Most learners like it when they can do something novel, but hands-on learning has to develop mathematical understanding for students to stay engaged. That is, concrete materials have to allow students to grapple with mathematical ideas they are in the process of developing. When that happens by design, students’ thinking becomes visible to each other and to their teacher. When students can see each others’ thinking, community develops, and when hands-on materials make it easy for teachers to see student ideas, being flexible and responsive to developing thinking becomes second nature.
High school teachers recently began a five-day workshop designed to deepen their understanding of functions with a lab involving water and vases. Their task was to predict the graph of the height of the water in a given vase as they added increments of water, and then test their prediction.
This hands-on math got the workshop off to a very strong start. Everyone, no matter their background, understood water and vases, or measuring, or graphing. No one could accomplish the task alone. Someone had to measure, someone had to fill and another person had to record. Soon the multi-dimensional nature of the task itself got people talking and laughing and helping each other and puzzling over results.
But more than that, the task’s focus on the relationship between an input and an output, landed participants in the center of grappling with ideas essential to understanding functions. Nearly everyone’s predictions improved as they repeated the task with vases of differing shapes, and issues of accuracy, rates of changes, and the relationship between independent and dependent variables became rich sources of conversation.
In fact, hands-on math can make visible the relationships between quantities, through relationships between people. This is at the heart of powerful mathematics teaching and learning. Once teachers get a taste of it, they want more.