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.

Response to "Muddying the clear waters": Teachers' take-up of the linguistic idea of revoicing

The theme of this article is based on the idea of "revoicing," which is defined as "the reuttering of another person's speech through repetition, expansion, rephrasing, and reporting" (Herbel-Eisenmann p. 268). On the surface, the concept of revoicing seems like a clear one. As a teacher, I find myself doing it very often. The existing literature in mathematics education on the concept, sees it as pretty clear as well. Of course, that is the purpose of academic writing; to clarify complex phenomena in our journey toward understanding. This brings me back to last week's article, where we defined a simpler model to understand something larger, and more complex.

The article examines a "study group" of eight working teachers, who have all been involved in professional development opportunities. The teachers met in a "study group" to discuss particular academic research papers and their implications in the classroom. Although the topics were varied, the author's were concerned with any mention of "revoicing" during the discussions. What the authors found, after examining the recordings of the sessions, was that although revoicing seemed to be clear on the surface, when teachers considered it in their own classrooms, they found that these experiences "muddy" the clear water. Teacher's spoke of how they were concerned that the revoicing of a particular student's idea, might turn the student's idea into the teacher's. That is, the teacher revoicing the student might be somewhat controlling and authoritative. I thought to my own classroom, and my habit of frequently revoicing. I was particularly struck by the idea of having other student's revoice, rather than the teacher. This removes the authoritative nature (somewhat) of the teacher, and might lead to a more mathematical conversation in the classroom. This not only requires students to be particularly attentive during class, but could also lead to "not only revoicing student thought, but also extending upon it" ( Herbel-Eisenmann, p 273). Something that I aim for in my classroom is a mathematical conversation, but with the time constraint on my classes, I find it extremely difficult to do so. I am going to try to make this a goal in my teaching next week.

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.

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