Trig Investigation: Planning for a Productive Struggle

My first attempt in trying to introduce Trigonometry to my MFM2P class didn’t get off to a good start. Yesterday, however, things were looking much better.

My main issue was in trying to plan for a lesson inline with one of my learning goals for this year —designing a student-led learning experience. Much easier said than done, but trying to choose the right problems/activities that will allow me to comfortably step back and watch as my students learn through productive struggle proved to be more challenging than I expected.

In the process, I went through some productive struggle of my own, but it was well worth it! The idea of learning a new concept by allowing yourself time and space to first struggle with it is something I strongly believe in, and hence the reason why I’m keen on learning how to incorporate it in my designs.

I decided to go with my AT’s (Associate Teacher) advice and instead of trying to put together a plan from scratch, why not try out some of the lessons she previously designed to allow myself room to focus on the experience itself and what the targeted outcome should feel like first. This proved to be extremely successful because I was able to focus my energy on planning and visualizing how I will engage the students in the lesson, orchestrate the discussion so that it would last longer than the usual question-answer period, support them in the learning process without actually interfering, and consolidate key concepts through feedback on their own work.

For this lesson, the curriculum expectation was to be able to determine the measurements of a missing side-length and angle in a right triangle using Pythagorean theorem and trig ratios.

First, students logged onto Desmos to complete a short activity that allowed them to come up with the targeted questions on their own and then estimate the measurements before solving for them. The entire activity is based on a lesson design by Laura Wheeler. Here is a screen shot of some of their answers:



I gave them some time to work on it then highlighted the questions they came-up with (length of missing side + measure of angle theta).

Next, I instructed them to go to their boards with their groups to find the answers. I didn’t want to separate the question into two parts (first this, then that) because I wanted them to practice problem-solving on their own and come-up with their own reasoning for which part of the question they wanted to solve for first. Most of the groups started with the missing side-length using Pythagorean theorem —they were applying prior knowledge for this part, but some experienced a little bit of trouble when solving for it. I guess this is what’s great about spiralling; you introduce a concept and keep referring back to it throughout the semester to build on it. Another group started solving for the missing angle by subtracting the 90 degrees from the total of 180 and then assuming that it’s an isosceles right triangle (which it was), but I instructed them to justify that assumption.

I circulated the room a bit more and engaged in some conversations with the students about the thought processes they used. One thing I planned on trying out differently in this lesson, was to pick-out which groups I wanted to call on to share some of their strategies and then provide them with a little bit of a heads-up and choice for sharing. I picked up on this idea from Jo Boaler and Cathy Humphreys book on Connecting Mathematical Ideas. Before, I would usually just throw it out as a general question during our discussion, “who would like to share?” and it didn’t really get much conversation going. But this time, I strategized better and asked the groups I was interested in if they would like to share their strategy/process with the class (hinting that it would benefit their peers) and left the choice of the ‘speaker’ up to them.

The lesson was going great so far; I wasn’t doing any direct teaching, yet still watching them learn. I called on my group and they shared their thought process for their work on finding the missing side-length.


I reiterated the points they shared and then instructed the class to polish their work—if needed—and move forward to the next step (finding the angle).

Trig tables and course-pack notes started coming out and most of the groups were able to find the missing angle accurately. I was glad to see that they used different ratios (and some used all) to find the angle because it added to our discussion later. Had it not been for a previous discussion with my AT about how I wanted to approach this part of the question, I probably would have directly asked them to find the angle using all three trig ratios and wouldn’t have had this much diversity in their answers. Pro-tips are awesome 🙂

Again, as I walked around talking to students, I picked out my groups and was happy to see that some went as far as re-polishing their work to be able to explain their process better.


We shared solutions from three different groups that used different ratios to find the angle and then highlighted what this ‘ratio’ they found and looked up in the trig table actually represents. I asked them to incorporate the meaning of this number as they explained their answer. Looking back on the lesson, however, I would have liked to ask them if this ratio would be equivalent in a similar triangle.



I got some thumbs up at the end of the lesson for how they’re feeling about trig ratios, but what really counted was how much they engaged with the lesson and in our discussion. The feedback I picked up on their learning is what distinguishes my good from not so great lessons.

Overall, I was extremely satisfied with how this lesson went and glad I had the chance to experience the process of designing and running a lesson from a problem-based approach.

I don’t expect that one great lesson will provide me with an ‘ah-ha’ learning moment for all my future ones, but this will definitely serve as an important motivational point of reference for when I need it.

Flipping Bottles

Today, for our intermediate level math course at uOttawa, Cassandra Mclean and I took our first shot at designing/presenting an activity station. Ok, it’s not really our first shot, we kind of did it last year as well for another course, but I think building off from previous experiences, it definitely felt a bit more developed this time.

The activity itself targeted Grade 9 linear relations. We had our ideas down but then two days before our presentation we came across Dan Meyer‘s  #BottleFlipping lesson and Jon Orr‘s successful implementation of it via Twitter and decided to scrap our original plans.

We modelled our (15-20 minute) activity around Jon Orr’s lesson design.

Set-up the table with an instruction sheet, “contest” sheet, an iPad to watch a quick video, some devices for Desmos, and of course …the water bottles!

Each group started off with watching this quick video, then they logged onto Desmos for the first part of our activity. Here’s a sample from some of their responses:


Next, was the actual contest:


Some groups got a little bit more competitive than others and started to have conversations about slight differences in the bottle design 🙂

Our timing was a little off because most of the groups didn’t get around to the final step (which is kind of important) and we also had to change it into only 2-3 one-minute trials. But, overall, I think it went pretty good and we even got some nice feedback.

Full credit goes to Jon Orr and Dan Meyer.

Planning for a 15-20 minute activity station was definitely a nice experience, kind of allows you to regroup your thoughts around lesson planning and your strategy for picking out main ideas/activities. It was nice to participate in them as well, felt like a mini in-class PD session on lesson ideas .

My final thought –if only all my lessons went this smooth 🙂  

Technology in the Classroom: Defining its Purpose

Technology use in the classroom largely depends on how teachers choose to incorporate it, but that’s not all.  One of the biggest takeaways from my first two weeks of practicum was reflecting on the purpose behind integrating technology in the classroom. On thought, I came to ask myself, why is it possible that one could walk into two different classrooms using the same tech tools, but see different results in terms of student engagement?

Having the fortunate opportunity to be able to observe how an experienced teacher capitalizes on their benefits allowed me to highlight the difference and dig deeper into thought about what makes their success vary from one classroom to the other.

From my personal observation and reflection, the key factor that stood out for me was the reason behind choosing to integrate technology –the one beyond learning that “special” skill. It’s first and foremost about taking the time to define the purpose I want it to serve in my classroom. Asking myself as a designer/facilitator the most important question when lesson planning—why? Why is this important for my students? Is it just about being tech-savvy, or can I further define its purpose and allow it to help me create a more meaningful and enjoyable learning experience? If my goal is to improve student engagement in the learning process, then how will they be able to engage in conversations with their peers, showcase their thoughts, share different strategies, and collectively engage in group discussions as they use this technology?

Walking into my placement two weeks ago, I was already curious to learn how technology can be integrated into a Grade 10 Math class. From previous experiences, I still had the idea in my head that when it came to technology vs. group-based activities, student engagement wouldn’t be as high.

Having that expectation and the imbedded image of students sitting on separate devices totally disengaged from the environment around them (only talking to their peers or teacher when they have a question or answer) definitely paved the way for one of my favourite experiences so far. For the first time I had the chance to experience what it truly felt like to be a facilitator/designer in the classroom. Albeit I was just an observer and active participant, but having that opportunity to experience the deep meaning of the former mentioned terms was truly inspiring.

For the entire 75-minutes students were leaders of their own learning. From the design of different activities that triggered curiosity, to the use of Visibly Random Groups (VRGs) and Vertical Non-Permanent Surfaces (VNPS), and the idea of showcasing and extending the learning with Pear Deck and Desmos, students were engaged and stimulated throughout the entire process. What struck me the most was that at no point was the technology isolated from the hands-on activities students were working on. They both served the purpose of complimenting each other. More specifically, the tools selected:

  • Created a safe space for students to communicate their individual thoughts, collectively;
  • Provided an extension to the hands-on learning with the added feature of personal/group reflection;
  • Provided students with a platform to analyze and present their own learning;
  • Offered room for feedback vs. direct teaching
    • Instead of highlighting key ideas by writing them on the board or reading them off a slide developed by the teacher, key concepts were conveyed through feedback on student work/thoughts projected through Pear Deck or displayed across the room (VNPS). The content of the “teaching” material was derived directly from student work.

Overall, more than just a tool to help students learn a new skill, technology served a much bigger purpose of creating an interactive community of thoughts.

It’s The Thought That Counts -Literally!

Imagine handing out a math test to your students with the following instructions:

Communicate in writing or in drawing:

  1. The process you will use to find the answer;
  2. Reason(s) to support your choice.

Please note: marks will be deducted if you write down the actual answer 🙂

This is the idea I got while scribing for a student during a standardized math test. Listening to him think aloud and watching him contemplate on ideas of how to go about in solving the questions was truly an exciting experience—one that would interest me, as a teacher, so much more than looking at the final answer.

The feeling of excitement was almost the same as that of reading a good book where you’re eager to flip through the pages to find out what happens next, but at the same time don’t want to skip over or miss any of the important details.

manipulativesI could tell that the student was enjoying the process of entertaining his thoughts and deciphering the puzzle (question) as well. Watching him jump from one thought to the other, as he experimented with his manipulatives, was like watching a detective trying to solve a crime scene. But suddenly the climax fell to an end. It was interrupted by his own voice as he shouted “The answer is …”

It fell to an end because instead of taking the time to follow through on his own thoughts that were heading in the right direction, he rushed himself to a different path as he was more interested in the final answer and whether it was right or wrong.

But it’s the journey in piecing together the puzzle that makes looking at the final piece all the more enjoyable—not just for a teacher interested in understanding the students’ thought process, but for the student as well.

By nature, the brain likes to play detective. It becomes bored when things get too predictable. Nora Volkow, M.D., Director of the National Institute on Drug Abuse (NIDA) says that:

“Neurons really exist to process information. That’s what neurons do. If you want to anthropomorphize neurons, you can say that they are happiest when they are processing information.”

Figuring out the solution or reaching a goal in general is definitely rewarding, but from my perspective, not as rewarding as the journey, memories, and adventures created during that process. After all, your brain will always be left asking, “what’s next?”

Focusing on the thought-process and understanding how students think was not only an enjoyable experience overall, but I believe it can actually help me become a better teacher by allowing me to pinpoint where the focus needs to be. I definitely  look forward to implementing this experience in my future classroom.


Learning Connections

Research has proven over and over again that the brain works best when many different areas are simultaneously activated together. Without integrating subjects and allowing students to engage in the process of learning, we as teachers are simply transmitting information, and focusing only on developing few areas of the brain —mainly language processing and memorization. When we allow students to learn through storytelling, for example, instead of the mere act of lecturing, they can form an emotional connection to the material, and thus will be able to understand and retain the information much better. Vittorio Gallese, one of the key members who discovered mirror-neurons explains that:

“when we read fiction or see a movie or a play and even when we see a painting, we map these fictional humans’ actions, emotions, and sensations onto our own brains’ visceral, motor, and sensory representations. That accounts for our emotional experience, which comes before our cognitive experience.”

This Is Your Brain on Culture, 2011.

During the past couple of years, I came across many readings which support integrating arts into learning, especially math. At first, this idea seemed a bit unconvincing, mainly because I never had the opportunity to experience this during my studies. When I signed up for Math Camp at uOttawa before school started, I had the impression that it would be just like any typical math class I previously experienced, where we would remain seated for the entire class and work on problems individually. Attending this camp was truly an eye-opening experience as it changed my perspective on teaching. Hardly ever did we remain seated or even use a pencil during that week. The focus was on expressing our thoughts, engaging in discussions, integrating arts into the inquiry process, and developing visual representations for our solutions.

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As one of the professors placed it,

“we learn so much more about how students think by asking them to explain how they reached their solution instead of looking at the final answer.”

Focusing on how students express their thought process allows them to self-asses their own understanding, helps them build self-confidence, develop a sense of ownership to the material, and learn how to articulate their thoughts in general.

Personally, my goal is to introduce students to the art of knowledge, and hopefully get them to fall in love with the process of learning and discovering. I strongly believe that by integrating the arts and the idea of story-telling into other subject areas, will help my students develop a sense of emotional connection to the learning and provide them (and myself) with a more engaging and stimulating learning/teaching experience.

RE: Talk About Assessment; Chapter 10

To be honest, this chapter left me a bit confused. Cooper addresses assessment reform on the school, district, and individual teacher level, but to me it sounded like standardizing assessment. I agree that teachers should be collaborating on issues that directly address specific needs or learning outcomes for students, and that they should be consistent in their grading and reporting practices; however, introducing a standardized assessment practice across the school or district doesn’t really resonate with me. How could the assessment strategies used in an above average class also be used in a high needs classroom?

In the case study presented in this chapter, Jennifer Adams, Superintendent of Curriculum, Ottawa-Carleton District School Board states that:

“Through this policy statement, the Ottawa-Carleton District School Board has encouraged teachers to find creative ways to allow students to demonstrate their understanding of curriculum expectations.”

To me, this is what assessment comes down to. In almost all the success stories I hear about including the most recent one at Rideau High School, success emerged mainly because teachers and leaders used their professional judgment and found creative ways to address issues and improve student achievement. Hardly ever do I hear of major success stories, specifically in urban schools, that are due to properly implementing “the system.” I really believe that teachers, especially those working with high needs students should be given more room and time to find new innovative ways that work best for their students without worrying about being held accountable for implementing standardized strategies. Yes, of course there should be a guiding practice and essential collaboration between teachers on all levels, but it shouldn’t be as time-consuming and demanding.

Juggling-quote-20141I felt the ideas Cooper presented in this chapter would be more practical in schools that already have some sort of base level to work from and are interested in “improving” rather than “establishing” student achievement. All the ideas Cooper presented seem logical and backed by strong research, but when it comes to working with students who are struggling with extreme learning disabilities and at times their learning targets are pretty much all behavioural based, I feel that these strategies are a step ahead of the current situational demand. Earlier in his text, Cooper mentioned that instructional design should not be based on a “one-size-fits-all” approach because students have different needs; similarly, I feel that no one-way assessment approach could possibly address the different needs of all the classrooms in one school, equally. In some classrooms teachers could afford to invest time working on the assessment reform demanded by their school or district, while in others, the bulk of time should be dedicated to establishing some sort of base level to even begin working from. As a teacher, I should be fully aware of all the current changes occurring in the assessment practice, but I also should be able to prioritize my time in accordance with the essential needs of my students.

Everybody’s Children

In Congo, the saying goes: “When a woman is carrying a baby, it’s her baby. But when the baby is born, it’s everybody’s child.” Unfortunately, this reality only exists on a minor scale in today’s society and most likely confined to one’s inner circle; which is why this documentary was really hard to watch. The refugee crisis is closer than ever and one can only begin to imagine the tales of struggle they are going through. Although I had some sort of previous understanding of the problems faced by refugees, this documentary definitely opened my eyes to details I never might have even considered about their daily struggles.


First, I must admit that I was one of those people who thought that Canada had some sort of system put in place to take care of refugees. I was shocked and confused as to why and how could there be no such thing. In the words of Ann Woolger-Bell, “Most Canadians presume that anyone that comes into Canada asking for asylum as a refugee, that there is a system in place where they are sheltered, welcomed and assisted. There is nothing. They are numbered among the homeless!”

They are numbered among the homeless?!! This is truly heartbreaking! I mean, even from a political and economic perspective, if we don’t guide them to finding the right path now, we will have to face the consequences that surface as a result after. It’s really just delaying the problem.

Second, it was pure sadness to watch how Joyce and Sallieu, two unaccompanied minor refugees, live such lonely lives at such a young age. Even when they have tears of joy, there is really no one around to celebrate with. Although I was happy to see that they are both resilient and able to maintain an optimistic look on life, I couldn’t help but think that the hug Joyce received at church from a stranger might have been the only affection she received all year. Similarly, throughout the documentary, all Sallieu hoped for was someone to talk to and share ideas or even just a meal with. No one from his school even knew that he lived alone or had any idea of the struggles he was facing. I believe that much more support could and should have been provided to these students. In an indirect way, teachers could have also provided support by incorporating essential skills/knowledge and truly authentic tasks tailored to address their current struggles directly into their lesson plans.

Finally, Joyce and Sallieu’s view on school is that it is some sort of bridge that just needs to be crossed in order to find success and happiness. They believe that one-day everything will be ok once they have a degree because it will enable them to find a job and thus be happy. This is great and all, but in my belief, not the right approach. First, “Canada” was that bridge to happiness, and now “school” has taken on that role. By the time they started applying to college they really had no idea what it is they wanted to do. Although Salliue was interested in becoming a firefighter, I felt he chose this path only because he wasn’t really exposed to other interests at school. Instead of providing them with a platform to discover their interests and abilities, school was just something they wanted to bypass in order to find happiness.