Graphing and the coordinate plane

Dan Meyer’s latest post is on an exercise involving using a gridless coordinate plane to place fruit along two dimensions. The goal is a worthy one: To give students the opportunity to explore what the coordinate plane is without getting tied down by its rigid formalism. However, the nature of the exercise highlights that there are at least three ways of seeing the coordinate plane in mathematics:

1. The statistics view: The coordinate plane is a way of demonstrating the distribution of data along two continua. This is how Meyer’s exercise works. The two dimensions are arbitrary and linked to the type of data assessed (in this case, characteristics of fruit). Because there is no numerical scale, the locations are relativistic and touchy-feely, which is fine for an informal introduction to statistics.
2. The algebra view: The coordinate plane is a way of relating the numerical input (x-axis) and output (y-axis) of data. While functions can be represented using a different layout (polar coordinates, for instance), by far the most common, at least during formal education, is the canonical one.
3. The geometry view: The coordinate plane is a way of locating specific geometric objects in space in order to discuss its properties; most often, to describe a geometric object in algebraic terms.

I have spent most of my teaching career as a geometry teacher first, so that’s how I tend to see the grid now. I spent around a decade in market research, so there was a time when my first explanation of the grid would have matched the statistics view. And during most of my high school experience, I was fairly focused on the algebra view.

This leads me to some questions. We could ask about which view is the “best” for mathematical or pedagogical purposes, but the answer to that is fairly obvious: It depends on your usage. The larger questions are: Should we be highlighting the differences or the similarities? and Does teaching one fossilize or prejudice students against the others?

In the statistics view, the placement of the origin is key to an understanding of the data. There is often a valuation placed on “negative” and “positive” in such grids: Anything above or to the right of the origin is “good” and anything to the left or below is “bad.” This isn’t always the case, of course: Political charts usually rely on the traditional left-liberal/right-conservative distinctions, with libertarian/authoritarian typically being the other access. And some other scales have no clear preferred direction, so the selection of direction is arbitrary. But as the XKCD chart linked by Meyer shows, if there is a good/bad distinction, the default is to align with the positive/negative scales of the standard gridded version.

In the geometry view, the placement of the origin is arbitrary. Objects and scales are set to maximize ease of working with the grid, not based on some absolute origin. When discussing the coordinate grid to my geometry students, I point out that, if we’re talking about the relationships between students in the classroom, it makes more sense to place the origin at some easy-to-identify place in the room (say, the door, or the teacher’s desk) than at some distant place such as the principal’s office or the local sandwich shop. But that doesn’t mean that there’s some zero-hood associated with the teacher’s desk; it’s a location that was chosen to make discussion easy, that’s all. In the same way, choosing units for distances between students, it makes more sense to use feet or meters rather than miles or centimeters. But we could create a coordinate grid with centimeters as units, and the local sandwich shop as the origin. It would just mean that we’d be working with very large numbers if we’re plotting out the classroom.

Once we start talking about the arbitrariness of the grid, we can also talk about its bias. This enters the realm of philosophy, and my students tend to shut down at this point. This reinforces to me (confirmation bias, perhaps) that we overemphasize the distinction between “positive” and “negative”. For accountants, the difference between “positive” and “negative” is key. For an abstract mathematician, not so much. Indeed, one thing that bothers me is that we teach that |-4|=|4|=4, with the idea that the 4 inside of |4| is “positive four”. We are sometimes even overt about it: “The absolute value of negative four is four, and the absolute value of four is itself.” No, that’s wrong. The absolute value function takes a signed value and returns an unsigned value. It just happens to be a historical accident (tied to how very long mathematicians took to even accept negative numbers as a thing) that we rarely write the positive sign in front of a signed positive value, so it looks identical to an unsigned value. But, in mathematical reality, 4 is ambiguous. It’s just ambiguous in a way that typically doesn’t matter in practice, only in concept.

When dealing with objects, geometry doesn’t really care about any of this. A line is a line is a line. Two intersecting lines intersect at some angle. If we want to talk about that angle, it’s probably most convenient to orient the grid so the origin is at the vertex and one of the lines is along the x-axis. Geometers even give this a name: Standard position (Heineman’s Plane Trigonometry with Tables, 4th ed [1974], p. 7). If we don’t like how the grid is positioned with regards to the objects, we reposition the grid, not the objects. Algebra speaks of “function families”, but geometry has the option of speaking in terms of functions being moved around and stretched on the grid, or the grid being moved around and stretched under the object.

When statisticians recalibrate the grid to better show particular aspects of data, they’re accused of manipulating the data. And typically, that’s exactly what they’re doing. With statistics, it’s important to know not just how data relates to itself, but also how it relates to some benchmark. Moving the benchmarks changes the meaning of the data.

Humans already have a tendency to place too much emphasis on arbitrary locations on a number line. Midnight is where it is for fairly arbitrary reasons (and time zones have arbitrary lines), and the slicing of a day into 24 hours of 60 minutes of 60 seconds is due to the peccadilloes of a long-dead culture, but people give specific time a lot of meaning. And, again, pure geometry doesn’t really care that much about any of this. The location and orientation of a line only becomes important when we then want to use algebra to describe it.

So on the one hand, I can see an exercise like Pomegraphit being used to emphasize that we choose grid models that are most ideal for the data, not the other way around. That’s a very important detail in geometry, as well as in statistics. The difference is that, in statistics, the arbitrariness is in the meaning of the axes, while in geometry, the arbitrariness is in their location. (Scale can be arbitrary in both, while algebra tends not to care that much about absolute scale at all.)

On the other hand, I am hesitant to further reinforce the notion that the grid’s orientation is meaningful throughout mathematics: It isn’t. I want to help students improve their ability to liberty this bias in geometry (and its application in carpentry, architecture, computer graphics, cartography, and so on). If I were to use a lesson such as Pomegraphit, which I might, I would want to find a way to reinforce that we can rotate the grid without changing the underlying data.