Digital technology continues to rapidly transform all aspects of life and work, even (and perhaps all the more so) in the developing world. It is designed, and presumed, to bring great benefit and empowerment to its users, as well as profit to its developers. Yet, as it opens new and even unanticipated possibilities, it poses as many problems as it solves, some new, and some techno-versions of classical problems, all of them important and interesting. And technology, for its novelty and glamorous aspirations, is greedy for our attention, liking to take center stage in every arena it enters. Education, and mathematics education in particular, is the context in which this panel is examining these transformations. I find it helpful here to distinguish three broad kinds of roles that technology can play in mathematics education. They are of course not disjoint.
I. Transmission: Use of technology (web, video conferencing, etc.) to transmit, perhaps interactively, instruction and/or instructional materials that are conceptually of a traditional genre – lectures, demonstrations, problem sets, assessments, etc. These are the kinds of uses that fundamentally support distance learning, for example.
II. Power, speed, and visualization: Use of technology to carry out quickly and more accurately and completely, mathematical processes of a traditional nature – perform large or complex calculations, solve equations, approximate integrals, exhibit function graphs, study effects of variation of parameters, produce vivid and accurate images of geometric figures, etc.
III. New ways to explore the (mathematical and experiential ) universes: Use of technology to do things we have never previously been able to do. Such capability affects mathematics itself, not just mathematics education. Examples include the study of long-term evolution of dynamical systems, and the images of fractal geometry that emerge there from. (This had an effect on dynamics comparable with that of the telescope in astronomy and the microscope in biology.)
Software development gave life to the field of computational complexity, with its applications to coding and cryptography. Mathematical modeling and computer simulation supports a virtually empirical study of physical systems and designs. Dynamic geometry offers unprecedented opportunities to visually explore and analyze geometric structures, and to produce evocative imagery of dimensions three and four (using time). Computer algebra systems furnish unprecedented resources for solving equations. Much of this new technological power is now within reach of many students, and this raises possibilities of thereby expanding the horizons of the mathematics curriculum.
At a pragmatic level, technology thus offers resources to address two fundamental challenges of contemporary education – distance and demographics. Distance because many learners in need are physically remote from the sources of quality instruction and materials. Gilda Bolaños offers us an excellent survey of diverse modes of distance learning formats. Demographics because class sizes, particularly in introductory level mathematics courses, are too large to afford adequate instructor attention to individual student learning. (Bounding class sizes is often done at the cost of using instructors of highly variable quality.) In this case, technology affords various interactive formats for student work and assessment. These include the “virtual laboratories” described by Ruedi Seiler, and the interactive online materials (lectures, automatically graded homework, etc.) discussed by Mika Seppälä.
But independently of these practical needs, technology also offers possibilities for improving mathematics instruction itself. And the fundamental questions about the quality of teaching and learning do not recede when the instruction is mediated by technology; they only change their form.
Instruction : By “instruction” I mean the dynamic interaction among teacher, content, and students. I rely here on the “instructional triangle” that Cohen and Ball use to depict the set of interactions that they call “instruction” (Cohen and Ball, 1999). Viewed in this way, instruction can go wrong in some simple but profound ways, for its quality depends on the relations among all of these three elements. When they misconnect, students’ opportunities for learning are impaired. For example, if a teacher is not able to make the content accessible to students, framing it in ways that are incomprehensible to them, the chances that they may misunderstand are great. If students’ interpretations of a task are different from the teacher’s or the textbook author’s intentions, then their work may be misrouted or take the work in unhelpful directions.
It may seem slightly strange, in the context of this panel, to propose the above representation of instruction. For, if you think about it, most descriptions of instructional uses of technology appear to reside exclusively on the bottom edge of the instructional triangle, absent the teacher. A tacit premise of some of this thinking is that somehow, the technology, with its interactive features, actually substitutes for the teacher, or renders the teacher obsolete, except perhaps as a manager of the environment. The viability of this view is a deep and important question, one that I shall not enter here except to make a couple of observations. One is that, in the most successful models of distance learning, it was found to be essential to have a tutor or facilitator available at the remote sites of reception of the materials, to respond to the many questions and requests that students would have, and that were not adequately responded by the technology environment. In addition, it was found to be important to have real time online questioning of the primary source available at certain times. In other words, prepared and transmitted material alone no more teaches a learner than does a textbook, unmediated by a teacher. The other comment is that interactive technology formats can at best provide well-prepared instructional materials and tasks, and respond to the student productions and questions that the software developers have anticipated and for which they have programmed responses. There are many domains of procedural learning and performance where this can be somewhat successful, though the software, no more than a skilled teacher, cannot completely predict and prepare for all of what students may come up with. Moreover this uncertainty is all the greater once one enters into territory that is less procedural and involves more conceptual reasoning and problem solving.
In what follows, I identify five persistent problems of mathematics instruction and discuss ways in which technology can be deployed to address these. How these are actually used, however, would affect the degree to which they were helpful, so for each case, I point out its possible pitfalls.
1. Making mathematically accurate and pedagogically skillful diagrams. One problem faced by mathematics teachers at all levels is how to make clear and accurate diagrams that make the essential mathematical ideas plain to learners, and how to do so in ways that are manipulable for mathematical reasoning. Doing this by hand is often no easy task, whether the sketch is of slices of an ellipsoid in calculus, or sixteenths of a rectangle in fifth grade. Mathematical accuracy is one dimension of the challenge; featuring is a second - that is, making the instructionally key features visible to learners. In addition, instructors must manage these challenges fluently, using class time effectively. An instructor who can make diagrams accurately and helpfully, but who must use 10 minutes of class time to do so, loses effectiveness. Diagrams are also used for a variety of purposes: explorationally, to investigate what happens if certain elements are allowed to vary, or presentationally, to demonstrate an idea, an explanation, or a solution. This means, sometimes, the need for dynamics - translations, rotations, rescaling, variation of parameters. Often diagrams must be made in ways that map clearly to algebraic or numerical representations. Drawing software, or other design tools, can help. Important is the capacity to produce carefully-scaled diagrams, with the capacity for color or shading, and to be able to move elements of a diagram. Its use must be fast and flexible, helpful both for carefully designed lectures and for improvisation on the fly, in response to a student’s question. Such software or tools can provide significant support for the use of diagrams in class, by both students and instructor. Making such software accessible to students increases their capacity for individual explorations and preparation for contributions in class. Students can quickly put their diagrams up for others’inspection, or support a point in class, in ways that are difficult to do when students go to the board to generate representations by hand. Using software tools to support the visual dimensions of mathematical work in instruction can significantly alter a major dimension of instruction and do so in ways that are mathematically accurate, pedagogically useful, and sensitive to the real-time challenges of classroom instruction where class periods are finite and time is a critical resource. Software tools to support the making of diagrams can create problems, too. For example, if the tools are rigid or interfere with the purposes for making diagrams, or cannot be manipulated as desired, the representations may not be as useful as needed. Another problem may be that the use of such tools inhibits students from developing personal skills of appraisal and construction. If the tools quickly make correct diagrams, students may not develop a critical eye with which to inspect them. If they never have to make a diagram themselves, they may remain entirely dependent on the software and not develop independent capacities for drawing.
2. Making records of class work and using them cumulatively across time. A second pervasive problem of mathematics instruction can be seen in the overflowing blackboards full of work and the slippery sheets of transparencies filled with notation and sketches, generated in class, and that vanish into weak memory when class ends. The record of class work (not just text or prepared materials), whether lecture, discussion, or exploration, is an important product of instruction. Under ordinary circumstances, this product vanishes and is thus unavailable for study or future reference, use, or modification. So acute is this problem that, too often, even during a single class, such work is erased (in the case of chalkboards) or slid away (as in transparencies). The work of that single class period is weakened for not being able to secure its place in evolution of ideas in the course. Moreover it is not available for students who may have missed a class. When the work done in class is created or preserved in digital form, an archive of the mathematical progress of the class can become a resource for ongoing learning. It can then be easily accessed and transmitted remotely to others. Doing it “live” in class requires skill and dexterity on the part of the instructor. Making records of classwork afterwards (i.e., photographing the board with a digital camera) is easier but possibly less manipulable for subsequent class work. Important, too, is that everyone who needs to access these records can work on a common platform or that the format will work reliably across platforms.
3. Alignment between classes and textbook. Instructors, perhaps in response to student ideas or productions, may choose to depart from the text - in topic treatment or sequencing, or even topic coverage, and in the design of student activities and tasks. If the instructor creates these variations and alternative paths in electronic form, then a new text is created based on the instructor’s design. This affords students access to
the substance and course of the lessons. This gives license to flexible and innovative instruction, by affording the means to do so without disadvantaging students through disconnection from a text to be perused and revisited over time. 4. Ease of access to the instructor between classes. In the developed world, it is hard to imagine university instructors who do not maintain email (and web) connection with their students. This has made much more fluent and elastic the traditional functions of “office hours.” Most student questions can be handled expeditiously, in timely fashion (though asynchronously), by email (perhaps with attachments), thus greatly reducing the need for face-to-face meetings, with their scheduling difficulties. And, as with the discussion above, these exchanges can contribute significantly to the record of the student’s work and progress. When appropriate, an exchange between one student and the instructor can easily be made available to other students, thus changing an individual “office hour” into a group discussion. Pitfalls can exist with electronic communications, of course. Misunderstanding is frequent when communication is restricted to text, without gesture, intonation, and the ability to demonstrate or show.
5. The repetitive nature of individual outside-of-class sessions. One feature of traditional office hours, or help sessions, is that they tend to be repetitive, processing over and over again the same questions and assistance with each new student or group of students. When such assistance is administered electronically, and it is seen to be germane to the interests of the whole class, it is an easy matter to copy the whole class, or perhaps selected individuals, on such exchanges. This puts to collective profit the considerable instructional investment made in one student, or group of students, and everyone gains, not least the instructor. An important consideration here is sensitivity to privacy issues and confidentiality. In particular, making individual
student communications requires prior consent.
Technology continues to transform all aspects of our lives and work. It is already difficult to imagine how we once functioned without email and the web. We are still at the early stages of trying to understand and design the best uses of technology for mathematics instruction. I have pointed to some promising uses of technology to address some endemic problems of even traditional instruction. I have also tried to signal that the fundamental problem of developing quality teaching does not disappear just because instruction is mediated in technological environments.
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