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:

• 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.

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:

• 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

What if you had a 4 gram weight?

• 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

What if you had a 6 gram weight? Then,

• 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

What if you had a 10 gram weight? Then,

• 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

With 28 grams:

• 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?

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