And when fear is experienced over a long time, as it is with many of these students, elaborate defence mechanisms are built up and used to protect the self esteem. Most students that I see are afraid of mathematics, afraid to speculate and reach into themselves for mathematical ideas when they are perfectly happy to do this type of speculation, in visual art and music. Further difficulties arise because students taking a particular course are in different stages of mastering the basic material and they also tend to be secretive about what they know and what they don't. What is even more inhibiting for them is that the addition of fractions is a very boring topic for those in the class who already know 'how to do it'. Any student needing Bridging Mathematics represents a failure of earlier mathematics teaching to adequately communicate the ideas involved.
Problems arising from a failure of understanding are curable.
The Bridging Mathematics community should have as one of its goals to find ways to eliminate itself. This suggests to me that, especially in the early years, mathematics should be taught very broadly and with less of an emphasis on compute-ability and more of an emphasis on acquiring an understanding of general principles and the interconnectedness of ideas. This would allow quicker students to work through the same material in greater depth, with more computation and excursions into related topics. Most teachers are trained at universities with few if any research mathematicians and most research mathematicians are not involved in teacher training. Young children frequently come up with ingenious ways to solve mathematical questions, but teachers who are often uncomfortable with anything off the beaten tract, will discourage non-conventional approaches.