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The Oersted Medal Lecture 2002 by David Hestenes from Arizona State University. It discusses the connection between physics teaching and research at its deepest level and how Physics Education Research (PER) can illuminate this connection. The lecture introduces a comprehensive language called Geometric Algebra (GA) that simplifies and integrates classical and quantum physics. The document also discusses the challenges of introducing research-based reform into a conservative physics curriculum.
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David Hestenes Department of Physics and Astronomy Arizona State University, Tempe, Arizona 85287- The connection between physics teaching and research at its deepest level can be illuminated by Physics Education Research (PER). For students and scientists alike, what they know and learn about physics is profoundly shaped by the conceptual tools at their command. Physicists employ a miscellaneous assortment of mathematical tools in ways that contribute to a fragmenta- tion of knowledge. We can do better! Research on the design and use of mathematical systems provides a guide for designing a unified mathemati- cal language for the whole of physics that facilitates learning and enhances physical insight. This has produced a comprehensive language called Geo- metric Algebra, which I introduce with emphasis on how it simplifies and integrates classical and quantum physics. Introducing research-based re- form into a conservative physics curriculum is a challenge for the emerging PER community. Join the fun!
The relation between teaching and research has been a perennial theme in academia as well as the Oersted Lectures, with no apparent progress on re- solving the issues. Physics Education Research (PER) puts the whole matter into new light, for PER makes teaching itself a subject of research. This shifts attention to the relation of education research to scientific research as the central issue. To many, the research domain of PER is exclusively pedagogical. Course content is taken as given, so the research problem is how to teach it most effec- tively. This approach to PER has produced valuable insights and useful results. However, it ignores the possibility of improving pedagogy by reconstructing course content. Obviously, a deep knowledge of physics is needed to pull off anything more than cosmetic reconstruction. It is here, I contend, in addressing the nature and structure of scientific subject matter, that PER and scientific research overlap and enrich one another with complementary perspectives. The main concern of my own PER has been to develop and validate a sci- entific Theory of Instruction to serve as a reliable guide to improving physics teaching. To say the least, many physicists are dubious about the possibility. Even the late Arnold Arons, patron saint of PER, addressed a recent AAPT session with a stern warning against any claims of educational theory. Against this backdrop of skepticism, I will outline for you a system of general principles that have guided my efforts in PER. With sufficient elaboration (much of which
already exists in the published literature), I believe that these principles provide quite an adequate Theory of Instruction. Like any other scientific theory, a Theory of Instruction must be validated by testing its consequences. This has embroiled me more than I like in developing suitable instruments to assess student learning. With the help of these instru- ments, the Modeling Instruction Program has amassed a large body of empirical evidence that I believe supports my instructional theory. We cannot review that evidence here, but I hope to convince you with theoretical arguments. A brief account of my Theory of Instruction sets the stage for the main subject of my lecture: a constructive critique of the mathematical language used in physics with an introduction to a unified language that has been developed over the last forty years to replace it. The generic name for that language is Geometric Algebra (GA). My purpose here is to explain how GA simplifies and clarifies the structure of physics, and thereby convince you of its immense implications for physics instruction at all grade levels. I expound it here in sufficient detail to be useful in instruction and research and to provide an entre´e to the published literature. After explaining the utter simplicity of the GA grammar in Section V, I explicate the following unique features of the mathematical language: (1) GA seamlessly integrates the properties of vectors and complex numbers to enable a completely coordinate-free treatment of 2D physics. (2) GA articulates seamlessly with standard vector algebra to enable easy contact with standard literature and mathematical methods. (3) GA Reduces “grad, div, curl and all that” to a single vector derivative that, among other things, combines the standard set of four Maxwell equations into a single equation and provides new methods to solve it. (4) The GA formulation of spinors facilitates the treatment of rotations and rotational dynamics in both classical and quantum mechanics without coordi- nates or matrices. (5) GA provides fresh insights into the geometric structure of quantum me- chanics with implications for its physical interpretation. All of this generalizes smoothly to a completely coordinate-free language for spacetime physics and general relativity to be introduced in subsequent papers. The development of GA has been a central theme of my own research in theoretical physics and mathematics. I confess that it has profoundly influenced my thinking about PER all along, though this is the first time that I have made it public. I have refrained from mentioning it before, because I feared that my ideas were too radical to be assimilated by most physicists. Today I am coming out of the closet, so to speak, because I feel that the PER community has reached a new level of maturity. My suggestions for reform are offered as a challenge to the physics community at large and to the PER community in particular. The challenge is to seriously consider the design and use of mathematics as an important subject for PER. No doubt many of you are wondering why, if GA is so wonderful – why have you not heard of it before? I address that question in the penultimate Section by discussing the reception of GA and similar reforms by the physics community and their bearing on prospects for incorporating GA
Modeling Theory is about the structure and acquisition of scientific knowledge. Its central tenet is that scientific knowledge is created, first, by constructing and validating models to represent structure in real objects and processes, and second, by organizing models into theories structured by scientific laws. In other words, Modeling Theory is a particular brand of scientific epistemology that posits models as basic units of scientific knowledge and modeling (the process of creating and validating models) as the basic means of knowledge acquisition. I am not alone in my belief that models and modeling constitute the core of scientific knowledge and practice. The same theme is prominent in recent History and Philosophy of Science, especially in the work of Nancy Nercessian and Ronald Giere,^6 and in some math education research.^7 It is also proposed as a unifying theme for K-12 science education in the National Science Education
Standards and the AAAS Project 2061. The Modeling Instruction Project has led the way in incorporating it into K-12 curriculum and instruction.^3 Though I first introduced the term “Modeling Theory” in connection with my Theory of Instruction,^8 I have always conceived of it as equally applicable to scientific research. In fact, Modeling Theory has been the main mechanism for transferring what I know about research into designs for instruction. In so far as Modeling Theory constitutes an adequate epistemology of sci- ence, it provides a reliable framework for critique of the physics curriculum and a guide for revising it. From this perspective, I see the standard curriculum as seriously deficient at all levels from grade school to graduate school. In partic- ular, the models inherent in the subject matter are seldom clearly delineated. Textbooks (and students) regularly fail to distinguish between models and their implications.^2 This results in a cascade of student learning difficulties. However, we cannot dwell on that important problem here. I have discussed Modeling Theory and its instructional implications at some length elsewhere,2, 8–10^ although there is still more to say. The brief account above suffices to set the stage for application to the main subject of this lecture. Modeling Theory tells us that the primary conceptual tools mentioned in the 4th Learning Principle are modeling tools. Accordingly, I have devoted considerable PER effort to classification, design, and use of modeling tools for instruction. Heretofore, emphasis has been on the various kinds of graphs and diagrams used in physics, including analysis of the information they encode and comparison with mathematical representations.9, 10^ All of this was motivated and informed by my research experience with mathematical modeling. In the balance of this lecture, I draw my mathematics research into the PER domain as an example of how PER can and should be concerned with basic physics research.
Mathematics is taken for granted in the physics curriculum—a body of im- mutable truths to be assimilated and applied. The profound influence of math- ematics on our conceptions of the physical world is never analyzed. The pos- sibility that mathematical tools used today were invented to solve problems in the past and might not be well suited for current problems is never considered. I aim to convince you that these issues have immense implications for physics education and deserve to be the subject of concerted PER. One does not have to go very deeply into the history of physics to discover the profound influence of mathematical invention. Two famous examples will suffice to make the point: The invention of analytic geometry and calculus was essential to Newton’s creation of classical mechanics.^2 The invention of tensor analysis was essential to Einstein’s creation of the General Theory of Relativity. Note my use of the terms “invention” and “creation” where others might have used the term “discovery.” This conforms to the epistemological stance of Modeling Theory and Einstein himself, who asserted that scientific theories
sented in several different systems, but one of them is invariably better suited than the others for a given application. For example, Goldstein’s textbook on mechanics^12 gives three different ways to represent rotations: coordinate matri- ces, vectors and Pauli spin matrices. The costs in time and effort for translation between these representations are considerable.
major task for theorists is to construct a mathematical language that optimizes expression of the key ideas and consequences of the theory. Although existing mathematics should be consulted in this endeavor, it should not be incorporated without critically evaluating its suitability. I might add that the process also works in reverse. Modification of mathematics for the purposes of science serves as a stimulus for further development of mathematics. There are many examples of this effect in the history of physics. Perhaps the most convincing evidence for validity of a new scientific theory is successful prediction of a surprising new phenomenon. Similarly, the most impressive benefits of Geometric Algebra arise from surprising new insights into the structure of physics. The following Sections survey the elements of Geometric Algebra and its application to core components of the physics curriculum. Many details and derivations are omitted, as they are available elsewhere. The emphasis is on highlighting the unique advantages of Geometric Algebra as a unified mathe- matical language for physics.
A recent study on the use of vectors by introductory physics students summa- rized the conclusions in two words: “vector avoidance!”^14 This state of mind tends to propagate through the physics curriculum. In some 25 years of graduate physics teaching, I have noted that perhaps a third of the students seem inca- pable of reasoning with vectors as abstract elements of a linear space. Rather, they insist on conceiving a vector as a list of numbers or coordinates. I have come to regard this concept of vector as a kind of conceptual virus, because it impedes development of a more general and powerful concept of vector. I call it the coordinate virus! 15 Once the coordinate virus has been identified, it becomes evident that the entire physics curriculum, including most of the textbooks, is infected with the virus. From my direct experience, I estimate that two thirds of the graduate students have serious infections, and half of those are so damaged by the virus that they will never recover. What can be done to control this scourge? I suggest that universal inoculation with Geometric Algebra could eventually eliminate the coordinate virus altogether. I maintain that the origin of the problem lies not so much in pedagogy as in the mathematics. The fundamental geometric concept of a vector as a directed magnitude is not adequately represented in standard mathematics. The basic definitions of vector addition and scalar multiplication are essential to the vector concept but not sufficient. To complete the vector concept we need multiplication rules that enable us to compare directions and magnitudes of different vectors.
Fig. 2. Bivectors a ∧ b and b ∧ a represent plane segments of op- posite orientation as specified by a “parallelogram rule” for drawing the segments.
From the geometric interpretations of the inner and outer products, we can infer an interpretation of the geometric product from extreme cases. For or- thogonal vectors, we have from (5)
a · b = 0 ⇐⇒ ab = −ba. (8)
On the other hand, collinear vectors determine a parallelogram with vanishing area (Fig. 2), so from (6) we have
a ∧ b = 0 ⇐⇒ ab = ba. (9)
Thus, the geometric product ab provides a measure of the relative direction of the vectors. Commutativity means that the vectors are collinear. Anticom- mutativity means that they are orthogonal. Multiplication can be reduced to these extreme cases by introducing an orthonormal basis.
For an orthonormal set of vectors {σ 1 , σ 2 , ...}, the multiplicative properties can be summarized by putting (5) in the form
σi · σj = 12 (σiσj + σj σi) = δij (10)
where δij is the usual Kroenecker delta. This relation applies to a Euclidean vector of any dimension, though for the moment we focus on the 2D case. A unit bivector i for the plane containing vectors σ 1 and σ 2 is determined by the product
i = σ 1 σ 2 = σ 1 ∧ σ 2 = −σ 2 σ 1 (11)
The suggestive symbol i has been chosen because by squaring (11) we find that
i^2 = − 1 (12)
Thus, i is a truly geometric
√ −1. We shall see that there are others. From (11) we also find that
σ 2 = σ 1 i = −i σ 1 and σ 1 = iσ 2. (13)
In words, multiplication by i rotates the vectors through a right angle. It follows that i rotates every vector in the plane in the same way. More generally, it follows that every unit bivector i satisfies (12) and determines a unique plane in Euclidean space. Each i has two complementary geometric interpretations: It represents a unique oriented area for the plane, and, as an operator, it represents an oriented right angle rotation in the plane.
Assigning a geometric interpretation to the geometric product is more subtle than interpreting inner and outer products — so subtle, in fact, that the ap- propriate assignment has been generally overlooked to this day. The product of any pair of unit vectors a, b generates a new kind of entity U called a rotor, as expressed by the equation
U = ab. (14)
The relative direction of the two vectors is completely characterized by the directed arc that relates them (Fig. 3), so we can interpret U as representing that arc. The name “rotor” is justified by the fact that U rotates a and b into each other, as shown by multiplying (14) by vectors to get
b = aU and a = U b. (15)
Further insight is obtained by noting that
a · b = cos θ and a ∧ b = i sin θ, (16)
where θ is the angle from a to b. Accordingly, with the angle dependence made explicit, the decomposition (7) enables us to write (14) in the form
Uθ = cos^ θ^ +^ i^ sin^ θ^ =^ ei^ θ^.^ (17)
It follows that multiplication by Uθ , as in (15), will rotate any vector in the i-plane through the angle θ. This tells us that we should interpret Uθ as a directed arc of fixed length that can be rotated at will on the unit circle, just as we interpret a vector a as a directed line segment that can be translated at will without changing its length or direction (Fig. 4).