In the spirit of UN Day this past November 2020, I designed a project that combines aspects of the Mathematician Project inspired by Annie Perkins from NCTM and also mathematics from different cultures and perspectives. We ended up implementing this project department-wide and I’d like to share it with you here (scroll down for an editable Google Slides link of the project).
The purpose of the project is to highlight a mathematician or maths concept that is underrepresented in popular culture. No offence to Euclid, Pythagoras, Pascal, and the like, but do a quick image search of “famous mathematicians” and you’ll most likely get a bunch of white-dudes like this:
Not to say that there’s anything wrong with old white dude mathematicians, but it’s high time we recognize some of the diversity out there. I think it’s important for our students to see examples of maths and mathematicians that they relate to. Bottom line, EVERYONE can achieve maths at high levels.
The second option gives students an opportunity to explore a maths concept or idea typically not covered in a regular K-12 curriculum. Some sample topics include:
Origins of Zero (India, China, Mayans, Babylonians)
Infinity and the aleph numbers
History of chess
Mathematics of Eastern countries
Islamic mosaic art
Construction of 4×4 magic squares
Topic ideas inspired by the Global Math course developed by Dave Ebert.
Students are given a brief outline of what to include for each option and must put together an engaging 3 – 5 minute presentation for their topic. For my Algebra II students, the best part of the project was the presentation portion. Each group of students (or individual, if they chose to work alone), was assigned a lower school classroom (grades 1 – 6) to present their findings to. Not only did the students have to research and learn about their topic, but they also had to find the best way to engage their specific audience.
The students had fun sharing their learning with the lower grades and were humbled by the experience. “Miss, they asked so many strange questions.”
“They were so excited when we said we had CANDY”
“I can‘t believe they don’t know how to solve one step equations.”
“They participated A LOT.”
These were some things I heard from my students. It was a valuable learning experience for them, and I was so proud to see my students shine. Added bonus: the teacher who’s classrooms the students visited were responsible for grading the presentation portion of their project (a simple checkbox style rubric), which helped me save some time.
What I Would Do Differently
Since it took a while to get the collaboration of lower school teachers, and approval from admin, I would start this process much sooner. This would also allow my students more time to tailor their specific presentations to the grade levels which they are matched with.
Next time, I would also had a specific section in the rubric just on the research and teach students how to effectively take notes and summarize information from their research. In addition, I would spend another class period on the anatomy of a good presentation and How to Avoid Death by Powerpoint.
I’ve been in Qatar for nearly two months now and am slowly making time and space for myself to just…exist. Quiet moments of thought and contemplation have been rare while balancing a full-time job that requires 2x the work, a masters, lots of home cooking (not my forte), and keeping up with a consistent fitness routine.
As an introvert, a lot of the social isolation that came with COVID earlier this year felt manageable (at first). Yet even the most extreme introverts crave social interaction, and so it came to be that by the end of my quarantine, getting to go to work was a welcome reprise indeed! Boy it has been an adjustment!
I have become the morning-est morning person that I have ever been in my life. 5 AM wake up and in bed by 9 PM at the latest. A disciplined sleep schedule aside, for the first few weeks, I was just barely keeping my head above water. I thought I had a decent handle on online teaching. I like to think that I’ve at least developed some level of proficiency as a face to face teacher… Then came the blended learning model, and BAM! Just like that I was back at ground zero. Like I said, it’s been an adjustment. I am still learning, and I’m okay with that.
Outside of work, though, I’ve been fortunate to have experienced a sampling of various activities and venues that Doha has to offer.
Traditional market with surprisingly clean bathrooms, an assortment of animals, foods, and various local goods.
Market at the W
Word on the street is that hotel brunches are a thing. They aren’t cheap, but compared to my experiences in China, not much is here. I went for a three course meal for the Market’s Supper Club menu at 120 qar per person, plus half off happy hour drinks.
I hear that all-you-can-eat-and-drink brunches are also popular at hotels since they are the only places that serve alcohol. I plan to hit up a few of those this year.
I’ll admit, I need to work on my food photography game. Above features the Tomato Soup (pre-soup), Blackened Hammour, and Market Cheesecake.
Check out some of these photos from @marketdohamenugraph for pictures that actually do the food justice!
A day of adventures in the sun. Camel riding, dune bashing, and lots of time by the beach!
Sunrise on the beach! What more could a girl ask for?
A small community of Capoeiristas who have been teaching me how to dance!
What a strange year it’s been! What began as physical distancing turned into social isolation, and just as I was beginning to integrate myself back into society, I decided to pack my bags and head to another country.
In a previous post, I reflected on my experience with online teaching for the first time in “China” and how different it’s been.
While I camped out back home in Canada I continued to teach online and wait for the Chinese borders to reopen. Then I found out that I no longer had a job for next year. My plan had been to stay in China and continue teaching at a different school, but coronavirus had other plans for me. So, I ended up working fast-food for a bit as I figured out what my next steps would be. What a humbling experience it was!
While I worked my part-time fast food gig, I continued on with the job search, and eventually landed on a position here in Qatar.
Qatar is an Arab country in the Middle East right next to Saudi Arabia. According to Wikipedia, Qatar ranks third highest in the world for GDP, and about 88% of the country’s population are expatriates (although I have not done much further digging on the statistics).
Is it safe? Yes.
Can you tell I get asked this question a lot?
At the moment, I am working in Doha, the capital of Qatar. It is very modern, clean, and welcoming based on my impressions so far. Though, to be fair I haven’t seen much of Qatar as I have been wrapping up my quarantine here.
My Quarantine Experience
Coming from a low risk country, I was eligible for a one-week home quarantine at my incredibly spacious school-provided apartment.
Quarantine in this apartment feels like I’m in a luxury jail, but with less perks than what Jeffrey Epstein had. I can still order delivery, have access to wifi, and plenty of space to work out, but at the end of the day I’m still stuck (“safe”) inside.
Upon arrival at Doha International Airport, you need to make sure you download an app called EHTERAZ, the covid tracking app for Qatar. It’s a simple colour-coded system that is linked to your identification. Green means a negative covid test result, and you are free to go about as you please, red meaning you have tested positive, yellow for quarantine, and grey for suspected infections.
They shuttle you off into a testing area where you sign some forms and get swabbed for your first covid test after landing.
I was able to be picked up by my Head of Schools, who brought me to my apartment to begin my one week mandatory quarantine.
On day six, you are supposed to get a call or SMS text message about going in to a designated testing centre for your second covid test. After results are processed, and if you get a negative test result the EHTERAZ app on your mobile phone turns green and you are free to roam about.
This was not my experience. What was supposed to be a one week quarantine has, by some unfortunate event, turned into two.
Frustratingly enough, on day six I received no calls or text messages. I reached out to my school HR representative to inquire, and was advised to continue waiting for further instructions, or for my code to turn green. The next day, I decided to call the number listed on the EHTERAZ app. The person who answered the phone also told me to wait for a call, even though I had already completed my mandatory one week quarantine at the time.
So I kept waiting… Eventually, it was communicated to me by word of mouth that I WAS in fact allowed to leave my apartment to get my second covid test since I was already finished my quarantine, which brings us to the present day. As I am writing and sharing my experience with you, I am also eagerly awaiting for the moment when I am officially allowed to leave the apartment and explore Doha.
Hopefully soon I can post updates about Qatar! (Sans quarantine).
I got this idea from Howie Hua! I read about how Howie used Student Autobiographies in his classroom to get to know his students, get them to learn about each other, and help build community in an online setting.
Here’s a link to the Google slides template that I shared with my students.
Last year I taught in China where this would have been difficult to do since I have not been able to find any other free platform that offers the same sharing and editing features as Google Slides.
It’s been really amazing being able to learn about my students and seeing them interact with each other despite being fully online at the moment. I was also able to share out the slides to parents where they could also browse the autobiographies and meet our community of learners!
Peter Liljedahl‘s work on Building Thinking Classrooms has been extremely influential in the education world. In his research, he discusses a collection of high-yield strategies that teachers can employ to help learners engage with work in the classroom and become better thinkers. They are sorted on a continuum of ease of implementation and “bluntness” (requiring less/more fine tuning).
When I first stumbled upon his research (hat tip to the Global Math Department), I immediately began to experiment with the two moves that were the easiest to implement: Visibly Random Groups and Vertical Non-Permanent Surfaces.
I created Seat Finder cards that I would hand out to my students at the beginning of every class as I greeted them at the door. The Seat Finder cards had various algebra problems on them whose solution corresponded to their group number.
The idea behind this is to help build a classroom culture where students are working collaboratively, and not just with the same people the entire year. In theory, this works great, however, I hit a few kinks along the way. The first kink was that I was not beginning my lessons with “good tasks” at the time (see here for some examples). Second, it quickly became exhausting and time consuming for students to find their seats in this manner each day. I also noticed that students would try and surreptitiously swap cards so they could sit with their friends. Rather than building that culture of community and collaboration, I was battling my students about compliance issues.
So, to alleviate some of these problems, I switched to weekly seat changes, and eventually settled with a change once per unit. Rather than assigning each student their individualized Seat Finder problem, I gave each student a playing card instead that corresponded to a group number. Rather than writing the group numbers on the tables, I opted for math problems whose solutions corresponded to group numbers instead. I created a set for different units of study that we looked at in our classes, which are meant to be solved relatively quickly.
I found that having one problem per table helped students get into their seats more quickly, and they often helped each other if some students were unsure of how to solve the problem.
Note that while students seats stayed the same for the course of a unit, they were still expected to work with others on various tasks I assigned during class. This gave them the comfort and consistency of knowing where they were sitting from day to day, but also the ability to interact with a variety of classmates.
For a link to my downloadable Seat Finders and templates, click here.
I remember sitting at a meeting table at my school in Suzhou, China late January earlier this year when a colleague said, “Heads up, there’s a virus going around in Wuhan. Very contagious,” and thinking something along the lines of “Well, at least we’ve got our Chinese New Year holiday coming up. Let’s cross that bridge when we get to it.” I was, absent-minded to say the least.
Two weeks later, nobody could wander into a public space without a mask, long distance buses stopped running, temperature checks were imposed at major transportation hubs, and cars were restricted entry into locations outside the jurisdiction of their licensure.
2020, I imagine, has not turned out to be the year many has expected it to be. I write this four and a half months after the World Health Organization declared the Novel Coronavirus disease (COVID-19) to be a pandemic.
This dramatic turn of events has had me feeling like the entire world is just holding its breath. For a while, all we could do was observe and wait, not fully willing to settle into what was slowly shifting into our new normal in hopes that we can just pick up where we left off pre-COVID.
By the time Semester 2 began mid-February, I was “temporarily” residing back in Canada. For safety reasons, the school re-opening date was to be pushed back two weeks. Teachers were told that online learning would only be temporary (lasting no longer than 2-4 weeks) to accommodate for rapid changes in travel plans and quarantine measures.
Our school eventually re-opened in May, but by that time, a majority of our teaching staff were out of country and had no way to get back in. China had already closed its borders and our visas suspended. All we could do was wait. I ended up teaching the entire semester online, with many changes and adjustments that had to be made to teaching style, content delivery, and assessment along the way.
Despite the many barriers that were imposed upon us, I remind myself that I still have a lot to be grateful for. I didn’t chance to say goodbye to my students in person, but we found new and different ways to connect online. We missed out on a ton of live in-school events and activities like Pi Day, the graduation ceremony, the school-wide lip dub, sports competitions, but that didn’t stop us from celebrating student achievements through their virtual counterparts. I lost my job too, but at least now I have an opportunity to start fresh. When one door closes, right?
When I started the school year, one of my professional goals was to be able to get into more teacher classrooms, of all different subject matters, to learn from and observe my colleagues, and to get teachers in my classroom as well. It was my way of taking small steps towards making #observeme more of the norm at our school.
As a department, we worked on re-vamping the way we structured our classes and assessments using principles from cognitive psychology to better help our students learn and retain information (I wrote about it here).
We experienced many successes in our first semester, but still had a long way to go in terms of finding our groove. When COVID hit, I knew we all needed a new goal: find ways to help us and our students successfully navigate the world of online learning.
Starting from ground zero
If I’m honest, for a long time it felt as if we were just keeping our heads above the water, struggling to balance between the uncertain timeline imposed by COVID, as well as expectations from the school, students, parents, and ourselves. Even though we were working long hours, none of us truly felt like we were operating at 100% capacity. A jumbled mess now laid in place of the clear path (or so it seemed) that was once before us.
Several cheating and/or plagiarism incidents took place with course assignments
High student absenteeism rates, especially at the beginning, which led to snowball effect of select students continuing that trend to the end of the year
Inconsistent scheduling (due to various factors)
Difficulty communicating with some parents and students
Developing a consistent school-wide plan for scheduling, parent and student communication
Keeping platforms for communication and e-learning consistent.
Incorporating interactive and collaborative elements in online synchronous lessons (e.g. Padlet, Nearpod).
Eliminating heavily weighted timed assessments (such as unit tests and final exam) helped alleviate pressures associated with adjusting to an online learning environment. Students had more time to work on fewer graded assessments, thus increasing the quality of work to be handed in.
High success and engagement experienced with implementing open tasks on Flipgrid.
Moving from weekly to bi-weekly quizzes to help with student workload.
Maintaining a positive mindset and staying flexible and adaptable to changing circumstances (school policy, overall outlook of global health crisis… etc.)
SUGGESTIONS FOR IMPROVEMENT:
Keep graded assessments to a minimum.
Opt for more open tasks, collaborative projects, or project-based and/or problem-based learning
Develop, communicate, reinforce and continually PRACTICE norms for successful online learning
May need to rethink mandatory synchronous live lessons. Issues of access may make this a non-equitable practice that may hinder the success of certain students. Providing equitable asynchronous learning options is ideal for ensuring equal learning opportunities for all.
For a long time, I was in denial. The practical me jumped in headfirst and did her best to adjust and adapt to changing circumstance, whereas my less practical self refused to accept that this is really happening. Yet, neither of those selves are helpful. In thinking about the past, or worrying about the future, I forget to live in the present.
This week I’ve had some great lessons, and some awful ones. Looking back at what I had done differently in the good versus not-so-good lessons, I realized that one of the biggest differences was the amount of “telling” I was doing in one class versus another. It didn’t matter that I had amazing visuals and was super enthusiastic about the content I was teaching; if I talked too much, students would start to zone out. Compound this with the fact that we are distance learning and all of my students are English language learners, we now have wi-fi/connectivity, audio, and language learning issues all thrown into the mix.
The one who does most of the talking, is doing most of the learning.
(Something I’ve heard from multiple sources throughout my teaching career)
At this point, I have to slap myself on the wrist because I know better, so I need to do better.
In yesterday’s class, I consciously made an effort to talk less and ask more questions. I also explicitly told my students that my goal as a teacher is to never tell them an answer, but to just show them the way. Classic case of easier-said-than-done.
I realize, with a sudden mixture of nervousness, trepidation, and excitement as I’m writing this, that this might be the first time in five years of teaching that I have really made a conscious effort to take Cathy Fosnot’s advice to heart. She writes,
Don’t try to fix the mathematics; work with the mathematician. The point is not to fix the mistakes in the children’s work or to get everyone to agree with your answer, but to support your students’ development as mathematicians.
On the surface, I’d like to think I was doing my best to project a calm, neutral tone as I jotted down notes while students shared their thinking. I wrote everything down, regardless of whether their strategy was “right” or “wrong”. Meanwhile, it felt like Hermione Granger was living in the back of my mind jumping up and down going “Pick me, pick me!! I know the answer!!” Talk about my “rescue the student” instincts being on overdrive!
In the past, I would have eventually given in to those instincts and immediately correct any mistakes that came to my attention. I tell myself that this is okay, because if I don’t, my students will continue to make those fundamental math errors, divide by zero, and initiate the end of the universe as we know it. I also think that deep down, that hidden behind this instinct is fear, fear that I can’t help them get where they need to go without just giving them the answer. Although, impatience is an equally guilty accomplice here in my crime of robbing students of a perfectly good learning moment.
This time, however, I tell myself a different story. I learn to trust myself and my students a little bit more by just letting them get where they need to go, in their own time and in their own way. This too, is a little scary.
To add some context, here’s a bit about how my lesson went.
The goal of today’s lesson was to introduce the idea of trigonometric identities, collect some strategies that may be helpful in identifying whether a given statement is true or false, and then work on moving towards what it means to rigorously prove the truth or falsity of a mathematical statement.
I began the class by doing a modified version of an “Always, Sometimes, or Never True” activity with radicals (trying to introduce some interleaving here) from the Mathematics Assessment Project and called it “Truths and Lies”. I asked students to tell me which statements they believed to be truths and which they thought were lies, and to share their thinking on Padlet.
After about ten minutes of individual think time, I selected a few student strategies and had students explain them to the class. Here’s what we came up with:
Plug in a number for x and check to see if both sides are equal
Start with LS or RS, use algebra to show it is equal to the other side
Assume the statement is true. Square both sides of the equation. (If both sides are equal after squaring, then the statement is true).
Next, I showed them this image about the different Levels of Convincing from Robert Kaplinsky’s site.
We then revisited each strategy and I asked students to mentally place each of these strategies fell on the spectrum of least to most convincing. Ideally, I would have given more time for students to really think this part through, but since we are doing distance learning, I was eager to get to the real meat of today’s activity, which was to prove trigonometric identities. From there, I took on the role of prosecutor and started to stir up some trouble.
For instance, in statement 1), we can demonstrate the statement is false by finding a value of x that shows LS does not equal RS, however, I argued that x = 5 worked, so wouldn’t that make the statement true? What I’m getting at here is that I want students to be able to articulate what exactly are we asking when we ask whether or not a statement is true? That it must be true all the time? Or only some of the time?
Then, I attempted to tackle the “squaring both sides” strategy… Couldn’t we also use same reasoning to show that 1 = -1?? (Can you see why?)
At this point something really amazing happened, and that was when a student interrupted me and said, “Ms. Soo, I just thought of another way to explain this!” The student was able to connect what we were doing to our study of transformations of functions from a previous unit.
I couldn’t —
keep my poker face, that is. This was me:
For the remainder of our lesson I had students work independently on the following:
My goal for them was to use different strategies and methods to try and “prove” or “justify” which were truths and which were lies. Students always have the option of messaging me privately for hints or advice if they were stuck, very few did.
After about 15 minutes, I asked students to send me pictures of their work and we could start talking about some strategies they used. The one mistake I see students make when “proving” trigonometric identity is to start by assuming the statement is true and start manipulating both sides of the equation.
Instead of telling students WHY they can’t do that, I referenced my earlier example of how, by the same logic, we can prove that that 1 = -1 and asked them, WHERE did the mistake occur?
Let’s assume. the statement 1 = -1 to be a true statement.
Next, let’s square both sides of the equation.Doing so, I get
Therefore, it must be the case that 1 = -1. (End of proof).
Getting students to where I wanted them to be was really challenging because many were focused on the math, and not the logic of the argument itself. They focused more on the operation of ‘squaring’ and how we need to keep in mind both positive and negative square roots, which is certainly a valid piece of mathematical insight, but again not where I needed them to be.
Since we only had about 5 minutes of class left, I decided to pause the discussion there and ask students to write me a 3 – 5 sentence of the strategies we used to justify whether a statement is true or false.
What Went Well
I stuck with my goal.
Where I Need Help
Right now, students still don’t understand what a proof is. I want students to be able to articulate that while plugging in values, and graphing both sides of an equation are helpful strategies to show why a statement might be true, they don’t constitute enough rigour to show that a statement is always true.
How do I get students to this point without just handing them the answer? How can I do this effectively in an online setting? They also have a common assessment (assignment) coming up in which they will be asked to prove trigonometric identities, and the quickly approaching deadline makes me feel anxious to default to just tell students the answer.
Any tips, suggestions, or feedback would be greatly appreciated! Please leave your comments below.
I’m no expert, but the COVID pandemic has given me the prerogative to scour the interwebs for useful tidbits on maintaining lively and engaging online lessons. In the last three or so months, I’ve created at least half a dozen new teacher accounts on educational sites and platforms; some of which I use moderately (EdPuzzle, Padlet, Flipgrid…etc.), a few that I use religiously (Zoom), and still others that I’d like to experiment with some more (Nearpod, Brainingcamp).
Transitioning to full-time online teaching has been a process of repeated trial and error, and a test of patience and flexibility in learning to adjust to changing circumstances. I’ve definitely made more than my fair share of mistakes, but here I’ve compiled list of tips and tricks that I’ve found useful for teaching online. Some of these are obvious and are good practice in general, and others are things I’ve learned along the way or things I wish I had done sooner. If there are any tips here that you think I missed, I would love to hear about them in the comments!
Delivering Live Lessons
Look AT the camera, not your screen. An easy one to remember for more formal settings, like online interviews, but also at an important one not to miss with your students too. It shows them they matter, you care, and gives them a sense that you are watching.
Left: Even though I’m looking at my students on screen, I appear less engaged. Right: Takes some getting used to, but this one has more of the right feel to it.
Display your daily agenda, and deadlines on your screen like you would in your regular classroom. This develops helps consistency and create routine for you students.
Always pair visual and verbal cues. If you want your students to respond to a question in a group chat, or complete an activity, make sure they can hear and see the instructions, as some may experience audio or internet connectivity issues. (Good practice in the regular classroom too).
Allow longer than normal wait times. Again, expect a lag between the time you pose the question to when your students actual hear it.
In general, I’ve found that live lessons take MUCH longer than a regular class. Plan more than you need but expect to cover less.
Engage students as much as possible. Q&A sessions can get tricky in an online setting and plain old cold calling… well, gets cold. In the next section, I’ll take about some low stakes methods to ensure that students aren’t just tuning into your lesson, only to be playing League of Legends off screen.
Start easy. Rather than dive right into the deep end with new content, a class-wide discussion…etc. why not begin with a warm up question? I like to start class with an attendance question that each student will answer (a tip I picked up from my VP). This gives me a quick and easy way to check-in with my students, they get to learn some information about each other, and it allows time for mentally transitioning into learning mode.
Keep in mind it DOES take time to check in with each student individually, so think about the type of question you want to ask, and whether or not you will give each individual student air time, or have them all type their answers into a shared document simultaneously.
Record your lesson and upload the video for later access. Zoom does this automatically, but there are plenty of free software out there for you to record your lessons. We are an Office 365 school, so I upload my recorded lessons onto Sharepoint for any students who missed a class.
Get a drawing tablet! Perhaps this goes without saying, but it is really difficult (in my math class at least) to pair visual and verbal cues when I can’t draw or write on the board. Having a tablet helped alleviate that issue.
Something I wish I had done is model to students how e-learning works. Sara Van Der Werf talks about this in detail here.
Set the expectation that students need to turn their video cameras ON right from the get go. This one may not work for everyone due to issues of access, but I found that in my classroom engagement is much higher when my students and I can all see each other. Not to mention this gives me a better way to gauge how they are responding to the lesson.
Don’t just lecture. If you are having a live session, use this time to build in as much interactive elements into the session as possible. Information heavy content can be recorded and made available to be accessed later.
Make PARTICIPATION, not evaluation, the norm. I thought that I would need to incentivize participation with marks (like marking Flipgrid responses), but looking back I don’t think this was the right move. Whatever platform(s) you are using for online engagement, use these early and often, and keep them low stakes.
Assessment is not the same as evaluation. Assessment is timely, and gives us a way to gauge where our students are at and for us to figure out how to get them to where they need to be. Assessment needs to happen early, often, and BEFORE evaluation.
Prioritize the learning itself, not the marks. I know from personal experience, this can often feel like an uphill battle, not only against whatever policies that have been set, but also against yourself. We’ve been teaching and learning for marks for so long it is easy to forget that the goal of knowing the Pythagorean theorem, or understanding transformation of functions is not so our students can pass the test, but because there is genuine enjoyment to be had! (This point deserves its own post).
Less > More! Really. This one was a biggie for me. Trying to do too much will only drive you crazy. While there is a ton of useful tech out there that can dramatically up our teaching game, it can also be time consuming to learn a new tool. Start small.
It’s not about the resource, but how you use it (check out this podcast, episode #70). Contrary to the last point, don’t let the fact that there is so much tech out there stop you from exploring a new tool. Yes, choice paralysis is real, but at some point simply sending your students links to Khan Academy videos ain’t gonna cut it.
Remember that kids have lives outside your classroom. This one is so important, even in a non-COVID situation, but nothing is easier to forget. I often get offended when kids don’t remember deadlines or to submit work, but the reality is that my class is NOT the centre of their universe and I have to be okay with that.
I love it when professional development is purposeful and practical. I’ve been following Robert Kaplinsky for some time now and finally decided to enrol in his Empowered Problem Solving Workshop.
“I don’t have time for problem solving in my classroom.”
TRUST me, I’ve been there. The first time I ever taught Calculus, my talk time during an 80-minute block was probably at 50-80%. It was awful, I was so dehydrated. It also didn’t help that I did not have a strong enough grasp of the material that I could deliver problem-based lessons with any sort of confidence. I was simultaneously teaching and relearning the material myself so how could I expect my students to develop these deep understandings when I was barely keeping my head above water?
Looking back, I realize that trying something is always better than nothing. Problem solving isn’t something you do “if you have time for it,” like at the end of a unit. Because you know what? You’re never going to have time for it. You’ll always feel like the time could be better used for review, a project, to reinforce a skill…etc. Problem solving is not something you should “make time for”, it needs to be integrated into the content we teach. I would argue that the heart of mathematics is problem solving. The sooner we realize that math isn’t just about getting the right answer, passing a test, or even getting into university, the sooner we can teach in a way that honours what doing mathematics is truly about.
“My students lack the basic skills and understanding to do these types of problems.”
If that is what you’re thinking, know that I too, have had this same thought. Herein lies the beauty of problem-based lessons: students don’t need to be pre-taught skills or content, they can learn them along the way.
“I’ve tried problem solving before and it doesn’t work. Students just want to be told what to do.”
Guilty. I’ve been there too. It’s not going to be perfect the first time you do this. Students WILL resist, and you need to persist. If you don’t genuinely believe that problem solving is worth the time and effort, your students won’t buy into it either.
When I first tried problem-based lessons, I did not spend enough time anticipating student responses and was taken off guard by solutions or strategies I hadn’t thought of. I tried to lead meaningful discussions about student work, but because I wasn’t getting the engagement I wanted, sometimes ended up making the connections for the students (I’m still working on scaling back my “rescue the student” instincts). Success, however, comes in small doses, like getting a student who normally never raised their hand to try a problem on the board, or maybe just seeing a decrease in off task behaviour.
Teaching problem-based lessons takes effort, from the student AND the teacher, but that is precisely why its so awesome. Students aren’t just passive recipients of knowledge, and teachers don’t need to spoon feed their students.
My Biggest Takeaways
Problem-solving: Just DO IT!
2. Be deliberate about how to facilitate meaningful discussions in math. Often, we get to an answer and that’s it. Full stop. Getting to the last act of a 3-Act Math Task doesn’t mean that the learning stops there. Here is a wonderful opportunity to discuss various approaches to the problem, potential sources of error, limitations of our mathematical models, and to make connections between different solutions. This is an area where I feel I need the most practice, and it is also most difficult to implement during this time of online learning due to COVID-19. I’m limited by the fact that I cannot circulate the classroom or peep over students’ shoulders to see where they are at, but I am trying to find alternative ways to connect.
Here’s a snapshot of me working out a selection strategy for sharing student work, and anticipating questions that might be helpful to ask:
3. You can always add information, but you can’t take it back. Dan Meyer refers to this as turning up the Math Dial. Robert Kaplinsky talks about “undercooking” our students (like you would a steak). Ask questions in a way that ranges from least helpful to most helpful to give your students a chance to make connections for themselves.
4. Ask yourself “Why” more often. Why am I doing this problem? To introduce a new concept? Get my students used to productive struggle? Problem completion? Be intentional about the purpose of your lesson and what can be realistically achieved.
5. Ask better questions. Shallow questions tend to lead to false positives. A student may appear to have procedural knowledge, fluency, and conceptual understanding, when in reality they are just good at replicating the work that you do (me in school…). You might be asking, “How do I really know if my students have the components of mathematical rigor?” Check out Robert Kaplinsky’s Open Middle problems and Depth of Knowledge Matrices.
I’ve been taking an online workshop to learn more about practical ways we can implement problem-based lessons in our math classrooms called Empowered Problem Solving by Robert Kaplinsky (#mathhero #teachercrush). In one of the workshop modules we troubleshoot various issues that may arise throughout the process of teaching a problem based lesson, for instance:
What happens if students don’t ask for information that they need to solve the problem?
What do you do if a student shares a strategy that you don’t understand or did not anticipate?
What do you do when students submit low quality or low effort work?
That last question really had me thinking a lot about assessment. When students submit low quality work it is often because they don’t know what the expectations are. Something I do quite often in my classes is share student work samples after an assignment or test to address common errors or mathematical practices. Here’s a brief overview of my journey in providing feedback for my students:
“What Should My Answer Look Like” Posters from MathEqualsLove, Examples are from my class 🙂
I don’t make enough time for level 3 work, and I should. Within a single semester, my goal is to give students at least two opportunities to do meaningful peer assessment. Of course, I anticipate this to be a gradual process, and it might take some time to get to a point where students can comfortably and confidently do peer assessment.
Assessment is difficult; even with a simplified assessment scheme one two teachers may assess the same student work slightly differently depending on their interpretation of what is “correct” or what qualifies as “sufficient reasoning.” Unfortunately these discrepancies will arise no matter what, but I think there is a lot of value in putting the students in our shoes and giving them opportunities to assess each other’s work.
Inspired by the Empowered Problem Solving Workshop, I’ve created a Mathematical Peer Editing Checklist and Feedback Form with practices I value and that I think is general enough to be used with most and/or all problem-based lessons. I’ve also incorporated an “overall feedback” section in the form based on Kaplinsky’s Levels of Convincing (originally inspired by Jo Boaler #mathhero #teachercrush) that asks students to rate each other’s mathematical writing based on how convincing they think their argument/work is.
We asked 101 high schoolers the following question:
There are 125 sheep in a flock and 25 dogs.
How old is the shepherd?
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?
The data was collected via an online survey (on account of school closures due to COVID-19) and was given to students in China earlier this year. Our initial goals were:
To collect some data regarding sense making in mathematics amongst our high schoolers (grades 10 – 12)
To analyze the data and assess what this means for us as educators. Are there differences in responses amongst the different grade levels? Are there gaps in student learning that we need to address? If so, how might we begin to address those gaps?
To use this activity as a way to begin a dialogue with students and teachers about sense-making in mathematics
The survey was conducted via Microsoft forms.
Of the 101 student who were surveyed:
25 stated there was not enough information to answer the problem, or did not supply a numerical response based on the constraints identified in the problem
74 gave numeric responses
2 students did not answer the question (e.g. one student wrote “nice question”)
At first, this data seems to be consistent with the results from Kaplinsky’s experiment with the 32 eight graders, in which 75% of them gave numerical responses by using random addition, subtraction, division, or multiplication of 125 and 5. Upon closer examination, however, we see that of the 73 that gave numeric responses, 28 used random math procedures, thus not making sense of the problem, but 45 of those students gave some sort of reasoning independent of the problem to support their numeric responses.
Students that gave numeric responses by combining 125 and 5 via random math operations (not making sense):
Note that a couple of students pointed out that there seemed to be an issue with the problem, but proceeded to give an answer anyway.
Students that did not provide a numeric response (making sense):
One response in particular really blew me away (click to expand the image):
This superstar response really blew me away.
Not only did this student state that the question did not give enough information to provide a specific answer, they used what information was presented in the problem, along with sources to support their thinking, to deduce an age range for the shepherd! Wow. How can I get the rest of my students here??
This is the point where I began to see another category emerge… Students who provided a numeric response, but justified their answers outside the range of expected range numeric responses such as:
125 + 5 = 130
125 – 5 = 120
125/5 = 25
Or other such random combinations of numbers. Note that no one responded with 125*5 = 625 as they seemed to realize this may be ridiculous age for a shepherd to be. Although one student did provide a response of 956000 through a series of random additions and subtractions.
I referred to this new category of responses as “supported guesses.” To be honest, I had difficulties categorizing some of these responses as “making sense” after seeing the superstar example from above, but ultimately decided that anything other than random math was a step in the right direction, although you will probably agree with me that some responses seem to employ more evidence of reasoning than others:
I was quite blown away by the number of students that treated this as a “trick” question and thus gave a wide range of responses, which ranged in creativity and depth of thinking. Like I mentioned, I found it difficult categorizing some of these responses, and found that after reading that Superstar response from above, my expectations rose (not necessarily a bad thing, but definitely made categorizing more difficult).
Some factors worth considering:
The responses were collected via an online survey. Would results have differed if this was done face to face? Did students try looking up the problem before attempting to solve it?
Students feeling like they did not have intellectual autonomy; not wanting to question the questioner due to respect for authority.
The old “my math teacher is asking me this, so I must calculate something” trick.
Cultural factors may be at play here. Perhaps students have seen some version of this problem before, thus accounting for the varied supported guesses observed.
It is also worth noting that in a follow-up reflection activity with students, some pointed out the need for teachers to ask less ambiguous problems, a few attributed their responses to poor understanding of the problem due to language barriers, while a fair number mentioned the importance of practicing different kinds of problem solving to develop critical thinking skills. This, I think, is a step in the right direction.
Teachers are getting a bad rap these days. To put it in perspective, my own mother — to whom I am and shall always remain eternally grateful for — expressed her annoyance at the fact that Ontario teachers were, yet again, going on strike. (I will also add here that she is also very supportive of the fact that her own daughter chose teaching as a profession). Like others, she felt that the strikes are an unnecessary waste of time, making teachers appear selfish and lazy. When asked what information she had to support her claim, the figure “100K” came up in conversation. WHAT? How much are teachers making a year?
$100 000. One hundred THOUSAND Canadian dollars. The supposed “average” salary Ontario teachers make a year.
Reported source? “The government.”
Had it not been for the fact that a) my mother has a tendency to exaggerate the truth and b) I am a teacher myself, I may have been inclined to side with her claim. To add a bit of context: I have been teaching for five years internationally and making nowhere near that figure. For me to be earning 100K a year, I would need to have my masters degree and an additional fifteen years of full time experience.
Let’s Talk About Averages
“Average” is a misleading term; it can refer to the mean, median or mode. In statistics, we call these “measures of central tendency.” Let me borrow an example from Wheelan’s book (Naked Statistics) to make a point.
Suppose five people are at a bar, each earning a salary of $35k a year. Undisputedly, the average salary (by all counts) of the group would be $35k. Typically, when we hear the word average, we equate it with the mean, which is the sum of all the points in a data set, and divided by the total number of values within the set.
Suppose Bill Gates walks into the bar, with a salary of $1 billion a year, bringing the average (mean) salary to $160 million. The reported figure, while still accurate, is not a fair representation of the average earnings of the majority of individuals in the group.
In this case, the knowing the median (middle value when all values are arranged from smallest to greatest) provides a bit of context. After all, the difference between 35 thousand and 160 million is no small sum.
This is a classic example of how precision can mask accuracy. Think about any time you’ve heard a number or figure reported in the news, consider the following statements, for instance:
Statement 1: “99% of statistics are made up” (Ha!)
Statement 2: “I have here in my hand a list of 205 — a list of names that were made known to the Secretary of State as being members of the Communist Party and who nevertheless are still working and shaping policy in the State Department” – Joseph McCarthy, a US previous senator (1950)
Don’t these it seem to bring credibility to whatever claim the person or organization is trying to assert? The first statement is, of course, made up. As for the second statement, it turns out that the paper had no names on it at all. Statistics is a tool that helps us bring meaning to data, but can be abused for nefarious purposes if wielded irresponsibly.
While math may be infallible, we are not. No matter how convincing the data may be, there is always more than one way to interpret it. It’s a little like telling your friends and family that the guy you just met “has a great personality,” which almost always implies that there is some other flaw or red flag that has not been said (Wheelan 37).
So, back to the this 100k salary I’m supposed to be making… How did they get this data? What are the demographics of the teachers being surveyed? (It makes a difference if the majority of teachers who have been working full time in Ontario have at least 15 years of experience under their belt). Are they including retired teachers? Teachers who have recently been laid off?
I tried to trace the origins of where this figure of 100k came from. After a bit of digging, I think its likely that my mother mis-reported the figure she heard from sources that gave out misinformation.
[NOTE FROM THE AUTHOR: 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.]
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.
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.
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).
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.
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.