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!