Entry: Micro-teaching Reflection

   The micro teaching was an interesting experience. Complications arose that I did not foresee (one student asked many more questions than I had anticipated) and the students' prior knowledge was more layered than I had hoped for (among the 4 students, there was 1 who was familiar with the game, 1 who was vaguely familiar, 1 who was not familiar but knew a similar game, and 1 who had 0 prior knowledge). This really complicated my 10-minute lesson as I had to relate the game in 4 different ways.

   I was pleased to read the feedback and discover two things: my personal feedback was the most critical and everyone's negative feedback focused on the one aspect that I also focused on. This is the most important part of the reflection because it confirms that my lesson did have a problem (I ran out of time and therefore my conclusion was very weak, but my introduction and lesson were OK), but that I understood the problem and know how to improve. I will incorporate what I have learned into my next micro-teaching.

Entry: Battleground Schools

   Public opinion of mathematics has had several radical shifts in the last century which correlate to the global narrative, as described in the article. In brief, the 3 significant focuses on mathematics were:

1. immigration and industrialization in WWI/WWII -> focus on math inquiry and science
2. technological race during the Cold War -> focus on abstract math and science
3. conservative shift -> back-to-basics mathematics with standardized testing

   As a teacher, the global (or often national) view on mathematics does play a significant role in various ways. It can affect funding of programs (depending on the importance that society places on math), structure of classrooms and curricula (relating to the current prevailing theories), and specifically the methods and goals of teaching (pertaining to the objectives set out by the curriculum). For example, a new teacher may need to focus on preparing students for a specific test which would greatly limit their breadth of scope. A teacher earlier in the century may be allowed to focus on aiding the student in a myriad of math inquiry projects. Perhaps a teacher in the future would have a flipped classroom where they are simply there to help the student when they are stuck. It is quite dependent on the global, national, and local goals and perceptions of math education, in conjunction with your own abilities and flexibility.

Entry: Micro-teaching Lesson Plan

Micro Teaching Criteria

Name: Jacob Brunner

Topic: 象棋 (Chinese Chess - Xiàngqí)

Objectives & Goals	The purpose of this session is to teach the students the basic rules of the game, so that they are able to go and practice amongst themselves.
Hook	I will introduce the lesson by showing them the game board and asking them questions about it. This will hopefully spark some curiosity, as well as test prior knowledge in preparation of the activities.
Materials	To teach this lesson, it is required to have 1 complete Chinese Chess set.
Prior Knowledge Check	First, it is necessary to check if the students are already familiar with the game. If several students are familiar, then it would be best to have them help demonstrate with you. Instead of simply explaining the rules, you could have the rules explained by having them answer questions. You could also have two of them demonstrate a game (slowly) while you point out each move and how it is made. If they are not familiar with Chinese Chess, but are familiar with Western Chess, then you would switch the rules explanation to be in relation to those rules (which are quite similar). Instead of simply explaining how the soldier moves, you would explain it and additionally ask which piece it is similar to in Western chess, as well as how they differ.
Activities	2 Minutes: I will explain the rules for each piece (there are only 7 and some have very simple rules). 2 Minutes: Then I will go over the rules of the game (again, these are very simple). 4 Minutes: Then I will have a demonstration game which will end quickly to show them what it looks like. 2 Minutes: Lastly, I will answer any questions they have.
Ideas and Skills	They will learn the basic ruleset for a game that involves a lot of mathematics. The idea is that they use these skills to practice the game on their own time, hopefully building an interest in it, which will improve their logical thinking skills.
Closing	The closing section will be the end of the activity. That is where I answer any questions that they have about it. In a 10 minute teaching session, there is not much time to have a summary of events, so I think that the Q&A segment will be sufficient.
Assessment	I will assess the students’ learning by seeing if they understand the rules of the game. This is quite simple as I can simply let them make a few moves and see if they make any errors. I am not worried about their strategy, only that they know the rules and are able to play outside of the classroom.
Applications	This game is a good example of Zero-Sum Game Theory and logical thinking. It helps prepare the student for other games that they will encounter in their life and get them thinking about more complex strategies which can loosely be applied to programming, economics, political science, biology, etc.

Entry: Campbell's Soup Can

   A regular soup can is 6.5cm wide and 9.8cm tall, while the average bicycle is 90cm tall. Judging from the picture, the circumference of the enlarged soup can is roughly 2 bicycle heights. However, due to the downward angle of the camera shot and guessing  how much of the can is buried, I will add an extra half of a bicycle.

   Therefore this can has a circumference of 225cm. A regular can has a circumference of 6.5cm. Therefore this can's circumference is 225/6.5 or 34.6 times bigger. If we extrapolate that, the can should be 34.6 times taller. Since 34.6 x 9.8 is 339.23, the large can should have a height (in the picture's case, a length) of 339.23cm.

The volume of this tank should be π x 112.5cm^2 x 339.23cm or 13.481 million cm^3. This equates to 3,561 gallons. This seems smaller 
than I would have guessed, so I will try a new method using the bicycles length, which might be more 
consistent between bicycles. I would judge the can to be 2 and four fifths bicycle lengths long. The 
average bicycle is 180cm long.

Therefore this can has a height (length from the picture's perspective) of 180*14/5 or 504cm (this is much larger than the previous method's height of 339cm). A height of 504cm compared to a normal can's height of 9.8cm making it 504/9.8 or 51.43 times bigger. The circumference of the can would therefore be 51.43 x 6.5cm or 334.3cm. The volume of this large can would be π x 167cm^2 x 504cm or 44.12 million cm^3 or 11,655 gallons.

This is an interesting question because the answer will depend on how interested you are in the question.
You could simply eye-ball the size of the can and try to work it out from that (which is acceptable), but
you can also factor in the angle of the camera, the distance that the bike is away from the can, how much
of the can is buried, etc. No one will have the same answer.

I think the most interesting follow up question is to ask: "If the can is a normal can, then what are the
dimensions of the bicycle?

Entry: Letters from future students

Dear Mr. Brunner,

  Thank you for being such a supportive teacher in high school! I always struggled with math and, even though I didn't fall in love with it after your class, I was able to get through it. I did end up going to university in the end and was very thankful that you encouraged me to take Math 12. Don't stop being quirky and a great teacher!!!

    Sincerely,

        Your old student


   "Hey Mr. Brunner! Funny bumping into you here. I remember your old math class. It was awful! You always gave us too many homework assignments, even though some of us were on the football team and had practice. And I remember all of those lame jokes!!! I hope you are cooler now or your students will be so bored! Catch you later!


   I think that by writing these letters, I clearly see a reflection of what I hope to achieve and the pitfalls that my old math teachers hit with some of their students. I wish to inspire a love of mathematics within my students, giving them the tools they need to pursue their own paths outside of the classroom. But I also fear that my enthusiasm and effort will be translated into something that the net generation can't connect to, turning me into a goofy teacher that the students don't take seriously and roll their eyes at.

Entry: Presentation/Project Feedback

   I had an enjoyable experience in re-creating hyperboloids using wooden skewers. There were several methods available to us through the internet which offered us varying levels of success. It was quite a struggle simply following the directions and in the end I had placed one of the skewers on the incorrect side, which caused the entire structure to require amendments. If I were to teach this in a classroom setting I would definitely create a clearer set of instructions, as well as blunting the ends of the skewers (which caused several unfortunate incidents).

   The mathematics involved were fascinating to observe, as adding each layer of rubber bands altered the shape that it expanded into. In a similar manner, the amount of skewers used played a role in the rigidity and lattice work of the finished structure. By modifying the method used to build it and by stretching it in different ways, we were able to create a wide range of hyperboloids using the same set of straight lines. I think this would be a perfect activity in one of two classes: an elementary school class to introduce the simplicity and beauty of various mathematical shapes or as a high school class to augment a lesson on geometry or hyperbolas. As an elementary class activity it would require a great deal of supervision, as they were not simple to build. I think that a highschool class would be fine with an explanation/printout of the instructions, followed by the teacher wandering the class and assisting where required.

Entry: Math That Matters

   I believe that in a perfect world, mathematics would be a pure entity, devoid of any cultural relevance: a universal language if it were. However, in our realistic world, mathematics is connected to our social and cultural identities. We don't simply use it to calculate and display data, but to sway public opinion and affect our culture.

   I agree with the author's intentions and feel that math should be used as a tool to understand the world, instead of using the basic problems of the world to understand math. We need to help students understand how companies and governments use math, be it in pricing, statistics, data, etc.. This won't simply help students understand the world and social issues, but by putting it in a relatable context, it will actually help students learn the math.

   I don't feel that there is any reason why these ideas could not be incorporated into a secondary mathematics class. It is simply a matter of taking the mathematics that the students are learning and applying it to real world problems. Obviously, different types of math would have varying levels of difficulty in applications and with the complexity of the math, the more difficult it becomes. With simple math, like algebra, you can apply it to a myriad of basic social problems, but with abstract or theoretical math it's a bit tougher to find relevant social topics to relate it to.

Entry: 孫子算經的『How Many Guests』?

   I know that everyone had half of a dish of rice, a third of a dish of broth, and a quarter of a dish of meat. I first want to see how many people I need for this to be possible. I will add together these dishes with pictures until it makes sense. If I start with 1 person I will have:

   Now I know that this is only possible with groups of 12 people. If I have less than 12 people, there will be leftover food. If I have 12 people, I can count and see that there would be 6 bowls of rice, 4 bowls of brother, and 3 bowls of meat. This is 13 bowls. We do not have enough food, so I will need to invite 12 more people and prepare 13 more bowls of food, bringing us to 26. If I repeat this 3 more times, I will have 65 bowls of food and 60 people to eat them.

   I don't feel that cultural context has much impact on this type of question, unless it was taught to a culture that was not familiar with sharing food (in which case they might not understand). Whether the items being shared were rice, soup, french fries, etc. typically would not change how the students would approach the question. However, I feel that this type of question is a great way to make foreign students feel included and relevant in the classroom and I would encourage using word problems that are influenced by global cultures (using ethnic names, foods, animals, etc.).