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.

Flow in Education

 My experience with flow is deeply associated with music. As a musician of 15 years, drumming is the only creative outlet in my life where I regularly am able find meaningful, creative challenges that inspire my best. In drum circles, I experience the timelessness that Csikszentmihaly associates with flow. Also, because I believe in my ability to create the rhythms I want, my consciousness is focused entirely on the ‘what’, not the ‘how’.  

Earlier in this semester, I had learned about Csikszentmihaly’s ‘flow-state’ while writing about authentic assessment. I hadn’t come across any satisfactory definition (for authentic assessment) in the literature, and so proposed (to myself?) that it be “any assessment in which students have the opportunity to demonstrate the full depth of their ability and knowledge.”. This reminded me of my own experiences with flow-states, particularly as a drummer. I know that the most ‘authentic’ demonstration of skill and capacity happens when I am in ‘flow’, and so was wondering how this might apply in education. Undoubtedly, it would be desirable to help students enter flow-states (for individual work, group work, or assessments). I was able to identify (as we did in class) that as a teacher, we can help students enter flow by giving them challenges proportional to their skill. I also found several resources which explain the importance of ritual when entering flow. This struck me as a less obvious way we might help students: by helping them intentionally create rituals to enter flow-state.

During class, I found it interesting the difference between Liljedahl and Csikszentmihaly’s flow diagrams. Namely, Liljedahl’s diagram suggests that flow can be achieved at all skill levels, while Csikszentmihaly’s diagram implies that flow is only possible with a high degree of skill. Does Liljedahl not believe ‘Apathy’ is possible? How do we characterize this ‘low skill’ zone where Csikszentmihaly believes we cannot achieve flow? 

Intuitively, I believe Liljedahl’s diagram is most accurate. For example, as a novice musician, one will often find that even with 'low' skill, there are some songs you can play. Playing songs is deeply engaging and allows ‘low skill’ musicians to enter flow. Consider ‘low skill’ mathematicians, mountain bikers, readers… the more I think about it, the more ridiculous is the idea that low skill disqualifies one from access to flow.

I posit that there is a dimension of engagement that is not quantified in these diagrams. When skill and challenge are high, engagement is somewhat implied. However, if skill is low and an appropriate challenge is presented, I posit that engagement determines whether or not someone can enter flow. Consider someone learning to read. When deciding what book (challenge) to offer them, we can find several which are appropriate. So how do we choose? We offer them the book with the most engaging content.

I’m deeply interested in how to enrich my teaching practice by intentionally cultivating a classroom in which ‘flow’ is common. I expect it will be a source of inquiry later in the year.

Campbell Soup Can Problem

First, I decided to use the wheel of the bike as my reference point. They look like older mountain bike wheels, which I know to be roughly 650mm. I then added 40mm to account for the extra width of the tire. Finally, the bike is leanings (although very slightly), which led me to correct for the height in the picture by a factor of cos(θ). I estimated θ to be π/24. This all leads me to believe that the vertical height represented by the wheel in this picture is roughly 684mm.

Using PowerPoint, I drew one line equal to the height of the wheel, and then stacked the lines to the top of the can. This gave me an estimate for the tank diameter to be ~3.75 ‘wheels’, which translates to 2,565mm.

I measured a can of Campbell’s soup to be 10.8cm high and 6.6cm wide. This gives a Height / Diameter ratio of 1.63. With this, I scaled my estimate for the diameter to get a tank height of 4,194mm.

Although we could estimate the tank’s volume with the volume of the entire tank, I thought it would be best to incorporate a correction factor. To determine this factor, I compared the theoretical volume of some commercial storage tanks with their rated volume. I found that generally, once a tank exceed 300-400L, the correction factor is roughly 0.98 – 0.99. There is higher variance with smaller tanks because volume lost to curvature at the ends is relatively meaningful. If a tank is holding thousands of liters, this curvature is insignificant.

Using a correction factor of 0.99, I estimate the capacity of this tank to be 21,200 L.

A bit of research shows that the amount of water required to put out a house fire can vary wildly depending on the size of the building, how much is (and is not) on fire, and what other hazards are nearby. There is also no set volume of water that will put out a fire – any one fire will have some minimum flowrate of water that must be used to extinguish it. For a small fire, 600 – 800 LPM is typical.

Supposing there is a 600 LPM fire, our tank would suffice for roughly 35 minutes. Given that fire fighters typically take 15-20 minutes to put out a house fire, this is adequate.

Teacher Bird

Once I started digging into this question, I realized much of the task involved making and justifying estimate. I chose to include estimates for the bike-lean angle and the tank correction factor, however even these were pretty insignificant. It did make the task of organizing my data much more difficult, although the ability to do this well is extremely valuable.

In all, I found that I spent almost no time on the use of ratios for this question. I can’t say how a student would have felt completing this question, but I worry that they might spend too much time invested in insignificant details (this might also be a good thing!). Generally, open-ended questions like this serve as excellent inquiry questions but are probably not best suited for teaching a specific idea (such as ratios).

There is a lot of additional geometric complexity one can incorporate in this question. Suppose we want to use time of day & lengths of shadows to more accurately determine a reference length? The top of the can is not flat in the picture – how can we best account for this tilt? Is the ground the bike rests on level? Suppose we know the required tank thickness – should we use that to determine a correction factor?

Arbitrary and Necessary - Reflection

 I am very grateful to made aware of this line between arbitrary and necessary. I recall a lesson I taught during my short practicum.  

The objective of this class was for students to understand how to sum geometric series, as well as how to use (the very arbitrary) Sigma notation. To start the class I explained to students that we were trying to calculate sums, gave them a formula to calculate the sum of a geometric series, and finally provided an explanation of Sigma notation. Although I gave them some necessary information (the formula for the sum of a geometric series), I judge that these students did not have the awareness to arrive at this formula independently. Explaining Sigma notation was required by me as a teacher because it is arbitrary.

After working through problems, I gathered the students to review. Of particular interest was a question I wrote asking students to us Sigma notation to represent a geometric series. At this point, students had a decent understanding of both Sigma notation and how to generate a geometric series. Slowly, we were able to leverage their existing knowledge to ‘derive’ the relevant formula. I had not realized the significance of this question when I wrote it, but afterwards I felt that this was a very powerful moment in the class. I know recognize this as students leveraging our arbitrary notation do deduce something necessary. As a teacher, I feel a ‘correctness’ when I am able to guide students to independently discover the necessary.

As I develop future lesson plans, I suspect acknowledging what is necessary and arbitrary will inform what information I provide students at the start of class, as well as how to minimize scaffolding required for students to arrive at the necessary. I will perhaps even include a section on my lesson plan which explicitly states the relevant arbitrary and necessary information. I am excited to develop my next lesson plan.

Oct 25th Professional Development Day

 For the Province-wide professional development day, I attended AOEC’s conference: “The Anti Oppression Classroom Toolkit: From Theory to Practice”. This was a rich day, though admittedly not explicitly in mathematics. This day involved a keynote speaker and some workshops, all which aligned with the philosophies I associate with anti-racism and SOGI.

My first workshop was about SOGI – Activism as Pedagogy; it was my first exposure to SOGI in an explicit educational context. It was helpful to have discussion around how ideologies associated with SOGI manifest in an educational context. In my second workshop, ‘Navigating Complex Conversations’, we worked with strategies to improve one’s ability to fully engage in conversation. In particular, we discussed some techniques which help one hold space with the intention of helping others share their story.

This conference ‘walked the talk’ as it relates to creating safe, welcoming environments – in this way I most clearly saw the value and importance of the pedagogies being discussed at this conference. This is work that I continue to integrate into my perceptions of the world, as a teacher. That being said, as junior math teacher, I intend to pursue a more math-oriented conference next pro-d day.

Lesson plan for Oct 16th Mini-Lecture

 

Non-Curricular Micro-Lecture Reflections

 Attached below are my peer-feedback forms. 

I think my micro-lecture went well, however I feel that I tried to fit too much content into 10 minutes. In doing this, I wasn't able to really interact directly with my peers, nor did I give them a chance to contribute / interact. I felt that the group was reasonably engaged, but I still felt like I was talking to / at them for 10 minutes. If I could do this micro-lecture again, I would included a Think-Pair-Share involving speculation about one specific archetype.

My peers agree that the lack of activity / interaction was an area for improvement. Additionally, from my own reflections, I realize that I used very academic language for this lecture. As such, I suspect this would have been difficult to access for English Language Learners. . 

This micro-lecture taught me about timing. In a classroom I expect to be able to use my time freely - I want to leave some 'space' which will allow the class to pursue interests that might specifically attract them. For this reason, I'm apprehensive to try 'timing' my lesson plan. That being said, completing this activity showed me the value of associating 'times' with certain ideas. Knowing how long it takes to explain some new concept is valuable information and does not restrict the flow of the classroom. Rather, it will better inform me as I guide the class towards certain ideas.

In short: this activity showed me timing is a tool, not a structure. 

Thinking Mathematically - Average Speed

While completing these, Q1 refers to the question in which we are given a 'number of minutes going at 60mph'. Q2 refers to the question in which we move at 60mph for a certain distance. 

At first, I didn't have any major problems. However, when I completed the Q2, I noticed that the generalization for Q1 and Q2 had the same coefficients! Given that the variables we were plugging in are different (a 'time' for Q1 and a 'distance' for Q2) something felt wrong. See below for my investigation of this. 

Q1:

Q2:

After converting the Q2 generalization to 'minutes', we see identical coefficients. This was confusing and warranted further investigation:

Looking through my work, I realized that the '10' in Q1 is assumed to have units 'mph'. This balances with the units on the bottom of the fraction, meaning that the 'time' units for t2 will be the same as t1. 

In Q2, because we are multiplying t1 by our initial speed (60mph), the final value for time is ALSO in hours. This yields my original equation for t2. However, when I convert this to minutes, I see that we still get the same coefficients. This was confusing! 

Finally, I realized that because our initial speed is 60mph, 25 min = 25 miles. This means that when we are travelling at 60mph, the magnitude of minutes that passes is equal to the magnitude of miles we travel. We are travelling at 1mile/min. This explains why we have identical coefficients in both equations. When we re-write the Q2 formula to be any unit of time other than minutes, the coefficients will be different. 

Interesting twist!

Non-Curricular Lesson Plan - Jung

In my experience writing lesson plans, I generally focus almost entirely on context. Although I was able to organize, in one page, all of the content I thought I would want to know while giving the lesson, I felt as though I would be detracting from it if I tried to fit information on the FPPL, Core Competencies, and Curricular Competencies. As such, I thought it best to have a separate page for this information. In my mind, I would print a single sheet, double-sided. 

Maybe this is too much? I hope this layout encourages me to give appropriate consideration to the FPPL and how they each tie into any one lesson. 

Lastly, I designed this template with math courses in mind. The sections for Curricular Competencies will be populated appropriately, depending on the Subject being taught. 

Art Project Reflection

 Our art project expanded on Katelyn Owen’s “Her,” a piece centred around the Fano Plane (or PG(2, 2), where “PG” stands for “projective geometry”).  Owen's work features paintings on each edge of the Fano Plane, with each edge representing a theme from her life.  Each of the seven art pieces corresponds to the intersection of three themes, encouraging introspection and the exploration of connections that might not have been considered without the geometric structure of the Fano Plane.  We took this concept further, both artistically and mathematically, by working with PG(3, 2), the three-dimensional extension of the Fano Plane.  This allowed us to explore 35 themes through 15 art pieces, each representing the intersection of seven themes.  We also categorized these themes by colour to simplify brainstorming and connection-forming, though this symmetrical and aesthetically pleasing approach isn’t necessary for constructing PG(3, 2).

My experience with this project reminded me of being an undergraduate, researching mathematics that initially felt out of reach. That being said, once I was able to re-familiarize myself with the basics of projective geometry, the research I did surrounding the Fano Plan was exciting. I found myself excited to convey the beautiful symmetries in this structure, though I I’m unsure if I was successful. The longer I spent time with it, the more my appreciation grew. I saw tremendous potential in leveraging this structure to explore many areas of life – political structures, math curriculum, introspection, and everything in between. Despite my excitement, I’m not sure how well I was able to present the deep symmetry of this structure to the class. I personally take a long time to process definitions in mathematics, and I’m unsure as to how well I explained the axioms in our short presentation. I hope to find better techniques in the future.

As a teacher, I fully expect to execute some variation of this project. However, I plan to place much more importance on ensure the art itself is meaningful to those who are working with/creating it. It is clear that there is value when we project mathematics through a creative lens, but unless you have a feeling for the mathematics, the art feels abstract and out of context. For example, when I initially saw Owen’s ‘Her’, although the symmetry of the Fano Plane was beautiful, the depth of the art was lost on me. It was only after understanding the interconnectedness of the Fano Plane that I was able to appreciate the completeness of the art.

I think given the abstract nature of math, in combination with the vast array of unique creative abilities and hobbies I hope to find in my students, any assigned art project will be highly open-ended. I hope to offer students an opportunity to leverage both art and math so as to better understand something they are already familiar with. I speculate this is most clearly possible after completing the ‘Combinatorics’ unit in Foundations 12.  

Battleground Schools - Article Response

 I wasn’t expecting to have as strong a reaction to this article as I did. This is the first time I have heard of the New Math, and it is the first time that I’ve read any account of the explicit style of mathematics taught to my parent’s generation. Frankly, this article made me angry. 

First, I was very impressed by the type of mathematical education suggested by Dewey. Given the quality of public mathematic education in North America for the past 60-ish years, I was surprised to learn that in the early 20^th century, Dewey advocated for unpredictable classrooms driven by student autonomy. Mathematics is returning to this style, demonstrating how modern Dewey was in his thinking. As suggested in this article, I can only speculate that the unbearably intense global politics of the mid-20^th century scared the public away from Dewey’s slower, more authentic approach to education. It makes me wonder what qualities of Dewey's education were infused in this generation of students. It is perhaps noteworthy that many of the young hippies of the 60s would have had parents educated under Dewey's system. Perhaps parents raised to see the value of student autonomy were more inclined to let their children pursue authentic ways of living. This is deeply speculative. 

My mom, born in 1969, was traumatized by her math class. This article gives me the understanding that the education she received was some abomination of ‘New Math’ and neo-liberal politics, completely absent of authentic problem solving, geometry, and critical thinking. For her, mathematics is instrumental calculation. This did not align with her strengths as a student – consequently, she has actively avoided any semblance of mathematics for nearly 50 years. It is terrifying to speculate on the number of elementary teachers who share this disposition.

Perhaps the most sinister quality of New Math mentioned in this article was its attempt to create a global, ‘teacher-proof’ curriculum. As has been discussed, teachers in the modern age do NOT exist as sources of content. It is only by cultivating authentic intellectual relationships with their students that are teachers able to facilitate the transmission of knowledge. A ‘teacher-proof’ curriculum blatantly goes against this pedagogy. If a 'teacher-proof' curriculum was possible, there would be no (human) teachers in the 21st century. 

Finally, I have a much greater appreciation for the role the NCTM has played in the current state of mathematics. The NCTM’s prophetic attitude towards standards enabled teachers to articulate their ideals, as well as to preserve those ideals against the static, fundamentalist world view of the religious right. Upon concluding this article, I feel a much greater motivation to improve my understanding of the NCTM standards. In particular, I plan to understand how they align with the BC Curriculum.

Lockhart's Lament - Response

 I am aligned with Lockhart. His lament has helped articulate the ‘wrongness’ that we’ve all felt in high school classrooms for so long. It is tragic that our schools continue to operate in the same way, despite us having writing like this in the world.

I am very drawn to the style of teaching he proposes. I believe that enabling autonomy in students will deeply enrich their learning, relative to what it is now. That we can provide resources and guidance, but ultimately, they will pursue the paths that appear to teach of them. In a separate class, I recently wrote about what qualities I believe describe ‘authentic assessment’. I noted that generally, when authentic art is produced, it is done so without expectation – the artist is not attempting to mimic anything. I speculated that authentic assessment should be absent of as many expectations as possible. I feel this would align very well with Lockhart’s conceptions of math class, since students would be pursuing their own problems without some concrete destination. This feels Authentic.

Lockhart speaks about giving students the opportunity to discover math. This resonates hard – as I’ve mentioned earlier, a book on Greek Mathematics provided me with this experience first-hand. If we could cultivate these experiences in students, math class would become a favorite for many.

Lastly, I want to note when Lockhart states: “Teaching is not about information. It is about having an honest intellectual relationship with your students”. This sentence gives me confidence that Lockhart’s ideas are grounded in humility. He understands that a teacher’s role is to engage with students, listening and guiding them to math’s natural wonders. What’s more is he offers (general) methods of accomplishing this - rich questions, student autonomy, and minimal expectations. These are qualities I hope to instill in my classroom.

The Locker Problem

 Teacher bird here. 

I'm not sure why, but when I first approached this problem I was expecting primes to play a role. Not sure why this was, but I initially thought both primes and squares would be open, until I saw that 5 was closed, and realized I had no justification for primes being open...

Initially I wrote out a sequence of 10 to get a feel for things. I only wrote out the changes in position, and then a final line which showed the position of the first 10 lockers after 10 students. I didn't notice that it was only the squares at this point, but had a feeling it was related to the factors of numbers. I wrote out the factors for 60 and was reminded that they come in pairs. This helped me realize that every locker would be closed except for those with an odd number of factors. Since only squares have an odd number of factors, the problem was solved. 

I love this question, though I'm not sure if the ceiling is as high as I originally thought. Once someone understands the algorithm, all higher cases are trivial. This might be good for a Math 9 class. If was to present this to a Math 9 class, I would ask questions in this order:

Given the problem, how many lockers are open if there are:

a) 10 Lockers

b) 50 Lockers

c) 100 Lockers

d) N Lockers

I would make sure to only show them one part of the question at a time so that they are forced to realize brute force will not work because of their own experience with the question, NOT because they are anticipating a harder case later on. 

Favorite and Least Favorite Math Teachers

 In truth, this exercise had me believing that I didn’t have a favorite math teacher. That being said, there are several moments in time which I know cultivated my joy for mathematics.

1.       Solving countless 2D and 3D spatial puzzles in video games as a child.

2.       My Grade 9 math teacher, Mrs. W, who regularly offered us geometry puzzles to solve (with and without the use of trig. Functions).

3.       As an engineering undergraduate, reading a book about the history of Greek Geometry. This book presented the history in a mostly chronological way and offered the reader many opportunities to prove Lemma’s for themselves using tools derived in the book.

Of these three sources, I may point to the book of Greek Geometry as my greatest teacher.

As for my worst math teacher, Mr. R from Pre-Calculus 12 comes to mind. He read directly from the textbook and frequently was shouting at students who repeatedly got wrong answers (generally after repeatedly receiving the same explanation). There was a silent tension in the class, though it often broke with laughter from the absurdity of it all. The knowledgeable students were frequently pointing out mistakes during lectures, and I do not recall feeling as though he could help me learn anything. Fortunately, his tests we’re predictable, meaning my final grade was not harmed (something I was deeply concerned about at the time). I later learned that Mr. R was an ex-NFL player. Hopefully helmets are better now.  

Reflecting further on my greatest teachers, I see that they are actually activities which cultivated a sense joy in mathematics and problem solving. In particular with the book of Greek Geometry, I felt that the incorporation with history offered authentic motivation, which in turn gave me the feeling that I was personally discovering the math. I consider this to be my richest learning experience and is one I hope to recreate for future students.