Micro-Teaching Lesson Plan - How Slugs Mate, Oct. 1st

How Slugs Mate - Lesson Plan, Oct 1st. 

1. Lesson objective: To learn about one of the two major ways which slugs mate (the overhanging style)

2. Structure:

    ① Ask students to share any prior knowledge they have about slugs and their mating (1-2 min)

    ② Explain what does "hermaphrodites" means, and the different ways for slugs to reproduce. (1-2 min)

    ③ Show an illustration of overhanging mating of slugs, and ask students to come up with description and interpretation. Then briefly explain the illustration. (2-3 min)

    ④ Explain when and how apophallation happens (1-2 min)

    ⑤  (Optional) Talk about how slugs may control when to fertilise themselves (1 min)

3. Materials: 

    ① Illustration(s) that shows the mating process. (Graphic representation is crucial for this lesson, while students may feel uncomfortable with pictures of slugs. Therefore illustrations are used.)

    ② My passion

Assignment 1 - Personal Reflection

In my past engagement with mathematics, I was mostly focused on the arithmetic side. This project was a great opportunity for me to look into a different facet of math. Designing and creating the artwork with Tiffany and Ross was a lot of fun. In this personal reflection, I would like to go into some thoughts I had during this process. 

As we finished preparing all the triangles we wanted to have on the faces of the flexagon, we were going to glue them onto the stripe. And how we orient these triangles was crucial, since we had to make sure the patterns come together when we fold the stripe into a flexagon. We (mostly Ross) experimented on a marked up flexagon, to figure out the orientation of the patterns in stripe form. The product was the stripe shown in the pictures below. Looking at the rules we found, we were trying to form interpretations about why the patterns has to be in this way (e.g why when the rhombuses point up and down on the stripe, they can form a star once folded into a hexagon). I found this process of seeking rules, and then gaining insights by looking at them was very much like what we do in math. 

The next step was to glue the triangles onto the stripe, and I was really concerned about the spacing between each triangles. The first flexagon we made, was made of thick card paper, and turned out pretty tight and difficult to flip (until we beaten it up). In a video I watched on Youtube about making a dodecahexaflexagon (so embedding 12 faces into a hexaflexagon), considerable spaces were placed between each triangles, to allow covering up the thickness of embedded layers. Having layers of fabrics glued together would similarly create some thickness, hence I proposed to space the triangles as we glue them. We chose this space to be somewhere around 1 cm (without any effort to calculate or justify the number), as shown in this picture:

The final product we got, with some disappointment, was very loose. (The positive side was, it was indeed very easy to flip!)

Looking back to Borcherds’s work again, I was so amazed by the precision of their work. I wonder what kind of math and planning have Borcherds done in order to achieve that precision. 

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Overall, I think my biggest take away from this assignment, was how we can gain enthusiasm with playful puzzles. I found that enthusiasm from my peers playing with flexagons, and I also found it inside me when I engage in the presentations from other groups. I am quite confident that most student would enjoy these playful activities we had in our classes this week, if offered in a non-evaluative & in-class manner. But I think the challenges would be, how do we draw connections to the BC curriculum, and how do we manage to fit these "extra" activities into the limited lesson hours.

I think even if we cannot afford a fantastic project/activity, it's still very nice to simply show students some artworks related to math. For example, when we talk about radical numbers in class, we can show students the artwork which JJ, Snow, and Katarina's group have chose and presented, and ask students what they think about the art. Or, when we talk about irrational numbers, we can show students the artwork which Juma, Jimena, and Shannon have presented. (However the enthusiasm you get in an art/craft-creating activity would be missing.)

Assignment 1 - Group Documentation

Members: Sissie He, Ross Kagna, Tiffany Williams

Original artwork: Fabric Hexaflexagon by Jill Borcherd

We chose this art work because we thought it would look visually appealing to the kids and would be easy for the kids to recreate with their own twist on it. For example, for us, we tried to make it our own by choosing different patterned fabrics for each face (Tiffany was especially proud of the side with apples and oranges because my friend made a joke earlier that week about comparing apples and oranges!). We also used fabrics Sissie already had, which might make for a good project on upcycling fabrics for the kids to find materials they already have but that might have gone unused and forgotten otherwise. Other than fabrics, papers, stickers, or other materials might also be used, as long as they are thin and gluable. 

We first started by making samples using paper. This first sample got beaten up pretty quickly as Ross was mapping out the sequences. In this first sample, we had to make sure our trig (!) ratios were right for our equilateral triangles, which might serve as a warm-up reminder of what the kids might already know about trig for us to evaluate how familiar they are with trig concepts (ie. the 2 to sqrt3 ratio). 

Here are some other samples we made, both for ourselves and for our peers to play with:

Sissie made the brown paper samples for the interactive activity of our peers playing with the faces and trying to figure out Tuckerman’s Traverse—the fastest way of getting to all 6 faces by mapping out the sequences of the faces they get to. We hoped this would be a gentle introduction for the kids! 

A struggle we initially encountered as a group was finding specific instructions to the strip of triangles that the kids would be able to follow and understand; we were kind of just folding it until it looked right at the beginning! Ross was folding it a bunch of times to work this out.

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Then, for the replication of the artwork, Sissie found a strip of fabric, on which we pasted the different patterned fabrics for the various 6 faces. Since the orientation of the faces mattered, we wanted to be thoughtful in choosing which patterns we used for each of the 6 faces, and careful in the way we glued them to the longer strip of fabric. For example, in the mapping of the faces, since 1, 2, and 3 occur the most frequently, we chose to focus on making the patterns for those the most visually elaborate and interesting. 

We used lace, floral, and polka dots for Face #1, apples and oranges for Face #2, and apples and gingham for Face #3. Then, we laid them out to make sure they looked appealing. After experimenting with a sample on which we drew different coloured stars to determine the correct orientation of each equilateral triangle on the long strip of paper, we carefully stuck the patterned fabrics in the correct position on the long fabric strip. After this, we folded the long fabric strip, as we had done with the paper samples! And we were done!

The Locker Problem - Sept. 29th

 I started tackling the puzzle by thinking about what happens to a specific locker, for example locker #1 or #1000, since I thought these numbers on the end are easier to figure out. I then realised that locker #1 is only going to have their state changed for once (by the first student), whereas locker #1000's state is probably going to be altered for many many times, including by the first and the second student. 

I then tracked down the first few lockers, what happens to locker #2, #3 and #4. I realised that a locker's state will be altered when, and only when, the count of the student is a factor of the number of the locker. The first student were to alter all the locker's state, was because 1 is a factor of every integer from 1-1000. 

Thus I could deduce that, by the end, a locker will be closed if it has odd number of factors (e.g. locker #1), and it will be opened, if it has even number of factors (e.g. locker #2). 

To figure out which numbers have odd/even number of factors, I considered a few numbers, such as 8, 9, and 12. I found that the factors mostly comes in pairs, with the exception of squares, which introduce duplicated factors. 

Thus, I could conclude that, from locker #1-1000, any lockers with a perfect square number will be closed by the end, that's #1, #4, #9, #16......, and will be opened otherwise. 

With the aid of technology, I figured that the largest perfect square within a 1000 is 31^2 = 961. Hence there will be 31 lockers closed by the end. 

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Overall I think my strategy was to start with considering specific examples, particularly the easier ones, to figure out the pattern. 

Favourite and Least Favourite Math Teachers - Sept. 17th.

It’s quite difficult for me to come up with my favourite and least favourite math teacher. For my least favourite, I would collectively say the math teachers I had in elementary school, which I think there was 2-3 of them, and I only have the blurriest memories about them. One episode I remember from then is when we were learning the multiplication table. To make sure we were memorising the table, we were assessed on the table every class. The things goes like, we start with a student sitting in the corner of the class, students are asked to stand up and recite the few rows from the table that come after what the last student recited. And we go around the class. I was very shy and reserved at that time, as a result, although memorising the table wasn’t too challenging for me, being assessed in front of the whole class was a very anxious experience. Especially since the teacher was being intimidating.

I think I understand the reasons behind her intimidation. Actually, the majority of teachers I had in my elementary school was employing intimidation as a strategy for classroom management. To me, I am pretty sure I have always been on the very sensitive end of the spectrum, their intimidation was a form of aggression. These experiences are definitely the reason I want to be a friendly and approachable teacher, but also the reason I struggle with drawing boundaries. 

When it comes to my favourite, I would say is my math teacher who taught me from grade 7-12. At least she’s the person came into my mind when I think about “my math teacher”. She wasn’t a very warm and friendly type of teacher, but she was very prepared and responsible - those are some qualities that I really admire about her. I very much want to be as prepared to teach as she has always been.  

Reading Response - Three Curricula, Sept 15th.

 This was a really condensed reading. What I really resonated with, from the article, was the questioning towards the exclusion of certain studies, especially law. For example, we all know a thing so called "marriage" since we were in the elementary school, and that most of us were told that it is something virtually everybody is going to come across in their lives. But marriage is actually, to a large extent, a legal contract, which the young are very often not taught about what exactly is that contract about. It was quite upsetting, when I read Eisner wrote "We teach what we teach largely out of habit". In BC, we have Law Studies 12 as an option of social studies 12, which is great. But I think the importance of law education, is likely more than just an option which students may take in their grade 12. 

There were also quite a few thing that I've rarely thought about, highlighted by Eisner in the chapter. One of them was the effect of fixed timetable education to student's working habit. I've never really reflected upon this fixed-timetable-scheme in education system, and I can hardly comment on it yet. But indeed, sub-dividing a day into multiple fixed sections to work on multiple subjects, does not sounds necessarily like the most efficient way of learning or working in general. And this connects to Eisner's argument of, how the fixed forms and patterns we teach in school might constrain students instead of inspire them. This reading made me ponder about, how may I encourage students to learn creatively in my class, instead of having me "teaching" them how to use the "tools". To some extent, aren't all assignments essentially constraining the way students learn?

Reading Response - Instrumental vs. Relational Teaching, Sep. 10th

I will start with a few things that I enjoyed about reading the text. I really liked how Skemp started with introducing the idea of faux amis. Two faux amis I found very interesting in math, while difficult to explain to student, are "general" and "always". Also, I found the music class analogy which Skemp made very interesting. It makes me think about how in junior level of math class, almost all kids seems to be capable of doing those arithmetic. But once "x" and "y" are introduced, to some students, the same operations somehow seemed like a new set of operations to learn. 

An idea that I always kind of had in mind is that, often mathematical insight comes after performing mathematical methodology repeatedly. Because, this is the case for me. But as I entered university, I learned from my study buddy that, some math learners really struggle on solving a problem unless they have understood the methodology offered relationally, which is totally not the case for me. I was so good at imitate whatever was done in an example on the textbook and get the right answer without understanding any principle underlying. After realizing that some people need principles before methodologies, I start to believe that both instrumental and relational explanation are needed in teaching. I recognize the value of "instrumental teaching" as described by Skemp, because as a prevalent idea, "math is a tool". When students ask me why they learn "pre-calculus", well, I answer, "because calculus is very useful in many discipline, so you might want to prepare for it". But at the same time, mathematical insight is, almost self-evidently, what we really appreciate and eager to share with others. And of course as Skemp has pointed out, it benefits students in long-term. Combined with all the situational factors Skemp has outlined, I think a mix of instrumental and relational teaching is the direction that I would like to go for.