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:

Here is a blogpost that outlines lesson that centers around a discussion of dimensions.

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.  

Area of a Rectangle Model in Grades 3 - 6

This blogpost accompanies some work we're doing in a grade 5 - 6 PLC group.  We've been looking at how the area of a rectangle model progresses in the middle grades.  See below for videos and links to instructional plans.

Progression of Additive/Multiplicative Models:

• Modeling in one dimension (review of grade 1 and 2 additive model) - Mini-lesson on using Cuisenaire Rods to represent addition
• Modeling fractions in one dimension (review of grade 3 additive model) - Fraction Machine lesson plans (adding unit fractions) or Cuisenaire Linear Fraction Models

Tool Building - Counting with Fractions 

The Fraction Machine (Part 1)

The Fraction Machine Part 2

• Modeling in two dimensions (review of grade 3 and 4 multiplicative model) - Mini-unit on the area of a rectangle model for multiplication, and mini-unit on division models and lesson on divisibility rules. (In fifth grade, apply this model to find surface area of rectangular prisms.)
• Modeling fractions in two dimensions (fifth grade) - Mini-unit on multiplying fractions and mixed numbers.  (In fifth grade, apply this model to find surface area of rectangular prisms.) 

• Extend multiplicative model to three dimensions (fifth grade) - explore and generalize volume and surface area of rectangular prisms.
• Addition and subtraction of fractions with unlike denominators (fifth grade) - Instructional plan for adding and subtracting fractions. 
• Division of fractions in two dimension (sixth grade) - Mini-unit on dividing fractions 

Live from EMES - Low Floor High Ceiling Makeovers

This Prezi accompanies a workshop at East Montpelier Elementary School on October 5, 2016.

Pumped Up Math for Grades Pre-K- 2

Meg Dawkins, speech and language pathologist at Berlin Elementary, threw down the gauntlet (but in a sweet way, because she's Meg, and she's really sweet). She's got a garden brimming with vegetables, so she challenged me to come up with some ideas for how to integrate the veggies with lessons that target our WCSU Performance Indicators for math from kindergarten right on up to grade 6.  In this post, I'll share what I've come up with in the primary grades.

Math Reporting Guidelines for Grades 1-6 SY 2016-2017

FALL 2017 NOTE FOR WCSU TEACHERS: Please note that this blog post provided guidance for last year (2016- 2017). It provided a work around with the old report card that used the reporting labels: Exceeds Expectations, Meets Expectations, Progressing, etc. Now that the labels on the report card are changing, the work around described here will no longer be necessary. I will edit this information when I get more information about what the report card will look like.

The purpose of this blog post is to provide guidance for WCSU teachers in grades k-6 on how to communicate and report student achievement in relation to our grade level performance indicators for mathematics.  I’ll provide some background information and then explain how to proceed going forward this school year.  If you have questions or suggestions, please let me know (email me at edorsey@u32.org or comment below), and I will update this post to incorporate your feedback.

Getting Fancy with a LFHC Telescoping Task

Blog readers know that I have a definite "thing" for low floor high ceiling tasks as a way to unite a class and engage all students in high depth problem solving in a heterogeneous classroom (click here if you're not convinced).  Low floor, high ceiling problems are also an excellent means of formative assessment and a way to emphasize math norms that encourage growth mindset in a math classroom.

As we start off the year, we take stock of our students.  What do they do well and what do they need to work on?  We see students at different places, and we wonder how to create a structure that includes, engages and challenges all learners.

In this blog post, I offer up a telescoping task as a low floor, high ceiling instructional tool.  It is linked below, and for lack of a more creative name, I'm calling it: 

"Creating Fancy Duck Tape Problems"

This task could be used for any grade level from second grade (although maybe the number would need to change) up through algebra.  It can be used as a formative assessment to gauge your students' ability to apply math concepts along a progression ranging from place value to additive and multiplicative reasoning to fractions to ratios and percents all the way up to algebraic systems.  Teachers and students can assess based on conceptual mastery as well as use of these math practices:

• CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
• CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
• CCSS.MATH.PRACTICE.MP4 Model with mathematics.

I would launch this task with the whole class after (or maybe while) introducing students to Jo Boaler's "Positive Norms in Math Class" (presented in the Prezi below or you can use the 8.5 by 11 posters linked here). 

I would use math mindsets to frame the task and explicitly focus on these positive norms:

• Norm 3. Questions are really important.
• Norm 4. Math is about creativity and making sense.
• Norm 5. Math is about Connections and Communicating
• Norm 6. Value depth over speed.

To kick off the task, students are given some information about the amounts of (yes, you guessed it) fancy duct tape left on rolls in an art classroom.  The low floor, high ceiling prompt is to ask students what they notice about the data: