Exploding Dots, Spy Codes and Minicomputers

When I was in fourth and fifth grade, my school used a math curriculum called “Comprehensive School Math Program” (CSMP). CSMP is one of the infamous “New Math” curricula developed in the 1960s. I expect I was one of the last few classes to use this curriculum.

I had occasion to revisit the CSMP materials after becoming fascinated by the phenomenon of Exploding Dots. In particular, I was struck by its similarity to some of the representations or “languages” of CSMP.

There are two points of convergence between Exploding Dots and CSMP: the abacus and the minicomputer. (It appears there is not a direct lineage according to statements by James Tanton on Twitter). The abacus is essentially isomorphic to Exploding Dots, while the minicomputer is related, but used much more thoroughly in CSMP.

The first is a representation in CSMP called the “abacus,” which comes in different forms (bases). For example, this is a task from the first semester of fifth grade:

This task shows addition of fractions with unlike denominators. At the time of writing, while Exploding Dots contains a decimal experience, and there is a discussion of division on other machines, no equivalent to the above task seems to be included. The process of “trading” or “exploding/unexploding” is equivalent in the two systems. The abacus uses a grey bar instead of a point to separate the ones from the negative powered place values. This has the advantage of not needing the word “decimal,” for numbers in other bases, which is a bit of a contradiction. (Dozenal advocates use the semicolon as the separator to distinguish base 12 numerals)

CSMP introduces the ternary abacus with a very Cold War story about spies and sending secret codes.

This script has an interesting interpretation of base 3 numbers: as functions from a finite set to a set of 3 elements. The “encoding” of the function is essentially the conversion process from base 3 to base 10, while “decoding” is the reverse.

The second representation that has similarities to Exploding Dots is the minicomputer. This is actually the representation that I recalled from my elementary days, as it was used consistently in the curriculum, whereas abacuses are more occasional.

The Papy Minicomputer is a chimera: a blend of base 2 and base 10. It is often introduced using the Cuisinaire rod colors. Checkers on each box are worth that value.

A single minicomputer is base 2. But after you continue, the values increase as base 10. The values of the second set of boxes is 10,20,40,80.

For example, the value of the above on the minicomputer is 800 + 20 + 4 + 1 = 825

The interesting thing about the minicomputer is that there are multiple correct ways of representing a single number, without using multiple checkers on a space. Similar to “antidots” in Exploding Dots, there exist negative checkers (notated with the “hat” symbol). Minicomputers also allow checkers with different values.

The archive of CSMP materials can be found at http://stern.buffalostate.edu/ Of particular interest is the videos of Frédérique Papy teaching children using the minicomputer and other languages of CSMP, found on this page: http://stern.buffalostate.edu/Movies/index.html