Oct 4 Math Art Project Follow-up Reflection

 

    My group decided to analyze and extend upon the artwork "Between the Devil and the Deep Blue Sea" by Gauthier Cerf, a series of hexagons embedded within each other, where their side lengths correspond to the Fibonacci numbers. Different tasks were broken down and we discussed ways in which such an artwork may be used as an active way to engage students in understanding how the Fibonacci numbers are related to each other through its recursive definition. We extended the Fibonacci pattern using the square, that generated the golden spiral (which is seen in many natural objects) and created a colourful aesthetically pleasing butterfly (credits to Esther). Linking patterns to other mathematical ideas visually allows students to see how some fomulae came to be/makes more sense, such as explaining an infinite geometric series with equilateral triangles.

    This project was initially very entertaining to do because I am intrigued by how Fibonacci numbers are implemented in beautiful artworks. The work distribution was also fair and figuring out who does what wasn't difficult. I loved the analysis by the artist on their artwork and I had to think deep about how it was constructed. Taking the time to reconstruct the artwork was also a fun process, since I love building big things from tiny things ground up. In this case it was the construction of many hexagons with two hexagon blocks. However in hindsight, one hexagon block was sufficient. Tracing objections, joining line segments with a straight edge, and cutting pieces are all activities that I don't do regularly. I also spent a lot of time preparing for my math extension part with the Golden ratio. I thought it was a good way for, say, older high school students to review/utilize some math topics, to help them use what they learned like fractions/ratios, pattern recognition, roots of quadratics, system of linear equations, and limits. Unfortunately, I got to talk about none of that due to time constraints which was frustrating as I put a fair amount of work into it.

    There was a lot of advice I took in from presenting this project. For example, preparing more materials in advance just in case some papers weren't cut to be big enough. It was fortunate to have the art supply room right next to our door, but that won't always be the case for all schools. Giving clear instructions on creating an artwork is crucial too. Many students will have a hard time following the steps if they just hear "this" and "that". It was mostly a stress issue from time constraints, but it was also my first time giving a formal presentation in front of a classroom for over four years. I have been tutoring and TA'ing, but those were less formal. I came into this program to improve my public speaking skills as well, and having some critiques during my UBC-presentations humbles me as I'm able to learn and improve for future presentations/practicum teachings. Teachers should also be taking advice and growing via feedback. For this artwork, I may construct it in front of the class (via screen projection) to let the students see how it relates to the Fibonacci sequence, but I probably will just let them try it out on their own time at home, since time is so limited and there is a rigorous curriculum to go through. I will mostly connect the Fibonacci numbers with the Golden ratio with the set of notes I have written. Visual proofs are very intuitive for students, and I use a lot of geometry to prove certain trigonometric properties, like why exactly is sin(30°) = 1/2. Physics is one direct way to apply math, and in a grade 12 calculus class on the integration topic, teachers can show students how their science 10 kinematics formulae are derived (from constant acceleration) using integration.

Sept 25 The Dishes Problem

The number of guests must be divisible by 2, 3, 4. You take the number of rice dishes (r) and double it and that gives you the total number of guests. You also get the total number of guests if you triple the number of broth dishes (b) or quadruple the number of meat dishes (m). So 2r=3b=4m. There is also the information that the sum of the dishes is 65, so r+b+m = 65. If we relate the broth and meat to rice, we get b = 2r/3 and m = r/2. Substituting those into the r+b+m expression, we get r + 2r/3 + r/2 = 65 = r(1+ 2/3 + 1/2) = r(6+4+3)/6 = 13r/6 ⇒r = 65(6/13) = 5(6) = 30 ⇒ total number of guests = 2r = 2(30) = 60.

Instead of using latin variables to set up an equation, one can use pictorial diagrams of the food dish that is being described in the question. One may also use words to describe the situation instead of using written equations. However there is an even better method to solve this problem without a system of equations that involves finding the total amount of dishes directly. If we know that 2r = 3b = 4m = n, where n is the total number of dishes, we can say n/2 + n/3 + n/4 = 65 = n(1/2 + 1/3 + 1/4) = n(6+4+3)/12 = 13n/12 ⇒ n = 65(12/13) = 5(12) = 60. This method is much more efficient. 

I believe using examples of puzzles from other cultures will spring interests into the students, so that they are working with something new. They may also feel better to see that mathematicians from their cultures were able to come up with problems as advanced as those studied here. Imagery is helpful for students to better visualize the problem and is more interactive. They're more familiar with whatever interesting character is introduced in the problem, for example a physics question involving R2-D2 catching Luke Skywalker's lightsaber. 

Sept 20 Letters from Future Students

Dear Mr. Zhang, 

I hope you're doing well. It has been ten years since you taught me high school math. I was happy to see you again at the restaurant and greet you briefly. However, I didn't get the chance to have a full conversation with you. Your math teachings were very helpful for me, since they prepared me well for the difficulties in university. Your rigor and proofs were genuine and didn't take too much time. They were intuitive and made me understand the topics on a deeper level. I loved your reminders that university does things a little differently. My friends from other classes were so shocked, but I wasn't! Your lessons were stepping stones to enter post secondary. Some topics are hard to grasp initially, but you broke it down very nicely. Thank you for your explanations!. . . 

Dear Mr. Zhang, 

I was happy to see you at the teachers conference. I'm teaching English/history right now. It was nice having a small chat with you regarding some of the pedagogical differences we have regarding math. To start, I'm not strong with math and not fluent. Sometimes your proofs are too long to follow, and I couldn't keep up. Your lessons on mathematical/symbolic logic was also not necessary, though it may be interesting to some. People like me are more interested in how and in which situations to use certain formulae. The tangents you went into about some cautions to take for the future weren't beneficial as I didn't take math afterwards. . . 

I'm hopeful that I can explain topics on a deeper level but I hope to not digress too far and focus on the main topics. The balance on how much rigor I should implement is something I have to keep in mind as well. Sometimes I may not have to introduce some concepts since it may not be useful.

Sept 20 Lockhart Article Response

I agree with Lockhart in that math taught in schools right now focuses too much on rote procedures and facts, an algorithm of sorts. Student's don't ask "why" or even care to ask the underlying principles behind such compacted formulae. A lot of it comes from very beautiful proofs that teachers don't bother spending time on, hence diminishing possible opportunities that gives rise to some intuition for the students. However I do think rigor is a necessary component in mathematics as it is the foundation block that builds up other fields around us. Treating it too much like an art is too flowery. While teacher enthusiasm and creativity is important, one can incorporate context to topics being taught, engaging in fun discussions while maintaining the rigor so that we know math is math and music is music. We can allow math and art to rhyme, but there is still a clear distinction between them. Compared with Skempt (who was relatively more neutral on this issue), Lockhart greatly emphasizes on the need for a more open, curiosity-driven approach, where students and teachers alike can experience the joy of mathematical discovery. Skempt gives an outline of relational vs. instrumental approaches to teaching. I believe Lockhart greatly emphasizes on the relational aspect, though not in the context of rigor but in creativity and personal exploration.

Sept 18 Favourite and least favourite math teachers

My favourite math teacher is my grade 12 math teacher. He was very jolly and had a huge sense of responsibility. Being the head of athletics as well as teaching a computer science class, juggling between all of these tasks and maintaining a good teaching attitude made many people look up to him. He explained the math problems using different colours on the smartboard, so it captivates people's attention. Telling stories as well casually in class and jokingly says "thank you, I was just testing you" whenever someone points out his mistake. When things did get out of hand he would tell us to just do what we want, sit down, and we would realize we misbehaved over the limit. He comes back up and explains he genuinely cares for our learning, but if we don't give him the chance to speak/present, then we might as well do it ourselves - which isn't feasible. But this rarely happened and the class was in good shape most of the time. So he was a very reasonable person overall, caring for the well-being and the learning engagement of his students, which inspired me to continue in his work. My least favourite math teacher was a math professor. For many profs, they are very smart in their own eyes because they have had many years of training and accumulated lots of knowledge. He would speak demeaningly to students. When we struggled, he wouldn't really explain the concept to us, but just tells us to refer to the textbook he wrote. Overall he wasn't helpful and didn't build strong relationships with the students. This is a red flag for me since having good communication with students is a good start for further engagement and exploration for them.

Sept 13 chalk board photo

 

Sept 18 The Locker Problem (Updated Nov 6)

This problem was very intriguing, as I never really enjoyed these sorts of math puzzle problems. But after four years of doing math in university and taking many discrete math courses, this problem can't be harder than my math assignments right? To begin, I initially wanted to shorten this problem from 1000 to 10 and brute force my way through the 10 students. However brute forcing by using a smaller sample and then later extending it to a larger sample misses the point of being able to solve this problem without any brute force. So I thought about it a bit more and came up with a solution. Initially every locker is open. For each locker number n ranging from 1 to 1000, the question is how many times the state of each locker is changed after all 1000 students had their turn. That turns out to be the number of positive factors n has. For example, for locker 12, the students who will affect it will be student 1, 2, 3, 4, 6, and 12. So that will be 6 changes. Since 6 is even, the locker will in the end be open, since the doors alternate from open and shut, and here we initially started at open. Students with numbers greater than 12 will not affect locker 12 as they'll start off with locker numbers bigger than 12. So this method can be done with every number n from 1 to 1000. If the number of positive factors n has is even, locker n will be open in the end, and if odd, locker n will be shut in the end. I could go in more detail on how to find the number of positive factors n has, but not going into too much detail right now (the explanation requires prime factorization and the fundamental counting principle). In the end, the lockers with numbers that are perfect squares will be closed as when you write out the factors, the square root of that perfect square will be repeated twice. However that counts as one factor, which means in the end a perfect square will have an odd number of factors. Thus it will be closed. For example, the number 36 has factors 1,2, 3, 4, 6 (6 only counted once), 9,12,18,36. 36 has 9 (odd) distinct factors so it will be closed in the end.

Also interesting to note: my SA does math riddles at the start of her lessons, and on my second day of practicum she showed the class this riddle from Ted Talk.

Sept 11 Relational Understanding and Instrumental Understanding

    The words faux amis really resonated with me. Mathematics uses a lot of English words for topics that are nowhere close to the main definitions. When I took courses in Ring theory, in my head I always pictured actual golden rings. Whenever fields and ideals are mentioned, I picture a green flat field and a perfect suitable concept. Radicals too aren’t advocating complete political/social change. They’re just surds. I believe, however, there’s a reason for those words to be chosen. Another mention that stood out to me was how rigorous a student should understand what they’re learning. Personally I don’t enjoy just throwing out formulae. I always try to give a proof so that the students may see how the derivation utilizes previous/current/new (but not out of their scope) knowledge. Stating A = L×W [for a rectangle] without explaining the definition of area (how much two dimensional space is taken up by the object) won’t help students apply the concept of area to other non-rectangular shapes like circles, triangles, or area under graphs. Finally, I collect from the latter parts of the article how much rigor should be taught. It's important to have a balance between instrumental and relational learning. Personally I encourage students to ask "why" and have an intuition of what they're learning, without just glimpsing over statements taken for granted.

    I did enjoy reading this article since it addresses a problem I talked about with a smart junior back in high school. How much emphasis is needed on rigor in the curriculum? Where is that balance between proving statements and throwing them out at the students? Skemp gives pros and cons to both instrumental and relational learning. At times students don't necessarily understand the real meaning behind a concept/formula. Other times students will try to focus too much attention on the background derivation than the equation's formal usage in everyday life. I lean more towards explaining topics on a deeper level, proving some things, but not everything. At times a lot of the background knowledge comes from what is already taught in the current unit, so it is just another form of applying/extending. Skemp's goal is for the students to problem solve on their own, even when somewhere in the middle of the process things don't go their way. That was how I was taught and I hope to present myself in a similar way.