Final 342 Reflections

 Looking through all three of my blogs, I feel strongly that the writing I’ve done for 342 is some of my most inspired. I’m very grateful for the material selected – many of the articles have helped me develop lenses for knowing/seeing that I regularly leverage to help make sense of the educational world. Some of the most notable:

Relational V.S Instrumental

Arguably one of the most influential – within our cohort, I noticed many of us naturally adopting these terms to speak about and make critiques of content. For me personally (and in addition to enriching my vocabulary), these terms helped me characterize what type of learning works best for me. In particular, I realize that I generally need some sort of instrumental understanding before I can make sense of relational details. It was a big ‘aha’ moment when I realized that this may not be the case for all students. Although it seems deeply natural for me, it was suggested that other folks (potentially those are neurodivergent) would learn best if some relational information was provided first. As a teacher, an awareness of this difference is invaluable. I’m very happy this was one of the first pieces we read.  

Arbitrary V.S Necessary

I love this lens almost as much as the ‘instrumental & relational” lens. As a teacher, I felt a ‘correctness’ when I (unintentionally) led students to a ‘necessary’ conclusion with arbitrary information. This language enables me to more clearly understand why this felt ‘correct’, as well as how to reproduce it. By being intentional with what information we give our students, we can create very authentic opportunities of discovery for them.

Teacher Bird & Student Bird

This type of thought exercise helped me to realize and delineate my student and teacher minds. Not much else to say… just another lens I am grateful to have in my toolbox!

Flow

Our lecture on Flow helped me to more explicitly tie my musical experiences to what I hope for in my classrooms. Perhaps one of the most important realizations I came to (which was affirmed by Susan) is that students of all skill levels have the ability to achieve flow-state. I was never of the opposite opinion, yet having this idea affirmed gives me hope that cultivating a culture of ‘flow’ is possible within my classrooms

In general, I am overjoyed with my experience in 342. The pace of reading was thoughtful and I am grateful to have had the opportunity embrace the role of student under such an excellent educator.

My only advice relates to the garden. I am a deep lover of the garden, however it was clear to me that my comfort in this environment stems from my experience in this world – I camp, hike, bike, and generally feel as comfortable outside as inside. This is not the case for everyone. If one is physically uncomfortable, they cannot fully engage in learning. As such, I would place greater emphasis in teaching and maintaining comfort in this space.

I would also like to speak length about my peers who were also instrumental in cultivating a culture that made this semester such a success – however, I reserve this discussion for my 450 blog post.

Thank you Susan!

Unit Plan Final

Unit Plan for Pre-Calculus 11, Angles and Ratios, is here: 

https://drive.google.com/drive/folders/1oR823o7436OMqfgYjDfYk3YF_TnZSiv7?usp=sharing

In this unit, we are developing proficiency with trigonometric ratios. There is an introduction to some advanced topics, namely Sine Law and Special Triangles, as well as a lot of arbitrary information. 

In Lesson 1, I will be introducing trigonometric functions as coordinates of the unit circle. This will involve a lot of estimation. Most of the class will involve students working through this worksheet at their desks, in groups. Although this isn't ideal, the worksheet is designed to provide Arbitrary knowledge in the hopes that students can uncover the necessary independently. The class will end with a collaborative estimation game, as well as an inquiry based consolidation in which I will try to guide students towards some necessary formula. For an elaboration on the origins of the worksheet, refer to my previous 'Unit Plan Draft' post. 

In lesson 4, I'll be providing some historical context for geometry and trigonometric functions. This will feed nicely into the project introduction, which will hopefully engage students and encourage them to think more generally about applications of trigonometric functions. This lesson will conclude with board-work in which students will work collaboratively on a few iterations of one of my favorite puzzles. 

In Lesson 5, students will be going outside to take angle measurements so as to measure structures on top of the school. The purpose of this will be to have students appreciate the value of Sine Law. 

I've also included the required worksheets (for Lesson 1 and 5), and the project handout / rubric. 

Excited to deliver this unit!

Curricular Micro-Lesson Reflections

 Generally I felt that the curricular micro-lesson went well. For this Reflection, I will summarize and respond to the common themes discussed in our peer-feedback. To more meaningfully interpret the feedback, I tallied up how many responses in each category were not 3/3.

1.       “Clarity of Presentation & Activities” & “Learning Objectives Addressed and Met” – several non-perfect responses.

These two categories complement each other – I suspect that if we had more explicitly stated the learning objectives at the start of our class, the purpose / clarity of the presentation would have improved. Perhaps we could have explained these objectives before jumping into the algebraic definition of things? I don’t think it would have been wise to speak about them before the drumming – I think having it as a cold open was a good way to get students engaged.

2.       Definition was a bit unclear

When we presented the definition, I think there was a bit of confusion regarding the use of ‘n’ as an indexing term. Introducing the relevant terms and their importance is not trivial; students likely haven’t seen any sequences before, and so there is a balance between explaining what a sequence is, how each term can be indexed (with n), and then how we can define / generate geometric sequences.

3.       Bongos was a good hook!

I’m happy that folks were able to engage with the bongos / drumbeat at the beginning! This part of the lesson went well, however if (when) I do it again, I would alter it somewhat. In a class setting, I would take more time and also introduce to a ‘tripling’ drumbeat (the one I showed was a doubling), and then finally a ‘halving’ drumbeat. Or perhaps, in high school, I could choose one drumbeat to start each day of the unit. Generally, I think that exposing students to a wider range of examples will help them ‘hear’ what is making it geometric (for me, it is the ‘exponential’ increase/decrease).

I had also hoped to have more time at the end to explore the relationship between these rhythms and geometric sequences. In particular, I want students to discover that the first term of the sequence correlates to the number of ‘bars’ required to generate each term. In our example I chose 2, but it could just as easily been 1, 3, or anything else. This is a nuanced idea – I’d likely introduced it on Day 2 or 3 of this unit in a high school setting.

In all, I am pleased with how this went and am excited for the opportunity to teach this unit next semester!

Micro-Curricular Feedback Post

 Let me know how it went! 

Lesson Plan: https://docs.google.com/document/d/1MBxwKz5iexRaJV0zTaUb-8RIJEwPlx25/edit?usp=sharing&ouid=114344616397971363305&rtpof=true&sd=true

Updated Draft Unit Plan

Unit Plan is here: 

https://drive.google.com/drive/folders/1oR823o7436OMqfgYjDfYk3YF_TnZSiv7?usp=sharing

A few notes: 

During class today, conversations with Carson led me to revised (again) the structure of my lesson plan. This is a template I've developed myself, but I am feeling much happier with it now. 

When I teach this unit, I intend to take 3 days to go through the unit circle. This is as-per the recommendation of Michelle Kovesi, the teacher who developed the worksheet I will be going through. It offers a radical new way to teach trigonometric functions and has many advantages to the methods generally offered in textbooks. In general, many necessary qualities of trig functions can be derived by the students. In particular: 

The maximum and minimum value of these functions are 1 and -1, respectively

Which quadrant one would expect to find positive or negative values for these functions

How many angles on the unit circle yield some value for sine or cosine 

Sin^2(A) + Cos^2(A) = 1 (!!!!!!!!)

The value of Sine and Cosine at 0deg, 90deg, 180deg, and 270deg. 

If this sounds too good to be true, I recommend you look at the pdf file in my unit plan titled: "BCMAT Trigonometry Presentation". 

I have the support of my SA to implement this method - they are as excited as I am. 

Unit Plan Draft

My Unit Plan Draft can be found here: 

https://drive.google.com/drive/folders/1oR823o7436OMqfgYjDfYk3YF_TnZSiv7?usp=sharing

Included so far is: 

My Unit Plan - Mostly finished, lesson order is subject to change

Project Handout - I wrote a draft of the handout I plan to give students for my project. Includes assessment rubric. 

History of Math Presentation - Will be used for Lesson Plan 2

Lesson Plan 2 - Very rough, incomplete

Lesson Plan 3

Hand out for Lesson Plan 3

I've not included Lesson Plan 1, however I am very excited to develop it. It will be based on a method of teaching trigonometry developed by Michelle Kovesi which she shared in Whistler. This method relies heavily on estimation and 'Arbitrary' knowledge which enables students to discover many 'necessary' truths. Students develop strong intuition and an understanding of how trigonometric functions are related to the unit circle. 

Math Textbook Reflections

 As a student, I've found myself intimately engaging with very few math textbooks. That being said, “Fundamentals of Complex Analysis – with Applications to Engineering and Science 3e” by E.B. Saff & A.D. Snider holds a special place in my heart. It was/is clear to me that it was written for students. Often ideas are presented as stories. The authors frequently use “we” and “us” to indicate that they are taking you, the student, on a journey. Provided you engage with this text as intended (reading without skipping), one finds that the dissemination of knowledge is paced very deliberately. It gives one time to digest and question new ideas before throwing the reader to the (mathematical) sharks. In this way, I suspect that those familiar with the ideas presented would find very little value – the pacing is intended for a new learner.

In the relevant article, I was interested in comments relating to first person pronouns. It reminds me of Drakulic’s ‘Café Europa’, in which she discusses her tendency to use the first personal plural ‘we’, as well as her hatred for this tendency.  For Drakulic, the use of ‘we’ is associated with anonymity – it is the movement of a massive, automatic, submissive puppet. Conversely, ‘I’ is associated with the development of individuality, responsibility, democracy, and initiative. Despite this, Drakulic often speaks with the first personal plural because she acknowledges a common denominator between members of all formerly communist states and herself.

Drakulic shows us that when the use of ‘we’ is natural when we are speaking on behalf of a community. Textbooks which use ‘we’ and ‘us’ speak on behalf of the mathematic community – a community that both the author and the reader is a part of. With that in mind, the use of ‘we’ also suggests a submissiveness on the part of the author – by using plural pronouns, one doesn’t get the sense that the author is speaking about their own ideas, but rather is escorting the reader through some well defined (mathematical) reality. Perhaps ‘submissiveness’ is too harsh… Regardless, there is enormous value in texts infused with the first-person singular. Such text would reflect on the author’s personal experience with the material, un-abstracting concepts from the distilled realm of elites. Paul Lockhart’s ‘Measurement’ is a fine example of this.