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  • 1.  Vectors in Calc II

    Posted 01-07-2025 01:14:00

    I've heard from a number of students in past years that when they start Physics I, they are bombarded with vectors and vector operations.  Most (if not all) of these students have never been exposed to vectors.  Now, we teach it in Trig, but most Calculus students test right into Calculus and have only taken precalc at their high school, and for most, that's no exposure to vectors.

    So, I've tried to "combat" this by starting Calculus II with vectors.  So, they're at least getting it mostly at the same time, and the students continually say how seeing it in Calc really helps reinforces the content knowledge.

    Now, I only teach the basic operations, dot products, and cross-products.  No vector-valued functions.  I'll come back to that.  And what I then do is save Polar Calculus to Calc III (that also means cutting/trimming a few things in Calc III).

    What are your thoughts about teaching vector basics at the beginning of Calc II?



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    Michael Caparula
    Professor
    Kankakee CC (retired)
    Kankakee IL
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  • 2.  RE: Vectors in Calc II

    Posted 01-07-2025 10:48:00

    Yes, introducing vectors at the beginning of Calculus II makes a lot of sense. I've incorporated this approach in the past, based on recommendations from physics and engineering professors.  However, I must admit that I eventually stopped doing so. The constraints of a four-semester-hour Calculus II course made it challenging to cover all topics in sufficient depth.  As I reconsider my approach, I wonder if emerging technologies might justify de-emphasizing certain traditional topics, such as techniques of integration.

    What are your thoughts on the future of integral calculus? How do you see it evolving in response to technological advancements?



    ------------------------------
    Robert Cappetta
    Florida SouthWestern State College
    FLRobert Cappetta
    Florida SouthWestern State College
    FL
    ------------------------------



  • 3.  RE: Vectors in Calc II

    Posted 01-09-2025 13:08:00

    The questions I would have are: What are the math prerequisites for the Physics class?

    If Calculus I is the prerequisite, then it feels like it is more on the Physics department to not bombard the students with vectors, and provide the support the students need. Having vectors to start Calculus II feels like a disconnect in the "story" we are trying to tell in Calculus. Especially if students are placing directly into Calculus I and skipping Precalculus/College Algebra.

    As for Bob's question, I feel like the integration techniques have their place, but we don't need every approximation approach. 

    I like the idea of introducing Riemann sums to explain how you could approximate an integral. With the integration techniques we need some for differential equations, plus I wouldn't be surprised if some engineering courses need them.



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    Matthew Lee
    Oakton Community College
    Des Plaines IL
    ------------------------------

    Original Message


  • 4.  RE: Vectors in Calc II

    Posted 01-10-2025 07:34:00
    I can't resist throwing out some thoughts on this thread. Rambling I'm afraid. I have been retired for five or six years and confess to not having taught calculus for probably 30 years, but it doesn't seem to have changed much.

    Obsolescence:

    In grade school (late 50's) I was presented with a pencil and paper algorithm for extracting square roots. I won't say I learned it, as I thought it was a waste of time, and I think I was right. (My point: some things become obsolete.)

    Vectors:

    I agree with the statement "it is more on the Physics department to not bombard the students with vectors" from the point of view that a physics course should own vectors.

    STEM students should have seen vectors before any calculus course. They should be given a good introduction wherever a student is introduced to trigonometry, and vectors are undoubtedly covered in a high school physics course. (But "According to the American Institute of Physics (AIP), the percentage of high school seniors attending schools that offer physics annually dropped to 84% in 2021".)

    Online I can find this recommendation for a first semester course entitled "Classical Mechanics/Math Methods" at the University of Colorado Boulder; the course should cover "Vectors, curvilinear coordinate systems. Quick review of vector addition, dot and cross products. Spherical and cylindrical coordinate systems, simple derivatives".

    Without trying to find more backup, I would argue that vectors should not be completely assumed in a college beginning physics course. The concept of vectors is fundamental to physics (mechanics) and the physics teacher should incorporate an introduction to vectors as needed within the introductory physics course. 

    On integration techniques:

    I learned calculus in the early 70's from a book by "Thomas" which was the standard text then (not gonna research it) and in the insides of the covers were many models of anti-derivatives–I was told that that's how the engineers do anti-differentiation, by seeing if there wasn't a model in the long list.

    At some point in my career I heard about computer algebra systems (CAS), then fairly new. I found this intriguing. In this context I remember hearing about ... let me quote from Wikipedia:

    "A procedure called the Risch algorithm exists that is capable of determining whether the integral of an elementary function* is elementary and returning it if it is." 

    *(a function built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations.)

    Might not that cause one to question how much time to spend on methods of integration for general STEM majors?

    Many years ago, perhaps 25 or so, I heard about a professor at a good mid-west state university that taught calculus exclusively through Mathematica, then fairly new.

    Caveat:

    Of course, two-year colleges must consider the four year institutions to which their students may transfer. We all know this can limit and dictate course content.

    I have not been working in isolation:

    Haven't taught calculus in 30 years, so thought I need to shore up some credibility if I can. 

    I first taught high school senior math and physics in 1972, and have thought a lot about curriculum over the years. In that time I did write a published text on algebra and trigonometry and a published text on trigonometry (long ago) and did two-thirds of a masters in computer science (not finished because I left the country for a year on sabbatical). Long ago I wrote an article for AMATYC on how a CAS might factor, as an example, an 8th degree polynomial over the integers. Have been an active member of the MAA since the 70's, and of AMATYC since the late 70s.

    Apology:

    To quote Blaise Pascal, "I would have written a shorter letter, but I did not have the time."


    Philip Mahler

    Retired Professor of Mathematics and Computer Science




    Original Message


  • 5.  RE: Vectors in Calc II

    Posted 01-10-2025 13:32:00
    I can't resist throwing out some thoughts on this thread. Rambling I'm afraid. I have been retired for five or six years and confess to not having taught calculus for probably 30 years, but it doesn't seem to have changed much.

    Obsolescence:

    In grade school (late 50's) I was presented with a pencil and paper algorithm for extracting square roots. I won't say I learned it, as I thought it was a waste of time, and I think I was right. (My point: some things become obsolete.)

    Vectors:

    I agree with the statement "it is more on the Physics department to not bombard the students with vectors" from the point of view that a physics course should own vectors.

    STEM students should have seen vectors before any calculus course. They should be given a good introduction wherever a student is introduced to trigonometry, and vectors are undoubtedly covered in a high school physics course. (But "According to the American Institute of Physics (AIP), the percentage of high school seniors attending schools that offer physics annually dropped to 84% in 2021".)

    Online I can find this recommendation for a first semester course entitled "Classical Mechanics/Math Methods" at the University of Colorado Boulder; the course should cover "Vectors, curvilinear coordinate systems. Quick review of vector addition, dot and cross products. Spherical and cylindrical coordinate systems, simple derivatives".

    Without trying to find more backup, I would argue that vectors should not be completely assumed in a college beginning physics course. The concept of vectors is fundamental to physics (mechanics) and the physics teacher should incorporate an introduction to vectors as needed within the introductory physics course. 

    On integration techniques:

    I learned calculus in the early 70's from a book by "Thomas" which was the standard text then (not gonna research it) and in the insides of the covers were many models of anti-derivatives–I was told that that's how the engineers do anti-differentiation, by seeing if there wasn't a model in the long list.

    At some point in my career I heard about computer algebra systems (CAS), then fairly new. I found this intriguing. In this context I remember hearing about ... let me quote from Wikipedia:

    "A procedure called the Risch algorithm exists that is capable of determining whether the integral of an elementary function* is elementary and returning it if it is." 

    *(a function built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations.)

    Might not that cause one to question how much time to spend on methods of integration for general STEM majors?

    Many years ago, perhaps 25 or so, I heard about a professor at a good mid-west state university that taught calculus exclusively through Mathematica, then fairly new.

    Caveat:

    Of course, two-year colleges must consider the four year institutions to which their students may transfer. We all know this can limit and dictate course content.

    I have not been working in isolation:

    Haven't taught calculus in 30 years, so thought I need to shore up some credibility if I can. 

    I first taught high school senior math and physics in 1972, and have thought a lot about curriculum over the years. In that time I did write a published text on algebra and trigonometry and a published text on trigonometry (long ago) and did two-thirds of a masters in computer science (not finished because I left the country for a year on sabbatical). Long ago I wrote an article for AMATYC on how a CAS might factor, as an example, an 8th degree polynomial over the integers. Have been an active member of the MAA since the 70's, and of AMATYC since the late 70s.

    Apology:

    To quote Blaise Pascal, "I would have written a shorter letter, but I did not have the time."


    Philip Mahler

    Retired Professor of Mathematics and Computer Science




    Original Message


  • 6.  RE: Vectors in Calc II

    Posted 01-14-2025 10:23:00
    Edited by Kelly Spoon 01-14-2025 10:24:04

    I wish there was an upvote feature. Totally agree with @Matthew Lee's post. When we had to create a corequisite support for Calculus there were two camps, those who felt we needed to review everything for the other STEM disciplines (meaning throwing a ton of trig and precalc topics into the course) and those who felt we just needed to support their calculus journey. I'm firming in the latter camp, believing that teaching students skills to remediate topics they may need to brush up on is more valuable than introducing them to topics that they won't remember when they see them again in those future classes. 



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    Kelly Spoon
    Associate Professor
    San Diego Mesa College
    San Diego CA
    ------------------------------

    Original Message


  • 7.  RE: Vectors in Calc II

    Posted 01-12-2025 12:56:00

    Hi, I teach Physics 1 & 2 periodically, usually the Physics with trig- PHY2053, which is different from Physics with Calc- PHY2049. Even in the Trig course, I've seen only a small discussion of vectors. Some students get it right away, but many of them are shaky about the concepts. I think vectors is a cool topic and students will enjoy the basics (add, subtract), distinguish better between vectors and scalars. But without the prep knowledge if they are thrown into Newton's laws, they'll get confused. I fear professors are also very frustrated with the lack of knowledge of vectors while starting Physics 1.



    ------------------------------
    Manisha Ranade
    Associate Professor
    Santa Fe College
    Gainesville FL
    https://portfolium.com/manisharanade
    ------------------------------



  • 8.  RE: Vectors in Calc II

    Posted 01-13-2025 07:31:00
    1. "What are your thoughts about teaching vector basics at the beginning of Calc II?"

    2. "Even in the Trig course, I've seen only a small discussion of vectors. ... I fear professors are also very frustrated with the lack of knowledge of vectors while starting Physics 1."
    I assume below that Physics I of any flavor is a Fall course, and Calc II is a Spring course. Also, talking about STEM students. 

    Quote 2 above indicates (a) a knowledge of vectors is needed in the first semester of physics, typically Fall I presume, and (2) right now we don't do that well enough.
    Quote 1 talks about doing a better job with vectors in the Spring (Calc II in the Spring, I assume).

    So ...
    Do math educators want to take responsibility for vectors for STEM students, as opposed to physics teachers taking responsibility? I will assume YES, else we don't need this discussion.

    I assume we do take responsibility for trig for STEM students. ... but when do we do that?
    • A separate course in Trig (and perhaps exponential functions as well) or
    • Just a review in Calc II when we start the calculus of transcendental functions
    I would argue that wherever students see trig they should get a good foundation in vectors. If we do that in a presumably Fall course (to be taken while taking Calc I) then we need to bang away at vectors in that course. This may mean removing some other content, presumably around Identities. Perhaps a lighter treatment, and no need to memorize identities.

    One problem I think may exist is that we generally do the transcendental functions in Calc II, so Trig may not even be required for Calc I, and in fact weak students may be taking Trig in the Fall along with Calc I, assuming a Trig course is offered at a given institution.

    I can see that I could write for a while speculating on all the possibilities, including co-requisite trig courses for Physics I. But I just looked and there is the AAPT, American Association of Physics Teachers, which has a "Committee on Physics in Two-Year Colleges". https://www.aapt.org/aboutaapt/organization/tyc.cfm

    Perhaps there should be (and perhaps there already is) some interaction between this AAPT committee and an AMATYC ANet https://amatyc.org/page/AMATYCANets , perhaps the Math Intensive or the Pathways Anets.

    For myself I conclude that trigonometry needs to be a prerequisite or corequisite for Physics I, and this does not mesh well with doing it in Calc 2, though for our math purposes it's fine of course–just in time math is a good idea. If we went to review vectors in Calc II, diminish treatment of methods of integration since a CAS can do that.
    Also in a trig course I would diminish a heavy treatment of identities, including developing them and requiring memorization of them, in favor of a good treatment of vectors. 






    Original Message


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