Final summary

So much happened with this course, where do I even start? If I were to pick out one thing, it would be critical thinking as pertaining to mathematics education. If anything, teaching really is the utmost art of subtlety.

• The readings really expanded my horizons in ways I did not know possible- from the recent history of mathematics education (I kind of knew that during the post WWII boom, seeds for putting some reason and inquiry into the industrial era grinding were already sewn, and that Reagan and Thatcher planted the seeds of modern reckless privatization; it was very reassuring to know that mathematics educators foresaw it, and tried to give them as little opening to attack public education with as possible) to critically analyzing the linguistics of mathematics textbooks to shortcomings of Euclidean geometry with a First Nations lens.
• Do you need a last-ditch five-minute filler? Are you preparing students for a competition? Do you even have aspiring lawyer types in your ranks? How about IT workers (Facebook, Amazon, Google, Microsoft, etc. are notorious with their puzzles during technical interviews)? No problem, here are puzzles! Also, I believe that dogmatically insisting on one approach to solving a problem is a hallmark of incompetence, and having been exposed to equally correct approaches, hopefully I am better equipped to practice what I preach!
• Mathematics for visual art. Interdisciplinary connections and syntheses are where it is at, and considering that students typically associate "art" with "fun", here are some project ideas to run with!

Dr. Ed Doolittle reflection

The "grid" concept in itself was an epiphany for me. We build these neat, rectangular, discrete boxes over a continuous universe; not that structure is not useful, but sometimes, we may need different shaped lines and boxes- those that can deform to the variability.

Ultimately, the Earth is a curved surface. Thus, at a certain scale, Euclidean geometry is inadequate- for example, navigation. It would appear that Euclidean geometry can "fail" much sooner than that if Hamilton, Ontario is any indicator. Tilings provide a unique solution in that at least some of them can be constructed from straight, rectangular (i.e. Euclidean) primitives.

I found the allegorical interpretation of the "grid" to be both pause-worthy AND insightful on indiginization. The way I see it, escaping the grid, as I would like to call it, is, at its root, about promoting metacognition, from which flexibility and humility, the virtues that seem to be advocated by concepts such as the "native American summer" and "only the Great Spirit is perfect", follow. Ultimately, might really does not make right: sure, you managed to produce cookie-cutter workers with at least some kind of competency, and you even managed to vanquish the natives, but does that really make everything about your education system inherently "above" the native American approach? They may be lacking in military might, but they may have a good point in other aspects of life- get off the grid and open up your mind a bit.

Reflection on Chris' students visiting

I cannot exactly recall how positive and negative integers were taught to me, even after a while of brain-squeezing. If anything, I found that they "made sense" to me, although I could not exactly say why.

A big theme for the students would be their discovering of advanced mathematical concepts and pedagogical techniques.

Fractions can be readily visualized and demonstrated, and one pair of enterprising boys who run a shoe painting business used that to great effect, using a gridiron football field, which conveniently comes with yard lines in multiples of 10. 

When it comes to students taking on signed arithmetic, by far the most prevalent was that of a "yin/yang" analogy- "chocolate/milk"; "pepper/milk"; "fire/water"; et cetera. On that note, one pair of girls, who used chocolates and milks along with flow charts, particularly impressed me because they were using Boolean logic by themselves without being taught about it. Using "chocolate" for "positive" and "milk" for "negative", they had four flow charts depending on the signs of the numbers:

Chocolate -> chocolate -> chocolate

Chocolate -> milk -> Not chocolate

Milk -> chocolate -> Not chocolate

Milk -> milk -> Not milk

One pair of boys had apparently discovered the concepts of "additive identity" and "subtraction as the inverse of addition" without using such words. They said that 3 can be like 3 + 1 - 1; 4 spots of fire, but one of them cancelled out by 1 bucket of water. It is one thing to be like "zero plus anything is zero"; rewrites like this are what fuels algebraic proofs, and they showed themselves capable of grasping that.

While using the coordinate axes to plot all solutions to a linear Diophantine equation is a classic approach, these boys had found a way to turn it into a game by throwing some Sponge Bob and treasure hunting into it; after finding the "right" pair of coordinates, you discover where you are supposed to "go to next", and the problem goes on. Have they been frequenting teacherspayteachers.com or something, seriously!?

Reflection on the West Point Grey Academy math unfair

I thought that I should add some pictures before publishing, but then I got swept away. Bad form on my part. Finally, here it is.

West Point Grey Academy definitely looked, felt, and smelled like privilege, although I am sure that whereas children of wealthy corporate and organized criminal types (I am not exaggerating or kidding!) probably pay full tuition, children of professors, teachers, and the like are on some kind of scholarship or student aid. Knowing how political and corporate elites have been conspiring together to gut public education, seeing the visible reminder of subsidized private education, at the higher end of scale at that, left me with a bad taste in my mouth.

Each group of students opted to run a game with theoretical probabilities configured such that the chance of winning (or at least scoring opportunities to retry) was either equal to the chance of losing, indicating a fair game, or strictly lower, indicating an unfair game.

I appreciated the practical lesson on deception in magic and carnival/funfair/amusement park games. For example, one group of students had a ball-throwing game where to "win", you had to hit the bull's eye... except the box was unevenly divided so that if your ball landed in the larger section, you lost anyway; to win, your ball needed to land in the smaller section of the box:

Another group of students had a draw-the-winning-ticket type game... except they purposely included more "din (lose)" tickets than "win" tickets in the draw.

For those who opted for a "spin-the-wheel" type game, there existed a group that purposefully gave bad draws larger sections on the circle:

Speaking of which, another group had a theoretically fair game whose implementation could potentially be disputed due to them not drawing the sections precisely enough. I pointed this out, considering that the circle appeared to have been freely drawn, without the aid of a ruler or a protractor; their insistence on its fairness was astounding and exasperating all at once. I would think that with a $20,000 tuition, you would at the very least be taught some metacognitive skills from a young age. For instance, the Harkness method at Phillips Exeter effectively drills those skills into you.

The wine and the rats

There are 10 rats. Each rat can be either live or dead. Therefore, there are actually (2^10) - 1 = 1023 different dead/alive combinations of rats available which you can use to pinpoint which wine was poisoned.

• Number the rats 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512. Notice that they are numbered after significant bits.
• Number the bottles from 1 to 1000 using binary.
• For each bottle: 
• Inspect the binary number assigned to it
• For each bit of the binary number, starting from the least significant bit:
• If it is 1, then have the rat with the corresponding significant bit taste it
• For example, if the LSB is 1, then have bat #1 taste it; if the second LSB is 1, have bat #2, taste it; if the third LSB is 1, have bat #4 taste it.

Claim: this algorithm can discern precisely which bottle was poisoned.

Proof: since it is known that exactly one bottle is poisoned, it does not matter if the binary encoding of two bottles share a bit or more. 

EDCP 342A unit plan second draft

https://drive.google.com/file/d/1G7ylF-GU9vLU1jznJFUO687VQb7W0yJI/view?usp=sharing: unit plan

https://drive.google.com/open?id=10D1hsqB2J38_uQdJHVzElW6B0h8J5QmN: history-themed lesson

https://drive.google.com/open?id=1g5AvLQkAJ7aocECMmYandnfyS9zCg15F: open-ended lesson, which conveniently doubles as one of the "work on project" times

https://drive.google.com/open?id=1dUYNyHQrrJsRRuq9UD6RoAZXw3Fa3znp: social justice-themed lesson

My unit and lesson plan

https://drive.google.com/open?id=1BVK2EUn_fWkCy_S2obX7YroTuzvbWzYV
https://drive.google.com/open?id=1Oe5rUKrhduEYMOPaZfhn8-8y2TKCEMcd
https://drive.google.com/open?id=1p1yBPucDPpVPUhoExsqrDidc6TUKTpb5: unit plan
https://drive.google.com/open?id=1dUYNyHQrrJsRRuq9UD6RoAZXw3Fa3znp