5.1 To answer …
Developing a system of positional notation to represent numbers beyond those we can count with our fingers and toes was a great intellectual feat of mankind. And it took quite some time coming. (Positional notation wasn’t fully adopted in the western world until about thirteenth century.)
And ever since then folk have been acutely aware that ten \(1\)s make \(10\), ten \(10\)s make \(100\), ten \(100\)s make \(1000\), and so on. This is the mechanics of the \(1 \leftarrow 10\) machine.
For many decades (centuries?) mathematics educators and curriculum writers have explicitly modeled these \(1 \leftarrow 10\) mechanics: representing numbers in tables with columns to separate place values (and carrying digits between columns from right to left when needed), or representing numbers with ones, tens, hundreds number blocks (where ten blocks of one kind combine – explode? – to make the next sized block), or by using physical boxes and chips to mimic place value. The dots and boxes model is precisely just this, of course. Nothing new. And of course an abacus too can be seen as a physical model of a \(1 \leftarrow 10\) machine.
As a mathematical thinking person I was fully aware that there is nothing special about the number “ten” in our representational the system: one can work in base two, or base three, or even base one-and-a-half, for example, if desired.
I didn’t think more of this until I learned from Dr. James Propp, while visiting the Boston Math Circle program one Sunday morning (www.themathcircle.org), that there are actually interesting unsolved research problems on the matters of base one-and-a-half. (See question 5 in lesson 3.1). My interest in base arithmetic was piqued and ever since then I have been reflecting not only on the mathematical depth of the base-arithmetic, but also the pedagogical depth of the subject. (I am so grateful to Dr. Propp for turning my attention in this direction.)
The specific imagery of exploding dots comes from the work of German educationalist Arthur Engel in the 1970s. He used it in a “chip firing” model to explain elementary probability to school students. (See Dr Propps fabulous essay on this – and more!) Dr Propp has done a great deal to popularize Engel’s work.
Taking matters from integer bases to fractional bases, irrational bases, negative bases, and abstract bases (base \(x\)), seems a natural thing for mathematical thinkers to want to do. Mathematics is all about playing and messing around with ideas in fun and quirky ways. Even the most “elementary” of topics – place value, for example – can serve as portals for fabulous deep thinking and intrigue. There really is no such thing as being “done” with a mathematics topic.
Intellectual play is joyous, and every topic holds within it little droplets of curiosities to ponder upon, invitations for further thinking, first steps to new paths of thinking and discovery.
Let’s all be open to seeing those joyous droplets!
Dr. Jim Propp’s essay MATHEMATICAL ENCHANTMENTS Polya’s Urn.
Lionel Levine and Yuval Peres’ piece Laplacian Growth, Sand Piles, and Scaling Limits contains (advanced) relevant mathematics to deep analysis of dots machines.
Also, an internet search on the terms “beta expansion” and “non-integer representations” yields a wealth of related content. Also search the term “cluster algebras” for contemporary thinking on issues of polynomial division. (My thanks to Dr. Jim Propp to alerting me to this work.)
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