Battleground Schools

From reading this article, the main argument that it is addressing is dichotomies surrounding mathematics education in North America. It is certainly interesting (and a bit scary!) to know that there are math teachers out there, either elementary or secondary, who are successful in mathematics learning and teaching, and yet the reason they are so successful in math purely because they have been "memorizing" algorithms and concepts in their brains without much sense on "why" they worked or why they are as they are. Interestingly, if a math teacher carries this sense of learning in his or her classrooms, there would be a chance that their students would also carry this mindset of math learning. This can be shown in their ability to educate themselves on how to interpret math. Most of the math education reforms, the Progressivist movement, the New Math Reform, and the "Math Wars" of NCTM, gave me new insights into their different ways of teaching and learning in that specific period of time. But I think that math learning over time is starting to focus more on the importance of real-life applications and what students can do with math in the future instead of just simply memorizing everything that deemed important.

From looking at the table between Conservative and Progressive stances in mathematics education, I think that I fit under the Conservative view more than Progressive view, depending on the level of schooling I had done in the past as a student. When I was younger, I attended part of elementary schooling in China and part in Canada. My perspective of math learning in China was highly conservative. Teachers would be drilling concepts and math formulas into my brain without necessarily telling me why that is. Interesting fact about me is that I used to really dislike math when I was younger because I was just not getting the concepts! However, as I came to Canada, this level of learning and assessment changed slightly. Math is no longer just about working by yourself, timing your arithmetic abilities in under 60-seconds and getting all the right answers. Some of my Canadian math teachers do focus more on the "why" and the "how" instead of focusing a lot on getting the answer. There were more collaborations and group work involved, which is what I want accomplish as a math teacher in the future as we live in the time of change and advancements.

Reflection on Micro-teaching (evaluation)

I enjoyed our first micro-teaching topic on something that is non-academic! The topics from our presenters were quite diverse and very interesting, though I wished I could have stayed longer to listen to all of the presentations.

My micro-teaching topic was on tea, more specifically on the history of tea and how tea has influenced and shaped the meaning of our cultures through time. From my reflection of my own presentation and how others evaluated my presentation, I had a clear objective in mind, the materials were well organized, and the scope of tea and its impact on societies from different parts of the world were thoroughly covered in the 10-minute presentation. One thing that I need to work on would be to include more participation from students. I need to include more ways to assess their learning and understanding by asking questions that are more challenging to get their minds moving. I should have brought in more fun activities for them to work on, but I guess there were so much materials that I wanted to cover but couldn't in a short period of time. Another thing I need to work on is to control the volume of my voice. I need to speak louder so that everyone can hear me clearly (especially when there are other groups presenting at the same time).

Overall, I had an enjoyable time learning new materials from other teacher candidates! Thank you all for sharing your knowledge with me!

Micro-teaching: Lesson Plan

Presenter: Ying Ting Lu
Topic: Tea

Task	Checklist
Objective of the Lesson	- Teach the historical and cultural influences of tea. - Make aware of the structure and different types of tea available (including its health benefits). - Indicate the cultural and social influence of tea in other countries.
Opening	-Opening up by asking if anyone drink tea, and what kinds of tea they like to drink.
Materials	- Laptop to present the slides. Each slide: aim for 2-3 minutes. - Various tea samples.
Check for prior knowledge	- Check to see if students know the essentials of tea and tea-making.
Activities	- Ask them which tea they drank before, and which they enjoyed the most (hot and cold). - Let them look at the sample teas. - Time: < 1 min.
Ideas/skills developed	- Attain strong knowledge on the history of tea, and its impact on human society.
Timing	Entire presentation: aim for 8-10 minutes!
Closing	- Conclude the usefulness and influence of tea all over the world. - Relate the facts to students.
Check for Understanding	Ask: which three countries have the biggest influence on tea. Who “invented” the first tea and where? What are the four common types of tea? (orally)
Assessment	- Allow more time for students to participate in the presentation. - Encourage students to ask questions for clarification any any point of teaching.
Indication of Future Application/direction	- What other tea traditions do they know? - How can tea influence anything beyond culture and society? - Can tea improve the overall well-being of humans?

Estimating the Dimensions of Campbell's Soup Water Tank

For this week's math problem, we are asked to find the estimated dimensions (length, width, volume) of a Campbell's soup water tank! This is a very interesting problem. I had fun doing this activity, except there were a lot of researching and estimating involved!

To get started with the problem, I needed to create a visual tool to help me estimate the water tank, and in the picture, the bike definitely helped with the problem! Looking at the picture as above, I began using the measurement of the bike to help me find the dimensions of the water tank. Several factors were involved in this process...

Susan's bike looks like a hybrid bike for commuting (road + mountain). I researched the standard (recommended) hybrid bike size for women who are 165 cm, which is Susan's height. The biker's inside leg length, according Cycle Experience, is 76 cm (http://www.cycleexperience.com/getting_the_right_size.php). This length is important because I figured that the length of the inseam is 76 cm for her height, and usually when bikers stand while sitting on the bike seat, their feet should still touch the ground without the need to tiptoe. Now, looking at the handle bar, according to MEC, the height for a road/mountain bike, is adjusted at 2.5 cm to 5 cm. Judging by the picture of her bike, it looks like it's raised 5 cm above the seat. So, the total length, labeled in green, of the bike should be, roughly, 81 cm. As well, since the tank is slanted, I have to take the top bit of length into account. I estimated it to be roughly 20 cm. Finally, the total length, or the diameter, of the tank is 81 cm + 81 cm + 20 cm = 182cm.
(http://www.mec.ca/AST/ContentPrimary/Learn/Cycling/Bikes/AdjustingYourSeatAndHandlebars.jsp). 
From there, we can calculate the radius of water tank is 182/2 = 91 cm (roughly).

Now, as for the length, labeled in yellow, I researched the estimated length of a standard road/mountain bike, and it is 1.8 m (http://safety.fhwa.dot.gov/ped_bike/tools_solve/fhwasa12018/). For the remaining blue label, I used the measurement of the diameter of the bike tire. According to Harris Cyclery (http://www.sheldonbrown.com/rim-sizing.html), the diameter is measured to be 21 cm roughly. In total, the height of the water tank is 180 cm + 180 cm + 21 cm = 381 cm.

From these estimated calculations, the volume of the water tank is...
Pi * r^2 * h = Pi * 91^2 * 381 = 9,911,916.459 cm^3.

This is equivalent to roughly 2618.451 gallon!

I think this is definitely a fun activity for students to work on and get their brains thinking! I can sense that as students are working around this problem, they are probably very curious as to who ultimately gets the right answer! Working at this problem has allowed me to explore the realm of bicycles as well. Who knew there are so many types of bicycles out there. And in all cases, math is so important because we need measurements of our body to pick the best bike to ride. As for the volume of the water tank, we needed to estimate the dimensions, such as the diameter and the height of this cylinder water tank, based on the bike shown in the picture.

The Imaginary Letters

After teaching for 10 years, I had received two letters from my past students with both positive and negative comments about my teaching. The positive letter from my student was very encouraging and motivating. However, the negative letter raised a lot of concerns on my part. The student didn't really enjoyed my class as much as others because it was too easy and not challenging enough. I wished I knew before so that I can make the course more challenging for students to stretch their brains than just simply giving them the information that's required to be learned for the curriculum. I should have created more complicated math problems that they could solve, such as a fun math brain teaser or an interesting math riddle that get students thinking.

The student also commented on my lessons telling me that the materials I provided was too superficial (or not enough information provided). I should provided them with more background information or the history of the contents covered in class than just telling them what it is we are learning for the today, how to get the correct answers, or what teachers are looking for. I feel that just by simply telling them these concepts without challenging them might be too superficial.

One more comment made was that I gave too much homework. I guess not all students like to do homework after class. Maybe I could give them a bit of homework time in class (say, for the last 15-20 min of class) so students could feel more motivated to do work since they were already learning the topic anyway. It would also be a good opportunity for them to talk it out, walk around the class and seek help from myself and others. Perhaps some students just felt less motivated to take out their textbook and get started right away - they might have been distracted doing practices at home or not have enough resources. I thank my students for their emails and letters! It's still a good time to modify my teaching strategies and some of the other things I hope for as a teacher. Thank you!

Math/art Project Reflection

For our first assignment, I actually had fun doing this project with my team! At first, it was a little difficult to have everyone agree on one specific idea on what shape to make because there are so many shapes that we can build with just binder clips! Our ideas are constantly changing as we weigh in on the pros and cons. We started building the stars with binder clips, since star shape alone has quite a few math structures that we can talk about with our class. However, we thought wouldn't it be more fun to expand on that idea by connecting the individual shapes together and to create a sphere?

I think this project would be great for secondary school students, particularly those in Gr 9 or Gr.10. It really helps them to be creative and build whatever shapes they want so the end result would be a 3-D sphere. Our team took many trials before deciding on the shapes that would be strong and durable to hold the structure together, so I can imagine the Gr.9 and Gr.10 students getting their hands dirty and exercise their brains on creating a strong and durable sphere made out of binder clips! It's all trial and errors. If a shape is too weak and unstable (not solid enough), they will have to take the shapes down and rebuild it.

As you can see, creating the desired structures with just binder clips need a lot of patience, effort and teamwork, which helps students with their communication skills. Communication, which I see in my team, was crucial because there were incidents when we weren't informed on what shapes to build and how to build it. Problem-solving is another area that can help students develop through this project. Students may ask “How can we build this shape so that a specific pattern follows to ensure durability?”, or “What are the ways (patterns) to link all the shapes together so that the sphere won't fall apart?” For math problems, students may wonder “How many different patterns do we see in a particular shape as this?”, or “How many clips do we need to invest so that all vertices are connected equally without having an extra vertex hanging unused (and all shapes are used accordingly)?” These are just some of the many questions that students may think with others.

In all, this project definitely helps Gr.9 and Gr.10 students to develop their sense of spatial visualization and spatial reasoning, as defined by the BCMT in high school level. Making connections and finding patterns to shapes is very important in this project because they need to know what 3-D shape to build upon, so the rest of the steps would be easy. It is also useful for students at Gr.9 and 10 levels because it can help them with their visualization between and among 3-D objects and 2-D shapes. If they find it difficult to visualize in 3-D, they can always draw 2-D shapes to get them started thinking (as many of our team members have done also). Doing so can help them visualize and interpret new ideas and helping each other. As well, because not all students can draw 3-D shapes on paper, so by using and manipulating concrete materials as binder clips, we hope that they can think abstractly and deeply about the process of building; possibly to reason with each other if certain steps make more sense than the other, if not try with different methods.

Maththatmatters: Beyond "Pizza Party" Math

David Stocker's book on Maththatmatters is certainly quite interesting book to dissect. One thing that resonated me while reading his article is that "we should be using the language of mathematics to help students understand the real world of race, class, gender, sexuality, ability, power and oppression", and not simply use things in real life to do or solve math. I think that is very true because, come to think of it, a lot of my middle school and high school math problems were about using real life examples to solve math; some may be impractical, but seems practical enough. For instance, as a popular internet meme says, "Only in math is it okay to have a person buying 1000 watermelons and not get judged." (very true!)

In this case, it is important to actually use math concepts and apply it to real life - to solve real-life problems that benefit the world. We should always be interested in asking questions, to challenge our brains and to think about things that "have real meanings", and only then can we gain deeper insights on how math can actually help us and the world around us. For this reason, I personally think that mathematics is connected to some form of social/environmental justice, one way or another, because we need math to reason the world. The link may be weak, but I feel that it is there. For instance, math, particularly probability and statistics can be effective in science and social research. Scientists may use math to find causation between smoking and cancer, between amount of sugar intake and diabetes. Math can also be used in business and law to see if companies can make profit by optimizing its operations while minimizing expenses. Math can be used for calculating insurance policies to recommend the appropriate policy for their clients. Math can also be used for calculating stocks so holders can have the acquired knowledge for stock options, and help them to forecast stock trends for profitable trading that is fair. Some of these topics, in my opinion, are connected to social justice one way or another.

The author may be right, "numbers only tell half-truth", since just by looking at numbers don't really tell us about anything. But, if we analyze those numbers in a given context, such as law, medicine, finance, or the society, they can mean so much more; it makes students and teachers wonder and think. And it is the responsibilities of a teacher to engage students with lessons that actually address concerns of social/environmental/global issues. As well, I think it can be beneficial for secondary class teaching because we need to have young kids think about these issues critically - to let them think about how math is such a broad field that doesn't just begin and end with calculating the area of a pizza box. To let them think critically about math as they transition from elementary to secondary level can potentially help them grow and develop ideas later in life (I'll never know what they are capable of!).

Chinese Dishes Problem

Without using any algebra, how can we find how many guests are there if there are 65 dishes, and that every rice dish is shared by 2 people, every soup dish is shared by 3 people, and every meat dish is shared by 4 people?

Surely there are many methods to do this, but the one I chose mainly involves fractions (and cross-multiplication).

If we know that...

• 1 dish of rice is shared by 2 people, then each person eats 1/2 dish of rice.
• 1 dish of soup is shared by 3 people, then each person eats 1/3 dish of soup.
• 1 dish of meat is shared by 4 people, then each person eats 1/4 dish of meat.

Then, to put all together...

We we know that each person eats 1/2 + 1/3 + 1/4 = 13/12 dish of rice AND soup AND meat. (We can also consider this summation by looking at that "each person" as a unit of measurement.)

Since each person eats 13/12 dishes of rice AND soup AND meat, and we know that there are 65 dishes of food (rice AND soup AND meat), we can use unit conversion to find how many persons are there.

The final answer is...

65 dishes / (13/12) dishes/person =  60 persons.

The culture context, in my opinion, is quite universal for a problem/puzzle like this one because food is universal across all cultures. For instance, this similar problem can be used for any large feast in any culture, such as a potluck, a wedding or a birthday celebration if any dishes are shared. However, not all events have shared dishes, so the way people think about this could be different. Regardless, I can imagine any restaurant owners who is feasting a large number of guests can undergo this kind of problem in their lives, either to maximize profits or to serve guests with an economical (appropriate) amount of food so no food can go to waste.