Saturday, 27 July 2019

Turning Pages

This is a submission to the first week of The Virtual Conference on Humanizing Mathematics, hosted by the amazing Hema Khodai and Sam Shah.

The prompts of this The Virtual Conference on Humanizing Mathematics were:


  • How do you highlight that the doing of mathematics is a human endeavor?
  • How do you express your identity as a doer of mathematics to, and share your “why” for doing mathematics with, kids?


And there are also mini-prompts, too! “Please share a time when doing mathematics was a dehumanizing experience for you.” Responding to that won’t be as coherent with the rest of this post, but I’ll try to weave it in.
These are all great prompts worth thinking about, and I hope to unpack some thoughts over a few posts by participating in this conference (which reminds me of the #MTBoS initiatives I participated in back in 2012).

My responses in this post is more about the first big prompt.

“How do you highlight the doing of mathematics is a human endeavor?”

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It’s about halfway through the summer break for me, and I am still dreaming about school. I’ve written before about a dream I’ve reflected on (2016 post), how I often have these irrational emotions when schools out (2013 post), and how a lot of my efforts just feel like it’s not good enough (2013 post).

Maybe it’s a bit different this year, because I decided to transfer schools. There were a lot of reasons and considerations that led to my move that I won’t get into. It ended up being a very sudden decision, with a really fast turnaround - like deciding to apply, applying, interviewing, getting to job, all happened within the span of 3 week days. Thank you to all my friends that helped me think through a lot of things. You know who you are.

While I don’t generally control my dreams, or mind dreaming about school, I felt that maybe my subconscious is telling me that I really need to write or do something in order to let go. I am always terrible at goodbyes, and definitely made a mess of the final staff meeting and subsequent get-togethers where there were opportunities to say something. I guess I was just really at a loss for words and couldn’t think of anything to say. I loved the kids at the school that I was at, and maybe what I really needed was a way to start the grieving process.

It’s a bit late, but this post is inspired by my good friend’s thread here:



I tend to focus on the negatives, and so starting with some wins might be a way to breathe better.

So I’m going to indirectly be touching on “How do you highlight the doing of mathematics is a human endeavor?” by reflecting on some survey responses that my students wrote.



This was from a student who made some great gains with how she saw herself in mathematics. It wasn’t significant to the point that she would gush about doing mathematics or anything, but the positive culture (I wrote a bit about foundations of culture here and here) that I attempted to build with #ThinkingClassroom seemed to have helped her find herself a bit more in the walls of the school.

But aspects of the thinking classroom, of course, were not embraced by all students in the beginning.



As this student pointed out, he hated working on the boards. (of course, #thinkingclassroom is more than just #VNPS or #VRG, which I’ve written a bit here, and will likely write more later) Well, what he meant was actually that he disliked the collaborative aspect of working on the boards. Social hierarchies as well as insecurities and traumas about mathematics surface really quickly in a discursive environment like ones we try to establish in the thinking classroom. In PD sessions where I talk about #ThinkingClassroom, I often reference Peter’s motivation of wanting to break institutional norms, and expand on the metaphor a bit: it’s like breaking open the dry wall or floorboards of old broken systems - when we break open the surface by drastically changing the collaborative experience, all of the crap comes out and we are forced to deal with it.

Some students also saw it as explicit opportunities to develop skills beside mathematics, as this student points out:



Although I’d probably argue that these transferable skills (in Ontario we summarize them as ‘mathematical processes’, and include problem solving, communicating, reasoning and proving, connecting...etc. Sample document here)

I was pleasantly surprised that this student saw these learning structures as related to equity and breaking up existing social groups:



I was also heartened by the statement about how the learning of mathematics was naturally picked up throughout the activities. This is the main reason why I spiral the way that I do (which I think requires more elaboration at a later date…), focusing on students uncovering curriculum through their own wonderings and subsequent explorations.

But I think none of these would have been possible had my central goal not been to humanize our mathematics classroom. The following were hardfought wins -- that I had mostly considered as losses until I read the surveys at the end.









Caring. It’s not something to be said, or a particular action to be done.

It’s a ‘doing’, a ‘being’ - present participles that demands genuine and sincere attention to my students.  It's an ongoing process that needs to be the core of every consideration, action, and reflection - not a neatly packaged movie with a climax and resolution.

Humanizing mathematics, in a way, is all about humanizing our considerations and interactions with students.  It's about seeing them as living breathing beings.  It's about building meaning into everything we do and don't do.

It’s also not easy. I struggle all the time with not being able to do more. When students don’t have positive experiences with their peers in a random group for that day, despite my many efforts to redirect, prompt, support, revoice, or amplify, I beat myself up over and over in my reflections. Of course, I often gear the next day activities so that student might find more positivity in the next day, but it still weighs heavily on me.

It was an eventful school year for me. I had some emotionally challenging classes. It wasn’t emotionally challenging because they were defiant, they weren’t - certain students seem to be defiant in other classes but were fine in mine. It wasn’t emotionally challenging because they hated mathematics, though they did - but I accept this as a common curse for mathematics teachers. It was emotionally challenging because I cared, and a large number of them let me in - despite their dislike for mathematics, for school, for life.

Also, I had a lot of responsibilities to manage, including a preemie newborn that made ‘sleep’ into something I can only dream about. That certainly upped the emotions bit.

But reading these surveys made me realize that there were triumphs.

There were a few students that I encouraged and helped to go after leadership conference opportunities. Black, 2slgbtq+, and indigenous students that didn’t see themselves as capable in the beginning of the year.

There was a student that had a hard time with life (and mathematics), and I ended up offering a space for them to come chat early in the morning. They didn’t always come. But the occasional arrivals seemed to help, and we even ended up doing some mathematics casually.

These ‘bigger events’ were the ones that come to mind, but I think the little things we do during class are what really matters. Who we select, sequence, connect, and how we do it. Whose voice we value in the classroom, and how we amplify their sayings and doings in ways that support their mathematical identity and agency.

I truly believe these are the real work of us as teachers.

I remain somber because I still don’t think what I did was enough. But in the spirit of following Idil’s example, I will count these as wins for now.

Also… hopefully by slamming down the final period at the end of this blogpost, I hope to turn to a new page.

New school. New colleagues. New students. New learnings. New possibilities.
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Wednesday, 6 February 2019

The Thinking Classroom - Note-Taking (Part 2)

So in my last post I wrote about note-taking, as a partial response to questions about #VNPS and notes.

Specifically, that our practices need to be coherent, and that we can "only work toward coherence in [our] teaching practice by thinking deeply about what [we] do, how [we] do, when [we] do, while constantly considering what [we] learn from [our] students moment-to-moment, day-to-day."

I also hoped to spark some reflection on the 'permanence' of notes.  I tried to emphasize that it isn't whether students need something permanent.  Rather, "It is about whether they have created personal meanings through the act of note-making, in a way that may consolidate, structure, and sequence their thinking, and perhaps also prompt further wonderings."

I also briefly wrote about studenting, before really trying to reiterate the idea that, as teachers, one of our tasks is to try to build meaning into what students do.

That was a quick re-cap of Part 1.

In this post, I'd like to share some ideas that might help build meaning into note-taking (or note-making).  I have to admit that I tried structuring this post into similar categories but didn't really have enough time to do it justice.  Instead, I'll just write about them in no particular order (although you might see that they are all related -- hence the 'coherence' point I attempted to stress in the previous post).

1. Using Pictures
In the class we often take pictures of our work.  Sometimes we're tricked into mistaking the storage of the images for a wealth of knowledge.  Certainly students might.  But these just get lost in a sea of selfies or fun memes -- collaged and compressed over time, drained of meaning.

One way that I help students add meaning to these, is by setting up explicit activities that ask them questions about the images.  It can be their group's work, or it can be another group's work.  The key is that students have had the experience during class of solving these problems, and now we ask them to make sense of those experiences, and to reflect on possibilities.

1a. Printed images with additional prompts.

I might give prompts that relate to general 'mathematical processes' (general overarching skills), like in these example from a few years ago:

I try to take specific pieces of work from their experience, and then build prompts that are structured under broader skills like representations, connections, and so on.

I might also include something like this:



Where I ask students to think about this very interesting mistake.  There are several intentions here.  First and foremost, I am attempting to place value in these mistakes, and framing them as important opportunities to learn from.  Second, I am being specific about the ways through which we might value these mistakes.  Not only as learning opportunities, but recognizing that they all stem from useful and interesting logic.

(of course I recognize that there are different ways of framing the word 'mistake' that can also be powerful, but I won't go there for now... I'll go with elaborating on this old document of mine)

These are done collaboratively, in visibly random groups, on #vnps.  When I go around to manage 'flow', it's a bit different than a problem that students solve, because the discussions are more meta.  They're about broad ideas, and may involve a large variety of examples.  Some kids will have lots of words written down, and others will have lots of discussions, and others will play with their thumbs.  This is often something that I work toward - in building student capacities for doing this.  At the same time, number 5 also allows me the opportunity to extend and change problems (or they can do it on their own) so then they can create infinitely many exercises (e.g. go around and add x's or constants, or different constrains and restrictions to number 5).

1b. Using Desmos Activity Builder to encourage reflection

Another way I might do this, is to throw it on Desmos.

This strategy was inspired by Thach-Thao Phan @MathPhan

She made a desmos activity specifically for homework in a thinking classroom.  She only tweeted about it once here



I liked the idea, and so I subsequently worked on my own alterations (we have to make them our own, right?) here:


And here's an example of what one question might look like from the kids.




.You might recognize the problem from Peter Liljedahl's website as the Pirate and the Diamond problem (which I usually change the story and use a different context for)

But the idea is that the template can be used for any board work (as long as there's enough redundancy around the room), as long as you are specific about what you put in.

In fact, I prepared the document and edited the desmos activity during class *while* managing flow in the room.  So I promise it can be super quick.

2. Parallel Problems

When we are managing flow in the thinking classroom, it's common strategy to think about the kinds of prompts we might give.  Whether problems we provide students (when they are done) are extensions, parallel, or gear-shift problems.  I will write more about my thoughts about these two ideas later, because I do believe there's quite a lot of nuance in that.  For now, I will  briefly describe extensions as pushing deeper into the concepts or broader around the concepts, parallel as similar problems that provide an additional opportunity for students, and gear-shift problems are ones where we switch the focus slightly to a different detail.  These are done on the fly (but rely on experience, knowledge, and imagination on the part of the teacher), and are focused on responding to our students in the fleeting moments of the classroom.

But beside during the problem, I've also used parallel problems to help me set up for 'notes.'

Sometimes during the day after an activity (which may take a day or several days), I begin the class with a parallel problem.  This might be a more succinct version of the problems that students have experienced on the previous day.  Then as they work on the problem, this time I am managing the class in a different way.  I am now focusing on aspects like precision, mathematical vocabulary, and really pushing for detailed understanding.  I am also asking students to verbalize, and, yes, write down these details.  So in other words, they're using this short problem to create notes with.  

Once students have all solved the problem, which should be quicker because it's not the first time, I gather the kids to remind them about our learnings from the problem, and then I send them off to annotate the problem/strategy/solutions with definitions, other examples, and other related concepts.  Putting their own names to the concepts and ideas.

3. Summary

Sometimes I also explicitly ask kids to work together on creating mindful notes. Just kidding, I don't use the word 'mindful' with them - I know that Peter's framework has 'mindful notes', but with kids, I actually tend to go with 'meaningful' notes or 'purposeful' notes.  I find that it's an adjective that I can play with, and one that they can grab onto a lot easier.

These can come in a lot of different forms.  Kids can work on these on their own, like in Alex Overwijk's @AlexOverwijk example here:


Here are some of my kids examples as well:






These are without explicit prompts of what they should be writing down.  And the last two pictures are electronic files (with one of them being a google doc that contained a picture of her work as well).

There are others that have done this, too, like this recent tweet by Lam Nguyen @NguyenMath




4. Structured Sheets

You may also prepare something in advance for students to collaborate on, think about, and then write in for themselves.  Laura Wheeler @wheeler_laura has these course packs, for example, that she provides her students.  I think of them as shelves that you create that students to put stuff in.


As for me, I tend to like to shape the shelves with a bit more questions.  See a sample file in this PDF, and the images below:


Usually I also have students work on these collaborately on the boards, as well.  That way I can listen to their conversations about these concepts.

5. Quizzes, Tests, and Reviews

Do your kids throw out quizzes and tests into the trash after you give them back?  I'm not saying it doesn't happen anymore, but it's certainly a lot less than before.  I believe that it's also important to build meaning into those experiences as well (as I tried to write about here).

Along with activities, these can also be amazing opportunities for collaboration and note-making/sense-making.

It helps a lot that I spiral through my courses.  Which means that we revisit the main concepts of the course several times throughout the year.  (To me, it actually means a lot more than just hitting the same topics multiple times...  it also has to do with responding to students and recognizing paths.  but I think that's another post for another time).

This can either be built INTO the quiz (e.g. questions in the quiz containing things like encouraging an elaboration on 'mistakes', the 'why' questions...etc), or it can be structured alongside the quiz, or after they have been returned (as an activity).

Group tests can help this along as well. (oh man, another topic I should write about at some point, too).

~~

How do we help students make meaning out of notes? What structures do we build in order for thinking to flourish?  Meanings are personal.  So ultimately, that's something each one of us would have to answer for ourselves, within our own circumstances, leveraging our own experiences, and acting on our own goals.  My examples are things I've tried and that has, at different points in time, worked for me with some kids, and bombed with others.  And so, how might you help students make meaning out of notes?

I'd love to hear about it


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