14 - For April 2 - Swetz, To Know and to Teach

Swetz describes the origin of several modern pedagogical tools used in mathematics teaching. One example of these is that of the instructor teaching to a neutral (the adjective Swetz used) student the information gained – a discourse, as he puts it, where there is a two-way (or in some cases, one-way) stream of information.  Although it originated as (what comes across as) an oral dialogue, it eventually became recorded – at first, it was recorded in verse, and depended heavily on reader intuition to solve (sometimes aided through the use of colour), and evolved into a set of collected problems with detailed solutions.

There were two comments in particular that Swetz makes. The first is the following:

“A collection of mathematical problems does not, in itself, preclude the possibility of pedagogical designs. While some ancient texts may appear to be merely collections of problems, those problems were carefully chosen and through their conception and organization, attest to their author’s ability as a teacher” (p. 77).

I found this interesting, and it reminded me, as a beginning teacher, how often I used to get bogged down by finding information to teach in a class – we were always warned not to re-invent the wheel, as there was a wealth of information “out there. While this may be true, sorting through all sorts of information takes a long time. It also occurred to me that it would be difficult to have mere “collections of problems”, because (I assume) there wasn’t as much collective mathematical content as there is today. I imagine it would have been the author’s or teacher’s personal research that they were sharing with their students.

The other quote that caused me to think was the off-hand comment at the end of page 80, when Swetz explains that authors of Babylonian, Chinese, and Italian pedagogical background “were, quite simply, concerned teachers writing for the learning needs of their students”. Beyond all the politics around the purchasing of textbooks and cost of individual textbooks, which was my first thought, re-reading the sentence narrowed my focus on “learning needs”. It is interesting that so early into pedagogical development and sharing of knowledge, there was already a determined “learning need”. What learning needs did people have then? Who was it that needed to learn? I was under the impression that only the social elite would have gone to school to learn mathematics (but Socrates teaching a slave boy about doubling area in a rectangle by manipulating its dimensions made me reconsider).

For March 25 - Healy and Kynigos

When I began reading this article, I have to admit I had little understanding of what was being discussed in the article – I was grateful for the definition of “microworlds”, provided by the author, as “self-contained worlds where students can ‘learn to transfer habits of exploration from their personal lives to the formal domain of scientific construction’ “ (Healy & Kynigos, citing Papert, 1980). The evolution of microworlds in the scientific age continued in a new way when computational environments were introduced into mathematics education. The interaction with these environments would allow students to receive immediate feedback on programs. These microworlds would be interpreted through “body syntonicity” and “ego syntonicity”, according to Papert (1980). The former of these would relate knowledge to the physical, while the latter would relate knowledge o the mental/personal, basing this on the idea that students develop schemes of understanding based on personal interactions. Another approach for interacting with microworlds is an instrumental approach instrumental approach with respect to a microworld, which allows for both the student and the instrument being used to develop. Microworlds, for example, may be computer apps with which students interact; this was explored by using digital media as a method for displaying rationality vs irrationality of integers.

I find one line in this article interesting, in particular: the authors explain their reasoning in introducing microworlds where, “it is not that the microworld substitutes the teacher” (pg. 65). Why not? Is the author saying that the construction of a microworld is there to facilitate the zone of proximal development and to facilitate the microworld? Is this because the authors are afraid of backlash, or because the authors believe that a teacher is truly important in a classroom?

Another thing that is interesting is the concern around the preference or ability to use the different syntonicities. For example, the use of a visual instrument to investigate a virtual microworld would be limiting to students who are visually impaired.

Overall, I found this article really difficult to follow before the example was provided; understanding what a microworld looks like before going into theory would’ve helped, but the image provided a lot more context. Some of the themes brought up here remind me of a conversation I had at the beginning of the year at a “meet the faculty” event. I spoke with one professor in the digital technology faculty in EDCP (I don’t remember the exact name of the faculty); we discussed something called “technology death”, or the rejection of technology out of fear of something unknown. What I recognized in that discussion is that the fear of a new, unknown technology (and the implications of using it immediately and readily without much consideration for the consequences) could have been eased much more gently had more articles like this been discussed. Meanwhile, many teachers do not use technology in their classrooms anyway, whether due to a learning curve or a hesitation to use an unknown medium.