The Spark

(Cue Music:  Sparks by The Who)

We started a letter writing project today.  The purpose is to persuade somebody to change something.  I started off by asking the kids what was bothering them?  What would they like to change?  At home?  At school?  In the community?

At first, the kids were all centered around sibling rivalry stuff and family issues.

I want my sister to stop taking my things.

I want my mom to give me more video game time.  

I want my dad to take me to Disneyland for my birthday.

The usual kid stuff.  Alright, I said, we can write a letter about those things.  You could write a letter to your sister asking her to stop hitting you.  Or, to your Dad, for a cool birthday present.  We talked about how to make a mind-map to organize our ideas and our letter.  Main topic in the middle, three reasons why, and then for each reason, two examples.  I sent them off to their tables to work.

And then.... THE SPARK!

One boy asked, in loud voice so everybody could hear, if he could write a letter to Mr. M in the high school asking him for more time to play Dungeons and Dragons during after school activities. An avalanche of ideas exploded out of every head in the room.  They all started talking about what teacher they would write to and how they could do something to change the school.  A new brainstorming session broke out, this one much more profound than the fighting with my sister conversation.  Kids broke off into groups and started working together, jointly pursuing their passions and interests.  Here are a couple of the ideas that are being kicked around so far:

  • A letter to the headmaster asking for a class sleepover in the school

  • A letter to a different teacher asking for help setting up a dance club

  • A letter to me asking for more meditation time during the day

  • A letter to the headmaster asking to expand our school garden

It was amazing to see the kids feel so empowered to act and make a change in their environment.  They looked like little activists  plotting a peaceful protest.  They are writing about these things because they want them.  Really want them.  That passion will take you far in life.

My Question

When a spark like this explodes in your class, what do you do?  How do you react?  Personally, I get too excited to sit back, and I jump right in and start throwing ideas around.  I try to listen to every voice and once and help as many as I can.  Sometimes, I just direct traffic, and ask people to think about this, or have you ever thought about that.

The ultimate goal is, of course, to tend that spark until turns in a blazing inferno of awesome.

How do you do it?



Visible Thinking; Chalk Talk

Context: We are about to start a new unit in reading that is all about making connections to literature.

I don't think I did this right but it was a great experience!  That is one of the things I love about the Visible Thinking activities, they are so versatile and open to different interpretations.  They expand the space of what is possible, rather than delimit it.  This activity is basically a big group brainstorming about a topic or sentence.  To me, the most important aspect of this activity is threefold; time, shared space and focus.

The kids need time to process what they are doing.  That goes without saying.  Yet, in an activity like this, I found that some groups were ready for the next stage before others.  We had different groups doing different things.  I was able to tailor it to each groups needs.  Second, the shared space is key.  The piece of paper in front of them is a shared artifact that they take ownership of and use together to create ideas.  Teachers often tell students to not worry about the formatting, the color, or the font when they are making something.  Just get the ideas down and fix it later.  However, the sense of ownership that comes with the retooling of their own personal space sets up a positive feeling in the group that allows them to work together more harmoniously   For some groups anyway!  Other groups, maybe not.  Mindfulness is key.  Finally, focus.  There are many different thinking modes we could use for this.  Elaborating, commenting, connecting, questioning, criticizing, etc.  Choose one and be explicit about what you are asking the kids to do.


[caption id="attachment_1282" align="aligncenter" width="300"] This group struggled at first, but eventually they focused on the importance of text-to-self connections.[/caption]

[caption id="attachment_1283" align="aligncenter" width="300"] First time around, write down your ideas silently[/caption]

[caption id="attachment_1284" align="aligncenter" width="300"] As a group, connect the ideas and explain the connections[/caption]

What I did better this time as opposed to my last session, was I had a discussion before and after of the thinking skills that we need.  As a class, we had a great discussion on what connections are.  We also were able to pull all of our individual connections into a meta-interpretation.  This emerged sponstanouelsy while we were on the carpet.  I asked them sum up what their connections were about in two-three words.  We wrote those down and then made connections between them.  What I find really interesting, is that there meta-interpretation is pretty close to the specific reading strategies we will be discussing!  Text-to-World, Text-to-Self, and Text-to-text.  Between the whole class, they got all of them.

[caption id="attachment_1281" align="aligncenter" width="300"] Meta-Interpretation[/caption]



Teapots update...

The project is going very well.  We have used this idea to invesigate the following mathematical concepts and ideas:

  • surface area

  • perimeter

  • characteristics of 2D shapes

  • characteristics of 3D shapes

  • google sketchup

  • measurement of 2D and 3D shapes

  • volume of 3D shapes

  • 3D drawing

  • building 3D models

And we still have to investigate....

  • 3D perspective

  • 3D nets

  • artistic design

  • company logos and sustainable packaging

  • shipping options and cost analysis

  • shipping routes and measuring distance on a map

It has been a fantastic way to start off the year.  We are zooming in and out of the project, working on it for a while and then stopping to fill in the blanks.  When we are zoomed out, the kids are super engaged because they know that what we are studying has a purpose and a goal; to get this box made and shipped.  They make the connections themselves and they explicitly figure out why we are doing what we do.

This weekend I bought another teapot.  We have three groups working and I decided that they all needed to ship to somewhere.  The newest teapot is going to Indonesia, to @jasongraham99.

Almost there!  Expect a blog post when it is all done....


Reading Time

This is something new I am trying this year.  I am trying to encourage independence, time management, and a love of reading.  Each week on Thursday morning, after our reading period in the library, the kids get 45 minutes to an hour to work on the following reading related checklist.

The first couple weeks I found were scattered and they had difficulties talking about books.  I wanted deeper conversations than, I like it, or it was really good because it was funny.  So, last week I introduced four colored boxes to talk about books.

For each interaction, Voicethread (oral self-reflection), talking with friends (oral sharing), and summary writing in journal (written reflection) I expect them to choose a box and keep their discussion limited to that one theme.  So far, it has been working and I find the conversations much deeper and focused, and the writing is much more personal with more connections.

But, by far my favorite part of this has been the do whatever you want graffiti wall.  Their is no rule here, just that you have to capture a snap shot of your book that makes sense to you.  It is out in the hall outside my room and it looks beautiful!


Mathematical Habits of Mind - Conjecturer

We had a nice emergence today in class.  We were practicing the habit of pattern sniffing.  Everybody has been assigned a 3D shape and they are trying to become experts in that shape.  Today, they started to count the vertices, edges and faces.  When we got around to the cone, a disagreement arose.

How many faces does a cone have?  

Some students said 1, others said 2.  Same problem with a sphere.  How can something have no faces?  It must have one, right?  We switched courses and started to conjecture.  A conjecture is a theory that you have before it has been tested.  It is like a hunch, it hasn't been proven.  We wrote three statements on the board and each child filled out their responses.

1) I think a sphere has _ faces because....

- 1 face because I feel something, so there must be a face

- 0 faces because faces are flat

- 0 faces because it rolls, and things with faces cannot roll

- 0 because a 2D circle has a face, but a 3D circle is not made out of 2D shapes

- 1 face because there can't be nothing

2) I think a cylinder has _ edges because...

- 2 edges because it is not smooth

- 2 edges because a circle needs an edge

- 0 because I can't see any points

3) I think a cone has _ vertices because...

- 1, because I can touch the sharp part

- 1, because there is only one pointy part

- 0, because there are no straight lines, and vertices are only on straight lines

- 6, because there are shapes inside that make the cone round

- 1, because a vertex is a point and it can't be round


Great conjectures.  We need to investigate further.  Where should we go next?  I was thinking of consulting an expert....


Product vs Process

I often write about the learning that happens in my class, but I rarely mention the learning that goes on in my home. My son, who is 3, continually amazes me and teaches me new things about teaching, learning, and knowing. Many of the lessons I learn from him, I am able to apply to the class, and vice versa.

Today, he colored a picture and it got me thinking about the process versus the product. Anyone who looks at this picture will think that he scribbled entire page in pink with no regard to lines or the whole picture. They would assume that this is not a thoughtful work of art with little to no cognitive thought behind it. They would be wrong.

During the process of coloring this photo, he painstakingly took his time on each component of the picture. The eyes, the windows on the train, the leaves on the trees. He narrated what he was coloring and created a story of how the pandas were learning to drive the train (well, he said it was a truck). Each piece of the whole picture was done individually. It was just all done in the same color, and the final product gives the illusion of messiness. Looking at the white bar at the bottom of the page, we see that he did not color that part in, because that is where he wanted to sign his name, T. There are clues to thoughtfulness, but you have to look closely to see them, because you were not part if the process.

Thinking about my class, I need to remind myself to be more aware, to see what is happening and to watch the entire process unfold. If I miss parts, I will miss thinking, and it will just look a pink page. These are all ideas that I already know, nothing is new with what I am saying. Yet, sometimes I need a reminder to keep me alert and mindful.



Visible Thinking - See-Think-Wonder

Context: We are working on a one-kid show for our unit on Self Expression.  The kids have to choose an emotion and investigate it in front of the class for 30 seconds through the use of their voice, body, and face.  To get them thinking deeply about the scene they will portray, we did a See-Think-Wonder in groups.

Silence: I had the kids look silently for about a minute and not say anything.  I did not specify what they were looking for, which I think is an important aspect of this that the authors didn't really reflect on in detail in the book.  During this time, the kids are cognitively going through all the steps that will follow.  They are thin-slicing the picture, organizing their thoughts, and getting a feel for the whole image.  I believe this happening on a sub-conscious level, hence the importance of not orienting attention to anything in particular.

I noticed that some of my kids were locked on to the image and I could see their eyes scanning and darting across the screen.  Others looked for a couple of seconds and then looked down at the carpet.  Perhaps they were process with the image in their mind, or perhaps they were uninterested and had switched to something else?  I don't know.  But I do know this, silent observation is something we are going to practice more.

SEE:  This is where the learners are observing and describing what they see, not interpreting.  These are the things that are really in the picture.  A man in a suit, a silver bucket near his feet, a tree with no leaves, red light, empty chairs.  Try and explain the things in as much detail as you can.

The kids did well with this stage after some prompting to dig deeper.  Instead of writing simply chair, how about you describe the chair in more detail.  Again, we will need to work on this skill more.

THINK:  Finally, we get to start putting together the images and making some interpretations.  What is happening in the image?  Here is where we are making our interpretations, but it is important to note that these need to have some grounding in the observations we made.   The man is waiting for someone.  He is a preacher.  He is lost and looking for directions.  He is practicing a speech. Each one of these can fractal out into new directions and create new stories or possibilities.

Some of the kids nailed this stage and did a great job, while others went into an imaginative rant that was thinly connected to the picture.  They said that he was a vampire hunter and he was chasing a master vampire.  Others said he was a business man who was lost.   Others didn't make a whole picture, but simply made a series of interpretations.  We need to work on the layers of interpretation.

WONDER:  this is where we let our curiosity go and ask questions.  Wondering is similar to thinking, but more expansive.  It speaks to a broader interpretation of what the themes, issues and ideas are.  My kids, for the most part, just took their thinking and turned it into questions.  This will be a big focus of our de-brief next time we do this routine.

Share and Compare: We walked around and compared what our maps looked like and paid attention to the similarities and differences.  We wondered why they thought the way they did, and how they looked at the same thing so differently.  I asked the kids to try and figure out why their interpretations differed, or to postulate a theory about it.  One boy commented that his group was focused mainly on the bucket, and the mystery of the picture came from that, but another group basically ignored the bucket in their interpretations and focused on the tree.

 Final Reflection: This was a really great activity, and it must be done over and over again.  Like the authors say, creating the culture of thinking is so important, and this needs to be become part of the collective brain.  Also, I need to choose REALLY rich and deep images.  This one was alright, but it could have been better.  There just wasn't enough stimuli in it.  Also, I discovered that the most important part of this activity is the sharing discussion afterwards.  In this discussion, I need to make the thinking moves (describing, interpreting, wondering) explicit and have discussions about how we engage with that type of thinking.  I modeled my thoughts to the kids and they found that very helpful, but the discussion slowly drifted back towards the image, and not the thinking behind it.  Put the thinking on the stage and See Think Wonder the thinking.....

[caption id="attachment_1261" align="aligncenter" width="300"] See Think Wonder[/caption]


Visible Thinking, Pt. 3

This is a quick little chapter, just setting up the routines section, which thankfully is the bulk of the book.  Still, there are many spots to stop and think in this short chapter.

  • I think to myself, I have no routines for handing in homework, or lining up.  There is no job list in my class, things just get done when they need to.  We are like an egalitarian hunter/gatherer society.  Will these thinking routines take longer to become ingrained in an atmosphere like this?  Or will they flourish?  Only one way to find out.

  • This is not a prescriptive list of activities to do in the class.  This will require work and effort on the teachers part to learn the routines and where they fit with what lesson.  I like this.

  • I am thinking I might make a poster for each routine and stick them up on the thinking about thinking board on the room; turn them into a monument or part of the class brain, so anybody can access them at any time

  • I love the idea of the lesson folding and folding back.  From a geometry of learning perspective, this is so true.  Our learning moves in and out and up and down.  It is dynamic and moving, like roots spreading.  Pirie and Kieren have a theory of learning that is a series of concentric circles.  We move outward towards the higher levels (back to our Blooms levels, which I don't like) but we also fold back onto our previous knowledge.  This folding back is something that teachers are looking for, because it means the learner is not pushing forward, and we need to provide a new stimulus to get the thinking moving forward.

  • There is a great quote by Paul Cobb.  We quotes an idea in the math classroom that happens when everybody agrees on a term, or a concept and they all use that shared knowledge to move forward and learn together.  He calls it Taken as Shared, and I would love to see these routines become a Taken as Shared artifact in my class.

  • A little unsure about the categorization of the routines.  I like how the authors admit that there could potentially be many ways to categorize them.  Still, I find it a very human trait to categorize everything, when maybe the best response is to just leave them uncategorized?

  • I need to visualize all these routines on pages 50 and 51.  I hate tables.  I need flowcharts and diagrams.  Great.  Another project.

  • ;)

Flipping ahead, the rest of the section is organized by thinking routine.  Not sure how I will structure my reflections, one thinking routine at a time?  Need to sleep on it.


A visual for Surface Area and Volume

A student discovered this today.  It is a nice little visualization to see and feel the difference between Surface Area and Volume:

Make a 3d object out of multilink cubes with lots of different colors.

How many cubes did you use?  That is the Volume (6 in the diagram).

How many squares do you see on the outside?  That is the Surface Area (22 in the diagram).

Putting kids in a Box

We had a spot of difficulty recognizing dimensions of 3D space today.  L x W x H is a simple enough concept to find mathmatically, but if we are going to be Visualizers in math class, we need to be able to picture the image in our head.  First, we started with a globe.  What size of box would you need to make to fit perfectly around this globe?

We practiced our guessing skills and constructed an imaginary box around the globe.  Then, we measured that box with our imagination and wrote our guesses on the board.  Then, we measured it and saw who was closest.  Repeat with a water bottle.  The students reflected that the water bottle was thinner, so the square at the bottom would have to smaller, and the height a little lower.

This lead to new guesses and new visualizations, and new conjectures.  We are starting to think like mathematicians and show these habits of mind.  But, we still have lots of work to do.  We need to challenge our conception of what we think math is.  It is not just calculation.

[caption id="attachment_1246" align="aligncenter" width="300"] Moving in my 10m x 10m x 10m box[/caption]

[caption id="attachment_1247" align="aligncenter" width="225"] Life in the 1m x 1m x 1m box is hard[/caption]

Next, we imagine we are inside a box.  The box is 10m x 10m x 10m.  There is enough space to live in this box.  We move in our box, we dance, we jump.  Suddenly, the box gets transformed to a 2m x 2m x 2m box.  We can still stand, lay down, and jump, but it is not as comfortable.  Suddenly, 1m x 1m x 1m.  Now we are unable to move.  We are trapped inside.  1m x 50cm x 50cm.  Only enough room to lay down.  50cm x 50cm x 2m.  We can stand again, put the walls are tight against us o the sides. 1m x 1m x 2m.  A bit more room, we can finally stretch.  3m x 3m x 3m, enough room to walk a couple of steps and jump.  10m x 10m x 10m.  Finally back to the wide open space!


What if... pt.9

**This is part of a series of weekly questions that are meant to act as conversation starters, or thought experiments.  Discussion and debate are encouraged.  I am not taking a position, merely opening the space of possibilities through discussion.**

Visible Thinking Pt. 2

Chapter 2 - Putting Thinking at the Center of the Educational Enterprise

  • I like their generalization about how policy makers tend to think that by changing curriculum, teachers will teach the content and the system will improve.  This view tends to think that the what in curriculum teaching is more important than the how.  Delimits the role of the teacher and makes our job look bad.

  • The distinction between teaching and learning here is wonderful.  Good teachers focus on learning, while ineffetive teachers focus on teaching.  I have always said that I don't think teaching is a real thing.  It is a word we made up to fit a role.  Learning on the other hand, is what it is all about.  Maybe we should change the name of the profession from Teachers to Learning Specialists?

  • Much of this chapter is about what I would call Mindful Awareness (look up Randa Khattar at the University of Toronto for some fascinating work in this idea).  Paying attention to learning that is happening and then continuing to challenge and push it forward.  As Brent Davis says, Teaching is making the familiar strange. You cannot do that if you are not paying attention.

  • Just realized I have a DVD with this book, but I don't have a DVD player.... ;(

  • Pg 28; restructure thinking; there are those mechanistic metaphors again....

  • The naming and noticing of the process is so important.  I find the same goes with skills in writing, math, reading, etc.  By making what skill we are working explicit (transitions in paragraphs, or punctuation and capital letters) the kids are able to notice what they are doing and pay attention to the specific skill.  The problem I have sometimes, and this is not criticism of the book but of myself, is that I sometimes get too focused on the details and I wonder how to harmonize that with also focusing on the bigger picture?

  • I teach reading with the 7 keys to comprehension, and in that book Susan Zimmerman talks about the important of modeling your reading.  VT also talks about the importance of modeling, and I agree.  Modeling is important, no doubt, but sometimes I wonder if that takes up too much of the instruction?  When is modeling too much, and when do we have students start doing it, rather than watching it being done?  I tend to think that the doing is more important and should be introduced as soon as possible.....

  • I like the Guess what the teacher is thinking part!  I have written about that here before.  Evaluative listening, interpretive listening, and hermeneutic listening.

  • I judge my kids not by the answers they give, but by the questions they ask, pg 32.  Great!  What is report cards were based solely on this type of thinking?

  • Nice distinction for the guideposts and goal posts, pg.33.  I tend to see curriculum as the fencing that surrounds our area, keeps us out of other places, but allows us ample room to explore.  Still, anytime you set up a fence, there are those whose first instinct is to break it down (guilty).

  • What makes you say that?  pg. 34.  I never considered how powerful this question is.  I am going to have to try and start using it more often.

  • Making the student an Active agent in the construction of their own knowledge; not only the construction, but the revisiting and (if we are continuing with the house building metaphors) the repairs from incoming damage.  See, these mechanical/construction metaphors just feel weak to me.  Man-made buildings and machines eventually turn to dust.  Evolution and the organic process of life are forever.  I would say, making the student an active agent in the evolution of their own knowledge.  This implies that the knowledge will change, and the learner will have to adapt.  Learning, Teaching and Knowing are evolving forms that involve ongoing exploration, and integration of new images  metaphors, and applications (Brent Davis).  I'm off topic. Back to the book.

  • Documenting.... ah documenting!  If anybody has any good advice on how to do this, please tell me.  I am not good at it.  I write comments in their books and I use that as a assessment data.  I have toyed with Evernote, but I never stick with it.  As I write this, I am thinking that narrative might be the best way to go for me..... how can I write the story of their growth in a narrative assessment, like a google doc story?  How would this help me?  I will play with this idea....

  • Lastly, I love the last paragraph on pg 38.  The collective access to brainstorming and knowledge is something that I investigating in my M.Ed research project

I must say I am fully impressed with the philosophy side of this book.  I am very much looking forward to jumping into the routines and seeing how my kids react to them.  I have a gut feeling that this book will be a practical methodology to the way I think.... but I am not getting ahead of myself yet.

Visible Thinking, Pt. 1

I have read so much about this book and how it is applied in a classroom from others.  I have always meant to get it, but there have always been so many other things on my mind that I kept putting it off.  Well, after the persistence of a good friend, I finally broke down and bought it.  In typical reflective fashion, I will use this blog to record my thoughts and opinions as I progress through the book and start implementing some of the ideas into my class.  Today though, will just be about the first two chapters, which are kind of like the philosophical base.

Chapter 1 - Unpacking Thinking

  • First though before I even began, unpacking is a term that is meant to imply an expansion of thought, not a reduction, which is how I see the geometry of learning

  • The whole section on beyond Bloom's is wonderful.  I remember introducing Blooms to a grade 6 class in my first year of teaching and they responded with so many questions about how to move up the levels, it drove me nuts.  I don't see learning as a linear process.  It is fractal, not sequential or hierarchical.  Even the new Blooms, it is just a replacement of the old nouns with new verbs, more of an emphasis on doing, but the steps are still in place.

  • Pg.7 the mind is designed to detect patterns .... I don't know if I like this word.  Maybe I am being a pedant, but....

  • I think the distinction between focusing on different levels of thinking (Blooms) and focusing attention on the levels or quality of thinking (Visible Thinking).  This is really the difference between the two for me; one categorizes thinking into a box, and the other analyzes it in order to expand.  Blooms is reductionist, and Visible Thinking is expansionist

  • A lot of the research I have been doing in my M.Ed is quoted in this book.  The authors are drawing directly from high level academic research.  I love this.  I hate the false dichotomy that teachers (and researchers) create between practice and research.  They are intertwined and dependent on each other.  You cannot have research with practice, and you cannot have practice without research.  I see them as one and the same thing, and I get the impression that so do the authors of this book.

  • All the stuff about high stakes testing (pg 9) are great quotes for teachers working in the public school system to argue and fight with bearocrats (I didn't spell that wrong).  Doesn't apply to me at this point in my life, but good to know for the future.

  • Gimmicks and Glamour; I love how they describe certain activities as fun and engaging, but still shallow.  The transmission model of learning is long gone, but education has this need to hold onto it.  We dress it up in a new dress and call it something new, but it is still transmission; taking something from my head and putting it in the students   Learning doesn't happen that way.

  • Great list of thinking skills (pg.11), but every researcher in the world has their own taxonomy of thinking skills, and to me, they are all the same.  The VT list in good in that it is languaged in verbs and actions, and it based on understanding.  Still, a list such as this is still reducing thinking to a set list.  I do, however, love that wondering is on the list!

  • Love the idea for Visible Thinking Portfolios.  Need to meditate on that one and see how I will set it up..... I am not happy with my current portfolio practice.

  • There is so much here that is just great practice.  Instead of saying, here do this, they present ideas and they reframe teaching as being attentive and mindful.  I can't say how refreshing this is to read in a book designed for classroom practice.  Usually, they are simplistic and reductionist.  This one is telling the teacher straight out, paying attention and listening is the core of your job.   Love it.

  • Pg 14; another list of thinking; at least they admit that thinking is not something that can easily defined (if defined at at all!).

  • Now we are getting into some things I can try.  So, I tried to this mind map with my students this morning and the results were really interesting.  I found that some of them are quite knowledgeable about thinking and are excellent at self analyzing.  Lots of talk of inner monologues and how to solving problems and asking good questions.  Others look solely to the mechanistic side, using my brain, getting ideas, and doing something with no real reflection on how it gets done.  There are lots of Emotional and Meta responses, and a few strategic responses.  This will be the goal of this program for me, getting the kids to think about HOW they are thinking, or HOW they could be thinking.  After we did the mind map, I had them wrote a tweet about what they think thinking is.  It will be interesting to come back to this later in the year and see how their conceptions have changed.

  • All in all, I am love with this book, and very excited to read the next chapters and begin to implement some of the thinking routines.  That does not mean I have no criticism of it, I do, but they are pedantic in nature and I feel that for the most part I am in agreement with the rational and reasoning.  I find there to be a real mish-mash of metaphors.  One moment they are talking about an organic process (fostering, growing, expanding) and then the next they are talking about a mechanistic one (the mechanical process of the mind, constructing understanding, drawing out knowledge).  One metaphors puts learning into the world of complexity  (and I offer up fully that I am biased and I see myself as a Complexivist) and the other makes it look like the transmission model of learning.  Still, I think when looking at it from a whole, they are way more on the complexity side than on the transmission side.  There are so many synchronicities in what they are saying and how I view teaching, learning, and knowing.



Mathematical Habits of Mind, Pt. 1

We are well into our teapots project at the moment.  In one of our brainstorming sessions we discovered  that one of the aspects of design we will need to be well aware of is the concept of surface area and volume.  Without knowing that, we cannot design a box that will be sturdy enough.

This year in Math class, I am attempting to focus on Mathematical habits of mind.  There are countless taxonomies of mathematical thinking out there (it seems every researcher in the field has one of their own, slightly different from the rest), but I have chosen to work with one developed by Al Cuoco, Paul Goldenberg, and June Mark.  I have chosen this one not because it is superior, but just because it is easy to follow and covers a broad spectrum of thinking skills (and it contains the word Tinkerers, which I LOVE).  This will be a recurring series of reflections on how this is shaping our math environment.  It is my hope that this changes their conception of what math is, and how we do math.

[caption id="attachment_1218" align="aligncenter" width="614"] Class Poster[/caption]

Part of how I use these habits is to orient attention to when they are being used.  At the moment, since the vocabulary is new and the kids are not used to them yet (beginning of the year and all) I am the one pointing them out, but as the year progresses on, I hope it is the kids who are bringing them up and realizing them on their own.  Also, I use them in my lesson design.  After only a couple of weeks, I find it is helping me bring different styles of mathematical thinking into our class.  Here is what we did today:

The Task

When the kids came into class in the morning, there were seven containers waiting for them with letters under them.


Each child individually inspected the items and ranked them from most to least (I didn't specify a unit, but most figured out I was talking about Volume).  Next, they described their reasons for choosing the order they did to a partner, and then another partner.  I kept interrupting and orienting attention to use of mathematical terms like volume, capacity, and surface area.  Finally, we had a class discussion about the choices, in which a mini-debate arose about the difference between the orange pan and the measuring cup with the handle (flat and wide versus narrow and tall).


After the discussion, we plotted all of our answers on the board and found the most popular choice, in fraction and percentage.

I then asked them to write sentences about this data.  What is it telling you?  What statements can you make about this?  They came up with things like; most people are sure about the smallest containers (81%), the biggest containers are guesses between two choices, and the middle three are very unsure.  This led to a fruitful discussion about the probability and likelihood of percentages.  We deconstructed, in language terms, what is the difference between 100%, 75%, 50%, 25%, and 0%.  Describing numbers using mathematical language was the habit of mind we were working on.


Next, I asked them to design an experiment to test and prove which one is the biggest, and then to present their experiment to the class in our math congress.  We had three different strategies presented:

  • Fill each container with water, then pour it into a large measuring cup and record the amount in ML.

  • Fill one container with water and pour it into another container; it is overflows it is bigger, if it doesn't fill it is smaller.  Continue this until you know all the sizes.

  • Put marbles inside each container and count the marbles

I originally intended to have them vote on one strategy that we would all do, but I changed my mind and let them each do their own experiment.

To finish it off, we all shared our data and compared to each other.  Each group got the same order, so we concluded that our methods were all sound.  Next, we compared our original guess to the actual answer we found.  We found that the group average was NOT the correct answer.  A few kids had guessed the order correctly, but most had gotten 2-3 wrong.  Finally, we had a discussion and each group made a T-Chart about what was effective about their experiment, and what was ineffective.  We talked about accuracy and how important it is be as accurate as possible.  I hope this type of hands on inquiry and reflection translates into a deeper understanding of Volume when we move to the calculation and mathy side of this concept.  At least we will have something to ground our calculations in, moving from an Enactive experience such as this, to a more abstract and Iconic situation of formal school mathematics (I have started calling it school mathematics since I read Street Mathematics and School Mathematics by Terezinha Nunes, which I HIGHLY recommend). 

Reflecting on the Habits

Finally, we reflected on the differences between the three habits of mind.  I asked them to write a definition of each one in their Evernote reflection journals.  Here are a couple of the reflections:

  • It is important to be a guesser because  if we use a do it, it is faster but it is not accurate

  • the more we practice guessing the more accurate we might be

  • I think it's important to experiment things all the time because if you learn a mathematical thing, you should always want to check it. It's always better to look at the thing, than just listening to details.  Do it yourself to find out.

  • It is important to describe because some one cannot understand you.

I am very interested to see how these habits affect the course of the year.  They struggled with the reflection process, and I will need to provide more scaffolding to help them to be become better reflectors in math.  Orally, we have some great discussions and they provide me with a lot of details about what they are learning, so I am hopeful that these habits will become part of the classroom culture, not only in math, but in all subjects.


Arts in the Classroom

I found this old post I did a while back and thought I would delve a bit deeper into this.  The following image is from Elliot Eisner and his model of Connoisseurs and Critics in an educational environment.

I love the message that this simple circle is suggesting.  It is saying that it is not just a fun way to work, but an integral part of how human beings adapt to and make sense of the world.  Art is important.  Not just to develop empathy and feeling, but for cognition.  It helps us grow more intelligent.

Part of my philosophy to the classroom is this; every student who comes into my room is an artist and how can I help them be better artists?  However, I am not just taking about the Arts with a capital A, I am talking about the art of science, the art of geography, the art of math, etc.  These all have an artistic side to them.  Is the type of thinking that a scientist does less creative than the type of thinking a painter does?  How about a mathematician versus a film maker?  Or a philosopher and a doctor?    They all need that creative burst of cognitive power that the Arts are so good at helping to develop.

It is not about the Art.  It is about the thinking that goes into the Art.

Download the PDF Arts in the Classroom, or watch the Slideshow below.


The Case of the Missing Corn

I came into school, as I always do on a Monday morning, filled with an odd mixture of tiredness and excitement.  Outside of my classroom window, I spotted the class garden.  Horror!  Shock!  Despair!  The class corn was gone.  All of it.  Destroyed.  Knocked down.  Pulverized.

It must have been the rain we got over the weekend.  It was a torrential downpour.  But wait?  Why were those flimsy pea plants still standing?  And the tomatoes are fine?  I went closer to investigate and I saw it.  Bite marks on a cob of corn.  An animal is responsible for this.  But who?

The kids started to slowly trickle in from the buses.  A boy was the first to notice.  He noticed the signs that foul play was involved, but he couldn't name the guilty party.  Like a pack of hound dogs on a hunt in the woods, he gathered friends and they began to investigate and sniff out clues.  Soon, the bell rang and class was about to start.

Once inside, before our morning journal writing, I told the class about the foul play that occurred in the garden, and apologized that we won't be able to make the roasted corn we were planning.  They were disappointed, but they held their chins high and kept their feelings in their heart.  There was nothing they could do, and they knew it.  Conversation continued about the scene.

Ss:  It was a bear.

Ss: It couldn't have been a bear, they are too big, they would have destroyed the tomatoes.

Ss:  Something small. Maybe there are tracks?

Ss: Or maybe we can find unchi (droppings)?

Ss: Why didn't it eat the peas?

Accusations began to fly about the perpetrator.  I gathered the group together and told them that they need to find evidence before they start accusing innocents.  Innocent until proven guilty.  They brainstormed a list of clues that they need to look for and then headed outside to investigate first hand and interview witnesses.  I stayed inside (with an open window) to take their notes and write them on the command centre board.  After, of course, we cornered off the crime scene.

Observations and notes:

  • a forest is very close to the school

  • whole stalks of corn were pulled from the ground

  • some cobs of corn were picked off the stalk, others were eaten still on the stalk

  • all of the corn was eaten, but only a couple of tomatoes were found crushed

  • there was a very hard rain the night before

  • there are bite marks in some of the corn cobs, the whole cob was bitten in half

  • there is a trail of corn leading away from the garden and behind the prefab building (that is also the way to the forest)

  • the husks were peeled off the corn cobs

  • no tracks

  • no droppings

  • no fur or hair or feathers

  • A friend of one of the students had their corn eaten recently

  • Mr Sato the school custodian saw a dead Tanuki on the side of the road two weeks ago, and recently had to build a shelter for the garbage because of animals getting into it

Once we had our observations gathered on the white board, we got down to the business of putting the story together.  It is a hard thing to do with so many passionate voices in a class, but we were able to make the following inferences (theories, conjectures)

  • must be an animal

  • must have sharp teeth to bite the corn in half

  • a lot of corn was eaten, so it is possible that this was a pack (or else we would have seen a lot of droppings)

  • must be able to stand on two legs to pull down the stalk

  • must have a small mouth because the bites in the cob were not large, but they were clean, suggesting sharp teeth

  • must be a small light animal because we cannot see any prints (however, the rain may have washed them away; same with droppings)

  • must hands dexterous to peel the husks off corn

  • it appears that they used the path between the prefab buildings to escape since we found corn shavings on the other side

We needed suspects.  Luckily, we have a few staff and students who live in the area.  We canvassed them for information.  They gave us a list of animals that they know live in the region.  We went through the list and ruled out our suspects.

  • Bears (haven't been seen in a while, too big to handle the corn delicately)

  • Foxes (can't grab because front legs are not used as arms)

  • Tanuki (likely suspect)

  • Wild Boars (again, hoofs make it hard to grab stalks of corn)

  • Kamoshika (same, hoofs)

  • Hakubishin (likely suspect, though hands aren't as dexterous as the Tanuki)

  • Crows or other birds (can't grab and pull with enough power to rip out the plant)

  • Human (they would have had to bend down and eat raw corn of the stalk and then bite the cob in two; possible but highly unlikely)

We had a list of two suspects, the Tanuki or the Hakubishin (both of which are noted tricksters in Japanese mythology).  Background checks.  I spoke to the Japanese teacher and asked if I could hijack a period for some intense research (in Japanese of course!).  The kids got the iPads out and went to work looking for anything and everything related to Tanuki and Hakubishin.

Finally, we were ready to mount our cases against the accused.  We made t-charts for each option

  • group 1 - the evidence for/against other animals

  • group 2 - the evidence for/against Tanuki

  • group 3 - the evidence for/against Hakubishin

Finally, after weighing all the evidence, we wrote a reflection about who we thought was the culprit and had a vote.  The guilty party turned out to be the Tanuki!  We have no way of knowing of course who actually did it, but we have a pretty strong case against to the Tanuki.  Here is our evidence:

  • loves to eat corn

  • has hands that are dexterous and can peel

  • lives in areas with old empty buildings

  • eats in packs

  • can stand on two feet

  • has sharp teeth

  • loves areas where humans live because of the garbage

  • light and would leave no tracks

  • nocturnal, hunt at night, and wouldn't have been noticed at all

  • The only really odd part that has us stumped is; why it left the tomatoes?

After all this hard detective work.  We decided to spend the last thirty minutes of the day making some art inspired by our day of investigation.  Some students wrote a song about the situation, others drew pictures, and some wrote fictional stories.

All in all, not a bad way to spend a Monday.