Each operation is described physically.
Execution is made possible either by using Digi blocks, with their special design, or by running an animation that mimics the blocks.
The key challenge is always: PREDICT!
Hypothesis: Child will master the algorithms, without any explanantion.
Perhaps the first product should just address three-digit addition and subtraction, using multicolored blocks. (No zero.) This gets around the difficulty of showing blocks of different orders of magnitude.
Create an animation of an odometer
User clicks on each “thing” in the picture. A mark appears, indicating it’s been counted.
An animated odometer responds, the right-most digit goes up by one.
Note: The transition from 9 to 0 requires a special explanation. Perhaps: Link to the Digi Counter. Perhaps: Make the digits of the odometer multicolored – corresponding to multi-colored Digis.
Challenge: Before clicking on a bunch of things, PREDICT the position of the odometer. Then run the animation and compare.
Animation can be run faster, e.g., to see that clicking on ten things increases the tens’ digit by one.
This teaches the Counting View.
Given a collection of blocks:
For example, a collection of 354 single blocks yields (photo)
3 Purple, 5 Blue, 4 Green.
This simple packing operation establishes a key property of the base-10 number system:
Every discrete quantity can be represented by powers of ten, with no more than nine of each size.
(If there were ten of one size, then they should have been packed to create a block of a larger size.)
The quantity is represented by means of the colored digits 1 – 9.
The digits need not be written in any particular order; there is no need for zero!
Addition and subtraction can be executed by kids without any instruction.
Multi-color Digis constitute a decimal model that is not positional.
It is instructive to understand the logic of the decimal system without being confused by the additional idea of positional notation, which requires the introduction of zero.
The introduction of positional notation is done separately, when moving to unicolored blocks.
From how many “things” to how much “stuff.”
The fundamental step: Choose a unit of stuff; then count how many units.
(This “trick” must be explicitly explained. It allows the use of discrete counting to quantify things that by their nature cannot be counted.)
(Many points have to be addressed: How to extend the base-ten representation; how to extend the operations; emphasize that now all numbers are relative to the chosen unit.)
Easiest to visualize when the stuff is liquid. Can then pour it into unit containers.
Obviously, the last unit container might not be completely full. What then?
For base-ten, it is natural to continue the pattern downwards: introduce increasingly smaller containers, such that ten of a smaller containers fit into one larger container.
Digi Block Math 