Journal Entry + Slides for Assignment #1

In working on Saara Lehto's incorporation of fractals in her art piece, I was given the opportunity to learn more about fractals and West Coast Indigenous art. On a basic level, we knew that the proportions aspect of fractals could be feasibly incorporated into the classroom, but I was surprised at the additional learning opportunities fractal art brought to the table. While working on the art recreation, I noticed that using artwork to demonstrate proportions was harder than it looked. Even after outlining how I was going to incorporate mini seahorses as part of a larger seahorse, I didn’t account for my pen width being a hindrance to the process. I also grew to appreciate that the requirement was “self-similar” and not “exact” because it made the project less stressful. Thinking about students, that would be a relief for them as well and it would allow them to be more creative. This project would make assessment more meaningful as well because no two students would have the same submission. Additional surprises came through working on the West Coast Indigenous artwork. Although on the surface it looked very easy, especially because we made use of templates as First Nations artists often did/do when using repeated shapes, it really wasn’t. The template had to be lined up and angled correctly to achieve symmetry. In hindsight, using a grid would have helped immensely, as well as created an additional opportunity to make use of math. A major flaw in our First Nations artwork was that we didn’t obey the rules of Indigenous art. I know that in the Northwest Coast, there are strict rules for colouring as well as proportions of spacing; however, I was unable to reach an artist who could provide that information. Not only would having those rules have made our artwork accurate in the artistic sense, it would have also been an asset in a classroom to have authentic proportion rules in a real-life setting so that students could see the value of learning about them. Following research or bringing in a guest artist, this could aid in Indigenizing our classrooms, while also adding an applicable layer to a lesson on fractals and proportions. Lastly, there was another opportunity to add different math concepts to the lesson through the use of Euclidean proofs. While creating the equilateral triangle shape in our Iterated Function System Fractal, we made use of the construction of Proposition 1 from Euclid’s Elements, Book I. 

 Edit: Here are our slides from the presentation:  https://docs.google.com/presentation/d/1infhLZ4Ow7wofXp-yRHw7PSjB_5yOkyrOllgYZgh5pY/edit?usp=sharing

Entrance Slip #3: Mathematical understanding and multiple representations

Image source: https://www.guangyangpri.moe.edu.sg/wp-content/uploads/departments/math/math01_diagram.jpg

An interesting note from the article that I hadn't thought about fully before was how a small child first learns the names of the numbers but doesn't necessarily understand that they are counting up to a total. It made me think about my friends who introduced their son to the symbols of multiplication and division along with the names before he was even two years old. They said their intention was to get their son familiar with the symbols, even if he didn't understand their significance.

Although I agreed with the authors encouragement of teaching both analytic and visual representations, I'm not sure that the idealistic case of a student using both always works. For example, in my case, I felt more comfortable with analytic methods when I was growing up. Even if we were taught geometric or other representations, I tended to gravitate towards the method I liked best. I have seen the same in some of the students I have tutored or helped - they tended to choose a method (analytic, geometric, or other) and use that whenever they could. That being said, the author does suggest an exploration aspect be incorporated in the classroom and maybe that was what was missing in my education growing up and in the classrooms I have worked in. 

The main types of representations listed were analytic and visual, and the connection of internal and external representations. There were also three stages mentioned which were enactive, iconic, and symbolic. Enactive (manipulating concrete materials) and iconic (pictures and graphs) fall under the visual representation category while symbolic (numerals) is more analytic. The article also mentioned combining analytic and visual, and I believe real-world examples fall into this category; specifically, translating a real-world problem into a math problem, solving it, and communicating the results. I think the idea of internal and external representations was interesting. I particularly liked the breakdown explanation of how we use numerical or language equivalents for visualizing a number set, and how it is easier for us to take in visuals that are organized in a certain way. 

The types of representations that I noticed were excluded were oral motion, and musical. Sometimes students learn more when they are explaining concepts to others or voicing their thoughts aloud. Those who are more inclined to move might benefit from activities such as using their strides as measurements in solving problems (i.e. the units become "5 strides of Jim" instead of metres). In terms of musical representation, the first example that came to mind for me was the clapping exercise that I had to complete during piano exams. Both listening to and having to parrot a pattern of beats was a form of representing fractions. I also had music directly used as a memorization strategy when I was learning multiplication. By either following our teacher's tune as she played the piano or by using bells and drums, the whole class sang along to the multiplication "songs." I can still recall the beat that goes along with the words "6 x 6 is 36, 6 x 7 is 42, 6 x 8 is 48." To implement music in my classroom, I might devise a similar strategy to help memorize important facts or formulas by creating a catchy tune to play once a day until it stayed with them (because, knowing teenagers, the only ones who would be singing would be those who were doing so ironically). Alternatively, it may be a good project idea if a student wanted to create their own song full of useful math.

My Teachers and My Future Students

Image source: https://i.pinimg.com/originals/6b/4d/6b/6b4d6b9f4921246afe813f8c4f309a1d.jpg

1. The difficulty in this exercise is that most teachers I have had over the course of my math journey have either been out of this world, or just not my style. Therefore, my least favourite teacher wasn't necessarily an ineffective teacher, but I didn't understand his methods. 

My least favourite teacher, let's call him Professor Brown, taught by writing notes and having us copy them. However, the notes were not structured in a way that made sense to me. Also, he tended to jump between pages without allowing us the chance to copy the notes, and creating confusion about what we were talking about. We as students were part of the problem because people seldom asked questions in class except to have the page flipped back, and sometimes we were told to copy off of our neighbours instead or to come see him after class. I often went to office hours because I was having trouble with class content but Professor Brown's explanations didn't make any sense to me. Over top of this, one time I asked a question and someone else at office hours tried to explain their method to me, which made sense to me. However, Professor Brown was angry with this student because he said it was his office hours and that the question was posed to him so he would answer it. This lack of flexibility made the professor intimidating, and was detrimental to my learning, causing me to seek outside help for the course. I did alright on tests but the scores almost always scaled up so my mark never reflected my comprehension.

My favourite teacher, let's call him Mr. Ezylryb, had a mixture of structure and exploration in his classroom. Mr. Ezylryb started his classes with an explanation of the topic. During the explanation, he would break to give us time to work on select problems. He would then go one by one to each student and have us answer those questions. If a student didn't know the answer, they could ask for help from the class or the teacher. After the lesson was over, we would have worksheets or an activity to do. Worksheets were generally straightforward and reflected what we learned in class. Activities, however, allowed us to play with concepts. For example, when learning trigonometry, we went down to the wood shop and used compasses and rulers to create designs. We made note of the different properties of each angle and realized certain combinations of angles were impossible to create triangles with. Another activity I remember was a tessellations project where we created a base design, cut it out, and used it as a tracer to create a repeating pattern (keeping the spaces even was especially difficult!). Aside from an engaging class, Mr. Ezylryb was always willing to help out. He helped anyone who came to him with questions during and after class, during breaks, or even after school. He also allowed retests as long as we completed a worksheet that demonstrated we had improved. 

Image source: https://blog.myheritage.com/wp-content/uploads/old-letters-e1552463906281-875x472.jpg

2. A letter from a student who wasn't too happy with my class:

Dear Ms. Sharma,

I hope everything is well with you and your family. I'm writing to you to express my thoughts based on my experience in your class. I'm sorry to say I didn't enjoy your class. I found your explanations confusing, and the notes didn't help much either because they weren't very organized. Particularly, it seemed that we were given several methods for one topic, but the methods weren't really explained well aside from the one that you were most comfortable with and used the most going forward. I also didn't feel comfortable asking you questions because your explanations didn't make sense to me so I just nodded along because I didn't want to look stupid. I don't mean to be rude - I just wanted to let you know my experience in your class in case there are others like me in the future. 

Sincerely,

Olivia

A letter from a student who liked my class:

Dear Ms. Sharma,

I hope everything is well with you and your family. I just wanted to let you know that I loved your class. I liked that we didn't just take notes for the whole class and were allowed to experiment with the content a little more. Your tests were really great too because they stayed on topic and even the challenge problems were doable with what we covered in class. I'd also like to thank you for always being willing to help. It was reassuring to know we could come to you if we were confused. 

Sincerely,

Tiên

Exit Slip #1: Response to class comments

Image source: http://www.weteachwelearn.org/wp-content/uploads/2016/05/Discussion.jpg

 From our discussion today, I really resonated with the comment made by Group 1 (Tyler's response). In their response to question 1, they talked about how they believe relational math makes for a strong foundation, even though it's commonly believed that instrumental creates a stronger foundation. I agree with this notion because I think having the big idea should be the focus and, as they said, "an instrumental understanding will usually follow." This feeds into a powerful analogy one of my group members made which was that a person can be a good writer even if they don't know the proper syntax or spelling of a language. Common formulas are useful to save time or create a cleaner solution in the same way grammar rules and spelling create for a more concise and easier read. However, in my opinion, the approach and process of solving a math problem and the overall story in writing are the real goals.

Entrance Slip #2: The Locker Problem

 When I first began this problem, I tried to parse the question. I needed to visualize what was going on so I draw out the lockers as squares with open squares representing open lockers and closed squares representing closed lockers. I drew out the status of the lockers after the first 10 students had gone through and made their changes. From there, I tried to find a pattern. At first, I thought it might have something to do with the difference of closed lockers at the end of each student's turn but moved away from that thought when I realized that lockers 501-1000 could only have one locker closed at a time (since the 501st student would only alter the status of 501, the 502nd would only alter the status of the 502nd locker etc.). However, I still couldn't tell what the status of the locker would be at these stages so I decided to take a closer look at what was happening for the first 10 cases:

I realized that all primes were open, and from there I realized that the status of the door (either open or closed) corresponded with how many factors the number had. For example, 10 was open because it has an even number of factors (1, 2, 5, 10) while 4 was closed because it had an odd number of factors (1, 2, 4). I wasn't able to see the full pattern right away so I used the property (# of factors of m) = (n_0 + 1)*...*(n_f + 1) for p^(n_0)*...*p^(n_f) = m, where m equals the number (also, if anyone knows how get sub/superscript on blogs, please let me know). Through this process, I realized all square numbered lockers are closed and the rest are open after all 1000 students go through and make their alterations (the specific square numbers are listed in the picture below):

Entrance Slip #1: Richard Skemp's Article

Image source: http://www.davidtall.com/skemp/

In reading Richard R. Skemp’s article “Relational Understanding and Instrumental Understanding,” I’m not sure I am clear on the changed view on viewing relational and instrumental math as two separate subjects. From what I understand, using his music analogy, instrumental is like music theory, and relational is like playing music - they are connected but both have their individual value. If that is the case, I don’t agree that both are separate. In my view, closer to Skemp’s original view, I believe relational mathematics is a deeper and fuller understanding of math, while instrumental is more surface level. However, I also believe instrumental math is the first step into the world of mathematics for many people and should not be discounted as useful. I agree with Skemp’s observations on pages 11 and 12 about why instrumental math is often the focus in classrooms. In addition to these, I think gaps in learning are another thing that prevent many teachers from diving into a deeper understanding of mathematical concepts with their students. For example, if a high school teacher is trying to derive a formula but there are many students in their classroom who struggle with basic multiplication, division etc., it may confuse the students more than it aids them. Having a set of rules to memorize would make solving problems in class accessible, which feeds back into Skemp’s comment on page 8 that students can build their self-confidence.


Another thing that stood out to me in this article was the map analogy. I thought it was very clever! To take it a step further, knowing a town has the same layers as instrumental and relational math do in my point of view. Instrumental math would be the same as walking the streets, knowing where the essential stores are, and it’s enough to get by. However, relational math would be getting to know the people, history, economy, culture, and community that lives in that town on top of knowing the streets. While all these extra things are not essential to get by in town, they enrich the experience and provide a reason to have interest in or love the place. In my teaching, I definitely want to derive concepts so students have some history or culture to connect with. Understanding that there are road blocks, such as the ones Skemp has mentioned in his article, preventing full relational learning in the ideal way, I would strive to pick and choose the ones I felt were most useful to the students. In other words, regardless of if relational or instrumental math are two separate things or not, I think a classroom can only be taught either fully instrumental or with a mixture of instrumental and relational. I believe a mixture of instrumental and relational is a better approach because chances that more students may resonate with math are increased when different perspectives are introduced. In the analogy, this would be that it may be impossible to learn all facets of people, history, economy, culture, and community, but learning any combination of those could potentially strike a chord with students who believe learning maps for a town they don’t care about is useless.

Hello!

Looking forward to entering the world of EDCP 342.