Dishes Problem Approch and Rough Thoughts

From question, we can get the information: the total number of guest is a multiple of 2,3 and 4, as no guests who share different number of same type of dishes compare with others, and obviously host cannot allow servants bring half plate of dishes to the guests. The least common multiples of 2, 3, 4 is 3×4=12. Now, we can count that: for every 12 guests, since every 2 guests had one rice, 6 dishes of rice; every 3 guests had one broth, 4 dishes of broth; every 4 guests had one meat, 3 dishes of meat; and hence, there will be 13 dishes served for every 12 guests. At last, as there are 65 dishes in total, so the total number of guests is: 65÷13×12=60.

It is an interesting question to think of without using functions. As a person who experienced entire Chinese K-12 education, I have seen many word questions like these in, while learning history, geography, and Mandarin literature. Which quite interesting to see that almost everywhere else we can see there are some math behind the content. So introducing real-life scenario problems is very nice for students to learn math. 

The imaginary of students would matter if a real-life example is far away from real life case, or not reasonable as normal people wouldn't do. So when we come up with some real-life scenario, we should think carefully would that be a good example? Is there some science background knowledge required a higher level of understanding for kids? I remember once I taught some content but spend a lot more time just to explain the physics, which unintentionally delayed my teaching. 

Letter from future students

 Hi You!

I wanted to express my appreciation for your class. I genuinely enjoyed it and wanted to let you know just how much it meant to me. Your teaching style not only made the subject interesting but also had a profound impact on my life.

I wanted to highlight how much I value the real-life philosophy you incorporated into your lessons. It's been incredibly helpful to me, especially when facing challenges and difficulties. Your insights have provided me with valuable perspectives that I can apply to various aspects of my life.

Thank you so much for your inspiring and caring class!

Best wishes,

A student who loves your class

Hello teacher,

I wanted to express my gratitude for your assistance. Though I may not particularly enjoy your math class, I truly appreciate the support you've provided beyond the classroom. Your guidance and help have been invaluable to me, and I'm thankful for your dedication. It's reassuring to know that I can count on your support even when the subject matter isn't my favorite. Thank you once again for being there for me.

Regards,

A student who dislikes math class

I'm writing these two letters to show that I wish I provide not only academical support but also emotional support. Teaching is more than just subjects; it involves guiding and nurturing students on a personal level. Teaching is about fostering both intellectual growth and emotional support.

Comments in Lockhart's Article

I wholeheartedly agree with Lockhart's perspective as presented in this dialogue. The current state of mathematics education often seems irrational, leading to frustration and anxiety among students. Instead of fostering a deep appreciation for the beauty and creativity inherent in math, modern education systems often prioritize rote memorization and technical skills. In doing so, they neglect the fundamental nature of mathematics as an art form, a discipline that should be explored and enjoyed rather than feared.

Lockhart's critique highlights the disconnect between traditional math education and the essence of mathematics itself. Which also mentioned in Skemp's article. By focusing on formulas, notation, and rigid procedures, we risk stifling students' natural curiosity and inhibiting their ability to think creatively. To cultivate a genuine understanding and love for mathematics, we must reevaluate our approach, encouraging exploration, problem-solving, and a sense of wonder. Math should be a source of inspiration and joy, not a source of dread and confusion. In reimagining math education, we can unlock its true potential as a tool for nurturing intellect and imagination.

While I appreciate Lockhart's perspective on the artistic and creative aspects of mathematics, I do have some concerns. Unlike music or poetry, mathematics often requires a structured foundation to effectively communicate complex ideas. Simply showcasing the beauty of mathematics does not effectively develop students' logical and abstract thinking abilities. Symbols and regulations in mathematics serve as a universal language, allowing us to express our intentions and convey intricate concepts concisely. While it's essential to emphasize the artistic and exploratory aspects of math, we cannot overlook the necessity of introducing these foundational elements early in education. Striking a balance between nurturing mathematical creativity and imparting the essential tools for effective problem-solving remains crucial for a comprehensive and functional mathematics education.

Math teachers in my memory

My favorite math teacher is my primary school math teacher. He made learning math fun and easy by using shapes to teach us about functions. Instead of feeling scared about these tricky math ideas, I felt excited to learn. He also always praised my progress, which made me even more confident. His way of teaching showed me that math doesn't have to be hard or confusing. Thanks to him, I not only got better at math, but I also started to really like it.

During my childhood, I was fortunate enough not to encounter a math teacher I didn't like. However, it was only later, when I worked as a math tutor, that I came across a teacher I couldn't agree with. This high school math instructor consistently taught incorrect mathematical theories and insisted on marking all of his students based on these flawed principles. It was disheartening to witness, as it not only hindered the students' understanding of math but also undermined their confidence. In my opinion, such an approach to teaching math is detrimental and can have long-lasting negative effects on a student's academic journey.

Locker Problem Rough Thoughts~

 It is obviously to see from the first will stay close until the last, as nobody can touch it after the first person closed it. And we can also easily to see the second locker will remain open after the second person opened it again and no one else could touch it. And the third locker would be open as well because the first and the third person are the only people are allowed to touch it.

If I label each locker by number and connect the locker to the number, then my thought turning to: "what matters the open and close to the each specific locker?" And my answer is: how many divisor of each specific number from 1-1000. My first quick thought leads me that: obviously, every prime number locker should remain open as only the first and the corresponding number student are the only two person who turned the locker close and open. So how many people can touch the locker is the key point to figure out each locker's state at the end.

Then, my brain suddenly related to number theory, or relative field, that there is a specific formula to calculate the number of divisor for each number, as long as we know the idea of primes, prime factorization, etc. And I also realized that, if I can figure out how many divisors for each number, then even number of divisors would lead to open and odd number of divisors would lead to a close, as a locker has only open and close state, if one person open it, the next person will close it, no matter what happens as long as under this game's rule.

So my goal turns out to: figure out the number of divisor of each number for 1-1000, and even number of divisors means an open state at the end of that number related locker, where odd number of divisors means a close state at the end of that number related locker. 

Now I am trying to figure out, what number's factorization would lead to a even number, and what would lead to odd.

By looking at number of divisor's formula, and I realized that only perfect squares would leads to odd number of divisors, and the rest will be even number of divisors, as all perfect squares have all even exponents in their prime factorization, and then by the formula, which is products of their exponents+1, they will be all odds multiplying odds, which leads odds products. And non-perfect squares will have odd exponents which gives even multiplier while calculating the number of divisors. Therefore, only perfect square between 1-1000, which are 1, 4, 9, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961 will be closed, and the rest will be open

Skemp reading short notes:

 The three interesting points that made me stop while reading are:

1. I appreciate how Skemp presents a vocabulary list that illustrates similar-looking words with different meanings in different languages. It reminds me of a previous experience when I watched a friend defend his PhD in mathematics. During the defense, an external professor asked about a notation in my friend's thesis and pointed out that the notation was not well-defined because it had a different meaning in statistics. This experience taught me the importance of ensuring that everyone is on the same page when discussing topics, or at the very least, that we approach them with a shared understanding during discussions; otherwise, miscommunication can easily occur.

2. An IQ 140 genius student struggled with math in his youth. I recall that during my high school years, my math grades were not impressive because I couldn't concentrate while sitting in a small classroom, just listening to the teacher. Additionally, my brain wasn't mature enough to grasp many abstract concepts until I entered my final year before starting university life.

3. The goal for students learning math may be merely instrumental, but teachers aim to help them understand math relationally. I often encounter questions like, 'Why am I sitting here and learning math? Will I ever use this stuff?' I always emphasize to my students that math is pervasive, happening everywhere, all the time. Math can take on various forms, and we unknowingly use it extensively in our daily lives.

Skemp also raised the question of whether understanding things relationally or instrumentally makes a difference and if one approach is truly superior to the other. I concur with Skemp's perspective that a relational understanding is superior and indeed crucial. I firmly believe that comprehending things relationally is essential for gaining insights into how our world operates. The instrumental approach to learning may lead to easier forgetfulness and challenges when confronted with changing circumstances, as it often lacks a deep understanding of the fundamental logic.

 Math is fun~