Today was all about spatial awareness and resilience.  Before the activity started, we shared strategies with each other on how we can deal with a situation that is very hard, and when we feel in over our heads.  Students made suggestions like;

  • Walk away from it and think about something else

  • Don't pay attention to other people

  • Ask others for help

  • Sing a song while you work

  • Try and switch between using your hands and your head

  • Let your hands think, and clear your mind

  • Set a reward for yourself if you finish

With that in mind, we started what was a very difficult activity.  The subject was Pentominoes.  Pentominoes are shapes made of 5 squares, where each side of one square touches another side of one square.  Here is what we did with them;

1) Using your blocks, find out how many possible shapes you can make with 5 squares.  Flips and rotations count as 1 shape, not two.  When you have found all the possible combinations, draw them in your math journal.

2) Using the shapes you made, make a rectangle of either 3 x 20, 4 x 15, 5 x 12, or 6 x10 (what do all these have in common?).  You must use every shape and have no blank spaces of leftovers.  When finished, draw the rectangle in your math journal.

This is as far as we got today, but I am sure there are tons of extensions to go on this activity.  How many squares can you make with these shapes?  Which pentominoes tessellate?  Can you make any other images?  How about Hexaminoes (six squares)?  What if we try using triangles instead of squares?  And so on....


A Reflection on Anxiety

I try to manage my classroom with as little stress and anxiety as possible.  I do this by:

  • No grades of any kind, just feedback

  • Almost no Homework

  • Collaborate activities and almost n0 competitive elements to the classroom

  • Freedom to opt out of doing anything at anytime (if students don't want to do something, they don't have to)

  • Weekly Free Time periods where they can explore and play whatever they want to

During the past three weeks we have been diving into the world of Graphic Novels and the Elements of a Story.  We have read a heap for pleasure, for analysis, and for research.  We have looked at the characteristics of GN, the art, the speech bubbles, etc.  We researched our own myth from an Ancient Civilization, and then planned and planned.  Plot, theme, character development, narration, setting, time, etc, etc.  We have reams of notes, and have gone through two rough drafts,  and finally are ready to get started on the good copy.  Everyone was very engaged, very interested, and very excited.

However, with that excitement comes anxiety.  Even a very engaging and exploratory style unit of study can lead to feelings of doubt.  My own personal take on it, the students invest so much of themselves, so much of their mind and intellect, that they create a fear of failure within themselves.  If the pressure is not coming from me, or from friends, then it must be self imposed.  I wanted to know where these feelings came from, and after we spoke about nerves as a class, I asked one of the boys in my class to go out in the hall and use my iPhone to reflect on why he was feeling nervous.  His response is wonderful, and made me reflect on what I can do to make this anxiety disappear (or if I can do anything? Or if I should do anything?).

Click on the link to hear the students reflection.

Emi Anxiety Reflection

Keep in mind while listening, that this phenomenon is not only this one child, but we all have kids who feel like this.






Finding Pi

Pi is an interesting mathematical idea for young kids.  The thought that a measurement can be same, everywhere in the world, every time, is a hard concept to grasp.  I came across this activity in the great Teaching Student Centered Mathematics by John van de Walle.

1. Ask the students to measure the Circumference and Diameter of 5 circles in and around the classroom.  Try and get a variety of sizes of circles.

2.  Make a Table to record your work

3.  Once finished measuring, find the ratio of Circumference to Diameter by dividing C by D.  Add this information to your table.

4.  Take you data and record it on a large, class sized Scatter Plot.

5.  Analyze data from Scatter Plot and Tables; look for conclusions or patterns

This should lead into a discussion about why all of the numbers for the ratios are close to 3.14.  Once we see this, you can introduce the concept of Pi.  From there, this lesson could fractal out into so many different dimensions and independent inquiries.

To further excite their curiosity, here is a website that shows Pi to four million places.  Another little tidbit of information, a team at Tokyo University have calculated Pi to 1.24 Trillion places!




Face to Face

This a nice little activity I picked up from a Paul Ginnis workshop.

  • Students sit with a partner and choose who is A and who is B

  • Person A,  grabs a clipboard and a pencil and sits with their back to the screen; person B sits in front of their partner, facing the screen

  • Teacher puts an image up on the screen

  • Person B will explain to person A what the image looks like; Person A will draw the image

  • RULES!

  • Only use words, no hand gestures of any kind; sit on your hands!

  • No looking at, or showing the paper until finished

  • Person A, no turning around and looking

  • Both A and B are allowed to talk, and ask each other as many questions as they want

Very effective for Geometry, shapes and oral explanations.  Also useful as a diagnostic to see what students already know.


Simple Math Assessment

Today was test day in my class.  We have been deconstructing Triangles and Quadrilaterals, and I needed to pause for a second and make sure they were getting it.  Instead of doing the standard pencil and paper test (which I have vowed to not do even once this year!), I used this little exercise;

Each child gets a Geo-Board and a pile of elastics and finds a quite spot in the room

Students are given a list of instructions
1.Make a right angle triangle

2.Make a triangle with an obtuse angle

3.Make a triangle with three acute angles; What type if triangle is this?

4.Make a Scalene triangle; why is it Scalene?

5.Make a quadrilateral with four equal angles; what shape is this?

6.Make a quadrilateral with two sets of equal angles; what shape is this

7.Make a quadrilateral with two obtuse angles; what shape is this?

8.Make a quadrilateral and two triangles and have them overlap; how many shapes can you now notice?

When each step is finished, students bring their geoboard to the teacher and check their answer

Teacher tracks progress of each student on a table (with whatever grading scheme you use)

For those who finished quicker, play time!

I felt this went well today, and as a teacher, I got to know exactly who knew what, and where they need to work.  It gave me all the data I need.  Also, it was fun, low stress, and the kids thought it was a game, not a test.

Am I being naive and idealistic?  Do we need to prepare kids (elementary aged kids, mind you) for high pressure tests that will face later in life (maybe)?

I say no.  How about you?


Sticks, Rocks, Housing, and Fun

We are currently studying Ancient Civilizations in our Unit of Inquiry.  To delve into this topic, we are creating our own civilization, and jumping around the history of past civilizations to better understand the imaginative one we are building.  I am trying to keep the learning as Emergent as possible, and letting the kids take their world in any direction they want.  One of the inquiries they raised was that of ancient housing;  How did people live back then?  What did their houses look like?  Why did they look that way?  To help the kids get a close up, hands on perspective into this topic, we undertook the following series of lessons.

Step One - Research Jigsaw

I set up four stations around the room and put the students into teams.  I told them that they were construction engineers and they were doing research.  At each of the four stations were pictures of housing in the ancient world.  The students then travelled from station to station, and made notes about the characteristics of each house.  I asked they to consider why it was built the way it was.  They talked about the geography and climate of each place.  They looked at the materials used, and the features of each house.  We analyzed why and how they were different, and similar.  We made a Venn Diagram with four circles in their groups, and then had a big debrief after, where they filled in the knowledge gaps for each other (or I contributed where I could).  Our findings were simple and clear; Each civilization designed their homes to fit with their natural environment.  Simple, yet a powerful big idea.

Step Two - Nature Walk

The next day we went for a walk in the nearby forest.  I asked the students to make note of the materials that exist in our natural environment (Northern Japan). If they were to build a house, what resources would they be likely to use, and what was in abundance?  They made notes on their clipboards, started to sketch diagrams, and had debates about what would make for a solid structure.  Sitting on a large rock near a slow moving brook, I told them that they were going to build their own house, and that they needed to design it.  I gave them a template and asked them to work as a team to gather the materials, draw a diagram, and write about why they made the design choices they did.  We came back to the school with giant garbage bags full of stones, sticks, and leaves.

Step Three - What are we learning?

We took a brief pause from the excitement and I asked them to consider why we were doing this.  What hidden curriculum was I trying to teach?  What skills would we need?  What attitudes and knowledge would be important in guiding us?  As a large group, we filled out the WAWL (What Are We Learning) template with the important elements.

Step Four - Construction

Each team was given a piece of wood for their base, a glue gun (we imagined that in ancient civilizations they would have had rope or mud to fasten their wood, and so we would substitute that bond with glue!) and a hefty pat on the back from me.  They were off!

(A little piece of advice, if you are going to let children use saws and sharp knives, go over safety rules beforehand!)

Some of the phrases I overheard while they were working, and I announced to them as I heard it (when they are doing group work I try not to help with the details, but rather as a narrator, helping them to see how they are using our key skills and attitudes).  It was music to my ears:

What would happen if......?
Could we try......?
It might work if......?
Is it possible to......?

They weren't using their plans, but they were adapting to the environment in front of them.  It was problem solving and critical thinking.  It was, in our Blooms Taxonomy, creating.  The room was an absolute disaster, but we loved every minute of it.

Step Five - Reflection

After the house was finished, we sat back and enjoyed our work.  We also took time to reflect on the process in two ways.  First, we group wrote a descriptive paragraph about the skills, attitudes and knowledge we needed to complete the task; what we did well, what we did poorly, what we learned, etc.  Once this was complete, we printed it off and posted it up next to our house in the hallway in front of the class, so everybody walking by would see it, and hopefully ask us questions about it.  Second, we sat down individually and filled out a personal reflection. This was a bingo card, and in each square there was a different question about the project; some of them related to the knowledge, but most focusing on the meta-learning we settled on before the project began (see WAWL above).  The reflection was then added to our portfolio, along with a picture, for future reference.

If we have a similar unit coming up, or you just like to do hands on stuff with your class, give this a try and share your story.  We would love to compare with other classes!


The Chessboard

I introduced the Polya Problem solving steps to the class and we talked about how we would implement such a framework.  As a class, we recollected our golf ball math problem, and applied these steps to the learning that took place during that problem.  Once we had a good idea of how it could help us, I introduced a new problem for the day, and asked them to keep this framework in the back of their mind while they worked through it.  At this point in our classroom growth, this is not something that I am going to force on them, but something that I will bring up and talk about after and before we embark on  complex math puzzles.

Today, we had a great problem that I picked up in The Elephant in the Classroom, by Jo Boaler.  The problem is simple, and the introduction is easy.

How many squares on a chessboard?

At first glance, this is easy.  64, you say!  Well, what is a square?  The board is a square.  2x2 squares.  3x3 squares.  And so on.  Also, they overlap!  This simple counting activity just got complex.  To solve it, you need some serious critical thinking, most importantly (and this is why I love it), organization.

Students were making tables, using unix cubes to plot out the center points of each square.  Finding patterns in their tables and making assumptions about what would happen next.  All in all, it was great learning, and a great problem that had they absorbed for almost an entire hour. I actually had to psychically carry one boy out of the classroom for lunch.  He didn't want to give it up!

The power of good problems, problems that are interesting and seemingly simple, is something that cannot be under-estimated in a math class.  This problem seemed so easy, but was so difficult because of the complexity.  This follows the video game model of learning that I read somewhere but cannot recall where.  Kids love video games because they are simple, yet they get incrementally harder as you progress.  Designing math activities that are similar to this model of understanding is something that I intend to play with over this year.

At the end of the problem, we went back to the problem solving framework and reflected on how it helped (or didn't), and how we used it to solve the problem (or didn't use it).  All in all, the framework was successful and got the kids thinking about the how and why of solving problems, but this will need to be something that is done and refereed to very regularly in order for it to become part of the classroom understanding.


George Polya and Mathematical Problem Solving

George Polya was a Hungarian mathematician who penned a book titled How to Get it.  In that book, we comes up with a four-step guide to mathematical problem solving

  1. Understand the problem.

  2. Make a plan.

  3. Carry out the plan.

  4. Look back on your work. 

I am going to try and adopt this to a grade 5/6 class, with level friendly language to help guide them through the process.  Here is what I have so far;

I hope start doing a weekly Problem Solving class, where we work through the steps and train ourselves to think like problem solvers.

I would love some feedback.  What do you think?  Is there anything wrong with what I have up there?  Anything I should add?  Anything that is unclear?  Put your thinking hats on and deconstruct it.


A Golf Ball Mathematics Inquiry

This series of lessons started with a simple premise, but with a deep desire to get my students thinking more mathematically; and reflecting on their thinking.  I have been trying to get more Meta-Cognition (or Meta-Learning, as I frame it with my students) into the classroom.  This was something that I tried to weave into the activity, and not necessarily tack it on at the end as something to think about.

As for the subject of golf, there was no particular reason to choose it.  I am not interested in golf.  My students are not interested in golf.  Their parents are not interested in golf.  The reason I choose it was because I found a bucket of golf balls in our school’s multi-purpose room, and it sparked a question in my head (which I explain below).  Here, I am trying to model inquiry in daily life for my students.  These things make me curious, what makes you curious?

The actual work that was done in this series of lessons took place over several days, but for the sake of distilling it down to the main essence, I will describe it as one continuous lesson (which I wanted to do originally, but too many external forces kept pulling me away from it).

Step One – Building the Environment

I brought in a small bucket of golf balls and let them all touch, feel, and make observations about the balls.  We made a list of characteristics that golf balls have, and tried to get some new vocabulary to use in the problem.  I asked them questions about golf balls and asked them to defend their answers; How many golf balls do you think the average golfer uses in a year?  Imagine a golf hole with a little lake, some dense trees and a gully.  Where would you find a lot of balls if you looked?  How many golf balls do you think a pro golfer would lose in a year?  A lifetime?

The point here is not to use the skills we are going to practice, but rather to embed ourselves in the environment, and to engage with the subject, even if we have no interest in it!  Golf is not something that kids are all that passionate about, but it was quite fun to step into the world of golf and imagine what it would be like.

Also, there are some hidden skills that will be necessary later on; things like making assumptions and backing them up, estimating with reasons, and multiplying out.

Step Two – Setting the Problem

Here I presented the materials we would use, and I set up the problem.

“Here are your tools, an empty shoe box, a bucket of golf balls, and a pile of rulers, paper, and pencils.  Using only these tools, we are going to try and figure out how many golf balls can fit into this classroom, assuming all the furniture has been taken out.  You will be in teams of three, and you may not ask the teacher any questions.  You may, however, ask the other teams in the class as many questions as you wish, but they can only have yes/no answers.  This is a race.  Alright?  Get to it.”

There are a couple of things to note here in the set-up of the activity.  First, the materials were selected to be a clue to one method of doing this problem.  They are meant to offer some sort of support, and a spark to get ideas flowing.  There are multiple methods to solve this problem, and as teachers we should encourage kids to use all the possible methods they can think of.

Second, there are some rules in here that seem odd, but they have a distinct purpose.  Brent Davis at the University of Calgary is a great researcher and educator who is bringing complexity science/thinking to the field of education.  He calls these Enabling Constraints, rules that prohibit students from doing one thing, but in turn, open more possibilities in a different direction.  They purpose of my rules is to prevent children from relying on me (hence, no questions to the teacher) and instead, communicating with each other.  However, I don’t want them to just copy the kids who they think will get it, so by specifying questions that have yes/no answers, I am forcing them to think before they ask, to be specific about their questions, and to focus the point of their questions.

Lastly, the concept of competition in not something I do often in my class, so the addition of a race in something that I experimented with.  My ultimate goal when I put competitive elements into the classroom is to have the students so engrossed in what they are doing that they forget this is a contest, because knowing the answer is more important than winning or losing.  Another reason for competition is too have the students under a certain amount of stress or pressure.  Too much stress negatively impacts performance, and too little stress has the same effect.  Trying to find that balance is what I am looking for, and it something I have not yet figured out in regards to my own practice.  This is just my own experimentation.  I will this say that when I did this project with my class, the idea of contest disappeared and was forgotten.

Step Three – Meta Learning

Before we get to our work, we are going to understand what we are doing, and what skills will be required to accomplish the task.  I always give them this template before we start, and either I fill it in beforehand, or we do it as a group.  The skills and attitudes we focused on during our investigation were:

  • Resilience

  • Teamwork

  • Organization

  • Communication

These are the areas that the students recognized as important to the problem.  I usually try and keep the list down to 2-4 skills that we directly work on or talk about.

After we have set our skills, have the kids talk to their group mates about what these skills look like, and how as a team, they can support each other and help each other improve in these areas.  Share any insightful ideas with the class.

Step Four – The Work

Here is where the students start to investigate and figure out the puzzle.  Some of them will pick up right away what the purpose of the shoebox is.  Others will need some time and will need to watch to get the ideas down.  From the teacher’s perspective, this is a great time to start wandering around and supporting students in their use of meta-learning.  Have them reflect on how they are communicating, what strategies they are using to organize their work, and how they are dealing with adversity (or whatever skills you choose to highlight).  Make notes for the carpet discussion later.

Step Five - Communication

Now that most groups have figured out the methods and are on their way to the answer, tell them that as an added element of math communication, they will be required to design a poster showing all their work and thinking.  It must be so clear, that any stranger could look at and know exactly what the problem was, how we did, and the solution.

Step Six – Math Reflection

Put all the completed posters up somewhere in the class and have a silent gallery walk.  The kids will walk around and take notice of the strategies that other groups used in silent concentration.  After everyone has had a chance to think about each poster, have them sit with a partner (from a different team) and debate methods.  This is where we reflect on our use of mathematics and our mathematical organization.   Which poster was more effective?  Why?  What is easy to understand about it?  What is difficult to understand? What would you do differently next time?  Switch partners as many times as needed.

Step Seven – Meta-Learning Reflection

This would be our final reflection for the activity, relating it back to the skills we discussed at the beginning of the lessons.  Bring the students back to those skills and attitudes and have them with partners talk about how they did.  I also have a large group discussion on the skills where everyone can make observations about themselves or about others.

Finally, the last step is a written reflection.  There are a ton of templates and ideas out there, but one that I tired with this activity was a PMI (Plus, Minus, Idea).  Remember these are related to Meta-Learning skills, not the math work.

Plus – What did I do well?  What am I proud of?

Minus – Where was I not successful?  Where did I struggle?

Idea – What will I do different next time?

An Idea that is Emerging

I don't usually give homework on the weekends (to be truthful, I don't like giving homework at all, but external.... well, you get it).  This weekend though, I changed it up a little and tried something new.  My instructions were incredibly simple, very clear, yet at the same time, incredibly confusing.  In their homework agendas, I asked them to write this simple task:


I had no idea what would happen.  They asked me what I meant, what they should do, can I give more clarification, what does that mean?  I resisted the urge to give ideas, and told them that the task was pretty self explanatory.  Just make something.

Well, Monday morning rolls around I go outside to pick up the kids as they stand in line and wait for me.  All of them, the whole class, is standing there in line with boxes, and shopping bags, and folders, and photos.  They are enthusiastically waving their creations at me trying to get my attention.  Sometimes, the things that children can come up on their own accord just floors me.  Here is how they interpreted my task (this is grade 5/6);

  • A cereal box cut open with origami furniture inside

  • A two foot tall robot made of paper cups and duct tape

  • A shoe box turned into a car, complete with rotating wheels and circuits/batteries inside for the headlights!

  • A stunningly beautiful paper art piece of a woman walking a dog (she said it took her 4 hours)

  • Photos of Moving Light (you open the shutter on the lens and move a flashlight in patterns and then when you develop the photo, the light all appears)

  • A Lego Ship that the child tried to make float with Styrofoam

Wow.  They could have taken the easy way out, drawn a simple picture and be done with it.  But they didn't.  They took time, they got involved in what they were doing, and they were proud of it.  This is something I will certainly try again next weekend.  I am already thinking of how we can display our creations to the school.  I don't want to impose any structure on it, because I don't want to kill the creativity.  I will just let it ride, and see what emerges from the journey.

The best thing about this activity was that the majority of them (not all) said they did the project with one of (or both!) their parents.  As a parent myself, I can think of no better way to spend my weekend than by making something with my child.  It so much more than just a shoe box car, it is a moment spent with loved ones, a memory that will last longer than cardboard.

PS - I would post pictures, but I forgot my camera