Math Art: Margaret Kepner

Our project was on Margaret Kepner’s Number Sequences on a Hexagonal Grid. We recreated her pattern with Happy numbers, Triangular numbers, Prime numbers and Fibonacci numbers on a smaller grid. As an extension, we tried these (and other) numbers on other grids (ex. diamond grid) and noticed interesting patterns that occurred. One such pattern was the diagonal pattern when colouring in perfect squares on a square grid, which we think is a fun way to introduce and visually show perfect squares to students. Thus, this art project would be a fun and creative way to introduce various number sequences (ex. factors/multiples, perfect squares, etc.) to students.

The project was quite fun to work on because it was an opportunity to test/experiment with different grid designs and types of numbers. I learned a lot of new things while working on the project, such as many new types of numbers, including "Happy Numbers," that I did not know about previously. I got really into the project and became really excited when I discovered the diagonal pattern created by perfect squares on a square/diamond grid. I think a lot of activities like these are most fun when you don't know what to expect and end up with something interesting that makes you want to find out why it happened. 

I think this project was very insightful; it gave many unique and interesting ideas and activities that can be used in a classroom to either introduce a topic or explore a concept further. When someone mentions math, people think of numbers and calculations, but I think using these activities with students would opens up their perspectives to see that there's more to it and unique applications. I also think it will make students more intrinsically motivated (by making it a fun activity) while learning something new at the same time (as they do research/play around with the ideas). Math and art was a combination I didn't expect but it turned out more interesting than I thought, and I think math and other subject areas could also be combined to bring more unique perspectives. For example, I've taken a math history course during my undergrad and I found it fascinating and I do want to incorporate some of those ideas (such as the origins of the number 0 and proof of Pythagorean theorem) in my own math classrooms too. 

The Dishes Problem

My though process (without algebra): 

At first, I took an initial 4 people and considered how many dishes they'd get in total. This came out to be 1 meat dish (1 for each 4), 1 and 1/3 broth dishes (1 for each 3) and 2 rice dishes which altogether was 4 and 1/3 dishes for 4 people. Since it's more tedious to work with fractions, I multiplied everything by 3 to get 13 dishes for 12 people. From there, I simply found multiples of 13 until I reached 65 dishes total. Since 13x5 = 65, then 12x5 = 60. Thus, there were 60 guests. 

I think it is a good idea to include problems from different cultures because lots of people grow up with a very western view of math, but math didn't just come from only Greece/Europe. Other countries and cultures (such as the Chinese, Indians, and Muslims) were also developing their own mathematics and mathematical techniques. 

I thought the story was nice to have because it makes the imagery much more concrete/vivid than if the problem were to be replaced with more abstract shapes or objects. Plus, I believe it's more interesting and engaging if the topic has some relevance to the students in some way, so that math problems don't feel "lifeless" with purely numbers and symbols. 

Student Letters

https://pxhere.com/en/photo/1093456

Hi Ms. Lam! 

How have you been? It's been a long time, but I was looking back and remembered how memorable your class was for me. The assignments and activities were really fun and interactive, and I really liked how you put emphasis on making the lessons have more relevance to the real world. I also really loved the weekly math puzzles/problems we had! They really made me think really hard, and it was super satisfying when my friends and I got the answer. I also appreciate how supportive you were to our learning and a fun person to talk to!

Cheers!

Student A

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Hi Ms. Lam,

I wanted to let you know I appreciate you doing your best to try new things and make math more interesting, but I didn't enjoy your class much because of how disorganized everything felt. Your activities and lessons were very different from other math classes I've had, but I felt I wasn't really learning anything in class, or I couldn't figure out the purpose of the lesson. Furthermore, I think it would have been better if you had exerted more control over the class and were more strict with the students misbehaving, because it made class uncomfortable when they were being disruptive and loud.

Sincerely,

Student B

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My hope as a teacher stems a lot from teachers I've grown up with or volunteered for (they're my role models!) My wish is to make math more fun and interactive by incorporating math puzzles/problems to develop students' critical thinking and logical reasoning. I also wish to be able to support my students as much as possible through providing thoughtful, honest feedback and additional resources/time after classes or during breaks. 

My biggest worry is lacking the credibility/respect as a teacher to take control of my classroom, given teenagers can get rowdy because of many factors. I tend to be on the more passive/timid side, so I worry I won't be able to assert myself well in such situations, but I'm aiming to work on that and classes I'm taking have very useful ideas. I also worry about students not liking the way I decide to teach my class (I want to try making lessons more unique and interactive), but perhaps I can get feedback from students themselves (they are the ones experiencing it firsthand) to make adjustments, or other colleagues. 

Lockhart's Lament

 So far, out of all the articles I've read, I really resonated with Lockhart's article the most and there are many points I agree with. 

There was one particular statement that I both agree and disagree with: (p.5)

"By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject."

I've always thought the monotonous lecture-y type lessons weren't that interesting and I wholeheartedly agree that being told a particular strategy/formula to solve problems simply becomes purely memorization/"plug-and-chug" without truly knowing what it is they're doing. I like logic puzzles, not because I can get the answer (I don't always get it!) but because of the journey to get to the answer --- it makes me wrack my brain to try different ways to approach the problem, be creative, and if I do get the answer, there's so much more satisfaction to it because I know it's my own hard work that led to it, not a ready-made algorithm given at the start to solve the problem for me. The same idea can be applied to mathematics. 

However, at the same time, I don't think there will be absolutely no engagement with the subject if a method/formula is given to solving a problem. What is stopping a student from working backwards or trying to explore for themselves how that formula/method is produced? Is that not a form of engagement and critical thinking with the subject? 

I can see resemblances between Lockhart's insistence on exploring the processes behind mathematics and Skemp's argument about relational maths (the "why") being very similar. However, what I found interesting was that Lockhart preferred a more unique type of relational math where students came up with their own reasonings compared to Skemp's teaching students relational maths which Lockhart was actually against (rigid proofs or so he called them) (see High School Geometry: Instrument of the Devil from Lockhart's Lament). 

Locker Problem Class Discussion

 It was very interesting and insightful to see the similar and different ways everyone approached and obtained the answer to the locker problem. I think it really shows that when solving math problems, there is no one single way to get to the answer --- mathematical thinking and problem solving can take on so many shapes and forms and I think there's a beauty to that. ✨

Favourite and Least Favourite Math Teachers

I still keep in touch with Ms. Chiang, my old high school math teacher. She's very friendly, supportive and caring. Her actions speak a lot on how much she cares for her students: she previously reached out to me to help tutor a student in her class because they were falling behind. I've also gone to her for advice and even talked to her about my undergrad math courses. She'd always get curious about the homework I'm working on and even want to try them herself which in turn influences me to try hard on them as well. That's why she's one of my biggest role models as a teacher; I hope to be as caring, supportive and passionate (about math) as she was to her students. 

Some characteristics of my least favourite math teacher would likely be the ones that were very lecture-y and monotonous in their teaching. I could tell they knew the material, but it was very boring. Thus, one of my goals as a math teacher is to make math lessons more fun, interactive and meaningful so that students understand there is meaning to what they're learning. 

The Locker Problem

My rough work for the locker problem.

 Explanation of my work:

I first tested a small sample set of 5 lockers and 5 students to see what the result would look like. Then I increased it to 6 to test to see if odd/even number of students/lockers made any difference (it did not). I noticed the C-O-O pattern (C for closed, O for open) was repeating so to test that, I tried a larger sample of 10. It did not repeat the C-O-O pattern but at this point, I realized that factors must be playing a role and made a list of factors for 1 to 10. I noticed that an odd number of factors closed lockers while even opened them. Finally, this led me to figuring out that odd number of factors occur on perfect squares because there is one "non-paired" factor (the squared factor). Thus, the lockers corresponding to perfect squares would be closed while the rest would be open.

The locker problem was really fun and made me use various skills/concepts, such as recognizing patterns and using knowledge about factors, to solve it. 

Exit Slip: Skemp Article Class Discussion

Photo taken by Jessica Williams
Images drawn were more of when our group discussed
our experiences with instrumental vs relational math.

Personally, I think that relational math should be taught first because I tend to see many students whom once they learn the "easy" way (instrumental), they lose motivation to learn the underlying reason why (relational). Letting them explore the relational way first encourages them to do some critical thinking for themselves. However, relational may be difficult for some topics, so it's sometimes easier to start with instrumental, but eventually both ways should be touched upon. 

I hope students will have fun and come to appreciate that there is meaning to the math they learn and take interest in it, or at least not come out of class hating it. After the class discussion, I think we can work towards this goal by expressing a positive attitude and not giving up on our students. If we, as math teachers, don't believe in them, how can they believe in themselves to improve in math?  

Richard Skemp on Relational and Instrumental Mathematics

(p. 2-3)

The examples of instrumental explanations used in mathematics made me pause and re-read them to try to grasp what instrumental explanations were and find how they were being used in the examples because this concept seemed so new to me at this point in the article.

(p. 4)

"All they want is some kind of rule for getting the answer. As soon as this is reached, they latch on to it and ignore the rest." This sentence was an epiphany for me because I finally grasped what Skemp was implying by "instrumental explanations." It also made me recall my experience tutoring students and when they simply wanted to know the formula to plug in values without understanding how those formulas came about and why they work.

(p. 13)

"between where I was staying and the office of the colleague with whom I was working." I read this twice and thought I missed a line and double backed to check only to realize it was indeed repeated twice. (a typo? or was this purposeful?)

My thoughts on the issue Skemp raises:

As the one teaching, I've gained an appreciation of "why" these instrumental methods work, and I agree with Skemp that relational understanding is more beneficial in the long-term. That's not to say instrumental math isn't good, but it should be used in moderation and not overly reliant (similarly with calculators!!!). I've witnessed the issue of using instrumental explanations a lot when tutoring. For example, a student sees an example of a right angle triangle and knows the area is 1/2×b×h, but when given an isosceles triangle, they would become confused because to them it's a different shape. So although instrumental math is easier to grasp, I believe it limits students' abilities to think outside the box, make connections and extend their knowledge further.

However, I realize there are some difficulties with achieving relational mathematics. Finding ways to not make this approach boring, frustrating or time-consuming will be difficult for some topics. Furthermore, many students learn these instrumental methods at an early age, so it will be very difficult to change their way of thinking, especially making them relearn something they believe they already understand but in actuality only memorized a formula. Perhaps the teacher shouldn't give out any formulas so quickly and let students try to brainstorm it themselves. 

I also wonder if other subject areas also have this issue of instrumental vs relational instruction?

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