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.