by Nancy Berkas and Cyntha Pattison (NCTM News Bulletin, November 2007)
Where do manipulatives fall in our understanding of interventions? Much has been learned about the teaching of mathematics since NCTM published Curriculum and Evaluation of School Mathematics in 1989, and a good portion of that learning is related to using manipulatives in the math classroom. Researchers frequently focus on manipulatives (or hands-on physical models) as tools for teaching students with learning disabilities. And they have based most studies of the concrete-representational-abstract (CRA) learning sequence on work with learning disabled populations. However, evidence is emerging that shows that manipulatives and CRA can be very effective tools for teaching certain concepts to all students.
The CRA approach begins at the concrete level, where instruction includes the use of manipulatives. Teachers who use this instructional approach determine whether students understand what has been taught before proceeding to the next stage. In some cases they allow students to continue to use manipulatives to demonstrate their understanding in the representational and abstract stages.
Other areas of research on the use of manipulatives show generally positive impacts when manipulatives are combined with (1) virtual manipulatives software, (2) reflective practices, (3) cooperative learning, or (4) learning activities that are exploratory and deductive in their approach. And we believe that manipulatives can indeed benefit student achievement in regular mathematics classrooms (as opposed to special education environments) when used in conjunction with instructional practices that develop a concept of a symbolic nature and don’t simply mirror a process or algorithm.
So manipulatives can be used as an effective instructional strategy in the regular classroom, but can they also work as an intervention in those classrooms?
Research that shows CRA and manipulatives to be effective in regular classrooms leaves us with a bit of a conundrum. If manipulatives are used in the classroom for instruction and students proceed to the representational and abstract stages before anyone is formally identified as “needing intervention,” is it still considered “instruction” if the teacher returns to the concrete stage and uses additional manipulative-based activities to develop understanding of the concept? Or has the teacher missed the boat on teaching that particular concept and moved into the realm of intervention? Let’s extend our boat analogy. Experience tells us that the boat leaves from a different place and goes in a different direction for each child and each concept. And we know that it is essential to check for concrete understanding and provide additional instruction immediately, especially when we are teaching concepts that have a natural concrete basis or are known to be difficult for students to learn. So yes, backtracking is intervention, but it is also a very good instructional strategy.
In our travels, we have found many states, districts, and regional service agencies that have developed a model of service to account for this type of built-in intervention. Their models posit that 75–80 percent of students will learn from the original lesson, 15–20 percent will need (and will develop understanding from) immediate classroom intervention, and 5 percent will need one-on-one intervention outside the regular classroom.
Let’s see what questions come up as we look at manipulatives through our “intervention lenses”:
- Learning Significant Mathematics. If every student should learn significant mathematics, do we have an obligation to extend our investigation of the situations and combinations that enhance concrete understanding of concepts through manipulative-based activities?
- Knowing the Mathematics. Can we identify concepts that are best developed with a concrete, manipulative base? Is our mathematical understanding sophisticated enough to know how to use manipulatives most effectively?
- Assessment and Data Gathering. Can we assess concrete understanding? Are our assessments and definitions of success limited by pathways that we perceive that all children must travel to learn mathematics?
- Quality Planning and Delivery. Do we have the ability to plan and deliver quality manipulative-based lessons?
- Alignment. Can or should we use standards and benchmarks to determine which areas are most appropriate for the use of manipulatives? Are there certain grade levels where it would not be appropriate to teach some concepts or skills by using manipulatives?
We have come a long way in mathematics education since Montessori and Piaget established the importance of concrete activities for the learning of mathematics. We now also know that manipulatives alone do not constitute an intervention; however, combined with intelligent minds and research-based approaches, they offer many possibilities for intervention for all of our students.