Response to "Mathematical Pedagogy from a Historical Context" - Frank Swetz

This article talks about the art of communicating mathematical ideas to novice mathematicians. Historically, those with the "privilege of  knowing" mathematics, often teachers, conveyed these ideas to an audience by tailoring and organising their thoughts in the form of the following sequence of instructional techniques: the use of an instructional discourse, a logical sequencing of mathematical problems and exercises, and employment of visual aids. The author states that current pedagogical practices in mathematics  are based on these historical techniques.

I really like the following quote from the article: "an initially passive observer who is gradually drawn into the train of instructional thought and hopefully led to the active realization or discovery of the mathematical concepts in question." It is interesting to note how a passive observer is assumed to be lacking in mathematical knowledge and that a teacher will be necessary to lead the observer to discover (if there is such a thing) mathematical concepts. This instantly creates a binary power structure between the teacher and the observer (clever/stupid, leader/follower, sophisticated/unsophisticated). The mere thought of leading someone to discover some new mathematical concepts is scary, especially in a classroom setting. Also, the phrase, "initially passive observer," implies that the observers are mentally lazy. What if the teacher fails to lead to the desired 'destination'? Or, more importantly, what happens if the students are unable to or don't want to realize or discover the euphoria of understanding mathematical concepts?

Yes, I like it when the art of educating our children hinges on 'hope'. Can we, as educators, rely on 'hope' in order to communicate mathematical knowledge to our students?

Response to "Charting the Microworld Territory" - Healy & Knigos

This article traces the historical evolution of digital microword in mathematics education from a theoretical perspective. Microworlds are generally classified as a form of learning environment (half-baked educational technology) linked to pedagogical methods that were based on Papert's conception of sense-making, Vygotsky's notion of zone of proximal development, and Piaget's individualistic approach to learning. More recently, microworlds are defined as educational computational environments embedded in technological tools and devices geared towards non-technical people to explore, construct, manipulate, and interact with programmable objects in order for learners to make sense of mathematical learning. Half-baked microworlds can be thought of as a communal design space where participants can redesign, reform, and restructure various aspects of the initial design to suit different scenarios. But, of course, despite improvements in every aspect of technological tools and devices over the years, the authors raise a critically important issue as to whether how relevant technological tools and devices are to learning mathematics today. The authors seem to acknowledge that "the practices in the world’s mathematics classrooms have changed rather less."

Their grim observations regarding the use of classroom technologies is rather discouraging. I am not sure if I would fully agree with their assessments targeting universal practices in mathematics classrooms. After all, how is it even possible to come to this conclusion based on their two half-baked examples from Brazil and Greece. I guess this is the drawback of conducting qualitative research. That is, it would be meaningless to extend local understandings to global understandings. 

Readings/Textual Analysis of FLM V17 - 2

It is interesting to note the frequencies of the following terms:

1. math* - 548 times (about 2.5% of the total number of words)
2. problems/examples -  274 times (about 1.18%)
3. educators/teachers - 231 times (about 1%)
4. student* - 198 times (about 0.86%)
5. research* - 106 times (about 0.46%)

Superficial observation of the word frequencies may suggest that the 12  articles published in this volume were mostly related to regarding, reducing, or treating mathematics education in terms mathematical terms, i.e. mathematizing. It also correlates well with Bingjie's findings on FLM (V29 - 1, 2, 3) where she found that about 32% of the articles were related to teachers' development, beliefs, knowledge, or thinking.

The same tessellation was used in both Vol. 1 #1 and Vol. 17 #2. I wonder if this tessellation mutated into other forms and appeared elsewhere on the front pages of other FLM issues.