Calculus and formal reasoning in intro physics

A physicist asked me, “One thing I noticed in most recent introductory physics textbooks is the slow disappearance of calculus (integrals and derivatives). Even calculus-based physics now hardly uses any calculus. What is the reason for that?” Here is what I replied:

Concerning calculus, I would say that I’m not sure the situation has actually changed all that much from when I started teaching calculus-based physics in the late 1960s. Looking through a 1960s edition of Halliday and Resnick, I don’t see a big difference from the textbooks of today.

More generally, there is a tendency for older faculty to deplore what they perceive to be a big decline in the mathematical abilities of their students, but my experience is that the students are adequately capable of algebraic manipulation and even calculus manipulation (e.g. they know the evaluation formulas for many cases of derivatives and integrals). What IS however a serious problem, and is perhaps new, is that many students ascribe no meaning to mathematical manipulations. Here is an example that Ruth Chabay and I have seen in our own teaching:

The problem is to find the final kinetic energy. The student uses the Energy Principle to find that K_f = 50 joules. Done, right? No! Next the student uses the mass to determine what the final speed v_f is. Then the student evaluates the expression \frac12 m v_f^2 (and of course finds 50 joules). Now the student feels that the problem is solved, and the answer is 50 joules.

We have reason to believe that what’s going on here is that kinetic energy has no real meaning, rather kinetic energy is the thing you get when you multiply \frac12 times m times the square of v. Until and unless you’ve carried out that particular algebraic manipulation you haven’t evaluated kinetic energy.

Another example: A student missed one of my classes due to illness and actually went to the trouble of coming to my office to ask about what he’d missed, so he was definitely above average. The subject was Chapter 12 on entropy. I showed him an exercise I’d had the class do. Suppose there is some (imaginary) substance for which S = aE^{0.5}. How does the energy depend on the temperature? I asked him to do this problem while I watched. (The solution is that 1/T = dS/dE = 0.5aE^{-0.5}, so E = 0.25a^2T^2.) The student knew the definition 1/T = dS/dE, but he couldn’t even begin the solution. I backed up and backed up until finally I asked him, “If y = ax^{0.5}, what is dy/dx?” He immediately said that dy/dx = 0.5ax^{-0.5}. So I said, okay, now do the problem. He still couldn’t! His problem was that he knew a canned procedure that if you have an x, and there’s an exponent, you put the exponent in front and reduce the exponent by one, and that thing is called “dy/dx” but has no meaning. There is no way to evaluate dS/dE starting from aE^{0.5}, because there is no x, there is no y, and nowhere in calculus is there a thing called dS/dE.

We are convinced that an alarmingly large fraction of engineering and science students ascribe no meaning to mathematical expressions. For these students, algebra and calculus are all syntax and no semantics.

A related issue is the difficulty many students have with formal reasoning, and here there may well be a new problem. It used to be that an engineering or science student would have done a high school geometry course that emphasized formal proofs, but this seems to be no longer the case. Time and again, during class and also in detailed Physics Education Research (PER) interviews with experimental subjects we see students failing to use formal reasoning in the context of long chains of reasoning. An example: Is the force of the vine on Tarzan at the bottom of the swing bigger than, the same as, or smaller than mg? The student determines \vec{p} just before and just after and correctly determines that d\vec{p}/dt points upward. The student concludes correctly that the net force must point upward. The student determines that the vine pulls upward and the Earth pulls downward. The student then says that the force of the vine is equal to mg! Various studies by Ruth Chabay and her PER grad students have led to the conclusion that the students aren’t using formal reasoning, in which each step follows logically from the previous step. Often the students just seize on some irrelevant factor (in this case, probably the compiled knowledge that “forces cancel”).

This problem with formal reasoning may show up most vividly in the Matter & Interactions curriculum, where we want students to carry out analyses by starting from fundamental principles rather than grabbing some secondary or tertiary formula. We can’t help wondering whether the traditional course has come to be formula-based rather than principle-based because faculty recognized a growing inability of students to carry out long chains of reasoning using formal procedures, so the curriculum slowly came to depend more on having students learn lots of formulas and the ability to see which formula to use.

Coming back to calculus, I assert that our textbook has much more calculus in it than the typical calculus-based intro textbook. This may sound odd, since we have had students complain that there’s little or no calculus in our book (we heard this more often from unusually strong students at Carnegie Mellon than at NCSU). The complaint is based on the fact that we introduce and use real calculus in a fundamental way right from the start, but many students do not see that the sum of a large number of small quantities has anything to do with integrals, nor that the ratio of small quantities has anything to do with derivatives. For formula-based students, \Delta \vec{p} = \vec{F}_{\text{net}}\Delta t has nothing to do with calculus, despite our efforts to help them make a link between their calculus course and the physics course.

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20 Responses to Calculus and formal reasoning in intro physics

  1. Rick Roesler says:

    Your comment on geometry and formal proofs is interesting. When I was in high school and college I did a lot of physics tutoring (I was a physics major). One of the things that I observed early on was a correlation between “i was really good in geometry” and “I’m pretty good at physics” AND “i was terrible at geometry” and “i don’t get physics at all”. I surmised that it might have to do with the ability to visualize. But your ‘formal proofs’ comment strikes a chord.

  2. It would be hard (not impossible, presumably) to tease apart the two issues, because it is certainly the case that visualization is very important. A particularly tough area is E&M, which is fiercely 3D. In our own teaching we use lots of VPython demo programs to help students visualize physics, especially in 3D. At matterandinteractions.org are videos of these lectures, which you might find interesting.

    • John Denker says:

      I agree 100% with the main point, namely that visualization is important, and that students often need help to improve their visualization skills. However, I would not have said that E&M is “fiercely 3D”. As I see it, electromagnetism is quite vehemently 4D. I don’t even refer to it as E&M, as if Electricity and Magnetism were different things; there is really only one thing, namely electromagnetism. Of course this doesn’t change the main point; indeed 4D is even harder to visualize than 3D.

      On the other hand, the cost of going from 3D to 4D is very small compared to the benefits. For one thing, in 4D as in 3D, it is often possible to deal with two dimensions at a time, which makes things very much easier to visualize. Also, computer graphics can help.

      The 4D point of view simplifies all of physics, even the most elementary physics, not just electromagnetism. For example, on page 4 of the book, compare Figure 1.5 to Figure 1.6. The latter looks like straight-line motion on a plot of X versus Y. However, on a plot of X versus T, it would be conspicuously non-straight motion. The spacetime view captures the essential physics. Specifically, it makes manifest the fact that the physics in Figure 1.5 is fundamentally the same as the physics in Figure 1.6.

      Plotting X versus T is not rocket science. Students “should” have been doing this for years, starting in grade school.

      As stated in the preface, the whole book is predicated on the idea that the modern approach to physics is /easier/ and in every way better than the premodern approach. This is particularly true when it comes to the post-1908 idea that X, Y, Z, and T cannot be separated. All physics is spacetime physics.

      For details, see

      http://www.av8n.com/physics/spacetime-welcome.htm#sec-straight-line-motion

      • You’re undoubtedly right, and we can hope that future intro textbooks that are widely used take a thoroughly modern perspective. In our own case (Matter & Interactions) and that of Tom Moore’s Six Ideas that Shaped Physics (and Tom is a general relativist) our goal was to bring intro physics at least up to date with the way most physicists see their discipline today rather than how they saw it in the 1800s. Many (most?) physicists who teach intro physics have not yet fully absorbed the 4D perspective. Even our textbook and Moore’s are considered quite radical and so far are not widely used. Evolutionary change is slow but steady, and I hope that growth in the use of our book and Moore’s might provide a foundation for the next textbooks, which ought to look at what we’ve done as quite out of date.

  3. johanna says:

    I think that this is connected to how calculus might be teached as something finished and true. It might help if students had an understanding of the process of calculus, that it was created together with mechanics and how it has been used even though it was not mathematically accepted.

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  5. Jane Jackson says:

    Regarding formal proofs in high school geometry: David Hestenes has a different vision of an effective geometry course. I quote the prologue to David Hestenes’ Primer on Geometric Algebra for introductory mathematics and physics. This was a resource at his full-day workshop at the AAPT summer 2005 meeting.

    Physics teachers are universally dismayed by the paltry understanding of mathematics that students bring from their mathematics courses. Blame is usually laid on faulty teaching. But I hold that the crux of the problem is deeply embedded in the curriculum. From the perspective of a practicing scientist, the mathematics taught in high school and college is fragmented, out of date and inefficient!

    The central problem is found in high school geometry. Many schools are dropping the course as irrelevant. But that would be a terrible mistake for reasons already clear to Galileo at the dawn of science.
    • Geometry is the starting place for physical science, the foundation for mathematical modeling in physics and engineering and for the science of measurement in the real world.
    • Synthetic methods employed in the standard geometry course are centuries out of date; they are computationally and conceptually inferior to modern methods of analytic geometry, so they are only of marginal interest in real world applications.
    • A reformulation of Euclidean geometry with modern vector methods centered on kinematics of particle and rigid body motions will simplify theorems and proofs, and vastly increase applicability to physics and engineering.

    A basic pedagogical principle: The depth and extent of student learning is critically dependent on the quality of the available mathematical tools.

    Therefore, we can expect a well-designed curriculum based on vector methods to produce significant improvements in the depth, breadth and usefulness of student learning. Further enhancements can be expected from software that facilitates application of vector methods.

    Whether or not the high school geometry course can be reformed in practice, the course content deserves to be reformed to make it more useful in applications.

    References for further study:
    D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics, “ Am. J. Phys. 71: 104-121 (2003).
    D. Hestenes, New Foundations for Classical Physics (Kluwer, Dordrecht, 1986, 2nd ed. 1999)
    Both are at http://geocalc.clas.asu.edu/

  6. John Denker says:

    This blog post raises several interesting and important ideas. By way of follow-up, let me start with the issue of “all syntax and no semantics”. Another way of saying the same thing is confusion between the symbol and the thing symbolized. I consider the student with the calculus confusion to be mostly just a symptom of this larger issue.

    I would say to this student — and to the gazillions of other students with the same problem — that this is not a new issue. It is pretty well understood. Take a look at these dolls:

    Each button is a symbol. It symbolizes an eye. You are supposed to look past the symbol to understand what it represents. This was pretty well figured out 2000 years ago. Let me tell you the story of Plato’s Cave.

    http://faculty.washington.edu/smcohen/320/cave.htm

    There is this abstract, ideal chair that you cannot see. You can however see shadows, i.e. projections of the chair. From this, if you are clever, you can figure out what the chair must be like. In the present case, there is an abstract, ideal thing called a derivative. We don’t get to see it, but we do get to see things like dy/dx. That’s not really the derivative, but merely a shadow thereof. When you tell me that you recognize dy/dx as a derivative, but do not recognize dE/dS as a derivative, it means you have not been watching the shadows closely enough, because on Tuesdays the derivative projects out as dy/dx, and on Wednesdays it projects out as dE/dS. I am not however saying that you should focus more attention on the shadows. Instead, go for the big picture! You need to wrap your head around the abstract, ideal, Platonic derivative. When you understand the derivative, you know that it projects out as all sorts of things of the form d(…)/d(…) with some variable in the numerator and some other variable in the denominator. The idea of derivative transcends the choice of variables.

    The nice thing about math and physics is that we are much better off than those old Greek guys. We are better off because our cave has multiple walls, allowing us to look at multiple projections of the same ideal thing. For example: On one wall the derivative projects out as symbols of the form d(…)/d(…). Meanwhile, on another wall, it projects out as a diagram with a tangent vector that indicates the instantaneous slope. On yet another wall, dE projects out as the gradient vector aka the exterior derivative, which we can diagram as a set of contours of constant E, such that dE = T dS can be considered a statement about proportionality between two exterior derivatives, in some subspace. Your job is to look past all these representations to see the referent, to see the thing being represented.

    Applying the same logic to our own discussion, calculus is not an important part of the story, but is a mere symptom or shadow of a larger idea. That is to say, the same story could be told using any other mathematical or physical abstraction, such as “vector” or “energy” or whatever. The key idea is to understand the relationship between the symbol and the thing being symbolized.

    At this point the students may be wondering whether they signed up for physics or metaphysics, but I make no apologies. It’s important to handle this issue correctly.

    BTW, here is yet another way to get across the point that students (and everybody else) must sometimes use their powers of abstraction, induction, and generalization to figure things out. Here is a way of explaining what a blue triangle is, without actually drawing the blue triangle:

    http://www.av8n.com/physics/causation.htm#fig-blue-triangle

  7. Krishnadas K P says:

    This is to Bruce Sherwood sir..
    I am a student in India and am really impressed with your ideas.Here every one thinks problem solving is all about picking the right formula and doing the math.But I really hate that.I want to analyse the problem and solve it from the fundamental principles.But I don’t know how to!Can you please show me an example from mechanics?How should I start from fundamentals?Please tell me.

    • An example is escape speed. Textbooks often give a formula for this quantity, and problems that ask students to calculate it have a problem with essentially zero steps in reasoning; just plug the numbers into the calculator. You should practice DERIVING such formulas, closed book, in this case starting from the Energy Principle and Newton’s gravitational force law. If you really understand the physics, this is a very simple derivation, so it should cost very little time. If you find that you cannot do the derivation closed book, that is a strong signal that you need to study the material more carefully and try again.

      Now suppose the question is not about escape speed but rather what launch speed is needed to get very far away with a final speed of 7500 m/s. The dead rote formula for escape speed is of no help in addressing this question. But if you start again from the Energy Principle and Newton’s gravitational force law you quickly arrive at a result that can be used to find the launch speed.

      At matterandinteractions.org you will find videos of lectures in a course that uses the Matter & Interactions textbook. These will give you lots of examples of this approach. There is a new Massive Open On-line Course (MOOC) that is largely based on the Matter & Interactions textbook. It has just started. See http://www.coursera.org/course/phys1.

  8. Krishnadas K P says:

    Thank you very much sir

  9. Krishnadas K P says:

    One more doubt is there sir…I tried to follow your method.But still what I am doing is ‘Starting from the equation of Momentum Principle etc..When I sit to solve a problem,How do I apply momentum principle etc. to it..Can we really solve all physics problems using these few principles?How should I study then.I have M&I text book with me..Please help..How to connect all principles to these fundamentals..I want to study your book thoroughly and be able to solve almost all physics problems..Please help..
    With respect
    Krishnadas K P

  10. Sorry, but I can’t say anything more in prose than what we already put into our textbook. Study carefully the worked out examples for how to start from a fundamental principle such as the Momentum Principle. However, you might find useful the videos of lectures based on our textbook, which are available at the bottom of this web page: matterandinteractions.org. The lectures on mechanics are given by Ruth Chabay, one of the textbook authors, and the lectures on electricity and magnetism are given by Matthew Kohlmyer, a colleague of ours.

  11. Krishnadas K P says:

    Oh okay…

  12. Jim Smith says:

    Pardon me for posting in a thread that’s over a year old, but I found your subject of great interest. I wrote a document to help address the lack of knowledge of formal proof techniques that you describe. I plan to translate it into Spanish, so I would appreciate any comments or criticisms that you might offer. The document can be downloaded at https://www.dropbox.com/s/xijrjbm86k91r1x/Proof%20techniques%20Props%20of%20equilat%20triangles.pdf?dl=0. I also have a document in Spanish on formal proofs of formulas for derivatives: https://www.dropbox.com/s/x7c1f9s8mn9qlvj/Dudas%20acerca%20de%20derivadas.pdf?dl=0 .

    Regards,
    Jim

  13. I’m neither a mathematician nor a logician, so I’m not competent to comment in detail on your presentations. However, I did notice the following points.

    “For our purposes, it says that if every object of a certain type has a certain property, then each object of that type has that property.” The wording makes this sound like a tautology. The words “every” and “each” sound to me like synonyms, and “certain” and “that” sound alike in this context, so I read the statement like this: “For our purposes, it says that if every object of a certain type has a certain property, then every object of that certain type has that certain property.” Is there other wording that avoids this flavor of tautology? In computer science jargon one might say that if a class has a certain property, every instance of that class has that property. Your Spanish version sounds to my ears a little less tautological: “If ALL objects (in the plural, instead of EVERY object, in the singular) of a certain type have certain characteristics, then each of these objects has this characteristic.” In fact, this formulation sounds a bit like the computer science class/instance statement.

    The marginal note beside “What do we want?” is broken; the left portion of the gray area is blank.

    In your Spanish article you claim that a cable hangs in the form of a parabola, but it’s a catenary, albeit not very different from a parabola: https://en.wikipedia.org/wiki/Catenary

    I would be pleased to have students in a physics class who had had some experience with the kind of formal proof you present. However, the way the issues play out in the physics course have less to do with invoking formal justification for a step than with taking the result of the previous step seriously and constraining the result of the next step to be consistent rather than an unrelated leap into the unknown.

  14. Leí con interés su ensayo “La Rebelión se Los Matemáticos”. Naturalmente no puedo oponerme a ninguno de los puntos en el ensayo. Era un placer ver referencias a libros de Trillas, porque Trillas está casi listo para publicar en versión española el primer volumen (mecánica) de nuestro libro para el curso de física introductoria, “Materia e Interacciones”, traducido por unos físicos voluntarios en Chile, Colombia, Puerto Rico, y México. El coordenador del trabajo es nuestro colega Genaro Zavala, jefe del departamento de física del Tec de Monterrey.

    • Jim Smith says:

      ¡Voy a buscarlo! También, veo que Ud. promueve el Álgebra Geométrica. A propósito, estoy solicitando comentarios sobre este documento:
      “Resoluciones de ‘problemas de construcción’ geométricas por medio de la geometría clásica, y el álgebra geométrica”

      https://www.dropbox.com/s/kbxw13lw4bpfcfu/AG%20y%20Geom%20Cl%C3%A1sica.pdf?dl=0

      Éste es un extracto del correo electrónico que le envie al dueño de un sitio web a la que pienso donarlo:

      “I have studied GA on my own for several years, as time allowed, in order to be able to promote and teach it where I live in Mexico. Toward those ends, I began to write a document (attached) that showed how to solve some classic geometric “construction” problems with GA, and — as a contrast — with ruler and compass as well.
      “You’re welcome to use the document as-is; I’ll be happy to supply the original text and a translation, plus the original figures. In addition, I think that the document could be used with benefit as a “good bad example”. By this I mean that I’d welcome having you make available a document in which experts have annotated my work so that students can see what I overlooked. In particular, I suspect that I did not make effective use of identities, or of GA’s techniques for translating and rotating rigid bodies.”

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