EDCP 342 Reflection

   Most of my math education has been at the high school and postsecondary level, and the primary teaching styles of these institutions have been the more traditional lecture style. As a result, when I started this program, I held a similar view on how math education should be approached, as that is what I grew up with and somewhat succeeded. However, this class has introduced me to plenty of reading that gave me a new perspective on what math education is, the kinds of math education that exist, and a variety of math teaching styles to consider. Of these readings, the Richard Skemp reading on "Instrumental and Relational Mathematics," Lockheart's Lament, and Battleground School all had profound effects on me with regards to thinking about my teaching styles, issues in math education and how math education ended up the way it is. 

   Of these readings, the very first reading we had to read, Richard Skepm on Instrumental and Relational Mathematics, was the one which had the most significant impact. This reading had me stop and think about what kind of math teacher I wanted to be, and I took an intense look at what I wanted to be as a mathematics teacher. The idea of relational and instrumental mathematics has plagued my mind as I try to better grasp the different teachings that I have experienced and had me review when I started feeling reasonably confident in my math abilities and when I was not. I linked many of the other readings with relation to relation and instrumental mathematics and pushed me to explore different teaching styles to allow for relational mathematics. This led me to explore the thinking classroom for my inquiry question, as I wanted to learn more about its impact on students. At this point, I am still thinking about what it means to be a mathematics teacher, but my views are drastically different from when the semester started.

However for this class I would love if some of the math art conent and garden content can fit the dynamic of highschools and the curricular conent for secondary schools. (A lot of the artwork for math art was a bit too advanced or required alot of aditional context to teach the students before they understand the beauty of it.)

Unit Plan - Final

PDF version:

 https://drive.google.com/file/d/1_ZVSasXwYTfr9x1T1-YF9LwoQ8kXwXfP/view?usp=sharing

Google Drive version: (Note - It was done on Microsoft Word so there are formatting issues)

https://docs.google.com/document/d/1IhvcuolnIjz2BZeSwQstJkLnu8o_lCaR/edit?usp=sharing&ouid=100646301109617629885&rtpof=true&sd=true

Unit Plan Draft

https://docs.google.com/document/d/1ImEQrbZTIEl7e2J0Oqy-ASjR5Cnz0wO3bdnaniXMsPk/edit?usp=sharing

Textbooks

    This article from Herbel-Eisenmann and Wagner covered quite an interesting topic on how textbook language and accompanying images could influence students, teachers and the world at large. The linguistic choices used in math textbooks tend to be more distanced from the students, making them feel disconnected. Words like "might" make questions too hypothetical, thus not applicable or realistic. Using the proposed framework, we question the phrasing of textbook wording to see if the content is relevant to the students. As a student back then, I rarely related to the textbook and instead used the textbook as a drill book to find practice problems or homework. I have never taken my time to read the examples or think too deeply about the questions as I find none of the content relatable or relevant to me. On the other hand, as a teacher, I see the textbook as no more than additional references for students and me to refer to on a particular topic. I would not bring attention to many textbook examples, such as the femur question, and instead would try to frame the questions more relevant to the students.

    Math teachers have three camps of supporters of textbooks: those who teach to the textbook, those who teach with the textbook and those who forgo the textbook. My school advisor does not use the textbook; they make their own notes package with examples using relevant questions applicable to modern times. The current textbooks in the schools are nearly a decade old, so many of the questions/problems need to be updated. My school advisor also dislikes how the textbooks emphasize small details, like it is a big issue, while specific topics are glanced over even though they are essential to discuss. (Also, textbooks tend to lie to simplify the lesson) Those teaching the textbook might follow it word for word or make note packages directly adapting from the textbook using similar or exact examples. Textbooks are more like an additional source of practice and should give further context with different wording on the topics covered in class if students need to look into it. I would create note packages with relevant examples so students feel more engaged in solving the problem themselves. If the students need additional help, the textbook is there to see another perspective on the topic while providing more practice problems if they want it.

Flow

Flow is an exciting concept. I have felt it before, but I cannot say with absolute certainty as the idea is so abstract yet familiar that I am unsure. I am familiar with flow because when I am doing maths that challenges my abilities enough, I am very focused and enjoy doing it. On the other hand, I have been bored with doing repetitive math or frustrated with problems that I find too challenging.

Similarly, I have also felt flow when I am doing team sports. When my team and I were up against an equally skilled team, we had a lot of fun as there was a back-and-forth between us and the opposing team. Like with my math example, when we are more skilled than the opposing team, we do not have much interest in playing, or when the opposing team is much more capable than us, we give up as it is clear there is a skill gap and winning is impossible.

Bringing out the flow in a math class will be tricky as the student's skill level would vary greatly. The big difference in math abilities makes it difficult for a math teacher to ensure everyone is in flow. One way is to offer a variety of difficulty to the entire class that will hit everyone's sweet spot so everyone can do it. Yet, it is also important not to give everyone the most difficulty as the students who struggle with the fundamental problems would feel discouraged if they can't solve the harder ones, but their peers can. In the end, it also depends on the grade of the class. In Math 8, our goal will be to build math confidence rather than ensure proper knowledge of all the fundamentals. Whereas precalculus 11 will likely be more of the latter with some of the former. As a teacher, we need to know our students, identify the students who are struggling and the exceptional ones, and work around them. When we do, we should capture the flow of most students and give them the best math experience possible.

Dave Hewitt's classroom teachings

Student-driven learning has always been something that I have struggled with understanding as I have never experienced it firsthand. Throughout most of my education, it has always been the teacher with the teacher as the driving force of my classes. Of the few chances that it was "student-driven," it was self-learning, which is different from how Dave Hewitt taught his class. Rather than giving the students a prompt to learn independently, Hewitt approached it by guiding them to discover something themselves. Throughout the lesson, Hewitt never told the students any facts; instead, he asked if they recognized any patterns. During the algebra lesson, it was clear that Hewitt was trying to elicit a specific answer from the students on how they solved the number. However, the students were clearly struggling, so he intervened with more guidance to get them to realize the pattern. Once the pattern was discovered, Hewitt did a few more examples with the students and got them to repeat the steps a couple more times. Then he went with a question that was more difficult than usual and required the need to write on the chalkboard. Hewitt seamlessly transitioned from a discussion to some chalkboard work.

While reading "Arbitrary and Necessary," I struggled with understanding how to make my teachings fit under necessary content and not give students the answers. However, this video on Hewitt teaching has brought to my attention how to guide students to develop their understanding of the content. Some of the techniques that Hewitt demonstrated while he encouraged students to understand algebra would be something that I would like to try and adapt during my practicum. While my experience teaching students math is a novice skill, these exciting techniques, which encourage authentic student-driven learning, are things I would love to explore and adapt into my teaching arsenal.

_______________ Nov 14 edit after seeing the post ____________________

Stop 1) How Hewitt leads the class to be "student-driven," where he tries to guide their exploration by providing hints and activities.

Stop 2) When students approach a roadblock or a hump that stops their exploration, Hewitt provides hints at varying levels of support to guide the students.

Stop 3) Hewitt transformed their exploration into something tangible on the chalkboard.

Stop 4) The paintbrush and the pencil introduction as I had to think, wait a moment, what am I supposed to say?

I believe that Hewitt created the fraction problems to encourage students to explore equivalent fractions in a way that does not require too much interaction with students, as this was during the peak of COVID lockdowns. The exploration had to be self-sufficient; thus, the student's chance of getting stuck must be minimal as the teacher could not help guide exploration. While these teacher-created math problem does not solve any particular problem, it gives the students a chance to be introduced to the idea without doing drill from a textbook. 

Hewitt likes to encourage students to explore their understanding of a topic, similar to how Peter Liljedahl wants students to think about it rather than copy the steps provided by the teacher. Because of all the self-sufficient learning students will do, students, in theory, can better understand how they learn and the concepts they just provided. So far, I am interested in how to encourage student-led learning while I am there to help students guide their knowledge to a correct understanding.

Arbitrary and Necessary

I have never heard about "Arbitrary and Necessary" in mathematics education. Digging into my memories, most of my math education was me listening to a teacher's lecture and receiving their wisdom from the teacher. The only times I had a more profound understanding was when I understood the reasoning behind a particular principle rather than memorizing it. While many facts in math are not arbitrary, you can find most of the arbitrary curricular content in elementary-level mathematics. This curricular content includes multiplication, fractions, division, addition, subtractions and many more. At that level, many students are introduced to these mathematical conventions for the first time and, most of the time, are asked to memorize them. In contrast, secondary math consists of much curricular content that falls under the "necessary." However, this requires students to have a good foundation of arbitrary content introduced in earlier years.

When teaching students at the secondary level, it is essential to realize that many students will come from elementary and previous years at varying levels of math abilities. To cater to a broad range of students, asking students to develop their understanding of everything would be manageable. As a result, some knowledge must be given to students as received wisdom as a guiding hand in hopes that they can better understand and develop their understanding. As shown in Figure 5 of the reading, arbitrary content generally consists of "words, symbols, notations and conventions," while necessary content consists of "properties and relationships." When doing lesson plans, content detailed as arbitrary will likely be given to students as received wisdom, along with a sprinkle of properties and relationships to guide students.

 An excellent way that I will try to implement is to start a unit by presenting the students with a problem that they can solve at the end of the unit. Asking the students to attempt it using their given knowledge first and then start the lessons to build the skill necessary will allow students to develop their understanding. Thus, by using these lessons, I hope that it will build up supplementary skills and understanding to solve the main unit question.