Thinking mathematically, response + problem

Response:

My first stop(s) would be all of the little bits of advice which are given for helping get through moments of being stuck on a problem. They mostly seem somewhat obvious, but I suppose the point is that it's not always obvious to students on how to handle being stuck. Just as a side note, it bothers me somewhat that so many words are capitalized. That might be my own fussy little thing, and I need to think more on why that bothers me... But I wonder if I feel like it's a little reductive to imply that there is like this formula of key-worded steps to properly think mathematically. I would however, counter those negative thoughts of mine by saying I think it's actually a great thing to celebrate getting stuck on a problem. It usually doesn't feel too good getting stuck, but we should all instead strive to be excited when faced with a problem requiring some careful and creative thought. Ironically, another stop for me was the "Threaded Pins" problem, which I found frustrating and didn't spend more than a couple minutes looking at because I thought it was a poorly formulated question. Now, it might just be me - I know I can be ignorant of very obvious things - but it was not at all obvious (to me at least) what was meant by a "clockwise gap." So flying in the face of celebrating getting stuck, I decided that spending any more time on that problem was a waste. The other problem I looked at was much easier for me to understand.

The Problem: Rational Divisors
Given that 14/15 divides into 28/3 a whole number of times (10 times), we
might say that 14/15 is a rational divisor of 28/3.
● Find all the rational divisors of 28/3.
● Find all the rational divisors of 1/2, and then find all the numbers that
are rational divisors of both 28/3 and 1/2.
Does it make sense to talk about the greatest common rational divisor of
two fractions, and the lowest common rational multiple of two fractions?

- I started off by working through the example division they gave. The idea was to get a sense of how dividing with fractions results in a whole number.

- Using the "keep change flip" method of dividing fractions, combined with "cross-cancelling," I realized that to get a whole number, the denominators (after the flip) would need to be factors of the numerators (after the flip)

- here I start to try and generalize that idea a little bit, to hopefully find a method to obtain all rational dividers of 28/3.

- I try to be a little bit more clear about the conditions required for a rational number to divide 28/3

- realizing that 3 has infinitely many multiples, it was clear that you could have as many rational divisors as you like. for example: 28/3, 28/6, 28/9, and so on, would all divide 28/3 nicely.

- now trying to answer the question, I set to expressing it as as set of numbers. 

- I saw that it would easy to show the infinitely many answers using a variable, and that it might be nice to include positive and negatives as well.

- I thought that I might be able express the numerators in a more compact manner, maybe with a variable as well, but I felt that this was more clear to simply list all of the possible numerators. especially since, unlike the denominator, there are not infinitely many choices.

- while I had technically solved the first part of the problem, I was still interested in rewriting this idea more generally, mostly for fun.

- I think this could still be refined, probably stated a little bit more cleanly. and in general I don't really like using the "divisible by" symbol. 

- It seems like it would be more useful to do something like "there exists a factor k for which m = a*k" but I think this gets the point across.

- using the method I had developed earlier, it was relatively simple to apply it to find the set of rational divisors of 1/2

- now looking to find the greatest common divisor, I started by thinking how to find the set of all common divisors.

- referring back to my method, I had a bit of an "Aha" moment, realizing that my conditions work just as well when considering multiple numbers.

- from there I just needed to find the common factors of the numerators, and the common multiples of the denominators. thankfully these were relatively simple.

 

- a key to this for me was knowing that all common multiples of 3 and 2 would be multiples of their least common multiple 6, which made it easy for me to write my answer in a similar looking set. I was enjoying the consistency of that format of answer at this point

- from there, it was straightforward to find the GCD, the only thing I stumbled over for a minute was that a greater denominator would actually correspond to a "lesser" rational number overall.

- this allowed me to realize in general, that to have the greatest possible divisor, we would want to minimize the denominator, but maximize the numerator.

- for fun, I generalized this for any given rational number.

- I also had the thought that this would work for any number of rational numbers, so I decided to use ellipses, m_i and n_i to indicate this idea

- I finished by answering the question that "yes" this does make sense, to me at least. I couldn't see any reason why this result wouldn't be applicable for any rational numbers. 

- I did however exclude 0 from the discussion. So I suppose in the interest of completeness I would add that it doesn't make too much sense to consider the rational divisors of 0, only because it is simply all numbers other than 0.

- here, I began to consider the LCM of rational numbers. and was assuming the traditional view point of multiples, which is they would be the result of multiplying by the positive integers

- I decided to work through some examples just to get some insight into the problem.

- here I saw that if the denominators were mutually prime, you would essentially need to find the LCM of the numerators.

- I did not formalize these ideas here, but I do think it would have been fun to play with that some more

- another example... it seems that if the numerators are the same, and one denominator is a multiple of the other, then the LCM would simply be the greater fraction

- what if the numerators were mutually prime but one of the denominators was a multiple of the other? (reverse of the first example)

- here I started to think about this problem as depending on cases of whether or not denominators/numerators were mutually prime

- at this point, I've reached the conclusion that it makes plenty of sense to consider the LCM of rational numbers, though some cases are simpler than others.

- I did also consider: what if weren't using the "traditional" definition of multiples, and instead, were thinking of multiples of rational numbers as being products of any two rational numbers? however, I had the thought that you could always choose to multiply by a rational number to obtain any rational number you like, so in theory the LC"M" of any two rational numbers is arbitrarily small. example 1/2 and 1/3... multiply 1/2 by 2/99999999... and multiply 1/3 by 3/99999999...

Microteaching reflection

 First and foremost, I was super impressed with everyone in our little group. They all had something very interesting to teach, were super engaging, and they all taught the subject matter so effortlessly. Personally, I was worried that my topic was A) going to be such well-trod territory that everyone already knew everything I wanted to say, and B) that I was going to struggle to fill the time and speak off the cuff. Both of these concerns were somewhat unwarranted, as my group was really receptive and a decent amount of what I had to teach was novel to them. Also, once I started talking, I found it difficult to stop. I wanted to keep explaining and adding on, and I think it turned out to be a bit of an issue in the end. Not only did I over plan, trying to fit too much content in the short amount of time, but I also struggled to find natural stopping points to shift from one part of the lesson to the next. As a result, I think the lesson as a whole suffered from rushed teaching and never reaching the conclusion or having made time for more class participation and inquiry. In retrospect, I wish I had maybe done the micro teaching lesson to myself in the mirror or something, to get a better sense of the timing and adjust my lesson plan accordingly. At the rushed pace I was teaching, I think it was more realistically 20 minutes worth of content, and if I had instead been teaching in a more calm and natural manner, it could have easily been extended to a 30 minute lesson. I’m sure eventually I won’t need to run through lessons “in the mirror” to know how much time they'll take, but while I’m still getting the hang of this whole teaching thing, that might not be a bad idea. Shoutout to Brandon, Taha, and Leon for the excellent microteachings, being a fantastic audience, and making that class a lot of fun.

 

non-curricular micro-teaching lesson-plan

 

Non-Curricular Lesson Plan: Basic Rules and Principles of Chess

Main objective: Learn chess, generate interest in its complexity 

Materials: board and pieces, chess timer

Probe for prior knowledge: “Show of hands: how many of you know how to play chess already?” (If large majority… spend less time on Lesson pt. 1)

Introduction (~1 minute): Brief history of Chess

Objective: promote idea of its ubiquity and as a long standing “tradition”

    - Chaturanga in India 1500 years ago, developing into many different games

Lesson pt.1 (~2-3 minutes): Objective of the game, and how each piece moves

Objective: Student will learn to play on their own 

    - In small group, explain how to set up board, explain each piece as you go

Lesson pt.2 (~2-3 minutes): Basic principles, openings, “advanced” moves

Objective: introduce complexity of the game

    - center control, piece development, pawn structure, protecting the king
    - kings pawn and queens pawn, four knights game, French defense
    - “pawn special moves” and castling

Activity 1 (~2-3 minutes): Timed game

Objective: class participation, fun

    - 2 volunteers to play a timed game,
    - set up chess timer for 1-minute of total time for each player

if there's time....

Lesson pt. 3 (~2 minutes): Analyze final position of the game

    - ask class about the position of each players positions while considering the principles covered earlier
    - what might be some possible moves for whichever student was next?

Activity 2  (~1 minute): Analyze position of a specific game

 *allow for questions between each part of the lesson/activity

art project reflection

Our group project began with Madison selecting the initial artwork, which gave us a strong foundation to build upon. When we first met as a group, we discussed extending the artwork by incorporating additional prime number concepts. Madison suggested using perfect squares as a visual anchor for viewers, and we explored adding elements like square-free prime signatures and Euler’s totient function to enhance the visualization. After discussing different shapes for clarity, we chose concentric circles to create bands of information, allowing us to visually represent our mathematical concepts effectively. Madison's early contributions and direction-setting allowed us to divide the remaining work: Sahl focused on recreating the original artwork, I delved into the specific mathematical extensions, and Andy worked on coding and visualizing these extensions using Python. Once Andy developed a proof-of-concept for the circular band plots, our group collaborated on selecting textures to highlight perfect squares and prime signatures, eventually finalizing the color and texture schemes. To complete the project, we incorporated Euler’s totient into the artwork, and Madison pulled everything together into a polished slide deck.

Firstly, I found this project very interesting, and I very much enjoy working with Andy, Sahl, and Madison. Overall, I'm super impressed with each of them as individuals and it was definitely a motivating factor for me to step up and contribute in any small way I could. That being said, the finished product and presentation wouldn't have been half as good without those guys. Sahl did an immaculate job of reverse engineering the original piece, Andy was really basically showing off his next-level ability to create computer-generated visualizations, and Madison absolutely crushed it with the presentation and slide deck. Initially however, it was challenging to try and think of ways to extend the original artwork, both in terms of the math as well as the artistic aspects. Once I actually started doing some basic research on prime factorizations, I found it hard not to get sucked into the number theory rabbit hole. Being in a mindset of "how can I represent this concept visually" was an excellent way to keep me extra engaged while poring over various wikipedia articles. It also made me miss my number theory class from my undergrad, and also kind of wish I could take some more math classes. There's so much math out there I've yet to learn! But we'll get back to that one day.

As a teacher bird, my big takeaway from this project is that there is so much room for creativity in how we engage with mathematics. Math and art in particular, is a super interesting interdisciplinary combination. Along those lines however, I think it would be cool to mix math with other subject areas in similar kinds of interdisciplinary assignments. Doing something with music would be interesting I think. The obvious exercise would be to look at a piece of music and analyze the fractions of beats and measures based on the given time signature. However, part of what I enjoyed so much about this project was the open-endedness of it. We were presented with some initial starting points, but then given free reign to develop ideas with our own creative ideas. So, were I to run with this idea of mixing math and music, I would try to do something similar: having some initial examples of ways to relate the two subjects and then open it up to the students to extend those further. As a teaching tool, I am 1000% going to steal this assignment idea and use it my own classroom (will give credit).

 

thoughts on battleground schools

The first stop for me was the bullet point descriptions of some cultural assumptions with regards to math. I especially appreciate how these are described as "socially sanctioned." As someone who thinks math is a really interesting subject, it irks me how it is not only socially acceptable, but also socially reinforced to dislike math. This ties into the 4th bullet point, which mentions how there is "lots of positive valuation" for students who vocalize their struggles with math. It's absolutely understandable for people to seek comfort with others when they are dealing with something difficult - I get that. I just wish it was framed differently. For example, why do people wear their struggles with math as a badge of honor? I'm not saying you should be ashamed, but it is frustrating because it propagates this idea that math is beyond understanding and has no value for most people (2nd bullet point). I think this is all basically a defense mechanism resulting from years of traumatic education experiences. So it's slightly forgivable on the parts of students commiserating with each other. On the other hand, I dislike it when parents say things like "Oh it's so funny how bad I am at math." Then again, I must acknowledge that growing up I was the beneficiary of at least some respect from teachers and peers as a kind of rare breed who enjoyed math class and was halfway competent with the subject.

The second stop for me was at the mention of "New Math." I've heard this referenced before, mostly derogatorily in popular culture, but I can't say I know too much about it. Just from reading a bit about it in the article and just now on wikipedia... It actually sounds pretty nifty to me. I'm sure its criticism are valid, but how can you not respect trying something different? Easy for me to say as someone who didn't actually experience it. The article points out that in many important respects, it was still quite traditional. And looking at a some other examples of "New Math problems" from google... I feel like I can see some of its DNA in contemporary math education (like Jump Math, for example). Were I a parent during that time period, I can see myself getting a kick out of it while simultaneously joining in on the chorus of people dog-piling on its obtuseness. If I'm understanding the history correctly, it seems like it was kind of an important development insofar as catalyzing changes to the curriculum. I think we're far from having perfected the math curriculum (it's impossible anyway), but I would love to see more and more "radical" changes to it, at least just to trying something different for a minute. It seems like such a long drawn-out ordeal to see any changes, I wish something like New Math would come out every couple of years just to keep us on our toes and have us thinking about alternative ways of teaching the subject.

The third stop for me was the point about the NCTM Standards being used as a model for the development of similar standards for other subjects. This kind of feeds into my egotistical opinion that math is the most important and influential subject in school. Along those lines... Nobody is out here comparing social sciences or language arts test scores between countries. It seems slightly contradictory how as individuals, most folks actually literally detest math, and yet as a collective, we all seem to look at success in math (and sciences) as the de facto barometer for the quality of education. This kind of ties into my thoughts on my first stop... In particular, how I can personally feel like a cool black sheep for actually enjoying math, while at the same time criticize "the masses" for not enjoying it. We're all a bunch of math fetishizing hypocrites is my point, but I might just be speaking for myself.