I recently completed a exploratory research project that examined how learners respond to a difficult but motivating math problem. I did a small case study, which compared adult math experts, advanced math students (one grade 1 and two grade 4), and two normal grade 4 math students. The problem is a famous problem from discrete mathematics and computer science called the map coloring problem: what is the least amount of colors needed to color in any arbitrary map such that no two neighbors (who share a side) have the same color? Neighbors that only share a point can have the same color.
I found that there were not significant differences in strategies between learners. The adult learners were more able to articulate their strategies. On the other hand, young students were much more motivated to work on the problem when they were told that no one had ever found a five color map before. (A five-color map would require 5 colors.)
I examined the five color map attempts of one of the student experts in detail. This student, without prompting, produced over thirty attempted 5 color maps over several months. When I looked at the progression of this map attempts, I found an interested combination of strategies. He combined old strategies with new strategies in different combinations in his attempts. See if you can color the example below with only 4 colors. Further research is needed to: investigate differences between students in this process, further analyze and classify the progression of ideas and knowledge, and determine the factors that make this problem so motivating. One thing is clear: the adaptation of rich mathematical problems is a highly motivating way to promote deep mathematical thinking and conceptual understanding with elementary students.
For more information, see:
http://kidsengineer.com/?p=620 Video of students working the problem
http://kidsengineer.com/?p=612 Study conclusions
http://csunplugged.org/ Web site with other rich computer science lessons for kids