Introductory physics course for physics majors
(Rex Adelberger)
Pedagogical points of interest
A beginning student might be introduced to the study of physics as having 4 parts
1. Learning something about the nature of physical reality,
2. Learning how to construct abstract models of physical reality (usually in mathematics and involving all sorts of approximations).
3. Working out the mathematical consequences of the models that are proposed to describe the situation.
4. Learning how to evaluate the validity of the models by comparing the results of the models over a wide range of parameters with the way that nature actually behaves.
Traditionally, the student spends much time and stress on part 3. To many students, the world of introductory physics is centered about working out the consequences of simple models: lots of time and energy is spent ‘grinding out’ the algebra, calculus and numerical results. Consequently, we find ourselves limiting the material covered in the class to those models (usually special cases) where the mathematical consequences are elementary computations that the introductory student can master in a finite amount of time.
The value of a symbolic computing package such as MAPLEV (or maple…) is that it provides a means for the student who might not be facile with mathematics the opportunity to work out the mathematical consequences of the models quickly and without significant errors. In turn, this allows the majority of the learning time of the student to be spent in parts 1,2 and 4 - the parts of the introductory physics course that are most interesting and most important.
A final point is that MAPLEV can be used as a wonderful tool for visualizing the dependence of one physical parameter on another. The graphics capabilities of the language can open a new world of understanding to the introductory student without adding significant time load to the course.
It is important that the introductory course be one that is about physics, not a course in learning to use a new computer language. MAPLEV should be introduced as a calculation engine, a tool to use once the mathematical relationships have been established that properly model the system of interest that is being discussed.
The introductory student does not need to know all of the powerful features that are available in MAPLEV. Care should be taken that MAPLEV is used in the introductory class in such a way that the vocabulary needed forms just a small set of commands.
It should be made clear to the student that MAPLEV is not a machine that replaces the brain and thinking, but a tool for doing difficult and tedious calculations.
MAPLEV needs to be used by the instructor of the course on a regular basis, students need to be given problems where MAPLEV use is necessary not just ones where they could just as easily use a pencil and paper. It is worth taking time to point out when one should be using a tool such as MAPLEV and when it is easier and more appropriate to use some other means of calculation.
Giving the students a set of well prepared work sheets and having them read and use them is an interesting use of MAPLEV, but might not be the most effective use of this tool. Students should be preparing their own worksheets as solutions to assigned problems. Students need to see some examples of well constructed worksheets so that they have models for preparing their own homework.
The worksheet is a wonderful form for handing in homework because it allows for easy modification from a rough draft to a final draft. It is very useful to have students hand in a rough draft of their homework that is examined and commented upon by the instructor. A final form of the homework, modified by the comments from the professor, should be handed in. This technique encourages students to properly discuss and explain the results they got for a problem rather than just turning in a set of equations with an answer.
The beginning student needs to learn how to find out techniques for correcting errors in MAPLEV commands and how to use the help files in MAPLEV. A clever way of doing this is for the professor to include mistakes when presenting calculations done by MAPLEV to the class and then explaining how to figure out what is wrong and how to correct what is written. If the student is shown only work sheets that work the first time around and have no mistakes, their first attempts at using MAPLEV can be very frustrating.
The introductory course is the best place to introduce the use of symbolic programs for solving complicated problems. By starting with a somewhat restricted set of commands and conceptually simple problems, the student gets the confidence to use it well in advanced courses.
Our introductory course for physics majors is not a survey course, but a course that covers a few topics in depth. The class meets 5 days a week for a 50 minute class and has a 3 hour lab. It is a 5 credit course. Our students come in with a wide background in both physics and calculus, from none to 2 or more semesters. A schematic listing of topics covered in the two semesters is shown below:
1. stability - radioactive decay
2. structure - photons and the atom - quantized energy levels, etc
3. conservation laws - particle physics… protons to quarks
4. dynamics - momentum and energy …. Newton’s laws
5. quantum mechanics - the model of atoms
We use MAPLEV as a calculation engine in all parts except 2.
The students are given a 1 page sheet as a guide to using MAPLEV. It tells the student how to open a work sheet, how to save a work sheet, and how to hand in a work sheet. (we have students hand in their worksheets electronically… we allow no printed copies). This sheet explains the difference between input, output and text statements, how to insert and execute statements. It also points out a few items which the student should know that are peculiar to MAPLEV.
The first classes are a review of algebra. During this session we introduce the ideas of assigning names to expressions and equations, plotting the expressions, and solving an equation or series of equations. In class we explore how expressions behave by changing parameters and looking at plots of these expressions. We show the students how to plot more than one expression on a graph. At the end of the exercise, the students are given the beginning of a vocabulary list which we encourage them to continue to expand. We give some algebra homework problems which they are to use MAPLEV to solve.
After a discussion of stability and instability, we begin to develop a model for radioactive decay by looking at pennies and dice. The objects are 'tossed' and the number of heads are counted, recorded and removed. This process is continued until all of the pennies are removed. The class discussion centers about how one can determine the rate of change of the number of objects left after a number of tosses. We use spread sheet plots of experimental data to develop some sort of intuition of how this system behaves. The derivative is introduced as an operator that calculates the rate of change (a number of students have not seen the derivative, so we are careful not to be too mathematical in the definition. We intuitively develop on what the derivative of the number of pieces left in the system depend. We introduce the MAPLEV diff() command as a way of describing this derivative operation. Some time is spent pointing out the difference between models and reality.
We ask the question as to how one finds the number of pieces left in the sample as a function of time. We point out that MAPLEV can solve equations which contain derivatives by changing solve() command to dsolve(). They follow the same steps as done solving an algebra problem and exploring the behavior of the of the calculated results.
We try to use this model to understand what is going on with the experimental data collected using dice and pennies. Some topics that arise in the class discussion include things such as when a continuous function can be used to describe data and when the data seems to be quantized and discrete. The MAPLEV solutions to the model seem to only work when the number of pieces is large. When are approximations ‘good enough’?
The first lab we do is a measurement of the count-rate as a function of time for a short lived isomer of barium. (approximately 2.5 minutes). Once the students see the count-rate vs time data, they are quick to model it using the same techniques as we used for the pennies. MAPLEV once again gets a workout.
At this point (almost 2 weeks into the class) our students have used a small but powerful vocabulary set for MAPLEV. They have been spending most of their time learning and discussing important physics ideas and only using the MAPLEV engine to do complicated and tedious calculations for them.
By the end of the first semester, they have been introduced to about 10 MAPLEV commands :solve( ); dsolve( ); dsolve( exp, var, numeric); plot( ); rhs( ); lhs( ); diff( ); int( ); evalf( ); simplify( ); assign( ). They also know how to use MAPLEV features such as: ( ); { }; [ ] ; x=0..10; x=1..infinity ; Pi . In general, their MAPLEV vocabulary sheet (which reminds them how to use the various features) is about three pages long. With this working vocabulary, our students are able to tackle a wide range of problems, from modern physics to coupled oscillating systems.