LEARN NC

3 Mathematically speaking

Mathematicians have rigorous ways of defining things like maps. As Wolfram MathWorld explains:

A map is a way of associating unique objects to every element in a given set. So a map ƒ:A|->B from A to B is a function ƒ such that for every a ∈ A, there is a unique object ƒ(a) ∈ B.

What does that mean? Say you have a bunch of things — a set, in mathematical terms — and you want to put them someplace or refer to them in a more convenient way. Call that set A. You need to map (verb) each element, or object, in set A to some other element in another set, which we’ll call B. And each object in A can only go one place. So, as Wolfram says, we need to associate unique elements in B with each element in A.

Those of you who liked high school algebra may note that this is the same as the definition of a function. In fact, mathematically speaking, a map and a function are the same thing. For the rest of you, here’s an example.

A classroom example

Consider the students in your class — a set of students. Each student needs a desk — a unique desk; one student can’t sit at two desks, although in a pinch they could share — and so you could map (verb) students to desks by creating a seating chart. Your seating chart would be the map (noun). You didn’t know that when you assigned seats you were writing a mathematical function, but there you go: Algebra in practice!

That’s a simple example. The original data set, your set of students, has just one variable, the student’s identity or name; and the set of desks can simply be numbered. You might say it’s one-dimensional.

Figure 4-1. Mapping students to numbered desks.

StudentDesk
Anastasia11
Billy Bob1
Candi5
Dora3
Elwin8
Fiona9
Grover2
Hargrove4
Ingrid12
Josepha6
Kaga10
Lerlene7

Of course, this is assuming you’ve numbered your seats — you’ve created a second map, by mapping the physical desks in your classroom to the whole numbers 1 to 12. If you actually want your students to be able to find their seats, though, you may want to do more than simply number them. You may want to map your list of students to the two-dimensional space of your classroom. (I say two-dimensional because I assume you don’t have a mezzanine. If you do, hang on, we’ll get to three dimensions later on.)

For simplicity, let’s say you have a simple rectangular arrangement of three rows of four desks each, numbered this way:

Figure 4-2. Numbering your desks by rows and columns: or, mapping a two-dimensional space to a set of whole numbers.1

Desk Row Column
1 1 1
2 1 2
3 1 3
4 1 4
5 2 1
6 2 2
7 2 3
8 2 4
9 3 1
10 3 2
11 3 3
12 3 4

Or, representing this visually:

Figure 4-3. Mapping physical desks to whole numbers (in other words, numbering your desks).

seating chart, numbered

Then you can give students this map of their seating assignments, representing students by their first initials:2

Figure 4-4. Mapping students to physical desks.

seating chart, lettered

Now this is starting to look like what most people think of as a map, which is a two-dimensional representation of physical space. Actually “a representation of a space” is a pretty good way to define a map more generally. A space, mathematically speaking, doesn’t have to be a physical space. It can be a coordinate space, which is the imaginary space defined by the axes of a graph. Remember Cartesian coordinates from high school algebra?

Figure 4-5. Two-dimensional Cartesian coordinate space.

two-dimensional cartesian coordinate space

Here you can see the “space” defined by the x and y axes. But of course we can map a physical space to coordinate space — that’s what we do when we make a map of the world or of a geographical region.

Figure 4-6. The Mercator projection uses a simple Cartesian coordinate grid.

satellite map of world, Mercator projection, with grid

You could add a coordinate grid to your seating chart, too, but that’s probably overkill — unless you were teaching in a lecture hall, in which case it might be useful.

Figure 4-7. Mapping the physical space of your classroom to coordinate space.

Seating chart (lettered) with coordinate axes

All this to tell students how to get to their seats!

  1. Those of you who enjoyed high school algebra may observe that if the desk number n is a function ƒ(r,c) where r=row and c=column, then n=4(r-1)+c. Pointing this out is of no real help in organizing your classroom, but it might be useful if you had a very complicated seating arrangement and needed to convince your principal that it was serious and important. [return]
  2. Note that in my hypothetical classroom, I have only one student per first initial. I recommend this in real life, as well. Students are often perfectly willing to change their names, and if you have more than twenty-six, you can use Greek or Russian names. (You may even be able to get away with calling this "globalizing your curriculum," but I don't recommend it.) [return]