Thoughts inspired by #MTBOS2015

Seeing the #MTBOS2015 hashtag made me think.

The questions from Ilana (@tchmathculture) are definitely interesting in their own right

#MTBOS: what was your favorite blog post this year? Which of your own blog posts do you want to share? Tag with #MTBOS2015

— Ilana Horn (@tchmathculture) December 29, 2015

And there are certainly lots of great posts that I've read this year.  With respect to the "own blog post" thing, I found it fascinating that some people jumped to their most popular posts, and I wondered quickly about whether that holds any meaning for me.

This post isn't about answering the two questions.  At least not directly.

Instead, the questions from Ilana made me wonder about two other things:

1. What kind of blog posts am I most interested by?  Most impacted by?  What conversations have I wanted to engage with the most?  What does this say about how I interpret the world around me?  What does this mean for how I interact with ideas I encounter?

2. Has my central questions about my teaching practice changed?  How does this blog reflect a change?  Did it even do this?  What are the things I find myself writing about, and are they things that I've been wanting to write about?  How do I stay true to the conversations I wish to begin, as opposed to the conversation I find myself in - and does a difference say something about how I question and what I pursue?

I am unsure how I might approach these wonderings.  It would be one thing to simply answer Ilana's questions and start listing things, but I get a feeling that I would be betraying something that I will be aiming to do for the future of this blog: which is to write things that matter to me.  What a weird statement, right?  Does this mean that I have been written things that don't matter to me?  No, I don't think that's the case.  Instead, I think it's more like I have been regretting not entering into certain conversations while I could - and feeling like I missed the boat on a lot of them.

The questions I posed are huge.  I certainly won't be able to tackle them all here.  And so these are the things I want to tackle in this post, in a corresponding way to the two wonderings inspired by the conversation over at #MTBOS2015.

A. Conversations I wish I pursued
B. Reflections on conversations I continue to have with myself

A. Conversations I wish I pursued

At the risk of being brief, I am going share some thoughts on the following posts and conversations that I wished I engaged at the time.  Not because I "missed the boat" on being able to continue the conversation with them.  No, it's more that - even if I continue the conversation now, I would be having a different conversation than the one I would have had if I engaged earlier.  I am referring to the fact that things have necessarily changed since I last considered the conversation. Some examples include context, state of mind, culture (related to the ever changing #mtbos present and what people seem to be interested in pursuing).

1. A conversation about standard based grading, growth mindset, and the outlook on "fads" of education (mainly with @math8_teacher, @mpershan)

Of course, this is not to lump all three of these guys together.  They all certainly come from different places, hold different opinions, and think differently.  But in my mind, they all relate to a similar conversation.  Dylan (@math8_teacher) wrote about his skepticisms concerning standard based grading here, and I recall tweets (and maybe even a blog post I can't find now) from Michael (@mpershan) about not only SBG but also growth mindset. Then Dylan also wrote about growth mindset here and here (gotta love that search function).

I wanted to engage with them in a more meaningful way.  I think my central theme would be around something like this  where I believe that it is more important to focus on what we take away from an idea, rather than how popular an idea is.  I don't think the ideas (growth mindset, SBG) are all perfect, and definitely not conceptualized the same way from one to another.  My conversations with others lead me to believe that most people continue to refine what they think these ideas mean to them, as they continue what they do.  Or, of course, some meet it with disgust and rejection, or attempt to consume them in some sort of pre-packaged form (which inevitably result in failure due to a variety of reasons).

Is SBG useful? it depends - on how you define it, how you implement it, how you contextualize it, and what it means in your classroom to your students.

Is growth mindset useful? it depends - on how you define it, how you implement it, how you contextualize it, and what it means in your classroom to your students.

To me, I think I like to better understand the ideas by first meeting with welcoming arms.  I critique, reflect, and comment after it has become my own - because then I am critiquing, reflecting, and commenting on something I have more control over - my own practice.  This was the way that I tackled something like the Ninja board.  I wholeheartedly embraced in implementing it in my own way as I spun it to focus on culture in the classroom while building functions into the structure (and thus play it out more in a 3-act-esque fashion), then questioned myself (and others) about many aspect behind why, how, when...  And maybe that's a good approach to have - examining something closely once I have made it my own.  But maybe not.

Are there educational fads that rise and fall?  I think the answers depend on who you are and what the fad means to you.  I still believe it's unhelpful to actively promote the idea of a pendulum or that there exists fads that come and go.  But I suppose I can only comment on why it's unhelpful for me:

It is unhelpful for me because it limits my ability to be open to ideas.
It is unhelpful for me because I would worry about trying something "simply because it's a fad."

On the other hand, I do see many people trying the same thing at roughly the same time.  This is the case in our board for ideas like vertical non-permanent surfaces or visibly random groupings.  Or the case where Al (@AlexOverwijk) and Mary (@MaryBourassa)recently spread the seeds in many people's minds about spiralling via #MTBOS and #TMC.  The fact that many more people are trying these ideas are helpful to me.  I am comfortable with the fact that everyone is taking and incorporating them in different ways, or reflecting and criticizing different aspects of them.  These conversations are helpful to me.  They are helpful to me as I continue to consider and reflect my own knowings of what these ideas mean to me - and how it is applied to my classroom.  It is unhelpful to me to insist that they are fads.  Ultimately, I ask myself: does thinking that they are "just another fad" help me?  No.  And so I stay away from that mindset.

2. Struggles with certainty, doubt, and roles of research.

Earlier in the year I attended a talk on the fallacy of certainty (not the right title).  It was about how misleading certainty can be in mathematics, and how it is important to help our students recognize the important role that uncertainty plays in mathematics.  This was presented to us from a speaker who had engaged in research in environmental science before move to educational research.  I missed the chance to reflect on this talk at the time, but this related to several other conversations over twitter - big or small.  Sometimes people talk about better (or best) ways to teach, better (or best) ways to write blog posts, better (or best) ways to do professional development.

Certainty is sexy.  Certainty sells.

It is unfortunate it is that way.

Maybe it's because certainty is often coupled with confidence, and confidence is an helpful ingredient for sharing and spreading ideas.  It is unfortunate because it gives the false impression that ideas (e.g. perhaps shared at a workshop or talk) are perfect.  That the ideas being shared are absolute.  That examples shared are the best examples.  That recommendations are scripts to be followed.  Unfortunate, woefully untrue, and even potentially damaging to the idea itself in the long run.  I can try to make an argument using SBG or Growth mindset as examples, but I think that's a longer post than I had intended.

Certainty is also something that we seek.  @mathcurmudgeon mathcurmudgeon ranted about research here where he argued that "the whole point of research is to answer a question, either to prove it or disprove it."  I read his (self-proclaimed) rant as a further indication of that people (principals in his case) take away ideas in different ways.  But then he called for research to settle debates, to solve problems - or else leave him alone.  This is another conversation that I wish I had time to flesh out - but will likely not be able to.

3. Dissecting spiralling - celebrating and questioning how we implement and what it means to us. (mainly with @AlexOverwijk, @MaryBourassa, and @BDMcLaurin - also many others).

I wrote about this before back in 2012, but have never delved into how I unpack spiralling for my own class - for my own purposes.  The conversations I've often missed out on, though, are opportunities to further explore the differences for how we implement it differently.

This might be a strange turn (though not in my mind), but I see people doing this very differently across the board.  I think that's great.  At the same time I also see people attempting to craft formulaic approaches to "spiralling the curriculum."  This is problematic for me.  Especially since what enticed me the most about the concept of spiralling, was the degree of freedom that we would have as teachers.  We would be able to follow students' intuitions, questions, interests, wonderings.  I couldn't do this before with a unit-by-unit approach.  It seems to me that attempts to standardize implementation would go against the core reason why I supported it in the first place.  Besides that feeling of unease, though, when I have conversations with others about this, I am genuinely interested in how people have shaped the idea into their own practice.

I don't want to make people feel vulnerable or uncomfortable by jumping straight into the problems they have with spiralling - but that's actually what interests me more.  I recognize the importance of celebrating the things that work, but I really want to find out what and why things don't work.  This is in a similar way to how I treated my ninja board.  Now that I've done it for a while, I want to split it open and put it through a gruesome set of interrogations.  I want to know what it still means to me, what it should mean to me, and what it will mean to me.  Furthering a conversation about spiralling is the reason why I began with this post outlining a shell of what I did with one idea.

B. Reflections on conversations I continue to have with myself

Looking back at the blog, I wonder how much of it represented things I've been interested in, thinking about, and reflecting on.  With the exception of a few, I rarely write about lessons.  I had been unsure why.  Maybe it just doesn't interest me as much.  I recognize that lessons may get more traffic, but that's not why I write in the first place.  I went so far as suggesting that maybe we shouldn't share act 2s back in 2013.  I think this was primarily spurred on by seeing people starting to make very detailed step-by-step plans to follow (which coincided with some of my own real life conflicts with others).

At the beginning of this blog, I intended it to be a place where I explore the three themes that fascinate me - mathematics, mathematics education, and philosophy.  Since that time I have found myself moving away from writing about mathematics or philosophy (or even at all).  Mathematics, in particular, had not been very prominent in what I write about.  Not to say that these aren't embedded in the things I write about, but my posts have often been about mathematics education.  It totally makes sense to me.

But I think I want to continue these conversations with myself (blog) in a way that allows me to explore some mathematics and philosophy as well.  I wonder if these might naturally take the forms of more elementary topics over the next couple of years as we expect our first baby soon.

There are many things that I've put on the backburner, though.  At several points in the past year or so, I've had many drafts where I recorded partial thoughts on a few things.  I think I am going to delete all those drafts.

It isn't because the topics aren't interesting anymore.  But more that how I think about them are slightly different, and set on a different background.

There are many conversations (with myself or with others) that I've been wanting to pursue.  Some were listed earlier, many have sunken deep into the sands of time.  But I don't think I will be able to pursue them all.

I have to focus.

I've taken on way to many things this year - despite taking a one year leave from teaching.  I don't even know how that happened.  But I don't want to lose out on using this blog as an opportunity to pursue conversations.  And so I think I will take things one at a time with respect to this blog.  I will begin with continuing to unpack this post that I've set up for further reflection.

In any case.  I went on a bit of a tangent, but these were things I thought about when I saw Ilana's post.  I am grateful for this opportunity to explore the few things that I've left on hold over the past few years - even if it just skimmed over the many thoughts and reflections I've had.

Lego: Star Wars with Friends

A few things have changed/happened since I last wrote.  Things that are not pleasant (e.g. family death, supporting family member with cancer...etc.) as well as things that are exciting (wife pregnant with our first!).  One of the biggest change was that this year I decided to take a year of absence for a few reasons (mainly to focus on my thesis work and to pursue other projects).

But in any case, let's skim over all of that and move onward.

When I saw this post from @MrOrr_Geek earlier this year in February, I immediately gravitated toward it.  I engaged Jon immediately after I saw that, as well as also mentioning this video that related to Lego and equality (which he didn't credit me! *shakes fist at Jon*).  I then used it a week or so later in my own classroom.  Below is a description of how I unpacked it.  You will noticed that my approach is actually quite different than how he approached it.  Afterall, there are many ways to cook with such a great ingredient right?  And what's good about a lesson if it isn't mine - something I paint with my own colours?  If you haven't checked out his post yet, it's a good one!

This post will mainly be about describing a series of lesson that related to the Lego activity.  The example I'm drawing from is from last year (that's how behind I am with my writing).  I had been wanting to reflect on a few things like spiraling, social justice, and attitude toward "fads" or "trends" in education, but I think I'll put all of those on the back burner for now, and just start with unpacking what I did.

I will describe a period of about 9 days that related to this activity.  I've actually never written a post quite like this before - where it is more of an overview of the whole thing.  The intention is to set up a broad overview so that I could later talk about the details.  Over the course of these 9 days, we unpacked 6 out of the 11 expectation, while connecting to 2 more within the course (so that's 8 out of 11 if it even matters.  Personally I believe it's more important to build on knowledge naturally rather than forcing it, but I think I will talk about that in a different post where I reflect on spiraling).  You may have heard about spiralling from Alex or alternatively read some of my thoughts about it back in 2012 here or here.

Note that I also recently lost most of my pictures... so I don't have lots of beautiful class work to share... :(

Day 1

The students come into class and picks up a card similar to the ones described here.  There were a few changes to the ones in the link, but basically: I greet the students at the door and pass them a card, they then work together to talk about where they should sit, and then settle in their seats.

After they settled in their seats, I showed them an image of the Lego Star Wars  and an image of the Lego Friends on the board using the files that Jon shared, except I cut out the words he had at the top.

I asked them to think about what they noticed and what they wonder, and write them down to help them remember.  I then sent them to their vertical whiteboards to continue their conversations. I asked them to share their questions with their group members and to write their questions down on part of their board.  A lot of good conversations about their questions ensued.  They cared about the topic.  They were interested to say more.  This was an important step to let the questions brew and the tensions mount.  I walked around and asked them about their question.  I asked them what kind of information might they need in order to answer their questions - whether it would be quantitative or qualitative data.  For some groups, I heard some great conversations about gender expectations and social justice.  For others, they began to look at price and number of pieces and started to talk about that.

I then gave each group a package of images.  Yup you guessed it, I gave each group a few images of Lego Star wars and a few images of Lego Friends.  Some were images that Jon had in his post, others were simply farmed and printed from the Legos website here.

I said nothing.

I didn't need to say anything, they were already engaged in conversations, and these images only prompted them to have even more conversations.  More than this, most of them have now converged to a type of comparison required more quantitative data.  Nice and handy to have more data, then. I encouraged those that continued to chat about social issues.  These issues are important to me, and they know that these are important to me.  Not to say that they were only talking about these issues because they knew that I was interested in them - but more that they were comfortable in exploring and discussing those ideas further.  Eventually though, all the groups moved into talking about questions that are quite similar to: "Which theme's the better deal?"

This movement toward the topic wasn't explicitly forced, although I can imagine that it was created somewhat naturally through the questions and actions spread from one group to another while they were working on their vertical boards.  For example, when one group started to make a table to organize their ideas, another group would see that and shift their conversation a bit.

I took an opportunity to listen to the students and to draw out their thinking on several related aspects of the curriculum that had to do with thinking about data, posing problems, reasoning, problem solving, reflecting... so on and so forth.

At this point it was nearing the end of the class and so I asked them to take pictures of what they've worked on so far, and then I debriefed what we all worked on.  I drew attention to some of the questions they came up with, as well as shared a few of their thoughts and ideas about the questions.  This then led to a discussion about seeking to set up ways of answering their questions, which became an opportunity to talk about quantitative and qualitative data.  Throughout this time, I invited several students to share their ideas and thoughts.

I then left them with homework which had to do with 1) continue to practice algebra skills that related to the cards in the beginning, 2) think about questions posed by everyone in the room and 2) create a sketchnote that defined and described various aspects of data.

Day 2 - 4

Students are broken up into a new group, and were asked to share their sketchnotes with each other.  At this time I floated around helping students with any issues surrounding the algebraic work.  I then briefly talked about quantitative and qualitative research approaches, and then sent them to their whiteboards again.

The bulk of day 2 was fairly straight forward.  Students worked on their questions - which at this point, all groups decided to work on a question related to "which theme is the better deal" as mentioned before.  Throughout this time, I encouraged various groups to use multiple representations, as well as tease out the concept of slope/rate of change a bit for some groups.  At the end of the class I also showed the part 1 video that I shared with Jon (mentioned earlier) from Feminist Frequency.  At this time I also shared some of my thoughts on the matter, while drawing in some of the points that students pointed out the day before.  Students were told that they'd share their work with others, and that they would be given a chart paper the next day to make a "good version" of their work.

Day 3 was fairly straight forward as well.  Students continued to work at coming up with a thorough representation of their answers.  I basically floated around and cause problems.  Through the process of listening to students and prompting them to refine their representations, we tackled lots of ideas within the class: concept of slope and rate of change, scatter plots and lines of best fit, intersection of lines, algebraic equations for predictions... and so on.

Toward the end of this class, I drew their attention to the groups that had established a few different representations of rates of change, and had them make some notes about those.  I asked the groups to share why they thought it was useful to have these representations of rate of change and how it helped them tackle the question.

Day 4 was more of the same where I worked with different groups on making sense out of various aspects of what they were doing.  This was also the day that I provided them with chart paper for them to share their work with others.

Day 5

Each group of students were then jigsawed into a larger group that had a member from each of the groups that had a chart paper made.  Students were asked to note down some general look-fors for every group they came across.  This was simply a sheet with plenty of space, and these two questions:
1)What did this group work on that was similar to yours and how did they tackle it?
2)What might you add to your own work now that you've seen this group's work?

The focus was on active listening and reflection.  They were also encouraged to ask questions.

At the end of the sharing, each group was asked to return to their whiteboards.  They were now asked to summarize the mathematical concepts that related to providing an answer for the problem they tackled.  This then was done as a whole class discussion where we created a more refined list of what the concepts were.  More specifically, we tied them to the expectations of the course indicated by the curriculum document, which I had been sharing and unpacking with the kids since the beginning of the course.

Day 6 - 7

Students came in and were randomly sorted with cards once again.

They then see these

And then also this

We go through the standard (Well, I guess not necessarily standard), what question do you have, what do you notice/wonder...etc.

Students then worked on using the models they've built before.  Note that this was a new group that they were put in, so they each had a slightly different model.  Their first task was to come up with a model that they were all happy with.  The focus here was to focus on the reasoning and justifications using their existing models while accounting for errors.  

The next day they continued working and we debriefed their strategies and different representations.  And then I gave them images of patterns:

*These are images from the amazing site Visual Patterns set up by the wonderful Fawn Nguyen.  And these submissions were from @nik_d_maths.

A side note, looking back I probably would have brought the actual legos for the students to use if I had them... but this time they actually just used cube-a-links.  With an incoming baby, I imagine I will be collecting legos in the future and so would be able to bring them to class!

The students worked on looking for a big step number (like step 100).

After getting this through multiple representations, though, I asked them about accuracy and confidence for their answer here versus the two examples before of the Taj Mahal and Millenium Falcon.  We teased out why they thought they were more accurate with these patterns, and talked about the differences between the data represented here.  For some groups, I just focused on the aspects related to their representations and understanding.

Day 8 - 9

Students are randomly sorted again

Students were shown this brief video that I made

We go through a similar process for estimation, commitment, and suggestion of questions.

Students were then given a bunch of linking cubes as I asked them how many they can put together in 5 minutes.  They pursued different strategies as I went around and get a better idea about their thinking.  *again, note that in the future I would hope to use Legos instead*

The day after the students had an opportunity to finish up the work from the day before. I then asked them how long they thought it would take in order to build the Millenium Falcon?  I asked them what models they might consider and how they would put it together, as well as how confident they thought they would be about those times.  I showed them the beginning of this video:

And had them come up with their estimations based on their models.  I pressed them to consider as many factors as they can - and be as realistic and logical about their estimations.  Do we consider breaks in between?  Would it take the same amount of time in the beginning versus the end?  What about the sizes of the pieces, does that matter?  In conversations with the students I also asked them why the guy decided to built components at a time instead of just putting it all together and so on.

Then we watch the end of the video toward the end of the class after we discussed their strategies.

And that was the end of the 9 days.

Throughout the process the students had multiple chances to hit various aspects of the course that we wanted to look at.  There were opportunities for them to explore concepts such as rates of change - but the beauty is I didn't hammer those ideas over any individuals' heads if they had a hard time understanding the ideas initially.  I pushed them to push themselves as much as I could, and then I looked for other opportunities to bring them up on the following day.

So as you can kind of see, I went a different route than Jon did.  His lesson was great and I am certainly a fan of it.  But I would never cook it the same way he did.

After all, we aren't the same person!

I am a firm believer that there are many many excellent ideas out there.  The most important thing is not about finding the best activities - it's about making them my own.  Making them ones that would work for my classroom.

I hope to dig into that thought a bit more in the future, but for now I'll leave it here :)

Post-it Mario and Pikachu

There are many delicious ways to cook with chicken, right?

I've alluded to this perspective of mine a few times, and wrote about some of it more explicitly in this last post.

Believing in that every teacher brings (and should bring) their own touch and their own personality to any raw idea, is why I love open-ended "idea sites" like 101qs or visual patterns or wodb...etc.  But in any case, that's another topic to be revisited another time.

(edit: since this draft, there's another post that touches on a similar discussion except with a more positive tone from @mathymeg07)

You may or may not have seen this post from @MrOrr_geek, which was inspired by @mr_stadel's cabinet problem, and supported by @nathankraft1's Van Gogh art.

Those are all great posts if you haven't seen them yet.

The following is my description of what I did.

I had it tentatively planned in my mind that I was going to do this, so a few weeks before the actual beginning of this, I had a separate activity with the students where we were discussing different types of data.  One of the discussion was for them to vote for their favourite out of the following:

We collected their votes and then the groups had discussions on what that data looks like, what it is, and what they can do with it.

Mario and Pikachu happened to have the same number of votes.

In my head, this was suppose to be a precursor to post-it art that I would be making.  and so I thought - why not make both.

A few weeks later, I made them.

The Monday when students came back, they saw this on the back wall - it's the first thing they see:

Before the bell even starts, they were filled with questions.  Since they were already asking such great questions I started this video as quickly as I could (as soon as all the kids came in)

At this point I get them to write down their questions on the boards.  Here's an example:

As they create the questions, I ask each group to think more about whether an answer that would resolve their question was quantitative or qualitative -- what kind of information would they need... what kind of data would they collect...  and pushing them to think more about exactly what we're doing.  getting them to plan, think, and plan some more.

(edit: when I ask students to generate questions, a lot of people ask me "what do you do with questions that have nothing to do with math?  A full response would merit another post, but basically I recommend honouring all their questions genuinely.  They're interesting questions to them, and so I encourage them to think about them.  Typically the ones that require more thinking time and can be solved more systematically - are mathematical ones)

So then I grabbed their attention again and had them tell me what kinds of information they might need in order to answer their questions.  

Right.  It's their questions.  

With everything I do, I always felt that was key - not only for getting them invested, committed and give them ownership - but also because it's genuinely more interesting that way (to them).

I only had a handful of information, and we had to figure some out on the spot.

For example

They ask me how much they cost -- I slip them this like it's the greatest secret in the world

They ask me how much space each post-it takes up -- I throw them a few post-its to play with

They ask me about time...  well this is where I show them this

It's not nice.

It's not the nice video from @MrOrr_geek where the numbers accumulate in the video.  This is primarily because I didn't know how to do that quickly with Videopad (I should probably ask him at some point), but also because I prefer to make things messy.

So they had the video, they paused, slowed down... paused, played... to get some data.

Here's an example of what one group worked on as they kept going

So you get the idea.

Over the next few days we explore different aspects of their question.  Including breaking down the timelapse app and figuring out how many pictures were taken to make the video (since timelapse was set at a picture every 2 seconds)

They also came up with different rates of their groups: e.g. would it be faster with 2? how much faster?  with three?... etc.

So throughout the few days that we worked on this (I think a week?), on their own they covered measurement and geometry concepts, as well as linear relations.  this is all from questions that they've generated - which is always big for me.  Students used Desmos as well to build their models and make better predictions.

Generally I just floated around, listened to their understanding, prodded for more thinking and throwing disasters in the forms of "what if" in their way - about scaling, covering the whole school, having the whole class do it, what if we were to cover the hallway...etc.

so they did a lot more things with it

And then at the end they shared with each other what they did.  They were encouraged to ask questions to clarify what the groups did (How I do these routines merit another post...  it will come later).

edit: a few weeks later, this article came out and so we had some more fun with extending our knowledge to something like this

So at the very end, I showed them this

Oh yeah, and don't forget throughout all of this, there was also Pikachu!

Here were the three videos I made for Pikachu as well:

A1 normal

A1 with the clock being bigger - for timing purposes

And the final reveal of how long it took in total

So some reflections.

Looking back: What worked for me and what I would recommend (for my future self or anyone else):
1. It was there, they could see it, interact with it, test it, look at it, touch it, feel it.  That was huge.
2. Having materials for them to get more info was helpful (measuring tape, post-its, something to write with...etc)

Looking forward: What I would change
1. One image would probably be enough.  Having two at the same time pulled focus away - and the video was only playing on the computer so it took time away from some groups while they waited to collect their own data.  Perhaps it would be better to throw it on my website so they can use their phones to view video and collect data
2. If there's enough interest - have them plan to make one.  It happened that noone had a similar question like this, but thinking back, I probably could have planted that question somewhere.


What are your thoughts?  What would you do with it?  What would you do to make this chicken dish delicious?

Our unhelpful metaphors: The Ivory Pendulum

If there is anything that I've learned over the last five years, it is that the idea of "the right way to teach" is unhelpful.

What exactly are the "right things" and what exactly is the "right way" anyway?  If we recognize that teachers are different people with different strengths and challenges, then employing a one-for-all approach without opportunities for variation can be devastating.

I have always maintained the opinion that I can, and should, learn from everyone.  This learning, ultimately is through my own lenses and connecting my own experiences.  For me, I am seeking coherence between my experiences (conversations with other teachers, reading of research, conferences, twitter, blogs...etc) and my actions (what I do in classes...etc)

At the core of every teacher that I have learned from, is the desire to promote learning in children - to motivate them in mathematics - to help them find success.  This means I don't only learn from those who "are doing inquiry methods" or "are following the latest in education" or "are on twitter."  I learn from the ivory tower, the "traditional teachers," #MTBoS, the students, and myself.  For example, I don't only learn from the amazing Fawn whose classroom involve intense problem solving, and filled with excitement from students.  I also learn from conversations with teachers who object to "exploration" types of lessons, and who prefer direct instruction (I will come back to this later).

As such, I think there has been two well-known metaphors that damage our conversation about effective mathematics education - at least from the standpoint of every teacher attempting to learn more about their practice.  (See how I wrote their practice and not just the practice)  The first is the concept of the ivory tower, and the second is the idea of a pendulum.

Let me elaborate.

The Ivory Tower.

A place that symbolizes the detachment of "academia" from the "real world."  Similar to how "pure mathematics" is perceived as separate from "reality," the ivory tower embodies, at least partially, the distaste many educators has towards education researchers (or any researchers in general).  In these metaphors, the ivory tower houses academics who have no idea what it's like on the ground -- who have no idea what it's like to communicate with or understand students -- who have no idea how to implement their fancy theories.

But wait.  Why stop at just academics? If this tower symbolizes an elitist detachment from the authentic experiences of teaching and learning, then its inhabitants ultimately include far more members than just academics.  It would include most of the world who are misunderstanding the process of learning.  It would even include some teachers.  It would simply refer to anyone who has lost touch with the realities of education.

Alongside teaching, I am also doing some graduate work in the area of mathematics education.  Upon learning that I am doing this, I am usually met with one of two reactions (of course, not always).  Either they are impressed and think that I have answers to everything (noone does), or they peg me as one of those teachers who is only into the latest "edu-fads" - who cares little about the realities of the classroom.  Untrue on both accounts (even though I often wish that I could do more about these realities of the classroom).

Perhaps it is because I seek a specific kind of research, but in my experience I have met primarily researchers who care very much about the "realities of the classroom."  They actively engage with the community - not to dispense knowledge, but also to provide conversation and a sense of community among teachers.

But let's pretend that wasn't the case, and that all the researchers are elitists.  I still don't think the metaphor of this Ivory Tower is useful or helpful.  The most important thing we should learn from, are the ideas.  They can be terrible human beings, but we can choose to pretend they are not, and learn from the words.  Not to take them as some sort of holy scripture, but to actively understand and reflect upon them, and seeing how we might make use of them in our own contexts.  This is how we should be treating all ideas.  Not to implement them as-is.  For example, a lot of people love the idea of #3acts.  @NatBanting revealed how some teachers assume that we just "wing it," but Dan reveals what he believes are important teacher moves in the whole process.

Besides not immediately dismissing ideas, we also need to understand them, think about them carefully, and then mold them within our own contexts and environments in ways that work for us.

This brings me to a discussion of another unhelpful metaphor to our practice.

The Pendulum.



This certainly isn't how it is portrayed in the media, with all the mention of reform efforts as "just another swing of the pendulum."  Unfortunately this also isn't how it is viewed by some teachers that I have had conversations with.
But regardless of whether the realities match the actions of this massive bob on a massless rod hanging from a frictionless pivot, this kind of metaphor is unhelpful for us as teachers who seek growth.

A weight hanging from a fixedpoint that swings back and forth, is used to describe the "back-and-forth" nature in educational research.  This massive bob hurts us as educators in more ways than one. Oh boy, where do I even begin?

It implies that there is only two directions for which educational research is going, when in fact there are many perspectives of, for example, how people learn.  Also not to mention, from what I can tell, researchers in the areas of education have not disagreed on the general ideas of, for example, how people learn.  Instead, they are trying to explore the nuances and details of our conception of learning as (for example) a process that is constructive, self-regulated, situated, collaborative, cumulative, and individually different.  Instead of a pendulum changing back and forth all the time, it's more like branches are growing from the tree trunk of our idea of learning.


We can, and should, learn from everyone.  Often by imagining this pendulum, we envision two sides to our conversations.  This sometimes even turns up as "you're either on my side, or you're on the wrong side" or "you're either on twitter and awesome, or you're not."  Which also gives a false sensation of something like an "ivory tower."  This time, we've walled up all on our own.  Let me draw some contrasts with a hypothetical example.

One, a teacher who teaches through direct instruction only.  This teacher tells stories and connects to students like you wouldn't believe.  She's animate and has the students grasping at her every word.  Students get excited and motivated by what she says, and they have a ton of fun.  On the other hand, another teacher sets up what she calls "inquiry-based" lessons throughout the entire semester.  Students often feel lost as to what to do, and is unengaged with what happens during class.  She believes that she's doing the right thing and sets up to do the same things year after year.

Who would you learn from?  Who would you imitate?

I would learn from both but imitate neither.  From the first teacher, I would recognize her ability to connect to students and engage students, but be careful about what and how the students learned.  From the second teacher I would learn from her ideas, but pay close attentions to the disengagement of the students.

Labeling ourselves and what we do is unhelpful.  Thinking carefully about what we do, recognizing our different circumstances, is helpful.

My objection is not that these metaphors are largely untrue, but that they are largely unhelpful.  To me, the concept of the ivory tower and pendulums are like unicorns.  It may be fascinating, beautifully constructed, or have similarities based in reality (a horse) -- but it ultimately does not exist.

So let's throw them at each other and break down both metaphors.

Student evidence record with Google Spreadsheet

Before you read onwards, you should probably read my other post about the different strategies for assessment (since I think it's more important than this sheet, because it's just a tool), as well as what I did before with excel.

I have to first thank @newmanmath for setting this up with me initially a year or two ago - that I've been using and modifying ever since.

Also if you're interested, @MathletePearce, @MrOrr_geek, and @alicekeeler have recently been working on something similar with Google Spreadsheet.  You can start reading about some of them through this post.

They have done thing slightly differently, but it's really cool to see that we have converged on similar ideas for student access.

But in any case this post is strictly technical.

First I should show the different features of what it looks like.

This is what students see

On the left side it indicates the major strands of the course, and then it is subdivided by expectations.

At one point I also had a "feedback" tab on the bottom

But the main idea is that I would have a master sheet where I load everything: scores for each assessment, feedback...etc...

So the first important thing is the setup of the mastersheet, where I call the tab "evidence"

Note how it is set up, down to the number of rows on the top.

All of the work is actually done on the second sheet.  And then you would need to set up a second sheet - where we will be using Doctopus to create duplicates with.  I won't get into the Doctopus part of things because I think that would take bit longer, but the Doctopus Add on has pretty good tutorials for it.  It's a convenient way to make duplicates for each student.

Okay so onto this second sheet I was talking about.  I will try to break it down to steps.  But first, you can take a look at this dummy sheet.  It has a lot of reference errors (since I removed the sheet that it was referring to), but all the major codes are there (minus the feedback tab)

1. On the second sheet you begin with this code on the top left:

What this is doing is calling the information from the master spreadsheet, and then using what I indicated in the "Gears" tab to call up each student's information.  Note that this calls info from "gears" which is a new tab you would have to create on the bottom.  This helps differentiate between different students for each one of those sheets.

2. The Gears tab has several elements to it

The first one is just the Google code that I placed, and the rest you can find the codes in the dummy file that I attached.

So once you navigate through all the codes, it will transport all the info from the master sheet to each individual one.

I was a bit more ambitious when I started typing this out... but then realized that it would actually take significantly longer to explain each of the codes.  However, with the dummy spreadsheet, it should help those of you who are comfortable with Google Spreadsheet to begin with!  Let me know if you need elaborations.

Randomizing groups

Other people have already talked about this, but there are good benefits to visible random grouping.  I am going to credit it to Peter Liljedahl as that's where I first heard of the term.  Visible random grouping was one of the three pillars that Peter described as supporting environments for problem solving.

But I'm not really here to chat about the research, feel free to check out this powerpoint where Peter goes over some of his arguments if you're interested.

I'm just here to share one way of randomizing!

Here are the goods:
Grade 9 - 1D
Grade 11 - 3U

The process is fairly simple:
1) cut out the different rectangles
2) Greet them at the door, give them a card each (this randomizes them)
3) have them discuss and encourage them to put up their work and ideas on the whiteboards (blackboards are fine,.. or anything they can write on)

Then from this point forward you can do whatever you'd like.  But I plan on doing:
4) Once they're at their groups, split them up again according to A, B, C, D, and get them to talk about the different ways of approaching their number.
5) Get them to return to their groups and chat about what they've just talked about, as well as similarities and differences between what they did.
6) Get them to come up with different representations of their number.  (emphasizing different things in the different courses.  e.g. the concept of equivalence, patterns...etc)

I will probably do 4, 5, 6 in different order over different days.  I've laminated each of those "cards" so I can reuse them during the same week.

(2015) where I was and where I am with AnE

It's towards exam season for a lot of us, and so I figured it'd be timely to share what I've been doing with my students with respect to assessments.  Initially I planned on writing about where I was, where I am, and where I want to go.  However, since this is turning out to be a monster post, I will focus on where I was and where I am.

If you are strictly here to get a copy of my instructions for project and portfolio, here they are:
Project
Portfolio
*note that a lot more instructions were given verbally, along with more student examples for clarification

Here are examples for the summative interview prep as well
Interview focus
Sample interview questions (school name and school crest was removed, which is why the formatting looks strange)

I wrote this back in May 2013: [AnE] SBG - where I am and where I want to go

It gave an overview of some of my thinking back then, which are still mostly consistent with my current thinking.  There is only maybe one exception which is clarifying how I feel about SBG.  I am still in favour of the direction that sbg allows us to go.  On the other hand, I came to realize that some people have begun to use SBG as some sort of end-all-be-all blanket shield for what they do.  And so I think I'd rather emphasize the spirit and the direction of the assessment process that I want to ideally facilitate - instead of championing a term that seems to have included some misuses from people.

Let me try to summarize the ideas from two years ago:
Where I was:

• Focusing on expectations according to the curriculum
• Evaluation with levels as a qualitative descriptor of student understanding
• Emphasizing feedback for promoting student learning

Where I wanted to go:

• Completely project-based spiral curriculum
• Focus on mathematical processes (described by Ministry of Ontario as including: problem solving, reasoning & proving, reflecting, selecting tools & computational strategies, connecting, representing, communicating)
• Incorporate more alternative assessments

Did I reach my goals?

So...  How did I do with these?  Since then, I have experienced some setbacks which I alluded to in previous posts but never fully explored (and for many reasons I won't go into details here either).  The biggest obstacles were different aspects of my personality.  While I have a lot of opinions (which I often question) and questions (which I often have opinions for), I dislike trying to convince other teachers of a particular way of doing things.  I've come to think that this is because: a) I prefer to listen and consider other views (while most of the time staying unconvinced), b) I believe that all teachers benefit from pedagogical approaches that celebrates their individuality, circumstances, and dynamic environments, and c) I hate conflict.

For a large variety of reasons, I haven't made much progress with respect to the three goals I made in the first few semesters after I wrote that post back in May 2013.  I've made a lot more of those steps in the past semester with the grade 12 advanced functions course, which I've alluded to here and here.

For a variety of reasons such as culture, I failed to have activities/projects every single day.  However, I was able to tie together the main ideas of the course through the concepts of equality, inequality, and identity, and we revisited all the strands several times within the course.  This meant that students had multiple opportunities to revisit and thoroughly understand the four main content aspects of the course:

See details from the curriculum document here.  I might elaborate a bit later on how I did this

While I had students work on a process portfolio back in 2012 - 2013 (which I realize now that I never elaborated on...), I was unable to continue that in the two following years due to a variety of reasons.  The past semester, however, I was able to re-incorporate it with the contents of the course, as well as several other assessments - including the summative interview that my students have recently finished with.

And lastly, this past semester I was also able to incorporate a large variety of assessments which I will elaborate on below.

I will divide these up with respect the types they are, as well as indicate the functions that they served (for me, but maybe for students as well).

Observations
Due to the way that I've been running my classes (groups tackling problems), I have been fortunate with the number of opportunities to observe what students say and do.  These include anything such as an activity with Desmos, what the students write on the boards (e.g. vertical surfaces) or in their notes, or what their manipulatives... but most importantly - what they say and do.

Formative functions: I would say that these primarily serve formative functions (although I would argue that everything we do should serve formative functions, otherwise why bother doing them?).  It allowed me to act immediately according to what the student (or students) need at the time.  Typically my action consists of immediate feedback (either posing more problems for students, or asking for clarifications...etc), but sometimes these also help me make decisions on what to consolidate with the whole class -- or what to do on the next days.  Not to mention it also has the value of letting me better understand what the students are understanding.

Summative functions: This is more rare, and usually is paired with other types of assessments that I am using.  Observations sometimes contribute to my understanding of what a particular student understands about a particular expectation.  These are often straws that, together with other assessments, make the weight of information heavy enough for me to "sum up" students understanding at a point in time.

Quizzes/Tests
As much as I'd like to minimize the stress surrounding these types of timed assessments, I still exist in our current culture where they have been traditionally emphasized.  I wrote a bit in some recent posts, but it's impossible to be comprehensive.  The format would often include open questions so that students would have both the freedom of being able to demonstrate their learning and being able to access this written work from a variety of levels, and also the structure which has been provided through in class activities and the wording.  Sometimes I may even build the process differently like this "ordered pair assessment" I described long ago.  Below I will describe what else I do with a typically written piece of work.

Formative functions: I try to use quizzes/tests more formatively than otherwise.  I will try to remember what I've done and list some.  Within the last post, I described: "As a separate activity (sometimes related to the third one), I get them to take a look at what challenges the group faced and what their next steps are." At the time I didn't mention the specifics of what I did, I will try to elaborate a bit.
With quizzes/tests, after they finish, sometimes I would:

• Give it back to them right away, along with a blank version.  Their goal was to collaborate to produce the best version among them
• Give a new version on the next day that consists only of incorrect solutions (typed).  Their goal would be to either a) describe what went wrong and how they would fix it, or b)describe a question for which the solution would be completely correct.  Usually b) is impossible because the solutions had each step typed out, and so the errors are within the processes.
• After a while they get to open their notes and finish it individually
• They get a chance to consult with others, no notes.
• With the quiz/test + their solutions, they identify how they have achieved the expectations that it was intended for.
• Let students have the opportunity of doing post-test interviews which was also described here.  I will likely elaborate on how these interviews work in the future.

I think there were more, but I'll stop there for now.

Summative functions: This one is kind of boring.  But, maybe surprisingly, I stand by my statement for observations in that these written ones are also  "paired with other types of assessments."  I consider these also as different kinds of straws that, together with others, allow me to sum up a students' understanding with respect to a specific concept at a specific point in time.  The post-test interviews also functioned summatively.

Project
The project, along with the following two things -- portfolio, summative interview -- were all related to each other during this semester.  So I will somewhat describe them together.  But first, here is the instruction again free for download(note that more verbal instructions were given in class, along with examples).  Students were basically directed to create a picture with Desmos using the different functions and concepts within the course.  Here are some staff-picked examples as creative art.  The format is fairly similar to what I described here except I've made several changes in order to accommodate for the contents and expectations of this particular course.  Along with the picture, students also had an opportunity to produce an analysis.  They may choose to focus on any expectation they'd like.  No, seriously.  Any one of them.  (they actually also have an opportunity to elaborate on connecting to the portfolio...)

Formative functions: Students had a lot of in-class time to do this.  The most valuable formative function that this served, was the amount of varied feedback they were getting - with respect to their understanding of functions.  They were getting immediate feedback from Desmos because they are constantly trying to re-shape their functions to fit what they want.  They were getting feedback from each other because they were discussing the different aspects of the functions.  They were also getting feedback from me in the forms of questions, suggestions, noticing and wonderings.

Summative functions: The primary summative function was to be served in the summative interview.  But also I have recently collected their final copies and will be matching their understanding (along with portfolio and their interview) to their work, as well as comparing them to what they have done during the term.  All of that to help me better put together a picture of how each student has progressed and what they have come to understand.

Portfolio
Students also had time to work on their portfolio during class, although a lot more time was provided for the project than the portfolio.  You can download the instructions here (note that more verbal instructions and examples were given).  The focus of the portfolio, as I mentioned before, were the 7 mathematical processes outlined in the curriculum.  Students were to demonstrate that they thoroughly understand the different aspects of these processes - using examples from any work that they've done.  Yup, any work.  They were free to include extensions in order to serve as examples for certain expectations that they've done.  The students knew that their goal was to provide the best incidences where they've demonstrated, for example, their ability to connect different representations...  or reflect... etc.

Formative functions: I've always maintained that the specific contents of high school mathematics is unimportant.  There, I said it.  It shouldn't come as a surprise to anyone.  How often would we use "factoring" in our day-to-day lives?  Probably never.  How often would we need to think critically about problems and solve them?  Probably all the time.  As such, the goal of this portfolio as an on-going assessment alongside their other work with the contents of mathematics, was to get students to think about these bigger ideas.  The portfolio functioned primarily formatively as a way of building these experiences for the students.

Summative functions: Similar to the project, the portfolio functioned summatively along with everything else - although it has close ties with the summative interview.

"Summative" Interview
I've always wanted to do this.  I was a bit jealous when Al was able to get there first.  But of course, our circumstances are different, so I won't complain.  While a variety of reasons prevented me from switching out the exam, I was able to do similar things with the "summative."  The term has actually come to mean quite different things for different people, so I am not too fond of it as a noun, but that's okay.  I am fairly certain that Al and I structured these interviews differently, but I wouldn't really know because I haven't had an opportunity to fully chat with him about it yet.
Okay, so this is how mine worked:

• Students submit their project + portfolio and at that time they sign up for several available days for their summative interview
• At the same time they also indicate which content expectation they would also like to focus on.  Here is the sign up sheet, along with a sample sheet which I make notes on during the interview.
• I take a week to read each one of their project + portfolios and come up with questions for each student.  The questions I come up with depend on a) what they've shown in their project + porfolio, and b) what they indicated was their focus for the interview.  You can download the example questions here.
• On the day of the interview, the first student gets the interview questions, and s/he had 15 minutes to work independently.  They may make any notes they'd like, and complete any calculations if necessary.  During this time I get the rest of the class ready to do other activities.  When their 15 minutes are up, I give the next student their interview questions, and then begin the interview with the previous student. (for which they get 10 minutes with 5 minutes flex time)... and it continues onwards until 5 minutes left in class and I consolidate what the rest of the class has done, and we finish up.

Formative functions: There is less opportunities for this to serve formative functions - primarily because it's meant to serve summative ones...  But this is not to say that I still had a brief chat with each student after their interview indicating where they can make improvements.

Summative functions: Along with the rest of the work that students have done during the semester, this served as a good way of "summing up" their understanding of the processes.

Exam
Nothing special here... blah.

Ok and so here I should end this monster post.

I know that a lot of people are interested in how I set up the visual portfolio of student achievements similar to the one I did in the past.  But I think I will stop it here and continue with the technical side of how I create the Google spreadsheet later on.