Whenever I chat to colleagues from physics or maths we tend to end up sharing our experiences and frustrations about teaching quantitative subjects to college students. Everyone, in all third level institutions, is exasperated. The same old worries surface: students struggling with basic algebraic manipulations; students having difficulty calculating numerical answers to problems, especially when scientific notation is involved; students unable to do unit conversions; students not having automatic recall of basic facts that free-up working memory and make problem-solving possible. The general consensus among colleagues is that third level students’ weakness at problem solving ultimately stems from their having a poor grasp of the basic language of mathematics; it’s not due to a lack of some ill-defined ‘problem-solving skills’.
It’s hard to know when all of these difficulties begin but I suspect it goes back to primary school. The primary school maths curriculum is undergoing a review but the current one (dating from 1999) is available here. In the review document, the 1999 curriculum is characterised as follows:
The 1999 mathematics curriculum has many strengths. With firm theoretical roots in Piagetian and radical constructivism, the curriculum promotes the development of children’s meaning making, mathematical language, skills and concepts as well as fostering positive attitudes to maths. There remains, however, scope for improvement. Contemporary thinking and research offers fresh insights into ‘how children learn’ and ‘why they learn in particular circumstances’. This thinking, which has strong Vygotskian influences promotes learning as a social and collaborative process where children’s learning is enhanced through active participation, engaging in ‘mathematization’, working collaboratively with others as well as children building positive identities of themselves as mathematicians. This shift in theoretical perspective demonstrates the need for revisiting the aims of the PMSC and identifying where improvements can be made building on the many strengths of the current curriculum.
While the current mathematics curriculum clearly demands a ‘progressive’ approach to the teaching of mathematics, it would seem that the view of the new curriculum designers is that the 1999 curriculum is not progressive enough. Worryingly, though, the review document contains not a single reference to the work of Daniel Willingham, Paul Kirschner, John Sweller and many others who have made substantial contributions to our understanding of how people learn, especially the roles that prior knowledge, cognitive load, long-term memory and working memory play in learning. Our education system is being driven by philosophy and ideology, not cognitive science.
To get a sense of what all of this ‘progressivism’ means, it is worth having a look at a few statements from the 1999 curriculum. These include:
The importance of providing the child with structured opportunities to engage in exploratory activity in the context of mathematics cannot be overemphasised. The teacher has a crucial role to play in guiding the child to construct meaning, to develop mathematical strategies for solving problems, and to develop self-motivation in mathematical activities.
In view of the complexity of mathematical symbols, it is recommended that children should not be required to record mathematical ideas prematurely. Concepts should be adequately developed before finding expression in written recording. The use of symbols and mathematical expressions should follow extended periods of oral reporting and discussion.
The child’s mathematical development requires a substantial amount of practical experience to establish and to reinforce concepts and to develop a facility for their everyday use. He/she develops a system of mathematics based on experiences and interactions with the environment. The experience of manipulating and using objects and equipment constructively is an essential component in the development of both mathematical concepts and constructive thought throughout the strands of the mathematics programme
The development of arithmetical skills, i.e. those concerned with numerical calculations and their application, is an important part of the child’s mathematical education. This mathematics curriculum places less emphasis than heretofore on long, complex pen-and-paper calculations and a greater emphasis on mental calculations, estimation, and problem-solving skills. Rapid advances in information technology and the ready availability of calculators have not lessened the need for basic skills.
For children to really understand mathematics they must see it in context, and this can be done through drawing attention to the various ways in which we use mathematics within other subjects in the curriculum
It is important that children come to see mathematics as practical and relevant. Opportunities should be provided for them to construct and apply their mathematical understanding and skills in contexts drawn from their own experiences and environments.
I’d love to know where the evidence is for any of these ideas becasue some of them sound truly bizarre to me, especially the idea that a child should develop a system of mathematics based on experiences and interactions with the environment. Seriously?
All of this reminds me of the classic pilot error, the one where the panic-stricken pilot, on finding that his aircraft is about to stall, pulls back even harder on the controls , ultimately ending with the plane plummeting nose first into the ground.