Measure the Blue Bucket

This was how we kicked off our Math class today.  I put the challenge up on the board and walked off to the side of the room and pretended to read a book, letting them deal with the problem as a group.  While they spoke, I took notes on how their ideas evolved.  Here the rough translation of their sequence of thinking:

  • How do we measure the bucket?

  • What unit should we use? (they choose cm)

  • Group worked together to measure around the bucket

  • Great teamwork

  • Got a piece of string and wrapped it around the bucket, cut it, then measured the string

  • Found that the string was 81cm

  • Hold on, that is just around, how about how deep it is?

  • Another piece of string to find the depth of the bucket

  • One student drew a 3D picture of a bucket on the floor and started to write the measurements onto it

  • Found that the bucket was 21 cm deep

  • Next, do 81 x 21 = 1701 (got a calculator for this)

  • Done!

  • What does this mean?  this doesn't look right

  • You can't times the around part by the deep part

  • It is not a square, its round

  • it is cm3?

  • No, it has something to do with pi, the radius and the diameter

  • Student draw a picture of a circle and labels it with pi, radius, and circumference

  • But that is a 2D circle, this bucket is 3D

  • If we times the circumference by the radius we get......

  • I'm confused

  • Wait a minute, What are we measuring?

Full circle.  Back to the initial question.  They started off down the wrong path because they asked the wrong question.  Not how do we measure the bucket, but what do you measure?  I interjected here and asked them to start their conversation over with the question of what.  Within a minute they had a list of ideas or concepts they could measure.  It strayed away from the bucket a little and was more of general measurement, but that is okay.  We wrote them all down on pieces of paper and then sorted these ideas into piles that were similar.  In the end, we ended up with four broad categories

  • Volume

  • Area

  • Weight

  • Length (all measurement of things that are straight lines)

Once we had our list, we grouped off into concepts and we will begin to research and inquire into each one.  Each group will do a different concept, and then bring their learning back to the larger group.  How?  I have no idea, but I'm sure they will come up with something.

And we forgot to actually measure the bucket.


Assessment as a Collective Activity

We often think of assessment as applying to only a singular child.  What that child knows at that brief snapshot in time.  However, our knowledge is not insular, we are a collective species that relies on the group to help us learn.  The collective (which brings up negative images of the Borg and communist Russia) is, I would argue, the most important part of a learning environment.  We learn more from the group than we could ever do on our own, and the sum of our knowledge is far greater than the individual parts.

Collective Assessment Activity

With that in mind, I decided to try a summative assessment that required the group to work together and use each others specific (and different) knowledge and perspectives to arrive at an insight or a conclusion that is bigger than the sum of its own parts.  Here is how I tried it, in groups of 4;

1) Each student gets a different colored pencil

2) Around the group there are 4 stations, and at each station there is a single piece of paper with a different question written on each

3) Each person will have 3 minutes at each station, and will rotate through the 4  stations 3 times each (they will get 9 minutes on each station)

4) At each station, students will answer the question in any way they wish (drawing, graphic organizer, text, poem, etc)

5) When they go to the next station, students are encouraged to add onto the answer of the previous person, continuing their style in answering the question, or trying a different style (disagreeing with the answer is also encouraged) making sure that all the new ideas you put down are not repeated from a previous person

6) At the end, we (as a group) will assess each piece of paper as a single answer

7) Reflect on what the process of working collaboratively (yet individually and silently) felt like

There are many different angles we could look at this assessment from.  I could assess the group understanding, since we spent the whole unit working collaboratively there must be some sense of a collective understanding.  How well did this group understand the concepts as a whole?  How much of their understanding was shared?  How much of individual interpretation did they bring to the concepts we studied?  This could also be an assessment on myself, and makes me look hard at how I presented and introduced ideas in the class.  Where were the gaps?  How could I have filled those gaps?  What part of our study was unclear, and in need of focus?

You will notice that built into the assessment is an exit door, a way for me to still assess each individual student (hence the different colored pencils).  I am looking at their individual contributions to each questions, but it could go beyond content area, and I can also look at how they used the previous information to build on their own knowledge.  Did they adapt to the thinking of the group, or did they take the answer in an individual direction?  Were they influenced by what was written before them?  Did they think critically about what they know?

Student Work

Student Observations from our De-Brief

- It made my rethink what I think about it

- It helped me to see pictures and words together

- Some of us aren't good at drawing, so to have a good drawer make an answer with us gave it more detail than I could do by myself

- I didn't know how I would have answer the question by myself, but when I read somebody elses ideas, it gave me more ideas

- The answer is not clear now and I feel confused

The last point is very interesting.  Most of the students agreed that the answer to the question is now cloudy.  The process of collectively coming to an understanding of the question has left everybody with a more complex view of the answer.  Coming to grips with the fact that not everything is knowable is a difficult concept to understand for adults, let alone for children.  The question; what is civilization? is a worthwhile question to examine, but an impossible question to answer.  Your answer is based on your perspective/perception, what you know, where you are from, how you express yourself, etc, etc, etc.




An Inquiry into Play, Pt. 1

I recently stumbled across an interesting article from a Linked In group that I am a member of.  The articles basic idea is that Hunter-Gatherer societies educated children in a very different manner to how we do it nowadays.  One of the important aspects of this early form of education was a child's sense of play.

This got me thinking.  What is play?  How do children play?  What do children think about play?  Is play an effective pedagogy?  What do the children in my class think of play?

With that in mind, I have decided to start a class inquiry into the nature of play.  I have no idea where this is going to go, or how it is connected to any of my curricular outcomes; but at the very least, I hope it will give me a better insight into the children I have with me on a daily basis, and how I can structure activities that they find interesting and engaging.

I am looking for other teachers around the world who would like to collaborate and try the same thing with their class; understanding that every class with have different dynamics, and will come to very different conclusions.  By sharing our journey with others, we can come to a more in-depth understanding of what it means to play (I don't know), how play affects learning (or does it?), and why it should be an integral part of our education system (or should it?).


Day One

We started our first day of the inquiry with a series of seemingly simple tasks.  However, it proved a lot more difficult that I expected, and the insights we came to more proved to be deeper than I had originally thought.

First, I handed out sticky notes and pencils and asked the students to write any words that came to their minds when they thought of the word play, and then stick them on the whiteboard.  This was simple enough, and soon we had a wall filled with stickies.  I noticed that this was a very quiet activity, as everybody was working independently, but still reading each others notes and getting ideas from the collective.

Second, I asked them (we did all this as a single group because my class is small) to go through their sticky notes and take out the ones that were repetitive, or have similar meanings, but not to throw them out, we want to know how many people wrote down fun or teamwork, just move them off to the side.  This lead to some interesting discussion about the meanings (and personal interpretations) or various words.  What does fun mean?  Is it the same as excited?  What is the difference between teamwork and collaboration?

Finally, I asked them to organize the sticky notes into categories that made sense to them.  This was where the difficulties arose.  The group couldn't really decide what words went together, as everybody had different interpretations of what the words meant in the context of play.  They were at a deadlock, a bit overwhelmed, and didn't know where to start.  Then something interesting happened.  One student made the comment that play doesn't have to be fun, and gave the idea of a video game, and how "you go crazy because you just want to finish the level, and you do it over and over again, and its not fun anymore but you keep doing it."  This transformed how they look at the word play, and the pieces started falling into place.  Their collective definition of the word play changed, and they formed a new definition (although that new definition has not been verbalized yet) that moved their thinking forward.  One of the first categories that emerged was the idea of Feelings.  We invest a lot of ourselves emotionally into play (as any Elementary teacher who has recess duty can attest to!), and this seemed natural to this group, who are no stranger to recess disagreements. Next, they came up with Skills, which I think was influenced by my incessant self reflection and focus on Meta-Cognition (already I am influencing how they view something as individual as play, which makes me wonder just how much a teacher changes the environment of the class).  Finally, as they were attempting to clean up the last bunch of stickies, they decided on Games (aspects related to how they play) and Actions (words related to how they do play).

This was a fascinating hour, and I learned a lot about the kids in my class that I didn't know before.  I also gained some more insights into how a collective of learners (in this case, my collective of learners) work together and perturbate each others thinking and move their ideas forward.

They also had fun doing it (were they playing?).  I have no idea where this is going to go next, but I am excited to find out.


Reminiscing about teachers of the past

My wife and I spent some time reminiscing about teachers we have had in our lives. The detail we can recall is astounding. We have such strong opinions, memories, and characterizations of our past teachers (both positive and negative). Neither of us had any difficulty in remembering any of our teachers, from Kindergarden up to University. I can remember each of one of them. I can remember their strengths, weaknesses, the times in which they helped, and the times in which they did not. I am sure you have similar opinions and thoughts of your former teachers.

Remember this the next time you stand in front of your students. They certainly will.


The Success of Failure

Making mistakes in an important part of life.  I make them daily. I tend to forget how old my students are, and I give them tasks that are way above their level.  I make assumptions about what prior background knowledge they are bringing to a task (Islamic Art is a great topic, but most of the kids don't even know what Islam is).

Today I gave my students (I teach grade 5/6) a very difficult task.  It was a task that puzzled a real Mathematician over 800 years ago.  His name was Fibonacci.  I read about the problem in Jo Boalers amazing The Elephant in the Classroom, and put in on my list to try.  Today, we dove in.

A man puts one pair of Rabbits in a place that is enclosed on all sides by walls. How many pairs of rabbits can be produced from that pair in a year, if it is supposed that every month each pair begets a new pair which from the second month on becomes productive themselves?

The real challenge of this problem, and the reason I choose to do it now, is that there are multiple strands of data that are changing with each iteration (in this case, a new month).  My students knew exactly what they had to do, and they go to work on a table.  I was quite pleased that they knew that a table would be the best way to organize this data.  Next, they experimented with the columns of the table (the rows were the months of the year, another point they noticed that made me smile).  This is where they struggled.  They could not see that there were not 2, but 3 levels of data here.  

A - Rabbits that can have babies

B1 - Rabbits that are just born and can't yet have babies (month 1)

B2 - Rabbits that are a month old and can't yet have babies (Month 2)

They just didn't see it.  I don't usually help with problems like this, I try and get them to do themselves by reviewing their problem solving process, or starting at the beginning, or listing what you know and what you need to know.  This time, I gave cryptic suggestions, hints, clues, and other little nudges.  Still, they didn't see it.  Finally, I sat them down on the carpet for a Math Meeting.  Once again, I tried to nudge them to the answer, but it just wouldn't go.  So, I told them.  I said and wrote on the whiteboard,  "there are three levels of data working here; A -Rabbits that can have babies, B1 - Rabbits that are just born and can't yet have babies (month 1 rabbits), B2 - Rabbits that are a month old and can't yet have babies (Month 2 rabbits).  Does this help?"

A sigh of relief when up from some of the students.  They ran back to their work and started adding third columns to their work.  There was still some confusion in what to do with the numbers after that, put the building blocks were in place, and the problem was staged.  More often than not, this is the most difficult thing about Math, not doing Math, but figuring out what Math to do.

We learned a lot through our mistakes today.


Art as Math Assessment

We are coming to a mid-point in our Geometry unit, and before we switch gears into dimensions, planes and 3D objects, I want to do a summative assessment of the first half of the unit (2D shapes, patterning, lines, angles, etc)

Note to Self: I have divided this unit into two sections, I would argue unnecessarily. The next time I do this unit, I would like the 2D world of Geometry and the 3D world to be intertwined, not segmented into distinct sections of knowledge. I need to marry them and teach them both at the same time, not two them in two seperate chunks. This is important for the kids to see how they are inter-related, and by teaching them as inter-related parts of the whole, we can make connections easier. That being said, shouldn't all mathematics be taught in the same intertwined manner? Why am I only doing Geometry once a year? Why can't I teach all strands and topics (Data Management, Geometry, Number, Probability, Algebra, etc) at the same time, simultaneously, not broken into different units scattered through-out the year? Is that even possible? Or beneficial? So many questions.....

To assess the topics we have covered so far, I have given them this project;

Through this project, they will do five things (and probably more....);

1) Show me (visually) what they know about the topics they have covered

2) Tell me (orally) on Voicethreads the reasoning behind their mathematical decisions

3) Think critically and solve problems (Meta Learning)

4) Work as a team and come to consensus and decisions with your partner (Meta Learning)

5) Use Math in a real and creative way (application and enjoyment)

Here is what they got done after one brief period;

I could have given a test that would have taken them a 40 minute period to complete.  Instead, I have chosen to start a project that, in all likelihood, will take an entire week.  Which way is more effective?  They both have pros and cons, and the answer will always be cloudy and unsure.  However, it is beneficial to have this discussion, with yourself, and other teachers (hopefully from around the world).

Debate appreciated.



Emergent Group Dynamics

I really love the website Math Pickle.  There are so many great resources, activities, videos, etc.  I wish there was more!

Here is the founder of Math Pickle giving a great talk about problem solving. Inspiring stuff.

On of the activities I used to today was their Symmetry search. The kids start with an image like the one below;

Next, they have to find all the lines of Symmetry in the shape (rotational and mirror). These include lines and rotations in each individual piece of the puzzle, but also as pices combine to make bigger shapes that have lines of symmetry and rotation. In short, there are a lot.

The point of the activity for my students today was not to find lines of symmetry. They know what lines of symmetry are. Today was all about organization. How can you keep track of how many lines you have found, and how can you communicate those lines to your classmates clearly? That was the task.

Many of them used tallies, different colored markers, letters to show rotational vs. mirror, etc. One thing that happened during the activity that I found very interesting was how the students self organized with the people around them. I made no reference to group work, nor did I say it wasn't allowed (this is on-purpose). I noticed that the students immediately went to work within the groups with whom they were sitting. However, since we all sit in very close quarters and noise travels quickly in a small space, they realized that they were using similar strategies for keeping track of their numbers with other people in other groups. At this realization, they began to hurriedly move about and readjust the groups to work with people who were using the same strategy.

What is happening here? I believe that they are searching for elements in a group that allowed them to increase their efficiency. They need an environment where their ideas can bump into each other, be challenged, and move forward. When we talked about it afterwards, they said that is was a simple matter of working with somebody who understands them. That is part of it, but there is more to the story, and I will tell in the coming weeks as I continue to hash out the details in my head.

For the time being, if you are interested in the topic, I highly recommend talking a look at some of the articles by Brent Davis (University of Calgary) website. He writes extensively about Complexity Science and Education.