"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.

"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

Response to "Proofs as bearers of mathematical knowledge" - Gila Hanna andd Ed Barbeau

This article is composed of two parts. The first part of the article delves into Yehuda Rav's paper, "Why do we prove theorems?" According to Rav, who completed his undergraduate and graduate studies at Columbia University, proofs can be used as "methods, tools, strategies and concepts for solving problems." Rav identifies (or differentiates) proofs to be either derivation proofs("application of rules of logical inference") or conceptual proofs. A conceptual proof "does not have a precise mathematical definition, mathematicians would readily understand its overall structure and could verify the correctness."

The second part of the article discusses about proofs as being expressions of mathematical knowledge. Numerous aspects of proofs, such as cognitive, intuitive, reasoning, logical, etc., are identified by Hanna and Barbeau in order to teach mathematical methods and strategies by mathematics educators. Several examples are provided as case studies to show how proofs may be incorporated into a secondary level for improving problem solving techniques. The authors acknowledge that their outlines (suggestions) were not tested in actual classrooms.

The authors of this article seem to approach the issue of proofs from a theoretical point of view, which may be in line with Kilpatrick's line of thought for an effective research program. 

Response to "Models and Maps from the Marshall Islands: A Case in Ethnomathematics" - Marcia Ascher

This paper focusses on Marshall Island navigators' (natives) use of stick charts to examine their experiential and representational knowledge with modern scientific and mathematical ideas and knowledge. These charts were made of palm ribs, coconut fibre, and shells attached and organized to display sophisticated geometrical models of ocean wave propagation. The abstract models represented shapes and motion of waves and land/sea features, which were used for navigating vast area and for training future navigators.

The stick charts were similar to diagrams on a modern-day blackboard. Information that was considered essential where represented by geometrical figures and objects. It is interesting to note how valuable this knowledge was when individuals learn to understand these representations. Marshall Island navigation techniques were kept secret and their future navigators, from noble lineage, were selected and trained by master navigators.

As knowledge seekers, today's students are selected, coached, and trained by educational institutions. How is today's tradition of gaining specialized knowledge in an educational institution (formal) different from other tradition?

Response to "A Linguistic and Narrative View of Word Problems in Mathematics Education" - Susan Gerofsky

In this article, the author looks at the nature of word problems in mathematics education by using three methods. The three methods are pragmatics (the branch of linguistics dealing with language in use and the contexts in which it is used), discourse analysis (approaches to analyze written, vocal, or sign language use or any significant semiotic event), and genre studies (a branch of general critical theory in several different fields, including the literary or artistic, linguistic, or rhetorical). Among other structures of word problems, the author describes the break-up of generic word problems into three components ("set-up", "information", and "question").

This is how I generally tried to solve word problems. I wasn't interested in the problem set-up. I would try to find all the variables involved in solving the problem and then identify the relationship to find the answer. Once I come up with an answer, I would go to the back of the book to check the answer. If my answer was identical to the book's, I would move on. Otherwise, I would re-read the problem to find where I went wrong. Whether the problems were word problems or abstract algebraic problems, I employed the same techniques. This could be why I hated word problems. Whenever I tried to come up problems to solve in the class, the problems were often times identical to those found in textbooks. The author cites John Lave to describe this phenomenon.

Finally, the author goes on to wonder why word problems in mathematics have persisted for so long. I am curious to know if the author has found some answers to the purpose of word problems in mathematics education.

Response to "Psychology and Mathematics Education" - Efraim Fischbein

This article primarily talks about cognitive psychology (the study of mental processes such as "attention, language use, memory, perception, problem solving, creativity, and thinking") and psychology of mathematics education. In the article, the author states two reasons for the inability of Piaget's ideas to provide a platform to link cognitive psychology and mathematics education. The reasons were that Piaget was not interested in the effects of teaching on children's learning and that Piaget believed that intellectual development was due to logical thinking, which implied that mathematical learning cannot occur from a local context to a global context. The author goes on to argue that employing scientific research strategies may have "played an important role in blocking the advancement of mathematics education." Thus, the author recommends that researchers in mathematics education identify new concepts and terms that are absent in psychology for describing interactions between psychology and mathematics education.

It is interesting to note that research findings based on psychological (scientific) approach to mathematics education may not be generalizable. There could be a connection to Kilpatrick's scepticisms about research findings in mathematics education. That is, Kilpatrick seems to suggest that it is not surprising why the research outcomes are ineffective.

Response to "Muddying the Clear Waters" - Herbel-Eisenmann

The  first stop occurred when researchers and teacher-researchers in the article seem to suggest that revoicing as "something that all teachers should do when they teach." Isn't this one of the things that we, as teachers, do with or without knowing much about research findings on revoicing?  Isn't this what we do on a daily basis when we interact with one another? The renewed research interests in "the idea of revoicing as a potentially powerful discourse" appears to be led by university researchers. In this case, teachers were involved as participants in the research and so, approaching from Kilpatrick's point of view, findings from this research may positively influence school practice.

"Context" seems to be in the way of many research findings. May be this the beauty of qualitative research when studying the lived-experiences of the participants in their day-to-day lives. That is, any interpretation of a study's outcome "depends" on so many factors.

Response to Kilpatrick's

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The Reasonable Ineffectiveness of Research in Mathematics Education
Author(s): Jeremy Kilpatrick
Source: For the Learning of Mathematics, Vol. 2, No. 2 (Nov., 1981), pp. 22-29
Published by: FLM Publishing Association
Stable URL: http://www.jstor.org/stable/40247734
Accessed: 07/01/2015 04:14

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By glancing over the title and the first paragraph of the article, I think that the author would be discussing about the existing state of research in mathematics education. In addition, the author will be using two assumptions, a) ineffectiveness of research in mathematics education is well understood and real and b) the understanding of the ineffectiveness is real, to justify why research in mathematics education is ineffective. My expectation is that the author would provide reasonable insights, which may be of interest when conducting research in mathematics education.

The article begins by questioning whether research in education is effective or has any value. The author uses Suppes' cautious optimism and Scriven's rigorous analysis to question the effectiveness or pay-off of research in education on classroom practice through Suppe's book, Impact of research on education: some case studies. The author extends this "crisis of faith" to probe whether researchers in mathematics education are trying to address questions such as:

1. Have we been doing the wrong things?
2. Have we failed to make contact with school practices?
3. Who, if anyone, is listening to what we have to say?

In trying to address these questions, there appears to be confusion between the impacts of pure (basic) and applied research. Basic research was assumed to be "up in the clouds" seeking generalization theories and seen to be higher in status, whereas applied research was thought to lack "specificity." The author goes on to suggest another model of research where "one cannot label a piece of research as either basic or applied." This is the lens model, in which the reader decides whether the purpose or usefulness of the research is basic or applied.

It was surprising for me to learn that the majority of dissertations in mathematics education do not come from the department of mathematics education or under the supervision of researchers in mathematics education. Even if this unsubstantiated claim is valid, the author seems to suggest that it is not surprising why the research outcomes are ineffective. Should research in mathematics education always come from the community of researchers in mathematics education for the research to have any value? I am suspicious of this line of thought, because Piaget was not a researcher in mathematics education. However, many researchers in mathematics education and mathematics teachers subscribe to Piaget's concepts of accommodation and assimilation, which ties to the Mathematics-Psychology-Philosophy (MAP) vertex on the MAPS perspectives of mathematical education.

Finally, it is startling to note that the author suggests that "lack of attention to theory" as a reason for the ineffectiveness of research in mathematics education. This suggestion implies that researchers in mathematics education hold knowledge-as-elements perspective. According to Ozdemir, novices hold this perspective of knowledge, whereas experts hold knowledge-as-theory perspective. I wonder if the author is vaguely suggesting that the majority of the research articles and dissertations in mathematics education are for or by novices. From the author's point of view, this may imply that the research findings may collectively "not add up to very much."

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An Overview of Conceptual Change Theories

Gökhan Özdemir

Niğde Üniversitesi, Niğde, TURKEY

Douglas B. Clark

Arizona State University, AZ, USA

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Family Math and Science Day at UBC

I had an awesome time at the Family Math and Science Day at UBC on Nov.1, 2014. Tons of pictures can be found here.