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We asked 101 high schoolers the following question: There are 125 sheep in a flock and 25 dogs. The question is an invitation to take a closer look at the kinds of mathematics that we are asking students to engage with in our maths classrooms today. What does it mean for us as educators when students give responses like 130 because 125 + 5 = 130 or 25 because 125/5 = 25? Moreover, what does it mean for us as educators when we expect these responses from students? I first heard about the shepherd question through Robert Kaplinsky though the question has its origins based on research by Professor Kurt Reusser from 1986, possibly sooner. |

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). |

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).

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.

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.

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:

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.

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.

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

Lies, damned lies, and statistics.

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

[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.”

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

- 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:

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:

**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).**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.**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.**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.

## 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: |

Sample Weekly Quiz | |

File Size: | 891 kb |

File Type: |

*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...

- 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.

**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"

"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.*

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.

**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.

- 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

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

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.

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.

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:

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).

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

*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).

*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:

*If you're going to fail, fail differently each time!"*

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:

-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

- 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!

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 |

<3

## Day 4 - Ancient Cities in Mandalay

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.

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!)

*Only 2000 ks for a photo with this beautiful foreigner! Anyone? 1000 ks special discount just for you!)*

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

**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!

On the way back, we saw a little boy and a dog at one of the ancient ruin sites.

But still, it was a fine friendship.

We decided to explore the area.

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...

What fine salesmen they turned out to be.

__NEXT UP__: Bagan!

## Day 1 - Mandalay

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).

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.

## A word of caution...

[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

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.

## Day 3 - 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.

*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**

## It Begins with a Question…

*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__:

- Algebra tiles - benefits and fall backs
- Picture talks
- Factoring method (criss cross or sum-product?)
- WODB
- Progress checks
- Taboo
- Human Bingo

## 1. Algebra Tiles

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.

**2**. Picture Talks

What are my photo prompts, you ask?

__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?

- Common factors
- Ex: 12xy + 3x

- Ex: 12xy + 3x
- Trinomials with leading coefficient of 1
- Ex: x^2 + 4x + 4

- 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

- 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.

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

**Which One Doesn't Belong? (WODB)**

__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"

"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 |

__Benefits:__

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

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.

I'm teaching high school math (grades 10 and 11) this year. My school runs on 80 minute blocks. Here's what I did.

**Algebra Seat Finders**

**and Visibly Random Groups**- Rather than making a seating plan or having students choose their own seats I greet students at the door and hand them each a card as they walk in. On the card are algebra problems involving one or two step equations and order of operations that are easily solvable via mental math. The answer to the question will tell them which table to sit at. I've arranged my tables into groups of four and have signs taped to the side of the desks so they can easily find the group number. (If you would like to download copy of the seat finder cards I used, they are available at the bottom of my post).

I do the same thing

**each day**, so that every day students will sit in different groups. I like this activity because students are doing math as SOON as they enter the classroom. Some students will cheat and trade cards with other people so they can sit with their friends, but you will come to notice this quickly. I tell students that in this class we are a community and that they will always be working with different people so they get to experience different perspectives and meet everyone in class. Even if certain students don't get along, it's low stakes because the seating changes every day. On Fridays I give them a break and tell them to sit anywhere they like. It was interesting for me to notice that given the choice, students tend to sit with classmates with similar level. Peter Liljedahl has done some cool research on visibly random grouping, check out his free webinar here.

**Day 1**

- Bell Work - Who I Am
- Start the class with low key student profile sheet from Dan Meyer as I take attendance. Gives students a chance to tell me about themselves. My favourite questions on this sheet are the "Self Portrait" and "Qualities of a good math teacher.

- Numbers Quiz
- Adapted from Sarah Carter. I beef this up a bit and use this as an opportunity to talk about test/quiz expectations (no talking, no asking a neighbor to borrow an eraser or calculator...etc.), and the consequences for cheating. I tell them that this is a difficult quiz and so far no one has been able to obtain a perfect score. All I ask is for them to try their best, and if they don't know an answer, guess. When I tell them to flip their papers over I usually hear a few chuckles or giggles. Again, I enforce that the room should be
__silent__and let them know I mean business.

- Adapted from Sarah Carter. I beef this up a bit and use this as an opportunity to talk about test/quiz expectations (no talking, no asking a neighbor to borrow an eraser or calculator...etc.), and the consequences for cheating. I tell them that this is a difficult quiz and so far no one has been able to obtain a perfect score. All I ask is for them to try their best, and if they don't know an answer, guess. When I tell them to flip their papers over I usually hear a few chuckles or giggles. Again, I enforce that the room should be

- Student Quizzes
- Next, I give them a chance to write ME a quiz about themselves. I take their quizzes and return it to them to be marked. Most students asked basic questions like "What is my favorite subject?" or "What is my favourite food?" Others were more creative and decided to have a bit of fun with the activity...

- Personality Coordinates (Dan Meyer)
- Originally planned to complete this activity the first day, but I was over-ambitious with my planning so ended up
*introducing*it and coming back to it later. First I showed students this diagram:

- Originally planned to complete this activity the first day, but I was over-ambitious with my planning so ended up

Student A: What do you notice about this picture?

Student B: I notice there are two perpendicular lines. What do you notice?

Student A: I notice the four dots are arranged in a square. What do you wonder?

Student B: I wonder what the teacher will ask us to do with this diagram. What do you wonder?

Student A: I wonder if this is a function.

..etc.

We discuss and review parts of the coordinate plan. I ask them a few questions about the dots. (

*Which two dots share the same x-value? Which dot has the lowest x and lowest y value? etc.)*

The next time we revisit this activity I start with an example:

- Name Tents (Sarah VanDerWerf)
- At the end of each class on the first week I asked my students to choose ONE question and answer it in their name tents:
- 1. One thing you enjoyed about today's class?
- 2. One question you have.
- 3. A suggestion for class.

- I write back to them every day. This is a big commitment but worth the time in my opinion.
- Some positive feedback I've gotten: Fun, engaging class, students enjoy group work and team activities
- Some things I need to work on: talking slower, writing bigger on the board
- Some questions I've been asked: When do we get the textbook? When do we have our first quiz? Is math difficult?

- At the end of each class on the first week I asked my students to choose ONE question and answer it in their name tents:

**Day 2**

- Syllabus Quiz
- Rather than giving a long speech about course expectations, school and class policies, I wrote a quiz. Even though I assign syllabus reading for homework most students will not do this. The quiz is open book and is graded (can be done in pairs), and I count it towards their "English proficiency" grade for the course.

- Talking Points
- This one MUST be modeled to students. It's a little complex, especially for EL Learners so it's important to explain clearly and minimize the amount of instructions given. The main point is to get ALL students talking and sharing their opinions. To model the activity, I pick three random students to do a "practice round" with me. This was less effective with my grade 10 students as they are new to the immersion program. Next semester I might film a teacher example of this activity to show students instead.

- What is Math?
- Share our ideas of what math is, give a common definition of mathematics that we will use for the course.

- Expectations for the Year
- Go over things like: cell phone policy, asking to go to the bathroom, materials needed for class, binder expectations, course evaluation...etc.

- Name Tents
- Again, end the day with student writing me some feedback.

**Day 3 - Day 5**

Teach some content and continue reviewing and practicing start of class and dismissal routines.

Algebra Seat Finders - Groups | |

File Size: | 16 kb |

File Type: | docx |

## It's personal.

Books like

*First Days of School*and

*Teach Like a Champion*have been invaluable reads, providing tons of practical advice teachers can implement right away. The issue is learning how to filter that knowledge so that it's true to your own teaching style and well-suited to who your students are. Teaching math in an academic classroom is way different than in a college or applied-level classroom, for instance, and not because the material is different per say, but because the students'

*attitudes*towards math differ tremendously. I found that students who are in applied or college-level math courses generally have lower confidence in their math abilities. Subsequently, each wrong answer means another failure added to the list and just reinforces what they already knew, "I'm not good at math." Here, priority #1 is to build a safe and welcoming classroom where a culture of error is the norm, and is celebrated as a vehicle for learning. Likewise, North American students and Asian students also differ in their attitudes towards math. Comparatively speaking, math anxiety seems to be a bigger issue in North America. On the other hand, students in Asia tend to be really

*good*at math, they

*respect*the subject, and they

*will work hard at it*, even when things get tough. In Asia, the norm is repeat and rehearse everything the teacher's taught, but the challenge is to get kids thinking

*independently*and

*creatively.*

Different mindsets on math, as told in memes:

## April's Tips on Classroom Management

**0. Plan a good lesson.**I'm echoing Fawn on this one when I say that having an engaging lesson solves soooooo many potential discipline issues in the classroom. Kids will act out when they are bored. I know this because I WAS this. I mean, I was an A student throughout high school and a MODEL student at that. One summer I took a physics and had a teacher who literally

*read the textbook*to us. I can do that myself, thank you very much. So, rather than sit in silence and boredom, I discovered that the reflective properties of light were pretty fun to play around with. I was particularly intrigued at how various angles of light rays from the window would bounced off the shiny surface of my watch right into - yup, the teacher's eyes.

**1. Learn names.**I always make it a point to know the names of all my students and connect with them in some way. To me, there's nothing worse than being called "you in the red shirt" or "hey you." Teach students, not the subject.

**2. Don't repeat student answers.**I first noticed this during my observations of a veteran teacher while I was student teaching, and it completely changes the way discussions flow in the classroom. If a student answers a question, and the teacher repeats the answer (usually in a louder voice or with elaborations), in the students minds this translates to, "

*Information is not important, unless it comes out of the teacher's mouth."*If a student says something really insightful, ask them to repeat it instead - you'll have reinforced two important messages to a) the student: "Your contributions are valuable!", and b) the class, "We have a lot to learn from our peers!" It is so vital for teachers to give students opportunities to be responsible for their own learning.

**3. No Opt-Out.**I got this one from

*Teach Like a Champion.*The premise is simple, if a student does not know the answer to a question, they cannot get away with "I don't know." You might ask another student for their thoughts, you might provide a hint, you might just say the answer outright, but you will

*always go back*to the student who said, "I don't know." "I don't know" is not an acceptable answer in my classroom. We want students to get from:

"I don't know" therefore "I don't have to try", to

"I don't know,

**YET" so "I'm going to keep trying."**

**4. Keep it simple.**If you're like me, you probably have a list of 20 different procedures, routines, and policies you'd like your students to do and maintain throughout the year,

__but this is not realistic__! I ended up flopping on most of them. My first two years of teaching have been chaotic, and I'm slowly coming to accept that it will be this way for a while. Focus on the five most important guidelines and procedures that your classroom cannot do without, then build from there.

**5. Document everything.**The biggest lifesaver for me last year was getting students to fill out "Action Plans" for whenever they made a bad choice. There are many variants of this on Pintrest. Student Responsibility Cards for homework were also cool, but didn't work out that well for my classroom because I didn't follow through on consequences. So I'll keep the first and toss out the latter. Links to some documents I used below.

**6. Give logical consequences.**When I was little, my punishment for making bad choices was always the same; mum would make me stand facing the wall with my arms up, and fingers pinching my ears. I guess it was supposed to make me feel ashamed of my actions, but it doesn't make sense. Let's remedy the

*behaviour*, and not punish the

*student*. Examples of logical consequences below.

I'm now going into my third year of teaching, and so far, each year has been in a different country, which has made each "first day" even more special.

## My First First Day

Subject: Math

Grade: 10

In my very first day of my very first full time teaching job in Kazakhstan (blog post here), I spent the first day getting to know my students, telling them a bit about myself, talking to them about my expectations for the class, taking selfies of all the students, and giving them some general advice about how to succeed in math class. I found that it was important and effective to start building those relationships with my students from day one, and by learning all their names as quickly as possible, I let them know that I notice them and care about them.

Prior to preparing my first day lesson plans, I soaked up as much information as I could with all the resources that were available to me. I had read

*First Days of School*by Harry and Rosemary Wong and a few other teaching books, browsed the internet for countless hours looking for ideas and inspiration, watched this entire video by Agape Management, and looked for elements of each that I thought would be suitable for my teaching style. What didn't work, however, was the fact that I did not start the year knowing where I wanted my students to be by the

*end*of the year. This was difficult because I didn't know much about the culture, the style of teaching that students were accustomed to, and I had never taught a class full of ELL students before (hence why nobody laughed at my jokes). Moreover, I did not have full autonomy over the classroom (it was supposed to be a co-teaching type environment but ended up feeling more like I was "guest teaching" a few times a week); my co-teachers were not fluent in English, and had different visions of how they wanted to run their classrooms, which made it difficult to have consistency when it came to expectations and rules.

What ended up happening was that the first day allowed me to start building relationships with my students, but it did nothing to help me manage my classroom (because nothing was consistently enforced). If I could re-do my first day, I would spend more time getting to know my co-teachers, and specifically, these are the questions I would ask:

1. What are your classroom rules and expectations?

2. What are your beliefs about learning in math? (i.e. How do students learn best?)

3. What are your beliefs about teaching in math? (i.e. How can teachers best reach their students?)

4. Describe a typical day in the math classroom for you.

I learned that it is important not to go in assuming that your teaching partner will have the same views about teaching and learning as you do, and not only that but that I needed to take the time to get to know and understand their views! Had I done so much earlier I would have discovered that hands-on activities, student investigations, or differentiated teaching and learning weren't a common tools in their teaching toolbox. The general style of teaching I observed included very fast-paced progression through the units, with lots repetition and mental computations, but very little time spent developing the concepts or looking at their applications. Knowing this, I would have modified my first day presentation to include some math activities that integrated both styles of teaching.

## The Second First Day

Subject: Science

Grades: 8 - 10

In my second year, I taught in South Korea and had full control over my own classroom, which made it significantly easier to plan and organize everything the way I wanted to. My first interaction with my students, however, was not on the first day of class. We had an "orientation day" in which both students and parents attended brief 10 minute presentations by all their teachers.

I began by greeting every student and parent at the door with a handshake. I called the students by name as they walked into the classroom, which took a lot of time for me to learn beforehand, but was so worth the reactions! Prior to meeting all of them, I borrowed the previous years' yearbook and memorized the faces and names of all the students I would be teaching in my classes. Some of them looked stunned that I knew their names already, when none of them had a clue who I was yet!

At the front of the room, I had copies of letter to parents and the course syllabus which I asked each student to pick up as they walked in. In my presentation, I talked briefly about who I was, my educational background, and what students can be expecting to learn this year. My primary goal was to let them know that I care about them and their learning, and that while this year would be challenging, they would also be supported by me.

Then, on the actual first day of class, I had students fill out a "Get to Know Me" form, we played an icebreaker game (two truths and a lie - my favourite to this day), I talked about the rules and expectations, and I ended the day by teaching them my class dismissal routine. What I DIDN'T do (but wish I did), however, was any science, and that's about to change for this year.

## This Upcoming School Year

Subject: Chemistry (?), TBD

Grades: 9 - 12 (?) TBD

As in the past, my main goals for the first day of school are:

1) get to know my students, and

2) set the tone for the rest of the year,

but how I plan to achieve them will change somewhat.

**1 - Getting to know my students.**

Ideally, I would like to learn student names as quickly as I can, before the first day, if possible. But regardless, I would still like to use name tents with feedback, an idea that Sara Vanderwerf talks about in her blog. I think this is a great way to connect with students individually and on a more personal level. I would also like to take pictures of the students with their name tents so I have a visual record as well. A modification I will make to Sara's version of the name tents is that I will provide some open-ended prompts that the students can respond to, so that they have a jumping-off point for organic thoughts to develop. For instance:

- I noticed ...

- I wonder ...

- I learned ...

- I wish ...

I also really like the Talking Points activity from MathMinds and plan to modify it to make it chemistry specific.

Another idea I've been toying with is some sort of homework assignment that addresses a few or all of the 5 Questions to Ask Your Students To Start the School Year from @gcouros but my problem with this is that I don't want it to JUST be about rapport building, it needs to address or be linked an aspect of science (or science learning) specifically... to be determined.

**2 -**

**Setting the tone for the rest of the year.**

We will, presumably, be doing chemistry so I would like to begin the first day with a demonstration, or an activity related to the nature and processes of science. Some ideas I would like to try:

Stacking Cups (Dan Meyer) - related to concepts of measurement, accuracy, precision, and estimation

Candle Light Activity (Art of Teaching Science) - importance of observation (qualitative and quantitative) in science, making inferences and predictions, chemical and physical properties

Ira Remsen Demo (Michael Morgan) - observation, predictions, inferences, chemical safety, chemical reactions

I believe that it is important to talk about my expectations and what students can expect out of the class, however, what I DON'T want to do is just read the syllabus on the first day. A prof once suggested just letting the students read the syllabus at home and talk about it the following day so they can ask questions about what they read, or doing a quiz if necessary about the content in the syllabus.

__First Day Plan (rough draft):__

1) Greet students at the door

2) Have an activity for them to get started with on their desk (either to quietly read the syllabus or fill out a Who I Am handout)

3) Introductions myself and the course

4) Student introductions + talking points

5) Do some science!

6) Dismissal routine

## Concluding thoughts

My first day experiences thus far have been pretty nerve-racking and exciting. I'm slowly learning to strike the right balance between talking about rules and procedures to relinquishing control, and giving voice to the students. This is particularly difficult in a room full of ELL students, but once they gain confidence in their ability to speak and be heard, I found that they had a lot to contribute. With international schools, it is usually the case that the students are well acquainted with each other already, so usually the introductions are more for the teacher rather than the students. Even though students may already know each other, however, my role as a teacher to facilitate a safe and positive community cannot be ignored. This was made prevalent to me in Korea when I realized that students still felt unwilling to work with particular classmates even though they had been in the same classes for years. Regardless of country, language, or culture, my biggest take away for the first day of the school year is to BUILD RELATIONSHIPS and ESTABLISH COMMUNITY. I will keep this in mind as I continue to plan for my first day of school in China this school year!

*"There's nothing wrong with a few dirty dishes. Right now, there are more important things to do*" - the means isn't really important so long as you achieve your goal. The second, more refined kind of BS is the kind that has a little more conviction behind it. It's the kind of high-tech BS machinery you never realized you possessed that only gets unleashed in the final hours before a major project, paper, or assignment is due. It is the BS that has the essence of utter crap and yet

*somehow*manages to surpass even your highest expectations. Let's face it, there is something tasteful about BS-ing with conviction; teachers do it and improvisers do it. There's an argument to be made about stepping up our BS game as teachers and helping our students do the same, and we have a thing or two to learn from improvisational theatre.

Improvising is wonderful. But, the thing is that you cannot improvise unless you know exactly what you're doing

## - Christoper Walken

The dream: foster greater teamwork, collaboration, and creativity amongst my students. The reality: a lot of reluctance, awkward silences, and miscommunication. Students were reluctant to participate because it potentially meant making a fool of themselves in front of their classmates. There were awkward silences because they were uncomfortable with the idea that they could control the dialogue rather than do what they were told. Miscommunication happened because students focused too much on making themselves look good at the cost of sloppy scene work.

Our first attempts at simple improv games failed tremendously. The foundation of trust wasn't there, and students had not yet learned the art of failing spectacularly. I challenged my students often and constantly pushed them towards more complicated tasks and scene work. What I didn't realize, however, was that my students probably needed a much gentler progression, and more scaffolding. If I were to teach improv again I would spend more time on the fundamental concepts and revisit them often. I would tell my students that you don't need to be loud to be heard, and I would focus more of my feedback on things that were going well rather than things that weren't.

Even though my little experimentations with improv failed in a lot of ways, I learned much from the experience and would do it again in a heartbeat. Not only did teaching improv put me out of my comfort zone in terms of teaching, but we had a lot of fun! In the beginning I did a lot of "telling"; I read up on the rules and common pitfalls and communicated these to my students in hopes that they would avoid them. This did not work. They did those things anyway. But they got really good at doing those things, so for half a class, we just practiced doing scenarios in which your partner either blocked or wimped* in a scenario and we talked about what that felt like and how to make it better. Then we tried adhering to the "yes, and" rule and discovered that it was harder than it seems! With repeated trials and errors, we slowly progressed to a point where a few students felt comfortable performing on stage, others began to feel a little less intimidated, and everyone learned a bit about what it means to work together effectively.

Teaching and learning about improvisation was like figuring out how to have a great conversation. When a scene goes poorly, it is usually because a player is too much in their own heads and not focused on their scene partner. Much like in life, this is annoying because we seldom have great conversations with people who try to make everything about themselves. As we continued to explore improv, both my students and I became more mindful of how certain responses can either help or hinder a scene. A friend of mine gave me some good improv advice, "You should always make your partner look good," which is another way of saying, "This isn't about you." For example, your scene partner makes an offer and says, "Wow, the view is amazing from here," and if you say, "What view?" then you are putting a lot of pressure on your partner to make something up to progress the scene. If instead you respond, "I have really bad vertigo, I need to get off this cliff," then you just made your partner look good by taking their offer, and adding some valuable information to it.

Wouldn't it be great if all class discussions could flow so smoothly? Where all students are equal participants, building on the ideas of one another, and each adding something of value? I see value in teaching and learning improv outside the drama classroom. This article makes some good arguments for why and how improv can be incorporated into all subject matters, and has some great sample games and activities that can be modified for all grade levels. If you are a teacher who is curious about how improv can be implemented into other subjects, I highly recommend reading this paper.

Below are a list of Tina Fey's four basic rules on improvisation, taken from her book

*Bossypants*. In my next blog, I will elaborate on how I believe these rules are connected with my teaching practice.

__Blocking__: Rejecting information or ideas offered by another player. One of the most common problems experienced by new improvisers.

Example

Player A: Look at that dog!

Player B: What dog?

__Wimping__: Accepting an offer but failing to act on it.

Example

Player A: We should go to the movies.

Player B: Yes.

More info here.

**1. Build a thinking classroom.**

This isn't a new goal for me, but something I'm always trying to do better. In teacher's college, I was introduced to the phrase "Explore First, Explain Later" in my Introduction to Biology Teaching class and this is something I try to incorporate into my math and science classes every single day. The concept is self-explanatory; students are given a chance to explore, investigate, and uncover ideas within a particular topic or concept prior to taking formalized notes. This teaching methodology is congruent to the constructivist theory of learning which states that "that learning is an active, contextualized process of constructing knowledge rather than acquiring it" (learningtheories.com).

"Exploration" can take many forms; investigation, experiments, noticing and wondering... however, something I'm keen on devoting more time to in my planning and lessons is developing the question. Daniel T. Willingham writes about this in his book

*Why Don't Students Like School*, "Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question." I've really been following Dan Meyer's lead on how to do this; his blog post on "The Three Acts of a Mathematical Story" are a good place to start.

Peter Liljedahl also hosts a free webinar on how to build a thinking classroom, available here.

**2. Get students to talk more.**

It is so easy to just fall into a routine of lecturing/note-taking followed by independent (usually textbook) work, but I eventually want to create an environment in which students manage themselves. This begins by getting them to talk more, exchange ideas, and share what they already know. Some things I'm excited about trying in my classroom are Stand and Talks (Sara VDW), and talking points (adapted from Lyn Dawes).

**3. Do fewer things better.**

When I first started my student teaching, it consumed my life. Go to school, plan for the next day, sleep, and repeat. I stopped exercising, watching TV, hanging out with my friends... and basically anything that was not work-related. I could've used an old lesson plan my associate teacher has taught before; I could've downloaded lesson resources online; or I could have picked one really good question and focus the class on that for the entire period. There were a million things I could have done better, but no. Instead, I scoured dozens of sites for lesson ideas, worksheets, and activities before creating my own unique cocktail using an amalgamation of the best ideas I had gathered. I made my own worksheets and presentations because I wanted things done in my own exact, particular way. Planning a single lesson would take me

*hours -*

__this is not sustainable!__

I know better, so I'm going to do better this year. Angela Watson's keynote presentation for the Build Math Minds Virtual Summit really helped me refocus and re-evaluate my priorities. I'm going to invest my energy in doing the stuff that matters, and NOT because:

- Of peer pressure "Everyone else is doing it, so I'd better do too!"
- It's tradition "We do this every year, so we must do it this year!"
- It's instagram-worthy "OMG this will look so cute when it's done!"

Instead I'll only commit my energy to doing something if:

- It will help me help students engage and interact with the subject in a meaningful way
- I believe it is the best use of my students' class time
- It is something I am genuinely excited about trying in my classroom

Three things I'm going to start doing now to achieve this goal:

1) Manage my time by setting a timer for the tasks that need to get done, and stick to it. Whatever gets done during that time doesn't have to be perfect or have beautiful fonts and layouts, it just needs to be good

*enough*.

2) Reduce my workload by only formally assessing student work if I believe it is a TRUE reflection of student learning.

3) Increase efficiency by delegating tasks to students, like self-marking formative assessments.

*Cirque du Soleil*? Much like the ideas presented in that chapter, The Upside-Down School is about questioning the traditional assumptions of schooling and education and flipping them on their heads – the same story with a different twist.

In science, one of the first activities I do with my students is have them sketch an image of a scientist. That's it. The activity is simple but reveals a lot about our preconceived notions of what science is and what exactly it is that scientists do. The stereotypical image of a scientist is presented as follows: a white male with wacky hair in a white lab coat working in a laboratory with chemistry equipment. We talk about what these stereotypes mean and where they come from. We talk about why these images are problematic and what we can do about it. And then, we revise.

__Materials:__

- Two identical Lego sets per group (I would recommend no more than 20 pieces per set as the focus should be on the procedure writing rather than the Lego building)
- Student Handout: Designing a Procedure
- Cardboard boxes (or some other material that can be used as a shield)

designingaprocedure.pdf | |

File Size: | 93 kb |

File Type: |

__Engage__

I started the lesson by asking students two questions:

- What is a procedure?
- When do we use procedures in our everyday lives?

__Explore__

**PART 1:**Designing the Structure and Writing the Procedure

*as*they built the structure or

*after,*all of them chose the latter option which I found interesting. I gave my students a time limit of 25 minutes to complete this part of the activity, which turned out to be a little rushed because it took them some time to settle on a design. In the future, I need to re-emphasize the main point of the activity, which is to write a clear, accurate, and reproducible procedure that someone else can follow. I ended up giving students closer to 30 minutes. Groups that did not finish within the time limit handed in an incomplete procedure.

**PART 2:**Following and Evaluating a Procedure

*What makes this procedure easy/difficult to follow? How can you tell which way the pieces are oriented? What would make this step clearer?"*

__Explain__

After students had a chance to follow another group's procedure, they completed question #5 in Part II of the student handout (above). Once the students have had time to write down their thoughts I had them share their feedback with the other groups.

Notes: Some students began to get overly critical and picky about each other's procedure. At this point, I reminded them of what constructive feedback looks like, and reminded them that mistakes are OKAY - they are part of the learning process.

__Extend__

An extension might be to have students follow-up this activity with a procedure about making a peanut butter and jelly sandwich, brushing their teeth, or some other common, everyday task.

Note: This would also be a good time to discuss how scientific procedures differ from everyday ones, and how the formatting may change depending on the class. For example, in my class, students will be expected to write their procedures in

*past tense*and include

*labeled diagrams*with figure captions written

*below*the diagram.

__Evaluate__

I will evaluate my students on their ability to write clear, accurate, and reproducible procedures as a part of their Mystery Solids Investigation performance task.

## April Soo

International math educator who writes, occasionally.

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