Mathematical Realizations

NOTE: S stands for students, it was not just one voice speaking but many different voices

Me: Look at the the two tables, one with a fixed perimeter and the other with a fixed area.  What do you notice?

S: The numbers on the right table are changing.  There is 9 in between, and then 7, and then 5, and then 3, and then 1.

S: Yeah, it is like a pattern.  But then the other side is ummm (calculating) 34, 10, 4, and 2 in between, so there is no pattern.

S: I see something weird.

M: Is it okay we change the subject and look at N's weird thing?

S: Yes.

M: Go ahead please.

S:  when the area is 36, the perimeter is 24, on the left table, and when the area is 24 the perimeter is 36 on the right one.  They are flipped.

Ss: Cool.  Why is that Mr. D?

M: Well, what do you think?  Look hard at the chart, you may see something special about those two shapes.

S:  Oh!  They are squares!  The others are rectangles and they don't flip.

S:  And look at this, in the area table, the square is the smallest number, but in the perimeter table it is the biggest.

M:  Wonderful observation!  Everybody think about that.

S:  (long pause) I don't get it

M: Imagine you are a farmer, and you're building a fence for you pigs.

S:  Well, the square will give us the biggest area, so the pigs get more room.

S:  It also uses the least amount of fence, so it is cheaper and and we can build it faster.

S:  That's why so many buildings are squares!  You get more space and use less wood!


What a great moment!  I had no idea that was coming, I was just simply hoping they would notice that when one value is kept constant, the other can vary.  It is moments like this that makes this job the best job in the world!


Collectivity in the Math Classroom

A presentation for our research proposal.  Looking forward to getting started!

The Learning Curve

I'm pretty adept with technology, but kids amaze me.  Our school recently purchased about 20 iPads to use in the elementary school level.  So, from time to time, I have a 1:1 iPad class.  A couple of them are equipped with iMovie.  I recently started a project where the kids are using the Trailer templates on iMovie to make a trailer about the story of their lives.

iMovie Trailer Storyboard

Once we started filming, I sat the class down and hooked the iPad up to the TV to give a short demo on how to use the basics in iMovie.  A couple of them said they already knew how, they use it at home.  The others brushed my tutorial aside and said they would play with it and figure it out.  So, I let them go, and I helped as we went.

By the end of the second day of shooting, I would say that the majority of the kids in the class had surpassed me, and were teaching me new tricks and shortcuts.  I felt like my parents must have felt when I helped them figure out how to send a link through Skype.

This got me thinking about technology.  I remember reading a great line somewhere (I want to say Prentsky.... but cannot recall) that technology is whatever was invented after you were born.  For me, the internet is technology.  For these kids, it is just part of the world they inhabit.  Like TV was to me.  They are so comfortable with it, like second nature.

I forget that sometimes the best way to learn something is to just play with it.  I need to remind myself of this more often, and I am glad this class is teaching me that lesson again.


Dancing with Systems

I love this.  I love it so much.  I cannot tell you how much this one short piece of writing has influenced me.  I can't remember where I read it, or who recommended it to me, or even when it was.  It was a while ago.  Before I had even played with the idea of becoming a teacher.  It send my mind down avenues I didn't know existed, and introduced me to ideas that are now very much a part of who I am.

http://www.sustainer.org/pubs/Dancing.html (click on the link for the full article)

The Dance

1. Get the beat.
2. Listen to the wisdom of the system.
3. Expose your mental models to the open air.
4. Stay humble. Stay a learner.
5. Honor and protect information.
6. Locate responsibility in the system.
7. Make feedback policies for feedback systems.
8. Pay attention to what is important, not just what is quantifiable.
9. Go for the good of the whole.
10. Expand time horizons.
11. Expand thought horizons.
12. Expand the boundary of caring.
13. Celebrate complexity.
14. Hold fast to the goal of goodness.


Teacher Inquiry vs. Student Inquiry

So, I've got this great unit planned.  I thought of an engaging provocation to hook the learners.  It worked.  They are loving it and they are following every step of the way.  It is engaging, fun, and full of opportunities for critical thinking, art exploration, and differing perspectives.  Along the way, I infused the unit with technology.  They are blogging, making Keynote presentations, and working with GoogleSketch.  I have scheduled in some reflection and meta-cognitive time at regular intervals.  They are putting their work into e-portfolios, and connecting with other students around the world.  I work my way through the initial points of the unit, going through my lines of inquiry one at a time, going back to check understanding, and moving forward when I feel they are ready.  I have structured the unit to scaffold the learners to the final summit, the understanding of the big idea.

But, it was my inquiry.  I planned it all out.  I chose the direction.  I chose the topics.  I selected from a very large field of possibilities, and ended up with a path that I made.  They ended up exactly where I wanted them to.

What about the students inquiry?  Is learning to inquire the most important part of inquiry learning?  How can you learn to inquire if somebody is always doing it for you?

Did I do my job as a teacher? Looking back at my first couple of classes, I would say not always.  Yet, if I could go back and tell myself this, I probably wouldn't want to.  It is something I needed to see, feel and experience on my own.  It was my inquiry, and it will never end.



Part of my personal focus for the upcoming school year is to improve my listening as a teacher.  As part of my summer studies, I was introduced to a taxonomy of listening written by Brent Davis.  His book applies it mostly to the mathematics classroom, but I think it is easily transferred to the all aspects of teaching, learning, and knowing.  I wrote a paper on the taxonomy that was a brief summary of his ideas geared towards the math class, but I would like to rewrite it here and expand on some thoughts to broaden the idea to encompass all of a teachers listening practices.

There are three levels in the listening taxonomy, and each level in nested within the next:

Evaluative Listening

The purpose of listening at this level is to assess what is being studied.  In other words, to check for right or wrong answers, or to find something that the teacher is trying to correct.  We do this all the time in a class.  Can anyone tell me what this is called?  Who can tell me what 6x7 equals?  In this context, the teacher is not listening to the student, but rather, they are listening for an answer that is already in their head.  They are listening for what they are thinking, not what the student is thinking.

It is important.  It is about checking on what was studied, or checking for understanding.  We need to be on the same page, or near the same page, to move forward.  There needs to be some redundancy in our group knowledge if we are going to continue to inquire further.  That being said, it is one part of the listening process, and if it becomes the main focus of the classroom dialogue, then it will limit inquiry, and it will give the teacher a partial (very small) slice of a students comprehension or understanding of a given concept.  It is useful, but it needs to be one of many ears that a teacher has to use.

Also, a teacher listening for something that is already in their head creates a perception of knowledge as being in the hands of the teacher, and access to that knowledge is only accessible through the teacher.  The role of the student is to guess (or know) what the teacher is thinking.

Interpretive Listening

In this level of listening, the teacher is looking for the interpretation that the student is bringing to their answer.  Through a constructivist understanding, the teachers main impetus here is to listen for an embodied history within that child and to make an interpretation regarding their meaning and implicit understandings behind their response.  Questioning and dialogue with students now moves from the right/wrong of evaluative listening, into how/why do you think that.  The main role of the teacher is seeking information.  This is done primarily through more elaborate explanations and demonstrations.  Can you tell me why you think that?  Could you give me another example?  Could you phrase that differently?  In the case of these questions, the teacher is attempting to get inside the child's head to interpret what they are thinking, and why they are seeing it in that way.  Once a teacher understands the rational and thinking behind a student response, they are better able to move them into a more fruitful direction, or to challenge them to think differently.

This opens new space for understanding and learning.  However, for the most part, the interpretive listening is still done purposefully to help a student with a specific concept.  It is limiting in the sense that we are still trying to achieve an end result.  There is a goal.  The teacher is listening in order to guide the student in the right direction (right being the grander understanding in the lesson or unit).  In this sense, it is very much like evaluative listening.  The knowledge still resides in the teacher, and access to that knowledge by a student is still through the teacher.  Instead of the student trying to guess (or know) what the teacher is thinking, they are using their past experience and collaborating to get down the path that the teacher has laid out for them.

Hermeneutic Listening

Hermeneutic listening is all about seeking variation and participating in making meaning.  In this level, the class members and the teacher are jointly exploring a concept and co-developing new ways of seeing the idea, rather than mastering already fragmented pieces of the concepts.  The first two levels of listening treated knowing and knowledge as an attainable thing with a path that was laid out.  Students had to find their way down that path, and the teacher listened in two different ways to get them there.

In the level of hermeneutic listening, it is an altogether different path and a different way of viewing the role of the teacher.  The concept or subject itself is more fluid, the unanticipated plays a major role, and intuition and feeling are noticeably present.  The class is deconstructing a given concept with the hope that they will put it back together in a way that makes sense to them.  The teacher is along for the ride, and is unaware what realizations or new knowings will arise.  Since this is unexplored water and a fluid environment, the teacher moves from being the holder of knowledge, to a co-participant who is also deconstructing their own understanding alongside the learners.

For a teacher to participate in this kind of listening, they must be willing to reconsider and rebuild their own knowledge.  It beckons for a more open way of being with the subject, and with yourself.  The learning might be messy and unstructured, and it might not lead to anything conclusive or insightful, but the fundamental knowledge and understanding is on the table and is being examined and dissected through a diverse set of eyes.  The class is bringing everything they know and is creating something new as a group, not as individuals.  It is an exercise in transforming, reforming, and complicating our individual and collective understanding of what the subject/concept is.

This year, I hope to be more mindful of these layers, and to notice when it is appropriate to move from one level to the next.  Each layer is important to a classroom, and is essential for learning.  Knowing when and how to listen in different ways is something I am going to reflect and pay attention to which much greater care.