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. 

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:

• 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?

I suppose what makes a puzzle geometrical rather than logical is the presence of measurement of space and shape. LSAT questions, for instance, emphasize decoding information precisely off of a deliberately convoluted text- something that requires some good logic chops, but not necessarily spatial reasoning per se.

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"!

Reflections on Eisner

Moments that made me think:

• 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.

So then, what is curriculum? On one hand, as educators, we would like to think that we have carefully distilled what absolutely must be passed along during compulsory education. On the other hand, is it merely a reflection of traditional biases and entrenched professional interests? Maybe, the real answer lies somewhere in the middle.

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.

15-min micro teaching lesson plan

Subject: Workplace Math 11

Topic: 3D objects

Duration: 15 min

Curricular Competencies:	Develop thinking strategies to solve puzzles and play games Think creatively and with curiosity and wonder when exploring problems Visualize to explore and illustrate mathematical concepts and relationships Represent mathematical ideas in  concrete, pictorial, and symbolic forms
Content Competencies:	3D objects: angles, views, and scale diagrams creating and interpreting exploded diagrams and perspective diagrams drawing and constructing 3D objects
Big Ideas:	3D objects are often represented and described in 2D space
Materials:	3 sheets of paper pencils/erasers White board/marker Object for introduction Object for drawing Materials for building

Lesson Outline:

Introduction	Teacher-lead, group participation What is a perspective drawing? Why does perspective matter? Do a drawing example with a side-distinctive object	2 min
Activity: Drawing	Split into 3 groups. Explain activity and introduce object. Have 3 groups sit around the object, each group with a unique side. Each group will draw the object from their perspective Teachers to walk around, monitor, and assist if necessary	4 min
Activity: Building	Take the object away Have the 3 groups rotate their drawings and attempt to build the object with the new drawing Teachers to walk around, monitor, and assist if necessary	6 min
Conclusion	Bring object back out Have groups share what their object looks like, see how different the 3 groups are Wrap up: so why does perspective matter? Because look how different all these drawings look, even though they’re all the same object.	3 min

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,

• 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:

• 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