Leap Year Math

It is always better to teach a concept when it is in a meaningful time and place.  There is a lot of Math around leap years, and this is one question that I gave to my kids today and let them struggle with it.  It took the entire group pooling thier knowledge together to figure it out.

The next chance you get to teach this in a meaningful context will be 2016!

Leap Year Math Problem



Name the Graph

Two quick resources I made last night for graphing and data management.  These relate to identifying the purpose of a graph, and understanding how a title can be a hint to what shape the data will take.  A good assessment piece to see if they get graphs.  There are two copies:

1) I have put titles at the bottom and the students have to match the title to the graph.  There may be more than one possible solution.  Great critical thinking skills required!

Name the Graph

2) the same copy as above, but with the titles taken out.  Now, the students have to think of their own original title for the graph that makes sense based on the data.

Name the Graph Blank


Try them out and share them.


Abstract Poetry

We had an elementary school poetry carousel this morning.  I was planning to do something based on sounds and rhyming fake words (like Dr. Seuss), but last night I changed the whole thing.  I was reading a book about creativity and I came across an interesting anecdote that made me grab my computer and change the whole plan.  It was a story about William S. Burroughs, and how he used to cut words out of the newspaper and arrange them into interesting sounding sentences (David Bowie later copied this strategy).  He would do this when he was out of ideas or suffering from writers block.  It was like abstract art with words.  He would focus not on the meaning of the sentences, but rather he would build words that form an interesting shape, sound, or feeling.  After he was done creating, then he would analyze the words.

I needed to try this.  I had different ages from Kinder up to Grade 6.  Each class had slightly different constraints put on them.  I asked them to look through a pile of old National Geographic and Newsweek magazines I had sitting in the corner of my class and cut out words that they think sounded interesting.  Next, we put all the words into the middle of the group and choose our favorite ones and then tried to put them into some kind of interesting order that sounded interesting or made you feel something.  The only other rule I put it was that each class had to insert one word that is related to the current Unit of study.

It was an interesting morning, and our final product is quite interesting:


Math Drama

Here is an activity that I have just finished over the last couple of days.  Feel free to download the PDF and share it.  You may not use the exact details, but I found the overall progression of the lessons to be well balanced and thought-provoking.

ABC International School Math Debate (PDF Version, if you would like the Pages version, or PPT, email me and I can send it along)

Slide 1 - Welcome to ABC International School

This slide is there to set up the story and give the students some perspective.  We spent a bit of time analyzing the message from the Head of School, and visualizing what the school looks like inside, how it smells, what it sounds like, and what it feels like to be inside.

Slide 2 - Choosing your role

Here we are trying to give the numbers and the math a frame, or a perspective.  Instead of analyzing the data, I want them to be thinking about how it affects their life (or the life of their alter-ego).  Once we worked out the groups and chose the roles in the fairest possible way (Janken!), we spend a bit of time brainstorming our new perspective and trying to think about what who that person is on the inside.  What is their life like?  What are they afraid of?  What do they love to do?  What are their parents like?  We got into role and did some drama to embody the characters.

Slide 3 - The Data

Here is where we did the math we have been practicing.  We are working on making graphs and tables and then reading them to make inferences and conclusions.  First, I give the students a data-set, and then ask them to find a stem and leaf plot (we also ended up making a Box and Whiskers on Geogebra).  I also leave it open with an open-ended question at the bottom in order for them to connect any other methods of analysis that we have encountered, or that they bring to it from their personal lives.

Slide 4 - T-Charts and Inferences

This is where we start to look at the data and use the numbers to make inferences, or draw conclusions.  At this point, I ask them to try and be objective and not choose a side.  That is why the T-Chart is an enabling constraint in this section.  It forces the students to think about from their characters perspective, and from both a positive and negative point of view.  Of course, this is impossible, since we are all biased.  However, the thinking exercise is a strong one.

Slide 5 - Debate

This is where I ask them to choose a side, and make a judgement about what classroom is more successful.  I give them some time to prepare their case, get their main ideas down, expand on their earlier thoughts, and just spend some time thinking about why they think what they do, and why their character or alter-ego thinks this way.  We try and imagine some personal connections that our characters could bring to the debate, since we all embody our history.  I encourage them to admit if they are unsure, and have difficulty choosing a side.  Several students went into the debate with the attitude that they did not know, and their contributions were very powerful for the group.

Slide 6 - Reflection

These questions could be anything really.  You could relate it back to the math.  You could relate it to the thinking strategies that were used.  Anything.  I choose to relate it to the social dynamics, to see if they were aware of any changes that may have occurred in their opinions.  How did the opinions of others change you?

Finally, I asked them to think critically about the practice of giving out grades.  Is this a good way to judge what students learn?  Call this the hidden curriculum of the lesson.  I was wondering if they would pick up on the fact that numbers are sometimes a difficult way (and dehumanizing) of analyzing a situation.  How do you choose a number to represent learning?  Is that number accurate?


How do you visualize inquiry and learning?

I am trying to hash out a couple of ideas in my head right now, so bear with me.

The book I am reading has me in a contemplative mood.  My question is; how do you visualize inquiry?  When you think of inquiry, what image comes to mind?  What is the organization of that image?  What are the implications of that image on teaching, learning, and knowing?  Here are three possible images (among a sea of endless ones):

Image #1 - The linear graph

This image brings to mind an input/output system.  The teacher inputs the information into the student and the student outputs it back in the form of knowing.  We then increase along the line, building knowledge in a straight linear fashion.  It assumes that the previous knowledge has been acquired, and it therefore is known.  Think of how prevalent this is in school; linear curriculum, direct instruction, and top-down management (not to mention schedule blocks, standing in line, desks in rows; if you really wanted to tease apart all of the linear metaphors we use in education, it would take forever).  This type of model also assumes that there is a desired destination at the end of the journey.  Perhaps the line can continue to infinity, but it always going along a straight trajectory.  This type of view is problematic.  Think of a group of students, and each student is expected to travel on the line, and learn the same things in the same way.  Or, perhaps each student has their own line, and each student builds their own knowledge upwards, one piece of information at a time.  My biggest problem with a model like this (and I have many) is that it assumes there is only way path to follow, and one way to know.  That is, the way of the line.  That is the only possibility.

Image #2 - The spiral

This image is a variation on the line.  It curls back over itself (review and relearning and rediscovering) and gradually gets bigger and wider.  Near the bottom of the spiral the partial circle is small, but as the learner grows the circle and the space in-between the edges of the boundary grow, suggesting their is more room to build.  Again, this line could theoretically continue off to infinity, with the space in between ever widening, and the students perspective of the world ever increasing.  Many curriculums are built on this model, as each year there is a gradual increase in the level of difficulty after a brief review of what was studied last year.  Assessment takes on this form as we build formative tasks that finish off in a culminating activity that is designed to widen the circle and encompass all that came before it.

Yet, it is still a line.  It still suggests growth towards a known goal (knowledge?).  It makes the assumption that there is only one possibility, and that possibility moves in this shape.  It is still, at its core, a simple model and simple view of how we know.

Image #3 - The fractal tree

This image is taken from fractal geometry.  It starts with a simple seed, in this case a Y.  There is only one rule to build this image; at the end of every branch of the Y, build another Y.  If the rule is allowed to iterate, we get a picture that looks very much like a tree (using this seed, other seeds may cause other images).  What is the seed?  The person, the idea, the curriculum, the content?  How this relates to inquiry is obvious; as you travel along the branches of inquiry, you are faced with an extraordinary amount of choice. Do I go this way, or that way?  The branches you travel along will bring you to different points, with different perspectives.  Students, and teachers, are free to choose a path that is interesting to them, based on their own perspective of what they are learning and how they are knowing.  This image too can continue into infinity.  Also, in the other two diagrams, what happens if you were to go backwards on the line or the spiral?  Well, it would result you going back on your learning, or your circle getting smaller.  In this case, going backwards should be encouraged, because it opens up more possibilities, and presents different paths and ways of knowing.  Then, if we accept this image as a metaphor for inquiry and learning; what is the role of the teacher?  To help their students see as much of the tree as possible?  To orient students to a particular branch?  To create an environment where there are endless possibilities?


Math Game

This is a fun little math activity I tried today, recommended by a colleague.

Flip the questions upside down and see if they can figure out the pattern.  It helps to tell them how many numbers are in the data set.  For example, the question above has 5 single digit numbers.  Can you figure them out?

We worked through a couple as a group, made a list of helpful strategies, and then we made our own and tried to trick our friends.  Great to see them so engaged and thinking about math for a full two hours.



Geogebra in an Elementary Classroom

Today I was fortunate to have a guest in my class; Sergio from Yokohama International School.  He is a high school math teacher and came to the school to work with our high school teachers on incorporating tecnhology into the math class.  As an Elementary teacher, there was no way I was going to let an opportunity like this slip by, so I made sure he stopped into Grade 5.6 to show us a thing or two.  I have always believed that there is a strong crossover and correlation between the strategies that go on in a dynamic high school classroom and the strategies that are used in an elementary classroom.  If there is a divide in how schools operate on these two levels, it is completely artificial.  We have more in common than you think.

Anyway, I'm getting off topic again.  Back to Sergio and Geogebra.

I know what Geogebra is, mostly through twitter and blogs that I follow, but to be honest with you, programming is something I am afraid of.  I don't know anything about it, and I have never really tried to learn even the basics.  Ask me to build a house, and I will go straight to it; but ask me to program a computer to make a box and whiskers plot, and I will stare at you blankly.  Thankfully, there are great people out there who are willing to sit down with people like me and help us.  Today, I got to sit back and observe my class working with another teacher.  More importantly, I got to be a student myself.

Sergio started his lesson with a simple little activity.  He led them through a discussion about what a reaction time is.  Why is it important?  When is it used by humans?  Animals?  And finally, how can we measure it?  After a brief conversation about the difficulties of controlling variables using something like a pen, or a ruler; we got them to work on their computers on a website that times their reactions to a changing traffic light.  Each child went through with two trials, recorded their data, and then gave it to him.  Lastly, he placed all the data into the spreadsheet in Geogebra.

Next, using a simple formula, he made it all come into a visual diagram, in a box and whiskers plot.


This was my students first exposure to a box and whiskers diagram, so he asked them a bunch of probing questions to see if they could analyze the data.  The beauty of the program is that you can go into the data and manipulate it and see the changes that happen as a result of the transformation.  For example, if we change the highest number from a .56 to a 1.3, they see that the outlying point on the arm will move outwards, and the box in the middle will elongate.  This lead to a series of fascinating discussions.  He led the group through some ranking questions; which is faster?  which is slowest?  which is most consistent?  which is most inconsistent?  and then had them create their boxes using the whiteboard marker on top of the screen; show me a box that was consistently fast but got really unlucky once.

What I love about the whole thing is that it was all based on something that they did, and data that they recorded.  Through all this talk of average, outliers, consistent, data; they were really talking about who has the fastest reflexes in the class.  It was great to hear that the data has a name, rather than just a numerical symbol.

As for myself, I spent the next hour afterwards trying to figure out how to use the programming and how to remake the models he built.  I found some great resources, and build a nice box and whisker diagram, but I will need continued practice.

I think it is a powerful enough tool that I might just take the time to learn it.

On a side note; it was interesting to see how my class interacted with another teacher that they had just met.  They were very shy, and different.  Sergio had a great demeanor in the class and makes them all feel comfortable and they laughed together, but they were hesitant to make mistakes and give an incorrect answer, while with me, they will always try.  There is no hesitation.

A powerful reflection on the coherence of a classroom community, and how hard it is to find yourself embedded in a new collective.


Repetitive Drills (with context)

I must admit, had my students doing some drills today.  They were combing through data finding the mean, mode, median, and range of different data-sets.  It was boring, tedious work.  However, they were engaged with the process.  Some background is necessary.

About two weeks ago we spent an entire day with Hot Wheels tracks, taking measurements about how the weight of a car affects the time it takes to go down the track.  We had three cars, and tested each car with four different weights.  Each car did ten trials (for those of you counting, that is 120 trips down the Hot Wheels track).  It was fun for the first hour, setting up the track, controlling our variables and then watching the car race down.  Then, it got dull.  We transformed into robotic bodies, meticulously dropping toy cars down a toy track.  However, at the end of the day, we had a ton of data to work with and, as a colleague of mine told my students, science isn't all ham and plaques, you know.  Even after the drudgery, their was a real sense of accomplishment and ownership.

Next, we took to graphing this data and then analyzing it.  We did several hands-on lessons on what a graph is, how do I make one, and what do they tell me.  After that, we jumped into terms like mode, median, range, and mean.  We explored those, what they are used for, what they tell us, why are they useful (or not), and how do I find them.  Finally, weeks later, we got back to our original data sets and went through them finding these numbers.

Yes, it was repetitive practice, but it was rooted in a history that they appreciated and that they felt connected to.


Fostering Emerging Ideas

Background:  We are studying forces and motion in our recent science, math, language unit.  As part of a summative assessment, I told the kids on the first day of the unit that they would be throwing an egg of the second floor of the school and trying to create a device (using their knowledge of forces and motion) to make their egg survive the fall.  I wanted them to keep this in the back of their mind as we studied different forces during the unit.  We are keeping track of our learning and reflections on a class wiki.

Emergence:  The other day I forgot to write our nightly ten minute writing exercise in their agenda.  Off the top of my head I said, “Go onto your personal page on the wiki and write about your egg.  Give it a name and tell me something about its history.”  The next day, they came back to school and one of them had drawn a picture and posted it on the wiki, another had written a short story, and another had done a biography.  During morning Tea, they were discussing their egg stories and they started linking them together, having their stories and biographies intertwine.  I told them that they could easily link it on the wiki and they showed them how to link one page to another.  Next thing I know, they had mapped out an entire universe on the whiteboard; they were from Egglandia, Eggartica and Eggistan; all different countries within the world.  They had links to TES (Tohoku Egg School) and their teacher was Mr. Dwegg (that’s me, and I will also probably write a page and stories about this character, joining the group).  They are linked to a human from the human world through some organization called the Egg-Human Friendship Group; and their humans friends are themselves.  It goes on.

I want to encourage this activity and let it grow, but I don’t want to get to involved.  I don’t want it to become school work, I want it to stay as their own thing, but this is so rich with opportunities to really target some of the aspects of their writing that they need work with.  How do I approach this?  Any ideas?

From a system perspective, I find this fascinating when this happens.  Ideas are born out of other ideas and they grow and evolve and change and eventually they take on a life of their own, going above the group and becoming embodied in something larger that the sum of our students.  It is amazing how brains, groups, and cultures all work together to create something new.


googledocs and the collective mind


This is an activity that I have done several times this year with my students, and I recently tried it with a group of adults (my cohort at the University of Calgary, and here is the PDF (EducationalShiftsinThinking) of the document we created as a group).  Every time I try, it is a fascinating experience and some interesting learning and observations come out of it.  I will describe the activity and then reflect on some of the more interesting aspects of it.

  • Choose a question of concept that you wish to brainstorm and gather information on

  • Make a googledoc and set it to public, anyone can edit

  • Put your question up

  • Have all the participants access the doc at the same time and start editing at the same time (the more the better; I have done it with as low as 4, and as high as 23)

  • Ask that they edit in silence and from their own ideas and experiences

  • Discuss the content of the document when finished  (maybe a synthesizing activity)

  • Reflect on what the activity felt like, and how the group contributed to the whole

What I find the most interesting is how this a great conversation starter for an interesting question; where do ideas come from?  During the rush of editing the doc (and when there are about 20 people editing a document at the same time, it is a rush!), ideas are born from other ideas and adjust our thinking forward.  We become, as it were, assimilated into the mind of the group.  The final result of this is a set of ideas that no one individual could have ever come up with, but also that no other group could have created.  It is unique to each learner, but it also supersedes the individual and creates a dynamic group system that is learning, sharing, and growing together.  It really is one of the most fascinating activities I have ever done, and resulted in some great dialogue and discussion.

If you try this, please let me know how it goes.  I am very interested in this idea of collective mind, and this is the closest I have ever seen it in action.


A Year is Not Enough

Why do students only stay with us for one year (sometimes two)?  Why can't we be responsible for them for 4,5, or 6 years?  Would be the benefits of this?  What would be the disadvantages?


Planning for emergence

I have been slowly moving to a teaching style where I hardly plan any learning experiences or engagements (sometimes I get ideas that are just so good I have to try it!).  My lessons usually start with a basic idea, a brainstorming session or a gathering of information, and then I wait and see what happens to it.  The ultimate learning experience is up to the students, or is shaped by the environment around us.  The activity emerges out of the information and the group.

Example: Today, we were making a list of sequencing words (after, then, before, next, etc) and then a looking up synonyms of common words used in procedural writing (take, make, get, put).  We put them all up on the class whiteboard.  I had no idea what we were going to do with this, but I was confident that something would come up.  About half-way through the gathering of data, a colleague wandered into my room while we were writing synonyms for take.  She helped us with a few more, and then said, "have you ever written crazy recipes?  With strange ingredients and funny verbs?".  The whole class got immediately excited and the mood changed.  I had my learning task.

This style works for me.  It keeps me on my toes.  I find the uncertainty and chaos of the unknown to be incredibly liberating, and strangely comfortable.

Thank you but no thank you

A great post over at the website "Letters of Note", which is a fantastic website for those who don't know it.  This letter was from Nick Cave to MTV, respectfully asking them to take his name of the ballot for Best Male Artist.  He says:

"I feel that it's necessary for me to request that my nomination for best male artist be withdrawn and furthermore any awards or nominations for such awards that may arise in later years be presented to those who feel more comfortable with the competitive nature of these award ceremonies. I myself, do not. I have always been of the opinion that my music is unique and individual and exists beyond the realms inhabited by those who would reduce things to mere measuring. I am in competition with no-one."

I love this.  Let's see what it looks like when we change a couple of the words.

I feel that it's necessary for me to request that my nomination for the student achievement award be withdrawn and furthermore any awards or nominations for such awards that may arise in later years be presented to those who feel more comfortable with the competitive nature of these award ceremonies. I myself, do not. I have always been of the opinion that my learning is unique and individual and exists beyond the realms inhabited by those who would reduce things to mere measuring. I am in competition with no-one.




Teaching Roles

I am trying something new.  Today in class I explained an activity that I wanted my class to do.  After the activity, one of the students asked me what I would be doing during the activity.  I responded with, "observing, and then giving advice at the end".  This got me thinking of the explicit role of the teacher in the lesson.  Of course, we float between different roles at different times under different circumstances.  As teaching is a complex phenomenon, we are playing roles at the same time while balancing many different ways of thinking.  My kids are used to me playing many different roles, though we don't articulate what I am doing in regards to the learning environment.  The reflection and meta-skills tend to be focused on what they are doing, rightly so.  Yet, at the same time, I am a part of the system (collective), and maybe that explicit understanding could be used to make new insights.

What if the teachers role was determined before the lesson began?  What if the students determined the teachers role?  Would this lead students to interact differently with the problem/activity?  Would an awareness of my role help them to understand how their learning system operates?  How they operate as learners? Could this be used to highlight the different ways that groups work together, and how we learn from and with groups?

During lunch I made a quick poster with only a couple of roles on it.  There are tons more, and please feel free to leave them in the comments below.  This list will grow with the concept.  I intend to use this to point out before an activity what my explicit role will be, and then to have the students reflect on how they feel the environment changes with my roles (not all activities, just once in a while).  At first, I will choose the role, but I am hoping that over time they take ownership and begin to start choosing roles based on their needs and levels of understanding.

Any thoughts?