Three pages into this reading and I could
tell that I was in way over my head. I
had to look up the definition of the words ‘epistemology’ and ‘ontology’. At first I thought that I could get by with
not knowing what the words meant, but after noticing that they kept on
reappearing I decided to look them up.
To my dismay, their definitions only further confused me.
Epistemology is “the branch of philosophy
concerned with the nature and scope of knowledge”. It discusses what knowledge is and how it can
be acquired. Ontology is “the
philosophical study of the nature of being, becoming, existence, or reality, as
well as the basic categories of being and their relations.” At least, this is what Wikipedia has defined
them as.
In this reading Sfard discusses how
mathematical “learning and problem-solving consist in an intricate interplay
between operational [as processes] and structural [as objects] conceptions of
the same notions.” (Sfard, 1991) In
order to better understand certain mathematical concepts, a student should
recognize and recall both the structural and operational description of said
concept. An example in the reading says
that the structural description of a circle is ‘The locus of all points
equidistant from a given point” (Sfard), whereas the operational description of
a circle is “{a curve obtained by} rotating a compass around a fixed point”.
(Sfard) In Layman’s terms (or at least
in David’s terms) it sounds like the operational description of a mathematical
term states how to perform a mathematical task to create a term (circle,
function, integer, etc), and the structural description states exactly what the
mathematical term is (a definition).
Most of our mathematical concepts were
created through a process (operational), however once we were able to prove, or
create something, that process was given a label or name (structural). For example, a child can learn that 1 + 1 = 2
by being able to count a single object and another single object and then put
them together and then recount the total.
This would be operational, whereas simply writing 1 + 1 = 2 would be
structural. In order for any
mathematical concept to come to fruition it must go through the operational
process before a structural definition can be applied to it. However, we often teach the structural
definitions to our students first, and then we will teach the operational
process later. Ideally, we want our
students to create an operational pathway to achieve a structural notion. That way, if the structural notion is lost,
hopefully the student can recreate the operational pathway again. This operational pathway is described in
three stages: interiorization (getting to know some preliminary processes),
condensation (being able to think of the process as a whole), and reification
(a higher-level of understanding).
I found this article quite difficult to get
a grasp of as there were so many new terms introduced to me. I found myself constantly stopping to take a
moment to understand what was just said (sometimes due to new terms, and sometimes because the author only ever used words that were 5 syllables or more).
Because of this, my reading of the article was disjointed and not as
enjoyable as I hoped for.
David,
ReplyDeleteI appreciate your honesty in your response and feel like I had similar feelings when I read my article. I found myself with my dictionary app open and close at hand during my reading and I read the article way more than once! You are not alone!
I find it interesting the difference between the operational process and the structural definition. I think this is often the process we take our students through when we use manipulatives at first to grasp concepts before moving them on to the structural notation. It is interesting to note the three stage process of the operational pathway, because I often feel as teachers we jump into the structural notation quite quickly, maybe it is time to spend a little more time ensuring the operational groundwork is more complete. I would love to know more about ideal timelines with regards to this, although I know this will differ for subject areas and individual students.
I feel somewhat responsible that you had such a challenging article to read. Since I was the second person to choose the reading, consequently, I made the decision for you as well. During the first class of my required methods course, we covered the terms "epistemology" and "ontology" and we have been using them repeatedly since. However, I can attest that most of the students are still struggling to make sense of the two words. On the other hand, you did a wonderful job encapsulating the main concept of the operational and structural ways mathematical concepts are created. This causes me to reflect on my own teaching method. Do I too hastily introduce new mathematical concepts in a structural way and expect my students to apply them operationally? Could failing to properly present the operational pathway contribute to why a large number of students have trouble extending the concepts to higher-level understanding? I really appreciate your response to this week's reading because it causes me to reconsider my mathematical pedagogy.
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