There’s a reason we keep track of quantities on our fingers. Our digits, always ready, map the correspondence between things and numbers. I remember, as a very young child, watching telephone poles and row crops fly by outside the rear window of the car. As they passed, I counted, keeping track on my fingers. I matched object to digit to number and found patterns in groups of three, four and five. When the grouping went to six, both hands came into play. Long before anyone told me not to do math on my fingers, I was connecting the world I saw to numbers through my body.
Manipulatives – from the Latin root Mano which means hand – build a foundation for understanding mathematical concepts in more abstract ways. And it’s never too late to map what we see with our hands.
In a recent workshop for high school teachers, participants used manipulatives to explore transformations of functions. They built a pattern of increasing squares with little square tiles. They built the pattern and then rebuilt it using transformations that would shift the graph of the pattern up, down, right and left. They built what it meant to narrow the squares, double them and divide them in half. By using their hands, they made core concepts in algebra visible.
Everything those teachers did, they could have done just using symbols – that is, digits and variables with procedures drawn from memory. As they worked, digits and variables came into play, as a meaning-making extension of what they did with their hands.
Counting on and with our fingers is not just for young children in the back seat of the car. For those new to working with the symbols of algebra, manipulatives fill the gap between what learners can see and understand, and what they are in the process of learning. Manipulatives matter in learning mathematics, no matter how old your students are.