"Research on Visualization in Learning and Teaching Mathematics" by Norma Presmeg

In this article, Presmeg talks about visualization as a significant research area in mathematics education. Here, visualization encompasses sight, hearing, smell, taste, touch, and their interactions and interconnections. And mathematics as a subject consisting of "diagrams, tables, spatial arrangements of signifiers such as symbols, and other inscriptions as essential components." It appears to me that the theoretical underpinning of visualization in mathematics education is based on the constructivist approach. Presmeg alludes to the rise in interest in constructivism, which is somewhat of a deviation from the influences of existing teaching and learning theories based on behaviourism, cognitivism, and communication.

Presmeg takes the following position in this article: when a learner creates a mathematical inscription, the learner generates, guides, and constructs visual images in their mind. This author prefers to use inscription rather than representation, because the term 'representation' was insufficient to provide an accurate definition. Visual images are assumed to be mental constructs depicting visual or spatial information. Presmeg identifies numerous types of visual images: Figurative (purely perceptive), Concrete (purely in the mind), Operative (operates on/with the carrier), Kinaesthetic (physical movement), Relational (transformation of concrete carrier), Dynamic (image itself is moved or transformed), Symbolic (formulas and spatial relations), Memory of formulae, and Pattern (pure relationships stripped of concrete details).

Presmeg's research on visualizers (learners who prefer to use visual methods when there is a choice) reveals that: 52/54 visualizers used concrete imagery, 32 preferred memory images, pattern imagery (18), and kinaesthetic imagery (16). It was surprising to note that dynamic imagery was used but rarely. This is a surprise because both pattern and dynamic imagery are thought to be involved with rigorous analytical thought processes. This implies that learners are capable of generating visual images, but are unable to use these images for analytical reasoning.

Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. Handbook of research on the psychology of mathematics education, 205-235.

"Constructivism: From Philosophy to Practice" by Elizabeth Murphy

Here, Murphy provides another view of constructivist perspective. According to her, within a constructivist perspective, knowledge is constructed by individual learners through their interactions with their environment. I think that environment, in this case, may include everything that is related to the physical, social, and cultural surroundings of the learner. In addition these, in a classroom setting, the term environment may also involve various interactions, interpretations, experiences, perspectives, and representations. Very similar to that of Jonassen, Murphy clearly differentiates between constructivist teaching approaches and transmission-type teaching models. Murphy goes on to suggest that "learners actively construct knowledge in their attempts to make sense of their world, then learning will likely emphasize the development of meaning and understanding."

One of the reasons often cited by skeptics of constructivist approach to teaching is that constructivism does not provide a teaching model for classroom implementation. Murphy thinks that this is good news for teachers interested in implementing constructivist approaches. This is primarily because teachers have the flexibility of designing and implementing innovative learning environments, including novel ways of exploiting technological tools and devices.

Before suggesting ways to incorporating technology in the classrooms, Murphy briefly touches on eight different types of constructivism (radical, social, physical, evolutions, postmodern, social, information-processing, and cybernetic systems). Then she goes on to summarizing the characteristics of constructivist learning and teaching as suggested by other researchers in constructivist theory. In addition, she provides a long checklist (18 points) of this theory can be applied to projects, activities, and learning environments

Murphy, E. (1997). Constructivism: From philosophy to practice.

"Investigation of interactive online visual tools for the learning of mathematics" by Jacobs

This article is related to designing online modules for learning and teaching, specifically for differential equations. These modules are designed in such a way that mathematical content is segmented into manageable chunks. Jacobs suggests that the advantage of breaking the content into smaller segments enable students to refer to, review, or jump forwards and backwards. According to Jacobs, the number of people embracing, studying, or appreciating mathematical knowledge declines each year. As a result, the author thinks that mathematical content should be presented in ways that maximize learning, enjoyment, and satisfaction.

Very similar to the previous article, this author seems to be in favor of constructivist student-centered approaches to teaching and learning mathematics. Similar to other researchers, Jacobs also provides a definition for constructivist perspective using three principles: 1) individual learners form their own understanding and representation of knowledge, 2) learning occurs when a learner experiences a dissonance, and 3) learning occurs within a social context. With this definition in mind, the author suggests that interactive online visual tools provide a platform for assimilating and accommodating existing/new mathematical concepts.

Jacobs points out that online learning tools, such as online instruction, applets, and graphical user interfaces, have the capacity to provide self-explanatory learning material. In these tools, there is a tendency to minimize textual details and replace these with colorful graphics. A reason for this seems to be that in constructivist learning environments, there is a tendency to provide tools that would increase active engagement. High quality interfaces with visual appeal (sliders, color, etc.) are deemed to maintain attention of the learners.

Jacobs, K. L. (2005). Investigation of interactive online visual tools for the learning of mathematics. International Journal of Mathematical Education in Science and Technology, 36(7), 761-768.

"Integrating Constructivism and Learning Technologies" by David J. Jonassen

This article talks about designing and integrating meaningful learning environments with learning technologies through the lenses of activity theory, distributed cognition, and situated learning. Jonassen appears to make a clear distinction between constructivist learning environments and transmission-type learning environments. In traditional trasmissive instructional models, improvement in learning was assumed to occur through effective communication of ideas from the teacher to the learners. That is, improvement of learning was embedded in communication, behaviour, and cognitive theories. According to the communication theory, knowledge flowed from one person to another, and thus communicational effectiveness and efficiency were the goals of transmissive-type teaching model.  Behaviour theory assumes that learning occur through observation and change in behaviour. Cognitive theory assumes that improvement in learning occur through practice. Unlike the traditional instructional-type models that were teacher-centered, Jonassen appears to call for learning environments that combine several learning theories (everyday cognition and reasoning, activity theory, ecological psychology, distributed cognition, case-based reasoning, etc.)  incorporating technological tools and devices. Jonassen identifies several technologies that may play a role in meaningful learning, which involve "willful, intentional, active, conscious, constructive practice that includes reciprocal intention-action-reflection cycles." Just to name a few, these technologies are:
-- Computer-Supported Collaborative Work (CSCW) - Can be used for group interactions to inspect, modify, and confer with the group members (blogs, social media, GeoGebra, etc.)
-- Electronic Performance Support Systems - Designed to provide interactive advice, demonstration, descriptions, feedback, etc. (manuals, spreadsheets, etc.)
-- Virtual Reality/Microworlds - These are exploratory and discovery spaces for simulating, observing, or analyzing results and hypotheses.
-- Videography
-- Multimedia
-- Knowledge-Building Communities
-- Mindtools

Jonassen, D. H. (1999). Designing constructivist learning environments. Instructional Design Theories and Models: A New Paradigm of Instructional Theory, 2, 215-239.

"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.