Friday, December 23, 2016

Fourth Dimension 101 for Three Dimensional Beings

One of my favorite books of all time is Flatland: A Romance of Many Dimensions published in 1884 by Edwin Abbot.  It's nerdy and cheeky and thought provoking, or at least it was for me when I first read it as an undergraduate.  Abbott was making a commentary on the social hierarchy in Victorian England, but he used the idea of spatial dimensions to do it.  Different social classes reside in different dimensions: Lineland is one dimensional, Flatland is two dimensional, and Sphereland is three dimensional.  One of the main themes is that perception depends on perspective - a central tenet in critical theory (when applied to social sciences) and relativity (when applied to theoretical physics).  It was an idea ahead of its time.

Abbott begins his narrative from the perspective of a two-dimensional being (A Square).  The dimension in which a character resides determines perspective.  For example, the two-dimensional being watching a sphere pass through their plane would perceive it as a line appearing suddenly, increasing in length (quickly and then slowly), and then decreasing in length (slowly and then quickly) before disappearing altogether.  Whereas, a three-dimensional being would perceive the same event quite differently.

Imagine how confusing the conversation between these the two beings would be.  Which perspective is "true?"  Both?  Neither?  Place yourself as a mediator between the two.  How would you get them to understand one another?  What language would you use?  What visual models?  What experiences would they need?  What dispositions would they need?  Math Practice 3 - Construct viable arguments and critique the reasoning of others comes to mind...  The interchange of perspectives between the two would stretch the perception of both beings.

How does this connect to teaching and learning?  How often have you asked a student a question and been completely confused by their response?  I have come to understand that students always respond in a way that makes sense - to them.  I can't think of it as me being right and them being wrong.  I am right from the standpoint of my own perspective.  In order to teach them my perspective, I need to be open.  I need to understand their perspective first.  I need to know how do they see it?  What sense are they making of it?  What experiences are they drawing from?  How do I provide them with opportunities to widen their perspective?  I've come to realize that simply asking, What do you see? and then following up with How do you know? is often a great, low floor, high ceiling starting point.

Just for kicks, ask your students (whatever their age) for their perspective on mathematics sometime.  What is math all about?  I've done this many times with students at various grade levels.  Most often they say that math is about answering questions involving numbers and equations.  They might mention the importance of efficiency, too.  They might mention the importance of "right" and "wrong."  They tend to focus in on the quantitative, closed-ended aspects of math rather than seeing creative and exploratory aspects of math.  Math is about discovering, describing, generalizing, and explaining patterns in structure, quantity, space, and change.  Efficiency is an important perspective, but it's not the only perspective.

There is a myriad of dimensions to mathematics (literally), each offering a unique perspective.  Dimensions are (and have always been) elemental to visualizing and conceptualizing math.  By exploring dimensionality, we can make concepts and patterns concrete and then extend from what we can experience to the realm of abstract ideas.  Dimensions provide us with a set of tools (hand tools, perhaps) that we can use to build understanding and communicate perspective.

So, what could the fourth dimension look like?  Is there only one answer?  What do you see?

Want more?

Here's a video about Flatland that I like:





Tuesday, December 20, 2016

A Peek in the Classroom - Stacey's Multi-dimensional Class

One of the things I love the most about my job is having opportunities to watch others teaching.  When we have done lesson studies in WCSU, it is one of the most valuable things teachers take away from that experience... the opportunity to see another teacher in action.

I am so, so, so grateful to Stacey Rupp (5th and 6th grade teacher at Calais Elementary) for being willing to allow me to share a peek in her fifth grade math classroom with you.  Rather than making one long video (which would make my iphone gag while uploading to youtube), I created a playlist of the videos starting with my introduction video (the first video below, the whole playlist is linked here).


Watch Stacey tackle:
  • A classroom management strategy to get kids to stop playing with those darn Cuisenaire rods when they're supposed to be learning math
  • Explaining dimensions
  • Efficient movement breaks
  • Using learning targets and ongoing formative assessment 

The first two videos give you an overview of what you will see, giving you the context for the lesson that I observed on Friday, December 16, 2016.