“Mathlets” for teaching elementary mathematics

Using visual models to help explain elementary mathematics

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, 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 ( 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 

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

·         Grouping by tens

·         Grouping tens and Hundreds

·         Tokens for Groups

·         Counting Powers of 10

·         Uniform Tokens

·         Base-10 Clock

                Addition as Continued Counting

·         Continued Counting I

·         Continued Counting II

            Laws of Addition

·         Commutative Law of Addition

·         Associative Law of Addition

                Train Track Addition

·         Click here to run.

                Base Ten Addition Methods

·         One Place Addition Method

·         Two Place Addition Method

·         Three Place Addition Method

                Base Ten Subtraction Methods

·         One Place Subtraction

·         Two Place Subtraction

·         Three Place Subtraction

·         Three Place Compound Borrowing Subtraction

·         Four Place Subtraction

·         Four Place Compound Borrowing Subtraction

                Skip Counting and Multiplication

·         Skip Counting with Pulleys

·         Multiplication with Pulleys


                Fractions and Fraction Wheels

·         Fraction Wheel

·         Fraction Wheel Addition

·         Fraction Wheel Multiplication

·         Decimal Fraction Wheel

Input/Output Machines  

·         Input/Output (I/O) Machine

·         Connecting Two I/O Machines

·         The Composition Operator

·         Order Matters!

·         Composing Three Machines: Associate Left

·         Composing Three Machines: Associate Right

·         The Associative Law of Composition

·         Identity I/O Machine

·         Properties of the Identity Machine:

o   Left Identity Law

o   Right Identity Law

·         The Inverse Problem

·         The Compound Inverse Problem

·         An Inverse I/O Machine

·         Properties of Inverse I/O Machines

·         Inverse of a Compound I/O Machine.