Papy’s Minicomputer for K-5 Math
Introduction to Papy and his Minicomputer
Georges Papy was a Belgian mathematician and mathematics educator who was a seminal figure in math education of the 20th century. For more of his background, see:
Link to IMACS article about Georges Papy
One of his best inventions for young math students was the Minicomputer, essentially a hands-on calculator, or a two-dimensional abacus that lets young children play with numbers in a concrete and powerful way. Young kids can add, subtract, multiply, divide, and learn different base systems all by moving around counters on a box with 4 different colors (you add boxes to do 10s, 100s, 1000s, and decimals). It is a perfect tool for K-5 students to practice the basics in a deep yet fun way.
Learning to use the Minicomputer
So, what does the Minicomputer look like?
I’ll include some links for online playing with the Minicomputer, but I highly recommend printing out a few sheets of the Minicomputer and using buttons or game pieces to play with it, hands-on is still the best way to learn for young children.
If you have a marker on the white box, that marker has a value of 1. If the marker is in the red box, it has a value of 2. The purple box is 4, and the brown box is 8. Sounds weird, I know, but stay with me. Put 10 buttons in the white/one box. Your value is 10, right? Is there any other way to represent 10 by moving the buttons to the other boxes?
The box on the left is also showing the value of 10 with a marker in the 2 box, and another marker in the 8 box.
The box on the right is also showing the value of 10, with 2 markers in the 4 box, and 2 markers in the ones.
Now, the next step is to put a second printout of the four boxes next to the first one. On this box, the values of the four boxes will be 10, 20, 40, and 80.
This arrangement of markers above shows 80, 20, and 10 in the left hand box (the Tens box), and the Ones box on the right shows 8, 4, and 2. All together, the diagram presents the value of 124. Notice that this isn’t the most efficient representation: you could move the 2 and 8 off the Ones box, and place one marker on the Ten box, and then, since you have two tens, you can replace those two with one marker on the 20 box. Oh, and then you can remove the two 20 markers for one 40 marker – fun!
If you put a third printout next to the second one (so that the ones box is on the far right), it will be 100, 200, 400, and 800. You can add a thousands page, a ten thousand page, and so on.
You can see how easy it would be gradually introduce the higher numbers as a student gets comfortable with the ones, the tens, and so on. It is remarkable tool that allows young children to manipulate big numbers in a concrete way.
And that is the core of the using the Minicomputer, but from very simple rules comes great possibilities, and I was amazed at all the things my boys and I did with the minicomputer. Not only can you go up as high as you like (I think we went up to the ten thousands), you can also put a page on the right hand side of the ones page to work on decimals. I used a large whiteboard to tape my pages onto so we could use magnetic circles for our markers, and to delineate the fact that we were using decimals, I put a strip of colored tape in between the ones page and the one tenth page. Again, the CSMP materials will walk through from simple addition to dividing decimals with their materials (see below).
Papys didn’t use binary to explicitly teach binary thinking to kids, he used it because it is probably the most efficient choice for representing numbers with physical markers, making trades and doing the arithmetic functions. For example, to represent the number 25, you need to put one marker on the 20 (the two box in the 10 column), one marker on 4, and then one marker on the 1 box. Let’s say you want to add the number 49, so you put the markers for 49 down (one on 40, one on 8 and one on 1), and you will quickly see that you would have to do a lot of mental addition to see the answer. However, if you trade your markers so that you use the fewest markers necessary, it is very easy to “read” your answer from left to right. In any case, using a binary representation makes doing trades a lot more efficient than say using base 10 (that would be 10 little boxes in each column – tedious and pretty useless), or base 3 (better, but you would be moving more checkers around).
To finish up, the Minicomputer is only one part of the CSMP math program, but it can be used on its own to enrich and add a lot of fun to a homeschool math program.
More Resources about Papy’s Minicomputer
For another look at how to use the Minicomputer, see this blog post: https://achildsplayorg.wordpress.com/2018/01/22/papys-minicomputer/
For the original CSMP math materials wich contain everything you need to play with the Minicomputer and much, much more: http://stern.buffalostate.edu/CSMPProgram/index1.html
Here are some PDFs from the CSMP currciulum that describes using the Minicomputer, along with starting activities and games:
http://stern.buffalostate.edu/CSMPProgram/String%20Games%20and%20MiniComputer/papy.pdf
Here’s a journal article (from 1972, but still worthwhile) that describes using the Minicomputer in a classroom:
https://www.jstor.org/stable/41188061
By the way, you can make a free account on JStor.org to have access to journal articles in all academic subjects, there are some jewels in there!
Late March, 2023 I removed the colorful header I inserted for this module but wanted to put in a screenshot here:
