If you are an elementary school teacher, I am sure that you are already familiar with skip counting. We want students to learn how to count by 2’s, 5’s, 10’s, etc. We think of this as preparing them to understand and efficiently multiply. I have recently learned from the research that if we add a new goal as we develop skip counting with students, we will more effectively help students with two very important constructions in their mathematical knowledge: understanding number, and conceptual understanding of multiplication. This would happen in two phases. We’ll use 3’s here to help us talk about the phases, but of course the same process would be done with all sorts of numbers.
The first phase would be having students segment their counting into 3’s, and then do the same again, but this time with a new goal of counting how many 3’s they segmented. For example, they would count by ones, segmenting 3’s: 1, 2, 3… 4, 5, 6… 7, 8, 9… etc. Afterward, they would do this again, while also keeping track of how many threes they segmented as they counted. They can make and count piles of objects, make tallies on their paper, use their fingers, or any other method that works for them. It might sound like: 1, 2, 3, that’s one, 4, 5, 6, that’s two, etc. This will help them to see three as a group of three ones, which itself can also be one thing, a composite unit. Also, it will help lead them to understanding skip counting not just as a string of words, but as “grouping” three ones together when learning to skip count, which would be the next phase.
In the second phase, the same new goal (keeping track of how many threes they segmented), is added to the skip counting we already do with our students in the classroom. After skip counting by threes: 3, 6, 9, 12, etc., we would want the students to do this again, but count the 3’s as they skip count. Again, they could use whatever method they are capable of using to keep track, but eventually we would want them to be able to say: 3 is 1, 6 is 2, 9 is 3, and 12 is 4, etc. As the teacher, you could help them move from needing to have actual groups of objects to count, to using tallies, to using fingers, and eventually using their number sequence itself to count the threes.
Guiding students through this process in the early grades will be an effective way to help them develop an understanding of what skip counting means, which will later make for a smooth transition into multiplicative thinking. Teachers are the most creative people I know, so I’m curious to hear any ideas of how you could make these phases fun for children, while keeping in mind this new goal. Feel free to share your ideas below.