The dishes problem

 If you don't use algebra, my method is the "try" method. Firstly, according to the question, we know that the number of guests must be able to divide 2 bowls of rice, 3 bowls of soup, and 4 bowls of rice. So the number of guests must be divisible by 2, 3, and 4. Then I started to list all the numbers that can be divided by these three numbers and worked out backward whether this number guarantees that all the dishes are 65. I started with 12 guests and found that there were only 13 dishes, which is still too small. So jumped straight to 48 guests and found it was 52 dishes, which I realised was close to 65. Most likely it's the next common multiple, so that's 60 guests, and then I found it was exactly 65 dishes.

I think it's very impactful, maths examples from different cultures give students something new and fresh, not only can they learn about the development of maths in other cultures it can also give them new logic to think about and feel the different modes of thinking and asking questions, it will make them want to explore more.

I think stories and imagery are important to enhance the enjoyment of problem-solving. For example, in this puzzle, I would imagine that people sitting in a restaurant eating would be more interested in knowing how many people are eating. Students may draw pictures to support their thinking, so the process of drawing pictures is also a way of exercising their ability to 'combine shapes' in maths, which is a way of exercising mathematical logic.

HW reading: Lockharts's Lament

 As a maths major, I feel a little bit aggrieved whenever some of my classmates are surprised to hear that I am studying maths, thinking that maths is purely logical proofs, hard and cold formulas, and boring lists of numbers. Often the rigor of logic, the golden ratio of geometry, as Lockhart said, the creation of mathematics is an art that requires experience, trial and error, and luck. In this way, I am reminded of a class I took at the end of college called knot theory, where the shape of a piece of thread tied into different knots, and the shape of lye-knotted bread, all have mathematical significance in them. I couldn't help but be shocked when I learned the deeper logical principles. At the same time, I agree with him when he says, "Maths is the art of explanation." It's in the proofs, it's in the explanations, such as, "Why is this line drawn here?", "Why do we have an unknown number here? ” The inner beauty of mathematics is something that needs to be thought about and felt, and if it were just told, it would be boring and rigid.

But the point I don't really agree with is that I think Lockhart is a bit too idealistic, and I admit that the way maths is taught in schools these days is too formulaic. But it's not very realistic for teachers to change maths lessons completely. After all, classroom time is very short, and maths at younger ages is more instrumental. It is not easy for many students to learn the tools in the classroom, let alone understand the "art" of maths. 

I think Lockhart's idea of the "beauty" of maths and Skemp's idea of relational understanding have something in common, in that they both express that maths is a subject that needs to be explored and explored more deeply.

Letters from future student

From a student who loved my lessons

Hi Ms.Yi, thank you for your teaching. I really like your style of teaching, you always introduce a small game related to the content before the start of the lesson, which instantly makes me feel engaged and makes Maths, which I think is a boring subject, very interesting. Your explanations of the lessons are also very clear and logical and you explain the topics in detail. What I liked most was that you accommodated a lot of my stupid questions, which made me feel comfortable enough to ask you questions whenever I wanted to.

From a student who didn't love my lessons

Hi Ms.Yi, I still want to write this letter after so many years to tell you that the experience of taking your Maths class at that time was bad for me. I felt that your style of lecturing was too gentle. You spent a lot of the time talking all about the knowledge points so I got tired very easily. You were too willing to mingle with the students so when some of them had problems you pointed them out to them and they didn't trust you very much, and this kind of atmosphere was brought to me as well.

My reason for writing two such emails is that I hope that in the future my maths classes will be lively and fun so that students can be engaged through mini-games. I also hope that I am a teacher who can always make students feel that I am willing to answer their questions. My concern about becoming a teacher is that in my practicum, I was very gentle with the students and they liked me enough to come and talk to me, and my SA told me that I was more like a big sister to them than a teacher. I am worried about this, I would love for my students to trust me and share their lives and troubles with me, but how to set a good boundary so that students like you and be authoritative at the same time.

Exit Slip （Sep. 13 class)

 What gave me a deep impression is Michael's group's point that they used a mathematical way (drawing a function image)  to represent the fact that a very small number of students in the class would pursue the goal of relational understanding, a very small number of students would purse instrumental understanding, and the majority of students would be the ones who probably could have built on their instrumental understanding and wanted to try to pursue relational understanding. I am skeptical of this view, although I think that the majority of the class will probably be composed in this way, the teacher needs to consider how to design the teaching methodology by putting themselves in the shoes of the students in the class and the distribution of their abilities.

Sep 13 Our group's blackboard

 

The locker problem（updated version)

 When I see questions that are usually very integrated and large numbers like 100, 1000, I first think that this is going to be a pattern question. I usually start experimenting with smaller examples. So I chose 10 lockers and 10 students, which are also integrated and relatively small, and started to draw a picture and observe it according to the requirements of the question.

From the result. it shows number 1,4,9 locker is closed whereas the others are open. We find that 1, 4, and 9 are all squares, so we can hazard a guess that for 1000 lockers and 1000 students, all lockers up to 1000 squares will be closed and all others will be open.

The following is going to explore why it would be perfectly square numbers that are closed.

Locker #1: only the first student will close it, so it will eventually close

Locker #2: It is rattled by students #1 and #2, so it will eventually open.

Locker 3: Toggled by students #1 and #3, so it will eventually open.

Locker 4: Toggled by students #1, #2, and #4, so it will eventually close.

The factors that cause students to switch lockers are a factor of one number. There are an odd number of factors that are not perfect squares. For example, the factors for 6 are 1, 2, 3, and 6. However, the total number of factors is even (4 factors), which means that locker #6 will be switched an even number of times and will end up in the open state.

In general, only perfect square lockers will have an odd number of factors. For example, locker #9 has factors 1 3 9. It will be toggled three times and will end up close.

Thus, after all 1000 students have had their turn, the perfectly square lockers (1,4,9,16,...) will be closed and the rest will be open.

Favourite and least favourite math teacher

My favorite maths teacher was my professor in my ubc maths class Math340. Math340 was about mathematical game theory. The teaching was about discovering mathematical explanations in games. Every class the professor would start with a game related to what we would learn and choose some students to come up and play the game. After the game is over the class content begins. Even though it was an afternoon class, I felt a little tired and sleepy at first, but I always felt energised by the little games before class, as well as having a great desire to learn more. What I have learnt from this teacher is how to capture the attention of students and increase their curiosity. In my future classes, I think I can also prepare some relevant games to engage my students before I start.

My least favorite teacher was a small classroom teacher in high school. This teacher's lectures were fast-paced and spent the first half of the lecture on knowledge and the second half of the lecture giving the students practice questions to practice themselves. When the students ask for help with something they don't know in the exercises, the teacher loudly judges the students for not paying attention in the previous class. What I have learnt from this experience is that in my future classes, I will welcome every question from the students and not blame them. I need to be careful and slow down during the lecture so the students can follow better.

Response to Richard Skemp

My first 'stop' comes from the author's desire to use the French-English confusion 'history' 'histoire', introduced in the previous question, as an example. The author's example of "We expect history to be true, but not a story." came up and made me start to think long and hard about the way I used to think about historical events, perhaps more often than not I defaulted to the idea that I was listening to/reading a 'story'. Like I just saw the film Oppenheimer two days ago, and after watching the film I felt more like I was watching the 'story' of Oppenheimer's life, but at the same time it was also a history. History is usually objective and true because it is based on facts and evidence. Stories, on the other hand, may contain fictional elements to convey a message for entertainment, education, or other purposes. Films have more of a "storytelling" feel to them, which is why I'm more likely to be moved by them. With this quote, I might want to re-read some of the biographies of Oppenheimer, etc. The second stop is “Well is the enemy of better" I read this with deep agreement. When we are used to thinking in a way that we are absolutely "good enough", it is sometimes hard for adults, let alone children, to accept that you can do better, even though someone who has your best interests at heart is telling you that you can do better. So I think it's important to teach children more about that, that it's good enough, but it could be better.
The third pause was in "But what constitutes mathematics is not the subject matter, but a particular kind of knowledge about it. ” This also prompted me to think about Relational Understanding and Instrumental Understanding. Teaching Mathematics is mainly about teaching students knowledge, and some students pursue only instrumental understanding, while others pursue relational understanding, but either way of understanding is a kind of knowledge. Although I do agree that relational understanding for maths can make it easier to solve problems by understanding the deeper logic of mathematical thinking. But at least both can teach students that knowledge is something we teachers can feel good about.

Hello World

Hello! This is my first post.