For January 25 (Sfard article) - "On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin"

         I was annoyed by some concepts introduced in this article, as well as some of the underlying statements made. This article, before I proceed with my sentiment, involved the discussion of mathematics learning as made up of what author decided to call "operational" versus "structural" conceptions. Although she admits she does not know how to define "understanding", the author writes that true understanding only comes from moving from beyond the operational conception of mathematics (requiring the five senses) to structural conception of mathematics (at an abstract level).

        The author outlined that the three stages of developing concepts are:
1) interiorization (in which a learner becomes familiar with a concept)
2) condensation (in which a learner conducts more work in manageable units, which requires concept synthesis)
3) reification (the sudden ability to se something familiar in a totally new light).

         One of my issues in this article is the lack of development/emphasis on how the author suggests we proceed from moving from condensation to reification. Although the process she described around waiting for a sudden understanding of a concept is one to which many mathematicians can attest, there is very little detail on how to come about this sudden revelation. However, it is this part of the process for which educators seem to come under fire in the article, since students are still stuck on the operational (if even that), while little/no structural understanding is developed.

         This article demands that we answer not only what mathematics is for, but also for whom does mathematics exist, and who does it serve (or, who does it fail to serve)? How many students are willing to stare at a problem and work it to exhaustion until they finally reach the "click"? How many have the confidence to continue manipulating the problem over and over until they finally have a revelation? Can "the click" be taught? Does it even need to be taught in a high school setting?

         It is also odd that the assumption in this article is that we begin operationally and then proceed to structural conception; Sfard uses the example of a child's counting being operational. I disagree that the set of three objects being counted aloud in place of a single answer yields an operational conception over a structural one, however: if I were asked to count fifty-seven objects, I certainly would not hesitate to count aloud in some form to fifty seven, even though I know that it is the product of two primes (which I assume would be the abstract form of understanding? However, this was a conclusion I made procedurally by being able to subtract three from a known multiple of three, 60, then knew that a number three smaller would also be a multiple).

        Finally, there were certain judgments about ability in the article that bothered me. For instance, I found the following quote irritating: "Being capable of somehow 'seeing' these invisible objects appears to be an essential component of mathematical ability; lack of this capacity may be one of the major reasons because of which mathematics appears practically impermeable to so many 'well-formed minds' " (pp. 3). Why is this an essential component? Does it always have to exist? How do you know how good your "understanding is", and perhaps Sfard could have defined what this is before claiming that having it is an essential ability in doing mathematics?

        Looking forward to discussing this in class.