The this I talk about has to be a thing, a phenomenon. Things lead us somewhere. Two and two are four, but why not also say ii & ii = iiii, a thought leading to a picture or an object; or i | ii, then ii | iiii, two or three facts leading to a system?
We pursue linearity, speed, and virtuosity in mathematics to a fault. Those most capable of abstracting information get the brass ring. Yes, a certain number of east and middle European émigrés have changed history with their gigantic strides in math and science. Yes, Alan Turing helped turn the tide in WWII. Yes, NSF scholarship recipients are going on to make our lives better by crafting stents and optimizing airplane travel. But if we make science, engineering, and mathematics about elitism, don’t be surprised if 99 percent of us don’t become alienated from these subjects.
Mathematical thinking, logical thinking, inductive thinking, orthogonal thinking need to be spread like a skin-to-skin virus. I propose to do my part by creating or distributing manipulable things that make people think, cognitive artifacts, that lead them to mathematical ways of thinking that lead other people to mathematical ways of thinking, and so on.
We have actual things that make us think, according to Donald Norman, alphabets and algebra and methods to set up tables. These are every bit as important as things that do things, like knives and watches and plows.
I want to make things like the sector, also known as the proportional compass, using everyday objects. We likely don’t see sectors. No one needs to use them. We use calculators instead. At the dawn of enlightenment they were in the hands of astronomers, surveyors, engineers, and gunners.
The sector, also known as a proportional compass or military compass, was a major calculating instrument in use from the end of the sixteenth century until the nineteenth century. It is an instrument consisting of two rulers of equal length joined by a hinge. A number of scales are inscribed upon the instrument which facilitate various mathematical calculations. It was used for solving problems in proportion, trigonometry, multiplication and division, and for various functions, such as squares and cube roots. Its several scales permitted easy and direct solutions of problems in gunnery, surveying andnavigation. The sector derives its name from the fourth proposition of the sixth book of Euclid, where it is demonstrated that similar triangles have their like sides proportional. It has four parts, two legs with a pivot (the articulation), a quadrant and a clamp (the curved part at the end of the leg) that enables the compass to function as a gunner’s quadrant.
A scale is a this to that. As an adult education teacher I often introduced the concept of the table. The simple act of making a “T” shape on a black board or a piece of paper calmed the problem, at least for me. On the left side of the “T” you had a number, on the right side you have the number transformed, whether by the addition of two or the multiplication of eight or the application of an exponent — a powerful idea that you can put on a stick.
You can go to my Instructable to find out how to create and use a sector. In this case, you can imagine a series of equilateral triangles, the 1s spaced 1 cm apart, the 2s, 2 cm apart, the 3s, 3 cm apart. Using a ruler and a set of dividers, you can begin scaling things. The Science Museum in London had my favorite explanation for using the sector to make proportional measurements.
Suppose you want to divide a line six inches long into five equal parts. Measure off six inches with dividers, and open the sector until the distance between the 5 and 5 on the linear scales is six inches apart. By the principle of similar triangles, the distance between 1 and 1 on the linear scales is then one fifth of the distance of the distance between five and five. Measure the distance 1 to 1 with dividers and the five equal lengths can be measured off the original six inch line.
And that’s just one function. What if you had a bunch of coffee stirrers and a way to easily connect them — a way that’s easier than my current method of poking a hole and adding a fastener? What if you had a specialized housing and a way of printing up coffee stirrers or card stock? Thousands of thisses to thatses in pockets on coffee tables, constant reminders of the comprehensibility of mathematics.
Let’s just call that my theses, my theses of thisses, “This to That for Everyone.”