Reflection of On the Foundations of Mathematics Education

Pre-reading impression:

Upon my reading of the first few paragraphs of On the Foundations of Mathematics Education, by author William Higginson, I was particularly struck by Gulliver's visit to the "Academy of Projectors at Lagado" where "professors contrive new rules and methods" with positive intentions, but, as these methods are incomplete, these positive intentions lead to Lagado lying in a state of distress. One could argue, as many do, that the continued development and change of the mathematics curriculum in United States is causing an analogous situation to that of Lagado, in mathematics classrooms across North America. I suspect that the core of this article will be in regards to the continued change and major shifts in curriculum within mathematics education, as well as a call for further development and consideration of the curriculum before releasing it on our (as mathematics educators) citizens of Lagado.

Reactions:

My initial impression to the article were somewhat valid, but the body of the paper wasn't as centred on it as I had expected. Higginson's main focus is that of the foundation of mathematics education as a field. When I first read "foundation," I immediately thought of how the field was developed; its initial beginnings and the history of the field. Higginson, on the other hand, considers the foundation of mathematics education in terms of the foundation of a home; the major building blocks contributing to the field. He states that in order to answer some of the major questions in mathematics education, one must "fully acknowledge the foundations of our discipline" (Higginson, p. 3). Of course, the foundation of mathematics education is a complex system, so Higginson turns to a basic model to better understand the "essential aspects of its foundation."

The model Higginson chooses to use is a tetrahedral model, called MAPS. Each letter represents what he considers to be the fundamental dimensions of mathematics education: M-Mathematics, A-Philosophy (arbitrary?), P-Psychology, and S-Sociology. I was immediately unsure of this model upon its introduction. To me, viewing a field as multi-dimensional as mathematics education in terms of a tetrahedral, seemed not to do the field justice. But, I grew to take it just as that: a model. Nothing more, nothing less. A model exists to represent something. Certainly, what we are trying to represent may be much more complex than the model suggests, but one would hope that the chosen model may reveal some aspects of the object which were invisible beforehand. Higginson's statement of mathematics education = psycho-philo-socio-mathematics brought me a slight giggle. It seemed somewhat arbitrary, especially realizing that this implies education = phycho-philo-sociology. I brought it back to mathematics though, to topology. I recalled how I would view complex spaces in terms of simpler ones, so that particular attributes might come forward.

Higginson goes on to explain how we might apply the MAPS model. What I found most intriguing was his mention of a "centre of gravity" for the American MAPS tetrahedron from 1965 to 1980. The extreme shifts between traditional mathematics and "new" mathematics has been a topic I have been grappling with recently. But why can't we have both? Why not didacticism and discovery? Why not intrinsic reward and utilitarian purpose? Curriculum is constantly changing to find an "alternative" because one or the other isn't sufficient. In order to gain a full-bodied appreciation for literature, one studies many different types of literature. One comes to gain an appreciation for the complexity of the field. The same goes for mathematics. Although I myself am a purist, I have been exposed to the technical uses of mathematics and have a great appreciation for what mathematics can do in that context. We need balance, so that the full beauty of mathematics may be revealed. I was pleased to read, although a brief statement, that Higginson agrees.

Through the MAPS model, one can being to appreciate the dimensions of mathematics education which one may not usually consider. From personal experience, I recall attending my first mathematics education conference, and at many of the talks wondering to myself: "Where's the math?" I reverted to my research mathematician-self who wanted the mathematics in psycho-philo-socio-mathematics. But, there are people who want the sociology, philosophy, psychology, or some other combination of M,A,P, and S. Together, as Higginson states, we are a "collection of responsible and concerned scholars looking for assistance with important questions affecting the day to day lives of large numbers of people." Together, we may learn from each other about all of the combinatorial pieces of the MAPS model of mathematics education.