Entrance Slip # 7: "Flow”, engagement and the Thinking Classroom

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Throughout Mihalyi Csikszenmihalyi's talk, I was reflecting on moments where I have felt "flow." I found the chart introduced around 15 minutes into the video accurate because I have felt those variations before, and I have witnessed their manifestation in students as well. I have often witnessed students disengage when a problem presented was above their ability, or at least perceived to be out of their reach. Their unwillingness to try shows they are in the zones of worry, anxiety, and apathy; the work is unattainable so they choose not to try or become too overwhelmed to start. One strategy I believe we as teachers could implement to move past this is by making math more tangible for students. That is, we could introduce manipulatives where possible so that students can work to solve problems with concrete visualizations. In her talk during the BCAMT conference, Kathleen Jalalpour spoke about the importance of starting out with physical objects, moving to pictorial representations, and then shifting to formulas and pattern recognition, which starts with the students' own understanding instead of memorization. She spoke of how some students may be able to memorize well and find the patterns themselves while others struggle to find the patterns, are told to just memorize a "shortcut," and never really understand so they declare math too difficult and become opposed to the subject as a whole. While the talk was geared towards younger grades, Kathleen spoke about the usefulness of tools such as base ten blocks at a high school level as well. I will admit I was not a fan of the blocks from my own school experience, but I became one by the end of the talk. Of course, Kathleen also mentioned how this transition from concrete to pictorial to abstract can be a long process, but I think if it's possible to do, it will be worth it in the long run for students who are stuck in a cycle of not being able to engage with material, and hopefully work towards control/interest, and eventually "flow," as seen on Csikszenmihalyi's chart.

Another way in which I have seen Csikszenmihalyi's chart in action is the range from boredom to arousal or interest. I remember having a conversation in which the person I was speaking with said something along the lines of, "I find that you math people get so caught up in a concept that you can't fathom how someone would not think it was cool. You really need to introduce things to your students in a way that makes them understand why they should care and why it's cool." In that way, I believe the introduction of a concept is quite important. The introduction should include linking to previous content and concepts so that the student understands which skills are required and can shift from a worry/anxiety/apathy position to interest/boredom. Additionally, the introduction should convey why the students should care so they are shifted from boredom to interest. Providing historical background and real-world applications are two strategies that I believe would help in achieving this. After setting up this atmosphere that could lead to flow, we also need to provide appropriate problems and extensions just outside the students' current abilities so they continue to engage and enter that state of flow as they work. 

In Peter Liljedahl's slides, he discusses some more strategies to use when a lesson begins to steer away from achieving flow. I have previously encountered giving hints or extensions based on if the students were having trouble or finished early, respectively. However, I hadn't thought about rearranging groups. I'm not sure I necessarily agree with this method because I think it could lead to more disruption and potential feelings of inadequacy if students are say, paired with a group that has higher skills than them. They may feel that they're "not smart enough" to solve the problem on their own. Instead, I think I would have different expectations of different groups and guide them towards an attainable goal for their current level. There may still be feelings of inadequacy if the students compare themselves to what other groups were doing but hopefully I would be able to keep them preoccupied with the task at hand by making it interesting enough to maintain their attention.

Homework: Geometric/Numerical Puzzle

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In my process of solving, I first tried drawing simply cases of 2 points, 3 points etc. so I could visualize the problem. Then, I realized that the point directly opposite the last one was half, so for the 30 case, all I had to do was write 1-15, then 16-30 underneath it and find the pair to 7, which is 22. Generally, this would be T/2 + N for N =< T/2, and N - T/2 for N > T/2, where T is the total number of points, and N is the number you're trying to match with its partner.

I tried to extend this with a square: A square is drawn with 32 points (numbered clockwise) spread out so that each side's points are equally distributed across the line. If the square was folded so that point 1 and 17 overlapped, what point would overlap with 3?

The solution I got was 15, but I wonder if there would be misinterpretations. That's why I said the square was folded in half in the first place - if I tried to say the point across a given point, there could be a lot of misinterpretation. Also, the numbers on the vertices would have multiple answers (it could be 1 (across the diagonal), 2 (the vertices on either end of 90 degrees),  or more because "across" isn't very specific). 

When trying to make an impossible answer, I chose the circle with 3 points and asking what number was across 3. None of these points should have a partner across their diagonal so there would be no solution. I honestly don't like impossible puzzles unless the student knows in advance that that is a possibility (e.g. when solving system of equations, students are prepared for and can confidently check if there is no solution). The benefit of impossible puzzles is that it builds confidence and deeper understanding in the students to be able to stand by their answer and explain why it is impossible. 

I would say a question is geometric if you're solving for a geometric quantity (e.g. measurements, angles etc.). In this case, even though there are geometric shapes and we're solving for the numbered point, it really has more to do with the organization of numbers than the shape itself.

Entrance Slip # 6: The new BC curriculum & secondary math course pathways structure

 While reading the glossary, I was surprised to find that there were so many different inquiry-based approaches. Before this, I has assumed inquiry-based approaches included things like research essays, projects that students would be guided towards in terms of questions posed and of course allowed to ask the teacher for help, but otherwise left to do the work on their own. I was surprised to see Project-Based learning was actually more structured than this to ensure students are learning the class content. The case method also interested me because, although it is about the individual learner, it is also an exercise of group work and realizing not everyone will agree 100% all the time. Design-based learning is an interesting distinction from project or scientific inquiry (although I'm sure they could overlap) and I particularly liked the idea of finding the restraints in a design. For example, maybe students could create different representations of Pi and discuss the limitations they encountered (one idea I had was maybe playing music so that each note is held for 3.14 seconds, for example, and discuss the limitations in accuracy or with technology that they encountered. In this case, the "product" would be the song. I wonder, if the song was 3.14 minutes long, would it count as a fractal as well?).

Mapping out the pathways of math was more difficult than I thought it would be. I thought back to the presentation I was given in high school for the different paths we could take and I remember there were allowances even within the chart they showed us so I took a similar approach of creating the chart but keeping in mind that further explanation would be needed. For example, I decided to have the Workplace Math 10 route lead to Geometry 12 and Computer Science 11 as well as Workplace Math 11 and History of Math 11 because I believe students would still be able to succeed in those courses with their background. I drew on personal experiences working in Workplace Math 11/12 as well as my experience taking Computer Science 11 and looking at the Workplace Math 10 curriculum when doing so. I took the recommendation to make Statistics 12 follow FOM11 or PC11 based on other school recommendations found online and by looking at the curricula for Workplace Math 11 and Statistics 12. If the school offered this option, it would be a good idea to mention to students they could inquire about Stats 12 through teacher recommendation if they were on the Workplace Math path. In terms of the three main pathways (Foundations, Pre-Calc, and Workplace math), if a student decided to switch pathways, I'm sure they could work out the best course of action with their school counselor.

Entrance Slip # 5: Elliot Eisner on Three curricula all schools teach

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 Eisner points out how the more rigid hierarchical structure of school likely continues to exist because it's good training for those who end up in hierarchical workplaces (p. 91). In this case, I believe the idea to create initiative in students through allowing them to map their own learning (p. 89) is actually more useful than this hierarchical system. A complaint I often hear from employers in my hometown is how a lot of their employees "slack off" for the rest of their shift after they finish the list of tasks given to them. I think if people have been taught initiative, they are more likely to self-motivate and look for ways to improve their workplace themselves. This is just part of the equation, considering there could be a myriad of reasons why an employee isn't working, but I think it would be more useful, not only for the job, but for the employee themselves. Taking initiative requires being present and, in my opinion, if you're working 35-40 hours a week, it's better to be using that time actively engaged and thinking, rather than shutting your brain off and following orders until there are none left to follow. 

Another interesting idea in Eisner's article was that of how much time is allotted to each subject and the choices of subjects available to students as core subjects or electives. I was always aware of this to some degree but I just assumed it was a convention we followed and that we could make choices once we left high school. However, I hadn't thought about how these decisions fed public perception of each subject. I also imagined what it would be like if everyone learned their basic language, math, science etc. skills in elementary/middle school, and was allowed to start exploring subjects they preferred in high school. Would that start to shift public perception or would it just switch to people viewing the basic skills classes as superior? Also, as far as I understand it, electives can be cancelled if there isn't enough enrollment. I thought about all courses effectively becoming "electives" in high school, and if Math 9 had been cancelled, I would have been devastated. But that's probably the same devastation students who want to pursue arts feel when their classes don't run, yet we've just come to accept that as normal now. As I don't think this system is going to come to be for a long time, if ever, I thought about how I could work within the system. I think it would be interesting to see if I could cater lessons in my own classroom to try to allow students to explore their own personal fields of interest in a way that still fulfills the curriculum goals of the subject. 

Although the newer curriculum includes ideas of incorporating social emotional learning, Indigenous knowledge, expanding knowledge, and taking risks in discussions, which are all important aspects of learning apart from the subject, they are still presented as add-ons to an existing system that continues to place unconscious weight on some subjects over others, doesn't address the fact that subjects are sorted into logical or emotional categories, and allows for a lot of variation among teachings. To me, it seems like the curriculum allows those who are more traditional to comfortably continue in their style while introducing one or two lessons in each of the boxes they need to check, while at the same time, it allows teachers who want to push past the rigid structure to expand the minds of their students and take perspectives like Eisner's into consideration. Using this freedom, I think it is important that we as teachers emphasize to our students that there is more to math than logic and facts, and that art also has it's advancements and techniques that they haven't necessarily been exposed to. Also, showing the overlap between subjects could be useful in getting this point across.

As a side note, I have recently been trying to learn Kathak dance, which requires intricate technique and precision. Even for basic footwork, it is easy to become lost if you aren't concentrating enough. Because of its mathematical nature (a pattern repeats precisely within a 16-beat cycle), becoming lost throws off the entire dance as it is hard to rejoin unless you are close to a checkpoint.

Lesson Plan

 

Introduction (1-2 minutes):
I would like to start the lesson with a greeting and by asking if anyone in the group has heard of Bhangra. Following this, I will give a brief historical overview and introduction to what we will be doing for the day:

Bhangra is a folk dance that originates from Punjab. It was said to have started out as a dance performed during the harvest festival Vaisakhi and has since changed over time. As is the nature with folk dances, steps aren’t required to be precise like you might see in classical dances such as ballet, and many fusion versions have been emerging in modern times. That being said, keeping in time with the beat does help to make your dance look cleaner so we will be aiming for that.

What I want to introduce today are steps that you can take with you to perform at events or parties during which there is an open dance floor. The steps I have chosen are ones I’ve learned from my dance instructor, Binder Basi, as well as YouTube videos from various dancers over the years.

The Beat (1 minute):
Bhangra music tends to follow a predictable and continuous beat. If anyone is in music, it’s in four-four time. To start, let’s try clapping in beat with the music. When you’re ever in doubt, clapping along is a perfectly acceptable dance step.
Dhol Jageero Da

Before jumping in (1 minute):
Some Bhangra 101 tips to remember while dancing are:
- Try to bounce by going up and down on your tip toes. This will help your energy.
- Try to shrug your shoulders as much as possible. It helps to make you look (and hopefully feel) natural.
- Open your arms as much as possible. This will add a presence to your dance.
- The most important one - Smile!
And finally, remember to have fun. The point of dance is to express yourself and Bhangra is specifically a joyous folk dance so don’t worry too much about the details.

Teaching (interactive; 5-6 minutes):
3 steps (depending on time) demonstration and practice:
Step 1 description:
Snap down to the right, snap down to the left, snap up to the right, snap up to the left.
Step 2 description:
Clap to the left, clap to the right, bring your hands down for two.
Step 3 description:
Swing to the right for two and swing to the left for two.
Try with music: Dhol Jageero Da

Try the steps without me + share links (if we have time):

Older style:
https://www.youtube.com/watch?v=5y0iLOfjdaQ

Traditional and fusion:
https://www.youtube.com/watch?v=vD-LFksC1Nc

Fusion:
https://www.youtube.com/watch?v=S2TDtA0Hk_M

 

Reflection:

Based on comments and my own feelings of the lesson, I thought the biggest problem was that I went a little "off-script" and introduced all three steps instead of stopping at 2 like I had rehearsed. This cut into my question/wrap-up period which would have been a better use of time. I was also unexpectedly scatterbrained when switching between the beat counting and doing the steps. That isn't exactly something I could have planned for but allowing for more wiggle-space would have helped with that too. Other than that, the teaching of the actual steps went well and everyone was more in sync than I would have expected over Zoom! Like Ivan pointed out, it would have been more fun and interactive if I had used videos as well but considering the time, I didn't want to risk technical glitches.

Microteaching Lesson Topic

I would like to talk about and teach basic Bhangra steps that people can use during events or parties. As it is a Punjabi folk dance, it is more applicable to Punjabi events/parties but Punjabi music is often played at other South Asian events so it may be danced there as well (or just used as a workout exercise, especially during COVID times!). 

Example of Bhangra teaching at Canada Day 2016 by Gurdeep Pandher and Manuela:

Entrance Slip #4: Battleground Schools

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I honestly had no idea there was so much controversy over math education before reading this article. The first thing that made me stop was the dichotomies table on page 392. I definitely know of teachers who are a mix of both sides, or on a spectrum in between each option so it's worrying, but not surprising, that opposing sides described each other using extremes. It made me question some of the advice I received when talking to teachers in the past. I didn't realize there were people who took such strong stances but if that is the case, then unconsciously or consciously, teachers would be conveying those stances to their students as well. 

While I agree with all the reasons listed for why math education tends to gravitate to a conservative system, the one that stood out to me was the lack of qualified teachers in some areas (p.393-394). Growing up in a small town, I only had one math teacher who had a background in math. Most others were science or wood shop teachers. That being said, these teachers were all very enthusiastic and made a clear effort to teach us math properly. While I had a good experience in math, the same can't be said for my science experience. I had a teacher who wasn't qualified and repeatedly told us, "The answers are in the textbook," whenever we had a question about the worksheets we were assigned. It wasn't helpful at all and made the entire class feel like a waste of time. I can imagine how quickly students would turn from math if they had the same sort of experience with an unqualified teacher who was not making the effort to learn how to teach it effectively.  

The view that students were being "shortchanged" by teachers "experimenting" was another that stood out to me (p. 399). To a certain degree, I believe all teachers have to take some risks when trying to teach something meaningful. Sometimes those experiments fail miserably but at least the teacher can take that information and improve in the future. In my opinion, sticking to the comforting idea that calculations and right or wrong answers are the only things that exist in the realm of math is the real situation in which students are being shortchanged.

The Dishes Problem

How could you solve this puzzle without algebra (or at least, without the algebra we are used to)?

Because this problem deals with smaller numbers, it would be feasible to list numbers and visually "assign dishes" to the appropriate people. In my case, I listed numbers representing people, circled every second number to represent sharing of rice, circled every third number to represent sharing of broth, and circled every fourth number to represent sharing of meat. Afterwards, I counted all the circles in order until I reached 65 dishes.

Note: 5 and 7 aren't circled - I just got carried away with my circling!

 

In the case with bigger numbers, the LCM could also be used. For example, had the number of dishes been 975, we could use the LCM of 2, 3, and 4, which is 12, to determine the number of people present. The objective here would be to find a multiple of 12 that adding the results of dividing that number by 2, 3, and 4 would give the correct number of dishes. It's similar to our normal algebraic way to solve but more of a guess and check method that focuses on the LCM as a guide to the answer instead of adding fractions, using an unknown as a placeholder, and making explicit use of the overall number of dishes in the calculation like our algebraic method generally would. Here is an example of the LCM/guess and check method with 975:

Does it makes a difference to our students to offer examples, puzzles and histories of mathematics from diverse cultures (or from 'their' cultures!)
I believe it can make a difference to represent cultures if presented correctly. That is, it's not enough to simply give the problem without context. If this was just a problem on an exam or worksheet, most students would likely extract the numbers from the problem and solve it. I myself visualized the dishes but didn't really make the connection to Chinese culture because I ignored the context. To bring the cultural significance and representation to our students' attention, it would be more useful to introduce the problem as one from the 4th century CE in China from the Sunzi Suan Jing (Sun Zi's Mathematical Manual). We could also talk about how the Sunzi Suan Jing has various problems and interesting methods of solving them in case students are interested in looking into it. Having this short introduction would place a higher emphasis on the origin of the problem and potentially engage students more than if they just went straight to solving it. For those of Chinese background, it may have been a nice tidbit to see the problem had reference to their culture, but stopping to recognize its origin would be more meaningful because they could feel more connected to the historical significance and feel a sense of pride as well. I remember when I was in school and saw names like "Raj" and "Padma" in textbooks, I would think it was neat, but would move on quite quickly. It would have been more meaningful to me if the problem was reflective of a real life situation or something more relatable to me so that I could potentially talk more about it with my peers who were not of Indian background and have an opportunity to share my culture with them or read more about my culture's history with others who were interested.

Do the word problem or puzzle story and imagery matter? Do they make a difference to our enjoyment in solving it?

I do think one of the benefits of word problems or puzzle stories is the visualization aspect. For instance, before even thinking about the math, I imagined the plates (not the food on the plates for some reason) being shared among different people. I think it gives a tangible aspect to the question that would be missing in simple "Solve for x" questions. I personally enjoy word problems because of this tangible aspect, but certain word problems can be tricky for the same reason. For example, there was a word problem about a longhouse in one of my math classes in high school. I grew up in an area where everyone knew what a longhouse was either due to our exposure of them through school, because our local museum has a performance longhouse in town, or because they are Indigenous themselves and it's part of their culture. However, if that same problem was given to my cousin in India, she would have to look up what a longhouse was first to be able to visualize it, or simply skip over the context and solve the question. 

Another part that makes word problems or story problems enjoyable is that solving them is like solving a mystery. In this case, how many guests did the chef serve? There is a game-like aspect in solving the question that makes you strive to figure it out just so you can know the answer.