In any case, I used to be a strong believer in the concrete-to-abstract continuum but the more I read the more convinced I am of the pedagogical dangers that arise by taking this progression too literally. I also get the sense that 'abstract' in the context of these kinds of discussions is sort of code for 'hard to understand'.
This post is not intended to discuss or argue these points in full, only to share some thoughts I've run across in the last few months in order to illustrate why this topic is not as straightforward as some might believe.
First, from Uri Wilensky, a seriously brilliant take-down of the whole issue in his 1991 article Abstract Meditations of the Concrete and Concrete Implications for Mathematics Education. Here are just a few excerpts. I recommend reading the whole thing to fully understand his points.
Second, in all my reading there is a theme of caution: the adult can see the math in the materials but this may not be evident to the child. Piaget himself cautioned that the materials themselves cannot and do not hold mathematical meaning. As Kamii says in Children Reinvent Arithmetic (2nd Edition):
"If we adopt the standard view [of concrete], then it is natural for us to want our children to move away from the confining world of the concrete, where they can only learn things about relatively few objects, to the more expansive world of the abstract, where what they learn will apply widely and generally.
"Yet somehow our attempts at teaching abstractly leave our expectations unfulfilled. The more abstract our teaching in the school, the more alienated and bored are our students, and far from being able to apply their knowledge generally across domains, their knowledge displays a "brittle" character, usable only in the exact contexts in which it was learned ...
"...I now offer a new perspective from which to expand our understanding of the concrete. The more connections we make between an object and other objects [not always physical, could mean ideas], the more concrete it becomes for us. The richer the set of representations of the object [idea], the more ways we have of interacting with it, the more concrete it is for us. Concreteness, then, is that property which measures the degree of our relatedness to the object, (the richness of our representations, interactions, connections with the object), how close we are to it, or, if you will, the quality of our relationship with the object."
"Base-10 blocks and Unifix cubes are used on the assumption that they represent or embody the 'ones,' 'tens,' 'hundreds,' and so on. According to Piaget, however, objects, pictures and words do not represent. Representing is an action, and people can represent objects and ideas, but objects, pictures, and words cannot." p31In the introduction to Mathematics Their Way by Mary Baratta-Lorton (a classic activity-based math curriculum for K-2 first published in 1976) the author provides an interesting take on this idea:
And this is the conundrum in which we find ourselves, isn't it? Physical objects not only create sensory memories but also help mediate what's in our minds to expression through our bodies. However, it seems that things are further complicated: Our bodies are strong, capable and skilled but, for some reason which is fully outside my area of expertise, it seems that our physical selves require augmentation in order to fully realize and express our individual and collective visions. I think that's why it makes sense to think that physical objects could be an effective mediator in math learning, especially when math appears to be an extremely abstract subject...
...and now we are full circle, back again to Uri Wilensky, who said:
In other words, it is the process of building multiple and varied relationships with an idea that makes something 'concrete' to the learner, not the properties of the physical objects themselves."...any object/concept can be become concrete for someone. The pivotal point on which the determination of concreteness turns is not some intensive examination of the object, but rather an examination of the modes of interaction and the models which the person uses to understand the object. This view will lead us to allow objects not mediated by the senses, objects which are usually considered abstract - such as mathematical objects - to be concrete; provided that we have multiple modes of engagement with them and a sufficiently rich collection of models to represent them."
It's nice to have some of this thinking out of my head for now. Any thoughts?