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?”

---

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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Tuesday, 27 November 2018

The Thinking Classroom - Note-Taking (Part 1)

So I came across this tweet seeking conversations about #VNPS




This question, of course, is an important conception to address.  More than just this question, though, it reminds me of many related questions that I have heard (and given suggestions to) many times before.  But there is so much nuance to unpack that I only briefly considered a twitter thread - before thinking that I should probably just put it in a blog instead.  I probably won't end up talking about all that I've thought about notes and the thinking classroom, but hey, it's a good start.

Here are some thoughts:

Coherence

I truly do not believe that it is helpful to only take #VNPS in isolation.  Not only do we risk oversimplifying the #thinkingclassroom and reducing it to a shiny 'fad' with extra pizzazz (see here where I argue the importance of going beyond labeling strategies), we also lose out on the complexities that we can learn from as teachers.  Worst of all, we lose out on coherence with the rest of our practice.  Of course, we build coherence in our own unique ways, as demonstrated in moments that @JudytaLarsen captured here in Ottawa last year.  This is also part of my focus in my Ignite last year.

Instead of looking at #VNPS in isolation, it's helpful to take a look at the #thinkingclassroom framework

In a way, it nullifies the rejection of #VNPS (even though I prefer to relate to #thinkingclassroom instead of just one element) on the grounds that kids need notes.  By considering the framework, it also shifts the conversation from "this won't work because kids need to write" to "how do we incorporate meaningful note-taking in our practice."

And, of course, a teacher can only work toward coherence in their teaching practice by thinking deeply about what they do, how they do, when they do, while constantly considering what they learn from their students moment-to-moment, day-to-day.

Permanence

As @sig225 attempt to illustrate:


If we're talking about permanence, then technically a picture can be considered as 'permanent.'  But permanence is not the focus.  Cal continued to share this:


What he is saying here - which was a bit sidetracked by the technology question after this - is extremely important.

It's not whether students write notes in the class.  It's not whether they take picture of their work on the boards.

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 (e.g. I often also ask students to write down questions in their "notes")

Studenting

Sure, if students are writing things down permanently - provided that they don't lose their notes or burn them when they get home - you can consider these physical forms 'retained.'  But if we are talking about knowledge retention, then this is a completely different ball game.

In a traditional classroom, teacher talks, writes, dances around in a monkey suit, and the student writes.  But these ways of 'studenting' (sample work from Liljedahl & Allan (2013) comes to mind, Allan also has more written on the topic) does not guarantee learning or retention.

Notes are only meaningful if the students have made them meaningful.

The same can actually be said about pictures, too, in a #thinkingclassroom.  After a while of working in the #thinkingclassroom environment, it is possible for students to want to 'game the system' by appearing like they are working by taking pictures. It is not enough to simply take pictures.  It gives a false sense of knowledge that can be held in our hands.  It is about what the teacher does with these pictures.  Is it a gallery walk? Is it an activity based off of annotating their work? What are the prompts? what directions do students take? What hints and prompts emerge?

So it's not about #VNPS or note-taking, as if the two are at odds with each other.  It's also not about 'let's just do both.'  It's about building meaning into what students do.
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Sunday, 2 September 2018

The Thinking Classroom - Beyond Labels

This is part of a series of reflections on The Thinking Classroom.

In this first part, I think out loud about what's in a label.

We have a toy abacus from Melissa and Doug that our 2-year-old (wow, she’s already 2) enjoys playing with.  Sometimes I point at a bead and ask her what colour it is, and she enthusiastically responds with “yellow!” or “red!”



She often tries to quiz me too, and asks “what colour this, daddy?”  To which, I sometimes respond with the wrong colour or something categorically wrong like “dog” or “rain.” 

“Noooo….  Silly daddy, it’s blue,” she’d say, and we’d laugh.

I remember the first few times I counted the beads.  I’d announce loudly as I moved the beads from left to right, one by one: “1, 2, 3…”  I did it carefully and watched her reactions. She watched attentively as I did it with the different colours.

I did this with each of the colours - something repetitive that would be boring with my high school kids.  I didn’t mind.  It was lovely and kind of soothing to count like this, and to watch her tilt her head slightly to the left.

Time sped quickly as I finished all ten beads of different colours.  All of the beads are now to the right side.  I moved my fingers to begin moving the beads from the right to left this time.  Immediately after I said “one…,” she yelled in protest

“Nooo…” She grabbed my right hand with her left and moved the beads back.

I was surprised, but I had a faint idea why she might have done this.  I wanted to find out for sure.

“Why?” I asked - something I try to do often with her, even though her responses don’t always make sense.  In this case, her response didn’t make sense.  But she explained enthusiastically, and made many gestures to the beads, while making several statements confidently (even though I couldn’t understand those statements).

“What is this one?” I tried a different strategy to understand what she meant and pointed at the bead closest to the left.

“Ten.”  She announced with a beaming smile on her face.

That was illuminating.  It seems that she had associated the numbers with the individual beads.  The words with the location of the bead, instead of the quantity.  I wondered briefly about how to best approach this.  Then I continued onto trying different things.

Words have meaning because we give them meaning.

We have a small dishwasher that we bought a few years ago.  It works well for us, since we’re in a small apartment downtown, and there’s just the three of us.  Instead of establishing regular chores, my partner and I just do the dishes when we see them (or at least that’s the plan).

Early on in our dishwasher-using life, we decided to use a little magnet to indicate when the dishes were clean or dirty.  Magnet on: dirty.  Magnet off: clean. 

This worked out well.  We could put dishes in whenever and be able to do them all together when the little dishwasher is full.  Magnet on; magnet off.  Dirty; clean.

Inexplicably, I tend to treat the little dishwasher somewhat like a dish rack.  I’d still wash most dishes by hand, and then put them in the dishwasher.  My partner, on the other hand, treats it more like a dishwasher.  She’d rinse the dishes quickly and simply place them in.  On; off; dirty; clean.  There’s no reason to designate who’s running the dishwasher or not.  Or is there?

I was playing with Sophie when my partner loudly exclaimed.  She was surprised by the dishes being dirty when she started to unload them.  “Did you run the dishwasher?” She came and asked.  “No, did you?” I quickly responded, since I recognized that I’ve been terrible with dishes lately. 

“Why isn’t the magnet on?”

I tried to think.  I probably took it off by accident.  Or maybe I put in the first few plates and forgot to put the magnet back on.  The reason was lost on me.  The magnet was a symbol, a label, for our dirty dishes.  Like the bat signal or an ‘SOS’ spelled out with rocks on a lost beach – a cry for help.  ‘On’ was dirty and ‘off’ was clean.

But labels have meaning because we give them meaning.

The thinking classroom has been sweeping across the mathematics education community for a while now.  Certainly through #MTBoS as well.  Alex Overwijk, I, and a few others from Ottawa encountered these ideas from Peter Liljedahl many years ago and began to incorporated these ideas immediately.  The elements from the thinking classroom framework has been powerful and transformative for us, and continues to evolve our practices. 

A year or two after, the ideas blew up in various places through different forms (twitter, for example).  For many, the ideas about the thinking classroom came second hand.  Either through Al’s sessions at TMC that one year, or elsewhere. 

For some, it was met with enthusiasm: another strategy to try; more stories of success to replicate; can’t be hard to start.  For some, it was met with skepticism: another silver bullet; more so-called experts ramming ideas down our throats; can't work for my students.

I find both enthusiasm and skepticism important, but the attitudes can be troubling.

Enthusiasm is good, but it requires that we put ourselves in it.  It’s good to take risks and try different things in our practice.  It’s not good to think of ourselves as soulless puppets that channel, for example, 'the thinking classroom'.  In passing, I’ve often heard similar sentiments:  “Oh, I’m doing a Dan Meyer today” or “That was a Marian Small lesson.”  You cannot be anyone but yourself.  I attempted to illustrate this with a spoken word piece during my ignite.  You cannot, and should not, be anyone but yourself.  I mean, what exactly does it mean to be doing a 'Dan' or 'Marian' anyway?  Detailed answers would probably differ from person to person anyway.  Let's not reduce their work to broad strokes.

Skepticism is good, but it requires investigation and exploration.  It’s good to reflect on the things we try and to think critically about the why, how, what, when, and where’s.  It’s not good to reduce other people’s stories only to ideas we recognize.  “Spiralling,” for some, has been reduced to simply doing random topics throughout the year.  The “thinking classroom,” at least for some, has been reduced to students using whiteboards.  This is a shame.  Since if that’s what you believe it to be – then that’s what it will be.  Personally, I've found it helpful to try to draw connections no matter what ideas I encounter.  Whether something is useful to my practice - is up to me.

Similar to what I wrote before about the damages caused by the metaphor of the pendulum, I shout this from this virtual rooftop:

Word have meaning because we give them meaning.  

 Labels have meaning because we give them meaning.

The thinking classroom, as Peter has pointed out every time he facilitates a workshop, is brimming with complexity and diversity.  It is a structure, from which we build ourselves, pose our questions, and involve (and evolve) our experiences and expertise.

Yes, students use VNPS in VRG every day, but that’s only the beginning.  These are designed to shatter institutional and (seemingly) non-negotiated norms in our classrooms.  But there so much more than this.  I mean, what happens when you smash these norms?  Of course we would need to rebuild!

Our teacher moves still matter.  Our teacher moves are still what makes the difference as we build our worlds with our students.

Our world has meaning because we give it meaning.

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Thursday, 15 February 2018

How do you bungee? A PD Session from last year.

What are labels good for, anyway?

This has been a wondering of mine for quite a while.

I think I had a vague feeling that labels are unhelpful to our teaching practices when I initially began blogging, but I couldn't really put it into words.

And so, even in the very beginning of my blogging life (which has been greatly overshadowed by my being-a-dad life now), I chose to mostly share 'act 1's.  Like this compilation of act 1's in 2012.  I also specifically explored the differences between those two act 1's that I put in the same post.

A year later, I had a better feeling about this vague idea of mine, and so I argued that, "maybe we shouldn't share act 2's" when we blog.  I went on to write:
It is important for teachers to develop their own approaches and mold ideas into ones that will work for them.  In other words, just like how we want our students to construct their thoughts and understanding - why not do the same with teachers?  Makes sense to be constructivist through and through.  Emphasizing act 1 and act 3 may be the best way to approach this.

I've heard people say that "the three act idea doesn't work!" "Students get tired of always seeing these pictures and videos!" "I can't do this every day!"... so on and so forth
In my mind, there is an important relationship between what I was attempting to argue back then, and a recent conversation about #thinkingclassroom from Nat and several others




Pierre is completely right in that it is important to have conversations about more than just labels.  We need to really explore what it is that we're doing, instead of stopping short at labeling what we do - devoid of ourselves and our students.

Not to mention that labels are easy to dismiss.  "Oh, that 3-act stuff doesn't work." "Oh, I tried that #thinkingclassroom thing, it doesn't work for my kids"

But how do we have these conversations?

I attended Canadian Mathematics Education Study Group in 2016, and I was blown away by the format of one particular session.  It's a 'working group' that tackles a problem (in math education) together.  Participants in the same group talk and exchange ideas, and facilitators draw out ideas and help discussions.  We did this for several days.  The intention is not to direct, but to truly facilitate conversations - respecting and valuing the places that people are all at with respect to their lived experiences and expertise (participants include teachers, teacher educators, mathematicians, researchers).

It was refreshing.

These weren't presentations.  Or fake-facilitation where there's actually a secret agenda.

It seemed genuine, and I loved it.  It reminded me more of my own classroom activities, where students have/develop their own agency.

I then spent the next little while after attending CMESG, thinking about how I might be able to bring a similar kind of experience to our provincial wide math education conference that I typically attend - the OAME annual conference.

So I made up a rough idea of what the session might look like, and grabbed Al Overwijk and Bruce McLaurin to join me on this experimental session.

We then met on several occasions trying to figure out the details:
  • What is our purpose of this session?
  • What might our prompts be?
  • What's something we can focus on?
  • What can we debrief?
  • Should we debrief?
We toiled over each one of them (not to mention other questions that came out).

To start, we needed a common task that has been popular enough for people to talk about.  This serves as the background on which we might have important conversations about: our particular teacher moves during class, our reasons for deciding on moves, our interpretations of the happenings of our classes/culture/lessons.

We know we really want to get to the meat of the session - our decisions for doing or not doing certain things, and under what circumstances - within lessons.

For us, these have emerged as important considerations - regardless of what tasks or activities we're running.

We wanted to create a space where people not only feel safe enough to share, but brave enough to really dig into why they do what they do.  A tall order for a class that we get to keep for one semester - a monstrous order for a 1 hour session.  We had a lot of discussion about whether we should share some of our own experiences.  If so, how we might do that.  Since we want to disturb the power dynamic of presenter vs audience, we decided not to.  

But then a different difficulty emerged: what might we debrief or consolidate?

We settled on 3 slides

The first slide is simply the title of the session:





The second one was



and finally


With the "five practices" in the back of our minds, we worked on anticipating the possible responses that people might have.

As people began to register, we also took note as to who had joined our session (at least on paper).

This was definitely a group of impressive thinkers and educators.  There were also a few well-known (on twitter) educators joining the session (e.g. Jon, Mary, Dan, and many more).  Since there were so many great people, we were excited about the possibilities of in-depth discussions about our practice.  At the same time, we also worried a bit about power dynamics a bit more.  Could the presence (of e.g. Dan Meyer) intimidate others from sharing their ideas? Or perhaps alter what they would have said in his absence?

But we can't control how people react, and so we went on with our initial plan.

People walked in, we sorted them in random groups with some cards, and then sent them to some temporary whiteboards to work on the two prompts.

Throughout the whole time we watched and listened to people's ideas and responses.  There were certainly a few themes that we were hearing.  Bruce, Al, and I passed each other frequently and exchanged quick words about "this group's talking about this in this way" and "this group is talking about that in this way".

Toward the end of the session, I stuck to our original plan.  I didn't share ideas about what we might do with barbie bungee (e.g. I don't use barbie).  Bruce and Al also held their tongue as we discussed.

Instead, I summarized some of the ideas that groups have come up with.  And in the spirit of the 5 practices, did my best to connect some of their ideas together (similarities, differences).

We then highlighted that the whole point of this session was to - from start to finish - focus on the different ways that we approach things, do things, think about things, explore things, make decisions about things.

I made a reference to the work on Flow from Peter Liljedahl (based on Csikszentmihalyi's work)

This was to emphasize that - what we do during classroom time - matters.

It isn't simply about setting up a well-known activity and everything will work out great.  It isn't about starting what has been a successful "3-act task" and calling it a day.  It isn't about simply having vertical whiteboards, and all the kids will suddenly love school.

It was to emphasize that our actions are what matters.  (I subsequently made a similar point during the ignite in the same conference linked here and here)

I also shared a google doc to invite people to read more or to discuss more - but also referencing the fact that barbie bungee had been brought up on several occasions by others in the past - when they noticed that it seemed to work for some, and not others.  (coincidentally Dan Meyer also talked about this at his keynote later that day - and gave us a quick shout-out)

As a side-note, Al and I subsequently did an impromptu session about the #thinkingclassroom the very next session, because their presenter didn't show up.  Al mainly led the session and he did an excellent job as he always does.  When I tweeted out that we (mostly Al) did an impromptu session, it was confused with our bungee session.  It was a bit regrettable since we put so much thought into that session for it to be something 'impromptu.'

Some Reflections After the Session

Bruce, Al, and I spoke about this session afterward.  I enjoyed it, but was unsettled.  I wanted more.  I wanted to dig more at what surfaced.  I felt that we could go so much deeper into our teacher moves - and worried that we didn't go far enough.  Bruce and I chatted a bit about this - and he thought it went well.  He made a good point that it was difficult to build that environment that quickly to begin with.

But I suppose it's a consistent theme with me - where I always want more discussions, deeper debates, and further explorations.

This upcoming OAME we have a session that also aims to explore how we do what we do.  Bruce, who had retired since then, will unfortunately not been joining us (he is absolutely amazing, and certainly had a quiet exit like Tim Duncan).

Instead, I'm excited to have Nat Banting join us for this upcoming session, which will use another popular task/activity as a background: Visual Patterns.

There are some obvious ties to the amazing work that Fawn Nguyen has been doing, but we also wanted to open the doors to other ways of using and having visual patterns beyond the website.

In any case... stay tuned!

What are your thoughts?

As people that respect agency and learning, how do you facilitate adult learning in the brief conference sessions that we often have?  How do you navigate power dynamics as well as perceived 'right answers' or 'silver bullets'?


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Monday, 9 October 2017

Exploding Dots - for Fraction Division (operations)

So... I came across this tweet earlier tonight




While I have typically leaned on this resource from Marian Small, I was curious about using exploding dots for something like this.

If you are not aware, this week is #GlobalMathWeek, lead by Global Math Project's work with exploding dots.  I highly recommend signing up if you have no already.

The idea of exploding dots leverages the deep concept and structure of place value, and use it to basically explode your mind :)

In any case, let me get on to doing what I came here to write about.

Fraction division.  The dreaded topic of many.  Again, there are many great resources out there using area and number line model.  Here, I am mainly playing with the exploding dot ideas.  I dug around for existing work using exploding dots for fraction division, but I came up short.  So I thought I'd play around with it and see what comes out.

Before I begin, I'd like to highlight that I am just playing.  I am unsure if it completely follows the logic that has been previous established by James Tanton, although I certainly tried hard to preserve the concepts/structures there.

I'm going to begin with the idea of representing 3/2 with a box.  Note that for my purposes, I only need two boxes.


As you can see, when the box on the right fills up, it becomes a '1' which goes in the left.  So I can use a similar idea to 'un'explode the dots to make 3/2

Ok so now I move onto something like (3/2) times (2)

Well that's just taking the dots and then duplicating each one.  All at the same time, remaining in the (1/2) box, which means I have 6, as seen below



Which can, in turn, also turn into 3 groups of 2 that 'explodes' into the 1's.  In other words, equals 3.

But wait, I'm not doing multiplication here.  So let me move into playing with division.

So (3/2) divide by (2) is what I want to do next:



So here is where it gets tricky.  In order to 'divide' by two, I am splitting the boxes into two.  Each of my 'sub'boxes now need to be filled before the entire box is filled and I can explode it into the 1 box.

And so since I now require '4' dots to fully explode, I have (3/4)

Ok so that was alright.  Let me do (3/2) divide by (3)

In the same vein, I split the existing box into 3 pieces


Each 'sub'boxes require 2 dots filled in order for the whole box to explode.

And so since I need 6 dots in total to explode, and I only have 3, I end up with 3/6.

Ok so now I need to add a new idea of what happens when I divide a fraction by a fraction....

(3/2) divide (1/2)

In this case, I split each dot into two.  But each half is now counted as a full one (since we're counting by halves).  So I can think of it like mitosis or the fact that I am counting differently now.  This is similar to, but slightly different than, how I multiplied by 2 earlier.



And so once I count by halves, I note that I now have 6 dots in that box, where every 2 can explode into the next one.  And so, once again, I end up with 3.

So putting everything together...

If I have (3/2) divide by (3/2):


So here, I begin by both splitting the dots into 2 (doing the 1/2 aspect), as well as splitting the boxes into 3 (doing the divide by 3 aspect), which then leaves me with 6 dots but in 3 'sub'boxes.  Since now I have each box being full, it fulfills my requirement of the entire box being full, and so I get 1.




As I mentioned in the beginning... this isn't really polished, and as I mentioned here, it isn't as intuitive as I would like it...




And so this is more of an open invitation for others to think about this with me!

Let me know your thoughts.

Let's have lots of fun exploding dots this week!






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Tuesday, 3 October 2017

Foundations for Culture (Part 2)

So building a positive classroom culture where ideas are respected, voices are heard, collaboration is valued, in-depth thinking is encouraged, then, is of utmost importance.  (perhaps more so, now that society's polarizing beliefs and, often, failures to communicate with + listen to each other are ever so prominent)  Speed and precision, while still may be of use somewhere, are no longer primary pillars that support learning and deep understanding of mathematics.
The above quote is from the part 1 blogpost where I shared some thoughts and strategies about how I work on a positive learning culture in the classroom.  The two main categories were "learning about students" and "Problems + Tasks"

In this post, I continue to share some thoughts and strategies.

Listening to students

No, like really listen to them.

Davis (1994, 1997) differentiated between 'listening to' and 'listening for' as well as establish the ideas of evaluative, interpretive, and hermeneutic listening.  At the risk of being brief and simplistic, I will offer some definitions of these three ways of listening:  Evaluative listening is where we look for specific things that the student is saying.  Seeking our own opportunities to say "correct" or to check off a box (a good reason to be careful with observation checklists).  Interpretive listening is when we attempt to make sense of what the student is saying, and consider the depth and complexity of their understanding.  Hermeneutic listening is when we listen and participate in the same mathematical exploration that students are involved in.

Of course, how we listen also has implications for assessment (which I see, at least in part, as practices and strategies for generating, interpreting, and responding to student thinking).  However, in this post I will focus on the implication for culture.

Vertical Non-Permanent Surfaces (VNPS), along with other thinking classroom elements, are valuable structures and strategies in the classroom for providing me with opportunities to listen to students' mathematical thinking, and to encourage students to listen to each other's ideas.  As students continue to interact with different peers each day (through visibly random grouping [VRG]), they earn valuable lived experiences solving problems with one another.  Over time, this often helps facilitate opportunities for the development of a positive classroom culture.

When I listen and participate with students at each board, I am also mindful of what I say and how I say it.

Because it matters.

It matters whether I stop their thinking with my prompts or suggestions.
It matters whether I choose words that paint mathematics or learning in a particular way (e.g. growth mindset).
It matters whether I follow up on their ideas and suggestions or use them as talking points.
It also matters who I pay attention to in the group when I enter into their world of problem solving.

Listening often also include what students are 'doing' with their gestures, diagrams, and more.  For this reason, I also provide manipulatives for students when it makes sense to use them for the activities.  As I watch what students are doing with the items, it can often be helpful for providing me with ideas as to how to better support their learning.

Value their thinking
Along with listening carefully and responding in a way that isn't dismissive, I am also talking here about valuing their mathematical thinking.  When students don't feel like their work is valued, it works counter to the positive culture that we are attempting to build throughout the semester.

**as I prepared this post, I actually had a conversation with many others over twitter about a particular aspect of what I wrote regarding defronting, student notes, and board work.  As a result, I rewrote this section to incorporate aspects of that conversation instead.

It began with Laura's tweet and my subsequent wondering.  I had my own thoughts on this, but I wanted to know how she dealt with this tension of power:




Throughout the conversation, Anna and Peter also jumped in and shared some of her thoughts.

The concern was about how we leverage technology in a way that does not counter the idea of 'defronting' the classroom.

It seemed that many suggestions aligned with some of my own practices.

Notably I summarize two thoughts

1. Letting students have a role in presenting and summarizing ideas










These were actually going to be my responses to a previous question from Joe Herbert:




I often encourage students to take images of their work from the board because the boards are erased from class to class.  But images are useless when it's swimming among million other pictures of my dog or my daughter.  Using google photos, the images I take of their work on the board is often incorporated into tasks for the next day.  These are synced automatically, which allows me to easily transfer and cut the images to a different document. Similar to Anna's work, I provide them with another activity involving these notes, often involving the mathematical processes that we have in Ontario.  These activities often involve several images being stitched together and then provided for students in the same way that Laura had put up.

2. Having the teacher building a narrative through what students have done




At the same time, if I need to personally debrief activities, I either make use of what they have already worked on their boards, or I make use the images and debrief the concepts that way.

So in general, with the images I take of their work, I may structure another activity that allows them to continue thinking about the problem, engage them in reflecting on what they have learned (as well as the math processes), or utilize their work in a debrief of the concepts.

With respect to tech, perhaps this contribution from Michael was the most telling




Thoughts? Ideas?  How do you build culture?

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