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.

Thanks for this.

I’m just preparing a lecture on techology in maths teaching and this post really endorses the view I have that sometimes technology can block good maths teaching. For all the Khan Academies, Education Cities and other tech maths tools that are out there, nothing beats a good teacher who can unpick the misconceptions of a child and see exactly where they need a concrete example, a manipulative or a pictorial representation.

As you say – for many students, this continuum can be very complex and dynamic – students confident with abstract concepts in one area, may well need concrete examples in another.

Do you mind if I share this post with the students in question (all are current teachers doing a post-graduate maths teaching qualification)?

Absolutely feel free to share. I would also be very open to hearing what others have to say around this topic. This post represents my thinking right now but I’m always open to new ideas, and suggestions.

Thank you for your comments as well. You are so right that a good teachers is skilled in identifying misconceptions and helping students correct them. Manipulatives can be a powerful tool to support thinking.

Thanks for your comment.

Jenn,

Thanks for sharing your thinking. With our new Common Core we have moved back into more thinking with manipulatives. This shift has helped students develop an understanding of number. I need to remember to spend more time in the drawing our thinking stage. I am sure it is true that students progress through stages again when concepts are new.

Much to think about,

Cathy

I’ve long wondered whether (or rather, how many) students can leap to the abstract without going through all the stages? Maybe they are actually just going through the stages quickly and internally, perhaps even subconsciously. We know that pre-attentive learning exists, so maybe it is something like that.

Anyway, a good friend of mine was a math ed prof here, and he wrote a bunch of textbooks on math ed. He endorsed the use of manipulatives, and I always thought it was because they just made good common sense. You’re helping me understand why. I wish he was still on faculty — I’d like to talk to him about your post. Thanks for making me think!

At what stage can the three stages be applied.the age of the learners.

At what stage can the three stages be applied.