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.

The Giant Soup Can of Hornby Island

Some of the terminology used here is how my SA taught the Math 8 class as a way to be consistent with the terminology. I used what I was comfortable with during my practicum here in this question.

As a question, there were a couple of things that I needed to research, the average dimensions of a bike, as it was the primary thing that I was given to figure out a scale. Another thing that I needed to figure out was the amount of water required to put out a house fire.

After calculating the dimensions of the bike and converting that to real life using a scale of 1cm to 26.25 cm, the volume of the tank came out to be about 54.5 L. Since we only see 80% of the can, the volume is about 43.6 L. Doing my research, I discovered that an average housefire requires 1291 L. Thus, our tank does not have enough water for a house fire. However, we assume the tank is only what we here in the photo. There might be more underground.

After my short practicum and experiencing many things, I see this question as a way for students to help with their math communications. Rather than the traditional approach of asking the students to solve a question, here we will ask students to "formally" prove to me the volume of the water tank. All they get is the average size of a bike and the image to use as a scale. We will ask students to organize their work in a much more excellent way than I have done. And justify how they concluded, such as the scale and how they got their volume.

A way we can extend this problem is: Given that the recommended size for a fire department's water tank should be about 10,000 L, can you draw what the rest would look like? Provide its dimensions.

CUEBC Pro-D post-conference

For my provincial professional development day, I attended the Computer Using Educator of British Columbia conference at Fleetwood Park Secondary. 

The conference was exciting, and we had an emotional keynote. The keynote speaker is Carol Todd, a teacher at School District 43 and the mother of Amanda Todd. For those unaware, Amanda Todd was a teenage girl who committed suicide in 2012 due to online harassment and blackmail. Many might have seen the YouTube video Amanda posted right before her death, and it was a powerful video of her telling her side of the situation despite all the harassment everyone was directed to her. The keynote was very powerful, and I have nothing to say about it. It brought about the importance of digital tattoos and how once something is on the internet, it will forever be on there.

The first workshop I attended was on building a successful esports program in your school district. The presenter details the benefits he has seen in implementing such programs and how he did it successfully. It gave me a lot of things to think about, as I love playing video games, and the idea of esports slowly becoming more legitimized in Canada is cool.

The second workshop was on robotics for all ages. I learned much about the different robotics products on their market and their pros and cons here. While the presenter had some biases, he showed many cool creations other students made.

The last workshop I attended was about AI and how we can introduce it in a classroom setting. This one was the most enlightening as teachers all chimed in with their experience with AI and how they have tried to work with it so that students can use it but not abuse it. Some of the interesting AI that I have learned is, for Work, there is a text-to-speech function that works wonders, elevenLabs a voice AI which can use your voice and convert text to speech. We also explored the difference between ChatGPT 3.5 vs 4.0 and how to use it with BingChat.

Overall, the conference was a lot of fun, and I had a blast meeting all these teachers.

Three Curicula that all schools teach

At this point in my life, school has been the main thing I have been doing, including elementary school, secondary school, and post-secondary. At this point, I will need to spend another 15 years before I have more time spent as not a student vs as a student. It is impressive and shows how ingrained school is to me and many others, so much so that I feel weird when I have work during winter, spring and summer break. Due to how stuck I am with school, I have also realized how schools tend to ingrain specific ideologies from students. 

Of these ideas, the idea of a competitive world. I spent a lot of my time with people around me asking me what I got on my exams and having my grades compared to one another. To not be too embarrassed, I tried hard to maintain a specific grade point so I did not look like an utter fool. While this has helped me get into post-secondary with ease, I also need to realize that I also trained students in a more negative way than we believe. I have a couple of friends who went into Sauder School of Business, and very quickly, by the end of the second semester, I heard them complaining about how you can't trust your classmates and how most of them are snakes. Due to how competitive everyone is in Sauder, many tend to backstab or ruin their classmates' marks so they can look better in the eyes of employers.

With the change in the curriculum in BC, we are slowly allowing the students to learn more than just content to be competitive in post-secondary. Teachers should now focus on the soft skill that leads to understanding the content rather than teaching the content that leads to developing soft skills. This shift makes students more rounded to think for themselves, be better at communicating and also know how to learn something new best.

Group Microteaching Reflection.

Our group teaching went pretty well as we tried a different approach than having the students sit and listen to us talk the whole time. By having the students come up to the whiteboard, we were able to get the students to practice some of the skills we discussed with the class. With everyone on the board, all the students had to try a simple problem out with one of our methods or a method they felt comfortable doing. Having the students come to the whiteboard allowed us to see if any students were struggling with the content. The cons were the class ended up being a bit too chaotic with everyone on the board, which was a bit hard to control. Yet, trying new things is great as I explore new ways to teach rather than staying safe with everything.

Group Microteaching lesson plan

 

Post Microteaching #1

It was a very successful first microteaching in EDCP 342A, and I am pleased with how it went. I pulled the card trick off without anyone realizing I did a sleight of hand. But, with all my microteaching, I always choose topics that are hard to fit within the time limit. As a result, it is a very tight fit, making it hard for me to slow down with my lesson. A common issue with all my teaching so far is that I need to rush too much. This rushing is usually the case as I try to fit in with the time limit given to me, so I feel flustered and rush through my lesson in hopes I reach the end by the end of the time limit. For future micro-teaching, I suggest I do a more manageable topic that does not require the need to reach the end of the time limit. This choice of topic will allow me to slow down and better understand what is doable in a specific time frame.

Battleground School and Change

I have always known that politics is significant in determining schools' curriculum. However, I have yet to realize the extent to which global issues pushed specific reform in mathematics education. Given many of these issues' distance, it is hard to feel their effects thoroughly. Despite only experiencing schooling in the mid-2010s, I have seen a drastic change in education. One of the most significant changes I have felt is the reduction in standardized testing. My year was the last batch of students to write the pre-calculus 10 provincial exams and the social studies 11 provincial exams. Since I have been graded on all sorts of things, I never really thought much about it, as up until then, my teacher made us write the final exam. However, I recently discovered that teachers no longer administer final exams to grades 8 and 9 students. Writing these high-pressure final exams has prepared me well for post-secondary as they helped me develop good study habits to ensure I can finish my exams successfully. With this shift, we are not assisting students to prepare for a high-stakes exam, as they will not be used to it. As a teacher candidate with only experience as a student, I sometimes need help understanding the purpose of changes. Until I have more experience as a teacher, my bias with a lack of experience will push me to prefer the old style more than the new. However, change is sometimes good as culture changes, and everyone, even the educational system, must adapt.

---------- Nov. 6, 2023 (Add-on) ----------

As a recent student, I understand the stress of balancing a load of homework and the need to relax. Rather than hammer my classes with homework, I allocate enough class time for students to finish all their homework in class. It will allow the students to spend in-class time asking questions if they need help with the topic. Homework is only helpful for students who know how to approach the questions. If we assign homework and the student already has issues doing them, then what's the point in giving them if they need help solving them? Thus, if students spend class time working on these questions, then it will be apparent to them if they have any questions, and they can ask right away.

While rereading the "Battleground School," the point of math phobia steaming from elementary never really crossed my mind during my initial read. However, after discussing with my SA about why many students coming from elementary have such below-average maths ability, it clicked. Of the points brought up, one of the main reasons why many students struggle with math phobia is their elementary teacher's inability to engage the students in math properly. They either avoid teaching maths altogether or hand out drills, which lowers the student's math confidence. As a result, most of Math 8 should not be used to assess student's knowledge of curricular content but instead should help build their math confidence.

Another exciting point brought up in the reading was related to the need for specialist math teachers at the secondary level. So far, I have yet to see this issue affecting any of the schools that I have seen, but I can assume that for more rural parts of the country, there will be a lack of math specialists. However, I can also say the same with topics like computer science and specific sciences, as many do not want to move to rural parts of the country to teach.

TPI reflection

After finishing my TPI, apprenticeship takes the lead with the most dominant trait as a teacher. The reason apprenticeship is such a dominant trait is due to how I want to teach computer science. I aim to introduce my students to coding and get them interested so the new generation is more aware of the technologies they use. As a computer science teacher, this ideal will likely influence my ideas as a math teacher, where I want the students to apply what they have learned to some real-world application. My two traits, developing and nurturing, are close to apprenticeship, but looking at the BIA, some apparent gaps need to be addressed. Unlike an apprenticeship where the BIA is within 2, here, the BIA is greater than 3. This gap indicates that these traits are likely to fluctuate over time since actions, in most cases, are low compared to belief or intention. On the other hand, looking at my recessive trait, social reform, I notice how the BIA are all very close together. The tightly packed BIA tells me it will be hard to change my perspective on that trait.

Social reform, being my recessive trait, is surprising, yet at the same time, not. When I think logically as a mathematics teacher, not much social reform will come out of maths. Most of the content in the classroom is meant to help students build more skills for more advanced mathematics classes. Yet, I want to change how my classroom is run from a traditional classroom. This change includes changing the questions being asked on exams, how students are assessed for their knowledge, how lessons are taught, and classroom activities to help reinforce learning (not only through homework). These changes entail some form of social reform, as the primary goal here is to change how classrooms are run and the end goal of specific math classes. However, social reform can be more easily seen in other subjects such as english, humanities, and sciences, as many of the discussions in class can be more related to global issues that teachers can bring up and start a conversation with the class.

On the other hand, most math classes need to complete a packed curriculum so that students can proceed to more advanced maths. This makes it harder to go off-topic to discuss global issues if there is a way to tie that social content with math content. I would love to discuss more in-depth how a teacher can better implement social reforms in a math classroom setting.

Microteaching #1

 For my microteaching I will teach my group how to do a card trick.

Lesson Plan for MicroTeaching 1

Updated lesson plan

Math Art Reflection

 

Our group recreated Melissa Schumacher's "Counting with Knots" by extending the original artwork with the counting from 0 to 15 using base 4. Because we changed from base 3 to base 4, we needed to create a new symbol to represent it. We each created a unique symbol representing the number 3 in this case. Allyssa made 3 correspond to an X symbol, meaning that the crossing has no restriction, but the line does not pass in this area. Asiya made 3 represent a crossing that has either 2 strands or 4 strands. I made 3 to correspond to both horizontal and vertical barriers. Of the three, my idea caused the final artwork to represent something other than a Celtic Knot as the horizontal and vertical barrier cut into my knot. Whoops

While working on this project was hard since each of us made our variation, we also knew the pain and frustrations the students would have when creating the artwork. The easiest part of the presentation was creating the PowerPoint and the actual presentation. Since most of the topics we were going to discuss were somewhat familiar to us or easily accessible online. The artwork, on the other hand, was very tedious. Initially, it took a lot of work to figure out how to start this art project. As we worked on it, I came across a grid with dots that I modified and used Photoshop to create the 3 grids to represent each part of the artwork. Even with the grid, I had to make multiple photocopies of one as I feared my inexperience in art would ruin my one good copy, resulting in a total restart. Out of the three parts, creating the knots with the over-under pattern was tedious. The other 2 parts were quite therapeutic as I worked in a set rhythm. However, since there is a precise over-under pattern with the knot, I had to constantly pay attention to what I was doing, which took a lot of energy. In the end, the Celtic knots were a lot of fun, and it took me about 5 hours once I had the grid to finish it. With an added 3 hours of thinking about how I would start this project, the artwork took about 8 hours from start to finish.

As a teacher, we must try to attempt the assignment ourselves to know what was difficult and how long it took us to finish. Given how different this math art is to many math classes, students would need help completing part 3 of the artwork as it was the most time-consuming and energy-intensive. Knowing the difficulties of that part, I have learned to modify the art project a bit so that students would create a smaller knot that is less vertical and slightly more horizontal, like a 5 by 7. Here, students will explore 7 consecutive numbers, but the numbers are larger in base 3 or 4. As we create Celtic Knots, this artwork will tie nicely with a Social Studies class if they ever cover the Vikings. This project would be a tremendous cross-disciplinary assignment with Math 9 and Socials 9 as we look at various aspects of the knot. For math, we would look at the underlying pattern used to create the knots, while in Socials, they would look at the underlying history of the knots to Celtic cultures. Overall, this was an enjoyable project, and I will keep this art project in mind as I head into my practicum and future teaching career.

The Dishes Problem

 Due to all the influences on me, it was hard not to consider the problem in a non-algebraic method. Thus, I had to solve the question algebraically to find the number of guests that generates 65 dishes... is 60. After doing it algebraically, I modelled and graphed the number of dishes created, given the number of guests attending. Here, I had to make some assumptions to proceed.

My algebraic work as well as working out how an x number of guests affect the number of dishes.

Suppose 5 guests showed up, and following the logic the cook provided, I assumed the following. 3 dish of rice was used as 2 guests used a dish of rice. Then, with 5, 3 dishes are needed, so the 1 remaining guest has a rice dish. Then, 2 dishes of broth had to be used. Once again, this is so the remaining 2 guests who still need to get a dish get one. Similarly, 2 meat dishes are used, so the 1 remaining guest gets something. Thus, we took all the dishes' mathematical ceilings to ensure all the guests got adequate food. With this assumption, one will realize that there are 2 correct answers to this question. 59 and 60 guests are valid, except one will have leftovers. Since the question did not mention leftovers, it is safe to assume both are correct. Instead of using pure algebra, graphing the question has yielded some interesting results, as I realized after seeing 59 guests with 65 dishes is also a valid solution.

Number of dishes needed given the number of guests.

My assumptions came with the idea of the image as I thought what would happen when this many guests showed up. As a host, I would rather have leftovers than a hungry guest because I failed to make enough food to satisfy everyone. Thus, providing imagery to the question can also get students to think outside the box, assuming it is allowed. At the same time, if we restrict students from thinking creatively by adding a handful of restrictions, what is the point of giving the students an image if the stated rules shackle them?

It is suitable to bring up the history of math as it demonstrates to students how/ancient some of the mathematical techniques taught in class are. With a question posed back in the 4th century CE, the student can realize how some of the topic being learned was once used to solve a problem. Similarly, math techniques did not only originate in 1 culture, and I like to believe that people with diverse experiences with different cultures can be a better citizens filled with love and not hate.

To future me

Hey Mr.Tse,

I appreciate your attempts to make math more fun and accessible for all students in class, but sometimes, we don't take math to enjoy it. Instead, some of us take it as a graduation requirement. I did not feel engaged in any of your classes, and in the end, I felt more lost than anything else. I preferred if you taught like all the other teachers, how we can solve the questions and that's it. All the extra fluff caused me not to understand the concepts next year, which caused me to fall behind in math. Thanks for ruining my chances of getting into university.

Student X

One of my biggest fears is that because I wish to teach math more relationally than instrumental, I would end up with students not caring about my lesson as they treat math as a mandatory class to enter university. Also, as I want to vary up my class and not heavily rely on pure classroom lessons and on some outdoor teaching or vertical class time to teach lessons, many students might not be ready for a shift, especially if all they have experienced in math is the idea of copy notes, doing homework, and doing exams.

______

Hey Mr.Tse,

I want to let you know that your class has really changed my views on math. While most teachers still teach in the traditional ways, I really appreciate your varied teaching methods to encourage more learning from us. This made me always want to show up to class as I am no longer always sitting in a stuffy classroom listening to the teacher go and go about things. You made math to be more than just memorizing numbers. While I did not score the highest marks in your class, I appreciated the use of math/logic puzzles in your class. I never got many of them on the first try but I did try my hardest and it helped me develop a more logical thinking method which I thought only smart people did.

I hope you keep up the good work best wishes,

Student X

As I change how we approach math in classrooms, one of the biggest things I want to accomplish is encouraging more students to explore different ways to solve problems. I also want to help students foster a growth mindset in my class and never feel too down when they struggle with something. While I understand that most people take higher levels of math in high school for university, I also want students to be willing to try and fail yet finish the course like they have learned something regardless of the mark they finish the class with.

______

As I look at my strengths and weaknesses, my worries stem from a sense of imposter syndrome about whether I should be qualified to teach in a classroom. After I start my teaching career and build confidence, my response to this activity would be very different as I would better understand how students react to certain things. As of now, a lot of what I imagine is what I have experienced as a student in high school and post-secondary. Yet, my idea of changing the way students are taught math will be a tough challenge to reach, so I hope that I am prepared to start slow with the changes. As I have been told, year 1 is for survival, year 2 is for change, and year 3 should be smooth sailing.

Lockheart's Lament

Lockheart proposes some exciting ideas regarding math education in schools, which I wholeheartedly support. Looking back at my high school math education, it has always been a monotonous cycle of lessons, reviews and exams. This rigid way of teaching has sucked away and made math boring to many students. As Lockheart points out, most students take math to improve their college application or get college credits to get math over with. Coupled with the need for marks or standardized testing in the US, I can see how the current education system emphasizes instrumental mathematics in the classroom, as detailed by Skemp.

While Skemp's argument of relational mathematics may seem similar to Lockheart's way of changing the mathematical education system, Lockheart's methods are more aggressive. Skemps's idea of relational mathematics still relies on the fact that there is a set curriculum that a teacher should follow, but rather than teaching the "hows of math," we as teachers should teach the "whys of math." Meanwhile, Lockheart argues that the rigid curriculum restricts the creative juices of math teachers. To truly embrace a new way of teaching math, Lockheart argues that we should thoroughly teach math without a proper curriculum so students can embrace mathematics and allow teachers to explore various topics.

While I appreciate Lockheart's view of dismantling the curriculum, he eventually still tied the dismantling of the curriculum to lead the students to take calculus in a formal setting. Yet, calculus is not all of math. Many different math disciplines still need to be more represented in our current math curriculum, such as discrete mathematics, elementary number theory, rings and fields, and many more. These math disciplines can be taught at a secondary level, albeit more conceptual, to introduce students to the abstract topics of these different math disciplines. Regardless of everything tying back to calculus, as a new teacher candidate, the best way to make math somewhat not dull is to break out of the current shell of classroom lessons and add more interactive activities for students to engage in and learn. Overtime, as we push for change in how math is taught in school, only then can we consider a proper reform, as mentioned by Lockheart.