"Applying Constructivist Theory to Practice in a Technology-Based Learning Environment" by Patricia Forster

This article uses the constructivist teaching approach to implement and analyse the effectiveness of technology-supported lesson plans in four secondary mathematics classroom settings in Western Australia. The author uses Von Glaserfeld's article (An exposition of constructivism: Why some like it radical, 1990) and Noddings article (Constructivism in mathematics education, 1990) as a premise in order to design this research study for enhancing students' learning of matrix, exponential functions, and descriptive statistics. According to the author, technology "could relieve the burden of calculation and allow the concepts involved to be approached in multiple ways: visually, numerically and symbolically." (p.82)

The research was carried out over a five-month period in several stages. The research involved students learning about algebraic and geometric properties of matrices with pre-designed worksheets using technology. Initially, through classroom observations, students' written documents, fieldnotes, and interviews, the author found that a) more support was required by the students when operating technologies, b) knowledge gaps were hard to patch-up, c) students tended work individually, d) reluctant to seek assistance from peers, and e) adopting mechanical approaches to completing the tasks. However, in the later stages, the study shows that students can become actively engaged in collaborative learning when technology is used in the classrooms.

It is not surprising that the findings from this study also support that competitive atmosphere (generated by assessments) may not be conducive to a collaborative teaching and learning environment.

Forster, P. (1999). Applying constructivist theory to practice in a technology-based learning environment. Mathematics Education Research Journal, 11(2), 81-93.

Response to "Editors’ Introduction: What Is Mathematical Visualization?" by Zimmermann and Cunningham

After reading this introduction, I am not sure that I fully understand what is meant by visualization in mathematics education. The definition and interpretation of the term "visualization" depends on who is describing this term. First, mathematics educators interpret visualization to describe "the process of producing or using geometrical or graphical representations of mathematical concepts, principles or problems, whether hand drawn or computer generated." Second, for scientists, the process of visualization enhances "scientific discovery and fosters profound and unexpected insights." Last, for psychologists, this term merely represents an individual's "ability to form and manipulate mental images."

Psychologists' narrow definition of visualization has major impacts on mathematics education. Psychologists often test their subjects' capacity to "form mental images" through answering a series of questions. These test results are then analysed, repeated, and observed in order to provide their insights to the process of visualization. However, in terms of mathematical visualization, the editors, suggest that manipulating mental images without the use of paper-and-pencil or computers to be an artificial experience. When curriculum policies are based on the so-called objective findings of psychological studies, psychologists’ interpretations and recommendations may be implemented. What may not be taken into account when curriculum decisions are made is the fact that the objectives of the psychologists' and mathematics educators' research interests. In terms of mathematical visualization, the objective is to be able to generate "an appropriate diagram to represent a mathematical concept or problem and to use the diagram to achieve understanding." This implies that visualization in mathematics education can be thought of as a tool for forming mental images (not an end) and for "mathematical discovery and understanding."

Zimmermann, Walter, Cunningham, Steve. Editors’ introduction: What is mathematical visualization 1991

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.

Response to "Why you should learn geometry" - Walter Whiteley

This article is a response to another article that was published in the print edition of the Los Angeles Times. The published article, “Why you should learn algebra”, was written by an English professor, David Eggenschwiler. Dr. Eggenschwiler's article was addressed to the Times readers' complaints about the usefulness and necessity of algebra in high school curriculum.

In Dr. Whiteley's response, he notes how algebra is associated with mathematics. The notion that studying algebra fosters rational, abstract, and systematic ways of thinking, reduces the significance of other equally important areas, specifically geometry, of mathematics. Several prominent figures, like Michael Faraday and James Clerk Maxwell, used alternative approaches to provide effective reasons for their pioneering work.

Dr. Whitely seems to suggest that exclusive promotion of traditional views on learning mathematics may be counter-productive. This could be because of potential talents and worthwhile contributions, from learners who excel through alternative ways of doing or understanding mathematics, may go unrecognised or disregarded.

Contact Info. . .

Hi Murugan.


Can you send me your e-mail?

Mine is dharris@sd44.ca


Cheers,
David Harris