Mathematics as Problem Solving, Mathematics as Communication, Mathematics as Reasoning, and Mathematical Connections are critical items throughout the NCTM Standards. They appear at every level because they form the core of what it means to do mathematics.
The questions below are inherent in the map coloring activities. Many of them will occur to students as they are thinking about map coloring. All of them are accessible, even to the youngest students, yet their answers--which are not necessarily singular or simple--and have inspired profound discussions among professional mathematicians for over a century. When students think, talk, and write about these questions, all of the four Standards at the foundation of doing mathematics are reinforced.
How many colors do you think it will take to color this map?
What is the fewest number of colors you can use to color this map? What is the greatest number of colors you could use?
How could you change this map into one that needs fewer colors? More colors?
How do you figure out which colors to use when you are coloring a map?
How can you tell if a map will be easy to color?
What makes a map hard to color?
What advice would you give to someone who was trying to color a map with the fewest number of colors?
Spatial Sense
At all levels, there is an NCTM Standard that talks about the development of Spatial Sense. The meaning of spatial sense that springs to mind most quickly is generally the sense of geometric space, particularly patterns and relationships that have to do geometric figures.
If you experiment only a short time with map coloring you will see that deciding how to color a map using a minimum number of colors involves a type of spatial reasoning which is not strictly geometric and is actually quite complex. Yet any child who can manage a crayon can become engaged in the spatial processes of map coloring.
This is only one kind of non-geometric spatial reasoning which engages mathematicians. Other examples of different types of spatial reasoning are found in the activities with knots, graphs, finite state machines and dominating sets.
