On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin

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. 

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

“Stops”:

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      My first “stop” occurred when the term “revoicing” was first defined (see below).  I had never heard of the term, revoicing, and I had to reread the definition a couple of times to engrain it more concretely in my head so that I could recall it as it was discussed throughout the reading.  I also stopped to reflect on how, or even if, I use revoicing in my own classroom.

. . .the reuttering of another person’s speech through repetition, expansion, rephrasing, and reporting.

·        My second “stop” occurred after reading the excerpt between Sammie, the teacher, and Charlie.  Upon reading this, I gained a clearer understanding of what revoicing meant, as the teacher essentially reworded what the students were saying.  However, I noticed that the teacher also supplemented the students’ comments with specific terminology that the students did not use.  On one side, I see this as being useful to connect the students’ thinking with what is being learned in the classroom at the time, and on the other side I was thinking that we need to be very careful that we’re not putting words into the students’ mouths.  Too often, as teachers, we make assumptions as to how a student came about finding a solution to a problem.  Perhaps if the teacher were to ask more guiding/prompting questions, the students would have come up with the terminology themselves. 

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          My third “stop” was when a teacher mentioned that they had used revoicing to catch a student who wasn’t paying attention, by asking them to revoice what another student had just said.  The teacher had said that it helped keep students paying attention during discussion time.  I would be wary to use revoicing as a consequence for not listening, as it could develop a negative connotation.

Questions to ask the author:

• ·      Would you suggest asking students to revoice what other students are saying (rather than the teacher doing the revoicing)? ß You answered my question further into the reading. . .Yes, it does seem to be a good idea!
• ·      Would you recommend that revoicing be used school-wide to help reinforce a common language, so that over the years the students begin to use the language themselves?
• ·      As a study group, did you practice revoicing yourselves when discussing readings, and your own classroom practices?

A Research Programme for Mathematics Education (I)

Upon reading the title and the first paragraph of this reading, I predicted that it would discuss the potential avenues that mathematics research may present itself in, and what said research may produce or suggest, in terms of pedagogical practice. 

I found the response from Pearla Nesher interesting.  She seems to question why learning mathematics, as a language, should be any more difficult than learning “ordinary language”.  It would appear to me that one possible reason why learning mathematics as a language is more difficult than learning English, for example, is because for many people, speaking/listening, and writing in English is something that we practice almost all day long.  We even use our dominant to make sense of other “languages” such as mathematics or science.  If we were to use mathematics as our primary language, and learn English and all of our other subjects within the parameters of the language of math, I would no doubt believe that as a society we would be much more proficient in it.

Alan Bell discusses how, as teachers, we must redirect a student’s thought process, if a mistake is being made, immediately, rather than the next day in order for the child to learn effectively.  As I read this, I reflect on my own teaching practices and how I could implement this strategy into my classroom of 29 students (half of which are grade 4’s and the other half are grade 5’s).  In a typical split class, during math class, I must teach to one group while the other works.  Once my instruction is complete with one group, I will assign some work for them, while I teach to the other group.  Due to the nature of this type of class, it can be very difficult to find time to observe every student’s thought process.  Typically, we mark our work the following day.  According to Bell, this may not be very effective.  While I agree that it’s not ideal, I wonder how his mentioned strategy could be implemented in a split class.