Response to John Mason's Article: Questioning in Math Education

1. Do Mason's ideas might connect with inquiry-based learning in secondary school mathematics? (Why or why not?)

From reading through the article, I do think that Mason's ideas on math teaching can connect with inquiry-based learning in secondary school mathematics. Because high schools can be quite challenging and even competitive for some students as they are wanting to get through each grade and to be able to graduate on time, students can often feel pressured into knowing everything that's being asked, or feel overwhelmed with the expectations from their math teachers. With Mason's teaching strategies, teachers are not seen as an authoritative figure that's standing in front of the classroom. Rather, they are seen as someone who really value the opinions and questions of students, who care for student learning, and who respect students without having them to feel excluded of being incapable to think beyond their abilities or doubt. These forms of math teaching allow students to think and inquire for themselves while actively collaborating with their teachers and classmates.


2. How might Mason's ideas about questions in math class be incorporated into your unit planning for your long practicum?

Mason's ideas about questions in math class can be incorporated into my unit planning through giving students more time to ask questions continuously through the lesson. If I give a word problem during my lesson that students can work on to reinforce the content, I can ask students to work on their own first, then pool their answers together regardless if they are right or wrong, instead of telling them the answer right away or instead of telling them that these are wrong answers. I can then work through the problem step by step, so students can keep track of what they did and compare it with my solution (as if I were also a student). I can also ask the students to come up to the board and ask them to solve it step by step. Even if we made a mistake, we will know that it's okay and so we try again to find the right answer, to become "stuck then unstuck" (Mason) allows students to see that there are possibilities to get to the right answer from where they are.

Reflection on Dave Hewitt's Video

Our class watched two parts of the video that focuses on Mr. Hewitt's teaching strategies on mathematics for elementary and middle school children. The video was very inspiring as it allowed me to think of some the practical teaching methods that I could incorporate into my classrooms. From the video, I really admired the various scaffolding techniques that Mr. Hewitt used in bringing movement, sound, wait time, repetition, and discussions into his teaching method. I could see that students were more engaged from implementing this strategy. I also noticed that Mr. Hewitt didn't speak much or jump in right away when asked a question, but rather he let students to think through the problem and work out themselves, little by little, until they get the right answer. Even though his presence and authority is still in the room, it seemed as if he gained students' respect and trust in creating a more comfortable environment for students' learning, regardless if they made a mistake or not. I had learned a lot from Mr. Hewitt's teaching strategies, and I need to keep them in mind when I teach math classes in the future!

Reflection on Micro-teaching 2

For our second micro-teaching this Monday, Etienne, Deeya and I were presenting our lesson on the Sine Law. I reflect that our teaching topic was interesting, and the materials were a definite bonus to help students visualize. However, the lesson was a bit rushed. I think the reason is that we had three people in the team, with each presents a different sub-topic of the Sine Law, there were a lot of information that we would like to cover for each case. Having 3-4 students per team and let them learn in less than 3 minutes might not have been enough before rotating them and starting on a new sub-lesson. I found that our discussions were cut short due to lack of time, even though there were still a few more ideas to discuss. Furthermore, we should have had a more well-rounded closing/conclusion before ending the lesson.

Micro-teaching 2: Lesson Plan for Sine Law

Title: Learning the different cases of the Sine Law
Date: November 30, 2015
Grade Level: Pre-calculus 11
Prescribed Learning outcomes: ·         Be able to understand how to apply Sine Law to solve problems. ·         Introduce oneself of 3 different cases of triangles in applying the Sine Law.
General purpose: The class will look at how the Sine Law can be used to find the unknown length of the side of a triangle given one angle and the length of the other two sides. We will look at three (3) cases: 1.       Where there is no solution, 2.       Where there is one solution, 3.       Where there are two solutions.
SWBAT 3.5    Sketch a diagram and solve a problem, using the Sine Law. 3.6    Describe and explain situation in which a problem may have no solution, one solution, or two solutions.
Probing of Previous Knowledge ·         The class has already been introduced with the notion of Sine Law (the relationships between the sides and angles) and the proof of Sine Law. ·         The class understands basic properties of triangles and knows how to draw a triangle, such as labelling the angles, sides, and the height, respectively. ·         The class understands the concept of SOH-CAH-TOA; students are expected to know how to calculate the height of a triangle using SOH-CAH-TOA.
Objective	Time	Activity	Materials
Introduction	1 minute	·         Ask students where they see the use of trigonometry in their everyday life. ·         Provide more background info on trigonometry that appeal to them.
Summary	2 minutes	·         Ask students if they can recall what sine law is (just in case not everyone does). ·         Draw a generic, 3-sided triangle. ·         Review the concept of Sine Law. ·         (With a diagram) Ask students if they can find an unknown angle or length of a triangle given the other two lengths and an angle. Is this always possible?	Whiteboard Markers
Inquiry Project	10 minutes The instructor will spend 3 minutes with each group, then rotate for a total of 9-10 minutes.	·         Introduce to class the three (3) different cases of the Sine Law. ·         Divide the class into three groups. Within each group: ·         The instructor presents a different case. ·         The instructor gives each student a set of 3-4 sticks, where students have to make as many different triangles as they can think of. ·         Once students are finished making their answers, they have to explain their answers to the group. ·         The instructor will help to guide the students, writing the problem on the board and explaining why there is indeed only one, two, or no solution.	Measured sticks with which to make triangles. Multiple sets for each of the three cases.
Handout + Summary	3 minutes	·         Provide the class with handout on the three cases of Sine Law. ·         Go over the handout briefly. ·         Use it to show students that the three different cases necessarily have one, two, or no solutions. ·         Illustrate this on the whiteboard. ·         Show the implications of this when solving the Sine Law.	Students may need pen and paper for this.
Summary Evaluation: By having the students learn from doing hands-on activities with the sticks (and other visual tools), they can understand better the numbers of triangles that can be made given certain angles and lengths.

2-Sided Puzzle: "Ladies Luncheon"

What is arbitrary and necessary in teaching (math)?

1. What does Hewitt mean by "arbitrary" and "necessary"? How do you decide, for a particular lesson, what is arbitrary and what is necessary?

According to Dave Hewitt, in a learning setting, "arbitrary" means that students are informed of the learning materials through external sources, such as teachers, books, the internet, and many others; "necessary" means that students do not need to be informed of the learning materials through external sources. Often times, they can work through the questions and figure out themselves. As Hewitt writes in his article, what is "arbitrary" is in the "realm of memory", and what is "necessary" is in the "realm of awareness". I have encountered this kind of thinking and reasoning during my short practicum. There were times when students asked why it must be this way and not that, in which I hear my teacher say, "Sometimes, things are just easier this way." I think many of us new and experienced teachers have asked ourselves this kind of question many times during our teaching, and many of these examples have been cleverly brought up by Hewitt.

For my teaching, personally, I would consider math symbols, notations, or any naming conventions as something that is arbitrary. I am unsure of how I can explain why 2 is bigger than 1, for instance. Number symbols (e.g. 1, 2, 3,...) are universal across the world, so explaining that to students would be difficult for me because we have been taught and memorized this way since the beginning.

I would consider proofs, properties and relationships to be necessary because in many of these cases, they can be derived and worked out by students; students within "awareness" that is. For example, understanding why a parabola is shaped a certain way can be understood by graphing and interpreting the formula because parabolic formulas can be revised. Similarly, one can probably also work out why Speed = distance / time, or why chemical formula is written this way because these formulas can be derived and solved from real-life applications, whereas symbols and notations do not.

2. How might this idea influence how you plan for your lessons, and particularly, how you decide "Who does the math" in your math class?

Hewitt's idea of what is arbitrary and what is necessary really blows my mind on thinking to myself and asking "why". There might even be questions that some university professors and teachers cannot answer because they themselves learned concepts arbitrarily. As a teacher, I will try to dig deeper on the materials being taught in class for the curious minds. I might explain, on the side, why we do math a certain way while showing and explaining that there are alternative methods to do math differently, but it would just be harder for some students to grasp (although they are not limited to solving math their own way as long as they derive a right answer in the end). I know that not all students are capable of understanding the "necessary" component to math, but as long as teachers have an answer to "why", they can use simple hints so students can really ponder on it and "think about it".

Exit slip: Reflection on SNAP Math Fair at M.O.A.

Today, our EDCP 342A class had the opportunity as guests to participate the SNAP Math Fair at the Museum of Anthropology. I had a great time meeting talented students from elementary school in Vancouver. What surprised me the most was their ability to reason and think critically on the questions they were presenting. Most of these students were (possibly) in Gr. 6 or maybe even younger, and at that age they could think "beyond the box" with alternate answers and multiple solutions to solve any math problem had even opened my eyes on my own thinking of the questions being presented.

Some of these students were quite eager to present their question. They were able to explain the concepts well that I was able to solve it on my own, and with occasional help from the hints at times. The games were not overly complicated and can be understood easily for their level of learning. As well, I was amazed by how creative and talented these young students were at designing their own math games. Many students had gone out of their way to explore and find common (household) materials and turning them into beautiful and creative art forms!

SNAP Math Fair Response

Ted Lewis, the author of the Math Fair Booklet, poses some very interesting and realistic key points on learning math. Some people fear mathematics because they think that they are incapable of solving problems successfully, so that discourages them to continually pursue math questions. I know a fact that I, for one, feared doing math when I was a younger, especially when I was faced with more abstract math problems. For this reason, problem solving is a great focus to counter the fear of math, especially in Math Fair events because these skill sets can give everyone an opportunity to think and reason beyond the abstract level, into a more concrete and practical area of their lives. Now that the math fair is not been "graded" or awarded, students and visitors will feel less overwhelmed by failures as they are given the chance to perfect their learning continually and gradually.

If I want to incorporate SNAP Math Fair in my practicum high school at Point Grey S.S., I would first start with the younger grades, perhaps the Gr. 8 math classes because students at this age and grade are beginning to step into a more advanced level of math learning, and they are beginning to grasp math even in a more abstract sense. But of course, I should align complexity of math problems that is appropriate for students in Gr. 8 or 9.

The puzzles or brain-teaser math activities such as the ones proposed in the booklet would be a great starting point for those students to work on, and they should be easy enough for students to teach the visitors, such as families and friends, so they can understand the materials as well. I am also a firm believer that math learning should not be dealt alone. Students can think alone first to process the information and try independently, but ultimately, for more difficult questions they are best solved when students work/collaborate with each other. Perhaps, I can mix the Gr. 8s and 9s math together in groups of 3-4 members. For older grades, I can let Gr. 10s and 11s work together on solving the problems and presenting them to others. Finally, the Gr. 12s will have a chance to solve too, maybe with people within their own grade. Students from the mini schools can work independently within themselves too.

Because this school runs on a tight schedule, with each block of class being only 75 minutes long, time can be a constraint that may not work so well in a SNAP math fair (which can run as long as a day!) Because of the time constraint, I may need to simplify problems and puzzles or reduce the numbers of problems given and making them into groups of 3-5 to solve so problems can be solved quicker as students all pour their skills together. Another way to go about doing this is to combine all the students (from all grades) who have a math block at period 1 day 1, for example, together within this 75 min, then divide them based on their grade levels to avoid unfair math competencies. Once that is done, we move on to the next block. For presentations, students can continue to work and perfect their work on a day 1 again so visitors can come and solve themselves with guidance from the students.

November 13: End of Week 2 Reflection

I am writing this blog to describe the summary of my 2-week practicum. Overall, I have to say that I am very thankful of the opportunity to teach at Point Grey secondary school under the guidance of my English and math sponsor teachers and my faculty advisor. The students at PG are always full of energy, either it is the first block of the day, after lunch, or the last block of the day where students are dying to go home, they are always enthusiastic and approachable when it comes to learning.

The second week was a lot busier than the first week because I had to teach multiple of classes. I felt I was more nervous to teach the English classes because I have never taught English in a large setting before. In the past, I had mostly worked with individuals to teach grammar and ESL, so the opportunity to teach a class of roughly 30 students was an eye-opening experience.

I realized that in English, there is really no absolute right and wrong answer - as long as there is evidence to support the argument. This style of teaching is unlike teaching and doing math, where there is always one right answer for every problem, even though there may be more than one way of solving the problem. I understand that while teaching, the questions that I bring to the class should be open questions that give everyone a chance to think and discuss. I also realize that I need to be constantly coming up with real-life questions to stimulate students' thinking process on the spot, and those questions should always tie with the topic of my lesson. In other words, to be able to improvise is important to all teaching professions.

Ultimately, as a teacher, I am here to let students learn, think, and reason. I should be giving students the chance to think for themselves, under the proper guidance and tools. Just as a Bachelor of Education student myself, my faculty advisor, my sponsor teachers, and my Education professors are all here to make me (and the rest of us) a better teacher, for now and the future.

November 9 Reflection

Today, I was supervising and helping English classes in the morning and math classes in the afternoon. From my supervision and assisting the teachers and students, I realize that speaking with a higher volume is very important as a teacher - especially if we are standing in front of 30 something students! We need to make sure that students in the back can hear us clearly so they don't fall behind. Speaking loudly also shows a certain level of assertiveness and confidence in the subject we are teaching.

As well, having a clear instruction for all students is equally important. This ensures that students are all on the same page and clearly know what to do. They would know what is expected of them and what they should accomplish in class.

November 6: End of Week 1 Reflection

In the past couple of days during the week of my practicum, I have been busy preparing lesson plans, meeting the students and teachers, going to various non-math and English classes to learn about different teaching/class management styles.

I have attended a Chemistry 12 class, a Japanese 11 class, and a ELL class. I find that the language-based classes are generally are more interactive and involved than the math and science classes. The math and science classes are more subject-based (subjects and contents are more important), and in language/art classes are more analytical (analysis and thinking are more important).

I have two classes that are repeated for this short practicum, both math and English. I realize that even though they are the same classes, the students' response, thinking and behaviours in that same grade and subject area are all different. In addition, I realize that teachers have slightly different teaching strategies and methods for every class they teach (regardless if they are repeated or not).

For this one week being at a school, I learned about...

• Students are all different at their own levels - their competencies, abilities, personalities, etc.
• Teachers are all different in terms of their teaching and practices.
• The amount of discipline that's needed to control the classrooms are almost consistent from the youngest grade (9) to the oldest grade (12). E.g. some Gr. 9s need more discipline than the Gr. 12s, and some Gr. 12s need more discipline than the Gr. 9s.
• Students at Point Grey are very passionate about what they learn and what their hobbies are. From the clubs that I have attended and observed, students are not only passionate about the big things (e.g. humanitarian clubs) but also about the littlest things that I often don't think about, which amazes me!
• Students do notice me and acknowledge my presence even if they didn't say Hi to me in the classes I observed. (They always come up to me and tell me that I was in his/her's this class and that class, and I didn't even know!)

November 3 Reflection

Today is a very interesting day. My FA, Susan, drop by Point Grey s.s. to resolve some administration situations. I have a new math SA now, which I am very happy about.

I met up with my SA and the two of us talked for awhile about my lesson planning and what content materials her students are learning for each class. I realized that even though she is teaching multiple of the same classes every week, the class atmosphere, environment, structure and the students are very different. For instance, the morning Gr. 11 morning class is very different from her Gr. 11 afternoon class.

Today, I also had the chance to interview the librarians at the school. I never knew they had so much responsibilities to do at the back-end. While I was interviewing them, I never realized how difficult it is to recieve tech support (i.e. fixing computers, software programs, etc.) at this school. They openly admitted to me that tech support is very low, and they needed more staff who can help them resolve any computer/programming issues. The staffs aren't trained on how to resolve tech issues because most of the problems are fixed by and at the central school board. So most of the time when something is down, they have to wait until the school board tech staff fix the issues remotely on their end - so it's really a waiting game.

November 2: First day of short practicum!

Today is my first day of our short practicum at Point Grey Secondary School. I met a few other UBC and SFU teacher candidates who will also be working with sponsor teachers in various subject areas. I am the only math TC at this school.

The morning began with a detailed orientation and breakdown of work procedures in the classrooms. We toured around the school and met different teachers and staff members along the way. Point Grey is a very big school! It is quite old as it was built in the 1920s, but the school has maintained itself very well since then. The staff was very friendly and knows what's going on at school. Our secretary is our go-to person if we ever needed help on admin.

Some other interesting things that I have learned today:

• The school alone has over 60 student clubs! I noticed that there isn't a math club... maybe I can start one!
• I'm really interested in their Comp Sci Club and the Dance Team. I want to check them out! They sound really fun.
• A slight change of plan on who my sponsor teachers are going to be. Hopefully we can resolve this by the end of this week.
• The students in the math and English classes that I have attended are very energetic and friendly. I dropped by a pre-calculus 11 class with fewer than 15 students. It was a quiet class and majority of the class time was devoted to homework time and extra practice. I was able to help out the students whenever I could.
• Students in the AP math class are very ahead and the teaching pace is very fast. Teachers allocated 5-10 min work time before taking up the questions.
• For those who are in enriched or AP classes (especially in Gr.11 or 12), I'm curious as to what they want to pursue in post-grad - maybe I can help them if they want to study math!

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.

Pro-D Day

For this year's professional development day, unfortunately I will not be in the province to attend due to my convocation on the same day. However, I am looking to see if there are other PD-related courses as part of BCMT.

My Best and Worst Math Teacher

From my past experience, I have had decent teachers that taught me mathematics. My best teacher was my high school math teacher who taught me math throughout my high school. I have to say that she was the one who motivated me and inspired me to become a math teacher. Even though she was a tough teacher, she really pushed me to work hard in class. In fact, she pushed the entire class to do really well. I guess it's the perk of being in an enriched class than the regular academic classroom. Sometimes, if I answered questions wrong, or take longer to understand certain concepts, she gets disappointed in me or almost felt like ashamed of me for answering the questions wrong, which I took it personally. It is almost as if I'm letting my parents down. This kind of "failure" really impact me in class. Perhaps, she expected a lot from me, and that not getting the correct answers isn't something that I normally do? Sometimes, I get intimidated by her, and the questions she asked in class or on the tests were challenging. However, if it weren't for her tough love and push I wouldn't be able to do well in math as I am now. So, I thank her for that.

In all, I think my worst math teacher is also my best math teacher. If it weren't for her tough love, her lack of patience and her constructive criticisms, I wouldn't be able to push myself to pursue math as my undergraduate study. I think she definitely improved my understanding in math (the technical aspects) for sure. There is another female teacher in high school who taught me Gr.10 math. She has not only taught me the "technical" aspect of math, but also taught me the "soft" side, the personal side, of teaching math. She was so awesome as a teacher that I decided to work with her as a volunteer teaching assistant at her new high school. She was more patient with her students than the first teacher I wrote about in this blog. Both were funny math teachers, even though one is more serious than the other, but they have all pushed me hard to help me reach my goal as a student.

Blog 4: My TPI Reflection

As seen in this TPI Profile Sheet Table, my result seems pretty consistent throughout. On average, I scored 32.6 on the scale, with a standard deviation of 2.65. I scored high on Apprenticeship teaching style and lowest on Social Reform, while Nurturing, Developmental, and Transmission teaching style are fairly consistent. From looking at the results, I would think that I scored high on Transmission because when I was tutoring or teaching students, I often value the subject matter (the knowledge) more highly than the teaching styles. As a student, I would think knowledge is the ultimate key to understanding because without it, there would be nothing to teach the students or us teachers! However, as I read more on other areas of teaching, I start to realize that there are more areas than just simply teaching the contents. As a teacher, we are no longer teaching the subjects, but really teaching the students. The high-scored Apprenticeship area tells me that, as a teacher, I "translate" knowledge into "accessible language" for students. I am here to provide "guidance and direction" to engage learners with ways that they feel more comfortable of learning. I find this result of teaching very true.

I remember when I was tutoring students on math areas that I felt comfortable with sharing, mostly because I learned and understood before as a high school student, I noticed that some students weren't getting the concepts, even if I explained to them in most simplest terms. I realized that how my teacher taught me was way different (and a bit more complicated) than how their teachers had taught them. I guess I didn't expect teachers within the same school board to teach students differently. As a result, they just weren't getting the concepts. To resolve this, I had to adapt my ways of learning to their ways of learning, to "translate" my thinking into a language that is "accessible" to them that they are more familiar with. It is important for me, as a teacher, to read and really understand students and their needs (i.e. strengths and weaknesses) so I would know what best strategies to teach them - to see whether they needed support or additional remedial assistance.

I am quite surprised that I scored lowest on Social Reform. Social Reform, as written on the TPI website, is the "effective teaching [that] seeks to to change society in substantive ways." I had always thought that mathematics is really practical in ways that can build our society. The world we live in is immersed with the concept of math - it can be found in science, engineering, and even arts - they all have math as a fundamental basis. I really encourage students to associate what they are learning to the world around them; to let them apply math in a practical setting is the sole purpose of math studies. As a teacher, it is also important to let students take critical stances on what they are learning (to mature their brain activities). As the guest speaker at one of my other class said to us earlier, "the most difficult child is the one who learns the most." I guess I will have to improve my my social reform teaching style!

Blog 3: How Many Squares Are in A Chessboard?

For this fun activity, we are given an 8x8 chessboard. How many squares are there?

Before we get excited and jumping into conclusions, if you are a visual person like me, you can start by either having the chessboard in front of you, or you can start by drawing an 8x8 grid on a piece of paper.

1. Once you have the visual in front of you, let's start by looking at 1x1 grid.

Think:
How many 1x1 grids are in an 8x8 chessboard?

Visual example.

Answer:
Since there are 8 pieces of 1x1 grid horizontally and 8 pieces of 1x1 grid vertically, we can find the total numbers of 1x1 grids by calculating the area.

There are 64 pieces in total. (You can count them all to be sure!)

2. We now look at 2x2 grid.

Think:
How many 2x2 grids are in an 8x8 chessboard?
Hint: think about how many 2x2 grids are there horizontally and vertically. Include any (overlapping) combinations of 2x2 squares, as seen in red.

Visual example.

Answer:
If you continue to find 2x2 grids, we can see that there are 7 pieces of 2x2 grid horizontally and 7 pieces of 2x2 grid vertically.
There are 49 pieces of 2x2 in total.

3. We now look at 3x3 grid.

Think:
How many 3x3 grids are in an 8x8 chessboard?
Hint: think about how many 3x3 grids are there horizontally and vertically. Again, include any (overlapping) combinations of 3x3 square, as seen in red.

Visual example.

Answer: 
We can see that there are 6 pieces of 3x3 grid horizontally and 6 pieces of 3x3 grid vertically.
There are 36 pieces of 3x3 in total. 

Do you notice any patterns?

See how as the grid size increases by 1, the numbers of pieces of square decreases by 1? As the pattern continues, we can create a handy-dandy table that tracks how many squares are in each case. You do the calculations!

So now that you have counted or calculated how many squares there are from 1x1 grid to 8x8 grid, what's next?

We can add the numbers of grids of all sizes together!

Why?

Let's look at a similar (simple) example: How many squares are in a 2x2 chessboard?

A 2x2 chessboard is consists of 4 pieces of 1x1 grid and 1 piece of 2x2, since the chessboard itself is a square. In total, there are 4+1=5 squares.

Final answer:
Similarly, in an 8x8 chessboard example, we have to add all the numbers of squares together because we are looking for the total number of squares.

In total,
There are 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares in an 8x8 chessboard.

Check: Did you do the calculations right?


This puzzle can be extended if we ask how many rectangles of a certain unit sizes are in the 8x8 chessboard. For instance, we might ask: Find the numbers of 2x3 rectangles are in the chessboard, or find the numbers of 3x2 rectangles in the chessboard. To make this activity even more interesting, we can extend this chessboard to a 3-dimensional objects, like a Rubik's cube! This might pose a challenge to some students, but it is certainly more fun as we get our brains moving!

Blog 2: Personal Reflection on Instrumental vs. Relational Learning

The choice of having an instrumental or relational understanding for students is very subjective among teachers, and I think it really depends on students' learning abilities to see which choice of learning is the best or preferable. In our debate, I think many of us teachers candidates used our learning strategies when we were students and how we think should be a good learning method for our future students. In my opinion, depending on the grades we teach, the overview of math concepts should be introduced first. For example, teachers should introduce the signs of addition (+) or subtraction (-), multiplication, and division. Once it is introduced, or it has been taught "instrumentally", students can then dwell deep into why addition exists, or why division exists, and how these concepts are practical in real life. Provided with these questions, teachers can then explain things such as "When you are at a supermarket, you may want to calculate the cost of each apple, and see which one is the cheapest!", for instance. Teaching students the concept of the order of operations is another example. We first introduce the concepts to students "instrumentally", then we can explain to them why this order of operation, BEDMAS, is important in real life, "relationally".

Furthermore, as we go on to teach at higher grades, complex math concepts such as the Pythagorean Theorem or the vertex form of a quadratic equation, in my opinion, should be taught first "instrumentally" as well. Once students understand fully these concepts and are able to, fluently and correctly, solve questions using the concepts as tools, we can further let them question why and how certain theorems work that way. Of course, the real meaning behind these formulas, proofs and theories is the foundation to a better understanding, but It doesn't necessarily mean that students with instrumental way of learning is wrong or are less knowledgeable than students with relational way of learning. It just means that they both understand the same concepts and that they both know how to apply those concepts as a tool to manipulate them in order to fit their learning framework. In all, I think students have various preferred learning strategies, and it is how they choose to learn that are ultimately important to our math teaching.