The Map is Not the Territory

In the past week or so I've had run-ins with very specific views on concrete-ness and abstract-ness. There are many people who feel very strongly about the notion that concrete objects create a bridge to more abstract ideas especially for young children. And, even if the discussions sound different, this viewpoint is not reserved exclusively for the little ones.

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.

"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."

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): 

"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." p31

In 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:

At the same time, she points to the reality that "children learn easily and joyfully from real things":


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:

"...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."

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.

It's nice to have some of this thinking out of my head for now. Any thoughts?

I've been thinking quite a lot lately about the role of physical objects in math education.  Sometimes called manipulatives or, more generally, tools, I've discovered conflicting opinions and strategies around the use of such objects. In her book Young Children Reinvent Arithmetic, Constance Kamii helpfully sums up some of the issues with which I've been wrestling:

"Manipulatives are thus not useful or useless in themselves. Their utility depends on the relationships children can make..." p25

"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." p31

So, it is not the object itself that holds the math, but rather the process in which the learner uses the tool that creates the meaning.  But, of course, when we use this kind of language we are talking abstractly about hypothetical objects and generalized characteristics of 'the child,' not any specific object or individual learner in particular.

Too much generality and abstraction drives me crazy so imagine how pleasantly surprised I was when this showed up in my mailbox the other day:


What is it? Well...it's an object. And a beautiful one, at that. An object that can be "manipulated" (the triangle comes out and can be turned). A thinking tool. It was designed and created by Christopher Danielson to investigate symmetry and group theory with his college students. Not only are parts of this tool moveable, but it also has the potential to help "facilitate [mathematical] conversations that might otherwise be impossible." (Christopher on Twitter, Jan 17, 2014)
 
What was even better than getting a surprise package in my real life mailbox containing a real life manipulative (not a theoretical one) was my (real) eight year old's interest in and reactions to said object.

She spotted the envelope and said, "Hey! What's that?!" I told her that a math teacher friend of mine had sent me something he made for his students to use. I took it out of the envelope for her to look at.

First thing she noticed was the smell -- lovely, smokey wood smell which we both loved.  She investigated the burned edges, tried to draw with them (sort of like charcoal). This led to a discussion about laser cutters (heat, precision) and the fact Christopher had designed it.

I pointed out the labeled vertices on the triangle, showed her how you can turn it, and mentioned that the labels help us keep track of how far the shape has turned. She immediately took over this process.  

She repeatedly asked if she could take it to school! I asked her, "What would you do with it?"  She said, matter-of-factly: "Play around with the triangle...and discover new galaxies."



Then, she turned the triangle 60° and said, "And make a Jewish star..." Then she put the triangle behind the the opening so it (sort of) made a hexagon.  I asked, "What did you make there?" She said, "A diaper." Ha! 

I hope Christopher's students are just as curious about and enthralled with the "object-ness" of this gorgeous thing as they are with the idea that it helped them talk and think about things that might otherwise be impossible to grasp.  I know that the objects themselves hold no mathematical meaning but watching how intrigued my daughter was with Christopher's gift, I am left thinking about what we miss out on if we consider a tool simply a bridge to the 'real' goal of mental abstraction.  

Beautiful and intriguing objects, I think, have a role in inspiring the whole of us, all our senses, kinetics, and curiosities, not just our minds, to engage in the process of math learning.  An object doesn't necessarily have to be tangible; narrative contexts are highly motivating 'tools' when working with children. As I blend math, dance and basic art making I see over and over again how presenting the object (idea) first pulls my learners in -- they are curious about what this dance is, how they might weave their own wonderful designs using math, what does she mean "growing triangles" and why are these pennies on the table?