In this month’s Mathematics Teacher magazine is an article about prime factorisation by me. It discusses an idea for teaching and learning about prime factorisation that minimises ‘telling’ and maximises students mathematical exploration.

Dave Hewitt, a lecturer in Mathematics Education from the University of Birmingham (and my PGCE mentor a few years ago), has written a series of articles about separating the arbitrary (or contingent) and the necessary and mathematics, and teaching by ‘telling’ only those things which are arbitrary. The idea is that students need to engage and discover for themselves the necessary connections and patterns in mathematics, but the arbitrary are not discoverable in the same way, and so need to be told to people.

This position has influenced my thinking about mathematics education, particularly with respect to algorithms. My conjecture is that using an algorithm involves no mathematical thought; at best, it is an exercise in arithmetic. However, where an algorithm exists there is likely to be a kernel of really interesting mathematics, and creating an algorithm to perform a particular function involves a great deal of mathematical thought. The goal of the investigation was to capture the interesting maths in an interesting way, that the students can engage with, and which they can learn from.

The accompanying software can be found in the primitives section of this website.

Daniel Tammet has an extraordinary relationship with numbers. Born with Asperger’s Syndrome, Daniel has an extremely kinaesthetic relationship with number, and has a host of other impressive mental abilities, such as his extreme aptitude for learning languages rapidly. His book Born on a Blue Day is a wonderful memoir of his life to this point, and well worth a look both because it is a great read, and because it offers an interesting insight into Asperger’s Syndrome.

Mathematically however, it is even more interesting. Daniel suffered at school because his mind was creating associations between numbers and other concepts, and between the numbers themselves in creative and unexpected ways. He and his parents had the courage - or perhaps just the necessity - to persist in working in and with those associations. They have served him well; his numerical mental dexterity is far beyond what almost anyone else could muster.

Schools are inevitably ‘one size fits all’ institutions to some extent. Schools are mass-education institutions; it is not possible to ‘personalise learning’ precisely for individual students. That is not to say that the Personalised Learning initiatives from the government are bad things, far from it; merely that because often a teacher is dealing with about 30 students, it is not possible to tailor learning exactly to the needs of every student.

In mathematics classrooms, the tendency to make uniform what is not becomes more apparent. In maths teaching there is a tendency towards drill and rote. It can be seen clearly in textbooks, and it is confirmed by asking any reasonable cross-section of society to recount their experiences of maths at school.

It would be absurd to argue that allowing students more freedom to explore the associations between numbers and to create their own understandings of them will most students become as adept at maths as Daniel. Nevertheless the root of his understanding of number comes across as creatively based rather than based in algorithms and routines. To me, Daniel’s story is compelling evidence in support of the theory that creativity and imagination are central to the development of young mathematicians.

As a foot-note, after reading Daniel’s book I began to reflect on the ways in which I perceive numbers. This directly led me to expand my own concept of the structure of numbers by creating the Primitives concept.