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The Design of Notational Systems

We were introduced to three notational systems to represent a concept like "the sum of 5 and 4":

  1. + 5 4, called prefix notation or Polish notation
  2. 5 + 4, called infix notation
  3. 5 4 +, called postfix notation or Reverse Polish notation

Prompt

Your goal is to come up with one new — to you! — observation or insight about how these notational systems intersect with the wider environment and how that relates to their design.

Think about where, how, and under what constraints we might use these different notations. What media are we using to express these ideas? What tools are we using to encode them? What tools are we using to read / interpret / evaluate the expressions?

What if you write on paper with pencil or pen? What if you chisel in stone? What if you type on a typewriter? What if you are entering the information into a calculator?

Put yourself in these situations as concretely as you can and play with each notation. For example, write down 20 simple postfix expressions by hand using pencil and paper. Is there any way someone could misinterpret what you wrote? How? Is that same type of misinterpretation more or less likely using infix notation?

How do the different notations fair when entering numbers on a simple, single-line calculator, like a TI-108 calculator, the calculator on your phone, or this online calculator?

Constrain The Investigation

Your job isn't to come up with a grand theory of notational design. The best possible outcome is noticing something new and interesting (to you!) for the first time. Constrain your investigation to maximize the likelihood that this happens.

There are many variables in the whole environment that can vary. Try to pick one variable to vary, fix the rest, and see what you notice.

Some of the variables in the whole environment:

  • The notation under consideration
  • The media in which we encode our expressions (e.g., paper, code, calculator, stone tablet, etc.)
  • The tool(s) we use to encode our expression (e.g., pencil, pen, keyboard, stylus, chisel, etc.)
  • Who or what is interpreting our expressions (e.g., other people, a computer program, a calculator, etc.)
  • When and why we're motivated to write down arithmetical expressions and others are motivated to read them
  • The extent to which we need edit expressions that have already been written down

Try to focus on 1-2 variables. Keep it simple. Go for those variables you can actually experiment with and observe directly in the classroom.

For example, there might be something interesting to say about the design of our notational system and the way it intersects with the economics of textbooks, but that's going to be hard to experiment with. But if you want to think through how easy or hard it is to edit expressions you've written down on a piece of paper, well, that's something you can play with directly.