For a long time now most teachers and researchers have agreed that effective learning takes place when learners actively construct meaning. For example, Mills, O’Keefe & Whitin (1996) suggested that learners are active constructors of their own knowledge and are the “meaning makers”. Children aim to make sense of any new situation by connecting it to their existing experiences. This is true whether they are ‘learning’ about the expectations placed upon them when eating in a restaurant, or developing the skills to solve a specific mathematical problem.
In the past there have been many attempts to break mathematical concepts down into separate ‘bite-sized’ parts. The logic seems to be that children should be able to develop an understanding of a small ‘bit’ when the whole concept is too unwieldy. In some respects this strategy has shown itself to be useful. However, in others it can cause more confusion and uncertainty than necessary since the size of the ‘bits’ is based on the teacher’s or (worse) the Maths Scheme book’s notion of what is reasonable, rather than the learners covert needs. This artificial ‘bite-size-ation’ can also remove any aspects of realism from the ‘teaching’, and it is this which drives an impassable wedge between ‘school maths’ and real life maths. In this respect this is closely related to Dan Meyers’ notion of “impatient problem solving”:
What is much more important than an externally imposed menu of ‘bits’, however, is the ethos in which the maths concepts are housed. The amount of risk that children will take when developing their mathematical understanding is closely related to the nature of the ‘community’ in which they learn. Therefore, creating a supportive community that encourages them to voice and to share their mathematical ideas is vital.
I’ve been studying and thinking about the effective use of collaboration in mathematics for a number of years now. Getting the ethos right is of paramount importance, since children respond best in a safe, secure and accepting classroom. However, the nature of the communication itself is a key factor too. A few years ago ‘Questioning’ was reassessed. Recently there has been a drive towards dialogic learning and a more appropriate use of discussion. I also see a move towards a more open form of written and symbolic recording. The use of graphics, writing and talking all help to build a collaborative community that leads children to generate their own ideas ideas, develop a confident communication style, and reflect upon their own learning.
I have recently seen all of these factors coming together in the use of mathematical journals. These, when used effectively, appear to support learning by offering children creative ways to come to terms with, reflect on, record and share aspects of their learning.
I’ll return to these ideas later. In the meantime, if anyone has had success using journals in mathematics please contact me.