David Hestenes wrote these comments about basic particle models
in 2011 to high school teachers:
“I have enclosed a slide* about basic models that I use in talks about Modeling Instruction.
Let me add some comments for teachers.
Mechanics can be divided into kinematics (description of motion) and dynamics (causal explanation of motion). Accordingly, I have two sets of names, depending on which aspect is of interest. Newtonian theory says that all causes of motion are forces, so causal models are classified by types of force.
More about this in "Modeling Games in the Newtonian World," where Kepler’s laws are identified as kinematics, which is explained causally by the gravitational force.”**
Notes by Jane Jackson:
David Hestenes asks that teachers introduce your students to his name of each particle model, even though you start out with a different name. You should eventually replace the “balanced force particle model (BFPM)” with “free particle model”. The “unbalanced force particle model (UFPM)” should be replaced with “constant force particle model”. These names are standard terminology, and your students should leave your course using accepted terminology, David Hestenes said.
David Hestenes has written that each basic particle model should be correlated with the appropriate basic mathematical model. His list of mathematical models is in two documents on the ASU modeling website.
1. in "Modeling Instruction for STEM Education Reform", a major proposal by David Hestenes (2009) (at http://modeling.asu.edu near the bottom of the webpage). I quote him on page 6:
Basic Mathematical Models:
1. Constant rate (linear change): graphs and equations for straight lines (proportional reasoning, constant velocity, acceleration, force, momentum, energy, etc.)
2. Constant change in rate (quadratic change) graphs and equations for parabolas (constant acceleration, kinetic and elastic potential energy, etc.)
3. Rate proportional to amount: doubling time, graphs and equations of exponential growth and decay (monetary interest, population growth, radioactive decay, etc.)
4. Change in rate proportional to amount: graphs and equations of trigonometric functions (waves and vibrations, harmonic oscillators, etc.)
5. Sudden change: stepwise graphs and inflection points (Impulsive force, etc.)
These models characterize basic quantitative structures that are ubiquitous not only in physics but throughout the rest of science. Their applications to science and modern life are rich and unlimited. Accordingly, we regard skill in using these models in a variety of situations as an essential component of math and science literacy.
2. in the original PHS 542 syllabus. I quote David Hestenes:
"These are the basic models at the foundations of the Modeling Instruction Program. This is shown explicitly in the syllabus for PHS 542, a course we developed to get physics and math teachers talking together about common essentials. These models are employed over and again in different contexts throughout the Modeling Program. Modeling pedagogy is designed to actively engage students in using the models to characterize physical phenomena quantitatively and evaluate the results."
David Hestenes’ original PHS 542: Integrated Mathematics and Physics syllabus (2001) states:
Abstract mathematical concepts, such as variable, function and rate, are learned and studied within the context of mathematical models with concrete applications to physics and other subjects.
Statistical concepts, such as mean and standard deviation, are learned and applied within the context of matching models to data collected by students using calculators and/or computers.
(1)Constant rate, linear change
Graphs and equations for straight lines
(2)Constant change in rate, quadratic change
Graphs and equations for parabolas
(3)Rate proportional to amount, doubling time
Population growth and radioactive decay
(4)Change in rate proportional to amount
Trigonometric functions and harmonic oscillations
Other models will be studied as extensions and/or elaborations of the basic models.
References: A. A. Bartlett, Physics Teacher 34: 342 (1996), and many articles cited therein.
* The slide that David Hestenes provided (at http://modeling.asu.edu/modeling-HS.html) is slide #22 in his slideshow at http://modeling.asu.edu, (near the bottom of the page) in his 2011 invited talk in Taiwan, called “Remodeling Science Education”. The URL to download his slideshow is
** You can download “Modeling Games in the Newtonian World” at