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.

For March 1 - Bingjie Wang and FLM 50th edition

Table of contents: 
What levels of schooling or age groups do they have as their theme (if any at all!) 
One article mentions work with grade 2 students, but otherwise, there aren't many. It seems like the journal is mostly isolated from classroom education

What kinds of issues are addressed? How are these distinctive?
The issues addressed are ethnomathematics, classroom teaching, and existence of mathematics education. 

Articles:
How long are the articles? 
About 5 pages each


Are they usually illustrated (and if so, how?)
Geometry and ethnomathematics articles have illustrations if any; most articles end with a quote or an image. 


Are there a lot of references cited?
There are less than ten references for each article. What is the citation style? I looked into Turabian, MLA, Chicago, AMS... it doesn't seem like any of these! 

Are there subheadings on the articles? If so, are they the subheadings that you expect, or not? No subheadings 
What language is the article in? English

Overall:
What is on the front and back cover, and why? 
Front cover - big 50. 
Back cover - contributors, I believe. The 50 on this front cover is only in this particular journal; other journals have an image on them

What did you learn from the author identifications? 
More men than women contributed in the fiftieth issue, which is similar to the first issue. In the 100th issue, however, there were much fewer men than women contributing.

What about the material on the inside of the front and back covers?
Front - table of contents, I believe
Back - Suggestions to contributors

Is there any material between the articles? If so, what is it? What kind of tone might it set?
None

Additional notes after taking a look at Bingjie Wang's article:
- Relatively easy to read
- No appendices in the articles - all data used is cited in-text
- Some articles written in first-person
- Focus on old vs new of mathematics education
- According to Wang, 96% of contributors to FLM are university professors
- The writing is not as dense as I've read
- There is a mix of research, opinion, and reflection in the article

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Looking at Bingjie Wang's article, I was surprised to see that only 4% of FLM's contributors are graduate students, as was similar with ESM and JRME, with university professors making up the majority of the other groups. Compare this to an only 60% contributor percentage in JME, where a much larger portion (than compared to the other journals) of teachers contribute (20%). What does this mean about the standards set forth by a journal? What is the rejection rate for each of these journals for graduate students? Is it higher to FLM, or are students not submitting much to FLM in the first place? Is there an invitation to teachers to write in FLM? I would be interested to discuss access to publishing and the acceptable styles of writing/publishing, etc. This reminds me of a story I heard in another class about Dr. Peter Cole, for example, who currently is lecturing (and researching?) at UBC. He completed his PhD thesis at SFU because although he started out as a PhD candidate at UBC, there were far too many standards UBC had for the way a thesis had to be written, etc. I am curious about how standards are established for publishing, and if there are any leftover trends from science journals embedded in mathematics education journals. Certainly, the foundation for mathematics education as a research area was still brought up in some articles in the 50th issue, so I wonder how many other trends are leftover from prior journals.

For February 22 – Whiteley Response – Decline and Rise of Geometry

This article explored multiple topics regarding geometry, its declining presence at a secondary and post-secondary level, and the case to revive it in curriculum. Whiteley begins his article by describing the decline in geometry first, then explaining the different types of geometry. So it is difficult to say if the statistics he provided at the beginning of the article refer to a certain type of geometry (suppose, Euclidean) or if they provided an overall picture. There were multiple applications of geometry brought up by Whiteley and were a large part of his case for continuing geometry in the curriculum – geometry is used in establishing the structure of chemical formulas, designing vehicular components, etc.

I must admit, my initial reaction to this piece was mostly indifference. I hadn’t seen much geometry in math in school beyond basic Euclidean geometry and basic geometric proofs (which, granted, I still don’t remember and was never really taught, just expected in some form).

I think that with geometry has made its way out of the curriculum for two reasons. For one, geometry is no longer required in university unless you major in mathematics. It is easy to argue that this was taken out of the curriculum while things essential for calculus were maintained. The second reason is the ease with which one can now teach mathematics in the school curriculum. In my discussions with other teachers, many did not have a great deal of mathematics experience (much less a degree) but instead only had a calculus course. For calculus, little or no geometry is required. So this would then begin to reinforce the lack of experience with geometry and decrease the exposure to the subject beyond a surface depth.

If geometry is being used in specific applications as Whiteley says, then I would sooner see geometry taken out of the curriculum completely and replaced with some basic matrix algebra. Matrix algebra is a basic skill in a variety of backgrounds, yet it is not taught at any level (except in the International Baccalaureate Diploma Programme). Matrix algebra is also easy to understand, provides a whole range of depth in topics, number of solutions, whether or not solutions can be computed – in short, more immediately accessible topics arise with much further-reaching applications.

For February 15 - Borwein - The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof

It seems almost inevitable that a paper advocating for the use of proof and computer analysis would include at least a brief overview of the controversy surrounding the Four Colour Theorem. Borwein (2004) writes that the proof and computer analysis can act harmoniously to support mathematical development. With a constructivist point of view, he argues that mathematics is a mainly human subject with some room for empirical testing and knowledge. Like science, Borwein argues that mathematics is an endeavour which, while it can be self-sufficient, can also be found to be false or incomplete. Computer analysis and computation can help ease the process of proofing by hand, which is supported by the examples in number theory which he provides for the reader.

Perhaps this speaks to my foundation in mathematics, but I have always found comfort in fields of mathematics in which calculations/programming supported mathematical understanding. I found that one of my most enriching courses (and one which seriously made me consider re-trying computer programming in the future) was a course which involved mathematical computation, Math 210. Although seen as an easy-way-out for avoiding the computer programming requirements put out by the math department, Math 210 was challenging because I had to know what to program. The programming helped me verify lemmas, create graphical arguments, and show different results visually - some of these I would not have understood as well as I do now, had I not had the opportunity to program them in two easier languages, Maple and Matlab.

What confused me with Borwein, however, is his claim that "extraordinary speed and enough space are prerequisite for rapid verification and for validation and falsification ('proofs and refutations'.) One cannot have an "aha" when the 'a' and 'ha' come minutes or hours apart" (p. 12). Borwein is likely aware of the misplaced nature of intuition in an empirical. This is to say, Borwein seems to encourage that mathematics become a "serious replicable enterprise" (p. 14). The creation of a scientific enterprise does not require the same kind of intuition as mathematics may, and truly, these two fields are separated except by applications of mathematics (engineering, computation, proofing in statistics, etc). The very nature of knowledge generation differs in mathematics from the positivist one of science, as the former requires personal interaction and interpretation of the problem. The final product in mathematics - in this case, a proof - often does not include evidence of the doer of mathematics, only their interpretation. It is, then, not the construction of the mathematics itself, but the choice of inclusion/omission of content and material that constructs mathematics rather than allows one to discover it.

For February 8 - Response - Gerdes

The Gerdes article investigates the politics around mathematics literacy of persons in Mozambique and deals with issues around colonization and the eradication of “indigenous mathematics” (p 138), where an implicit mathematics was disregarded and replaced with rote learning and memorization in schools. Gerdes’ use of cultural methods for teaching geometry involves imposing European mathematics over top of cultural interpretations of geometry (i.e. the use of mathematics in everyday life).

            As a White mathematics educator of the dominant culture, I know I must tread carefully when discussing issues of colonization. Although it is critical to retain aspects of culture in teaching, it would be as oppressive to maintain only a form of indigenous mathematics, as it would be to teach the colonized mathematics curriculum. By this, I mean that reducing persons to indigenous mathematical knowledge and restricting what other mathematics they learn sustains a colonial dominance in the opposite direction. Gerdes’ suggestion is just as problematic, as his imposition of European mathematics to build “cultural-mathematical confidence” (p. 144), as he calls it, colonializes the everyday mathematics and shows that, “See, your mathematics is acceptable and correct in the way WE (White) do mathematics!” I will nod in Gerdes’ directions for briefly mentioning an in-class question that could be used to generate discussion around the choice in use of mathematics, “Which method has to be preferred for our primary schools? Why?” (p 149). This provides the opportunity to explore the ownership of knowledge, which is at the heart of creating change.

By exposing persons to mathematical ideas in “European” mathematics (that is, the current mathematical structure in which gender, culture, race are normatively White), if university education is truly to be the focus (again, White institutions), then we must prepare students for this institution. Building bridges, as Gay and Cole suggest, is also problematic, as the bridge-builder in the connection between indigenous and White mathematics establishes that there, indeed, exists a rift between the two, and hence also owns the divide between the two; I am sure those in Mozambique did not decide that their mathematics were “different” from the White form of mathematics.

But what, then, is the point of teaching ethnomathematics, if we are just to Whiten individuals in their respective institutions? It is then, my opinion that the institution must change. For if the expectations maintained in the institution are dominant and remain unchanged, then teachers will always hesitate to teach traditional ways of knowing. The defensive reason will be, “But you need to know this method for university mathematics”.  The types of mathematics that are decided to be “valid” and “invalid” in a university, as Burton (1995) writes, remains pre-decided and has social and cultural implications that remain ignored due to the “objective” characteristic of mathematics. This falsely marries the content with the interpretation, the latter of which is not objective.