I recently spoke about the challenges of teaching math with a good friend from college who is now a computer science professor at Yale University. I have fond memories of our freshman year sitting at a pizza joint doing math on napkins. Our lives took completely different directions, although we both got advanced, math-related degrees. He stayed in academia, where he has been very successful, and I went to work in Wall Street. We kept in touch throughout the years, and we now find ourselves with young daughters at similar stages of development. Our conversation focused on how to to spark interest in math and develop abstraction using topics accessible, yet absent from the traditional, elementary school curriculum. He said he had been trying a bit of graph theory with his daughter. This sparked my interest. While I never liked discrete mathematics, I now realize that it is probably the most accessible area in the discipline. Graphs are simple to explain, and, most importantly, many opened problems in the subject could be explained to a smart six or seven year old.
Let me give you an example. Show your kid the following star. Ask your kid to figure out if it is possible to draw the star without lifting the pencil or going over the same line twice.
Obviously, it is possible to draw a five-point start in the manner described above. That's not the point. Look at the following star-shaped graph.
Have you ever heard about the traveling salesman problem? The problem aims to find the shortest tour of a group of cities without repetition. Clearly, this is equivalent to draw the star without lifting the pencil or going over the same line twice. In other words, you are introducing your kid to a major area of graph theoretic research by asking how to solve a cute, little drawing problem.
Here are two additional graphs to try. Determine if it is possible to draw the following two figures without lifting your pencil or going over the same path twice.
If a solution is found, ask if there is a shorter way to draw the figure. Depending on the maturity of the child, you may draw little circles at the corners of the diagram and then explain that this is really a graph that can be used to represent lots of real-world problems such as how to travel to a bunch of cities along given roads in the least amount of time. Whether or not you choose to tell your kid that he is working with graphs and what they mean is irrelevant. The key here is to get the kid to think about optimal ways to draw these shapes. At the end of the day, you are teaching your child to think about optimal algorithms. Who cares if you achieve this by drawing silly figures or proper graphs?