Flow

Have you experienced a state of flow through certain experiences? 

I have experienced flow when rock climbing. I enjoy climbing the different routes and working out both my brain and body. I will run into obstacles both mentally and physically as I climb. But the challenge, as well as the music in the background, keep me going until 2 hours have passed and my hands are blistering.

What prompts it? 

My rock climbing flow is very inline with the videos we watched on the idea of Flow. I will keep trying harder and harder problems, but eventually I reach a point where I can go no further and don't have either the technical skills nor the expertise to solve a problem. I lose all flow as my self-efficacy of the task at hand drops. Once I feel like the task is so challenging that no amount of effort can solve it, my engagement drops.

Likewise, if the gym is ever so busy that I can only squeeze in to do the easy routes, I also don't enter into a state of flow. The task is not challenging enough and so I do not end up engaging with it.

Is it sometimes connected with mathematical experiences?

One can definitely use math while in flow while rock climbing. Recognizing the distances between holds and using spatial awareness to project one's self on the wall are two ways my mathematical mind gets activated in flow.

On a more general note, I often find that when I am facing a challenging math problem, I often enter into flow. The only issue with my current sources of math problems is that I find them either too challenging or too easy. But on the occasions where I find a perfect problem I can lose and entire day playing with it.

Class Summary/ Blog Reflection

 Looking through my blog posts, I have unlocked memories of the earlier days of this course when we were in the garden in the early fall days. I also got to take another look at the art projects that we did over the span of the semester. So much has happened that I can barely keep track of it all, so hard to process. But as I read the blogs I get to remember the things we did.

My favourite part of the course happened early on when we reading the Skemp article. By breaking the idea of knowledge into that of instrumental vs relational understanding unlocked something in my brain. It was a new way of thinking that had never happened before. Because of having this reading early on, I was able to see the lessons and projects through the lens of instrumental vs relational understanding, and that enormously helped me through the course.

I also liked the math art projects that were related to the curriculum we would be teaching in the schools. While we learned a lot of art ideas, it will be a bit of work to find the time to implement them in our classrooms. I would have loved more lessons like the GCF/LCM dances that I will be able to implement in my practicum seamlessly.

Unit Plan & Lesson Plans - Exponents and Prime Factorization

Sorry for the quality of the class notes, the pen did not convert to Google Docs well. During the classes, students will be given the blank notes at the start of class. As a class we will work through the notes together, with the green writing being the targeted information I want to convey with students.

Unit Plan

Lesson 1 - GCF/LCM

Lesson 1 - Class Notes

Lesson 2 - Prime Factorization

Lesson 2 - Class Notes

Prime Factorization Art

Lesson 11 - Encryption in Math

Lesson 11 - Class Notes

The way textbooks may position position mathematical learners

 This text highlighted some key examples of how different textbooks represent mathematical ideas differently. The ones that stood out to me were how absolute values, the binomial coefficient, and derivatives were all somehow represented with different lenses. It is expected that the arts will have different biases, but it is not expected for bias to emerge in STEM subjects such as math. Because bias is not expected, its presence is even more insidious.

I have talked about textbooks at great lengths with my SAs. In order to teach the current grade 10 curriculum, I require 3 different textbooks. This is because the curriculum keeps changing and the textbooks quickly become out of date. Because textbooks are static by nature they are not good teaching tools for the changing environment of learning.

My final thoughts on textbooks is that they teach material a certain way, and it might not line up with the way I view that topic of math. I will be biased in my own teaching, but I think that as an educator, it is better to teach well with your own bias than to teach a choppy, inauthentic lesson with someone else's bias. Students learn best from teachers who are passionate about the material, and that passion might disappear if I was forced to teach math in an inauthentic way.

Dave Hewitt & mathematical awareness

 In the videos we watched in class, the three things that made me stop and think were:

1) Hewitt's ideas that students will naturally find the correct answers, they just needed guidance to get there faster. That if left to their own devices, mathematical thinking comes naturally, it is only that the pace needs to be quickened

2) The idea of a group choir makes a lot of sense. I especially enjoyed that for tricky answers the unison would break and then the group as a whole would understand that further thought had to be put into this question.

3) I loved that Hewitt did not tell his students what was right and what was wrong, he was only a guide. He was constantly asking if the students thought they were right. In honesty, I remember being a student and being decently upset with teachers who did this. I remember feeling cheated, why was I doing their job for them? But i recognize now the significance of framing the relationship with students in this way.

I think that the fraction problem is a great inquiry problem that doesn't have any correct answer, at least not at the start. It is obvious that Hewitt has a particular fraction in mind, but by framing the question in terms of multiple steps, students feel much more agency. Just finding a fraction in between 5/7 and 3/4, there are infinite answers. By making the problem more concise, he is controlling which fraction the students give as an answer, but it still does not feel like agency is removed from the student.

I have already incorporated group choir into my teaching because I felt like it is a good way for students that have self doubts to more easily play along; they are just one voice in a group of 30. It make me happy to see that someone so professional as Hewitt approves of this teaching style.

Hewitt Arbitrary vs Necessary

 After reading this article, issues that I ran into during my practicum were voiced. I was tasked with teaching the equations of lines, but ran into difficulties quickly. I thought that I would have been able to just share my knowledge with my students. However, I ran into needing to know a bunch of different forms of the lines. For example, I needed to teach that "General Form of a line is 0 = Ax +By +C where A B C are all integers and A is non-negative."

I marked tests and found that many students did not exactly follow these rules, and felt bad that I had to take marks away for these mistakes. I am happy that I now know that these rules were arbitrary, and this exemplifies why there was so much friction between me and this issue.

The most annoying aspect of this experience is that students were so busy learning these arbitrary rules, that they did not understand the necessary concepts of lines being a relationship between x and y. Students would remove x and y just in order to make the equation look more like General Form.

When I teach my classes, I am going to focus on necessary knowledge and prioritize it. After students understand necessary concepts, I will teach them the arbitrary rules.

Oct 20 Pro-D Day

Last Friday I attended the BCMAT conference in Surrey and it was an amazing experience.

My first session was about a teaching style called argumentative thinking. Using this routine, students are pushed to argue their thoughts in front of the class. I like this routine because it promotes the competency of explaining ideas through mathematical language.

My second session was about Group Quizzes. This idea aligned well with my ideas of how teaching should be done. We have been taught that students learn well from each other, and when group quizzes are used they can supplement each other's knowledge. The only downside is that assessment becomes complicated because you are not assessing the individual. I think that a classroom breakdown should be group work at the start, then individual work to be assessed on afterwards.

My final class was about culturally relevant material. Unfortunately, this session did not teach what I was expecting from the summary. It did talk about the idea of a warm demander, which happens to be an idea that I had already researched myself individually. 

The best thing that I took away from this conference was the interactions I had with other teachers. I did not feel like a lesser person there. I asked questions that were relevant and did not stick out. My confidence increased drastically after this conference, which is perfect for the start of my short practicum.

What is curriculum

 Before this article, my definition of curriculum was confined to the explicit curriculum. This article has expanded my view on things, and connected ideas from many of my other classes.

These new definitions of curriculum were something of which I had a comprehension, but never such an articulate understanding. We have learned about the historical and societal role that school has had on our world. Many have argued that as a social institution, schools reinforce standards that reinforce society. A society of artists would prioritize art. A society of dystopian oppression would prioritize those that sat still and kept quite. A society of of invention would prioritize innovation. These hypothetical societies are examples of the three types of curriculum, and how each can be used to shape the society's members; Explicit, Null, and Implicit.

By explicitly stating curriculum, students are forced to choose between a handful of futures. I have learned that the majority of electives offered by many rural BC communities are trades. This rural society has determined that trades are important, so trades are offered.

Null curriculum is where I think things become interesting. In the above example, by not offering a fine arts degree these communities steal that potential future from their youth. It's unfortunate that there are finite resources to provide programming. I like to imagine all the potential societies that could exist if no knowledge was off limits to students.

Implicit curriculum made me think of all the little things that need to be taught to citizens in order to have a functioning society. In this category of curriculum would be molding of their personality. We now live in a world where everything is so connected and optimized that we could not exist without punctuality. What I like most about implicit curriculum is that this is where I will have the most agency to teach students. I think that collaborative problem solving is an important skill, so I will set my classroom up to implicitly teach this.

I think that BC's shift from content to competencies also indicates their shift in priorities from explicit curriculum to implicit. The soft skills that we teach are often so much more important later in a students career, and I am excited to work under a government body that understands this.

Cambell's Can

 This was my work estimating the volume of the can as well as it's effectiveness for fighting fires. I could not find the water required to fight a fire, but was able to estimate how long the water would last in a fire fight.

I also wrote a question that I remember thinking of years ago while camping in Squamish.

Micro-Teaching: Estimating Percents

I am so content with how this lesson worked out. I feel like we improvised a lot, but did it in a very professional way. It was strange to me, that some members of my group were unhappy with how much improvisation we did. This is strange to me, for I feel that all of teaching should be more improvisation than structured lectures. My strategy has been to learn the material enough that I can just talk about it with those in front of me, and I feel that it has been paying off.

Lesson Plan

 EDCP 342 Coteaching Lesson Plan

 

Subject: Estimating Percentages

Grade: 8

Date: Oct 16, 2023

Duration:  20 minutes

 

Curricular Competencies   https://curriculum.gov.bc.ca/curriculum	Estimating reasonably Using multiple strategies to solve problems Communicate mathematical relations
Big Ideas	The principles and process underlying operations with numbers apply equally to algebraic situations and can be described and analyzed Similar shapes have proportional relationships that can be described, measured, and compared

Core Competencies

Critical/Reflective Thinking: Using logic and reasoning to evaluate thinking Communication: Knowledge, skills, processes and depositions we require to interact with others Personal awareness and Responsibility: Understanding our strengths and catering to them

Materials and Equipment Needed for this Lesson

Markers for whiteboard

  

Lesson Stages	Learning Activities	Time Allotted
Warm-up	Assess understanding and connect to their previous knowledge of percentages by giving an easy “fraction to percentage question” Ex: 5/20 -> 25% Present this warm-up as review (because we did this last class) Present hard fraction as an intro to this lesson: 27/43 -> 62.79%	4-5 mins
Middle	Evan- multiply both top and bottom by a number to get the denominator as close to 100 as possible	3-4 mins
	Jessica- add or remove some amount from the numerator until the fraction is simple and can be simplified	3-4 mins
	Jacky- Use a calculator  *Con is when stuff is simple*	1-2 mins
Recap	Discuss pros and cons of strategies and which they prefer. In what situations will some estimates be better or worse? Encourage them to make up their own strategy All of class has to tackle 27/43 -> 62.79%	6-8 minute

 

Assessment/Evaluation of Students’ Learning

Exit Slip: What makes an estimation a good estimation? How would you judge the “goodness” of an estimation method

Was Pythagorous Chinese

By framing a knowledge as your own, it creates a perceived superiority over others. During the cold war, the ability for the USSR to put a satellite into orbit did not rouse cheers from the Americans, but despair. Cultures often compete with each other over which is more advanced, and this is often measured by which has the greater academic knowledge. We are now living in a global culture, which means that there should be collaboration, not competition, between nations. If we continue the false narrative that knowledge was only developed by "our" culture, then we lose the ability to work together. In competition there is no sharing of resources between sides. If our global culture is to overcome the challenges faced in the upcoming years, we must work together and recognize that all cultures have knowledge to bring to the table.

I am torn about the naming conventions of certain theorems. On one hand, I recognize that calling it "Pythagorean Theorem" emphasizes a the competitive approach to knowledge I mentioned above. However, I do believe that if it is taught with the emphasis that it was not only discovered in western society we can acknowledge that a name is only a name. Many cultures have discovered the concept of a chair, and thus many cultures have different words/names for the concept. It would be problematic if western students were taught that only their culture were smart enough to invent chairs and that is why they call it chairs. Hence, calling it Pythagorean is not a problem as long as the concept is introduced through an international lens.

Mircro Lesson Peer Reviews

 

**UPDATE**

I think this lesson went ok, but unfortunately I did not time out the lesson nor scaffold it particularly well. I think I was too ambitious, for I did not recognize how large the zone of proximal development would be for my peers. I assumed that an idea would have been common knowledge. Because of this, I had to improvise and begin teaching a different topic. I believe I taught the concept of levers well, but was not able to make the extension to swing stage scaffolding properly.

My peers saw that my timing was off, and the reason for this is outlined above. I saw that the content clarity was off because I had to jump between many different things to close the zone of proximal development.

Micro-Lesson Plan

 Here is the lesson plan for my micro lesson on swing stage scaffolding:

Battleground Schools: Reflection

 This piece was interesting because I had never thought of Math as a politically charged subject. We hear in the news about english and history being re-written due to political influence, but math seemed to be one of the untouchable subjects. I see now that this is untrue.

I saw many similarities to the concepts of instrumental vs relational understanding. As has been mentioned in class, instrumental understanding is more quickly grasped, while relational understanding leads to better applications of one's knowledge. It is apparent that society chooses which type of understand to prioritize depending on the society's needs. 

The most difficult part of this reading is that it is apparent that there is no right answer, and there is no pleasing everyone. It is humorous that parents who wish to protect their child's future will argue about which path forward is best. Parents will argue both for and against changes to the curriculum, both with the exact same end goals.

My big takeaway is that becoming a math teacher does not make you immune from politics. It will be my duty to fight for what I see to be in my student's best interests. 

TPI Reflection

 I am very happy that I was categorized as Developing and Nurturing. I related very strongly to these categories after reading their summaries.

Developing means that I meet students where they are and work with them towards their goals. My experience focuses on those who need concepts broken down into simple terms before being able to approach the complicated ones.

Nurturing means that I recognize the emotional aspect of teaching, keeping students engaged by supporting their abilities and congratulating their efforts.

I think this both of these qualities stems from my origin as a tutor for failing students. They required me to teach with a developmental mindset, for the were only in the emerging category. They also required me to teach with a nurturing mindset, because they thought they were terrible at math and therefore had lower self confidence. 

I am unsure how this profile will change after I teach in a classroom. I think that as my class size increases, and as the skill level of my students increases, my profile will change.

My Mini-Lesson Theme

 I'm going to teach the class the fundamentals about installing swing stage scaffolding. I used to do this work.

Reflection on Assignment 1

 Our group presented on Serpinski Triangles. We focused on two aspects their design, Cellular Automoton and the use of different cells. Cellular Automoton was taught by showing the class the different rules that generate different triangles, while the concept of creating fractal shapes one level up was taught by having the class play around with physical cells that we cut out.

This project was a great experience for me. Every time that we hit a road block, it felt more like a puzzle to solve than a challenge to overcome. I had genuine fun with this assignment. I understand how much more enjoyable math is when it is in the form of visual art, contrasted with its traditional form. The best part of this project was that I did not have to practice at all. Typically when presenting on a topic, I have to rehearse. For this project I was so engrossed in the subject that I learned everything on my own time. Once we began presenting, the ideas just flowed out of my brain to the class. This happened because I chose to learn the topic in side and out, as opposed to memorizing a script. I could improvise everything because I had relational understanding, not instrumental.

Teachers cannot prepare for every possible outcome. The tiling activity did not at all go according to plan, but went better than I could have dreamed. While they were not fractals, the patterns which the class came up with had an incredible mathematic beauty. When I am a teacher, I will be very open to ideas that students propose, and will improvise around them. I want to promote inquiry in my classroom, and that is best done when students are allowed to take the reigns.

I have no current ideas for projects at this time, but because of this project I will have the eye to see them in my daily life. Both math and art exist around us all the time, and I now that I can recognize them everywhere I will bring them into my classroom.

Letters from Future Students

 

Hi Mr. Braun,

 

I just wanted to send you a quick shoutout and say thanks. While math wasn’t always easy for me, you were able to help me get through. My mom was so happy that I was able to make it through high school, even though I never continued with math in academia.

The reason I’m reaching out is because I just finished an exam for my plumbing apprenticeship. It wasn’t the math that you are probably used to, but studying for it ended up being really easy. I remember what you said about pattern recognition and was able to build upon that to ace the exam. None of my friends in the program know anything about math, but I was able to help them out.

Thanks for never giving up on me!

 

Hi Mr. Braun,

I just wanted you to know the repercussions of your words. I remember how much you I spired me to follow skills and further my math knowledge. Well guess what? I graduated and got a job in a highly math related field and it is terrible. My life is terrible. You never told me math used in the real world is so soul sucking. Sure it’s fun in the abstract ways of beauty, but what good is that when I have to pay the bills?! And guess what? My job is being automated next week because it’s all algorithm based anyways. Going into a field which emphasizes social skills would have been a better investment, but because you showed me the “beauty of math” I got bewitched and am now here.

Thanks for ruining my life.

**UPDATE**

I worry that my excitement about math may inadvertently deify the subject to some students. My enthusiasm for math is one of my strongest assets, and I feel that I will make math more attainable to many students who would have otherwise been scared of the subject. However, I worry that my excitement could bring a level of fanaticism that is not compatible with our current society. In my experience in the corporate field, I found that math used in the real world really was soul sucking. I obviously don't want to tell this to students because I want them to be engaged, but I don't want to lie to them.

Math is an art, but when art and a capitalistic society merge the art can become tainted. My experience with engineering was exactly this. I will try to guide students down proper paths that allow them to play with the art of math more earnestly. 

The dishes Problem

 My first step was to reword the problem and understand what was happening. When the chef says “every 2 used a dish of rice” it means that each guest had half a dish of rice. Another way of looking at this is that, of the total number of guests, half of them had a dish of rice, one third had a dish of broth, and one quarter had a dish of meat.

Let’s represent the total number of guests by twelve groups of one twelfth. This is because we can easily figure out halves, thirds, and quarters of twelve. Half the guests had a rice dish, that is 6 twelfths. A third had a broth dish, that is four twelfths. A quarter had a meat dish, that is three twelfths.

Rice

Rice

Rice

Rice

Rice

Rice

Broth

Broth

Broth

Broth

Meat

Meat

Meat

 

You can see that we have an extra twelfth of dishes. Because there are 65 dishes served, there is an extra twelfth. The problem becomes “What number, when increasing that number by one twelfth of itself, becomes 65.” Traditionally, algebra would be used at this point. However, a culture could guess and check until they arrived at the correct number of 60 guests.


While presenting itself as a problem of Chinese origin, the Chinese culture does not exist in the problem outside of the choices of food. However, the existence of the problem as a piece of history, going so far as to provide the original historical text, is very important. It shows that the traditional western view of mathematics is incomplete by giving a concrete example of maths existing outside that realm.

The fact that this is presented as a story does make the problem more enjoyable to solve, almost as if we are being transported back to an ancient Chinese restaurant. It is still nonsensical, one would simply count the number of tables and average table size before counting dishes. But if one can understand the problem as a fun test of our math abilities with a flair of the dramatic, it becomes much more enjoyable.