"Applications GeoGebra into Teaching Some Topics of Mathematics at the College Level" by Ljubica Diković

This paper presents results from a didactic experiment in teaching calculus at a business-technical school in Uzice, Serbia. The objective of this study was to determine whether there were any significant differences in Students' math achievement before and after GeoGebra-based instruction in important concepts in a calculus class. With a sample size of 31 students (19 females and 12 males) and Cronbach Alpha of 0.784 for the instrument, GeoGebra applets were designed to provide alternative representations to already existing representations of calculus concepts. In addition to traditional lectures and problem-solving sessions, these participants also participated in mathematical experimentations with GeoGebra, i.e. working in a computer lab to carry out investigations, individual research, and group work. Pre- and post-tests were administered at the beginning and end of the course. A paired-samples t-test showed that the mean scores were statistically significantly higher at the end of course.

It is interesting to note that the author fails to mention whether the distribution of the data satisfied all the underlying assumptions of this test (eg. normality, variances, independent, etc.) Solely based on the small sample and the instrument, the author goes on to suggest that the use of GeoGebra applets had a "positive effect on the understanding and knowledge of students" when teaching differential calculus (slope of tangent lines, connection between slope of the tangent line and graph of the gradient function, continuity/discontinuity of functions, connection between continuity and differentiability, etc).

On a positive note, citing other research findings, the author suggests several advantages of using GeoGebra for teaching and learning mathematics: user-friendly interface, guided discovery learning through multiple presentations and experimentations, manipulate mathematical objects, cooperative learning environment through task-oriented interactive situations, etc. Some of the disadvantages are that the uses GeoGebra require basic skills in algebraic commands, minimal training, and that independent explorations and experimentations may be inappropriate for some students.

Diković, L. (2009). Applications GeoGebra into teaching some topics of mathematics at the college level. Computer Science and Information Systems,6(2), 191-203.

"Teaching and Learning Calculus with Free Dynamic Mathematics Software GeoGebra" by Markus Hohenwarter

This article talks about some of the advantages of embedding technology in the classrooms for teaching and learning mathematics by using GeoGebra. The author of this article, Markus Hohenwarter, is the creator of GeoGebra. As part of his graduate studies in mathematics education and computer science at the University of Salzburg in Austria, during 2001/2002, he developed GeoGebra with the support of Austrian Academy of Sciences. From 2006 onwards, development of GeoGebra continues at the Florida Atlantic University.

GeoGebra is an open-source dynamic software that is designed for teaching and learning mathematics for all levels. It is based on the principles of a constructivist learning environment, which connect geometry, algebra, calculus, and some other areas of mathematics into an easy-to-use software environment. Since this software is free of charge, students and teachers can use it in the classrooms and at home to learn and teach calculus, for instance.

This article briefly looks at other powerful dynamic learning environments such as Geometer's Sketchpad and Cabri Geometry. These two software packages are not only proprietary, but also require considerable amount of time commitment for mastering, which renders these suitable for only higher education. And also, these packages are geared towards addressing one area of mathematics (geometry, algebra, calculus, or statistics). In contrast to these commercial packages, teachers, educators, and researchers develop GeoGebra worksheets and methods. For instance, fully functional dynamic calculus worksheets can be created on the fly in the classrooms or customize existing worksheets, created by others, for classroom use. That is, dynamic and interactive figures, functions, constructions, or objects can be manipulated by moving points, changing graphs, or manipulating parameters with sliders.

The author goes on to suggest that by providing such a dynamic teaching and learning environment, the process of students' learning becomes mathematical experiments, i.e. a student centered approach to learning mathematics.

Hohenwarter, M., Hohenwarter, J., Kreis, Y., & Lavicza, Z. (2008). Teaching and learning calculus with free dynamic mathematics software GeoGebra. In 11th International Congress on Mathematical Education. Monterrey, Nuevo Leon, Mexico.

"Proof, Explanation and Exploration: An Overview of Mathematics learning and Teaching" by Gila Hanna

In addition to looking at the role of proofs in mathematics education, this article also talks about the importance of dynamic geometry software in teaching mathematical proofs. According to Hanna, as educators in mathematics, a key task for us is to first understand the role of proofs in teaching mathematical concepts and ideas so that we are in a better position to promote conceptual understanding of mathematics through proofs.

For instance, the use of dynamic geometry software can provide more effective ways of challenging students and also teaching sound and logical mathematical reasoning to produce valid proofs for geometrical propositions and objects. The author goes on to suggest that dynamic software tools can not only provide an avenue towards heuristics, exploration, and visualization of mathematical concepts and ideas, but these tools can also pose challenges to teaching proofs.

The author notes that the term 'mathematical understanding' is somewhat elusive. By this, I think that the author is wondering what it really means to gain an understanding of mathematics, here, through proofs. For instance, expert mathematicians view proofs to be most valuable when it leads to understanding. That is, understanding proofs assists them to clearly verify, explain, systematize, discover, construct, communicate, and explore mathematical concepts and ideas. However, students and novices view proofs as alien, unnatural, not useful, or simply too boring as a human activity. That is, proofs are not appealing to the learners. One reason that novices feel this way about proofs is because, according to Hanna, teaching of proofs in traditional classrooms amounts to rote learning of mathematical proofs.

In order to move away from learning mathematical proofs through lower order thinking skills, dynamic software tools are available in order to explore, discover, and construct mathematical proofs, properties, and conjectures through dynamic investigations, explorations, demonstrations, constructions, problems, arts, or puzzles. For example, these tools can provide students to "check a large number of cases or even ... an infinite number of cases" through dynamic manipulation of mathematical objects.

It will be interesting to see if students' views of proofs as intuitive collection of properties change to that of viewing proofs as a system of theoretical ones through dynamic geometry software.

Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational studies in mathematics, 44(1-2), 5-23.

"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.