Digital Manipulatives

Digital Manipulatives

They are the 21st century version of tokens, a computing tradition older than cuniform writing. Digital manipulatives permit one to move pictures on a spread sheet as one might move tokens on a board. They also permit one to neatly record the work, make comments, and create.
Open a spread sheet. Right click on an image at the left. Copy it and paste it in the spread sheet. Repeat the process for the other image. You've just created your first digital manipulatives, a set of negative and positive tiles for signed number computation.
Distinct from paper and pencil mathematics, different from concrete manipulative mathematics, this is about digital expression, represention, manipulation (MOVEMENT!), and CREATION.

Click, Drag, Select, Copy, Paste, & A Bit of Experimentation
Nothing moves unless the user makes it move. In some cases the digital manipulative spread sheet contains only tokens, pictures to be moved. In other cases it has instructions and "hot" cells which self-compute and enhance the learning environment. In some cases, the digital manipulative is ready for a teacher to present class-wide instruction. In some cases, it is simply a game or an abacus.
Faciltiy with basic computer commands (Azzolino, 2009) of select, click, drag, select, copy, and paste speed the movement of the pieces and may easily be acquired while performing digital manipulations.

Spreadsheet required.	Click. Select.	Delete or Copy.
Edit, create text, assign task.	DRAG & MOVE.	Complete task.
Summarize, document.	Insert automatic computation.

It is not possible for the "digital hand" to achieve the speed of the hand which moves a concrete manipulative, especially with a thing such as an abacus. Use of these commands does greatly increases speed of manipulation.
Consider a quick select/copy/paste of the graphic at the right into the digital manipulative spread sheet until your speed increases through practice.
Newer versions of Excel® have spinning and line drawing capabilities. Even old versions have stretching and shrinking capabilites (Azzolino, 2009).
Digital Manipulatives as a Math Class Language
"Math Class Languages" (Azzolino, 2009) is the theory that forced the creation of digital manipulatives. In brief ...

The Languages of the Math Classroom
© '98, '08, '09 Agnes Azzolino

MOTHER TONGUE & OTHER TONGUE(S)
Most Sophisticated and also the Most Basic

MOSTLY MATH TONGUES
Most Sophisticated, Most Basic

VERBAL / Auditory

WRITTEN / Symbolic

PICTORIAL / Visual

CONCRETE / Kinesthetic

formal spoken mathematics

written word

DIGITAL MANIPULATIVE

object

informal spoken mathematics

written symbol

moving picture

model

spoken symbol

semisymbolic

static picture

manipulative/token

symbol speak

calculator symbol

numeral

calculatoreze/computereze

graph

web speak

nonverbal body language

Digital manipulatives were created to:

Introduce in the concrete and debrief in the abstract.
Increase the number of clear, valuable, graphics to communicate an idea in class.
Increase visualization as a form of presentation, summary, debriefing.
Create math experts.
Restate a mathematical idea in as many math class languages as possible for increased understanding on the part of both the speaker and the listner.
Bring a cart load of manipulatives to class without having to bring a cart load of manipulatives to class.

Because of them teachers/students might:

Write, illustrate, print a worksheet or classwork.
Complete a lab, play a game, "use an abacus."
Take and store notes.
Create something beautiful.
Make a picture with tokens and write text on some mathematical aspect of the picture.
Create tests, quizzes, or projects.
Complete homework or teacher-written assignments.

Come play. Come create.

Digital manipulatives easily permit movement of pictures. Abstract ideas and symbols become almost concrete. One might:

Divide the circle and generate a parallelogram-like shape to compute its area using areaf.xls.
Verify a 1/4-bar with another 1/4-bar have the same length as a 1/2-bar using fract.xls.
Find a fraction greater than 1/4 and smaller than 1/2 and verify through decimal approximation using fract.xls.
Make multiples of x+2 to represent 2(x+2) and x(x+2) and (x+2)² using create.xls.
Simplify algebraic expressions and solve equations using tiles and tokens using create.xls.
Expand (x-y)³ using tiles.xls.
Play addition rummy with a deck of cards using deck.xls.
"Roll" a die or pair of dice using dice.xls.
Add and subtract fractions, decimals, integers, on a nomograph using nomogrf.xls.
Add and subtract fractions with multiple strips and fraction bars using strips.xls.

Complete vector addition using polrect.xls.
Reinforce mental computation and words like multiple, reciprocal, cube, double, and prime using 42.xls.

Download digital manipulatives at www.mathnstuff.com/math/xls/xls.htm#digitalm.

References

Azzolino, 2009, "DIGITAL MANIPULATIVES" at

www.mathnstuff.com/math/xls/xls.htm#digitalm

Azzolino, 2009, "DIGITAL MANIPULATIVES" at

www.mathnstuff.com/math/xls/digital.htm

Azzolino, 2009, "MATH SPOKEN HERE! Classes - Taking Notes" at

www.mathnstuff.com/math/spoken/here/2class/cbreak.htm

Azzolino, 2009, "MSH! DIGITAL MANIPULATIVE -- Use and Creation Spread Sheet Pg. b" at

www.mathnstuff.com/math/spoken/here/2class/cspredb.htm

Azzolino, 2009, "The Languages of the Math Classroom" at

mathnstuff.com/papers/langu/page0.htm