David Hestenes

Force Concept Inventory

Every student begins physics with a well-established system of commonsense beliefs about how the physical world works derived from years of personal experience. Over the last decade, physics education research has established that these beliefs play a dominant role in introductory physics. Instruction that does not take them into account is almost totally ineffective, at least for the majority of students. Specifically, it has been established that (1) commonsense beliefs about motion and force are incompatible with Newtonian concepts in most respects, (2) conventional physics instruction produces little change in these beliefs, and (3) this result is independent of the instructor and the mode of instruction. The implications could not be more serious. Since the students have evidently not learned the most basic Newtonian concepts, they must have failed to comprehend most of the material in the course. They have been forced to cope with the subject by rote memorization of isolated fragments and by carrying out meaningless tasks. No wonder so many are repelled! The few who are successful have become so by their own devices, the course and the teacher having supplied only the opportunity and perhaps inspiration.

The initial knowledge state of college physics students

An instrument to assess the basic knowledge state of students taking a first course in physics has been designed and validated. Measurements with the instrument show that the student’s initial qualitative, common sense beliefs about motion and causes has a large effect on performance in physics, but conventional instruction induces only a small change in those beliefs.

Common sense concepts about motion

Common sense beliefs of college students about motion and its causes are surveyed and analyzed. A taxonomy of common sense concepts which conflict with Newtonian theory is developed as a guide to instruction.

Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics

1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5. Geometric Algebras of PseudoEuclidean Spaces.- 2 / Differentiation.- 2-1. Differentiation by Vectors.- 2-2. Multivector Derivative, Differential and Adjoints.- 2-3. Factorization and Simplicial Derivatives.- 3 / Linear and Multilinear Functions.- 3-1. Linear Transformations and Outermorphisms.- 3-2. Characteristic Multivectors and the Cayley-Hamilton Theorem.- 3-3. Eigenblades and Invariant Spaces.- 3-4. Symmetric and Skew-symmetric Transformations.- 3-5. Normal and Orthogonal Transformations.- 3-6. Canonical Forms for General Linear Transformations.- 3-7. Metric Tensors and Isometries.- 3-8. Isometries and Spinors of PseudoEuclidean Spaces.- 3-9. Linear Multivector Functions.- 3-10. Tensors.- 4 / Calculus on Vector Manifolds.- 4-1. Vector Manifolds.- 4-2. Projection, Shape and Curl.- 4-3. Intrinsic Derivatives and Lie Brackets.- 4-4. Curl and Pseudoscalar.- 4-5. Transformations of Vector Manifolds.- 4-6. Computation of Induced Transformations.- 4-7. Complex Numbers and Conformal Transformations.- 5 / Differential Geometry of Vector Manifolds.- 5-1. Curl and Curvature.- 5-2. Hypersurfaces in Euclidean Space.- 5-3. Related Geometries.- 5-4. Parallelism and Projectively Related Geometries.- 5-5. Conformally Related Geometries.- 5-6. Induced Geometries.- 6 / The Method of Mobiles.- 6-1. Frames and Coordinates.- 6-2. Mobiles and Curvature 230.- 6-3. Curves and Comoving Frames.- 6-4. The Calculus of Differential Forms.- 7 / Directed Integration Theory.- 7-1. Directed Integrals.- 7-2. Derivatives from Integrals.- 7-3. The Fundamental Theorem of Calculus.- 7-4. Antiderivatives, Analytic Functions and Complex Variables.- 7-5. Changing Integration Variables.- 7-6. Inverse and Implicit Functions.- 7-7. Winding Numbers.- 7-8. The Gauss-Bonnet Theorem.- 8 / Lie Groups and Lie Algebras.- 8-1. General Theory.- 8-2. Computation.- 8-3. Classification.- References.

Toward a modeling theory of physics instruction

An analysis of the conceptual structure of physics identifies essential factual and procedural knowledge which is not explicitly formulated and taught in physics courses. It leads to the conclusion that mathematical modeling of the physical world should be the central theme of physics instruction. There are reasons to believe that traditional methods for teaching physics are inefficient and substantial improvements in instruction can be achieved by a vigorous program of pedagogical research and development.

Modeling games in the Newtonian World

The basic principles of Newtonian mechanics can be interpreted as a system of rules defining a medley of modeling games. The common objective of these games is to develop validated models of physical phenomena. This is the starting point for a promising new approach to physics instruction in which students are taught from the beginning that in science ‘‘modeling is the name of the game.’’ The main idea is to teach a system of explicit modeling principles and techniques, to familiarize the students with a basic set of physical models, and to give them plenty of practice in model building, model validation by experiment, and model deployment to explain, to predict, and to describe physical phenomena. Unfortunately, a complete implementation to this approach will require a major overhaul of standard instructional materials which is yet to be accomplished. This article lays down physical, epistemological, historical, and pedagogical rationale for the approach.

A modeling method for high school physics instruction

The design and development of a new method for high school physics instruction is described. Students are actively engaged in understanding the physical world by constructing and using scientific models to describe, explain, predict, and to control physical phenomena. Course content is organized around a small set of basic models. Instruction is organized into modeling cycles which move students systematically through all phases of model development, evaluation, and application in concrete situations—thus developing skill and insight in the procedural aspects of scientific knowledge. Objective evidence shows that the modeling method can produce much larger gains in student understanding than alternative methods of instruction. This reveals limitations of the popular ‘‘cooperative inquiry’’ and ‘‘learning cycle’’ methods. It is concluded that the effectiveness of physics instruction depends heavily on the pedagogical expertise of the teacher. The problem of cultivating such expertise among high school teachers is discussed at length, with specific recommendations for action within the physics community.

Generalized homogeneous coordinates for computational geometry

The standard algebraic model for Euclidean space E n is an n-dimensional real vector space ℝ n or, equivalently, a set of real coordinates. One trouble with this model is that, algebraically, the origin is a distinguished element, whereas all the points of E n are identical. This deficiency in the vector space model was corrected early in the 19th century by removing the origin from the plane and placing it one dimension higher. Formally, that was done by introducing homogeneous coordinates [110]. The vector space model also lacks adequate representation for Euclidean points or lines at infinity. We solve both problems here with a new model for E n employing the tools of geometric algebra. We call it the homogeneous model of E n .

Projective Geometry with Clifford Algebra

Projective geometry is formulated in the language of geometric algebra, a unified mathematical language based on Clifford algebra. This closes the gap between algebraic and synthetic approaches to projective geometry and facilitates connections with the rest of mathematics.

Observables, operators, and complex numbers in the Dirac theory

The geometrical formulation of the Dirac theory with spacetime algebra is shown to be equivalent to the usual matrix formalism. Imaginary numbers in the Dirac theory are shown to be related to the spin tensor. The relation of observables to operators and the wavefunction is analyzed in detail and compared with some purportedly general principles of quantum mechanics. An exact formulation of Larmor and Thomas precessions in the Dirac theory is given for the first time. Finally, some basic relations among local observables in the nonrelativistic limit are determined.