This post is about the paper Transitional Devices, written by @AlfColes. The abstract reads as follows:
Ascent to the concrete
The paper starts by questioning the widely-held assumption that we should move from the concrete to the abstract when learning mathematics. Coles cites Mason's suggestion that learners move between enactive-iconic-symbolic representations when learning, and related work by Davydov. As Coles suggests:
What seems significant in learning mathematics is that a link to the enactive or iconic meaning is maintained...
The paper also cites the work of Abraham Arcavi on developing symbol sense. Arcavi suggests that successful learners are able to tolerate 'partial understandings' until meaning emerges, which brings to mind Magdalene Lampert's call for teachers to give credit to students' partial understandings.
The corresponds with Gattegno's observations on how children learn:
Children can recite the number name sequence before being able to show recognition of the link between number names and collections of objects... In learning language, it seems we are happy to engage in playing with sounds without worrying about what they mean.
This is of particular interest to me at the moment. My 19-month old daughter is currently starting to experiment with number names. She does not always get them in the right order, or use them in the right circumstance, for example using the word two to describe 'more than one'. However, I think she is doing more than 'playing' with these words; I think she might be trying them out to see if they fit. I suspect she might be using them and gauging our (her parents' reaction) to see if she is using them correctly.
In recognition of this, Gattegno's (mathematics) curriculum introduces algebra before arithmetic, allowing students to gain fluency with the symbols before attaching meaning. This is intriguing. I had a conversation about something similar with a colleague recently, in which we both noticed that we sometimes introduced topics in a similar 'reverse' manner, starting with the abstract or complex, then gradually extracting meaning and order, allowing students opportunities to use their own awareness to identify structure.
Coles describes in the paper how he used these techniques in the context of the Gattegno chart to allow students to 'explore journeys' in multiplying and dividing numbers by ten. The children in the project were encouraged to 'follow their fascinations', to form their own lines of enquiry. He found evidence of children's playful exploration of symbols in line with Gattegno's theory:
What I observe is that they have become energised in gaining fluency in symbolic manipulations and have developed awarenesses linked to these symbols.
Coles describes a lovely and important moment when one of the students asks whether their journey can traverse columns in the chart, a question that he had not anticipated and something he had never used the chart for! As a result, students started using the chart for methods resembling multiplication by fractions using a unitary method.
Why the title 'transitional devices'? This is a reference to the work of psychologist DW Winnicott, who describes the 'transitional object' as the 'first possession' in his book Playing and Reality:
The object represents the infant's transition from a state of being merged with the mother to a state of being in relation to the mother as something outside and separate... The term transitional object gives room for the process of becoming able to accept difference and similarity.
An example of a transitional object is the rabbit my daughter wants to take everywhere with her.
I interpret that Coles is drawing a parallel between Winnicott's transitional object - which acts as an 'intermediate area' between the infant's inner world and its (perception of an) outer reality - and a mathematical device that 'gives room' for a students to make transitions between Mason's enactive-iconic-symbolic representations.
Children are not happy when the transitional object is taken away. I wondered how students might cope when the Gattegno charts were taken away?
Winnicott suggests that for the 'healthy' child, the transitional object simply loses significance over time, as the child forms a relationship between inner and outer reality through other means, such as play. [Interestingly, Winnicott suggests that 'no human being is free from the strain of relating inner and outer reality.'] This loss of significance is how Coles perceives the fate of the transitional device:
The Gattegno chart allows students, in the early stages, to perceive the relationships that symbols stand for and provides them a reference (a comfort?) to which they can return if necessary, to ground symbolic transformations. After some time the chart becomes unnecessary, it is abandoned, but can be invoked, for example as a mind image, if needed.
This brought to mind this video of Japanese students using 'virtual' Sorobans (from 01:20):
Finally, Coles leaves us with a challenge: what other transitional devices can we find that allow mathematical symbolism to be linked to relationships that can be perceived?