What Teaching During the Coronavirus Outbreak Has Been Like for Me

​ Shortly before the start of our Chinese New Year holiday at the end of January, news had started to spread about a new virus in Wuhan, China. By the time I actually left the country, virtually everyone was wearing a face mask and activity at all major transportation hubs (railway stations, airports) had basically stalled. For a while, it seemed like we were able to escape the mass hysteria that was beginning to ensue and enjoy nearly three weeks of worry-free traveling around the Philippines.

My travel companion, Jose, and I had been keeping a close tab on the coronavirus situation, and we were warned by my relatives in Hong Kong to stock up on as many face masks and hand sanitizers as we could while we were in the Philippines as they were virtually sold out everywhere in Hong Kong and China. We had originally planned to return to Shanghai on February 16th but had been notified by our principal that the start of physical classes in China had been delayed until at least March 2. At that point, the number of reported infected people had been raising still and we contemplated travelling elsewhere to ride out the situation. The problem was, we had, and still have, no idea how long this situation would last, nor have we been given any sort of certainty as to a specific return date for work.

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The streets of Hong Kong seemed emptier than usual.
Jose ended up returning to Suzhou, where we both work and undergoing a 14 day quarantine, which was monitored by the building management. I made a last minute decision to return to Canada. Both our decisions were spurred on by an unfortunate encounter with bed bugs (we suspect), and us having to deal with two very different sets of symptoms that caused us a lot of emotional stress and worry. Luckily for us, we’re on the path to recovery. As far as I know, majority of international teaching staff from our school are taking “extended vacations” (using this term loosely here) in various countries around the world. A few have opted to go back to Canada, some returned to China, and a few never left.
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A snapshot of my class WeChat group. As you can see most of the posts are announcements from me, haha!
With all this in mind, teachers, administrators and staff members hustled to get an online instructional plan in place for the start of semester 2, which began on February 19. Our main learning management system is Moodle and it is a platform that our school has been using for a few years. We use Moodle  to communicate information and share resources and lesson materials. Since we are teaching in China, WeChat (the Chinese equivalent to WhatsApp) has also been indispensable as a communication tool, especially since Moodle was unprepared to handle such a high volume of users, or accommodate our rapidly growing storage needs (our brilliant IT team has been able to curb many of these issues since then, but the server still undergoes regular maintenance causing minor disruptions in our workflow).
I know that many teachers and schools express issues with using WeChat as a way to communicate with students, and this was something I had a lot of hesitations with as well, which is why I’ve never created WeChat groups for my classes in the past. Over my last few years though, I’ve quickly realized that it is really the best and fastest way to reach students, and I’ve joined and created WeChat groups for sharing or keeping up to date with school-wide announcements, communicate with course teams or departments in the school, or get in touch with students who are part of extracurriculars I’m running.

WeChat is not just a messaging app, but also has social media features, payment options, and several other utilities built in. In short, WeChat is pretty much woven into the fabric and lifeblood of what living and working in China is like. That said, it is THE number one tool to utilize if you are looking for a stable and reliable method of communicating with people in China. No server issues, no need for a VPN… so while privacy is still a concern, it is now a part of my online instructional plan. (There is an option to limit communications with contacts to “chats” only so you can hide your social media posts).
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Students have been learning with a mix of interactive lessons, course notes, formative quizzes, and live sessions.
Moodle is, and remains, the MAIN communication platform for students to access course materials, view links to filmed live sessions, submit assignments…. And so on. A couple of other tools that my colleagues have introduced that I’ve found extremely helpful for my classes include Zoom, an online conferencing tool, and Loom, a video recording software that uploads any videos you make onto a cloud and sharing a video is as simple as copying and pasting a link.

Given that we’ve been fully online with our learning for about two weeks now, we’re addressing minor hiccups as we go, adjusting the pacing of our lessons, and working on finding authentic ways to assess student learning. We’re thinking about how to troubleshoot potential issues with academic honesty and ways to get an accurate and holistic picture of how our students are learning. The biggest unknown at the moment is when we will be back in the classroom, and how the coronavirus situation will pan out… Guess we’ll just have to wait and see.  ​​

Map Projection

We’ve been looking at map projections for my masters course and I continue to be blown away by how embedded mathematics truly is in our every day lives. As a self-identified directionally-challenged individual, geography and anything like it is to be avoided at all costs. I find myself at my wit’s end now and have to admit that even maps hold a lot of mathematically intriguing ideas that are worth exploring. The course I’m taking now is called “Math for Global Citizens”, offered at the University of Waterloo to MMT students, taught by Judith Koeller and is arguably one of my favourites in the program. 

 

The problem of the “flat earth” has been around for centuries. It is believed that as early as the year 354, pre-medieval scholars asserted that the earth was in fact spherical (University of Waterloo). The problem for map-makers, then, is to find a way to depict a spherical object on a 2D surface, and this is turns out to be an impossible task. Take a look at the animation below for what’s called a “Myriahedral projection” developed by Jack Van Wijk from the Netherlands. 

 

The idea behind a “Myriahedral projection” is to split the earth up into polygons, thousands of them, in order to preserve both shape and size of major land masses or bodies of waters (see article here). Map projections have not always been so advanced however. 

 

In trying to depict a spherical surface onto a 2D plane, one can try to preserve distances, shape, areas, or shortest distances between points by straight lines. It is impossible to have all these desirable properties in one map. For instance, the Mercator projection map is the one that we are probably all most familiar with as it preserves angular distances, making it easy for navigation, but it drastically skews areas the further away the land masses are from the equator. See this true size (thetruesize.com) comparison below, showing how large the continent of Africa actually is compared to  the US, China and India:

On that note, I would highly recommend checking out thetruesize.com and just playing around. 

 

Here’s another great video explaining “Why all world maps are wrong” that was recommended to me by Mr. Schwartz, a geography teacher and the humanities Department Head at my school. 

 

Ch 1. What’s the Point?

What is the point? The point is to do math, or to dazzle friends and colleagues with advanced statistical techniques. The point is to learn things that inform our lives.

– Charles Wheelan

[PREFACE: I purchased Naked Statistics by Charles Wheelan many years ago, thinking its an important book to add to any Math Teacher’s arsenal (and it is!) but had only gotten through the first three chapters before dismissing it for another read. It is not a boring book – quite the opposite in fact – but I felt that mere passive reading was not enough for me to really retain the important ideas and intuition that Wheelan is trying to impart to his readers. This time, I’m giving it another chance and plan to summarize material I am learning, relate it to my own experiences, and share that learning here on my blog.]  

I wrote about why statistics matters in a previous post. Here, I continue to elaborate on the point as I summarize my biggest takeaways from the first chapter. This chapter provided an overview of big ideas in statistics that we’ll be learning about throughout the book. 

Description and Comparison 
Descriptive statistics is like creating a zip file, it takes a large amount of information and compresses it into a single figure. This figure can be informative and yet completely striped of any nuance. Like any statistical tool, one must be careful of how and when we employ such figures and the implications it might have on the audience. 

So a descriptive statistic is a summary statistic. Let’s start with one that many of you may already be familiar with – GPA. Let’s say a student graduates from university with a GPA of 3.9. What can we make of this? Well, we might be able to discern that on a scale from 0 – 4.0 a GPA of 3.9 is pretty darn high. But some universities grade on a scale of 0 – 4.3, accounting for a grade of A+. What this simple statistic doesn’t tell us is what program did the student graduate from? Which school did they attend? Did they take courses that were relatively easy or difficult? How does this grade compare with others in the same program? Wheelan writes, “Descriptive statistics exist to simplify, which implies some loss of nuance or detail (6).”

Inference 
We can use statistics to draw conclusions about the “unknown world”  from the “known world.” More on that later. 

Assessing Risk and Other Probability Related Events
Examples here include using probability to predict stock market changes, car crashes or house fires (think insurance companies), or catch cheating in standardized tests. 

Identifying Important Relationships
Wheelan describes the work of identifying important relationships as “Statistical Detective Work” which is as much an art as it is a science. That is, two statisticians may look at the same data set and draw different conclusions from it. Let’s say you were asked to determine whether or not smoking causes cancer? How would you do it? Ethically speaking, running controlled experiments on people may prove unfeasible, for obvious reasons. 

An example Wheelan offers here goes something like this:
Let’s say you decide to take a few shortcuts and rather than expending time and energy into looking for a random sample, you survey the people at your 20th high school reunion and look at cancer rates of those who have smoked since graduation. The problem is that there may be other factors distinguishing smokers and nonsmokers other than smoking behaviour. For example, smokers may tend to have other habits like drinking or eating poorly that affect their health. Smokers who are ill from cancer are less likely to show up at high school reunions. Thus, the conclusions you draw from such a data set may not be adequate to properly answer your question. 

In short, statistics offers a way to bring meaning to raw data (or information). More specifically, it can also help with the following:

  • To summarize huge quantities of data
  • To make better decisions
  • To recognize patterns that can refine how we do everything from selling diapers to catching criminals
  • To catch cheaters and prosecute criminals 
  • To evaluate the effectiveness of policies, programs, drugs, medical procedures, and other innovations
  • To spot the scoundrels who use these very same powerful tools for nefarious ends 

(Wheelan 14)

Lies, damned lies, and statistics.

 – Mark Twain

Why We Should Care About Statistics

It’s easy to lie with statistics, but it’s hard to tell the truth without them.

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​[PREFACE: I purchased Naked Statistics by Charles Wheelan many years ago, thinking its an important book to add to any Math Teacher’s arsenal (and it is!) but had only gotten through the first three chapters before dismissing it for another read. It is not a boring book – quite the opposite in fact – but I felt that mere passive reading was not enough for me to really retain the important ideas and intuition that Wheelan is trying to impart to his readers. This time, I’m giving it another chance and plan to summarize material I am learning, relate it to my own experiences, and share that learning here on my blog.]  

A couple of days ago, my younger brother, who just started his first year in university in the Fall, was complaining to me about the woes of student life; in particular, the obsession with grades and the paradoxical lack of willpower to work for them. Having taken an accounting class together, his friend recounted to him that it was, “The sketchiest 90 I ever received.” Let’s break that down for a moment. Humble brag? Yes, but what he really meant was that his friend was blindly memorizing formulas, plugging and chugging without any idea how they were derived and why they are meaningful. 

Does that sound familiar? How many of you have had similar experiences in math class? I know I have. Not just math, but in science, language arts, history… sometimes it feels like we are just memorizing facts in isolation without an understanding of their greater purpose. To be fair, I’ve taken statistics classes that feel no different, a series of formulas that need to be applied to raw data. What makes statistics inherently different, however, is that unlike calculus or algebra courses, which often teach skills in isolation of their applications (to which I will argue there is intrinsic value in knowing and learning, another post perhaps) statistics IS applied mathematics. Every formula, number, distribution test…etc. is meant to clarify and add meaning to everyday phenomena (though, when wielded improperly, can have the opposite effect).

Statistics are everywhere – from which are the most influential YouTubers, to presidential polling to free throw percentages. What I love about this book is that it focuses on building intuition and making statistics accessible to the everyday reader. A quote by Andrejs Dunkels shared by the author, “It’s easy to lie with statistics, but it’s hard to tell the truth without them.”


Make It Stick

This blog post is about how the math department at my school in Suzhou, China implemented changes to the way we taught Math 10 and 11 to incorporate data and research from cognitive science to help our students learn better. I include a summary of what we learned, and some ideas for improvement.

 

It began last summer, at math camp. Yes, I attended math camp as a fully-fledged adult! Yes, there were other adults present. And yes, it was awesome! (Officially named the “Summer Math Conference for Teachers” but let’s not get into the nitty gritty). One of my favourite sessions was the one led by Sheri Hill, Arian Rawle, and Lindsay Kueh on the grade 10 course redesign they have implemented in at their school in Ontario. The course redesign is based on research and best pedagogical practices outlined in the book Make It Stick, The Science of Successful Learning by Peter C. Brown, Henry L. Roediger III, and Mark A. McDaniel.

Book Synopsis: Why is it that students seem to understand what is being taught in class but end up failing when it comes to test day? How does one progress from fluency to mastery over challenging content? Many common study habits like re-reading and highlighting text create illusions of mastery but are in fact completely ineffective. This books explores insights from research in cognitive science on learning, memory, and the brain, as well its implications on teaching and learning. 


THE WHAT

​After the session, I couldn’t wait to bring these ideas back to the math team at my school in Suzhou, China so we could start putting them in action too! We began by looking at issues we noticed our students faced:

 

  • Not knowing, understanding, or practicing math vocabulary
  • Low retention rates of material from one year to the next (in some cases from one week to the next!)
  • Lack of basic skills (algebra, numeracy)
  • Low perseverance
  • Low completion rates for homework

We made it our goal to address some of the issues above, taking many ideas directly from the session presented by Hill, Rawle, and Kueh.

Like Hill, Rawl, and Kueh, we removed unit tests, which freed up a significant amount time for other topics and activities. Instead, we moved to weekly cumulative quizzes that held students accountable to everything they have learned in class up to the Friday before quiz day (no skills expire!).

The weekly schedule looks as follows:

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Our school runs on 80 minute blocks, with Fridays being half days with 40 minute blocks.

 


​The HOW and WHY

Fast Fours. Four warm up questions printed front and back on half a sheet of A4 paper that’s ready for students as they walk into class. Each question relates to a different math topic that may be review from previous years, numeracy focused, or review of current material. By mixing up the problem types, we are introducing interleaving to students, the idea that we learn better when multiple topics or subjects are woven into the same learning session.

 

If you are interested in redesigning your course or looking for ideas on where to begin, I would say the Fast Fours are the easiest to implement. They work well because in all likelihood, every student is able to answer at least one question out of four, it is low stakes (not graded), gives you time to check in with students at the beginning of the class, check homework, answer questions, and it also gives students an opportunity to collaborate and help each other. (Scroll to the bottom of this post to see examples of these documents).

Weekly Quizzes. The quizzes themselves are one-page, double-sided documents comprised of three main sections. Part A focuses on vocabulary where we ask students to match key terms or fill in the blanks. Part B is review of previously learned material, and may include basic algebra questions from previous grade levels. Part C is new material that was covered the week before.

The quizzes only take up half a block and we drop the lowest two quizzes at the end of the year. If students are away for a quiz, they do not write the quiz and instead it counts as one of their dropped quizzes. 

Problem Solving/Project Days. At the start of the semester, we wanted to implement problem solving days that helped students dive deeper into the content they learned throughout the week and try some more challenging problems. Alternatively, our vision was to use these days as project days.

Fun Fridays. Our Friday blocks are shortened and many teachers find this time unproductive for teaching new material, which made having Fun Fridays built into our schedule a good fit. During this time, we may play a fun review game with students, have them explore an activity on Desmos, or one might even teach them something outside of the prescribed curricular content like coding.

Homework. In their original course redesign, Hill, Rawle and Kueh wrote customized homework assignments that introduced the ideas of interleaving and spaced practice to their students. That is, their homework assignments would begin a set of ten mandatory questions: five questions from previous material, and five questions from the lesson, as well as one or two challenge questions.

Unfortunately, our team was unable to implement so many changes at once, so we simply kept homework the same, and instead implemented randomized homework checks. Our hopes were to emphasize the importance of practice, and keep students accountable for it. 


​RESULTS

Some of my takeaways from this semester:

 

  1. Fast Fours. I’m definitely keeping the fast fours in my classes. In their of year reflections, students mentioned that this was one of their favourite things to do because it helped them remember content they had not practiced for a while, and they were able to get immediate feedback on it. One student suggested having a balance of easy and more difficult questions for those who finish early (perhaps a 3:1 easy to challenging ratio).
  2. Weekly quizzes. Since the weekly quizzes introduce interleaved and spaced practice, they reduce the need for large blocks of class time devoted to final exam review as we were continuously reviewing content throughout the entire semester. In terms of grading, however, it is important to keep up with it to ensure that students get feedback before the next quiz. An outcome we did not expect was that despite seeing several iterations of the same types of questions, students continued to struggle with the finance unit and were unable to identify the correct formula to use for the question.
  3. Vocabulary. As a department, we found it extremely valuable to teach, review, repeat, and practice math specific vocabulary to help students increase fluency and be better equipped to answer difficult problems. Many Chinese students arrive in our classes already having much of the essential background knowledge in math but lack the English skills to succeed, so we have found this to be a fruitful endeavor. We plan to begin our Math 10 classes with a mini vocabulary unit to give students started with some common terminology and foundational knowledge for the upcoming semester.
  4. Problem Solving/Project Days. Problem solving was a lot harder to implement, and we did not have a clear structure for it. As a result, Thursdays were mainly used for projects or as additional lesson days.

Overall, the teachers in my department felt the changes were worthwhile to implement and will continue with the same program for semester two, with a few new projects that we’ll be adding to some units that did not have one. In the future, I’d like to rethink how we might implement problem solving days in a more structured way. 

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Select responses from student feedback regarding Fast Fours in Pre-Calculus 11 (Nov 2019)

 

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Select responses from student feedback regarding Weekly Quizzes in Pre-Calculus 11 (Nov 2019)

 

AREAS OF IMPROVEMENT and NEXT STEPS

Finance Unit. I’m unhappy with the way finance is currently being taught to our students, and I think we can do better. I remember very little in the way of learning about finance when I was in high school. This was usually the topic my teachers skimmed over, and hence my dislike for it as a teacher now myself. In textbooks, it is usually presented as a series of formulas and how to apply those formulas, which is, I think, an area where we are doing our students a disservice. I’m a strong believer in getting students to first understanding the math behind the formulas, and some of these formulas (like the loan formula, for instance), does not lend itself well to building students’ conceptual understanding of it at the grade 10 level.

 

A useful analogy from Barbara Oakley’s course Learning How to Learn goes like this: a formula is like a summary, it describes several important ideas that mathematicians have packaged into a simple and elegant mathematical statement. Take Newton’s second law of motion for example, which is stated formally as, “The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object” (physicsclassroom.com). Simply put, the relationship can be condensed into the mathematical formula f = ma.  As such, we must understand the meaning behind each symbol and look at how they work together to tell a story.

My plan is to put together a rough plan for how we can revamp the finance unit, pitch these ideas to my team before the start of semester two, and see if we can collectively find a way to improve the way we teach this unit to our students (more to come in a later post).  

Active Retrieval. Next semester, I plan to pause frequently during lessons to quiz my students on material. I’ll ask them to put their away notes, and engage in some simple recall exercises. A useful analogy to think about this is described in Make It Stick; Dr. Wenderoth, a biology professor at the University of Washington tells her students to “Think of your minds as a forest, and the answer is in there somewhere… The more times you make a path to find it, the stronger that path will become.” This is exactly what happens in your brain as you engage in active retrieval to strengthen new neural connections as you gain new knowledge or learn new skills. 

Elaboration. Once a week, I will ask students to complete a written reflection or summary of ideas learned throughout the week in their own words, with added connections and extensions if applicable. It will be a five sentence summary of concepts learned, with enough detail to help recall important ideas when it is read it at a later date, not too much detail that students end up reciting the entire lesson. 

 

Sample Fast Fours
File Size: 38 kb
File Type: pdf

Download File

 


 

Sample Weekly Quiz
File Size: 891 kb
File Type: pdf

Download File

 


Teaching for a Math Mindset: A Not Yet Successful Study

So I was lucky enough to have the opportunity to teach in the Head Start summer program at my international school here in China. The program is intended to help students going into high school to gain exposure to full English immersion classes in Math, Science, Socials, and Language Arts. I taught four blocks a day for 70 minutes each. Each class had anywhere between 12 – 16 students. Ten days straight; on the one hand, no break (kinda brutal), and on the other, open curriculum (YES! Free reign).

I had lofty plans. I’d been refreshing myself on Jo Boaler’s work about mathematical mindsets (see my previous ramblings here). I was going to do a little study.  Please note that I do not have any experience whatsoever doing educational research. While I have a general understanding of the scientific method, I was mostly doing this out of pure curiosity and a desire to become a better teacher.

Like all good mathematicians and in the name of good science, it was perhaps inevitable that first time was not the charm, and rather than have a very successful, replicable study, I instead gained some knowledge about how I might proceed in the future. Nice.

Content that I had planned to cover in 10 days would have taken closer to 18. The students had an incredible range of English speaking ability, with drastically varied dynamics between groups of students. The schedule did not operate on a cycle, so I saw the same group of students at the same time each day, which definitely influenced their learning experience. For instance, Group C who were absolute angels and ready to learn each day in my first period class were exhausted by the time they got to third period, which led to more behavioural problems in the classroom.

STUDENT DYNAMICS
Group A: A challenging group. I saw them the period right before lunch each day and there was a group of four students who were unable to sit still and wandered the class during inappropriate times, such as in the middle of me giving instructions. I lost my cool on this group; shame on me because I wasn’t able to regulate my emotions and respond calmly to the situation. Just to clarify, a “losing my cool” moment for me doesn’t mean shouting or yelling, which is neither helpful nor productive. I simply raised my voice to get the students attention. But, in that moment,  I had lost my cool because I let the students dictate my response rather than carefully assess the situation and respond calmly and accordingly.

Group B: Did absolutely anything in their power to NOT pay attention. Would whine anytime I introduced a new activity. Would put their heads down and sleep in class. I saw this group after lunch each day, they were my last and perhaps most challenging class because of the incredible amount of sleepers and students who wanted to do absolutely nothing. There were definitely some gems in this class that would have benefitted from being in a group with other, more responsive students. Lots of patience and flexible teaching strategies required.

Group C: The first group I saw each day and by far the best group. Students had a decent command of English and I rarely had to repeat myself. They would listen and follow instructions the first time. Students would always do as they were asked. The challenge with this group was pushing them to work slightly beyond their zone of proximal development.

Group D: A diverse group with students who always wanted to be two steps ahead, students who needed a lot of personal assistance, students who got distracted easily, and students who were happy with just coasting along.

HOW I COLLECTED DATA
I used Boaler’s Mathematical Mindset Teaching Guide as a self assessment tool for how I was and was not strengthening growth mindset culture in my math classroom. I wanted to focus on changing students’ inclinations towards math learning, challenging those who believe math is a subject that defies creativity and passion, and pushing those who already saw themselves as “math” students to expand their definition of what math is. With the help of my math mentor, I settled on collecting data through a mindset survey.

Students took a before and after survey. I added two prompts on the after survey that required students to provide written answers to the following:
– What I think math is…
– How math class makes me feel… 

A source of error here is that for students with low English level, they may not have fully understood the meaning of the statements they were agreeing or disagreeing with. Another possible source of error (though unavoidable) are those students who “did” the survey by randomly clicking boxes just to appease their dear teacher.

HOW I TAUGHT
I chose content from YouCubed’s Week of Inspirational Math. I chose these tasks because they were all low-floor, high-ceiling tasks and were designed to build good mathematical habits of mind. For example, on day 1, we did an activity called “Four 4’s” which encouraged students to think creatively and work collaboratively to come up with as many expressions as they can that equal the numbers 1 – 20 using only four 4’s and any mathematical operation of their choice (see picture below).

Other activities we did:

  • Escape Room Challenge: A mixture of math puzzles, grade 9/10 content from trigonometry, polynomials, and simplifying expressions. Designed by me and was meant to last one period, ended up taking two.
  • Number Visuals: Students examined visual representations of numbers 1 – 36 and were asked to identify and describe patterns (prime v composite numbers, factorization…etc.).
  • Paper Folding: An activity from YouCubed that challenges students to slow down and justify their answers. (Meaning that, anybody who claimed they were “finished” after five minutes clearly did not understand the activity…)
  • Movie: Students complete an agree/disagree questionnaire and watched The Man Who Knew Infinity about an Indian mathematician named S. Ramanujan making waves in England. Great movie starring Dev Patel. We did a discussion circle afterwards that touched base on prompts from the questionnaire that students were interested in exploring. (E.g. “Math is creative”)
  • Pascal’s Triangle: Find and describe patterns hidden in Pascal’s triangle.

In terms of assessment, I wanted to stay as far away from tests or quizzes as possible. Instead, I focused on providing students with specific, written feedback on their journal entries, group quizzes, and one final presentation at the end. I wasn’t concerned so much with what they knew, but rather the process through which they were learning and engaging with the material.

The Four 4s Activity

Students working on the escape room activity.

Looking for patterns in the Visual Numbers activity

That time a puppy wandered into my classroom. Oops.

RESULTS

Before

After

Select responses to “What I think math is”
Interesting”
“The most important things we need to learn”
-“Have unlimited creativity”
“Magic”
“Subject between creative and and teamwork”
“is very interesting. make my brain growing”
“beautiful”
“fantastic”
“Math makes me hate and love”

Select responses to “How math class makes me feel”
“Better”
“Moer interesting than chinese class”
“It may not very interesting, but OK”
“happy that I learned a lot”
“I feel very good, I meet very good teacher also know the very good friend in the math class”
“exciting”
“I feel happy when I fiand the ancer”
“free”
​”Good! make me more confedent”

WHAT I LEARNED
​A majority of students already had tendencies towards a growth mindset in mathematics, perhaps as a result of the general high regard Chinese people hold for mathematics as a subject. For the most part, students liked math and saw themselves as capable of achieving if they worked hard enough. Of the 59 students I taught, a small number of students (three or four) were of the opinion that they were “just not math people” and were extremely hesitant in trying.

In the end, I can’t really say definitively which factors of my teaching influenced (or failed to influence) a stronger growth mindset towards maths. What I do know is that the switch to low-floor, high-ceiling tasks was extremely freeing — for me and for the students. It allowed us to take a concept or idea as far as we wanted to go. There was no script or prescribed problem set that the students had to work through in increasing levels of difficulty, but rather a greater depth of thinking, and the time and space for that thinking to happen. Despite (or maybe thanks to?) the lack of testing (there were none), students still engaged with the tasks and content at high levels, drawing conclusions they might never have done with a pre-made worksheet of the skills they were supposed to practice.

By building a stronger focus on increased depth of knowledge, it then follows that a necessary norm to advocate would be that math isn’t about speed. When people refer to themselves as not “math people”, that’s usually what they refer to, the fact that they aren’t fast at mental arithmetic. But math is so much more than that.

In all, while it is hard to say from the students’ perspective whether or not they appreciated a stronger switch to teaching with mathematical mindsets in mind, I know that for me it resonates as a noble endeavour. Yes, it is much easier to write a test and spend 70 minutes of your life making sure no one cheats. But take that same test, rip it up, and replace it with a diagram, an equation, a single question, a blank sheet… and possibilities begin to emerge. Some groups may reach a higher level of understanding and some may not, but then again, we teach students, not subjects.

Turn Your Classroom into an Escape Room

I recently attended a professional development session led by a colleague titled, “How to Make Any Worksheet into an Escape Room,” which helped us experience an escape activity from the student perspective. It was the bomb. Dot com. The session touched on ideas expressed in this article, which happens to share the same title.   

Two weeks later, I ran an escape room in my classroom. It was the most fun I’d had all year. 

Cue intro. Goal: Answer the question, “what is life?” Other than that, I gave my students VERY little prompting. I figure I’d let all the mysterious new locks that had been placed in my classroom do most of the talking.

In order to answer the question, they need to collect all four puzzle pieces, which eventually led to this:

The escape activity was designed to work in a linear fashion, so students had to unlock each combination in sequence in order to get to the next clue. 

Clue 1: Integration 
Students were given a numeric code that had to be converted to a word after correctly solving the given integration problem. 

The answer was “SNACKS,” which happens to be a location clue, leading to the refreshments centre where I provide students with water, tea, and snacks. The answer to the first clue was hidden under the snack basket. Many students got stumped at this point and wasn’t sure what they were supposed to do (I didn’t give them ANY other instructions). Once they got going, however, they really got into the flow of it.

Clue 2: Derivatives Matching 
I used a matching activity here from Flamingo Math (teachers pay teachers) and students had to find the four digit number code based on the highlighted boxes. (So they didn’t actually have to complete the entire matching activity).

Clue 3: Find the Mistake 
The answer: Students convert correct answer into letter code to unlock the letter lock. 

Clue 4: Calculus Crossword
The answer: Highlighted in invisible ink are the words TRIAL. 

A couple observations: 

  • DON’T set letter locks to be something obviously related to your subject. I stupidly set mine to be “MATH” and had students guessing random four letter words rather than actually engaging with the problem sets that I had worked so hard to create! (I later changed the combo to “BATH”) 
  •  On that same vein, you can set a rule so that students can only attempt one combination at a time. 
  • There’s always that one kid who examines everything with the UV light… so I ended up writing a few random messages around the class not related to anything but just for giggles. 

A great format for STEM OLYMPICS

The same colleague who lead the Escape pro-d was also part of the planning committee for our first ever STEM Olympics (shout out to my buddies Flower, Jeon, Im, Yin and Patel if you’re reading!).  

 

ROUND 1: Unlock one of three boxes

  • Event began with nine teams of four 
  • Students work in teams of four, they have a choice of which question set they would like to work on, however, once a box gets unlocked, then that box becomes unavailable 
  • The question sets corresponding to each box cover a different range of subjects (ex. Box A might cover Math 10, Science 10, Physics 11 and Chemistry 11 while Box B might cover IT 10, Math 10, Science 10 and Math 11). 
  • Inside each box are a series of “advantage cards” 
  • Only the teams that unlock the boxes proceed to the next stage of competition 

ROUND 2: Gain 5 points in a trivia style tournament 

  • Each box contained a specialized advantage card that can be used in round 2
  • Advantage cards may only be played after the question topic is revealed and BEFORE the question is revealed 
  • Examples of advantage cards: skip the question, make the question worth double points, invite an expert to answer the question 
  • First team to 5 points wins
  • Remaining teams compete for second place 

While it does take some time and planning, the escape room format is a great way to review and preview content for a unit or course. I like that it is completely student driven and there is a great deal of collaboration that happens. The novelty factor with the physical locks also played a great role in keeping students interested and engaged, although it is possible to adapt this activity to be completely digital (Onenote or Google forms). 

Since then, I’ve created two other escape activities with my classes. They’re a lot of fun to make and the possibilities for clues and questions are endless! This is definitely an activity I’m going to keep using in my classes.

Finding Culture as an International Educator

It’s amazing to think that I’m now in my fourth year teaching internationally. People often ask me what it’s like to work overseas. Friends and family back home are always curious about where I might end up next. This is my life now, I’m a nomad!

In all honesty, when I graduated teacher’s college, I panicked. Having been a part of the concurrent education program at Queen’s University, I was in a class full of driven and hard-working individuals who always had a plan. Everybody in the program (or so it seemed) knew they wanted to teach, and they pursued that goal relentlessly. By the time February rolled around, a lot of people had already gotten offers or had jobs waiting for them. By the time I graduated, I had nothing.

Knowing what I know now, finding yourself jobless after graduation is completely normal. What felt like weeks of unemployment was actually mere days. What seemed like dozens of personalized cover letters and job applications was probably more like five or six. In fact, it took me about two weeks to get a job. I wasn’t picky, knew I wanted to be overseas and it didn’t matter where. So when the opportunity presented itself to teach in Kazakhstan, I went for it. One job interview later, and I was preparing myself for life abroad.

I only stayed in Kazakhstan for a year. The contract itself was a dream (great pay, light workload), but my gut told me it wasn’t the right job for me. When I decided I wouldn’t return for a second year, many experienced teachers cautioned me I would never find another job with the same benefits and salary, and they’re probably right. But I left. Eventually I ended up in Korea. Long story short, a very different experience from Kazakhstan! The work hours were longer, the work was more taxing at fraction of the pay, in a city whose standards of living were much higher, but it felt more real.

Eventually, I left Korea too. That’s a whole other story. Now I’m in China… a place I never thought I’d end up working. A place I never had any desire to work in. I just felt like too much of an anomaly – “Who is this girl that looks Chinese but cannot speak the language and behaves differently from us?”

When I think about my experiences growing up as a Chinese-Canadian, I carry a lot of guilt and shame. It feels like there is this great burden to fit in and be accepted into different social groups, but also pressure to live up to your family’s expectations and pass on the culture, traditions, and language to the next generation. If I leaned too much to the left, I was too jook sing (roughly translated as “kid who betrays one’s cultural roots”), and if I leaned too much to the right I was considered too much of a FOB (“fresh off the boat”). Rather than living up to my cultural/familial expectations (whether spoken or implied), I tried to run away from them. I decided that being an outlander in a country where I am very clearly foreign would quench those weird notions that I had about fitting in once and for all. I would work anywhere but China, I decided. Oh the irony. 

I’m happy to report that these feelings of guilt and shame have mostly subsided, or at least, I have come to a peaceful cohabitation agreement with them. In fact, being in China has helped me feel more connected to my culture and my family. I’m even taking Chinese classes again! For me, that is a big frickin’ deal, and this time, a step in the direction I want to take. 

Visual Patterns and Mathematical Mindsets

This summer I enrolled in a course called, “How to Learn Math for Teachers,” taught by Professor Jo Boaler, a Professor of Mathematics Education at Standford University. The course brings together best practices from research on brain growth and classroom techniques for anyone who’s curious about engaging students in mathematics education.

One of the course modules talks about creating or giving students tasks with a growth mindset framework, which has the following components:
1. Openness
2. Different ways of seeing
3. Multiple entry points
4. Multiple paths/strategies
5. Clear learning goals and opportunities for feedback

​The example that is given from the course is as follows:

Without any numbers or formulas, describe how you see this shape growing.

A teacher might ask, “There are more squares in case 2 than in 1, where are they? There are more squares in case 3 than in 2, where are they? Describe what you see.”

Go ahead and try this task on your own first. Watch the video to see examples of different responses (skip to 3:50).

This type of task is referred to as a “low entry, high ceiling” task, as anyone, regardless of their skill level can engage with the question, “How do you SEE this pattern growing?” and the question can be extended to higher levels. youcubed.org has  tons of videos, teaching resources, and research papers that challenge the status quo on what it means to be “mathematically minded”. Check them out!

I decided to try a similar task with my Pre-Calculus students in China, and picked a pattern from Fawn Nyugen’s site visualpatterns.org

Based on my students with Chinese students thus far, many of them are quite baffled whenever they get an open task like this. They are used to the typical, “how many squares are in the next case? The 100th case? The nth case?”  type questions and so my challenge was really to get them to train their brains to operate different ways with respect to math. This took time. Two classes in fact, but it was worthwhile.

Here are some answers that students came up with (I posted 6 copies of the same image and challenged my classes to fill all 6 with different representations).

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(From top to bottom, left to right) 1. “Raindrop” method. Squares fill in from the top. 2. “Bowling Alley”. Squares being pushed up from bottom. 3. Squares pushed in from the left. 4. L-shape 5. Rotating Left/Bottom 6. “Negative Space” the missing squares form the same number of squares as the previous case.
After, and only after students have had a chance to visualize the problem, and see other representations of the same pattern in multiple ways did I have them attempt to come up with a formula for the n-th term.

Most students were able to set up a table and saw that the difference from one case to the next increased by 1 each time:

But only a few students were able to break it down further. A message I kept telling my students, “If you’re going to fail, fail differently each time!”
It turns out that most of these students had been exposed to Gauss’ summation before. Those that did were able to find a formula for 1 + 2 + … + n, but the challenge with this pattern is that we start at 3.

Another student used the “square” representation as a part of his proof but isolated the last row.
Looking at the diagram below, we see that the total number of squares can be represented by (n+1)^2.

Ignoring the last row, we see that the number of actual squares and “negative space” squares are equal. The total number of squares (excluding the bottom row) is therefore given by [(n)(n+1)]/2.

Putting both these parts together, we get that the total number of squares for case n is:

My favorite proof thus far, though, is this one:
-Take the square representation, ADD another layer
-Now we have a rectangle with equal amounts of actual squares and “negative space” squares
-The resulting formula is just the area of the rectangle divided by 2
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Wondering if there’s a way to embed LaTex code into my blog… (AKA me trying to be more tech saavy)
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(Please excuse the crappy graphics, I did what I could with Powerpoint…)
Even though this material isn’t explicitly stated in the curriculum documents for this course, it was a valuable exercise to have done with my students. I had a few students approach me after class, eager to show me their proofs and what they had discovered. Throughout our whole discussion, I never gave students any answers, but focused on process. This is a message I want all students to internalize when they leave my classroom.

First Days of Calculus: Grapher-Explainer Activity

 Never in my life did I ever imagine myself teaching in China, and yet, here I am for a second year at that! Below are images of welcome packages I put together for the members in the Math Department this year, which includes:
– A door sign with the teacher’s name, room number, and teaching schedule
-Stickers, ‘cuz duh
-Coffee, a key element in sustaining the life force of a teacher
-A pack of cards, essential in any math teacher starter kit 
-A math puzzle, fuel for the brain 

I’m super happy with the way they turned out, and I’m looking forward to a good year ahead!  

This year I’ll be teaching Pre-Calculus 11 and Calculus 12, which I’m both excited and nervous about! It’s been years since I’ve taken Calculus and this will be my first year working with twelfth grade students (I’ve been doing a lot of review this summer on Khan Academy). Here’s a fun activity that I found on Kate Owen‘s blog that I plan on using this week with my Calculus 12 students. It’s a great way to review concepts and vocabulary from Pre-Calculus to see what students already know and remember from the course. 

I’ve added some modifications and created an accompanying PPT that’s a full lesson, all ready to go. Scroll down below to access this resource 🙂 I’m a big believer in sharing teaching resources for free, and this is my way of giving back to the online teaching community that has given so much to me. Huge shout out to everyone in the #MTBoS, I love this community. 

The activity works as follows:  

1.Students it with a partner, shoulder to shoulder.
2.One person faces the board, the other person faces away.
3.The person facing the board will be the explainer.
4.The person facing away will be the grapher.

Warm Up: Teacher does warm up round with the students, describing a basic graph (ex. linear function) and students attempt to draw it in their notebooks. Discuss: What prompts were useful? Is there something the teacher said that could have made it easier? 

The Activity: (see above)

Exit Ticket: Given a picture of a graph, students are to write a description that matches it in as much detail as possible.

Extension: Students draw a graph and write a corresponding description. Scramble the results and have students match them!  

grapher-explainer_activity.pptx
File Size: 2261 kb
File Type: pptx

Download File


Myanmar in Ten Days: Day 4

Day 4 – Ancient Cities in Mandalay

For our final day in Mandalay, we opted to hire a private car and paid about 35 000 ks for a “three city tour”. As it is common for taxi drivers to advertise private tours of the surrounding area, it wasn’t necessary for us to book ahead. We had collected a few business cards from taxi drivers during our first few days in the city and opted to go with the driver who seemed the friendliest and spoke the best English. 

 

For our first stop,  our driver took us to a monastery in Mandalay where we had the opportunity to speak to his friend, a monk who teaches English there. We were shown around to various buildings (the dormitories, dining hall, study halls…etc.) and learned about life in the monastery. Becoming a monk is a well-respected and esteemed route to take for boys and men of all ages. A family’s status is elevated if they have a son who decides to become a monk. Of course, not many choose to stay one, some quit years, months, weeks, or even days into monkhood, which is not uncommon. At one point, both our taxi driver and tour guide (whom we would meet later in Bagan) had taken up monastic life.  

At the monastery, we met an especially charming and charismatic young monk who went by the name of “Drake.” Funnily enough, we would later run into “Drake” again three days later in a totally different city, at sunset, on the top of a temple, where he would re-introduce himself as “Maha Raja,” and add us as Facebook friends. To this day I am still not sure if he is using his real name. 

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Our friendly guide around the monastery.
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Notice to foreigners about proper etiquette and dress.
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We were told to stay for the monk procession, in which 1000 monks would line up according to rank and seniority for their second and final meal of the day. If I’m completely honest, the sight made us feel uncomfortable in comparison to our calm and quiet morning around the monastery. In an instant, the empty streets became crowded with tourists, with their big cameras, tablets, and cell phones; we witnessed a few elderly women handing out sweets and loose change to the younger monks, perhaps out of charity or something else, I don’t really know. It just seemed like such strange way to sensationalize their lunch time… It was good to finally get out of the crowd. 
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Heads bowed, bowls in hand, the monks walk in procession to get lunch.
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Tourists looking to get a shot of the action.
Next, our driver took us to a location where they made longyis, a long sheet of cloth commonly worn as a skirt by both men and women in Myanmar. We were shown how the longyis were woven and taken to a nearby store were they could be purchased. Sarah suspects we were taken to what is known as a “tourist trap,” but heck, it was cool and we bought one for ourselves anyways.
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Afterwards, we ate lunch at a restaurant of our driver’s choice. The food was pricey and not particularly noteworthy. 

 

Like Mandalay Hill, U-Bein bridge is a popular tourist destination at night time, as people like to go for the sunset. We decided to go earlier in the day to avoid the crowd. Here, we purchased some coconut ice cream (DELICIOUS) and walked about halfway across the bridge before turning back… on account of some uncomfortable cat calling. We weren’t dressed in scantily clad clothing by ANY means but my Sarah does happen to have strikingly blonde hair and fair skin which drew a lot of unwanted attention. We definitely had to check our privilege at that point. 

PRO-TIP #1: Please don’t do what we did and walk the entire length of the bridge! We missed out on exploring Amurapura city as a result, but we ended up having a great day regardless (read on to find out!)

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A tasty snack in the February heat!
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U-Bein bridge is said to be the world’s longest timber bridge (according to Wikipedia).
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Various vendors and stalls at the entrance of the bridge.
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Along the walk back… a sight for sore eyes. Two young boys flying make-shift kites out of plastic bags and string along the side of the bridge.
Now at this point we had been to half a dozen temples and seen a ton of pagodas so if you can forgive me, I do not recall the name of the temple our driver took us to next. The highlight for me, however, was watching the line that quickly formed as soon as Sarah agreed to have her photo taken. One, led to another, and then another… People wanted group shots and individual shots. Blonde, white-skinned, and beautiful, Sarah quickly became a hot commodity! (Only 2000 ks for a photo with this beautiful foreigner! Anyone? 1000 ks special discount just for you!)
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“Save me” she whispers without speaking.
Having my friend taken from me for photos would be a common occurrence throughout the entire trip. Me, on the other hand, being of Chinese descent, and having been told I have a face that can pass for a variety of Asian ethnicities, was able to (at times) conspicuously blend in with the crowd.

 

The next part of our journey would be my favorite in Mandalay. That was our brief tour of the ancient city of Inwa. 

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First, a pit stop in one of the smelliest, but by far not the worst, porti-potty I had ever encountered.
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Then, a short ferry ride to our destination, Inwa Ancient City.
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Once we arrived, we hired a horse cart and driver to take us around the Ancient City (~9000 ks). It’s possible to do on foot, but you’d need at least two hours and we were running short on time. He took us to a few notable locations before dropping us off for the last ferry back.

 

PRO-TIP #2: Keep in mind most places you visit will require you to go barefoot (temples, pagodas, ruin sites…etc.), so bring comfortable shoes that slip on and off easily! 

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A journey via horse cart.
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Bagaya Monastery, built entirely of teak wood in 1834 A.D.
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Beautifully cultivated green pastures.
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​On the way back, we saw a little boy and a dog at one of the ancient ruin sites.

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It looked like they were friends.
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(The dog was likely a stray).

 

But still, it was a fine friendship. 

​We decided to explore the area.

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Some post-card vendors preparing to close up shop for the day.
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But we noticed that someone kept showing up in our photos… 
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“Follow me!” he said.

 

And so we did.

And saw the most breathtaking statue.

There was something about the way the light fell, the little boy giggling and running around us, the other little one who turned out to be his brother, prodding us along, telling us to climb here, sit there, pose like this, not like that… Making faces at us when we did something they didn’t like and giving us the thumbs up when they deemed we had the perfect pose…

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Moon running to bring me a leaf!
Moon trying to hid behind the broken pillar so I would have the perfect shot. I liked it better when he was in it
We had so much fun running around with  those two that we didn’t even break a sweat when they eventually busted out their post-cards and offered to sell us some. 

 

What fine salesmen they turned out to be. 

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Sarah bonding with the little ones.

NEXT UP: Bagan! 

Myanmar in Ten Days: Days 1 – 2

This past February, I took a trip to Myanmar with my good friend Sarah. As we were both teaching in Shanghai at the time, we wanted to take this opportunity to explore Southeast Asia during the Chinese New Year holiday. We visited Mandalay, Bagan, and ended our trip in Yangon. 

Day 1 – Mandalay

We landed in Mandalay at around 3pm on a Sunday. The airport is fairly small and underdeveloped. Depending on the time you arrive, there may or may not be services available. From what I remember, the only place that was open at the time was a shuttle service and a money changer. 

 

Our priorities for the day: get to our hotel and get food! We exchanged what RMB we had in our wallets and took a taxi from the airport direct to our hostel, the Moon Light Hotel, which cost maybe 30 CAD (for reference, the exchange rate at the time of our travel was about 1 CAD to 1000 MMK). 

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Small but cozy. Our room was very neat, clean and tidy (until we arrived, that is).

 

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We stayed at the Moon Light Hotel for 3 nights, which cost us about 50 USD. The hotel is very new, staff are extremely friendly, and breakfast is included. 
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Breakfast featured both Asian and Western cuisine.

 

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View of the city from the dining room.

 

If you haven’t traveled Asia before, some of the imagery you encounter can be pretty jarring. While our hotel was tidy imitation of some Western hotels, just outside you can see signs of impoverishment; unpaved roads, large piles of garbage, stray animals, and the like. Not a vacation destination for the faint of heart. 

 

For dinner, we walked to Mingalabar, the #1 rated restaurant on Trip Advisor in Mandalay, and boy – did it live up to those standards! For bout 15 CAD, we had a beer, lime soda, soup, rice, a main of lamb curry, and dessert. The main course comes with all the side dishes you see below, the idea being that you can customize each bite according to your taste preference. The side dishes they serve vary from night to night, but ours featured peanuts, fish, potatoes, cauliflower, a shrimp paste, and some raw vegetables. 

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So good we went back the next day!

 

A word of caution… 

Our biggest mistake on this trip was not bringing enough CASH! We had read online that there have been many improvements in the big cities in terms of ATMs being available. Having come from China, both of us have UnionPay cards that are accepted at many ATMs throughout Myanmar, according to research. We did not, however, factor into account that these ATMS may not be regularly maintained, so many that we visited were out of cash! 

 

[For some mysterious reason, I was not able to withdraw ANY money on my UnionPay card, but luckily Sarah was able to to do on her Canadian bank card.]

Long story short, to avoid running into this issue, I would recommend bringing enough cash with you to last the trip. But beware of pickpockets, especially in touristy places! 

Day 2 – Mandalay Palace

Our second day was spent getting acquainted with the city, hitting up every ATM we encountered, and getting SIM cards. I would highly recommend getting a SIM with a data plan for your travels, as it makes life significantly easier (access to GPS, Trip Advisor, etc.) SIM cards are fairly cheap and top ups are easy to come by (most convenience stores will have them). Popular carriers include Oredoo and Telenor. 

 

In the afternoon, we asked our hotel to help us call a taxi to take us to Mandalay Palace. If you call a taxi through your hotel, the prices are usually set (though still very reasonable). If you choose to hail your own transport, usually there is a bit more room to negotiate. Keep in mind that these are not “taxis” in the Western sense, but rather random strangers you’re waving down in the streets who happen to have a car and want to make a few extra bucks driving people around. 

To get into the Palace grounds, you need a visitor’s pass. You’ll be asked to leave your passport with the guards in exchange for one. We did not have our passports with us, but luckily, they accepted Sarah’s drivers licence (phew!). In the area surrounding the palace there’s a park and some temples and pagodas. We just walked around and took our time exploring the area. 

At one of the vendors, a girl offered to paint our faces with thanaka, a yellow-white paste made from tree bark. We later learned that wearing thanaka is like putting on clean clothes; worn by people of both genders who may perceived as “unruly” if you did not put it on, though trends seem to be changing in the big cities. 
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Thanaka is used for both cosmetics and as sun protection.

 

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In the afternoon, we ate at a restaurant in town and freshened up at the hotel before heading out again in the evening. We went back to Mingalabar (which means “Hello” in Burmese) for dinner, and walked to the bar across the street for cocktails.

Day 3 – Mandalay Hill

Early next morning, we had breakfast at the hotel and took a taxi to the Lion’s Gate entrance of Mandalay Hill. Mandalay hill is a popular destination at night time, as many tourists often go to see the sunset. I found the views in the fresh morning air just fine, and seeing as there were hardly any people around during the hike up, wonderfully peaceful. 
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Entrance to Mandalay Hill.

 

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The majority of the climb is done under a covered walkway. The climb must be done barefoot, so we left our shoes by the entrance. (You pay 1000 ks for someone to “look after” them).

 

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You’ll meet many strays along the way. Cats, dogs, turkeys even!

 

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Lots of buildings, sculptures and pit stops along the way to the top.

 

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Part of the covered walkway up the hill.

 

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There are many scenic places to stop and take photos along the way!

 

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You’ll know you’re at the top once you reach the giant escalator that takes you down Mandalay Hill. We opted to take a shuttle down for 2000 ks instead. 
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Blue shuttle bus that took us down from the top of Mandalay Hill.

 

Final verdict: Mandalay Hill is a must! Definitely enjoyed our morning hike. We took our time, and stopped a lot to take photos and enjoyed the scenery. 

 

I can’t remember what else we did in the evening (ate food somewhere definitely), but the morning hike did take up a lot of our energy. A day well spent overall. 

​NEXT UP: A private tour to the ancient cities in Mandalay (click here for Day 4 details). 

Polynomials

Oh Polynomials. My least favourite unit by far in the Foundations of Math and Pre-Calculus 10 course I am teaching. Find the greatest common factor, least common multiple, factor these trinomials, collect and simplify like terms, the swimming pool has a width of 5x + 1 and a length of x + 2… YAWN.

The Challenge

How can I frame a boring, completely algorithmic and skill-based unit into something that’s relevant and meaningful for my students? I am borrowing Dan Meyer’s definition for relevance here.

It Begins with a Question… 

A colleague asked me today, “How much time do you have for homework at the end of class?” This was a surprising question to me, and as I thought back over the 10 day unit, my answer was almost none. The question sparked a great dialogue between us about our approach to teaching the same content in our respective classrooms. It really made me think. I realized that while I still dreaded teaching polynomials, I had found a way to improve the way I taught it from first semester that required less rote work and more thinking.

One thing that has not changed, however, is that I avoid teaching FOIL method like the plague. It only  works for expanding binomials and does not apply for polynomials with more than two terms. After I read this article I was convinced I would never need FOIL in my classroom:
http://www.makesenseofmath.com/2016/11/why-i-will-never-teach-foil.html
For a good laugh:
https://saravanderwerf.com/2017/04/01/why-ive-started-teaching-the-foil-method-again/

Some things that came up in our discussion:

  1. Algebra tiles – benefits and fall backs
  2. Picture talks
  3. Factoring method (criss cross or sum-product?)
  4. WODB
  5. Progress checks
  6. Taboo
  7. Human Bingo

1. Algebra Tiles 

Definitely a hate-hate relationship. As a math teacher, I am obligated to entertain this idea and I do admit it has its benefits, especially in lower level math classes when students are initially being exposed to distributive property and the like. The problem, however, was that my students were already armed with the skills and knowledge of multiplying and factoring polynomials. Moreover, the limitations of using tiles far exceeded the benefits, in my opinion. Algebra tiles do NOT work for: polynomials higher than degree two, multiplying more than two polynomials, and multiplying polynomials with more than three terms. This meant that it took more effort for students to understand how and why it works.

Nevertheless, we spent a few classes examining algebra tiles and their usefulness. Rather than approach it from the typical standpoint of using algebra tiles as a manipulative, I wanted students to see the link between the algebraic and pictorial representations of polynomials. This took work and was not as straightforward as it seemed. A big takeaway for me was that students gained much more out of the experience when they were able to physically manipulate the tiles and arrange them into their “factored forms.” Last semester, I “taught” algebra tiles by merely showing them examples and drawing them on the board.  It took a bit more prep, but this semester I printed eight sets of tiles (positive and negative) in my classroom and had students manipulate them instead.

If we were to spend any more time on the unit, or if this was a lower grade level, as an enrichment activity I would have students discuss the limitations of algebra tiles and look for ways to address them.

2Picture Talks

I like to use Sarah VanDerWerf’s Stand and Talks as a format for students to discuss picture prompts. I find that the buy in for engagement is much higher when the prompt is linked to physical movement. My favourite questions for photo prompts are: “What do you notice?” and “What do you wonder?”

What are my photo prompts, you ask?

That’s right. Algebra tiles.

Goals for students:

  • Make observations and ask questions
  • Use math vocabulary
  • Share ideas with their peers

​I like this activity because it is easy to differentiate and works well as a “minds on” for any topic. Asking students a general question like what they notice/wonder means that lower ability students can comment on ANY aspect of the photo (e.g. “there are blue and green rectangles”) while higher ability students can be pushed towards making observations based on any mathematical patterns or relationships they observe (e.g. “the green tiles represent positive polynomials and red tiles are negative”).

3. Factoring Method – Criss Cross or Sum Product?

I’ve had the great fortune of only having one prep and a spare block this semester (for friends and readers who don’t teach, that’s teacher jargon for FREE TIME, kinda. The details are not important). Anyway, I’ve been making drop-in’s to my fellow colleagues classrooms with my new-found “free time” and one thing I picked up was the importance of proper SEQUENCING. For instance, a natural progression for factoring trinomials might look as follows:

  • Common factors
    • ​Ex:    12xy + 3x
  • Trinomials with leading coefficient of 1
    • ​Ex:     x^2 + 4x + 4
  • Factoring when the GCF is a binomial
    • Ex:      x(x + 1) – 2(x + 1)
  • Trinomials with leading coefficient not equal to 1
    • ​Ex:     3x^2 + 8x + 4


That, together with a quick exercise on sum/products, helped me push students towards seeing the relationship between the factored form of  a trinomial, and the sum/product method.

I prefer this method over the traditional “criss cross” method for a few reasons

  • Focus is on noticing patterns
  • Less trial and error work
  • Allows them to answer questions like this:

Which One Doesn’t Belong? (WODB)

Fantastic activity for building up thinking skills and vocabulary. Each student picks one of the expressions and must argue why that one doesn’t belong.

Benefits:

  • Everyone speaks (I always encourage full sentences and proper use of math vocabulary)
  • More than one argument may arise for each expression

Sample answers:
“27x^2 doesn’t belong because it is the only expression that has a coefficient with a perfect cube”
“45x^2 doesn’t belong because it is the only expression that has a coefficient with 5 as one of its prime factors”

More WODB prompts can be found here.

Taboo 

How it works: One student is chosen to stand/sit at the front of class facing the audience, they are in the “hot seat”. Behind them, a vocabulary term is shown for the rest of class to see. Students in the audience must help the student in the hot seat guess the vocabulary word by miming, explaining the definition, or giving examples. They may not use any part of the word in their explanation.

Modifications: Differentiate by giving students the option of bringing a “cheat sheet” of vocabulary terms with them. Prepare students for the activity by giving them cross word or fill in the blank exercise reviewing the vocabulary words for the unit. An “expert round” can include vocabulary not on the cheat sheet. “Challenge round” can be facing a peer or the teacher. Can play in teams or as a class.

Human Bingo 

chapter_5_human_bingo.docx
File Size: 87 kb
File Type: docx

Download File


This activity was shared by a good colleague of mine. To get BINGO, students must find one “expert” in the classroom to answer each question on the bingo card until all the questions have been answered. The student who answers the question must sign their name. A student may not be asked more than once to answer the same Bingo card.

Benefits:

  • Gets students moving
  • Students can pick the question they want to answer
  • Fun review activity for a test or quiz!

Reflections on My First Semester Teaching in China

Semester one of my first year living and working in China is officially over! Since my last post about the first day of school, I realized haven’t blogged at all this entire semester. I am a little disappointed that I had skipped through all the middle bits, but regardless, here we are.  

This past semester I taught Math 10 and 11 of the British Columbia curriculum at an international school in Suzhou, China. With the exception of a handful of students, all of them are English Language Learners. Some might argue that this does not pose a big problem in mathematics, since the language of mathematics can be viewed as a combination of abstract signs and symbols separate from the English language.  The problem is, it is one thing to understand mathematical ideas and concepts, but another to be able to communicate them. Someone who is well versed in a mathematics should theoretically be able to describe the same concept in more ways than one – numerically, algebraically, graphically, and verbally. Mathematicians strive for precision in expressing ideas, and this is not always simple. Aside from students having to approach mathematics from an ELL standpoint, the issue is compounded when you consider all the ways in which ambiguity arises in the English Language. Take for instance the word “and”; conjunction in mathematics is commutative (A^B is the same as B^A), but you can see from the example below that “and” in everyday English is not commutative. 

The sentence, “John took the free kick, and the ball went into the net,”  would have a very different meaning if the conjuncts were reversed (Devlin, Introduction to Mathematical Thinking).

For my most challenging students, the issue wasn’t so much as getting them to communicate their mathematical ideas well, but getting them to communicate at all. For students with extremely low level English ability, being afraid to speak or ask questions in class was a huge roadblock in developing  a good grasp on the mathematics we aim to study. The most frustrating times were when students didn’t even bother to try. Perhaps this has something to do with being in a culture where “saving face” is important, but students were sometimes so afraid of being wrong that they left entire test pages blank, multiple choice even! (Yes, I know, I was stunned!) You’ve probably heard this a million times but I’ll say it again, mathematics is not a spectator sport! You have to do it to get it, like riding a bicycle. (Am I preaching to the choir here?)

My biggest goal this semester is to get students talking more. About mathematics. In English. A large part of my success will depend on how well I set up a classroom culture of trust and acceptance. This is huge. If I have any hope of getting students to share their original thoughts and ideas they need to know they are safe doing so. Luckily, I’ve got some ideas to help me get started, but the rest will be trial and error (as is most of my teaching anyway). I also plan on working in a slower progression at the beginning of the year to first get students acquainted with some of the language used to describe mathematical expressions before we dive into what exactly mathematics is. With any luck, every student will be able to describe, in English, what we are learning in any given unit. 

Things That Went Well in Semester 1
1) I finally found a groove! Lesson planning no longer takes up hours and hours each day (#win), and I also have a nice support network of experienced teachers to draw ideas from and borrow resources from. Establishing daily routines early on in my classroom (and enforcing them!) also worked wonders. 

2) Brain breaks. I was a little hesitant about these at the start since they seemed silly and unnecessary if the lesson is well-chunked. I learned early on though, not all lessons are made equally and some days really are a drag, especially when are teaching 80 minute blocks. Taking a short 5-10 minute break to stretch/play a game/go on your cell phone provides both myself and the students some much needed refuge from a long period of work. 

3) First week activities. As I mentioned earlier, setting up a warm and inviting classroom culture is key to being able to get students to talk more math, and learn more in general. I spent about a week doing activities and playing games related to math with my students last semester before I started diving into teaching any curricular content. I plan on spending about the same amount of time, if not more, this coming semester settling in with my new classes.