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. 

 

 

 

Imaginary, transcendental, irrational curriculum

    Eisner identifies paradoxes in educational philosophy. A stated goal of the curriculum could be to train initiative, an important trait for success and a sign of responsibility. At the same time, the structure of assessment and rewards encourages the opposite, blind compliance. This is an example of an unintended effect of part of the curriculum. The next side-effect is destructive competition. Pre-meds must go from cutthroat students to colleagues working together to save lives, and medical schools are starting to recognize this mismatch, and are implement assessment policies to encourage this cultural change. 

    Almost a century ago, Mumford wrote a scathing critique of materialism which echoes loudly in the AI-marketing blitz we're currently weathering. The forces that drive us towards vapid consumption and deplete our autonomy are more powerful than ever. Can I structure my lesson plans to reward initiative and reinforce personal autonomy? I can try. 

    The new BC curriculum breaks down its overarching goals into 3 synergistic categories. Knowing certain things, the ability to solve certain problems, and the understanding of relational concepts are given equal weight. Inquiry-based learning is a curricular policy to encourage initiative. Developing inquisitiveness has hidden benefits beyond just getting higher test scores. It's a fundamental intellectual orientation with profound consequences. 

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