The purpose of this website is to present animated visual models which (we hope!) 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 prehumans 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. Its 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 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 for new mathlets. Please send these to dave@CommonCoreMathlets.com. (Unless you say otherwise in your email I’ll assume that it’s OK to post the body – but not the email address – of your comments and suggestions on this site.)
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 works best on the latest Google Chrome browser because it has 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.
I. Base Ten Counting
II. Base Ten Adding
The fundamental idea of base ten arithmetic can be described as “compounded grouping by tens,” i.e., we attempt to group a collection of objects into groups of ten and then group the groups of ten into groups of groups of ten, which we call “hundreds”, and group the hundreds into groups of ten which we call “thousands” and so on until we can no longer form any more groups of ten. We can model this as a kind of packaging factory. Items move down an assembly line where they are put into packages at a sequence of stations. At the first station individual items are placed into packages that can hold up to ten items. When a package is full it is sent down the line to the second station. At the second station the packages of ten items are put into boxes, each of which can hold up to ten packages of ten. When a box is full it is sent to the next station where boxes are stacked into crates each of which can hold up to ten boxes, and when the crates are full they would be sent to the next station and packed into containers capable of holding up to ten crates and so on. The number of units at each station corresponds to a digit in the base ten representation of the total number of items that have been packaged. The “Grouping” mathlets model the first two stages of this process. The user releases items into the line by clicking one of the buttons next to a pool of individual items. As the items are released a counter keeps track of the total number of items that have been released and counters next to each station track the number of units at each station. The primary observation we hope the student will make of course is the correspondence between the digits in the totals count by the pool and the number of units count by the corresponding station. Of course, this correspondence will be subject to a delay until all currently released items have been packaged. We hope the students will be able to account for this as well. The students may also observe that the effect of adding ten or one hundred individual items to the collection is equivalent to adding one unit directly to the corresponding station. Another observation is how rapidly the number of items making up a unit grows with the position. Imagine how small we would have to make the items in order to fit the thousands units on the page.
A next step in the progression to the base ten numeration system is the realization that for the purpose of counting we can replace the groups with tokens. In the “tokens” mathlet we imagine a first step in which we use different kinds of tokens for different units. In this case, rather than moving a container to the next station when we have ten tokens at a station, we replace the ten tokens with a token for the next unit and move the new token to the next station. (When we learn the base ten addition method we will call this process “carrying.”) We again hope that the student will observe that the net effect of adding ten tokens at one station is equivalent to adding one token to the next station.
Having observed counting a power of ten number of items is equivalent to adding one unit at the appropriate position, we can make our counting machine far more efficient by adding separate pools for the different units.
We can further simplify out counting machine by making the tokens more uniform. We can now easily fit five place values in our screen space.
The base ten clock gives us another way of viewing the base ten numeration system. Instead of a collection of units representing each place value, we have a “clock” labeled with numerals ‘0’ through ‘9’. The hand initially points to 0. The hand advances one position each time a unit is added. The clocks are connected by pulleys with ratios of ten to one so that advancing one clock ten positions, i.e. one complete turn, will cause its neighbor to the left to advance one unit. Thus every ten items will cause the “ones” clock to return to 0 and advance the tens clock one position. Adding ten tens, i.e. one hundred items will cause the ones clock to make ten full turns causing the tens clock to make one full turn causing the hundreds clock to advance one position. The base ten numeral for some number of items can then be read off the clocks from left to right. This representation has a number of virtues. It makes it intuitively clear why adding a power of ten leaves the places at smaller values of ten unchanged. It also raises concepts that we will eventually call multiplication and division along with the concept of fractional change. Eventually this will provide a model for decimal fractions. (Put clocks to the right of the driving wheel.) As noted earlier, this mathlet looks best on browsers, e.g. Google chrome, which support the “dashed line stroke style, because the operation of the pulleys is illustrated.
The “continued counting” mathlets illustrate the concept of addition as sequencing by extending two of the counting machines to become adding machines. As before counting is modeled as releasing tokens into containers which accumulate the total. Clicking “+” resets the counters but retains the accumulators. The accumulators will thus represent the sum of the two counts.
The idea for the one and two place addition mathlets is to show the steps in the base ten addition procedure in conjunction with a model. Each time the mathlet is run a randomly generated problem is presented in both forms. Each click of the addition button shows one “step”, i.e. for one place value, in the process. So the first click shows the one’s place addition with a possible carry. The second show the ten’s place addition, if any, and the third click if necessary shows the hundred’s place addition if any. Between steps the addition button changes color to indicate it’s ready for the next step, so clicks will be ignored until the button has changed color. (I had some problems getting the model and the digit processes synchronized so there may still be a bug.) I wanted to show a one place addition method to illustrate carrying and the need to add an additional place value (column) in cases where the student already knows the expected result so that these steps would not seem so strange in the case of two digit addition.
The associative and commutative law mathlets use models to illustrate these laws. Ideally after observing a number of examples the student would be able to formulate the principle being illustrated and even make an argument based on the model for why it is true
As specified in the Common Core Curriculum the first step towards multiplication is “skip counting”. In this mathlet we use “pulleys” like those used in the Base 10 Clocks mathlet. Where the pulleys in that mathlet were used to reduce the rotation, here the pulleys are run the “opposite” way so that they magnify the rotation. The degree of magnification is determined by the relative size of the pulleys. In this mathlet the student can choose one of 2, 5, or 10. To turn the pulley you click on the “+” button. Each click results in one turn of the magnifying pulley which will result in 2,5, or 10 turns of the output pulley which in turn drives a “robot” along a number line. Hitting the “reset” button turns the pulley in reverse until the Robot is back at the beginning.
One way of looking at multiplication is as “compound skipping” – you skip and then you skip the skips. We can model this by connecting a second pulley to the first. A good way to think of this is to think of a pulley as a machine like a computer that takes “input” and produces “output”. The input to a pulley is a rotation. The output of a pulley is also rotation. Connecting two pulleys is like taking two computers and connecting the output of one to the input of the other. This creates a new machine. The process of connecting the output of one machine to the input of another is called composition. So multiplication is really a form of composition. This mathlet allows the student to experiment with multiplication of values between 1 and 12.