Micro teaching topics

In anticipation that some may be duplicates (if none or only one of them is a duplicate, I will pick one for Wednesday from the remaining at random):

• Card magic
• Gangnam Style dance
• How to sing high notes

Reflections on the math art project

The Fibonacci sequence- a classic, overused example for teaching recursion and dynamic programming in computing science. Considering that a slightly easier version of what gets taught in first-year university courses have made their way into high schools now, I suppose that a variation on the project can be used for a computer programming course (e.g. write a program that outputs the sequence up to a given user-specified limit; with some careful planning by the teacher, extend the program to draw the squares in their correct relative positions, and maybe even trace out the spiral too for bonus; et cetera). In addition, "history of mathematics" is making its way into BC classrooms now, and this project can probably be delivered almost as-is in conjunction with number systems of various civilizations (e.g. with Egyptian/Chinese/etc numbers).

I will admit, I did not know that the sequence actually appears in nature until we conducted research for the classroom activities portion. As much as life on universe entails consuming another life for survival and sudden, violent, unforeseen, unjustified tragedies, there are also moments of sheer awe and immaculate patterns, too. The Fibonacci sequence and the spiral recur in many different places if you know where to look. The pattern of bee ancestry just struck me because how it managed to be simple, fun, and relevant all at the same time!

We worked together well as a group. People immediately jumped in with what contributions they were willing to make. For the record, using the spirals as a variation/extension on the original art was my idea. The original work simply put the numbers in order, so (at least in my opinion) the spiral extension was right there. 

I must admit that cutting out and gluing felt pieces, drawing with a compass, et cetera all evoked some rather nostalgic feelings!

The Wordy Puzzle

"Brothers and sisters have I none- but that man's father is my father's son"

The writer denies having any siblings, yet one interpretation of "that man's father is my father's son" may indicate that the father has a son, other than the writer, who is "that man's father"- a contradiction on first sight.

If I were to hazard an interpretation that does not introduce contradictions, I would consider the following: the writer is an only son as supported by "brothers and sisters have I none", and the writer is in fact referring to himself in the third person with "my father's son". Then, "that man" is the writer's own son.

On the locker problem

Rather than sticking to 1000 and getting overwhelmed by the sheer magnitude, let us try smaller instances first. In order to do that, let us understand what the problem demands in general.

According to the problem description, there are n doors and n students. The nth student toggles every nth door. If we stick to smaller values of n, it is entirely feasible to do some simulations by hand and extract patterns.

We start with all doors being open, and start tracking from the first student onward.

• With n = 2, we have: closed closed; closed open.
• With n = 3, we have: closed closed closed; closed open closed; closed open open; closed open closed.
• With n = 4, we have: closed closed closed closed; closed open closed open; closed open open open; closed open open closed.

Let us pause for a moment to think about some properties of natural numbers, which are those typically used for counting in everyday life; no negatives, no rationals, nothing like that.

1. Every natural number is either even or odd.
2. Every natural number other than one has at least two numbers, called the divisors, such that dividing the given number by the divisor results in another natural number: one and itself.
1. Prime numbers are those that only have one and itself as divisors.
2. Composite numbers are those that have three or more divisors -i.e. at least one more divisor in addition to one and itself-. A composite number always has either an odd or an even number of divisors, but always at least three.

Let us number each door from 1 to n. Since we are working with natural numbers, knowing their global properties helps us predict which ones will be open or closed. In fact, we have already studied the following cases: when the number of the door is prime, and when the number of the door is composite with an odd number of divisors.

Recalling that all doors are initially open, we can make the following prediction: given any nth door, if n has an even number of divisors (two sub-cases: composite with even divisors; prime), the door will be open after all students have finished; if n is odd, the door will be closed.

Indeed, one can even write a computer program to simulate each student toggling the door. Here is code using Python 3 (excuse the lack of syntax highlighting):

'''

Using the command prompt, run this file by "python filename.py [0-9]+",

or "python3 filename.py [0-9]+", et cetera,

where the third argument, denotes how many doors there are.

USE PYTHON 3 OR LATER VERSION.

'''

import sys

#The python interpreter crops "python", so argv[0] = filename.py,

#and argv[1] = number of doors.

numDoors = int(sys.argv[1])

doors = list()

#Initialize the doors

for i in range(0, numDoors):

    doors.append(1)

    

#This variable is needed to control the toggling offset

#every door, every second door, every third door, etc

offset = 1

#Simulate how each student toggles every nth door

for i in range(0, numDoors):

    for j in range(i, numDoors, offset):

        

        #Use the exclusive-or operator to flip

        doors[j] = doors[j] ^ 1

    offset += 1

    

for i in range(0, numDoors):

    message = "Door #" + str(i + 1) + " is "

    if (doors[i] == 1):

        message += "open."

    elif (doors[i] == 0):

        message += "closed."

    else:

        message += "this should not happen."

    print(message)

Given any nth door, knowing that we can exploit properties concerning n to tell if it is closed or not, we can ask the following related questions:

• How do we know how whether a given natural number is prime or not? This is a worthwhile question to ask because primality allows for an immediate answer (i.e. two).
• If it is composite, how do we actually compute how many divisors it has, apart from 1 and itself?

As a degree-holder, I know how to answer these, using university-level mathematics. As an educator, this takes an interesting turn.

Both of these are an excellent opportunity to introduce students to the state-of-art and the sheer cosmic scale of mathematics. Primality testing is one thing; prime factorization in particular is a different story. By yourself, as the numbers get bigger (1000 is rather small), the computations may not finish within your lifetime; even grown-ups are completely stumped over how to perform prime factorization efficiently in general. This is such a problem that grown-ups are exploring a completely new architecture of computing- quantum computing.

Furthermore, this can show students how entire civilizations can hinge on seemingly innocent properties in mathematics. For instance, the fine art of mathematically obfuscating your information to make it inaccessible, called cryptography, relies on seemingly simple questions, like prime factorization, being nigh impossible to answer quickly. This is some real life sorcery right here.

Role of representations

In mathematics, knowing how to say the same thing in many different ways is a powerful asset; at the same time, jumping to algebra first, while an "elegant" approach, can hide details and fail to truly illuminate the "discovery" of the final product; thus, I must say that external representations done wrong can actually increase the cognitive load on the student. While I may seem to be slightly disagreeing with the article, both of those points are actually supported by the same Russian experiment on teaching trigonometric proofs seems to assert as much.


Speaking of jumping to the abstract first, a case in point would be starting a group theory course in university by dropping the four group multiplication properties out of thin air. When group theory was the hottest frontier back in the 19th century, the notations and concepts were nowhere near as slender as today; it took several generations until Cayley came along and cleaned things up.


When the authors discuss Hiebert's insights on developing competencies in working with mathematical symbols, it would appear that after the initial connection between the symbols and the underlying objects are made, pupils shall learn how to manipulate symbols, and then that is that. It is tempting to assert that understanding how the objects are manipulated underneath the said symbols cannot be neglected.






Assuming that "iconic" includes other kinds of multimedia in addition to pictures in static frames like ye olde portraits, I do not think anything in particular was "missed" per se.

Two letters: hope and worry

Positive letter:

Dear Mr. Park:

You showed how all the dots connect together, and then where all the lines lead to. You showed how relevant everything is, and just how much there is to learn out there. I may have been born into a poor family and lived most of my formative years as an at-risk youth, but now I am about to graduate from Caltech with a Ph. D. in physics and go into the warmhole research program by NASA that was announced worldwide on the news. All of this is because you were the first to show faith in me. I would not be where I am today without you.

Negative letter:

Dear Mr. Park:

Why did you not tell me that I should just give up? Why did you have to encourage my overzealous parents, and then play up my hopes, too? Do you not realize how much the world changed? The tech industry is not lucrative anymore, and I am collecting unemployment while my "dumbass" friend squeezed her way into public service and probably will not retire for the next 80 years.


I tend to emphasize, among other things, relevancy and motivation. Indeed, while the school system is designed to be flexible, willing individuals must find the drive to put in extra work by themselves.

Also, one of my biggest fears is to see my students suffer because of my outdated or irrelevant career advice, hence the second kind of letter. I also hate to see students give into despair.

Math and me

Sometimes, private education does more harm than good. If my memory serves correctly, Kumon taught some very haphazard tricks for doing linear algebra, which actually stumped me when I hit university because I underestimated learning Gaussian elimination properly due to overconfidence from Kumon.

First of all, I decided to become a teacher because I am jaded by just how much in tech is profit-driven and polarizing . Now, within there, mathematics is inherently a part of it, and results on polynomials or number theory are just fascinating on their own right. Still, I do not see myself as a "math teacher" per se.

Response to the Skemp article

    I find it that much of my pedagogy education so far consists of rigorously re-visiting, and sometimes challenging what can be thought of as "common" sense. Two of my personal three standouts relate to this theme:

• Instrumental instruction prevails in part due to the teachers themselves not having learned mathematics on a relational level themselves; indeed, it seems that engineering majors and mathematics minors are more common, and I know that in both of those programmes, you can mostly get by with an instrumental approach. To make matters worse, I have known many arrogant and cliquey mathematics majors who think they are bound to become the next Terry Tao.
• Perhaps it does take a special kind of pupil to actually appreciate the relational approach, which is not to say that you can help pupils acquire the marvelous taste of relational instruction. Considering how much emphasis is placed on the teacher being able to transform the pupil, and just how much onus and pressure can be placed on the teacher, it was validating to see Skemp emphasize the mindset of the pupil. 

Separate from this, I truly look forward to what is being researched in terms of this novel qualitative model for assessing mathematical thinking  mentioned towards the end of the article. There has got to be something beyond the traditional end-product-based evaluation; this is education, where  we are supposed to be thinking about how to make teaching better.

    As for where I stand with regards to this article, it takes so many words right out of my mouth that I cannot muster anything; indeed, anything I may write are merely interesting case studies in support of Skemp's points. For instance, in EDST 401, a curious case study from Newfoundland was presented in which a professor criticized the undergraduate students for not being able to do logarithms. 

    Still, I think Skemp does neglect something, which is addressing the parents. Speaking of pupils only being interested in the mechanics of mathematics so that they can crunch some numbers and get a job, there really are parents who encourage their children to treat just like mathematics like that; they just want their children to get out of their hair as soon as possible, and count every year of parenthood like a prison sentence. They are actually hampering their children's professional and academic potential by instilling such a lazy attitude because if you merely aim for "good enough" in adulthood, you will never actually be "good enough" in adulthood. At the professional level, somebody always finds something in your work to bring it down a few pegs even if you bring your absolute best; by setting your starting point as "good enough", you are almost guaranteeing your own failure.

Hello World

Real gangsters move in silence like Lasagna.