Final reflection

Looking back at my posts this term, it’s exciting to see how much I’ve grown in both my knowledge and teaching strategies through this course. I can also see how my confidence and sense of belonging have grown as I became more comfortable sharing my ideas with classmates.

Some personal progress that stands out to me includes (1) a deeper focus on inclusion for diverse learners (it is a special interest of mine at this point), (2) a willingness to experiment with new ideas and activities, and (3) finding ways to create meaningful extensions by connecting math to other fields, such as art and music. Inclusion and accessibility in learning are core values I hold deeply, and this class has equipped me with so many tools to support different types of learners. For instance, the Math Art project was an incredible opportunity. Although it initially felt overwhelming, collaborating with peers brought fresh perspectives and inspired ideas. The insights I gained from other classmates’ presentations further enriched the experience. I found immense value in expanding on ideas I had researched and workshopping them with classmates. This process left me feeling empowered, knowing I have the tools to make math more engaging and enjoyable for students.

The readings also challenged and deepened my understanding profoundly. Having clear definitions for concepts like "instrumental" versus "relational understanding" and "arbitrary" versus "necessary knowledge" provided excellent starting points for my own reflections and helped me approach lesson planning with a more critical and thoughtful perspective.

Overall, I learned so much from this course and am incredibly grateful for the supportive class community. The environment made it easy to explore, share, and grow together. Thank you, Susan, for fostering such a wonderful space for learning!

Unit plan + 3 lesson plans

Here's the link with all my unit and lesson plans. 

Please leave a comment if you have any feedback or suggestions. Thank you!

Unit and lesson plan drafts

 Here's the link to my unit and lesson plan drafts for Math 9 "Powers and Exponents".

Response to Herbel-Eisenmann & Wagner (2007): Textbooks

As a former high school student, I followed the command words in math textbook practice problems without question. I rarely examined the instructions unless they included an unfamiliar story, graphic, or a "things you might not know" section. I struggled with self-learning math, which may have been influenced by my exposure to textbooks that felt impersonal, as if the instructions were not written for me. In the U.S., most math classes I observed did not use textbooks directly; instead, teachers provided pre-prepared, hole-punched, fill-in-the-blank notes and worksheets for students to store in binders. While teachers referred to textbooks to inform their lessons, students rarely had direct access to them. At the time, I thought this approach was advantageous solely because it spared students from carrying heavy textbooks.

Currently, the practicum school I am teaching at faces budget constraints that prevent every student from having a textbook or workbook. Some classes, like Pre-Calculus 11, use individual workbooks, while others, like Math 9 and Workplace Math 11, do not. My experiences with these varied approaches, combined with our class discussions on textbook usage, have made me more critical and aware of their role. I now feel empowered as a teacher to influence how textbooks are used, ensuring they serve as valuable tools rather than rigid guides.    

When I was in school, math textbooks were the only type of mathematical book that I had access to. Our small school library did not have math story books for casual reading. Then, I got into more and more nonfiction books about math, such as "How Not to be Wrong: The Power of Mathematical Thinking" by Jordan Ellenberg, "Mathematics for Human Flourishing" by Francis Su", as well as other nonfictions subtly discussing topics in math like "Invisible Women: Data Bias in the World Designed for Men" by Caroline Criado Perez. They opened me up to a different approach to learn math - one that promotes more relevance, critical thinking, and storytelling. They made me question if math textbooks should be the "standards" and guidelines for learning math. Unlike most math textbooks, which focus on discrete skills, these books categorize content by concepts and broader topics, promoting connections and deeper understanding. This has shaped my view of textbooks as guidelines, problem banks, and secondary resources rather than definitive curricula. Textbooks are valuable aids for teachers, particularly for those just starting, but lesson content and progression should not rely solely on them. I see textbooks as a "box" that provides structure and security, but effective teaching requires thinking outside that box—designing engaging lessons, tailoring content to students' needs, and responding to the evolving demands of knowledge and skills in our world.  

The idea of "Flow"

I have experienced flow in various activities, such as playing music while tackling a challenging yet exciting piece, participating in sports when inspired by fleeting "visions" during the game, and solving complex math problems with the confidence that I can succeed and that the outcome will be satisfying. One specific instance that stands out was graphing a parametric "rose curve" with eight petals in my Calculus II class—a task I enjoyed from start to finish. Across all these experiences, the common thread was a combination of challenge and confidence. While the task was unfamiliar, I felt equipped with the necessary skills to eventually achieve the objective. Interestingly, flow could arise whether the stakes were low—helping to ease my nerves—or high, such as during the final game of a basketball tournament, where nervousness blended with excitement. This suggests that flow, at least to me, is not dependent on the stakes but rather on the balance between challenge and confidence.  

I believe flow is achievable in secondary math classrooms. While it might be more evident in small groups, educators can strive to create an environment where the entire class feels engaged, active, challenged, and confident. I particularly appreciate Peter Liljedahl’s insight that flow can be facilitated by dynamically adjusting tasks and their difficulty levels during a lesson. This approach fosters a growth mindset for me as an educator; even if the initial task isn’t perfectly aligned with students’ needs, I can adapt it in real time to better suit the class. For students, it’s equally valuable to understand that they have choices—there are always simpler or more advanced tasks available. Providing this autonomy empowers students to take ownership of their learning and work towards personalized goals in the math classroom.  

Soup Can puzzle

Based on a quick search, the average length of a women’s bike, measured from front to rear tire, ranges from 173 to 178 cm. For this estimate, I’m assuming a length of 175 cm. Additionally, typical dimensions for a Campbell’s soup can are about 10.8 cm in height and 6.6 cm in diameter.

By analyzing the photo, I estimated that the "height" of the cylindrical water tank is roughly 2 and 2/3 times the length of the bike. Given the bike’s assumed length of 175 cm, this suggests the water tank's height is approximately 466 cm. This makes the Campbell’s soup can a scale model, magnified about 43 times. Consequently, the water tank’s diameter would be around 285 cm. Using these dimensions, the tank’s volume is nearly 29,766 liters, or close to 8,000 gallons—sufficient to supply an average house fire, which typically requires around 3,000 gallons of water.

While none of the measurements were exact, I felt confident in my estimates because they were logically sound: there were no implausible values, like negative numbers or proportions that seemed unrealistic. My approach was straightforward, though it didn’t account for the depth perception of the water tank, which could mean the actual height-to-bike ratio is slightly more than 2 and 2/3. However, this level of estimation was sufficient for the purpose of approximating the tank’s volume.

To extend this problem, I might also ask students to estimate the weight of a filled water tank and brainstorm methods to move it if it were portable. This task would encourage not only estimation skills but also imagination and problem-solving. Estimation is especially valuable for helping students focus on logical reasoning and explanation rather than finding an exact answer, which builds their confidence in justifying their thinking.

Response to Hewitt (1999)

Hewitt (1999) prompted me to consider what is “arbitrary” versus “necessary” in math education and how I address these distinctions in my teaching. His ideas resonate with Skemp’s (1976) concepts of “instrumental” versus “relational” understanding: while “arbitrary” knowledge and “instrumental” understanding provide the language and foundational procedures for discussion, “necessary” knowledge and “relational” understanding facilitate higher-order thinking, allowing students to internalize and apply their knowledge. During my recent short practicum, I co-taught a Pre-Calculus 11 unit on radical operations and equations with my SA. We emphasized students working through some processes, such as applying the expansion (A+B)^2 = A^2 + 2AB + B^2, and treated other concepts as “arbitrary”—a practical decision due to time constraints and the fact that these concepts had been covered in earlier grades.

This article has made me more conscious of which ideas students can discover for themselves to enhance retention and understanding. In my lesson and unit planning, I intend to incorporate time for students to explore concepts independently or with guided prompts, allowing them to derive rules through their own reasoning. For arbitrary concepts, I plan to use visual aids and classroom posters to support memorization. Reflecting on Hewitt’s perspective also deepens my empathy for students who struggle to memorize arbitrary details. Building a classroom culture where students feel comfortable exploring, making mistakes, and supporting each other will be essential for fostering this balance between “arbitrary” and “necessary” learning.