Playing with Math

Report cards are looming on the horizon.  I came in early and sat down at my computer, turned on some Bach, and began typing.  I got into what some scientists call flow (the mental state of operation in which a person in an activity is fully immersed in a feeling of energized focus, full involvement, and success in the process of the activity), and the time flew by (I also got lost in the garden for awhile!).  Before I knew it, it was time to start class for the day.  Unfortunately, I hadn't planned a thing, and I realized this I was walking outside to bring the kids in.

They arrived in the classroom and got their bags unpacked and put away in their lockers.  I looked around the room for some kind of inspiration.  Luckily for me, I had just received a box full of math manipulative the day before.  I grabbed a box of fraction circles and just as everybody was sitting down to the communal table, I dumped them all on the table and walked away.

I'd like to say I was surprised at what happened next, but I really am not.  Play is a powerful learning tool, and when given creative freedom over a tangible item, kids will come up with amazing things.  Kids need to play (adults too).  One of the girls made a circles out of two halves.  Then, four quarters.  Soon, the rest of the class was trying to see how many ways they could make a whole.  I interjected for a moment, asked them to get their math journals, and then we got back to playing.  As they played I gave them challenges, and asked them that must do the following for each challenge; a) show it with the tiles, b) draw a picture of it, and c) show it mathematically.

With those simple rules, we started off on the challenges, which I will list below.  It is important to note though that these challenges were not being done as a list to get through.  Each student was working on something different.  Since I hadn't planned a thing, I was creating these challenges based on what I witnessed in their play.  I tried to let their natural play flow into a lesson.  By taking what they were playing with, and harmonizing it to the math we were studying, I think we provided a great deal of depth into this lesson (which lasted about an hour and half).  As we came up with new ideas and activities, I wrote them down on the white board, and as they finished one, they would go to another.

Make Wholes

How many different ways can you make a whole?  Write down all the different ways, draw picture of your circle, and show how the pieces add up mathematically.

Make Thirds

How many different ways can you make a third?  Write down all the different ways, draw picture of your third, and show how the pieces add up mathematically.

Make Wholes with.....

- only 3 pieces; none of them are the same

- only 6 pieces; none of them are the same

- using a fifth and an eighth piece

- Using eight pieces, only 2 of which can be the same

- And so on.... (the kids came up with some good ones which I can't remember nor did I write down)

Pizza Problem

Joy and Takumi walk into a pizza shop.  Joy orders 3/4 of a pizza, and Takumi orders 2/3.

Who ordered more?  How much more? (show with pictures, tiles and symbolically)

How do we make it equal, so each of them get an equal amount of pizza?  (make a poster for the Pizza cook informing of him of how to make 2/3 equal to 3/4, just is case this situation happens again, he can refer to it for help)

Stack it

Take a piece to act as your base, preferably something large (half, third, quarter etc.).  How many equivalent fractions can you make?  Stack them on top of the base and see how high your tower goes.  Keep track of your progress by listing each layer and showing how it works mathematically.

Having them play, and then allowing them the lesson to emerge out of that play..... I could have easily planned a lesson like this and then had them work through all the problems one by one.  But, this was more fun.  It was directed by them and even if I changed their activity, they could still see their original idea in the final activity.  They owned it.

It is interesting that my role as a teacher in this morning was just to dump a bucket of tiles on the table.  After that, they came up with the activities, and I simply oriented their attention to how this applied to their mathematical study (and altered the activities to make them more challenging in some cases, or to make them easier in others).  It required, from my part, an understanding of the pedagogy of learning, but also I had to have a sound working knowledge of fractions, and the myriad of ways they can be represented.  It is not just a matter of playing, but rather, orienting play in a purposeful direction that promotes growth and increased understanding, conceptually, and from a computational perspective.


Art in the Classroom

For my coursework, I had to do a presentation on Elliot Eisner and his evaluation model of connoisseurship and criticism.  While researching the topic I came across this image about using arts in the classroom.

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, but for cognition.  It helps us grow smarter.

That being said, this diagram is pretty boring, and not very artistic.  So, I spiced it up a little.  I tried to keep all the content exactly the same (made a couple little changes for flow and aesthetics) but the overall feeling is the same.  Download the PDF Arts in the Classroom, or watch the Slideshow below.



What if... pt 5

**This is part 5 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**

Part 1

Part 2

Part 3

Part 4


Play with it

A friend and classmate of mine made an interesting comment on our course discussion board:
Hey Craig...do you ever ask your students about what they are working on? What if we only asked them what are you playing with? Might we get a different answer or attitude towards it? I might try that from now on and see....

I like this.  I am trying to be more aware of the language I use in class and how it changes the context, especially since none of the kids in my class share a common country of origin and are all from different cultures and backgrounds.  Play might become my new verb instead of do or work.  Instead of saying, what are you working on?  or even what are you doing?  I think I may try something like, what are you playing with?  Did you finish your playing?

Play, even when it is not fun, is always more enjoyable than work.  Work is something that somebody else tells you to engage with, it is play when you chose to engage with it.


What if... pt. 4

**This is part 4 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**


Part 1

Part 2

Part 3


What type of learner are you?

I first came across this video (in an educational context) from a workshop I attended at OISE on gifted education.  We were to watch the clip while pondering; which one is gifted?  We wrote our rational and then had a discussion with our groups.  The only rule, was we HAD to chose one, and then back it up (even though choosing only one is next to impossible!).

I plan on showing this to my students later today.  I will give them the following graphic organizer, and ask them to jot notes about each character and how they are similar to, and different from, themselves.


Peanuts Learning Styles (Download PDF here)

Possible Variations or Extensions

  • Write a self-reflection from one characters perspective about the how the book report turned out

  • Compare each of the characters to other people you know

  • Imagine you are the teacher, and give feedback to each student on their book report

  • Which one do you think you would get along with best?  Why?

  • In groups of 4, create a situational drama presentation with each of you as one of the characters

  • What do you think these kids will be like when they are adults?  Why?

  • Rank them from 1-4 in the effectiveness of their strategies and debate your choices

  • Draw your own peanuts comic strip with all four characters in it

If you have more ideas please leave them in the comments!

My own personal reflection; I am (or I used to be) a mixture of each of them.  When I was young I was just like Charlie Brown, I used to wait to the last minute and procrastinate!  I grew out of that in part, because having a young child means I have to get things done when I have free time.  I am also very similar to Linus.  I like reading a myriad of different perspectives, and I like to complexify issues, not simplify them.  I used to be a simplifier, but my readings into self-organization and complexity science have expanded my view of the world.  As for Schroeder, I might not be musical, but I am a dreamer and I like to let my mind wander where it will, regardless of the task I am doing.  I reinterpret problems and questions in my own way, and take my learning where I want to go, which is not necessarily the same direction as the instructor or teacher intended.  I have always been this way, and I hope I always am.

I am nothing like Lucy.



Brainstorming in gDocs

1) Set up a googledoc with your questions or provocations

2) Set the sharing options to open to all, can edit

3) Assign each child a color

4) Let them loose


There will be a flurry of fighting and jostling for position at the beginning, but once the comments start rolling in, it self-organizes.  The first time you try, it will not be smooth.  Stick with it.  Once you get the hang of it, it is a great activity.  This is a powerful way to collaborate, and a fun tool that kids love!


What they told me

I had this great lesson today.  We went outside and measured out ten meters on the asphalt and then found the points where the crust would be, the mantle, the outer core, and the inner core.  First, we had to work out the percentage for each and then convert that percentage over to our 10m model.  The math was challenging, they were engaged and it was great to work with sidewalk chalk!

Afterwards, we climbed to the roof and took a birds eye view of our handy work.  It was beautiful.

[caption id="" align="aligncenter" width="300" caption="One of those days...."][/caption]

Unfortunately, none of it is true.  It was what I had imagined in my head would happen.  Here is what really happened.

About 20 minutes prior to Japanese class finishing, a big black cloud arrived over the school.  I knew that the outside sidewalk chalk would not be a good idea.  I thought about doing it in the hallway, but the grade 3.4 class had done something similar earlier in the year, and I decided to do something different.  We recently just got our hands on a set of iPads, so I thought, lets try that.  Instead of comparing to the measuring tape, we would compare the layers of the earth to our own bodies, and find out where the crust would be, the mantle end, and so on.  Then, we could take a picture with the iPad, and draw the layers right on top of the picture!

Scrambling, I got the iPads signed out, tried to download the necessary app (but the layers of security on them slowed me down), and then got my Keynote page ready to reflect the new changes.  Just as they came in, I got it done.  We started on time.

Well, the math part of the problem didn't really cause any problems.  They got the idea, needed some help from each other walking through the conversations from Earth to Me, I had to step in a couple of time and correct some errors that had gotten out of control and spread through the group (when measuring the layers on your body, make sure to start each layer from where the last one left off), but for the most part, they did well.  It would have been easier with the ten meter tape, since dividing into ten is a lot easier than dividing into 154cm.  But, they have calculators for that!

Then, the iPads struck.  We took the pictures, loaded them into app, drew our pictures on top of them, and then nothing worked.  It was hard to write on the pictures, the calibration was off.  I couldn't email them.  I couldn't save them.  I couldn't export them.  Several of them got deleted, and redone, and then deleted again.  A couple more got flipped around so that the picture we drew was at a different orientation to the picture we took!  It took us over 45 minutes to do the part of the lesson that should have been the quickest.  The kids did their best to stick with it and try and solve all the problems that came up, but they just got the iPads yesterday, so they don't know much.  It was frustrating for all of us.  In the end, we got it to work, but it was bittersweet.  We were tired and we wanted to go home.  We knew it was not our best.

The problems rest completely at my feet.  I took a risk.  I tried something that I thought would be fun, interesting and different.  It would have been, had I been more prepared.  I rushed it together and the results showed.  I sat the kids down before the day ended and apologized that today was not a smooth day, and that everything seemed to go wrong.  I told them it was my fault because I was not prepared.

They told me not to worry

They told me that sometimes things don't go the way you plan

They told me that sometimes you have to go with the flow

They told me that you learn more from mistakes than success

They said that it was fun trying to figure it out together

They said we will do better next time


I agree.


What if... (pt. 3)

**This is part 3 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**

Part 1

Part 2

Inquiry Beetles

The other day a co-worker of mine and I were digging some horse manure into a wheelbarrow (teaching is a great job).  He found a massive beetle larva.

[caption id="" align="aligncenter" width="223" caption="Not exactly like this, but very similar"][/caption]

He took it back to class and showed his kids.  They fell in love, and before he knew it he had an aquarium full of them, books about beetles, charcoal to kill the smell, spray bottles to keep the soil moist, and a host of other paraphernalia related to the care and raising of beetles.

[caption id="attachment_1058" align="aligncenter" width="300" caption="The Beetle Station"][/caption]

Kids are constantly giving advice, taking ownership of the care, and showing great curiosity at what will become of the beetles.  Oh, I forgot to mention the best part!  He (and I as well) have no idea what these larva will turn into to!  I am secretly hoping for Dragons, but it will more than likely be some kind of beetle. Their current unit is all about plants, so the link between this and the unit is there, but that is not important.  What is important is he, as a co-inquirer in the learning environment, saw something that made him curious and acted on it.  He knew this would light a fire under the class. That action has created a positive feedback loop that is amplifying through-out the system and growing.  When ideas like this grow, we may never know where they end up, but that is the beauty of life and learning.

Beetles are a wonderful metaphor for inquiry learning.



Just because...

I gave my students an outline to an optional project about three months ago.  The task was to build a model of a famous place out of 100% recycled materials (paint and glue were the exception).  The other catch, you had to include as many members of your family as possible in the construction.  Here are the final results.

I have had many happy emails from parents telling me how great it was to work with their child on a creative project.

This was not connected to anything we are studying in class, and it was 100% optional.

Sometimes we don't need reasons to do something.  Sometimes we do things just because they are fun or engaging.


Balance and Harmony

I read a lot of blogs.  Not just education related, but blogs about art, energy issues, politics, etc.  People tend to read blogs written by people who have the same viewpoint as themselves (but I strongly encourage everybody to read one from the opposite side) so you tend to agree with everything that you read.  It gives us inspiration and new ideas (the currency of education).  Yet, even when I respect the author, I find some things still grate on me.  Little things make a big difference, especially the use of words and the implications and hidden meanings that they bring with them.  One word which is used a lot in educational contexts is the word balance.  I do not like this word and I think it sends the wrong message than intended.  I admit with full disclose that I can a bit of pedant with terms and words, but that is only because those terms and words act as metaphors which do shape our lives and the way we interact and view the world.

Balance is literary things that are equal, or a state of equilibrium or parity characterized by cancellation of all forces by equal opposing forces.  2 on one side, 2 on the other.  In terms of education, this means that time must be balanced, and learning engagements should be equally distributed.  Taken literally, it would mean we have to have equal time dedicated to all subjects (what happened to dance, art, and music?).  If we try to find balance, we try to make ideas equal out in the long run.  Using tech and using your hands to make artifacts; both are valuable tasks, and in finding balance we are suggesting that we need to find equal time for both.

In my mind this is impossible, and highly ineffective.  The world does not operate at equilibrium.  It operates at disequilibrium.  If an ecosystem is put into equilibrium, it will die.  It needs disequilibrium to continue the process of change, adaptation, and evolution.  Complex systems operate far from equilibrium.  A classroom is no different.  It needs to flow and move.  It needs to evolve.  Learning is not a static entity that is characterized by the opposition of two opposing forces.  It is an ongoing exploration and integration of new images, metaphors, and applications.  If we impose the structure of balance onto it, we are doing it a disservice.

A more apt image is the Yin Yang.  When viewing the Yin Yang, it is not meant to be viewed is a static 2D form.  That view is simply a holistic view of balance.  Black on one side, white on the other, black dot in the white, white dot in the black.  It is the same thing as the scale!  The true nature of the Yin Yang is flowing.  It is moving.  Sometimes the white side is bigger than the black side, and sometimes the white dot is bigger than the white dot.  The point being that there is always white in black, and black in white.  It is harmonious.

I find this to be a more powerful metaphor for the classroom, and for knowing, learning, and teaching.  It is ongoing, always flowing and changing and adapting.  True, scales can move, but only up and down.  Classrooms move in every direction.  A harmonious synthesis of people, ideas and discoveries.

Or maybe I am being pedantic?  I am really not sure.



What if... (pt. 2)

**This is part 2 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**

Previous Entries

Part 1