Metacognition and Reflection

This is a post that I started pre-covid, but never finished.  I just came back to my blog, and saw it 'under drafts'.  Here it is, just for a record:  

Metacognition, the awareness of one’s own thinking processes, is something that some of my students lack. 

I saw it in my students from summer school.  There was a dearth of metacognition especially amongst my students who are weaker in mathematics.  

My fellow grade 10 academic colleague have tried to address this problem through a short series of assessment wrappers and assignments based on their most recent quiz/test.  

In the beginning of the year, we spent 3 days reviewing grade 9 mathematics and trying to look at the big picture of what was taught last year.  It is often easy when learning things for the first time to get lost looking at the tree when there's a whole forest to be discovered.  Looking at mathematics again a year later gives a chance to see the course as a whole. It also lets them reflect on what information they retained, and what was and wasn't so easy to recall.  We were able to discuss how the mind forgets, but can easily relearn what they learned well.  It is also worth reflecting on what concepts wasn't so easy to recall.  

We did a short half hour diagnostic of grade 9 material after the 3 days of learning.  As soon as a student finished, we completed something called an "assessment wrapper" - one of my favourite things I learned from my colleague 10 years my junior.  It comes in many forms (as you may google and find out), but the one I did was have students go to the side of the room where there were solutions and a red pen available and reflect.  

After a linear systems quiz, students were to make corrections and write a short comment for helpful feedback for themselves.  Then they need to choose one particular question they did wrong, and go into depth displaying some metacognition.  

Reflection is considered one of the core processes of being a mathematician.  I have to learn to emphasize it more.  And how do I emphasize it?  By evaluating what I value.  

Remote Learning vs In-Class Learning

The Covid crisis forced my teaching to change quickly. I was used to reading “data” in-class and adjusting to my students’ needs in real-time.

I miss using body language data to scaffold student learning to help facilitate their understanding. I enjoy reacting to this real -time data — giving wait time to their quizzical looks as they process a question, offering guiding questions in response to frustrated faces, and then celebrating their “aha” moments of understanding.

I cherish building a classroom community — talking about things other than mathematics and eventually looping this data back into creating a story or joke for our own unique classroom.

I appreciate being able to easily identify student disengagement — the ones who found an easier dopamine rush than a math problem in whipping out their phone to see if they got more likes on their last Instagram post.

I relish acting on this data — creating teachable moments — having students reflect on their behaviour and consider the opportunity cost of their actions. I miss it so much that I would recreate a lame version of it in the middle of my remote learning lesson.

I am used to reading such a rich set of analog data every day. Instead, I receive digital data through activities — inclusion activities, weekly reflections, and small surveys. I get to literally read how they feel, what they found effective, what they dislike, and how long they watched my videos.

Here is an example of a recent exit card survey where students were asked to ‘rank’ eight different aspects of the distance learning using 1 (least important) to 8 (most important).

Part 1 of the survey results asking students to rank the importance of aspects of distance learning.

Part 2 of the survey results asking students to rank the importance of aspects of distance learning.

This comes as no surprise — students find the video lessons to be the most important aspect, followed by their assignments, (AKA: summative google form assessments) and finally followed up by their homework.

This is the traditional way of teaching and learning mathematics — watch a lesson, practice the “home”work, and then get assessed on it. This is something I’ve been slowly getting away from the last few years — and resorted back to (for now) given the pandemic situation.

The students say the video is the most important aspect — but what do the numbers say?

Distance learning has produced 157 hours of watch time and 1400 views. In-class learning would produce 540 hours of watch time and 1624 views . (Calculations assumed the same 9 days, 30 minute lessons , and full attendance — 540 hours of watch time and 1624 views.)

It’s important to put some sort of context with these numbers so I decided to compare them with my ‘average’ in-class numbers:

• Watch time: Distance Learning (157 hrs) vs In-Class Learning (540 hrs)
• Views: Distance Learning (1400 views) vs In-Class Views (1624 views)

The disparity in watch time could be caused by the fact that students pick and choose what they want to watch in a video. The average view duration is shorter than the actual duration of the video lesson. (see below) You will see that students are watching as little as 30% up to 60% of a lesson. Students weren’t actually watching much of the videos even though they rank it as the most important aspect of the distance learning.

The average view duration of top viewed videos and the percentage of the video that’s been watched.

Another interesting tidbit in the above photo is the number of views. I only have 58 students, yet my views of these videos were well over 58. It supported multiple anecdotal statements about distance learning math:

This could explain why the average view time was shortened — were students going back to re-watch certain aspects of the video? Apparently — yes. This observation is confirmed on a larger sample size with Khan Academy.

To further probe how my students were faring with distance learning, I asked the students to rank how well they were learning math before the quarantine vs afterwards. Be aware that below is a sample size that only consists of the 38/58 engaged learners and is asked 6 weeks into distance learning.

Graphically, there is a lot more spread in terms of how students are faring with distance learning.

Taking a closer look — students on average rated in class learning one rating higher than distance learning. In other words, if a student rated a “6” for in-class learning of mathematics, they will on average reduce it to a “5” for distance learning.

One student went from a 10 rating (in class)down to a 3 (remote learning)

One student went from a 3 rating (in class) up to a 9 (remote learning)

Moving forward, there is much to learn for myself. In the short term, we may be continuing this distance learning for summer school, and I have to continue to find ways of helping students learn.

In the long term, there is talk of an increased use of a blended learning model as the advantages of technology are becoming increasingly apparent. The key is to find out what aspects of in-class learning are vital and identifying areas where technology can augment learning.

Setting Students Up for Success

Andrew Shin

May 9 · 3 min read

Foregoing an examination in mathematics education is very different. Replacing it with a portfolio can be especially daunting. Care must be taken as we try new things to ensure student success as well as authentic learning.

Since students have never seen a math portfolio, we must carefully design timely checkpoints and build in opportunities allowing students to adjust to feedback.

We outlined what we wanted to see from our students in a portfolio:

· Demonstration of their knowledge

· Annotation of their work to ensure understanding

· Reflection through curation of their notes and homework

We realized that these fundamentals are similar to our daily homework collaboration activity where students are asked to display one piece of annotated independent work over a two-week period. Students are also asked to reflect on others’ work and provide constructive feedback.

Students could upload either notes or their solutions to homework questions on a common Google Slide. The annotations help students reflect on the symbolic nature of mathematics.

Caption: Our expectation was that students give constructive feedback according to the hamburger model

Annotations provided formative assessment to gauge whether students were accessing the learning. There is a fear that students are using Photomath app or the video solutions to copy. However, when we read students’ annotations, we could assess the validity of their learning.

This activity satisfied both 1) the asynchronous restrictions in place by our board and 2) the collaboration and communication portion of the global competencies in place by our school. By annotating their learning and posting their work — not only was information and knowledge coming from us, but students were also teaching other students how to complete homework and take effective notes.

Friends are helping each other during this tough time as given by survey results, but it’s important for students to see work outside of their immediate friend group as well.

However, the activity induced much anxiety for a great number of students. Students suffering from math anxiety (and/or pandemic anxiety) could not participate. Some students opted to complete the activity privately and uploaded work directly to me.

About 60% were willing to fully participate in the homework collaboration. This is the result of the survey of 38/58 students asking what made them feel reluctant to participate.

Although not all of my students publicly participated, they all saw great exemplars. Because the homework collaboration had students demonstrate, annotate, and reflect– it served as a great starting point for our portfolio.

For the next unit, we decided to establish a checkpoint for our students. We will be assessing using a mini-portfolio where students would be curating two homework questions. We will give a mark, formal feedback, and allow them to use this feedback to improve. The final portfolio will allow them to include these exact same pieces with the improvements that they have added.

We released the mini-portfolio details and provided a video explaining our expectations, the marking scheme, exemplars from two separate grade 9 and 11 portfolios. There is a May 4th Easter egg at the end! =)

We hope that the homework collaboration and mini portfolio will help set students up for success as they complete their final portfolio in June.

All of this is a culmination of work from our department. I am proud of our department collaboration in creating this learning for our students. Thank you to my friends and colleagues who have given us important points to consider on twitter as well as on the google doc. If there are any further thoughts, let us know your questions and/or comments below. As this is very new to us, we are eager for ideas to help improve students’ online learning.

No Examinations, Now What?

So here we are, in the midst of a pandemic.  I know that I should be slowing down and taking a breath, but from my surveys - the majority of MY students are bored and are worried about next year.  I have phoned home and talked with the parents who are undergoing crisis. I have adjusted the course for those students.

For the rest of the students - we are trying to think of a final summative to replace the exam.  Here were some characteristics of the ideal assessment:

• fair 
• representative of students' ability 
• self reflective 
• help students make connections between big ideas of the course (especially quadratics)

Enter the portfolio.

I played with it last year because of @mrcedward's amazing guidance for me.  Here is the result of one student's portfolio last year, which only a few students decided to take on.

The students' work blew me away.  It showed that some of my students demonstrate knowledge better in this format than testing - which can create anxiety and muddles the accuracy of the assessment.

We decided (click here for a working draft) that the audience of the students' portfolio was their future teacher and their future selves.  They are to create a display portfolio that demonstrates their fulfillment of the expectations with a self reflection piece on what their strengths and weaknesses are.

I know the students technically could work with each other, but the annotation required should help reduce the copying that is done.

Now it's time to build in some early checkpoints for our students, as portfolios for mathematics might be a little different for our students.  I'll let you know how those go!

The Differences in Approaching 9 Academic and Applied Mathematics

From my various experiences teaching Grade 9 applied, I found an approach that worked and had the highest rate of success amongst my students. The method utilizes a two-pronged modern learning and mental health approach. By applying specific solutions to remove individual obstacles, students develop a more positive outlook as well as increased risk-taking towards mathematics.

The challenges often unique to Grade 9 applied students are one or a combination of the following:

• ·         a self-imposed mathematics learning limit
• ·         a fixed mindset
• ·         math anxiety
• ·         low metacognitive skills
• ·         learning skills need improvement

As a result, when exposed to learning new ideas or extremely interesting mathematics facts, the students often result in behaviour that ranges from extremely disruptive to complete withdrawal.  These challenging mindsets and lagging skillsets may undermine even the most engaging mathematics lessons. 

My approach to teaching Grade 9 Applied has been to devote time to breaking down negative mindsets and to increase assessment as learning as part of the evaluation set.  By focusing some attention to how their mindsets affect their learning, students can start to learn more effectively. 

We start every class using the community circle.  We discuss our experiences with learning;  both in and outside of mathematics classroom.  It is important to have students examine their successes in learning, regardless of subject, and to translate these successes to the math classroom.  We draw upon their successes in sports, music, and other learning experiences and connect it to learning mathematics. We watch Jo Boaler videos and learn about applying the growth mindset.  Most importantly, we also talk about anything that is on their mind for that day – and really listen.  Building these relationships of trust as well as removing any obstacles to learning prior to the lesson is imperative to a delivering each lesson successfully. 

By integrating aspects of Building Thinking Classrooms over the past three years, engagement of the Grade 9 applied students has increased.  Working with numerous online resources (desmos.com, 101qs.com, John Orr and Kyle Pearce’s work) has helped students grasp mathematical concepts while also improving their learning skills.  To conclude the learning cycle, we utilize assessment as learning tools such as assessment wrappers, reflective questions and goal setting to help develop the learning and metacognitive skills. 

In addition, spiraling through the Grade 9 applied course can also have dramatic effect in students’ learning.  By spiralling, students will have the opportunity to see concepts repeatedly in intervals to increase retention and confidence.  At the same time, it is vital to match one’s assessment policy with the growth mindset and Growing Success.  Students will not be penalized for “early not knowing” and instead be rewarded for their resilience and grit.  Combining a spiral curriculum with a flexible assessment plan will allow students to focus more on learning and less on grades.  Furthermore, it allows students to feel safe to make mistakes and to spend the time necessary to work on their learning and metacognitive skills; this in turn will will help solidify their mathematical learning and increase self-confidence. 

In addition to weekly quizzes, we have had students demonstrate their learning in the form of interviews, portfolios, assignments, whiteboard work, in-class observations, assessment wrappers, group performance tasks, journaling, and retests to get a broader evaluative understanding of a students’ comprehension of mathematics. By also incorporating Flipgrid videos, creative journaling, or projects – students can express their creativity and personality, which simultaneously makes them feel included in the classroom.   

To summarize, the following approaches to teaching 9 applied were suggested:  

• ·         community circle
• ·         Comprehensive Math Program
• ·         Building Thinking Classrooms
• ·         spiralling
• ·         varied assessment

Summer School 2019

Summer School 2019 - My Classroom Management Weakness

Grade 10 applied - 38 students - 33 failed the course previously - two already failed course twice.
Result - 2 failures, 36 passes - no drops.

Interestingly enough - I had students with numerous day school attendance issues.  Skipping over 15 classes was the norm for about 8 students.  Lates numbering in the 30s was the norm for a couple. However...

The structure of summer school and the shortness of the course (18 days) reduced the possibility for them to start skipping.  The threat of 2 absences and you're out of the course and 4 lates = to 1 absence was enough to keep most in line.  Minimal interventions needed - just a couple of reminders to the students themselves and they shaped up in no time.

Even more interesting - many of them showed up for exam review day even though they knew it wasn't necessary.

Relationships are key.  It is the only thing my TES teacher (SP7 - students at Risk at OISE) taught me - that's what he focused on.  Positive teacher relationships.

I had to fight off pneumonia.  It is difficult to do lecture based teaching with pneumonia and I don't recommend anyone try it.  As a result, some of my classroom management went to the wayside and the classroom was unruly.  It was embarrassing as I had one teacher ask me to keep my own class quiet.  I had another colleague friend come in multiple times and check in on me to see if I was ok given the ruckus that she could hear from her classroom.  It was a circus.

Looks like I was mixing up 'positive student teacher relationships' with letting the students do whatever they want.  Not positive at all.

There were some conflicts.  Two boys and homophobic slurs culminated in a slap to a face; I tried to do some brief conflict resolution but it failed as one week later, they were close to fighting again with lots of hurled insults between the two of them.  I noticed most students starting to back away; one little girl who was closest to them, looked up and then continued to work.  I found out later that she lived at Jane and Finch and this sort of altercation was considered 'light'.  Turns out her brother was taught by my wife.  I get to look at my wife with more newfound admiration.

I told my wife about the state of the classroom and the lack of classroom management.  She told me there is a fine line between classroom management and classroom community.  I still think my management needs work.  Hahaha - I'm 11 years in to the teaching profession...but I guess 38 students stuck in one room, with the same students, with the same subject for almost 6 hours a day will really test the limits of one's classroom management. Do I get a 'pass'? 

I had to get administration involved with this one after the latest altercation.  I decided to get them involved with some ring leaders of the misbehaviour in the class.  It certainly helped with the management of the classroom.  I'm not sure if I should have done it right away - as some of the 'roasting' I did back at them seemed to build some community.  Roasting your own students is a very dangerous exercise, as most of these students are here in summer school because of lack of self esteem and/or have math anxiety in class.  It also has the potential to turn my classroom into a routine bullying session and that isn't a safe place to learn - at all.

This was probably the most involved admin with my class that I have had in my history.  They were there to support and it was awesome.  I think I've always had a little fear with involving admin - as it may question my own effectiveness as a teacher.  But involving admin was awesome and a positive experience.

Dimensions and Exponent Laws

OLD DRAFT POST FROM 2017 that NEVER GOT POSTED

Some of you asked how I was able to spiral in algebra at so many times - and I completely forgot to share this with you.  I show a cartoon version of Carl Sagan's Flatlands to anecdotally show the difference between a point (constant), a length (x), a square (x^2), and a cube (x^3).  The video is thought provoking that we often get into discussions involving what the 4th dimension would look like. Having a discussion about ghosts, God, or whatever supernatural things...is what x^4 implies. For those that are super interested (usually my academic students) I usually point them to an xkcd and this discussion can really go on.