How Do I Love Thee: Modeling and Dimensional Thinking

M.C. Escher

Modeling: 

The purpose of a model is to represent an object or idea that is most often too large or too small to grasp.  Using models creates a common understanding between imagination, visualization, and physical representation.  

Dimensions of Addition:

The topic of addition seems pretty simple when discussion begins in a first grade classroom.  It all comes down to putting numbers together.  However, the broader terms of addition and its uses both inside and outside of the classroom can support student understanding.  

Patterns:

The foundation of first grade is that of patterns.  Students begin recognizing patterns all around and then represent patterns with a variety of resources.  Patterns can be found on the side walk, the bark of a tree, and the colors on a shirt.  Furthermore, students show different types of patterns that can be 2 dimensional and dimensional.  The use of colors, words, and blocks bring out creativity in designing patterns.  Patterns on a paper, patterns they will look...patterns on the playground, patterns in a book.  The topic of patterns directly connects to counting by 1s, 2s, 5s, and 10s on a number grid.  Coloring in a number grid to show different ways to skip count will reveal a variety of patterns and designs that mystify students.  

Subtract:

Skip counting and identifying number patterns on a grid are the early stages of addition as well as subtraction.  Both skills work hand in hand as students learn to count forwards and backwards, up and down, and from side to side.  The formal terms of addition and subtraction are not always enforced, but students apply known patterns to explore counting in all directions.  Problem solving begins to take shape as real life examples are used to give meaning to the use of numbers.  For example, I have 4 balloons in one hand and 3 balloons in the other.  How many balloons are there altogether?  OR the oposite...I have 4 balloons in one hand and 3 balloons popped!  How many balloons are left?

Opposites:

Opposites in math are a natural part of problems solving.  Students are taught that items can be given out and then taken away.  Addition is connected with words like, altogether, more, in all, and both.  All of these words tell a student to combine numbers in order to find the total.  The same can be said for subtraction: take away, minus, left over, and so on.  Over time students become comfortable with the rules that govern problem solving.  This leads to a flexibility between addition and subtraction where students use one and then the other to check their work within a fact family.  For example, 4 + 3 = 7 and 7 - 4 = 3 AND 3 + 4 = 7 AND AGAIN 7 - 3 = 4.  Eventually the boys and girls are adding to subtract and vise versa.  

Graphical Representation:

At first glance, M.C. Escher's artwork does not appear to represent the topic of addition.  This idea came to be after rethinking my topic.  I first began with a simple representation of a growing pattern...counting by 1s and the adding on of manipulatives.  Students do this task in first grade and become shocked and surprised that they have created stairs.  However, these unifix stairs do not fully capture the dimensions of addition.  Students are adding up and taking down, but the fluidity of positive and negative combinations is not represented.  This led me to M.C. Escher's complex piece of art where stairs are growing and changing depending on the individual's perspective.