How Do I Love Thee: Embodied Thinking

Embodiment:

Thinking with the body involves a combination of muscle movement and thought.  Sometimes our bodies move in such a natural way that it seems like there is hardly any thought at all.  

The task of finding a way that the body moves to accomplish simple addition problems brought me to the use of fingers.  As a young child, I remember using my fingers to add...but there was always a problem.  I only had ten.  My first graders have this same problem, and while some try to count their toes, there is a better way.  However, if I am being honest, I had no idea that this existed until today.  

Below is a video of a 3 year old girl using "Finger Math." Watch carefully to see if you can identify the pattern in her counting...

If that didn't impress you then here is another video of a young boy using the same strategy...

Embodied Thinking:

Engaging your body in the act of problem solving.  Embodied thought can happen whether you are solving the problem of art, music, or math.  

I'd like to say that I have been using Finger Math in my classroom for years, but I just stumbled upon the idea during my course investigations.  I first noticed the young boy on YouTube, and I thought it was a spoof.  I could barely keep track of the instructor's commands let alone the finger manipulations.  After some more video explanations, and a lot of practicing, I watched his video again and kept track up until he moved into two hands.  The three year old girl is more my speed!

Both videos have commonalities that I have noticed after several viewings.  The finger movements seem to happen naturally while the children are focusing on one command at a time.  When both children are adding or subtracting multiple numbers, they are starting from the number in their hands at all times.  (The three year old girl is given the problem, 3 - 2 + 5.  She makes the number 3 with her pointer, middle and ring fingers, puts down her middle and ring, and then puts up her thumb for a final total of 6.)  She never starts over, repeats a command or even counts her fingers until asked to do so.  Both children in the videos problem solve in this same way with finger math.  They don't seem to be keeping track until they count up the answer once the commands stop. It's as if their finges are moving at their own accord, and their brains catch up to read the answer at the end.

This seriously impacts my topic of addition.  Finger Math is a way to use the body without the limitation of ten finges.  The problem solving method also frees up space by allowing students to keep track of one number at a time and build or take apart that number.  This same strategy works for addition, subtraction, multiplication and division.   Once the strategy is understood, the possibilities are endless and the boundaries of age or grade disappear.  I can't wait to try it out in my classroom!

If you haven't figured out the pattern, here is a video with the concept mapped out.  

If you're STILL not impressed...you may want to keep a calculator on hand!

How Do I Love Thee: Abstracting

Abstracting: 
Addition is made up of several components.  The most obvious being numbers, symbols and words.

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, =)

http://goo.gl/1zKYi 

Others include the specific concepts of combining quantities and items as well as the actual sum or total.    All of the above are abstractions.  Abstracting is the process of narrowing down a topic or idea to a single, simple, important element.  

I will be taking a focused look at the concept of adding, or +.  Now the symbol alone is an abstraction for the idea of addition.  However, I'd like to further abstract the thought.  A new baby in the family is often thought of as a "new addition."  A family grows larger and changes with new life.

Furthermore, addition can be seen with a different take on "addition."  A train can grow and change by the adding on of cars.  This can be realized while waiting for a long train to pass at a train stop.

Abstractions:
Abstractions are representations that hold the same meaning.  The abstraction of a human could be a stick figure.  Both are humans, but range in representation.

I chose to represent the concept of adding through the use of "new additions."  Both images above represent growth and change.  When you add together two numbers like 2 + 3, you will observe a larger number that is different than both parts...5!  The addition of a baby into a family is just that.  A family may start with two people and grow to three.  This particular abstraction can be further represented with the number sentence, 2 + 1 = 3.  Relating the concept of addition to growing and changing families will support student understanding.  Furthermore, number models can be represented to capture student families in multiple ways.  For example, mom and dad (1+1) or my parents and I (2+1) as well as the people who live with me, my grandparents, and my cousins (3 + 2 + 5).

The second representation is another type of "addition."  This form of addition takes shape with the adding on of train cars to make a long extension that passes, one after the other.  The adding on of train cars can be related to the classroom with the adding on of manipulatives.  Boys and girls can use unifix cubes to build number representations and create a concrete visual that supports addition.  For example, 5 red cubes and 7 blue cubes equals 12 cubes altogether.  It appears that a "cube train" has been built with the connecting of colored cubes. The manipulatives demonstrate the same growth and change as the new addition to a family.

These ideas impact my topic of addition while both simplifying and clarifying the overall concept.  Whether you are adding numbers, objects, money or thoughts the process will result in a larger total that is different from its parts.

Below is a link to a company with its own play on abstract addition...
http://identitydesigned.com/addition/

How Do I Love Thee: Patterning

Patterning:
Patterns are one of the first mathematical concepts covered in elementary school due to their presence in most every learning domain.  First grade students are encouraged to seek out and apply patterns to better understand a variety of content including addition.

Counting in itself is addition since you are adding 1 to the first number to arrive at the second number in a number sequence.  If starting at 0, 0+1=1 and so forth.  First grade students master counting by 1s followed by 10s, 5s, and 2s.  This is often called skip counting, otherwise known as adding on, or the beginning stages of multiplication (repeated addition).  These counting patterns can be easily seen on a number grid with the help of a little color...

• Counting by 10s can be connected to a color and students begin to recognize that all of the numbers have commonalities.  There is a 0 in the ones place and the tens place is counting by 1s.  Students can connect this with addition and subtraction.  One hop straight down will add ten to any number, or hoping up will do the opposite, subtraction.  
• Counting by 5s with the addition of color is a back and forth motion on the number grid.  Students are encouraged to notice the overlay of 10s and 5s.  In doing so, students recognize the number pattern of 5, 0, 5, 0 in the ones place with the tens place counting by 1s yet again. 
• Counting by 2s on the number grid is a reaffirmation of place value patterns with additional connections to both the tens and the ones place.  Students also gain a better understanding of even and odd numbers as well as the addition of evens and odds.  
• Adding on with different numbers becomes exciting as students discover new patterns.  They look for new number grid designs that can be found when adding on by 3s, 6s, 12s and so on.  Students also notice what numbers are most frequently used in patterns, and those that are often left uncolored.  

Patterns on the number grid initially help cement learning and provide a resource for students to independently check their work while adding on.  However, this knowledge also provides a short cut that can be used ineffectively if a student is not secure in his or her foundation of number sense.  Those who try to fill out a number grid with strictly the pattern in mind often end up skipping rows and losing their place.  Other students absentmindedly memorize patterns without the understanding of how the patterns are formed by adding on the same numbers over and over again.  Furthermore, these patterns on the number grid will only help those when adding numbers from 1 to 100 (or as high as the number grid will go).  The limitations of the number grid can be harmful if students do not have other foundations in mind such as place value.  Adding together 2,o46 to 15,334 will take quite awhile on a number grid.  These misunderstandings can be dangerous and cause a large amount of reteaching.

Re-patterning: 
Patterns are a repetitive form or plan (Bernstein, Sparks of Genius).  They are discovered when understandings are layered to build new connections.

First grade students are originally guided to discover number patterns with adding on, or skip counting. This usually becomes a rote memory procedure with a sing-song rhythm.  Young children are taught to drag out the numbers before the new group of tens so that they will later recognize these stand out patterns.  When given a number grid, the oral counting is transfered to the written numbers.  This is when students recognize the visual similarities in the numbers they have been singing all along.  New patterns can be found when students color counts by 10s, 5s, and 2s.  This provides students with a visual representation of repeatedly adding on by the same number.  It is important that students take ownership of these patterns so that the number grids do not hold limitations.  Students who know how to count to 100 by 1s, 2s, 5s, 10s, and so forth should be able to count on and on, or backwards for that matter.

Later in first grade the number grid is deconstructed and put back together to support patterns of addition.  The numbers are no longer in order, but connected on a grid by combinations from 1 through 10.  The numbers on the top of the grid can be traced down while the numbers on the left side of the grid can be traced over.  The answer can be found to an addition problem by tracing rows and columns until they intersect at the answer.  (For example, by placing your right hand on the top 2, your left hand on the side 5, and tracing down and over until the lines intersect will bring you to 7.  2+5=7!)  The color coding on this new tool further supports patterns in addition and helps students recognize new ways to combine numbers.  (For example, 9+2=11 but one more over is 12, so 9+3=12.)

http://bte2.wikispaces.com/Addition+Timed+Tests

This impacts the topic of addition and the learning process.  Different students learn in very different ways.  Some students find it useful to organize information with pictures, words, colors, numbers and more.  This new representation of a number grid to support addition is simply another resource that will reach students on a new level of organization and pattern building.  However, it is important that the understanding of patterns within addition continues beyond the visual representation.  Such visual enlightenment should only be used to cement pattern recognition and inspire future pattern seeking as well as pattern creating.