- 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!
Thursday, December 19, 2019
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.
Sunday, December 8, 2019
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.
Wednesday, December 4, 2019
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!?
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.
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.
Sunday, December 1, 2019
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
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
Wednesday, November 27, 2019
My unit and lesson plan
Sunday, November 17, 2019
On Thinking About Math Textbooks
Textbook use is a tricky subject. On one hand hand, they can contain relevant academic examples that can be difficult to find, such as the equation of human height and femur length; also, at the grade school level, they contain plenty of exercise question for honing mechanical proficiency. On the other hand, as the authors have pointed out, textbooks can overvalue the formal and deductive approach to a fault (experiments and observations have their role in mathematics), give the impression of issuing commands to the reader in a top-down fashion, and promote detachment in spite of drawing from real-life examples (apersonification versus depersonification).
Speaking of which, when I went through junior high and high school, I do not think I saw much textbook use; custom handouts were already prevalent. Still, if the technology is inadequate or papers are difficult to come by, I suppose there is little choice.
As for the examples (height and femur equation algebra with Figure 2, hand dropping coins with Figure 1, word choice for problems),
as a student:
Speaking of which, when I went through junior high and high school, I do not think I saw much textbook use; custom handouts were already prevalent. Still, if the technology is inadequate or papers are difficult to come by, I suppose there is little choice.
As for the examples (height and femur equation algebra with Figure 2, hand dropping coins with Figure 1, word choice for problems),
as a student:
- Figure 2 would actually lead me astray, thinking that height is a scalar multiple of femur length, when it is not exactly so
- I actually visualized myself doing the experiment in Figure 1 without difficulty.
- Not that mechanical practice is unimportant, but I suppose I might feel like a soldier going through orders when it comes to how exercises are worded
as a teacher:
- I would actually highly value the height and femur equation- here is a Diophantine equation being used for anatomy in real life! I would try to present a diagram drawn by myself based on Figure 2 rather than presenting it as-is.
- Figure 1 could potentially be interpreted as "the teacher" or "the author" doing the experiment. Thus, it may do me well to prompt students with phrases like "visualize yourself..."
- Knowing that language is important for promoting engagement, as far as mandatory worksheets go, I would probably create my own with words chosen deliberately towards that end.
Tuesday, November 12, 2019
On the Scales Problem
You would have to get started somewhere; on that note, you must have a 1 gram weight. However, if you need a 2 gram weight, a 3 gram weight, et cetera, this gets nowhere quickly.
Let us proceed by a greedy approach, where we try to come up with weights such that the widest range of weight values are covered with each new weight, because why not? Besides, "higher" optimization techniques like dynamic programming are motivated by the failure of the greedy approach anyway.
What if you had a 2 gram weight: Then, with a 1 gram weight, you can measure:
What if you had a 3 gram weight? Then, with a 1 gram weight, you can measure:
At this point, maybe we should keep the 3 gram weight. Let us select another.
What if you had a 5 gram weight? Then,
Skipping a few, what if you had a 9 gram weight? Then,
At this point, we should keep the 9 gram weight. I think we see a pattern here: as we seek the next weight to add, keep adding the candidate weight by 1 until the "minimum" becomes untenable in a sense. For example, we abandoned the 4 because there was no way to make 2, settling at 3; we abandoned the 10 because there was no way to make 5, settling at 9.
I am going to conjecture that the next weight to add is 27 grams:
I do not think that any other solutions are possible; either you will have some weights that you cannot make, or you will end up needing more than 4. Thus, this is in fact optimal.
I suppose you can extend the puzzle and help them develop a more precise understanding by making the weight bigger, asking more questions, et cetera:
Let us proceed by a greedy approach, where we try to come up with weights such that the widest range of weight values are covered with each new weight, because why not? Besides, "higher" optimization techniques like dynamic programming are motivated by the failure of the greedy approach anyway.
What if you had a 2 gram weight: Then, with a 1 gram weight, you can measure:
- 2 grams: self-explanatory
- 3 grams: put both weights on one pan
- 4 grams: impossible
What if you had a 3 gram weight? Then, with a 1 gram weight, you can measure:
- 2 grams: put the 3 gram weight on one pan, and the 1 gram weight on the other pan
- 3 grams: self-explanatory
- 4 grams: put both the 3 gram weight and the 1 gram weight on one pan
- 2 grams: Impossible
- 3 grams: put the 4 gram weight one pan, and the 1 gram weight on the other pan
- 4 grams: self-explanatory
- 5 grams: put the 1 and 4 gram weights on the same pan
At this point, maybe we should keep the 3 gram weight. Let us select another.
What if you had a 5 gram weight? Then,
- 5 grams: self-explanatory
- 6 grams: 5 and 1 gram on the same pan
- 7 grams: 5 and 3 grams on one pan, 1 gram on the other
- 8 grams: 5 and 3 grams on the same pan
- 9 grams: 5, 3, and 1 gram on the same pan
- 10 grams onward: impossible
- 5 grams: 1 gram on one pan, 6 gram on the other
- 6 grams: Billie Eilish (duh)
- 7 grams: 6 and 1 gram on the same pan
- 8 grams: 6 and 3 grams on one pan, 1 gram on the other
- 9 grams: 6 and 3 grams on the same pan
- 10 grams: 6, 3, and 1 grams on the same pan
- 11 grams: impossible
Skipping a few, what if you had a 9 gram weight? Then,
- 5 grams: Put the 9 gram on one pan, and the 1 and 3 grams on the other pan
- 6 grams: Put the 9 gram on one pan, and the 3 gram on the other
- 7 grams: Put the 9 and 1 gram on one pan, and the 3 gram on the other
- 8 grams: Put the 9 gram on one pan, and the 1 gram on the other
- 9 grams: Duh
- 10 grams: 9 + 1 on the same pan
- 11 grams: 9 and 3 on one pan, 1 on the other
- 12 grams: 9 and 3 on one pan
- 13 grams: 9, 3, and 1 grams on the same pan
- 5 grams: Impossible
At this point, we should keep the 9 gram weight. I think we see a pattern here: as we seek the next weight to add, keep adding the candidate weight by 1 until the "minimum" becomes untenable in a sense. For example, we abandoned the 4 because there was no way to make 2, settling at 3; we abandoned the 10 because there was no way to make 5, settling at 9.
I am going to conjecture that the next weight to add is 27 grams:
- 14 grams: Put 27 on one pan, and 9, 3, and 1 grams on the other pan
- 15 grams: 27 on one pan, and 9 and 3 on the other
- 16 grams: 27 and 1 on one pan, 9 and 3 on the other
- 17 grams: 27 on one, 9 and 1 on the other
- 18 grams: 27 on one, 9 on the other
- 19 grams: 27 and 1 on one, 9 on the other
- 20 grams: 27 and 3 on one, 9 and 1 on the other
- ...
- 29 grams: 27 and 3 on one, 1 on the other
- 30 grams: 27 and 3 on one
- 31 grams: 27, 3, and 1 on one
- 32 grams: 27 and 9 on one, 3 and 1 on the other
- 33 grams: 27 and 9 on one, 3 on the other
- 34 grams: 27, 9, and 1 on one, 3 on the other
- 35 grams: 27 and 9 on one, 1 on the other
- 36 grams: 27 and 9 on one
- 37 grams: 27, 9, and 1 on one
- 38 grams: 27, 9, and 3 on one, 1 on the other
- 39 grams: 27, 9, and 3 on one
- 40 grams: 27, 9, 3, and 1 on one
- 14 grams: Impossible
I do not think that any other solutions are possible; either you will have some weights that you cannot make, or you will end up needing more than 4. Thus, this is in fact optimal.
I suppose you can extend the puzzle and help them develop a more precise understanding by making the weight bigger, asking more questions, et cetera:
- Suppose that you need to be able to measure up to 121 grams now.
- If you have a 41 gram weight...
- How far can you measure up to?
- How many more weights would you need to go up to 121?
- Repeat the questions for a 43 gram weight, 75 gram weight.
- What is the minimum number of weights needed to be able to measure up to 121? How much should each weight weigh?
Wednesday, October 23, 2019
Battlegrounds reflection
When I saw the point on the connection between incompetence and conservative teaching, I hit my knees. No wonder! When you literally do not know what kind of feedback to give to a student if the student were to give a "different" answer, you dread the possibility that you cannot spit the answer back at an instant; worse, rather than saying "let me get back to you on that" after writing it down somewhere, you would succumb to your own insecurity, and then try to shut it down with force in an attempt to "save face". Thank you so much for pointing out an answer other than "industrial era factory worker training" for once. It is an elephant that needs to be confronted.
Another "moment" would be just how ironic the end product of education innovation motivated by the Cold War turned out. On one hand, I cannot imagine living with the scare of nuclear weapons; on the other hand, I really do believe that the Cold War was a boon in some ways: the vigorous interest in STEM-related endeavours and the desire to "do them properly"- by the government, by the people, et cetera. Still, I find it so baffling and ironic that the end result was mostly the old-fashioned approach slightly rehashed- but with more abstraction in the interest of "elegance" (I have seen how university-level mathematicians' love for "elegance" often result in exceedingly abstract products that do not hint even remotely at what motivated their development in the first place).
Finally, the rise of neoliberalism in the 80s and "accountability". I do not know too much about the United Kingdom side of things, but as far as the United States goes, I know what kind of backlash against various human rights-related progresses, especially racism, contributed to Reagan getting elected. Furthermore, the Cold War was over. Oh did the Republicans get to work on dismantling the United States; what is worse, Reagan got especially lucky with the oil prices when the "Morning in America" happened too. As you pointed out, "accountability" in particular really was a call for dis-empowerment and infantilization of teachers; of course, this is business as usual for the self-serving neoliberal narrative- speaking of which, one of our readings for EDST 401 precisely concerns the creeping influence of neoliberal economics in public education. I truly appreciate that the NCTM tried to stay a step ahead and take matters into their own hands, trying to give no room for politicians to turn the parents against the teachers.
Monday, October 21, 2019
The numbers on a circle
Knowing that cos^2 (theta) + sin^2(theta) = 1, let 1 corresponds to theta = 0, 2 correspond to theta = pi/30. Then, 7 is 6*pi/15, 16 is pi, 30 is 29*pi/15. I took a very pedantic interpretation of "the numbers are evenly spaced"- that is, 1 and 30 do not both sit on the coordinate (1, 0).
Then, what is diametrically opposite can be recovered by 6*pi/15 + pi = 21*pi/15. This is 22.
I stared at it, and I realized that working directly in Cartesian coordinates would be inconvenient; thus, I turned to that trigonometry identity and polar coordinates, where I can actually spell out the coordinates in a convenient way.
Some extensions based off of this puzzle:
Then, what is diametrically opposite can be recovered by 6*pi/15 + pi = 21*pi/15. This is 22.
I stared at it, and I realized that working directly in Cartesian coordinates would be inconvenient; thus, I turned to that trigonometry identity and polar coordinates, where I can actually spell out the coordinates in a convenient way.
Some extensions based off of this puzzle:
- If you were to draw another diametric line such that it is 90 degrees with the one drawn from 7 to 22, which two numbers would it be?
- Choose any n you want where 1 < n < 15. Between which pairs of numbers would you draw the diametric lines if you want them all to have the same angle between them?
- Impossible puzzle: what if, rather than from 1 to 30, it was from 1 to 31? Suppose that 1 is theta = 0, 2 is theta = pi/15.5, 3 is 2*pi/15.5, ..., 31 is 30*pi/15.5. Now, 7 is 6*pi/15.5, so the "opposite" side would be 21.5*pi/15.5... what is this, halfway between 22 and 23?
Group microteaching reflection
I thought I would take a risk; rather than explore calculus, computing science, pre-calculus, or even the standard "mathematics" curricula in K-10, I thought I would explore workplace mathematics. I do not mean to be the gatekeeper of privilege and close doors in pupils' faces, but sometimes, higher academics is not the right path, and I need to be able to teach everyone, not just the ambitious types. Thankfully, Danielle agreed with my idea. That was just the beginning.
Danielle can drop encyclopedias at the drop of a hat; I do not know how she does it. While I was contemplating what area to select and teach, she already had something in the 3D shapes area completely mapped out and she signed us up; it would later turn out that she had a sculpture that she had crafted during her childhood.
So, the research began. I began to pour over what actual teachers would post on their blogs. Sketching 3D shapes from various viewpoints on a dotted paper (called perspective diagrams in the BC curriculum, in spite of having nothing to do with the one-point, two-point, three-point perspectives as taught in visual art) seemed to be a common enough task. I had a massive writer's block over how to cover exploded diagrams- it is not exactly handy to fetch Lego, IKEA, or industrial machinery parts at the drop of a hat! So, we ignored the exploded diagram part.
Thus, the plan was born: first, make them notice how viewpoints "reveal" or "conceal" different parts of a 3D object; next, have them sit around in a circle, staring at Danielle's sculpture, and sketch the sculpture as best as they can from their vintage- that is, based only on what they can see while in their respective seats, where the sculpture is to be placed in the centre of the table and not to be interacted with; finally, provide materials so that they can try to reconstruct Danielle's sculpture... without the original sculpture on sight, using only their own sketch.
The point is that inferring depth from paper sketches is difficult- after all, in two dimensions (i.e. on paper), there are only two axes! Furthermore, they would have had only one viewpoint to work from.
This is why people try to add shading lines that suggest lighting (as seen in dessin sketches), or draw sketches taken from multiple viewpoints within the same picture (as seen in exploded diagrams and industrial blueprints).
As a side note, I do not know how it feels like from "out there", but from my own perspective, I still feel like I am not comfortable with silence. It is so funny how it works- when I recite acting scripts, deliver presentations, et cetera, I have a way with using silence and pauses, yet when I am "explaining" something in front of the class, and I feel like I am "running out of words", it is so panic-inducing. This is incredibly ironic since I generally find myself to be a very terse speaker- I mean, I know that I am not the type to go on and on and on, yet I feel so uncomfortable when I cannot, and this just about only happens when I am in front of a "class"!
Danielle can drop encyclopedias at the drop of a hat; I do not know how she does it. While I was contemplating what area to select and teach, she already had something in the 3D shapes area completely mapped out and she signed us up; it would later turn out that she had a sculpture that she had crafted during her childhood.
So, the research began. I began to pour over what actual teachers would post on their blogs. Sketching 3D shapes from various viewpoints on a dotted paper (called perspective diagrams in the BC curriculum, in spite of having nothing to do with the one-point, two-point, three-point perspectives as taught in visual art) seemed to be a common enough task. I had a massive writer's block over how to cover exploded diagrams- it is not exactly handy to fetch Lego, IKEA, or industrial machinery parts at the drop of a hat! So, we ignored the exploded diagram part.
Thus, the plan was born: first, make them notice how viewpoints "reveal" or "conceal" different parts of a 3D object; next, have them sit around in a circle, staring at Danielle's sculpture, and sketch the sculpture as best as they can from their vintage- that is, based only on what they can see while in their respective seats, where the sculpture is to be placed in the centre of the table and not to be interacted with; finally, provide materials so that they can try to reconstruct Danielle's sculpture... without the original sculpture on sight, using only their own sketch.
The point is that inferring depth from paper sketches is difficult- after all, in two dimensions (i.e. on paper), there are only two axes! Furthermore, they would have had only one viewpoint to work from.
This is why people try to add shading lines that suggest lighting (as seen in dessin sketches), or draw sketches taken from multiple viewpoints within the same picture (as seen in exploded diagrams and industrial blueprints).
As a side note, I do not know how it feels like from "out there", but from my own perspective, I still feel like I am not comfortable with silence. It is so funny how it works- when I recite acting scripts, deliver presentations, et cetera, I have a way with using silence and pauses, yet when I am "explaining" something in front of the class, and I feel like I am "running out of words", it is so panic-inducing. This is incredibly ironic since I generally find myself to be a very terse speaker- I mean, I know that I am not the type to go on and on and on, yet I feel so uncomfortable when I cannot, and this just about only happens when I am in front of a "class"!
Reflections on Eisner
Moments that made me think:
Indeed, I think the BC provincial curriculum does aim to address the objections raised by the author. There are offerings of philosophy and law under social studies now, for example, which negates the "null curriculum" problem. Self-advocacy and self-regulation are explicitly stated as a part of the core competencies. Speaking of relevancy for students, financial literacy has been hugely emphasized. However, I do not think that I can comment on issues pertaining to implicit curricula without having access to an actual timetable.
- If you are not careful with how you use external rewards, extrinsic motivation will replace intrinsic motivation
- How the traditional school system actually emphasizes comparisons and competition, in addition to reinforcing obedience and mindless routines: extrapolating from this, the industrial era public school is a hegemonic structure that aims to produce compliant students that will never achieve the level of organization necessary to rebel against the status quo
- The subjects offered, or not offered, at schools may have more to do with tradition and political interests than for actual "greater good"; the lack of offerings in law, for example.
Indeed, I think the BC provincial curriculum does aim to address the objections raised by the author. There are offerings of philosophy and law under social studies now, for example, which negates the "null curriculum" problem. Self-advocacy and self-regulation are explicitly stated as a part of the core competencies. Speaking of relevancy for students, financial literacy has been hugely emphasized. However, I do not think that I can comment on issues pertaining to implicit curricula without having access to an actual timetable.
Tuesday, October 15, 2019
Monday, October 7, 2019
Mini lesson reflection
On one hand, I definitely like to plan out the material that I intend to cover; I understand that having a hearty lesson plan allows substitutes or successors to conveniently pick up where you left off, and it also helps with classroom discipline in that students can tell when you are unsure of what you are doing. Furthermore, just as athletes have a game plan mapped out so that they can focus on which complex movements are to be expected, having a lesson plan allows you to achieve the pedagogical equivalent of that. On the other hand, it seems that getting lesson plans done quickly and efficiently in itself seems to be a major skill, and I believe that excessive pedantry results in a certain fragility in your lesson where any student asking a slightly "interesting" question throws you off your course completely. Still, it seems that striking the right balance can be challenging.
For my preliminary part, I thought I would go into the history of house music- first appearing in Chicago and how it can be traced back to disco. Also, there is patriarchal theme in the language "Oppan Gangnam Style". "Oppa" is a language used by women and girls to refer to their older brothers, giving the kind of domineering vibes when a drill instructor refers to male recruits as "ladies" in order to degrade them on purpose. Also, Gangnam is an upscale area with expensive clubs and apartments, so, in a way "Gangnam Style" actually contains a rather boastful message. Still, most of the lyrics of the song are about what kind of ladies he likes and what kind of guy he is (temperament-wise), so the song can also be interpreted to mean "this is how we party in Gangnam".
As for PSY, PSY himself actually comes from an exceedingly privileged background and received some top-tier schooling in the form of Boston University and Berklee School of Music. So, I took a dance lesson and introduced some social critical elements.
The signature movement has many "building blocks" in it.
https://giphy.com/gifs/vevo-gangnam-style-psy-oppa-nYI8SmmChYXK0
It requires you to hop and drag your feet in rapid succession. Here is how one would scaffold the learning process: isolate them into simpler movements, slow practice until it really "sinks in" your brain, and put the isolated elements together. To that end, I first made them practice hopping on one foot, then dragging the foot that you are hopping on mid-air without moving backwards (a lot of slow, deliberate practice here), then I added the signature hand movement. Also, sharing the source of the inspiration helped greatly too: mimicking a horseback rider. The group members picked it up very well.
Do you know how clumsy you feel when you lift weight for the first time, because your muscles do not have the "practice"? Admittedly, I felt that a bit when I when it came to the choreography part. incorporating the signature movement among many others. I actually got nervous because I thought that the planned choreography did not incorporate the signature movement enough, which then began to affect my recall.. I thought I was running out of time, and then it turns out that I still had two minutes, so I pulled out what I had in mind for "extending" the lesson (just my opinion: it is better to have just a little too much material because you can always cut some in order to meet the time) and then the two minutes flew by. Checking the clock discreetly and quickly, and being able to accurately "feel" how much time has passed all seem to be actually non-trivial skills to develop.
In the future,
For my preliminary part, I thought I would go into the history of house music- first appearing in Chicago and how it can be traced back to disco. Also, there is patriarchal theme in the language "Oppan Gangnam Style". "Oppa" is a language used by women and girls to refer to their older brothers, giving the kind of domineering vibes when a drill instructor refers to male recruits as "ladies" in order to degrade them on purpose. Also, Gangnam is an upscale area with expensive clubs and apartments, so, in a way "Gangnam Style" actually contains a rather boastful message. Still, most of the lyrics of the song are about what kind of ladies he likes and what kind of guy he is (temperament-wise), so the song can also be interpreted to mean "this is how we party in Gangnam".
As for PSY, PSY himself actually comes from an exceedingly privileged background and received some top-tier schooling in the form of Boston University and Berklee School of Music. So, I took a dance lesson and introduced some social critical elements.
The signature movement has many "building blocks" in it.
https://giphy.com/gifs/vevo-gangnam-style-psy-oppa-nYI8SmmChYXK0
It requires you to hop and drag your feet in rapid succession. Here is how one would scaffold the learning process: isolate them into simpler movements, slow practice until it really "sinks in" your brain, and put the isolated elements together. To that end, I first made them practice hopping on one foot, then dragging the foot that you are hopping on mid-air without moving backwards (a lot of slow, deliberate practice here), then I added the signature hand movement. Also, sharing the source of the inspiration helped greatly too: mimicking a horseback rider. The group members picked it up very well.
Do you know how clumsy you feel when you lift weight for the first time, because your muscles do not have the "practice"? Admittedly, I felt that a bit when I when it came to the choreography part. incorporating the signature movement among many others. I actually got nervous because I thought that the planned choreography did not incorporate the signature movement enough, which then began to affect my recall.. I thought I was running out of time, and then it turns out that I still had two minutes, so I pulled out what I had in mind for "extending" the lesson (just my opinion: it is better to have just a little too much material because you can always cut some in order to meet the time) and then the two minutes flew by. Checking the clock discreetly and quickly, and being able to accurately "feel" how much time has passed all seem to be actually non-trivial skills to develop.
In the future,
- Be confident about what you have to offer.
- Speaking quietly with clear enunciation actually requires attention.
- There is a difference between minor adjustments and disrupting your own lesson stemming from a lack of confidence.
- Break it down, slow practice, put them all together- scaffolding cannot get any more literal or physical than this, I think.
- Figure out a way to check time accurately and discreetly. If you can tell time without looking at the clock, even better!
The cook math puzzle
"How many guests are there?" said the official.
"I don't know.", said the cook, "but every 2 used a dish of rice, every 3 used a dish of broth, and every 4 used a dish of meat between them". There were 65 dishes in all. How many guests were there?
Let us investigate some earlier ones.
The first three guests used a dish of rice and a dish of broth; that would be two dishes.
The fourth used used a dish of rice and a dish of meat; that would be two more dishes.
The sixth used a dish of rice, and a dish of broth; that would be two more dishes.
The eight used a dish of rice, and a dish of meat; that would be two more dishes.
The ninth used a dish of broth; that would be one more dish.
The tenth used a dish of rice; that would be one more dish.
The twelfth used a dish of rice, a dish of broth, and a dish of meat; that would be three more dishes.
I think we can in fact find an iterative summation algorithm here:
Asking if a given integer is a multiple of n is equivalent to asking if the said integer does not produce a remainder when divided by n.
Thus, we arrive at the following Python code:
dishCount = 0
numGuests = 1
while (dishCount < 65):
print("Current number of guests: " + str(numGuests) + ", current dishes served: " + str(dishCount))
if (numGuests%2 == 0):
dishCount += 1
print("Current number of guests: " + str(numGuests) + ", dish of rice served.")
if (dishCount >= 65):
continue
if (numGuests%3 == 0):
dishCount += 1
print("Current number of guests: " + str(numGuests) + ", dish of broth served.")
if (dishCount >= 65):
continue
if (numGuests%4 == 0):
dishCount += 1
print("Current number of guests: " + str(numGuests) + ", dish of meat served.")
if (dishCount >= 65):
continue
numGuests += 1
print("The number of guests is: " + str(numGuests))
The program prints 60:
"I don't know.", said the cook, "but every 2 used a dish of rice, every 3 used a dish of broth, and every 4 used a dish of meat between them". There were 65 dishes in all. How many guests were there?
Let us investigate some earlier ones.
The first three guests used a dish of rice and a dish of broth; that would be two dishes.
The fourth used used a dish of rice and a dish of meat; that would be two more dishes.
The sixth used a dish of rice, and a dish of broth; that would be two more dishes.
The eight used a dish of rice, and a dish of meat; that would be two more dishes.
The ninth used a dish of broth; that would be one more dish.
The tenth used a dish of rice; that would be one more dish.
The twelfth used a dish of rice, a dish of broth, and a dish of meat; that would be three more dishes.
I think we can in fact find an iterative summation algorithm here:
- dishCount = 0
- numGuests = 1
- while (dishCount < 65):
- if (numGuests is a multiple of 2):
- dishCount += 1
- Immediately exit loop if dishCount is at or exceeds 65
- if (numGuests is a multiple of 3):
- dishCount += 1
- Immediately exit loop if dishCount is at or exceeds 65
- if (numGuests is a multiple of 4):
- dishCount += 1
- Immediately exit loop if dishCount is at or exceeds 65
- print("The number of guests is: " + str(numGuests))
Asking if a given integer is a multiple of n is equivalent to asking if the said integer does not produce a remainder when divided by n.
Thus, we arrive at the following Python code:
dishCount = 0
numGuests = 1
while (dishCount < 65):
print("Current number of guests: " + str(numGuests) + ", current dishes served: " + str(dishCount))
if (numGuests%2 == 0):
dishCount += 1
print("Current number of guests: " + str(numGuests) + ", dish of rice served.")
if (dishCount >= 65):
continue
if (numGuests%3 == 0):
dishCount += 1
print("Current number of guests: " + str(numGuests) + ", dish of broth served.")
if (dishCount >= 65):
continue
if (numGuests%4 == 0):
dishCount += 1
print("Current number of guests: " + str(numGuests) + ", dish of meat served.")
if (dishCount >= 65):
continue
numGuests += 1
print("The number of guests is: " + str(numGuests))
The program prints 60:
Current number of guests: 1, current dishes served: 0
Current number of guests: 2, current dishes served: 0
Current number of guests: 2, dish of rice served.
Current number of guests: 3, current dishes served: 1
Current number of guests: 3, dish of broth served.
Current number of guests: 4, current dishes served: 2
Current number of guests: 4, dish of rice served.
Current number of guests: 4, dish of meat served.
Current number of guests: 5, current dishes served: 4
Current number of guests: 6, current dishes served: 4
Current number of guests: 6, dish of rice served.
Current number of guests: 6, dish of broth served.
Current number of guests: 7, current dishes served: 6
Current number of guests: 8, current dishes served: 6
Current number of guests: 8, dish of rice served.
Current number of guests: 8, dish of meat served.
Current number of guests: 9, current dishes served: 8
Current number of guests: 9, dish of broth served.
Current number of guests: 10, current dishes served: 9
Current number of guests: 10, dish of rice served.
Current number of guests: 11, current dishes served: 10
Current number of guests: 12, current dishes served: 10
Current number of guests: 12, dish of rice served.
Current number of guests: 12, dish of broth served.
Current number of guests: 12, dish of meat served.
Current number of guests: 13, current dishes served: 13
Current number of guests: 14, current dishes served: 13
Current number of guests: 14, dish of rice served.
Current number of guests: 15, current dishes served: 14
Current number of guests: 15, dish of broth served.
Current number of guests: 16, current dishes served: 15
Current number of guests: 16, dish of rice served.
Current number of guests: 16, dish of meat served.
Current number of guests: 17, current dishes served: 17
Current number of guests: 18, current dishes served: 17
Current number of guests: 18, dish of rice served.
Current number of guests: 18, dish of broth served.
Current number of guests: 19, current dishes served: 19
Current number of guests: 20, current dishes served: 19
Current number of guests: 20, dish of rice served.
Current number of guests: 20, dish of meat served.
Current number of guests: 21, current dishes served: 21
Current number of guests: 21, dish of broth served.
Current number of guests: 22, current dishes served: 22
Current number of guests: 22, dish of rice served.
Current number of guests: 23, current dishes served: 23
Current number of guests: 24, current dishes served: 23
Current number of guests: 24, dish of rice served.
Current number of guests: 24, dish of broth served.
Current number of guests: 24, dish of meat served.
Current number of guests: 25, current dishes served: 26
Current number of guests: 26, current dishes served: 26
Current number of guests: 26, dish of rice served.
Current number of guests: 27, current dishes served: 27
Current number of guests: 27, dish of broth served.
Current number of guests: 28, current dishes served: 28
Current number of guests: 28, dish of rice served.
Current number of guests: 28, dish of meat served.
Current number of guests: 29, current dishes served: 30
Current number of guests: 30, current dishes served: 30
Current number of guests: 30, dish of rice served.
Current number of guests: 30, dish of broth served.
Current number of guests: 31, current dishes served: 32
Current number of guests: 32, current dishes served: 32
Current number of guests: 32, dish of rice served.
Current number of guests: 32, dish of meat served.
Current number of guests: 33, current dishes served: 34
Current number of guests: 33, dish of broth served.
Current number of guests: 34, dish of rice served.
Current number of guests: 35, current dishes served: 36
Current number of guests: 36, current dishes served: 36
Current number of guests: 36, dish of rice served.
Current number of guests: 36, dish of broth served.
Current number of guests: 36, dish of meat served.
Current number of guests: 37, current dishes served: 39
Current number of guests: 38, current dishes served: 39
Current number of guests: 38, dish of rice served.
Current number of guests: 39, current dishes served: 40
Current number of guests: 39, dish of broth served.
Current number of guests: 40, current dishes served: 41
Current number of guests: 40, dish of rice served.
Current number of guests: 40, dish of meat served.
Current number of guests: 41, current dishes served: 43
Current number of guests: 42, current dishes served: 43
Current number of guests: 42, dish of rice served.
Current number of guests: 42, dish of broth served.
Current number of guests: 43, current dishes served: 45
Current number of guests: 44, current dishes served: 45
Current number of guests: 44, dish of rice served.
Current number of guests: 44, dish of meat served.
Current number of guests: 45, current dishes served: 47
Current number of guests: 45, dish of broth served.
Current number of guests: 46, current dishes served: 48
Current number of guests: 46, dish of rice served.
Current number of guests: 47, current dishes served: 49
Current number of guests: 48, current dishes served: 49
Current number of guests: 48, dish of rice served.
Current number of guests: 48, dish of broth served.
Current number of guests: 48, dish of meat served.
Current number of guests: 49, current dishes served: 52
Current number of guests: 50, current dishes served: 52
Current number of guests: 50, dish of rice served.
Current number of guests: 51, current dishes served: 53
Current number of guests: 51, dish of broth served.
Current number of guests: 52, current dishes served: 54
Current number of guests: 52, dish of rice served.
Current number of guests: 52, dish of meat served.
Current number of guests: 53, current dishes served: 56
Current number of guests: 54, current dishes served: 56
Current number of guests: 54, dish of rice served.
Current number of guests: 54, dish of broth served.
Current number of guests: 55, current dishes served: 58
Current number of guests: 56, current dishes served: 58
Current number of guests: 56, dish of rice served.
Current number of guests: 56, dish of meat served.
Current number of guests: 57, current dishes served: 60
Current number of guests: 57, dish of broth served.
Current number of guests: 58, current dishes served: 61
Current number of guests: 58, dish of rice served.
Current number of guests: 59, current dishes served: 62
Current number of guests: 60, current dishes served: 62
Current number of guests: 60, dish of rice served.
Current number of guests: 60, dish of broth served.
Current number of guests: 60, dish of meat served.
The number of guests is: 60
I think I can whip out a formula at this point:
d/2 + d/3 + d/4 = 65
6d + 4d + 3d = 780
13d = 780
d = 60
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