The purpose of this website is to present animated visual
models that run in a web browser and help explain concepts and methods from the
elementary math curriculum. (Such models
are sometimes called “mathlets”.) Models
play much the same role in mathematics that images and ideas play in
literature. Unless they are interpreted
in terms of mental images, the words in a book are so much nonsense. Unless a mathematical theory can be
interpreted in terms of models, the theory is essentially pointless. Models are the means by which we *understand *theories and the means by
which we apply those theories to solve problems.

In presenting a theory to a mathematically mature audience one may sometimes choose to present the theory without reference to its models. The audience will usually accept on faith that the instructor wouldn’t be presenting the theory unless it had interesting models and that the instructor will sooner rather than later provide examples of such models. This audience is familiar with the mathematical process. They know how theories are interpreted as models. They understand the process of deductive reasoning and that the results of that process lead to truths about the corresponding models.

But of course we have no such foundation to build on when we
are teaching elementary mathematics. The
problem of the teacher in this context is similar to that of the monolith
attempting to evolve pre-humans to dominate their environment and extend that
environment to outer space in Kubrick’s “*2001
A Space Odyssey”. *The monolith
didn’t have a lot to work with. Its
“students” had limited language skills and little if any concept of abstract
reasoning. The goals were not something
that could be met by training the creatures to perform a collection of
predefined tricks. As depicted so
beautifully in the movie the solution to the problem (see http://www.youtube.com/watch?v=mM6OIlreneA),
was to endow the creatures with the ability to see and manipulate images in
their minds. It is this ability, which
we call “imagination”, that allows us to explore a multitude of possible
solutions to problems before actually committing to one. It also allows us to compare the behaviors of
a variety of systems and discover patterns across those systems. These patterns become the basis for higher
level concepts including the concepts that are the basis for mathematics and
reasoning.

Fortunately our students come pre-endowed with imagination. So our idea is to seed that imagination with images tailored to evoke those patterns that are the basis of the elementary mathematics curriculum. Our hope is that those seeds will grow like the seeds of a crystal as the student observes these patterns in a variety of contexts arising from experience.

This is a long term project. We are beginning at the beginning with the concepts relating to numbers and base ten arithmetic. While the current set of mathlets touch on some of the other Common Core Math Standards (http://www.corestandards.org/Math/Content) the primary focus is on those falling under the “NBT category” (Numbers and Base Ten Operations) for grades K through 3. Even for this modest goal our coverage is far from complete. We will keep plugging away as time and resources permit.

We will be very grateful for any comments and suggestions users may have. We will especially appreciate bug reports (please give as much detail as possible including browser and computing platform), suggestions for improving existing mathlets and ideas for new mathlets. Please send these to dave@CommonCoreMathlets.com.

Though these applets were originally written in Java, I have since converted them to Java Script so that they should run on virtually any browser and platform. The “Base-10 Clock” mathlet and other mathlets that use pulleys and belts work best on browsers like the latest Google Chrome browser that have support for “dashed lines”. Since this feature is now part of the W3 spec the other browsers should provide support soon. The mathlets use a screen area of up to 600 x 600 pixels. Each mathlet has a simple set of behaviors and simple UI with a minimal amount of text. The input is all pointer-based (i.e. mouse or touch) and should be easy to figure out because there aren’t that many buttons to try. (I actually believe that figuring out what a mathlet does and “explaining it to your teacher” would be a useful learning experience.)

Here is a list of categories of mathlets implemented so far with links to the corresponding pages and mathlets. Clicking on the heading for each category takes you to a guide to the mathlets in that category.

**Place Value and Base Ten
Counting**

**Addition as Continued Counting**

· Three Place Compound Borrowing Subtraction

· Four Place Compound Borrowing Subtraction

**Skip Counting and
Multiplication**

· Fraction Wheel Multiplication

·
Composing Three
Machines: Associate Left

·
Composing Three
Machines: Associate Right

·
The Associative
Law of Composition

·
Properties of the Identity Machine:

·
The Compound Inverse
Problem

·
Properties of Inverse
I/O Machines

·
Inverse of a
Compound I/O Machine.