I love supporting students’ mathematical learning with manipulatives. Manipulatives provide students with the opportunity to physically manipulate objects as they think about and interact with a mathematical concept. On their own they are NOT mathematics but when used to support thinking and understanding they can be a very powerful tool. In order to use manipulatives to their full potential it is important to think about some key questions first:
- What mathematical understanding are your students discovering and learning?
- What would it look like (with manipulatives and without) if they understood the concept?
- Which manipulatives would be most effective to support student understanding of the concept?
- Do all of the students need to use manipulatives to demonstrate understanding of the concept?
There may certainly be more or different questions one should ask. The important point is for teachers to know what the math behind the manipulative is.
A tool that can help educators answer these questions is the Concrete to Abstract Continuum. The continuum shows how a students’ thinking might evolve as they learn in mathematics. It can also help teachers to think about and prepare to engage learners at different points along the continuum.
Initially children start out with the concrete…they need to see, touch, taste, and smell an object in order to get a complete understanding of what it is. One only has to think of baby to see that concrete, tactile is very important in sense making.
As their thinking matures children, can begin to represent one object for another. They can pretend that the disk is an apple…or the mud is a pie. It is this pretending that allows them to suspend their disbelief and use their imagination. This semi concrete stage is employed all of the time in mathematics classrooms when students are asked to use the manipulatives in problem solving.
The next stage is pictorial or semi abstract…a stage that I feel is often forgotten in the classroom. This stage is where students draw a picture to represent thinking. The pictures are less manipulative than objects and therefore, more abstract. However, pictures can be a great tool for helping students make sense of the math as the allow them to “see” it more fully.
The last stage is abstract. This is where symbols are used to represent quantity, functions, operations etc. It is the most abstract way of thinking about mathematics and often the hardest for students to grasp.
It is important to know that although my graphic is in a straight line it is rarely a straight progression for student thinking. In fact students may move fluidly from one way of representing to the next depending on the concept, the situation and their understanding. I have found that even the most abstract thinkers may need to use manipulatives or pictures from time to time especially when introduced to new concepts and ideas.
The goal of manipulatives in the mathematics classroom is help students develop a deep and meaningful understanding concepts. By representing their thinking and understanding with a variety of manipulatives students can become empowered mathematicians.