Microteaching Lesson Plan

Teaching Weiqi:

structure:

    -start with introducing some quick background about the game (1 min)

    -teach basic rules of the game (2 min)

    -give simple examples (2 min)

    -give puzzle pieces that group mates work together to solve (5min)

        (https://online-go.com/puzzle/60350)

plan B:

    -if time is not enough, consider shortening the example time or puzzle time.

    -if more time is left, introduce more difficult puzzle pieces, or talk more about 

 assessment criteria: 

     -do learners understand the rules    (basic)

     -can learners understand the examples

     -can learners solve simple puzzles    (intermediate)

Locker problem

The problem is asking the final state of each locker door after all students have finished their turn. 

Each student will change the open/close state of the lock when they touch it.

The locker door's final state is then essentially decided by the number of students who touched it, divide by 2 since there are only 2 alternating state.

Each student only touches the locker number that is a multiple of their own #.

So this is asking the factors of each whole number from 1-1000, if the number of factors is even the door stays opened, if it is odd then the door stays closed.

Take locker 999 as example, it is divisible by 1, 999 naturally, 

then /3 gets 333,

then /3 gets 111,

then /3 gets 37, and 37 is a prime number.

So the factors are 1, 3, 37, 111, 333, and 999, so after 6 state alterations the door will stay as "opened", at its initial state.

Unfortunately, I could not find a universal equation to describe the phenomena; I could only decide each lock's state by individually assessing its number of factors.

Math Art - Individual Response

The origami art we chose appeared to be a very complex structure, which I feared that we would not be able to recreate. However, it turns out to be manageable, with the expertise of Minami of course. The process showed very interesting relations in geometry concepts during origami steps. It required some specific angles and edge alignment to manipulate an initial square piece into the piece we need, and some valleys folded for marking purposes also demonstrated the concept of assistive assumptions used during geometric proofs. Learning these relations through origami art is a new experience for me, and it opened opportunities to maybe help others learn math concepts through origami.

Overall I find origami to be a value tool for teaching geometry and problem solving. The process natually encompasses the divide and conquer technique in problem solving, its hands-on process makes it memorable to people through repeatedly working on same pieces, and it is a great activity to teach teamwork and responsibility. Then the result shows complexity, so it brings sense of accomplishment to the individual or groups finished it. Last but not least, it is cheap and simple enough to be implemented in all levels of education, with adjusted difficulties all grade levels can benefit from origami projects.

Math Art - Group response

Group: Doreen, Minami, Yikang

Original Artwork: Neel Shrestha - Origami Snub Dedecahedron 

Remake & Issues: 

We aimed for a complete replica, but with more colors to better demonstrate the number of shapes required to enclose the volume. One major issue in our process appeared on our first few attempts at assembly, the dodecahedron required a very specific orientation of each piece, where mirrored or chiral pieces would not fit in. In the end we realized all pieces require the same orientation, and remade several to complete the assembly.  Below are pictures of the two mirrored pieces.

One interesting aspect of making this structure was seeing how the 2D papers linked together to form the final 3D structure.  The folds we made needed to be precise and creased thoroughly, otherwise the locations of connection wouldn’t align with each other perfectly.  You can see this in the pictures below.  With just two pieces, the origami still lies relatively flat.  When we begin to link them all together, it introduces the curve that we see.

Alterations: 

After the snub dodecahedron was finished, we realized that this piece would be too difficult for the class to make, both in individual making and the assembly stages. So we decided to use another polygon as the class activity topic for demonstration. During planning phases, we believed that adding names to everyone’s own pieces would be a nice addition to encourage group work spirit, so it was added to the plan.

This second structure that we made was an icosahedron, which was an interesting counterpoint to the dodecahedron that we had previously made, since their faces and vertices were swapped.  Additionally, it had faces that rose up in pyramids as opposed to being empty space like the snub dodecahedron.  Below is an example of the icosahedron we made.  Another interesting observation about this second creation was that it was structurally quite sound.  This is actually common for origami, but it can be strange to think that all of these pieces of paper interlock to form a ball that can be handled more roughly.

Interactive Activity:

We decided to use our interactive activity as our alterations to show the class what they might be able to do with their own classes in the future.  This included a walkthrough of how to fold the basic piece of a Sonobe unit, as well as a little exposition on how they might talk to their class about playing with origami.  When designing the activity, we tried to make sure that it would be fun and accessible to both beginners and experts, as well as creating a sense of community by contributing a small part to end up with one cool structure.  If we had more time, it would have been nice to give the class a chance to try and explore to put some structures together themselves.

Favourite and least favourite math teachers

A favourite math teacher I had was back in Grade 12. He was always in a light-hearted mood, would play basketball with students in the gym at times, and often expanded on topics in the classroom. Looking back at the teachings, he was likely not strictly assigning specific lengths that a topic would take, may even discourse into unrelated tangent stories at times. But he gave great related stories from when he worked in a company, showed us what some concepts could actually be represented in real-world settings, and how they actually cared about some details and invested less time in some aspects in projects. It made topics more relatable and less dull, attracting more focus from students, and invoked more interests.

My least favourite math teacher would be my 5th grade math teacher back in China. Although it was a totally different system, focusing on different goals, and using different methods, he was still outstandingly more strict in classroom discipline than my other teachers in China. Many students were scolded for being noisy including me, and overall the atmosphere was more effective in rejecting interests in math with its dull, feared silence. I hated math for a few years after his class, and overall was not passionate about math for a long time. Looking back, I believe early years need a more encouraging atmosphere to promote interest, instead of overly strict classroom discipline.

Eisner reflect

 One segment that caught my attention to stop and contemplate is that the teachers' gratification is shared by all the students in the classroom, and they soon learn to delay their satisfaction and share success with the classroom. I vaguely remember there was a period of time in elementary school when I was eager to answer questions, and it slowly vanished later as I advanced to higher grades. I thought it was a natural course since in lower grades, the answers are easy, so children are all eager to display their understanding, while in higher grades, the answers are not direct so many would take time to get an answer, and also become less confident in their correctness. The article brought a different view on such a transition, that it is rather caused by social behaviour adaptation instead of confidence in answers. It is an interesting angle, and teachers should certainly adapt to this behaviour change when teaching transitioning grades.

Another segment caused me to stop and think is the part about using payoff systems that give rewards to children outside of the process itself. From other sources I've read about, this is a widely discussed issue that many different schools of thought were involved. My personal take is that in principle, such a system could be harmful in the long run in guiding a healthy learning habit that would be beneficial in a life time. However, there are aspects and limitations to a person's learning journey, where at times, especially when we are facing critical assessments or disturbing effects in our lives, such a system could be beneficial in occasion. In an ideal society and education system that promotes equality, provides ample resources, and uses a full spectrum assessment system, students may have the time and resources to healthily continue their learning journey. But in our current state, we inevitably have students who do not have the luxury to have the time, environment, or resources to improve at a slow but healthy pace suitable for themselves. Instead, they must pass certain critical assessments to improve their life quality to have the resources to feedback to a healthy lifestyle where constant learning is undisrupted by life events. 

After finishing the article, I would have to say I agree that there are much more aspects where a school teaches children outside of the classes themselves. Eisner brought up many aspects where the teaching methods have more intrinsic subtle effect that people may not realize when designing or operating. However, school is naturally part of social structures and operated within society's value system, hence there are inevitably certain structures that follow our society's operation, therefore changes to these structures may face the resistence of parents, employers, and even governments. I believe B.C.'s current curriculum design must have considered many factors, but some may be outdated and inevitably out of sync with current student's needs or living environment. It would be difficult but also beneficial for teachers to adapt their day to day operations based on their observations when situations allow.

Reading response on: Skemp on two approaches to teaching and learning mathematics

 Finishing up on reading the article, I can't help to have an impression that this article is a well-constructed retelling of the same idea "the more you learn the better". The idea is inherently correct, but the article bringing up the new definition of relational and instrumental understanding offers very little new ideas on top of "learn more". The author used an analogy of drawing the schema of a city mentally as an example of relational understanding; however, in the eyes of a city planner, such drawing is merely an instrumental understanding of the city. To a city planner, the author does not understand why hotel would be allocated at a certain place, why one place would be closer to another, he can now draft routes but did not learn the logic behind why these buildings were issued permits to be built at their places. To sewage planners, the city planner's understanding of city plan would also be instrumental on a certain level. Sewage and power grids had to be planned due to geological or other factors, and these capacities in turn limited building permits, in their eyes city planner doesn't understand that. In their eyes city planner and author's understanding are all instrumental, just at different levels. One may argue that the author only wanted to draft routes and his understanding was sufficiently relational for his purpose, then such understanding is equivalent to the instrumental understanding defined by the city planner. The definition of relational understanding is now merely a deeper layer of instrumental understanding in a more knowledgeable system.

Taking the concept back to math as a tool, the example of one student doing division wrong because he extended methods from multiplication to facilitate the definition was very poor. The student did not fully grasp the definition of fractions application within multiplication's boundaries, this shows a lack of definition and precise description in teaching. If the student operates division within a complete strict definition, doing correct steps and get correct answers, the understanding could easily be defined as within instrumental understanding. Digging similar mathematical definitions or methods through all the layers, we can eventually reach to the bottom of the layers and discuss why define 1+1 equals 2, and discuss elemental definitions in number theory. Similarly, the mismatch in unit in area calculation is another lack of definition. In physics, dividing 10 newton by 2 square meters gives 5 pascals, math as a tool in such physics calculations abides by the definition and is a correct application. The terms relational understanding and instrumental understanding has no fundamental distinctions, relational can simply be defined as n-th layer of instrumental understanding in author' examples. For different subjects like statistics, physics, or chemistry they realistically require different levels of "instrumental understanding" in different topics, and at certain level the "instrumental understanding" is redefined as "relational" according to author's definition. The article is a convoluted retelling of the importance of building solid foundations with new terms, and to a certain level, a dangerous idea that implies learning up to a certain level is enough, instead of searching for more.

Hello World

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