Inquiry 1 Project and Final Reflection

    I learned through this project how the notion of proof is encapsulated in secondary math classrooms. This was my first time doing a research based project in university so the process was a bit difficult. However, navigating through google scholar and reading articles by recommended authors allowed me to see many people's point of views on this topic. There were also some findings that I never came across such as mathematicians' views on proofs, and reasons why teachers won't implement proof in the class. The concept of rigor and formalism isn't necessary for younger secondary students and proof is less deductive as they may see the bigger picture. I would like to touch on using using technology/software to address proof that may also enhance student reasoning. When can Desmos be used to be convincing for students when they don't have the sufficient background knowledge for other forms of proof. How many proving questions should I put on an exam, say an easy question like "show that the sum of two linear functions with no constant is a linear function with no constant?" I definitely should bring some logic into the context of student's work as well and bring in more social perspectives say from teacher's perspective: Although proof promote a deeper understanding of mathematical concepts by requiring students to justify their reasoning, teaching proofs might take longer to students and potentially affect the pace of covering the curriculum. As it isn't mandatory in the curriculum, students aren't expected to learn it in full great detail, nor do problems relate to it. . . besides questions like computations and finding errors in step X. My curiosity leads me to explore how proofs can be introduced effectively in secondary mathematics classrooms while still covering curriculum topics. I'm interested in understanding the challenges students might face when learning to construct proofs and discovering innovative teaching methods that make this abstract concept more accessible and engaging, either more visually, with more technology, or showing that they can come up with it using knowledge the students already possess. I believe this class has led me to further teacher inquiry, to ponder on issues society is facing nowadays and how it affects education. How to bring math into these issues may be a great way to help. It was fun exploring around in the garden too, learning how to make ropes. The first class topic was also memorable as it was on forming good relations with students. The notion of grades, competition, and cooperation was also interesting as I read articles and watched videos (like Kohn) on it to see such a different perspective. Nonetheless it is crucial to hear what they have to say inorder to better understand their viewpoint and why they believe it is so. This course made me delve further into the nature of this profession of being a teacher and how teachers affect students' lives and their own.

Nov 23 Entrance Slip Annotated Bibliography

Annotated Bibliography

Nov 9 Exit Slip

Currently found 8 sources pertaining to my topic: 

- Proof, Explanation and Exploration: An Overview

- Secondary School Mathematics Teachers' Conception of Proof

- Proof and Proving in School Mathematics 

- Mathematical Proof: From Mathematics to School Mathematics

- Who Should Learn Proving and Why: An Examination of Secondary Mathematics Teachers’ Perspectives

- Logic in Secondary School Education

- Review: Proofs That Really Count: The Art of Combinatorial Proof (Undergrad math book review)

- Support For Teachers - Proof (Website)

Oct 19 Exit Slip Grades/Competition Reflections

Today’s class primarily focused on two parts: questioning the accuracy of grades and a debate on competition/cooperation. Both are based on some thoughts made by Alfie Kohn, who is quite an extreme individual. This is a unique trait of his and it was interesting to see someone very fervent in these topics to express them out boldly. To begin with whether grades are accurate or not, I think it’s impossible to completely evaluate a student’s learning based on external assessments. Only the individual would know whether they know something or not once prompted with a question. I think grades give a general idea of how well students are doing, but there are so many other factors that should be taken into account that aren’t present within a numbered percentage. One can say someone who got 100% on an exam generally understands and mastered the concepts better than someone who scored 70%. However, what about two students who scored 70% vs. 71%, is the 71% student necessarily a bit more mastered than the 70% student? Sometimes there will be students who don't try and others who just have complications in life that hinder them from studying well. What were those 30% gaps that were missing? So we began to discuss proficiency scales, which is something I’ve never seen used in math growing up (it looks like a rubric for essays and presentations for English/social studies classes). Proficiency scales are helpful in that they can help identify areas students know and where they can work on. I think it’s quite interesting to implement in Alberta. I hear that this is used in junior high grades (7-9) in this province, and in high school they start using percentages (maybe along with proficiency scales). This method might be good since it allows students at younger ages to focus less on grades and more on improvement and developing skills to master them. Once high school hits, those who do plan on advancing onwards to post secondary may need some percentages for applications. Allowing them to also know their strengths and weaknesses in certain concepts will help them continue to develop good studying habits and metacognitive skills. Now, regarding Alfie Kohn’s comments on the negatives of competition, he has some good and true points, but are very generalized. Some people may find competition as a huge motivating factor to improve themselves in this world and how they can better themselves after seeing the improvement of other people. Cooperation helps build team/net work skills and allows one to figure out how to work with someone they don’t necessarily prefer to work with. This in turn can cause strife amongst the team and individuals blaming each other (lose-lose situation). But it's nice to weigh the pros and cons of each type and see where they can be implemented.

Oct 19 Inquiry Project start up

1. Define your question 

Search Topic: The Usage of Proofs in Math to Aid Student Learning 

2. Analyze your topic into concepts 

Concept A: The importance and value of having proofs 

Concept B: In which areas should proofs be implemented and how much rigor is necessary? 

Concept C: Different forms of proofs and their convincingness to students across stages 

When discussing the usage of proofs integrated in math, I want to draw upon existing educational theories, practices, and expert opinions. The role of proofs in math education stems from different philosophies on the purpose of education in mathematical understanding and the skills students need to succeed in the modern world. Including proofs can enhance students' logical reasoning skills and deepen their comprehension of mathematical concepts, while a lack of them might make the content more friendly/accessible but less water-tight/convincing. I happened to be exposed to multiple forms of proof in post-secondary. These may be utilized in teachings to allow students to not only get the right answer from a formula but also understand the inherent logic/structure that governs the notion. Without certain forms of proof, harder topics may be less intuitive. Allowing students to see different ways certain concepts came to be and how they are derived resonates with students in varying ways. The efficacy of a particular method often depends on the specific concept being addressed and the individual student's learning style. I was interested in proofs in latter years of high school. A lot of the equations are derivable based on previous knowledge students have encountered. This allows a build up of past ideas (or use them as a scaffold) into new ones. Certain definitions and evaluations were plopped onto the formula sheet, making it tedious for students to use them as they just take them for granted. Other students may feel indifferent but they lose out on some richness and value in linking information. Finding the sweet spot is hard and to balance how much time is spent on proofs in the classroom is something I have yet to experience.

Oct 19 Entrance Slip Reflection on marks, grades and their effects in schooling

I think grades and percentages are helpful for students in monitoring their progress to an extent. As a teacher it is very easy to assign percentage grades to assignments since it gives a ranking. Generally, grades can help indicate how well a student understands the course material. A high grade might suggest that a student has grasped core concepts, while a low grade might indicate confusion or gaps in knowledge. Sometimes, grades also factor in a student's participation, effort, or engagement with the material (though this might not directly reflect a student's mastery of the content). For my math classes back in grade school, every single summative assessment was a quiz/exam. This leads to little room for further improvement, if say, for mastering the concept or achieving a higher grade to apply for college/post-secondary. Indeed Kohn’s concern on the fact that the mainstream grading system needs revamping (since it raises negative effects on the desire to learn) is very evident in the classrooms I’ve been in. I think many of my classmates (myself included) were motivated to study/grind in order to get a higher grade, not necessarily to understand what was really happening. A lot of concepts were memorized or understood at an adequate level at the time, and then they disappeared after the exams. Going back to grades creating a ranking system or a social hierarchy, students labeled as "A" students are treated differently than those who typically earn lower grades. This labeling can affect self-esteem, peer relationships, and even teacher expectations. It also generates a lot of stress on students as they have to worry about their marks for tests. From the article, even high achieving students like Lucy are nervous about tests. Low achieving students may feel like they’re static and can’t improve, while high achieving ones will be pressured to remain at the top levels. All in all it creates bragging rights for the school so more parents can send their children there in order to get more funding. Grades are a conspicuous factor that students use to determine their success. But success for them need not to be expressed solely with a letter/percentage. The teacher may provide other projects/activities to evaluate a student, such as an creative math-art piece, a recorded mini lesson going over a problem, or presenting to the class. As long as they’re putting in effective effort in their learning with a positive attitude, the confidence levels should rise. I haven't experienced any course where a grade wasn’t present. Having categories like emerging, developing, proficient, and extending with goals/guidelines will help students find out which areas they should improve on. Although some students want grades due to competition, that notion of competing against oneself to do better in the future is just as effective. So for example assessments without grades like portfolios, socratic discussions, and class presentations can reflect the learning process. In a system without grades, the emphasis shifts from performance to learning orientation. It's also important for both the teacher and student to commit to whatever alternative assessment method is chosen. Parents and administrators also need to understand/support these alternative methods for them to be effective.

Oct 5 Entrance Slip Dancing Teachers Into Being With a Garden (Updated Oct 5)

I realized that teachers in schools are often regulated by the strict constraints of time, symbolized by the act of frequently checking a watch. I have to be careful, since doing this action can cause students to lose focus on the lesson, and more so on when to leave the class. It allows me as well to focus more on the time since I have to get through a set amount of things. I usually go on rambling about a topic, so time management is something I should work on. I should also pause and give students time to digest as well, since I’m so used to lectures that are continuously pouring out information/facts. I don’t want my students to feel constrained with lessons, so taking them outside to experience learning is a new thing. They would enjoy it more and catch a breather. Some metaphors I see include the squared-off boxes of classrooms, which represents the restrictive/linear nature of traditional education. The rectilinear rooms, tiled floors, and ceilings may also illustrate the structured, rigid layout of educational settings. From students’ responses on learning in gardens, the desks are overgrown and demonstrate the growth around the unstoppable nature of organic growth/curiosity. It could signify that despite standardized learning being present, students will always have an inherent urge to grow, learn, and explore. A lot of outdoor garden learning is helpful for a small junior high class (though it is much harder to get a large class to stay on task). Letting students in that age category step outside and experience wildlife while doing math art/projects is a fun way for them to get in touch with the foundational learning topics that will be used for high school, post-secondary, and work. However, I doubt it is feasible for older high school students. They have a rigorous curriculum to go through and many are, for lack of a better word, ignorant of university learning/living lifestyle. The gap between grade 12 and first year is large, so it is crucial to help their mindset grow towards being more mature/responsible/experienced. Back to my chemistry teacher, his exams and layout are similar to those of university. Though he may have seemed harsh, in hindsight it was helpful as we got a taste of the nature of university. Many alumni have constantly come back to visit to thank him, and ask him questions instead of their professors. I think a lot of experiences of living in/being knowledgeable about nature can be taken from outdoor education.

FIGURE 3: (Colour online) Japanese orthography worksheet got my attention when the article was discussing about grids. Grids are mostly used for organizing information such as a lesson plan and categorizing ideas. In math the small grid paper squares allows students to easily draw shapes and connect the dots. These grids are tools to help and aid us in certain practices. The orthography worksheet also uses grids in order that people are able to practice their writing neatly. Chinese orthography worksheets also have vertical, horizontal, and diagonal lines to help the writer. The question "why am I committed to using this method to practice my writing" can be answered in which it helps one able to know the boundaries for writing, how big a character should be so that you won't have this big radical on the left side of the character. The grid allows people to see their new work and compare it with their old work. Boxing the old work helps with the understanding that these are just practice trials and that they stay in this grid, and the characters I write outside of practice are the ones I chiseled to beauty.