Response to "On the Dual Nature of Mathematical Conceptions"

When I looked at the titles for the articles this week, this title immediately sparked my interest. The author, Anna Sfard, discusses what she considers two ways mathematics may be conceived: structurally and operationally. The structural conception is one in which mathematical entities are "seen" as types of objects. The ability to be able to "see these invisible mathematical objects," may be one of the reasons for which some very developed minds find mathematics to be inaccessible. The operational conception on the other hand, is that of algorithmic, procedural, and action oriented ability. The author notes that although the structural conception seems to be more prized within the mathematical community, there is a purpose and usefulness to operational conceptions. She argues that much of mathematics was developed operationally, before any abstract structural concept was endowed. 

The article questions how these conceptions play a role in students' mathematical learning and which might be more conducive to the development of mathematical ability. Sfard argues that these two conceptions, although "ostensibly incompatible" are in fact, complimentary. In order to develop a well-rounded mathematical understanding, one must have a structural and operational view of mathematics. Throughout Sfard's arguments, I was brought to think of the "Math Wars," which dominate much of the curriculum discussion in mathematics education. There seems to be a lack of balance within this argument. On one hand, we have the traditionalists that want the curriculum to be very operational, while the "new math" enthusiasts want to take a structural approach. As Sfard notes in the title, processes and objects are different sides of the same coin; if we are to consider mathematics and mathematics education as a whole, we cannot just consider one side of the coin. Both sides of the "Math Wars" need to find a common ground, as they are certainly arguing on the same one.

Ballet and dance has always been a major aspect of my life, and I attribute much of my success in mathematics to the many years I spent in a dance studio. If one would like to learn how to dance the salsa, one must have a particular sense of rhythm in their movement; a passion for movement. At the same time, one must be technically proficient and quick on their feet in order to execute the steps of the dance. Technicality and rhythm are two faces of a coin which we cannot split apart, as they work together to create one movement. This was always evident to me in the dance studio, and I have a deep respect for those who have such an incredible sense of rhythm and/or immaculate technique. Now, while much of my time is spent around budding mathematicians, it's exciting to surround myself with minds (like those of Poincaré and Hadamard) who have such a natural, structural view of such abstract entities. The fact that I have the opportunity to surround myself around such minds is something I am truly thankful for.