Monday 30 September 2013

“avoidable” and “unavoidable” misconceptions

In this article we deal with one of the most used terms for decades in Mathematics Education research, the word “misconception”, interpreted according to a constructive perspective proposed by D’Amore (1999):
"A misconception is a wrong concept and therefore it is an event to avoid; but it must not be seen as a totally and certainly negative situation: we cannot exclude that to reach the construction of a concept, it is necessary to go through a temporary misconception that is being arranged". According to this choice, misconceptions are considered as steps the students must go through, that must be controlled under a didactic point of view and that are not an obstacle for students’ future learning if they are bound to weak and unstable images of the concept; they represent an obstacle to learning if they are rooted in strong and stable models of a concept. For further investigation into this interpretation look D’Amore, Sbaragli (2005). 


This semantic proposal is analogous with Brousseau’s use of the term obstacle, starting from 1976 (Brousseau, 1976-1983), to which he gave a constructive role in Mathematics Education, interpreting it as knowledge that was successful in previous situations, but it does not “hold” in new situations. Within this interpretation misconceptions have been divided into two big categories: “avoidable” and “unavoidable” (Sbaragli, 2005a); the first do not depend directly on the teacher’s didactic transposition, whereas the second depend exactly on the didactic choices performed by the teacher.  We will analyze these categories within the theoretical frameworks upholded by Raymond Duval and Luis Radford.


According to Duval’s formulation the use of signs, organized in semiotic registers, is constitutive of mathematical thinking since mathematical objects do not allow ostensive referrals; from this point of view he
claims that there isn’t noetics without semiotics. "The special epistemological situation of mathematics compared to other fields of knowledge leads to bestow upon semiotic representations a fundamental role. First of all they are the only way to access mathematical objects" (Duval, 2006). This lack of ostensive referrals to concrete mathematical objects obliges also to face Duval’s cognitive paradox: "(...) How learners could not confuse mathematical objects if they cannot have relationships but with semiotic representations? The impossibility of a direct access to mathematical objects, which can only take place through a semiotic representation leads to an unavoidable confusion» (Duval, 1993). In particular, conceptual appropriation in mathematics requires to manage the following semiotic functions: the choice of the distinguishing features of the concept we represent, treatment i.e. transformation in the same register and conversion i.e. change of representation into another register. The very combination of these three “actions” on a concept represents the “construction of knowledge in mathematics”; but the coordination of these three actions is not spontaneous nor easily managed; this represents the cause for many difficulties in the learning of mathematics.
To better understand the learning processes it is suitable to integrate Duval’s theoretical frame with the one proposed by Radford who enlarges the notion of sign incorporating in the learning processes also the sensory and kinaesthetic activities of the body. Radford (2005) considers learning an objectification process that transforms conceptual and cultural objects into objects of our consciousness. This objectification process is possible only by turning to culturally constructed forms of mediation that Radford (2002) calls semiotic means of objectification; i.e. gestures, artifacts, semiotic registers, in general signs used to make an intention visible and to carry out an action. Like Duval, also Radford (2005) underlines the importance of the coordination between representation systems, when he claims that conceptualization is forged out of the dialectical interplay of various semiotic systems, with their range of possibilities and limitations, mobilized by students and teachers in their culturally mediated social practices. In the continuation of the article we will read “avoidable” and “unavoidable” misconceptions according to these theoretical frameworks.


“Unavoidable” misconceptions, that do not derive from didactical transposition, can depend on the representations teachers are obliged to provide in order to explain a concept because of the intrinsic unapproachableness of mathematical objects. These representations, according to Duval’s paradox, can be confused with the object itself especially when a concept is proposed for the fist time. These representations can lead the student to consider valid “parasitical information” bound to the specific representation, in contrast with the generality of the concept. This “parasitical information” for example can stem from sensory, perceptive and motor factors of the specific representation since Radford (2003) claims that cognition is embodied in the subject’s spatial and temporal experience and therefore requires to mobilize semiotic means bound to the practical sensory-motor intelligence. The “embodied” character of cognition and the use of semiotics makes these “misconceptions unavoidable” and interpretable as steps the student must go through in the construction of concepts. As we will show in the following example, these particular misconceptions can also be put down to the necessary gradualness of knowledge. In fourth primary school one day the teacher shows how the request that highlights the “specific difference” between the “close genus” rectangles and the “subgenus” squares regards only the length of the sides (that must all be congruent). After drawing a square on the blackboard, the teacher claims that it is a particular rectangle. The possible misconception created in the mind of the student that the prototype image of a rectangle is a figure that must have consecutive sides with different lengths, may create at this stage a cognitive conflict with the new image proposed by the teacher. This example highlights that it is unthinkable to propose initially all the necessary considerations to characterize a concept from the mathematical point of view, not only for the necessary gradualness of knowledge, but also because in order to propose mathematical objects, they must be anchored to semiotic representations that often hide the totality and complexity of the concept. These examples of “avoidable” misconceptions seem to be bound to ontogenetic (that originate in the student) and epistemological (that depend on intrinsic facts to mathematics) obstacles (Brousseau, 1986); the last are considered by Luis Radford related to the social “practices” (D’Amore, Radford, Bagni, 2006).


In the appropriation of a mathematical concept the pupil performs a desubjectification process, that leads him beyond the body spatial temporal dimension of his personal experience. The teacher has the delicate task of fostering a cognitive rupture to allow the pupil to incorporate his kinesthetic experience in more complex and abstract semiotic means. The student thus goes beyond the embodied meaning of the object and endows it with its cultural interpersonal value (Radford, 2003). In this perspective, Duval (2006) offers important didactic indications to manage the rupture described above, when he highlights the importance of exposing the student, in a critical and aware manner, to many representations in different semiotic registers. Nevertheless didactic praxis is “undermined” by improper habits that expose pupils to univocal and inadequate semiotic representations. These habits cause misconceptions considered “avoidable”, since they are ascribable to the didactic transposition. An emblematic example of inadequate choice of the distinguishing
features that brings to improper and misleading information relative to the proposed concept, regards the habit of indicating the angle with a “little arc” between the two half-lines that determine it. Indeed, the
limitedness of the “little arc” is in contrast with the boundlessness of the angle as a mathematical “object”. This implies that in a research involving students of the Faculty of Education, most of the persons interviewed claimed that the angle corresponds to the length of the little arc or to the limited part of the plane that it identifies.


An inadequate didactical transposition can in fact strengthen the confusion, lived by the student, between the symbolic representations and the mathematical object. The result is that "the student is unaware that he is learning signs that stand for concepts and that he should instead learn concepts; if the teacher has never thought over this issue, he will believe that the student is learning concepts, while in fact he is only “learning” to use signs» (D’Amore, 2003).


This misunderstanding derives also from the univocity of the representations that teachers usually provide students with, as is the case of geometry’s primitive entities. Researches aiming at detecting incorrect models built on image-misconceptions relative to these mathematical concepts show that as regards the mathematical point, some pupils and teachers ascribe to this mathematical entity a “roundish” shape that derives from the univocal and conventional representations they have always encountered (Sbaragli, 2005b). Moreover, some students and teachers are led to associate with the wrong idea bound to the unique shape of mathematical points also a certain variable dimension. From these results it emerges how often the choice of the representation, is not an aware didactical choice but it derives from teachers’ wrong models. And yet, to not create strong misunderstandings it is first required that the teacher knows the “institutional” meaning of the mathematical object that she wants her students to learn, secondly she must direct the didactical methods in a critical and aware manner. “Avoidable” misconceptions seem to be bound to the classical didactic obstacles (Brousseau, 1986) that originate in the didactic and methodological choices of the teacher.


From a didactical point of view, it is therefore absolutely necessary to overcome “unavoidable” misconceptions and prevent the “avoidable” ones, with particular attention to the semiotic means of objectification, providing a great variety of representations appropriately organized and integrated into a social system of meaning production, in which students experience shared mathematical practices.





Why Should Science and Mathematics be Integrated?

Plans to change schools fundamentally require that we face many harsh realities. First, schools resist change with a remarkable resiliency. Efforts to restructure mathematics and science curricula historically seem to have had little effect on conventional uses of the textbook and methods of delivery. Second, all students, especially many low-income and minority students, need continuity between schooling and the rest of their lives. The inclusion of science in a mathematics curriculum, and vice versa, is one way to provide this continuity. The key thought behind this process is to develop relevancy and applicability of the discipline to the existing student experiences. Students must see mathematics, as well as science, as relevant  components of their world. In other words, mathematics should no longer be seen as a discipline studied and applied for mathematics sake, but rather, because it will help make sense out of some part of our world. The “doing” of mathematics and the “doing” of science creates a new way for students to look at the world way that develops depth rather than breadth in a mathematics curriculum.


The expression “integration of science and mathematics” is used in different ways throughout the science and mathematics education community. Because integration has been a commitment of the School Science and Mathematics Association, teachers need to understand different ways in which the term integration can be used and how they apply to the teaching of science and mathematics (Underhill, 1994). School Science and Mathematics has taken the lead in presenting teachers with models for integrating mathematics and science (Berlin, 1991; Berlin&White, 1994). Two questions seem to emerge from this discussion:

l. To what extent can these integration efforts represent a bona fide integration of science and mathematics?
2.To what extent has the integration of science and mathematics been merely cosmetic?

Answers to questions such as these are critical in this climate of significant curriculum reform. We became
interested in exploring such questions as we sought to redesign pre service teacher education courses to integrate science and mathematics. The Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989) are radically changing views of school mathematics, and Project 2061: Science for all Americans (AAAS, 1989) and subsequent benchmarks present blueprints for changing science education. Few educators would argue about the need for an interwoven, cross-disciplinary curriculum, but to many, the nature of the integration in many interdisciplinary projects is not readily apparent. A mom pervasive problem is that integration means different things to different educators. The purpose of this article is to describe the method, type, and value of integration between and among the two disciplines and to discuss the meaning of such integration. Many topics in mathematics and science are touched upon at surface level, but few topics, historically, are covered or developed in much depth. Content coverage, rather than the provision of contextual understanding has been the valued mode in mathematics and science teaching. For these reasons, science and mathematics can be integrated to make disciplines relevant and meaningful to the learner. Mathematics, when integrated with science, provides the opportunity for students to apply the discipline to real situations, situations that are relevant to the student’s world and presented from the student’s own perspective. Integration deals with the extent to which teachers use examples, data, and information from a variety of disciplines and cultures to illustrate the key concepts, principles, generalizations, and theories in their subject area or discipline (Banks, 1993). 


Understanding different types of integration becomes necessary in order to begin to understand the integration found between and among science and mathematics. Five types of science and mathematics integration (discipline specific, content, process methodological, and thematic) can be used in interdisciplinary curriculum development (Miller, Davison, & Metheny, 1993). 


What does it really mean to integrate science and mathematics? Whether the integration of science and mathematics occurs within the disciplines or is infused with the disciplines, integration will provide for a more
reality-based learning experience. As national, state, and local curriculum efforts continue, closer links between science and mathematics will be explored, and thereby lead to more obviously integrated science and mathematics curricula. We have presented different meanings of integration in this article. Each of these interpretations provides a valid approach to integrating the disciplines. We believe that the most potent approach to integration is to focus, not on science and mathematics content, but on scientific processes. We have used this strategy as a guide in the redesign of a methods course in science and mathematics. In an atmosphere of curriculum restructuring which focuses on connections between the disciplines, the use of scientific processes as an emphasis is deemed appropriate Such integration between science and mathematics can serve as a model for process integration across the curriculum.


The teacher education community needs to more actively explore ways in which programs can be redesigned
to respond to this change.

Creative Mathematics - Exploring Children’s Understanding

Creative Mathematics is a delightful collaboration of mathematical curiosity, and liberal learning. The collaboration reported here is of the toughest sort – a classroom teacher, Phillips, and a university researcher, Rena , it is enriched by the further collaboration of Bill acting at a distance, contributed advice to the pair working in the classroom and commentary on their written account. Phillips and Rena worked with third and fourth grade children while Bill encouraged, advised, and commented.


Mathematical curiosity infected both the children and their teachers. It is clear from the start that Phillips is a fine teacher, but she had never before tackled the mathematics reported here. Rena is a music specialist and was lured into the project in part by promises of connections between math and music. Bill is something of a polymath mathematician, one interested in math and its connections to almost everything. The children, none of whom had listed math as their favorite subject at the beginning of the school year, became eager for math time and sorry when it ended. Everyone involved seems to have become more, not less, enthusiastic as the year progressed. The math learning was liberal in the best sense. The activities offered set the children free to investigate matters that excited them, and they learned a good deal more than the usual arithmetic. Their projects took them into art, music, technology, and invention. Along the way, their vocabularies grew impressively. Each author provides an introduction and commentary in each chapter. Rena's many conversations with Bill led her to explore the connection between music and mathematics. Confessing that she had been a proficient but not particularly interested math student, Rena reports being converted by the fascinating titles in Bill’s mathematics library. (I have seen that library and understand Rena's surprise and delight.) Rena notes:

“It is not as if these books do not talk about algebra and geometry – they do. It is that these books do not use algebra and geometry as the endpoints in mathematics, but rather, as tools for creation – tools that are needed to make beautiful tessellating patterns or to understand the allure of a snowflake” .

With Bill’s inspiration and the cooperation of university colleagues in Vancouver, Rena set out to find a teacher – collaborator. Phillips, in her introduction, expresses fears common to classroom teachers – concern about tackling unknown subject matter, uneasiness about working with a university researcher, and worry over an already busy teaching life. Besides, she was a successful teacher by all the usual standards. Why, then, should she complicate her professional life by plunging into an alien project? However, she was inspired and reassured by Rena and, further, she reports that something had been bothering her for a while:

“I felt that I was holding back. I was focusing too much on being accountable to people outside the classroom. I wanted to focus on the pupils and on what they might need to know. Even when I moved to a combination of textbook work with manipulative  materials and methods from my early days, I remained unsatisfied.” 

These remarks struck me as especially important for other teachers. With the current emphasis on higher achievement scores, many teachers feel that accountability is pointed in the wrong direction. Instead of being accountable to their students, teachers are forced to consider the demands of administrators and a public only partly aware of the ramifications of their demands. As I travel around my country these days, I hear teachers everywhere complaining that they have been forced to teach in a way that kills interest and does little to further understanding. At the same time, of course, they are urged to teach for understanding.


Phillips exudes a wisdom rarely seen in classroom teaching. Watching the pleasure and enthusiasm of her students, she becomes dedicated to new approaches and topics. But in response to the concerns of parents (“When will you teach real math?”), she wisely combines textbook exercises, work sheets, and projects. She knows how to accommodate a host of conflicting needs and interests. I found her attitude refreshing. After all, we really do not know how much practice is required for children to learn the skills they need to address significant mathematical problems. Alan Schoenfeld remarked on this issue that we do not know “the degree of fluency required to do competent work” (1994, p. 60). Thus, Phillips was not simply accommodating concerns that might have had a political impact on her work as a teacher, but she was also sensitive to the actual needs of her students. Neither Phillips nor Schoenfeld subscribe to the debilitating notion that students need to learn a mass of rote skills before tackling hard and interesting problems, but both recognize that many skills have to be acquired somewhere, sometime, some way, if students are to be successful in using mathematics. Indeed, when the children were working with a project on animation, one child commented, “It’s a good thing we know our eight times tables” .


Bill, in recounting his student days in mathematics, tells a familiar story. He pleased his teachers and out-performed his peers. How many of us could tell this story! (And how many simply gave up because they could not “out-perform” their peers?) It was not until graduate school that he became thoroughly engaged in mathematics. Now, of course, it is a matter of professional concern for him to find ways to make mathematics accessible and interesting to a great number of students.


Thus, three thoroughly motivated people launched an impressive project. In the following chapters, they describe how children worked on tessellations, animation, mathematical “jewels” made from paper, kaleidoscopes, and musical composition. They learned to look for patterns in the world (seeing tessellations everywhere), came to a working definition of tessellation (“no floor showing”), made important generalizations (e.g., “all quadrilaterals will tessellate”), used symbols to describe their patterns, applied the basic skills they had acquired earlier, and extended their vocabularies. Picture little kids easily using words such as tessellation, animation, trapezoid, hexagon, quadrilateral, and thaumatrope.


Phillips and Rena put considerable emphasis on mathematical communication. This is a familiar focus today,
but their approach is more sophisticated than most. The children were encouraged to describe their work in symbols so that others could re-create the patterns they had constructed, and they largely succeeded. I found this work impressive, and it contrasts sharply with many examples of mathematical communication found in various portfolios. The difference, and it is an important one, may be in the time spent on these projects. The children were not rushed, and the communication was undertaken as important in itself; it was not pro forma and, because they were asked to invent symbolic forms of communication, it added considerably to their mathematical knowledge. Typically, after Phillips and Rena describe what happened in the classroom, Bill offers a commentary loaded with helpful references. There is clearly a lot of help available for teachers who want to engage in the kind of activities described here. I was reminded, as I read these accounts, of the optimism and good sense that accompanied the best of Open Education. Other readers, too, might want to revisit that literature. (For a personal/historical narrative on Open Education, see F. Hawkins, 1997.) Interestingly, although the authors do list work by David Hawkins in their references, they do not mention Open Education. Instead, and I think this is a strategic error, they draw an analogy between their approach to mathematics and the whole-language approach to reading and language arts. This is a strategic error because whole-language has come under devastating fire in recent years , and anyone who wishes to use it as an example needs to defend it. I believe that a persuasive defense can be offered, but the best use of whole-language would probably be laid out in terms of accommodation and eclecticism – as Phillips described her own approach earlier.


The question of what constitutes “real” problems for students arises several times, and it is sensitively discussed. In the  initial stage they agree with my comment on “real-world” problems – that “a problem that is ‘realworld’ in the sense that adult human beings grapple with it may not be ‘real’ at all in the school setting” (Noddings, 1994; Pimm, xii). All ot them clearly locate the meaning of “real” in what is real for children – activities that engage them and from which they learn. Thus, the “real” presents a challenge worth meeting. It
can be fun, but it is not merely fun. Working on real problems induces growth. On this, they are demonstrably Deweyan, although they do not refer to Dewey. Bill’s comments on technology are eminently worth reading. They are even-handed and sensitive. He notes the promise of technology but also its downside. Students may bog down in trivia, pursuing this and that bit of information without studying anything in depth. They may feel compelled to use technology even when their  own styles suggest a different approach. On this, Bill’s report of an interview with the computer scientist Donald Knuth is heartening. Knuth finds e-mail counterproductive for his work. He acknowledges the usefulness of electronic mail for many occupations, but not for this own! Comments like these can be enormously liberating. On the positive side, Bill’s account of what can be found on the Web may send people scurrying for useful information. Consonant with his even-handed analysis and his love of books, Bill finishes by saying, “As we explore the Internet in the years to come, we would do well to remind ourselves to search the shelves of second-hand
bookshops as well” .


They frankly discuss a few things that didn't work. Among the unsuccessful projects was one on codes. I was surprised by this because I recall codes as one of the most successful topics I used with sixth graders many years ago. Maybe the children need to be a bit older. Maybe the teachers did not uncover the underlying interest that might have motivated such a unit of study. Maybe they were not all that interested themselves. I remember that my students were interested in secret languages, writing with disappearing ink, and that sort of thing. From there, we began to do cryptograms and then went on to read a Sherlock Holmes story, “The Adventure of the Dancing Men” and Poe’s “Gold Bug”, both of which involve codes. I remember the unit as great fun, and the children learned quite a lot about the frequencies with which various letters appear in English and about letter combinations.   Children differ in their interests; so do teachers. That
is an important message in the last chapter. There is no plausible reason why every group should be interested in codes or tessellations or animations. However, a teacher’s enthusiasm may infect the children. At least, that enthusiasm will tend to support the teacher through the hard work of introducing and successfully completing the kind of rich projects described here.


My own mathematical interests tend to lie in the logical and literary. I wondered, as I read this fascinating report, whether I would enjoy doing tessellations with children. Maybe. But I know that I enjoy sharing and sorting out the logical puzzles and anomalies in “Alice in Wonderland” with students of all ages. And I am always looking for stories that include mathematics in any form. Not long ago I discovered the short stories of the Japanese writer Kenji Miyazawa and found that numbers appear in almost every one. My favorite is the story of General Son Ba-yu. A physician examining the General learns that his patient has been bewitched “about ten times.” The diagnostic session proceeds from that disclosure:

“I’d like to ask you something, then. What does one hundred and one hundred make?”
“One hundred and eighty.”
“And two hundred and two hundred?”
“Let’s see. Three hundred and sixty, if I’m not mistaken”
“Just one more, then. What’s ten time two?”
“Eighteen, of course.”                                          (Miyazawa 1993, p. 25–26)

At this point, the doctor knows just what to do and, after treating the General, repeats the original questions. Now the General answers correctly, and the physician says, “You’re cured. Something was blocked up in your head, and you were ten percent off in everything” . Perhaps it is stories such as these that make Japanese students so good at mathematics!


My point in introducing my own interests in logic and literature is to remind readers that our interests differ.
Teachers and researchers who are not enthusiastic about one set of topics can surely find another, and successful completion of one project will almost certainly lead to further exploration. Teachers, like children, may build on their current interests and move well beyond them as their confidence grows. I may differ with them on one important issue. I’m not sure that students have to learn “to think like mathematicians.” I concede that it is the popular view today in informed circles. But why? If we were to take the advice of all the disciplinary specialists, our children would have to think like scientists, artists, historians, writers, linguists,
musicians, geographers, literary critics, dramatists, editors, and mathematicians. 


I’m not sure this is reasonable. It does seem reasonable, however, to analyze these ways of thinking for those elements that students might profitably adapt to their own purposes. I agree wholeheartedly that students should be able to use the mathematics they learn but surely the uses will vary with interests and talents. We may be deluding ourselves when we try to capture the complexity of learning to use mathematics by reducing it to “thinking like a mathematician.” This provides strong evidence that kids can use mathematics while doing work similar to that of artists, engineers, film makers, or composers. In none of these cases, however, do the children take on the complex modes of thinking, semi-permanent attitudes, and social networks characteristic of professionals in the field. For kids, doing real mathematics involves using mathematics on problems of interest to kids. Fortunately, we can almost always find such problems, and their mastery may well lead to new levels of reality. 

Connections Between Mathematics and Music

The two disciplines have been interlinked throughout history since Ancient Greek academics began their theoretical study; since antiquity, mathematicians have often been music theorists. The fascination that mathematicians have with music will then be discussed.


MUSIC THEORISTS AND MATHEMATICIANS : ARE THEY 
ONE  IN THE  SAME ?

For about a millennium, from 600 BC, Ancient Greece was one of the worldʼs leading civilizations. The ideas and knowledge produced at this time have had a lasting influence on modern western civilizations. The “Golden Age” in Greek antiquity was approximately 450 BC, and much of what constitutes western culture today began its invention then . Brilliant Greek academics contributed a wealth of knowledge about music, philosophy, biology, chemistry, physics, architecture and many other disciplines. With the Ancient Greeks came the dawn of serious mathematics. Before their time, mathematics was a craft . It was studied and used to solve everyday problems. For example, farmers might implement mathematical tools to help them lay their fields in the most economical way possible. In Greek antiquity, mathematics became an art. It was studied purely for the sake of knowledge and enjoyment. Philosophers and mathematicians questioned the fundamental ideas of mathematics.


Pythagoras, Plato and Aristotle were three very clever academics, and very influential figures when detailing the historic connection between mathematics and music . Pythagoras was born in the Classical Greek period (approximately 600 BC to 300 BC) when Greece was made of individual city-states. A dictator governed the island on which he lived, so he fled to Italy. It was there that he founded a religion (often called a cult) of
mathematics. Pythagoreans , the followers of his religion, believed mathematical structures were mystical. They had elaborate rituals and rules based on mathematical ideas. To the followers, the numbers 1, 2, 3 and 4 were divine and sacred. They believed reality was constructed out of these numbers and 1, 2, 3 and 4 were deemed the building blocks of life . Pythagoras was instrumental in the origin of mathematics as purely a theoretical science. In fact, the theories and results that were developed by Pythagoreans were not intended for practical use or for applications. It was forbidden for members of the Pythagorean school of thought to even earn money from teaching mathematics . Throughout history, numbers have always been the building block of mathematics .


Plato was a Pythagorean who lived after the Golden Age of Ancient Greece. Plato believed that mathematics was the core of education . He founded the first university in Greece, the Academy. Mathematics was so central to the curriculum, that above the doors of the university, the words “Let no man enter through these doors if ignorant of geometry” were written . From antiquity, many famous Greek mathematicians attended Platoʼs university. Aristotle, the teacher of Alexander the Great, is an example of a famous student of Plato. Aristotle was a man of great genius and the father of his own school. He studied every subject possible at the time. His writings had vast subject matter, including music, physics, poetry, theatre, logic, rhetoric, government, politics, ethics and zoology. Together with Plato and Socrates (Platoʼs teacher), Aristotle was one of the most important founding figures in western philosophy. He was one of the first to create a comprehensive system detailing ideas of morality, philosophy, aesthetics, logic, science, politics and metaphysics . A natural question now arises: why are these ancient figures so important in understanding the relationship between mathematics and music? The answer is simple. It was these early Greek teachers and their schools of thought (the schools of Pythagoras, Plato, and Aristotle) who not only began to study mathematics and music, but considered music to be a part of mathematics . Ancient Greek mathematics education was comprised of four sections: number theory, geometry, music and astronomy; this division of mathematics into four sub-topics is called a quadrivium . Itʼs been previously stated that the ideas and works of the Ancient Greeks were influential and had had a lasting effect throughout history. Those of music and mathematics were no different. The four way division of mathematics, which detailed music should be studied as part of mathematics, lasted until the end of the middle ages (approximately 1500 AD) in European culture. 


The Renaissance (meaning rebirth), a period from about the fourteenth to seventeenth centuries, began in Florence in the late middle ages and spread throughout Europe. The Renaissance was a cultural movement, characterized by the resurgence of learning based on classical sources, and a gradual but widespread educational reform. Education became heavily focused rediscovering Ancient Greek classical writing about cultural knowledge and literature . Music was no longer studied as a field of mathematics. Instead, theoretical music became an independent field, yet strong links with mathematics were maintained . It is interesting to note that during and after the Renaissance, musicians were music theorists, not performers. Music research and teaching were occupations considered more prestigious than music composing or performing . This contrasts earlier times in history. Pythagoras, for example, was a geometer, number theorist and musicologist, but also a performer who played many different instruments.


In the seventeenth and eighteenth centuries, several of the most prominent and significant mathematicians were also music theorists . René Descartes, for example, had many mathematical achievements include creating the field of analytic geometry, and developing Cartesian geometry. His first book, Compendium Musicale (1618) was about music theory . Marin Mersenne, a mathematician, philosopher and music theorist is often called the father of acoustics. He authored several treaties on music, including Harmonicorum Libri (1635) and Traité de lʼHarmonie Universelle (1636) . Mersenne also corresponded on the subject with many other important mathematicians including Descartes, Isaac Beekman and Constantijn Huygens .


John Wallis, an English mathematician in the fifteenth and sixteenth centuries, published editions of the works of Ancient Greeks and other academics, especially those about music and mathematics . His works include fundamental works of Ptolemy (2 AD), of Porhyrius (3 AD), and of Bryennius who was a fourteenth century Byzantine musicologist . Leonhard Euler was the preeminent mathematician of the eighteenth century and one
of the greatest mathematicians of all time. While he contributed greatly to the field of mathematics, he also was a music theorist. In 1731, Euler published Tentamen Novae Theoriae Musicae Excertissimis Harmoniae Princiliis Dilucide Expositae . In 1752, Jean dʼAlembert published works on music including Eléments de Musique Théorique et Pratique Suivant les Principes de M. Rameau and in 1754, Réflexions sur la Musique . DʼAlembert was a French mathematician, physicist and philosopher who was instrumental in studying wave equations. 



WHY ARE MATHEMATICIANS SO FASCINATED BY MUSIC THEORY ?

Mathematicians fascination with music theory are explained clearly and precisely by Jean Philippe Rameau in Traité de lʼHarmonie Réduite à ses Principes Naturels (1722). Some musicologists and academics argue that Rameau was the greatest French music theorist of the eighteenth century [4]. Rameau said:

“Music is a science which must have determined rules. These rules must be drawn from a principle which should be evident, and this principle cannot be known without the help of mathematics. I must confess that in spite of all the experience I have acquired in music by practicing it for a fairly long period, it is nevertheless only with the help of mathematics that my ideas became disentangled and that light has succeeded to a certain darkness of which I was not aware before.” 

Mathematicians have been attracted to the study of music theory since the Ancient Greeks, because music theory and composition require an abstract way of thinking and contemplation . This method of thinking is similar to that required for pure mathematical thought . Milton Babbitt, a composer who also taught mathematics and music theory at Princeton University, wrote that “a musical theory should be statable as connected set of axions, definitions and theorems, the proofs of which are derived by means of an appropriate logic” .


Those who create music use symbolic language as well as a rich system of notation, including diagrams . In the case of European music, from the eleventh century, the diagrams used in music are similar to mathematical graphs of discrete functions in two dimensional Cartesian coordinates . The x-axis represents time, while the y-axis represents pitch. The Cartesian graph used to represent music was used by music theorists before they were introduced into geometry . In fact, many musical scores of twentieth century musicians have many forms that are similar to mathematical diagrams. At the beginning of a piece of music, after the clef is marked, the time signature is marked by a fraction on the music staff . Common time signatures include 2/4, 3/4, 4/4. and 6/8. The denominator of the fraction, is the unit of measure, and used to denote pulse. The numerator indicates the number of these units or their equivalent included in the division of
a measure. Groups of stressed and relaxed pulses in music are called meters. The meter is also given in the numerator of the time signature. Common meters are 2, 3, 4, 6, 9, 12 which denote the number of beats or pulses in the measure . For example, take the time signature 3/4. Each measure is equivalent to three (information from the numerator) quarter notes (information from the denominator). The count in each measure would be: 1, 2, 3. The 1 is the stressed pulse, while the 2 and 3 are relaxed. The time signature 3/4 is common in waltzes . Besides abstract language and notation, mathematics concepts such as symmetry, periodicity, proportion, discreteness, and continuity make up a piece of music . Numbers are also very instrumental, and influence the length of a musical interval, rhythm, duration, tempo and several other notations . The two fields have been studied in such unison, that musical words have been applied to mathematics. For example, harmonic is a word that is used throughout mathematics (harmonic series, harmonic analysis), yet its origin is in music theory . Itʼs been discussed that throughout history, mathematicians have long been fascinated with music theory. This concept will be further developed in the, which suggest mathematics is, like music, a form of art.






Sunday 29 September 2013

The opposite of talking is not listening. The opposite of talking is waiting. - Fran Lebowitz

Mathematicians are accustomed in their professional discourse to conditions which are alien to all other disciplines: On any given issue, there is a universally recognizable correct answer. If there is disagreement, it is because one side or the other does not correctly understand the situation. Therefore, the proper response to disagreement is to attack ruthlessly until the truth becomes clear. Once that happens, those in error will admit it gracefully and move on. We sometimes make the mistake of expecting the same conditions to apply in arguments about mathematics education. Particularly damaging is the belief that there is no such thing as being half-right; there is nothing to be salvaged in the practices of one's opponents. Unfortunately, Fran Lebowitz's quip describes only too well much of the debate about mathematics education. One of the great pleasures of organizing this conference was to have witnessed some genuine listening. For example, the working groups on The First Two Years of University Mathematics and on Outreach to High Schools contained prominent representatives from opposite sides of the debate on mathematics education reform, yet
forged remarkably uni ed position papers after two days of intense debate. This is only a rst step, however. In addition to listening to each other, we need to take the next step and learn to listen to voices from outside our
profession. 


The fi rst group of people that we should listen to is our students. I recently had a very illuminating conversation with my daughter. Doing her homework one evening, she said:

"Dad, 14 sevenths is 2, right?" When I answered "yes", she said:
"Good, I just wanted to make sure that it wasn't division."
"But it is division; how else can you get the answer?"
"Oh, I was using fractions."
"How do you use fractions to show 14 sevenths is equal to 2?"
"Well. . . 14 is equal to 2 times 7, and the 7s cancel."
"Doesn't that mean that 14 divided by 7 is 2?" (Long pause.)
"Oh . . . yeah."


My student is good at both division and fractions. Without this exchange, I would never have guessed that she wasn't clear on the connection between the two. Our mathematical training does not equip us to make such guesses, because it is in the nature of mathematical progress to erase the missteps in our journey towards insight. By listening carefully to our students we can eliminate  the guesswork and detect the missing connections in their understanding.


We can also learn what makes sense to them and what doesn't. A business calculus class may not have much taste for mathematical abstractions, but can demonstrate great mathematical pro ciency when presented with the same ideas in a concrete context that they know. The article by William V elez and Joseph Watkins in this volume illustrates the power of presenting mathematics in contexts that mean something to the students.


The second group we should listen to is teachers and those who study teaching. (This includes, to some extent, ourselves. However, it must be admitted that university teaching has been pervaded by a dilettantish attitude which discourages serious discussion of teaching philosophy.) I include in this group anyone who takes teaching seriously as a profession; someone who can piece together with clever detective work the thinking of students, or who looks at a syllabus as more than just a list of textbook headings, or who delights in constructing homework problems that make students think about what they are doing. This group includes many school teachers and education researchers; Anneli Lax writes persuasively about what can be learned by listening to them. The group also includes mathematicians at the college level, on all sides of the educational debate, for whom, in Hyman Bass's words, the practice of teaching has become a part of professional consciousness and collegial communication, not unlike their professional practice of mathematics itself. Jerry Uhl and William Davis in one article and Hung-Hsi Wu in another write about their college teaching experiences in a way which goes beyond personal anecdote to provide valuable professional insights.

Others worth listening to are those who use mathematics, and whose students we teach. We often invoke the opinions of engineers and scientists to defend our positions, but how often do we bother to go back to the source? I once spent a couple of hours talking to a colleague in our other department. Although I obtained some useful examples , perhaps the most useful thing I discovered was the way he visualized functions of two variables. To my surprise, he did not regard the surface graph as the central geometric object, but rather the contour diagram. All his geometric reasoning about the behaviour of functions proceeded directly from the contours; it was only of marginal interest to him that the contours could be related to a surface in three dimensions. This insight fundamentally changed . The article by Dorothy Wallace gives more examples of what can be learned by talking to our colleagues in other disciplines.


Of course, we, as mathematicians, have a role as speakers as well as listeners. We must judge what are the important concepts, make sure that what is being taught is correct, and ensure a balance between technique, theory, and applications. Most of us, I think, have no trouble lling this role; it is, as Oliver Wendell Holmes said, "the province of knowledge to speak." He added that "it is the privilege of wisdom to listen."

Mathematics games: Time wasters or time well spent?

The use of mathematics games is often cited as an effective strategy for teaching mathematics. However, the researchers were unsure whether games are being used as supplementary activities for children who finish their ‘real’ work early, as busy work, or used with a real purpose. Although children may be having ‘fun’, the belief is that this is not sufficient reason for their inclusion in a mathematics program.


While it may be assumed that the use of games in the teaching of mathematics has been researched, a brief literature review revealed that surprisingly little empirical research into the use of games has been carried out. Often teachers assume that the use of games is an effective teaching tool. This may not be the case (Bragg, 2006).

Many authors have presented the use of games as a beneficial tool in the mathematics classroom (Bragg, 2006; Booker, 2000; Gough, 1999; Ainly, 1990). Also numerous authors assert that games should not just be restricted to practice and that they can be an effective vehicle for teaching new concepts to children (Bright, Harvey & Wheeler, 1985, Kamii & De Clark, 1985; Thomas & Grows, 1984; Burnett, 1992; Booker, 2000) Clearly, what is important is the structure of the games used (Ainley, 1990; Badham, 1997) and the literature does highlight that if this structure is not provided learning does not always take place
(Onslow, 1990; Burnett, 1992).


Games are used worldwide as a means of developing mathematics concepts. Often parents purchase educational games as a means of supporting learning in the home. There is a large educational games industry referred to as “Edutainment” that is designed to tap into the parent market. The games used in this research tended to be more ‘classroom games’, although games such as Battleships that cross over the school/home market were explored.  There are many games of educational value played in the classroom that help make some topics less onerous; however, children would not choose to play at home. If a game has proven educational value and children choose to play them out of school, then the children would be spending more time on task and it could be assumed become better at the task or in this case the embedded mathematics. The fact that Asplin (2003) noted in her study that simply playing Numero as a time-filler or casually was not
sufficient to improve mental computation skills. The authors argue that if a game is deemed worthy of playing then it should be elevated from the status of a time filler or activity for early finishers but rather be an integral part of a teaching sequence. Asplin went on to note that in the class where the teacher encouraged students to describe orally their moves, and where the children were grouped according to ability, gains in mental computation ability were much higher. However, Bragg (2006) who studied the use of two games in the teaching of mathematics, raised questions about grouping children according to ability when playing games. In this research the children within classes were not grouped according to ability but, in at least one case, the class had been formed because the children were less able. The larger research project explored, in some depth, the use of games in the teaching of mathematics, in particular number, and aimed to develop guidelines for maximising the effectiveness of games in the teaching of mathematics.


Criteria for Assessing Games

Classroom management issues

The most common criteria for selecting games as reported by teachers in the study were concerned with classroom management issues. Some of the comments related to trivial issues about pieces going missing or taking too long to set up and pack away. However, the authors do not wish to trivialise the impact on the smooth running of a lesson and the associated loss of teaching time. There were conflicting comments about number of players. In two cases the teachers preferred to play whole class games, while three teachers preferred playing games in small groups because it maximized the time children were thinking about the concepts embedded in the game. Short games appeared to be favoured over longer ones. This allowed for more flexibility as to when and how games were used in the classroom. Simple rules were favoured as less time was spent introducing the game and sorting out conflicts based on misinterpretation of rules.


Clear links to skills and concepts

While the teachers liked the idea of using games from a motivational point of view, they preferred the game to be linked to a specific skill or concept. Several factors appear to affect this concern. Teachers feel pressured to ‘cover’ a great deal of content and felt that devoting too much time to games without there being a direct link to specific content would erode their teaching time. The need to justify the use of games in terms of concept or skill learning was apparent. The teachers reported pressure from parents for their children to be seen to be completing some rigorous mathematics. In some cases, this translated to completing sets of algorithms on a page as evidence of  having ‘worked hard’ in the lesson. The extreme case is the reported pressure to complete all the pages in a textbook before the end of the year.


Motivation and Engagement

Most teachers would choose to use games as motivators for engaging in mathematics. Games are often employed to make practice more pleasant. Difficult concepts such as fractions may be embedded in a game format to encourage deeper thinking about the concept. Bragg (2006) noted in her research that once the challenge of a game is lost, motivation wanes. Children are less inclined to engage with the game. For example, once you know how not to lose in Noughts and Crosses (Tic Tac Toe), it seems pointless to even begin a game.


Globally education authorities are placing increasing emphasis on the development of literacy and numeracy in primary schools. This article reports on research designed to assist teachers to improve the numeracy of their students by making the use of mathematics games a more focused aspect of the teaching and learning experience in mathematics. Classroom experience and anecdotal evidence suggest that games are often used without really focusing on the mathematics involved in playing the game, and are justified simply on the basis of children having ‘fun’.



WAYS TO REDUCE MATH ANXIETY

Recently I was eating in a restaurant in Austin where I sat next to a table of five people who looked to be in their mid twenties. The waitress brought them their bill and I could not help noticing that they were struggling to figure out how to divide it up and how much to leave for a tip. This went on for a while when finally one of them said: “Isn’t it funny that we are all graduate students and we can’t figure out the bill?”


Now imagine this situation:

Five people walk into a restaurant and are seated. The waitress brings each of them a menu. They all sit quietly for a while until finally and one of them says: “Isn’t it funny that we are all graduate students and none of us can read these menus?”

Do you think that this is as likely to happen as the first scenario?

The point here is that everyone is aware of the importance of knowing how to read in our society, but very few view mathematics with nearly the same regard. Where it can be considered funny not to understand basic mathematics, it is usually considered to be quite embarrassing to be illiterate.


Math anxiety is an intense emotional feeling of anxiety that people have about their ability to understand and do mathematics. People who suffer from math anxiety feel that they are incapable of doing activities and classes that involve math. Some math anxious people even have a fear of math; it's called math phobia. The incidence of math anxiety among college students has risen significantly over the last decade. Many students have even chosen their college major in the basis of how little math is required for the degree. Math anxiety is an emotional, rather than intellectual, problem. However, math anxiety interferes with a person's ability to learn math and therefore results in an intellectual problem.


Following are the ways to reduce maths anxiety.


1. Practice If you’re having math trouble, practice a little math everyday. Do you think athletes improve their game just by watching? By practicing their craft everyday they improve both their skills and confidence. Repetition is important in math. You learn how to solve problems by doing them. So keep on practicing problems. But don’t do it blindly, make sure you learn how to recognize when and why you should use a specific method to solve a problem. When practicing, try to solve the problem on your own first, then look at the answer or seek help.

2. Know the basics. Be sure you know your math from prior grades. Maybe you missed something when you moved to a new school. Math builds on itself, so you will have to go back and relearn that stuff. Remember it’s never too late to learn. Besides, you’re older now. It’ll be easier and quicker to learn. This is an area where a math tutor can be of great help.

3. Don’t go by memory alone. Try to understand your math. Memorizing is a real trap. When you’re nervous, memory is the first thing to go. By understanding the material you minimize what you must memorize. This also makes it easier to apply concepts to new situations.

4. Make a list Make up a list with all of the formulas you need to know. Try to make sense of the formulas and understand the purpose of each formula. Use the list to help study and try to recall a problem that was solved by each formula.

5. Be well prepared Being well prepared is one of the best ways to reduce test taking anxiety. Begin to study for a test at least three days in advance. Review class materials often and practice homework problems. Be very thorough. Do not skip over large sections of material, and most important do not wait until the night before the test and try to learn everything.

6. You are not alone! Many people dislike and are nervous about math, so know that you are not alone. Even mathematicians are sometimes unsure of themselves and get that sinking, panicky feeling called “math anxiety” when they first confront a new problem. But successful people use nerves in a positive way.

7. Stay positive Maintain a positive attitude while preparing for tests and during the test. No loser-talk please.

8. Ask questions. Some people think asking questions is a sign of weakness. It’s not. It’s a sign of strength. In fact, other students will be glad. (They have questions, too.) Again, a tutor can be a big help here.

9. Do math in a way that’s natural for you. There’s often more than one way to work a math problem. Maybe the teacher’s way stumps you at first. Don’t give up. Work to understand it your way. Then it will be easier to understand it the teacher’s way. Remember, “every mind has it’s own method.”

10. Exercise and sleep Exercising for a few days before the test will help reduce stress. It will also help you get a good night’s sleep before the test.

11. Trouble with the text? Get another math book. Maybe a book in the library will explain things better. There are many good review books. If you can not find one ask a teacher or a tutor.

12. Get help. Everyone needs help now and then. Try to form a study group with friends (two heads are better than one), take a review course, or work with a math tutor.


In conclusion, math anxiety is a very real phobia for many people of all ages, gender and ethnicity. Research has shown that creating a positive environment and attitude toward mathematics and students understanding their own confusions and learning style contributes toward alleviating math anxiety.


Combined with positive reinforcement, students need to be given opportunities for experimenting, exploring, conjecturing, solution inventing and reflecting on work (Curtain-Phillips, 2003, and Martinez, 2003). Therefore, there is no stand-alone teaching technique in overcoming math anxiety but rather a whole range of techniques and strategies.

Math Anxiety-1

Math anxiety is a serious obstacle for many children across all grade levels. Math-anxious students learn less math than their low-anxious peers because they take fewer math classes and get poorer grades in the math classes they do take. Math anxiety has been studied for many years but has recently received renewed attention. Researchers now believe that implementation of strategies to prevent or reduce math anxiety will improve math achievement for many students (Geist, 2010; Mission College, 2009; Cavanaugh, 2007).


Math anxiety is defined as negative emotions that interfere with the solving of mathematical problems. It is more than just disliking math and leads to a global avoidance pattern - whenever possible, students avoid taking math classes and avoid situations in which math will be necessary (Sparks, 2011; Hellum-Alexander, 2010; Ashcraft & Krause, 2007). Tobias, often referred to as a pioneer in the study of math anxiety, described it as “the panic, helplessness, paralysis and mental disorganization that arises among some people when they are required to solve a mathematics problem” (Tobias & Weissbrod, 1980).


Physical symptoms of math anxiety include increased heart rate, clammy hands, upset stomach, and light headedness. Psychological symptoms include an inability to concentrate and feelings of helplessness, worry, and disgrace. Behavioral symptoms include avoidance of math classes, putting off math homework  until the last minute, and not studying regularly (Mission College, 2009; Plaisance, 2009; Jackson, 2008; Woodard, 2004).


A large part of learners indicate that they experience some level of math anxiety. Math anxiety can develop at any age. For many children, negative attitudes toward math begin early in life, sometimes even before they enter kindergarten. In fact, studies have found a negative relationship between math anxiety and math achievement across all grade levels. Some researchers have found that math anxiety is most likely to begin around fourth grade and peak in middle and senior high school (Geist, 2010; Legg & Locker, 2009; Sun & Pyzdrowski, 2009; Scarpello, 2007; Woodard, 2004).  Evidence suggests that anxiety is more of a factor in math than in other subjects. Studies have also found that math anxiety is more common in girls, especially at the middle and senior high school levels (Beilock et al., 2010; Cavanaugh, 2007; Woodard, 2004).


Students’ math anxiety is often based on years of painful experiences with math. Studies indicate that the origin of math anxiety is complex and that anxiety forms as a result of personality, intellectual, and environmental factors. Personality factors include low self-esteem, inability to handle frustration, shyness, and intimidation. The intellectual factor that most strongly contributes to math anxiety is the inability to understand mathematical concepts. Environmental factors include overly demanding parents and negative classroom experiences, such as unintelligible textbooks, an emphasis on drill without understanding, and a poor math teacher. Researchers agree that math teachers who are unable to adequately explain concepts, lack patience with students, make intimidating comments, and/or have little enthusiasm for the subject matter frequently produce math-anxious students (Plaisance, 2009; Sun & Pyzdrowski, 2009; Scarpello, 2007; Furner & Berman, 2004; Woodard, 2004; Brown, n.d.).


Research indicates that there is a strong negative relationship between math anxiety and test scores. In other words, as students’ math anxiety increases, their test scores decrease (Furner & Berman, 2004; Woodard, 2004; Brown, n.d.). Researchers concur that educators have reduced the diagnostic ability of math tests by administering them in stressful situations (Sparks, 2011; Geist, 2010; Ashcraft & Krause, 2007; Cavanaugh, 2007). Scarpello (2007) stated that over reliance on high-stakes tests has reinforced the development of negative attitudes toward math and increased students’ anxiety levels by turning math into a high-risk activity.


Math anxiety has been universally recognized as a non-intellectual factor that impedes math achievement. Some students who perform poorly on math assessments have a full understanding of the mathematical concepts being tested; however, their anxiety interferes with their ability to solve mathematical problems (Sparks, 2011; Hellum-Alexander, 2010; Ashcraft & Krause, 2007; Cavanaugh, 2007; Tsui & Mazzocco, 2007). Beilock and colleagues (2010) concluded that “the fears that math-anxious individuals experience when they are called on to do math prevent them from using the math knowledge they possess to show what they know.” A number of researchers have hypothesized that math anxiety disrupts performance because it reduces students’ working memory, leaving them unable to block out distractions and irrelevant information or to retain information while working on tasks (Sparks, 2011; Legg & Locker, 2009; Ashcraft & Krause, 2007; Cavanaugh, 2007; Beilock & Carr, 2005).


Math anxiety is the way in which students’ lack of confidence in that subject undermines their academic performance and is a serious obstacle for many children across all grade levels. Mathanxious students learn less math than their low-anxious peers because they take fewer math classes and get poorer grades in the math classes they do take. Math anxiety has been recognized as a non-intellectual factor that impedes math achievement. Studies have found a strong negative relationship between math anxiety and test scores, such that as students’ math anxiety increases, their test scores decrease. A number of researchers have hypothesized that math anxiety disrupts performance because it reduces students’ working memory, leaving them unable to block out distractions and irrelevant information or to retain information while working on tasks.


Researchers have found that both teachers and parents have a strong influence on students’ math anxiety. There are strategies teachers and parents can use to prevent or reduce math anxiety. For example, teachers should develop strong skills and a positive attitude toward math; relate math to real life experiences; encourage critical thinking and active learning; and de-emphasize correct answers and computational speed. Parents should avoid expressing negative attitudes about math; provide their children with support and encouragement; and carefully monitor their children’s math progress. There are strategies students can use to overcome their own math anxiety. Students should practice math every day, study according to their individual learning style, and seek immediate assistance when they don’t understand a particular mathematical concept.




Digital Game-Based Learning ( DGBL )

After years of research and proselytizing, the proponents of digital game-based learning (DGBL) have been caught unaware. Like the person who is still yelling after the sudden cessation of loud music at a party, DGBL proponents have been shouting to be heard above the prejudice against games. But now, unexpectedly, we have everyone's attention. The combined weight of three factors has resulted in widespread public interest in games as learning tools.


The first factor is the ongoing research conducted by DGBL proponents. In each decade since the advent of digital games, researchers have published dozens of essays, articles, and mainstream books on the power of DGBL—including, most recently, Marc Prensky's Digital Game-Based Learning (2001), James Paul Gee's What Video Games Have to Teach Us about Learning and Literacy (2003), Clark Aldrich's Simulations and the Future of Learning: An Innovative (and Perhaps Revolutionary) Approach to e-Learning (2004), Steven Johnson's Everything Bad Is Good for You: How Today's Popular Culture Is Actually Making Us Smarter (2005), Prensky's new book “Don’t Bother Me, Mom, I'm Learning!”: How Computer and Video Games Are Preparing Your Kids for 21st Century Success and How You Can Help! (2006), and the soon-to-be-published Games and Simulations in Online Learning: Research and Development Frameworks, edited by David Gibson, Clark Aldrich, and Marc Prensky. The second factor involves today’s “Net Generation,” or “digital natives,” who have become disengaged with traditional instruction. They require multiple streams of information, prefer inductive reasoning, want frequent and quick interactions with content, and have exceptional visual literacy skills1—characteristics that are all matched well with DGBL. The third factor is the increased popularity of games. Digital gaming is a $10 billion per year industry, and in 2004, nearly as many digital games were sold as there are people in the United States (248 million games vs. 293.6 million residents).


One could argue, then, that we have largely overcome the stigma that games are “play” and thus the opposite of “work.” A majority of people believe that games are engaging, that they can be effective, and that they have a place in learning. So, now that we have everyone's attention, what are we DGBL proponents going to say? I believe that we need to change our message. If we continue to preach only that games can be effective, we run the risk of creating the impression that all games are good for all learners and for all learning outcomes, which is categorically not the case. What is needed now is (1) research explaining why DGBL is engaging and effective, and (2) practical guidance for how (when, with whom, and under what conditions) games can be integrated into the learning process to maximize their learning potential. We are ill-prepared to provide the needed guidance because so much of the past DGBL research, though good, has focused on efficacy (the message that games can be effective) rather than on explanation (why and how they are effective) and prescription (how to actually implement DGBL).


This is not to say that we have ignored this issue entirely. Many serious game proponents have been conducting research on how games can best be used for learning, resulting in a small but growing body of literature on DGBL as it embodies well-established learning principles, theories, and models. On the other hand, many DGBL proponents have been vocal about the dangers of “academizing” (“sucking the fun out of,” as Prensky would say) games. This is partly the result of our experiences with the edutainment software of the last decade or so, which instead of harnessing the power of games for learning, resulted in what Professor Seymour Papert calls “Shavian reversals”: offspring that inherit the worst characteristics of both parents (in this case, boring games and drill-and-kill learning).Many argue that this happened because educational games were designed by academicians who had little or no understanding of the art, science, and culture of game design. The products were thus (sometimes!) educationally sound as learning tools but dismally stunted as games. Yet if we use this history and these fears to argue, as some have, that games must be designed by game designers without access to the rich history of theory and practice with games in learning environments, we are also doomed to fail. We will create games that may be fun to play but are hit-or-miss when it comes to educational goals and outcomes. The answer is not to privilege one arena over the other but to find the synergy between pedagogy and engagement in DGBL.


In this article, I will outline why DGBL is effective and engaging, how an institution can leverage those principles to implement DGBL, how faculty can integrate commercial off-the-shelf (COTS) DGBL in the classroom, what DGBL means for institutional IT support, and the lessons we can learn from past attempts at technological innovations in learning.


If we are to think practically and critically about DGBL, we need to separate the hype from the reality. Many who first hear about the effectiveness of games are understandably skeptical. How much of the research is the result of rigorous, controlled experimental design, and how much is wishful thinking and propaganda? A comprehensive analysis of the field is not possible here and, in any case, has already been done by others. Several reviews of the literature on gaming over the last forty years, including some studies that use rigorous statistical procedures to analyze findings from multiple studies (meta-analyses), have consistently found that games promote learning and/or reduce instructional time across multiple disciplines and ages. Although many of these reviews included non-digital games (pre-1980), there is little reason to expect that the medium itself will change these results. A cursory review of the experimental research in the last five years shows well-documented positive effects of DGBL across multiple disciplines and learners.


What accounts for the generally positive effects found in all these studies about games and learning? These empirical studies are only part of the picture. Games are effective not because of what they are, but because of what they embody and what learners are doing as they play a game. Skepticism about games in learning has prompted many DGBL proponents to pursue empirical studies of how games can influence learning and skills. But because of the difficulty of measuring complex variables or constructs and the need to narrowly define variables and tightly control conditions, such research most often leads to studies that make correspondingly narrow claims about tightly controlled aspects of games (e.g., hand-eye coordination, visual processing, the learning of facts and simple concepts).


As Johnson says in Everything Bad Is Good for You: “When I read these ostensibly positive accounts of video games, they strike me as the equivalent of writing a story abut the merits of the great novels and focusing on how reading them can improve your spelling.” Although it’s true that games have been empirically shown to teach lower-level intellectual skills and to improve physical skills, they do much more than this. Games embody well-established principles and models of learning. For instance, games are effective partly because the learning takes place within a meaningful (to the game) context. What you must learn is directly related to the environment in which you learn and demonstrate it; thus, the learning is not only relevant but applied and practiced within that context. Learning that occurs in meaningful and relevant contexts, then, is more effective than learning that occurs outside of those contexts, as is the case with most formal instruction. Researchers refer to this principle as situated cognition and have demonstrated the effectiveness of this principle in many studies over the last fifteen years. Researchers have also pointed out that play is a primary socialization and learning mechanism common to all human cultures and many animal species. Lions do not learn to hunt through direct instruction but through modeling and play. Games, clearly, make use of the principle of play as an instructional strategy.


There are other theories that can account for the cognitive benefits of games. Jean Piaget's theories about children and learning include the concepts of assimilation and accommodation. With assimilation, we attempt to fit new information into existing slots or categories. An example of an adult assimilating information might be that when a man turns the key in the ignition of his car and the engine does not turn over, and in the past this has been due to a dead battery, he is now likely to identify the problem as a dead battery. Accommodation involves the process whereby we must modify our existing model of the world to accommodate new information that does not fit into an existing slot or category. This process is the result of holding two contradictory beliefs. In the previous example, should the man replace the battery and experience the same problem, he finds that the engine not starting both means and does not mean a dead battery. Accordingly, our stranded motorist must adjust his mental model to include other problems like alternators and voltage regulators (although perhaps only after an expensive trip to his auto mechanic). This process is often referred to as cognitive disequilibrium. Piaget believed that intellectual maturation over the lifespan of the individual depends on the cycle of assimilation and accommodation and that cognitive disequilibrium is the key to this process.


Games embody this process of cognitive disequilibrium and resolution. The extent to which these games foil expectations (create cognitive disequilibrium) without exceeding the capacity of the player to succeed largely determines whether they are engaging. Interacting with a game requires a constant cycle of hypothesis formulation, testing, and revision. This process happens rapidly and frequently while the game is played, with immediate feedback. Games that are too easily solved will not be engaging, so good games constantly require input from the learner and provide feedback. Games thrive as teaching tools when they create a continuous cycle of cognitive disequilibrium and accommodation while also allowing the player to be successful. There are numerous other areas of research that account for how and why games are effective learning tools, including anchored instruction, feedback, behaviorism, constructivism, narrative psychology, and a host of other cognitive psychology and educational theories and principles. Each of these areas can help us, in turn, make the best use of DGBL.


The positive effects of DGBL seen in experimental studies can be traced, at least partially, to well-established principles of learning as described earlier (e.g., situated cognition, play theory, assimilation and accommodation) and elsewhere by others.This means that DGBL can be implemented most effectively, at least in theory, by attending to these underlying principles. How, then, can we use this knowledge to guide our implementation of DGBL in higher education?


A review of the DGBL literature shows that, in general, educators have adopted three approaches for integrating games into the learning process: have students build games from scratch; have educators and/or developers build educational games from scratch to teach students; and integrate commercial off-the-shelf (COTS) games into the classroom. In the first approach, students take on the role of game designers; in building the game, they learn the content. Traditionally, this has meant that students develop problem-solving skills while they learn programming languages. Professional game development takes one to two years and involves teams of programmers and artists. Even though this student-designed approach to DGBL need not result in commercial-quality games, it is nonetheless a time-intensive process and has traditionally been limited to computer science as a domain. It is certainly possible for modern game design to cross multiple disciplines (art, English, mathematics, psychology), but not all teachers have the skill sets needed for game design, not all teach in areas that allow for good content, not all can devote the time needed to implement this type of DGBL, and many teach within the traditional institutional structure, which does not easily allow for interdisciplinarity. For these reasons, this approach is unlikely to be used widely.


In the second case, we can design games to seamlessly integrate learning and game play. Touted by many as the “Holy Grail” approach to DGBL because of its ability to potentially address educational and entertainment equally, and to do so with virtually any domain, this professionally designed DGBL process is more resource-intensive than the first option. This is because the games must be comparable in quality and functionality to commercial off-the-shelf (COTS) games, which after all are very effective in teaching the content, skills, and problem-solving needed to win the game. The development of such "serious games" is on the rise, and the quality of the initial offerings is promising (e.g., Environmental Detectives, developed by the Education Arcade; Hazmat: Hotzone, under development at the Entertainment Technology Center at Carnegie Mellon University; Virtual U, originally conceived and developed by Professor William F. Massy; and River City, developed by Professor Chris Dede, the Harvard Graduate School of Education, and George Mason University). However, the road to the development of serious games is also littered with Shavian reversals (poor examples of edutainment in which neither the learning nor the game is effective or engaging). Consequently, fewer companies are willing to spend the time and money needed to develop these games, for fear of revisiting their unprofitable past, and so the number of games that can be developed is limited. Although this professionally designed DGBL approach is clearly the future of DGBL, we are not likely to see widespread development of these games until we demonstrate that DGBL is more than just a fad and until we can point to persuasive examples that show games are being used effectively in education and that educators and parents view them as they now view textbooks and other instructional media.


The third approach—integrating commercial off-the-shelf digital game-based learning (COTS DGBL)—involves taking existing games, not necessarily developed as learning games, and using them in the classroom. In this approach, the games support, deliver, and/or assess learning. This approach is currently the most cost-effective of the three in terms of money and time and can be used with any domain and any learner. Quality is also maximized by leaving the design of game play up to game designers and the design of learning up to teachers. I believe that this approach to DGBL is the most promising in the short term because of its practicality and efficacy and in the long term because of its potential to generate the evidence and support we need to entice game companies to begin developing serious games.


This approach is gaining acceptance because of its practicality, and research shows that it can be effective. Entertainment Arts (EA), a game-development company, and the National Endowment for Science, Technology, and the Arts (NESTA) in the United Kingdom have entered into a joint partnership to study the use of COTS games in European schools, and similar initiatives are being proposed in the United States. If the United States is like the United Kingdom, where 60 percent of teachers support the use of games in the classroom, the United States may be well-positioned to begin generating the evidence (through the use of COTS games) that the game industry needs to begin developing serious games.
Integrating COTS games is not without its drawbacks. Commercial games are not designed to teach, so topics will be limited and content may be inaccurate or incomplete. This is the biggest obstacle to implementing COTS DGBL: it requires careful analysis and a matching of the content, strengths, and weaknesses of the game to the content to be studied.


There are ways to minimize these drawbacks, some of which I will discuss later, but the elephant in the room is that in our conversations about DGBL, we rarely acknowledge that the taxonomy of games is as complex as our learning taxonomies. Not all games will be equally effective at all levels of learning. For instance, card games are going to be best for promoting the ability to match concepts, manipulate numbers, and recognize patterns. Jeopardy-style games, a staple of games in the classroom, are likely to be best for promoting the learning of verbal information (facts, labels, and propositions) and concrete concepts. Arcade-style games (or as Prensky and others refer to them, “twitch” games) are likely to be best at promoting speed of response, automaticity, and visual processing. Adventure games, which are narrative-driven open-ended learning environments, are likely to be best for promoting hypothesis testing and problem solving. Many games also blur these taxonomic lines, blending strategy with action and role playing, for instance.


It is critical, therefore, that we understand not just how games work but how different types of games work and how game taxonomies align with learning taxonomies. This is not a new idea. In perhaps one of the most ambitious and rigorous examinations of the use of games to teach mathematics, a 1985 study undertaken for the National Council of Teachers of Mathematics developed eleven games for different grade levels using 1,637 participants. The study authors intended their eleven separate game studies to examine if and how games could be used to teach mathematics at varying learning levels.13 Games, they hypothesized, might be better at promoting learning at some levels than at others. Further, they distinguished between three types of game use: pre-, co-, and post-instructional, based on when games were used in relation to the existing curriculum. The study authors found that there were indeed differences by learning level and by whether games were used prior to, during, or after other instruction and also that there were interactions between these two factors. They concluded that although drill-and-practice-type games at the time made up the vast majority of edutainment titles, instructional games could be effective for higher learning levels if designed and implemented well. Though this seems to support the development of serious games, the core principle—that games can promote learning at higher taxonomic levels—is as applicable to COTS games, which require and promote problem-solving and situated cognition before they are integrated with instructional activities or content.


It is important to understand how the theoretical issues outlined here relate to the use of games to teach. Although this section gives a practical description of the issues, it is meant more as a heuristic for understanding the issues involved than as a prescriptive tool. There are a wide range of other factors that must be considered, such as using the game outside of the classroom (as with all homework), balancing game play and other instructional activities, and rotating students’ use of the computers in classrooms where there is not a one-to-one student-computer ratio. Many of these issues are not unique to DGBL, however, and are adequately treated by authors of texts that emphasize integrating computer technology into the learning process.


Once we have chosen a game, and have analyzed it for content, we have to decide what to do about missing and inaccurate content. What content will have to be created to address gaps? Who will provide this content? Some believe that this is the teacher's responsibility, but current thinking in education suggests that the more students are responsible for in their learning, the more they will learn. Certainly, there is some content that will not be practical for students to address on their own, but wherever and whenever we can maximize student responsibility, we should.


The way we choose to maximize student responsibility is important. Because we are going to have to go out of the game environment and into the classroom, we run the risk of eliminating what is fun and engaging about the game. So, rather than simply providing additional reading or handouts with the missing or accurate information, we should strive to design activities that are logical extensions of the game world. Learning is integral to the story of the game world—players are never asked to step out of the game world to do something (although they frequently do so when stuck). The constant cycle of cognitive disequilibrium and resolution—the engagement—is what leads to the experience that Professor Mihaly Csikszentmihalyi refers to as flow.15 Flow occurs when we are engaged in an activity (physical, mental, or both) at a level of immersion that causes us to lose track of time and the outside world, when we are performing at an optimal level. Good games promote flow, and anything that causes us to "leave" the game world (e.g., errors, puzzles that require irrational solutions) interrupts flow. If we were to simply design “traditional” classroom activities (workbooks, textbook reading, teacher handouts, etc.) that addressed the missing, misleading, or inaccurate content in the game, we would be interrupting the flow experience. Granted, anytime we ask the players to stop the game and do something else, flow will be interrupted. But to the extent that we can keep these additional activities “situated” within the game world (i.e., connected to the problem being solved, the characters solving it, and the tools and methods those characters use or might use), we will minimize this interruption of flow. For the same reasons, we should make sure that students spend enough time in the game to promote flow and, correspondingly, significant time in the extended instructional activities. Even if these extended activities do not promote flow, the more frequently students move from the game to other activities (even those related to the game), the more frequently flow will be interrupted in each activity.


Although it is not possible to stay entirely within the game world (and therefore to keep students in flow) when implementing COTS DGBL, there is another reason we should strive to keep the activities we design situated within that game world. Malone and Lepper identify fantasy (endogenous and exogenous) as one of four main areas that make games intrinsically motivating. 16 Endogenous fantasy elements are those fantasy parts that are seamlessly integrated with the game world and story; exogenous fantasy elements are those that, though in the game, do not appear to have much relation to the story or game world. Endogenous fantasy elements not only help make games intrinsically motivating; in theory, they should also promote flow. So whenever we ask students to not be in the game, we should strive to keep the activities and roles they take on (the fantasy) endogenous to the game.


Thus, the roles we ask them to take on should be extensions of the roles they play in the game. These can be main characters, ancillary characters, or characters that could hypothetically be part of the game. The activities we ask them to perform as these characters should be authentic to the goals of the game world and the professions or characteristics of these characters. Some examples of endogenous activities might be to develop budgets, spreadsheets, reports/charts, and databases; to write diaries, scientific reports, letters, legal briefs, dictionaries, faxes; to design, duplicate, and conduct experiments; to conduct and write up feasibility studies; and to assess the veracity of game information or provide missing data. We should not be so naïve as to think that students will find these activities to be as engaging as the games, but given our need to meet curricular goals and our desire to tightly integrate the games with the learning process, this seems a good way to meet in the middle.


Of the several technology "learning revolutions" during the last quarter-century, most have failed to achieve even half of their promise. Although there are many reasons for this, the primary fault lies with our inability (or unwillingness) to distinguish between the medium and the message. Two examples of such technological learning innovations from our recent past are media technology and computing technology.


In the 1960s and 1970s, audio and video (and later, television) were touted as technologies that would revolutionize learning. We rapidly began implementing media wherever possible, regardless of grade, domain, or learners. Many studies were conducted during the 1970s to compare media-based classrooms to "traditional" classrooms, and some of the more sensational ones found their way into the public eye. By the 1980s, enough studies had been conducted to allow for meta-analyses and reviews of the literature. Most of these resulted in what has famously been called the "no significant difference" phenomenon—meaning that, overall, media made no significant difference to learning. This was not surprising to instructional designers, who argued that the implementation of media was not consistently of high quality and that the quality of the instruction in "media" versus "traditional" classrooms was not controlled. The key to understanding this issue lies in the difference between use and integration of media. Using media requires only that the media be present during instruction. Integrating media, on the other hand, requires a careful analysis of the strengths and weaknesses of the media, as well as its alignment with instructional strategies, methods, and learning outcomes. Weaknesses are then addressed through modification of the media or inclusion of additional media and/or instruction, and instruction is modified to take advantage of the strengths of the media. In cases where there is poor alignment, the media is not used.


Sadly, the history of the use of computing technology in learning parallels that of media use. The personal computer arrived in the 1970s, and predictions of revolutionized learning quickly followed. Schools spent hundreds of thousands of dollars on computers in the early 1980s, vowing to place one in every classroom. Studies comparing classrooms with computing technology and those without proceeded at the same pace as had studies comparing media-rich and media-poor classrooms. Once again, instructional designers and others pointed out that the quality of implementation varied greatly, making comparisons impossible. By the time there were enough studies to evaluate and review, the quality and diversity of the different implementations made it difficult to draw any meaningful conclusions. Once again, it seemed there was "no significant difference" between classrooms that used technology and those that did not. Once again, we had mistaken technology use for technology integration.


Eventually, though, educators learned from this and from prior experience with media. They began developing and testing better-integrated uses of computing technology. Since the early 1990s, educators have been moving toward technology integration and toward pre-service teacher training, emphasizing alignment of the curriculum with the technology. We must take what we have learned forward as we consider how, when, and with whom to implement DGBL in the future.