My son and I were watching Sesame Street, and an (updated) version of the The Ladybug Picnic. The ladybugs appear one at a time, with three on each of four sides of a picnic blanket. My son (who is 5, and about to enter Kindergarten) says “Did you ever notice that four threes is twelve?” After he did this he counted it out on his fingers to confirm it and proved to me (and himself) that he was right.
My son has no formal education in multiplication. He doesn’t know anything about times tables, and would not know what an “x” between two numbers (or a dot, or any other symbol) means. He was given a set of data (in this case an image of four sets of three ladybugs), made an observation and double checked his conclusion.
I was delighted by his observation because (1) I am a proud papa and (2) it exemplifies my own teaching. I teach by having students work in small groups of three or four on inquiry activities that I design. Each activity starts with data − a graph, a table, an animation, or a set of molecular models to play with. Each day in class, my students talk, argue, and discuss the meaning of that data, and develop a better understanding of the material than they would if I just told them about it. It was affirming and exciting, then, to see my son make the same sort of connection I help my students make in chemistry. If you give students the right data and a little prodding it is truly surprising what they can learn .