Conceptual Understanding in Introductory Physics V

This series continues with yet another question from introductory special relativity. I have seen this question asked many times in various places and unfortunately, have rarely seen the correct answer given.

After studying special relativity, students sometimes ask, “What really happens to a moving rod to make it contract?” and “What really happens to a moving clock to make it run slow?” Comment on whether or not these are legitimate questions and on how to respond to them. 

Special relativity has a reputation for introducing non-intuitive findings into physics and these are somehow treated as obstacles that must be overcome. I take the point of view that instead, they are opportunities to refine and retune our physical intuition. It’s important, I think, for introductory students to understand that our intuition isn’t necessarily innate, and that it can change to reflect our deeper understanding of Nature when evidence suggests something new is afoot.

The next post in this series will begin a string questions addressing vector analysis.

Okay, see what you can do with this question!


Building Up to Simultaneity (Activity)

Here’s a classroom activity intended to demonstrate the issue of simultaneity in measuring a stick’s length. Students need a calibrated metre stick (I’m trying to get into the habit of spelling it that way), another stick approximately 1/3 m long although the precise length is unimportant, two coins of the same denomination or two small pea-size balls of sticky clay or something similar, and a smooth tabletop. All references to “the stick” are to the second, smaller stick.

First, you students must establish an operational definition of what it means to measure a stick’s length.

1) Place the metre stick along the table’s edge, place the other stick on the table anywhere along, and parallel to, the metre stick and then write down an operational definition for “measuring the stick’s length.” Use this operational definition to measure the stick’s length and record it with an appropriate unit. Write your operational definition on a whiteboard, and label is as Definition I.

Answers varied more widely than I ever expected in today’s class, but eventually everyone should settle on something like “record the numbers from the metre stick marking the ends of the other stick and simply subtract these two numbers so as to get a positive number and this number is the stick’s length.” There are other ways of saying it of course, but all should be equivalent to this simple articulation.

Now, there is a hidden assumption in the above operational definition that we now need students to be lead to see.

2) Hold the stick with both hands such that the fingertips hold one coin (or clay ball) under each tip. Dropping a coin (or clay ball) onto the metre stick constitutes “taking a reading” or “recording the number” and of course we’re ignoring the duration of the actual drop. Move the stick slowly from left to right, making sure both ends stay within the ends of the metre stick. At a given moment, drop the left coin (or clay ball). Then after one or two seconds, drop the right coin (or clay ball). Using the operational definition articulated above, measure the stick’s length and record it with an appropriate unit.

If you ask students how the two measurements compare, they should find that the second one is greater than the first one. This may not seem significant to them, but ask them to consider which one is “real.” Most students at this point will say that the first measurement is the “real” one because obviously the stick couldn’t have simply lengthened because it’s the same length it always was.

3) Repeat step 2, but this time drop the right coin (or clay ball) first. Then after one or two seconds, drop the left coin (or clay ball). Using the operational definition articulated above, measure the stick’s length and record it with an appropriate unit.

If you ask students how this third measurements compares to the original one, they should find that it’s less. Again, students will say that this isn’t the stick’s “real” length because the stick obviously hasn’t shrunk. In a sense, they’re more correct than they realize. Students have stumbled onto the fact that there is something significant, as opposed to “real,” about the measurement from step 1.

Note the wording of the next step.

4) Repeat step 3, but this time drop the left and right coins (or clay balls) in whatever way is necessary to replicate the measurement from step 1. The stick must be moving!

Eventually, students will realize they must drop the coins (or clay balls) simultaneously or at least as simultaneously as possible given the situation. This is the assumption missing from the initial operational definition, which they now must refine.

5) Revise your operational definition in step 1 to explicitly include this new finding. Record your revised operational definition on a whiteboard, label it as Definition II.

Students should now see that simultaneity play a role in measuring what we have called the stick’s “real” length. Now, they must also be explicitly lead, by facilitating a class discussion, that all of these measurements are as “real” as they can be, and are as “real” as the walls of their classroom.

Now for a very important question.

6) Which definition will give a result that could give different results in different situations, depending on how fast the stick is moving and how great the duration between dropping the coins (or clay balls)?

Students should, perhaps after discussion, unanimously agree that it’s Definition I.

7) Which definition will give a result that will always be the same regardless of how great the duration between dropping the coins (or clay balls)?

Students should, perhaps after discussion, unanimously agree that it’s Definition II. Interestingly, one of my students tried hard several times to home in on this very question, but he tried to articulate it as an issue of whether or not there was something material between the two points marked on the metre stick. In other words, he was asking of “length” depended on something material, like a wooden stick, existing in the space between the two endpoints. His reasoning seemed to be that the presence of a material stick somehow made the measurement more “real” than otherwise. Finally, he articulated his question as in step 7.

Now another important connection.

8) What does step 7 imply about the stick’s motion?

Students should realize that step 7 is equivalent to saying the stick is stationary relative to the observer. Don’t go any further until this sinks in, and it may indeed take a while.

9) Now that students have an operational definition of “measuring a stick’s length” that always gives the same result, we can, in the spirit of Arons, give this idea a name: proper length. Because all observers will agree on the stick’s proper length, we call it an invariant.

My sense is that what students have been conditioned to call “the length” of a stick is what we call the proper length. They all seemed to collectively say “AHA!” at this notion.

There’s an even more important lesson here, and that is once we have a mutually agreed to operational definition of something (in this case, measuring a stick’s length), we must agree to use that definition even if it gives results that don’t match our intuition. I think this is the deepest lesson in this activity. If our definition is good, then we can’t abandon it suddenly when we’re faced with apparent inconsistencies. Instead, we may need to retune our intuition and accept the inconsistencies. In a way, this is what special relativity is all about. Our classical conceptions of space and time were so deeply ingrained that abandoning, or even modifying them, seemed out of the question.

This activity could be far less structured. You could define “measuring a stick’s length” as “subtracting the two numbers onto which the coins (or clay balls) fall” and then ask students to perform measurements that give results greater than and less than what they get if the stick is stationary. Then ask them to discuss the implications of these results.

Okay, now for the big finish.

10) What really happened to the stick as its moving length was measured?



Conceptual Understanding in Introductory Physics IV

I have tens of these questions ready to use here so this series will likely go on for a long time. Part of my strategy for posting each one separately is to get me into the habit of writing regular posts. While these first few are, or should be, relatively simple I promise the difficulty will increase. I have a full line of questions on vector algebra that I guarantee, or at least hope, will not be so simple. It’s not that they’re inherently difficult, but rather that we omit these issues from introductory courses where they should be present and part of a solid conceptual foundation.

This series continues with another question that ties together reference frames and special relativity. It’s a two part question, and the parts may seem almost contradictory given what we typically tell students, or better what they read for themselves, about special relativity.

Under what conditions could a person be observed moving at 0.99c? On the other hand, in what sense is it correct to say that a person, as well as any other material object, always moves at precisely c? Carefully distinguish between these two situations.


My Attendance Woes and How I Brought Them on Myself

I’m writing this to explain the details of some recent tweets about my attendance woes not to complain, but to explain what those of us in the North Carolina Community College System are required to do and the problems that result if you try to innovate. It’s not meant as a rant and I don’t intent to frame it as such. It’s a rather long story, but the background is important.

The story begins in August, 1992 when I was hired. At that time, all NCCCS campuses were on quarter systems (I seem to remember a few on trimesters). Every campus had its own course catalog. Every campus has its own course numbering system. Every campus was its own academic entity. This resulted in a great many campuses becoming hubs for certain curriculum areas. My own institution, for example, was known the world over for its furniture technology, furniture production, and furniture production management programs with visitors from as far away as Denmark. Not every campus had a “college transfer” curriculum, but mine did as it was one of the larger medium sized schools (enrollment of around 3000-4000). The problem with this system was that students could not guarantee that a given course at one campus would transfer to another campus or, more importantly, to four year schools in our state or anywhere else. In 1997, it was announced that the NCCCS would begin using a semester system as of, as I recall, 1999. NCCCS and UNC (when I refer to UNC, I mean the UNC System of universities, not any one particular campus) would hash out a Comprehensive Articulation Agreement specifying in great detail which courses would and would not transfer to UNC campuses. Private schools could decide for themselves which courses they would accept, but UNC campuses would be required by state law to accept all courses included in the CAA. A companion Common Course Library (CCL) was also developed for NCCCS campuses containing every course offered by every community college in the state. Campuses could select course offerings only from the CCL. This meant that campuses had to negotiate among themselves to solve problems of curriculum duplication (e.g. no two schools within X miles of each other could offer the same specialty programs like culinary certifications or law enforcement training, etc.). This also meant that community colleges had to negotiate with UNC campuses over which courses would be accepted by all (ALL) UNC campuses for either general degree graduation requirements or for elective credits. This new system also established a host of pre-majors (e.g. pre-engineering, pre-chemistry, pre-mathematics, pre-education, etc.) for students to take and easily transfer into those majors at the UNC campus of their choice (or to any other non-UNC college, public or private, that voluntarily chose to honor the CAA). UNC campuses had no choice and were mandated by state law to honor the CAA. That’s an important detail to remember. Of course along with all this came a transition to semester hours from quarter hours, but we weathered that change fairly well. One of the earliest problems was that the UNC campuses didn’t want to accept community college courses in the sciences and a few other disciplines because they perceived community college faculty as unqualified, which by their definition meant non-PhD. Some UNC people proclaimed that community college faculty were only required to hold Bachelor degrees and were thus not qualified to teach college or university courses. Well, that’s not the case for college transfer courses; the Southern Association of College and Schools, our regional accreditation agency, recommends (note that accreditation agencies cannot mandate this) the exact same criteria for all college level instructors in our region. The pre-education major wasn’t implemented for over a year after the transition because two prominent Schools of Education (Chapel Hill and East Carolina as I recall) couldn’t agree whose program was the best to use as a model for the poor little community colleges (…that sounded ranty didn’t it…).  The net result was that in the sciences, the university system basically dictated our course titles, ostensible content coverage, credit hours, lab hours, lecture hours, and pretty much everything else to us to compensate for having this legislatively shoved down their throats (…their sentiment, not mine…). So after the transition to semesters and the CCL, there were several tiers of college transfer physics (I won’t discuss the astronomy situation here…it was and still is a mess.). We were granted a conceptual physics course PHY110 and an accompanying lab PHY110A. We were granted a two semester introductory algebra-based sequence PHY151/152 and accompanying lab, but the lab wasn’t separated out as a separate course as it was for PHY110. To this day, I have never been told why. We were also granted a two semester introductory calculus-based sequence PHY251/252, again with accompanying, but not separate, lab. Their Highnesses (…there goes the ranty thing again…) also allowed us to offer a modern physics course PHY253 but most campuses refused to accept it for students majoring in physics despite that it was far more rigorous than any of their own similarly named courses; I taught it every time it was offered, usually during summers. To further get their point across, the UNC system also dictated that lecture sections would all carry three semester hours of credit and labs would carry. PHY110A got two contact hours per week for one semester hour of credit. Similarly, the lab component of PHY151 and PHY152 was dictated to be two contact hours per week for one semester hour of credit. Collectively, PHY110/110A, PHY151, and PHY152 each carried four semester hours of credit. However, PHY251 and PHY252 were dictated to have lab components of three contact hours per week, each for one semester hour of credit. So every course/lab combo was dictated to have a total of four semester hours of credit. Great. Fantastic. No problem, right? Wrong. Read on.

The problem came when we began scheduling our courses. NCCCS mandated that courses must be scheduled chunks according to how the contact hours were divided up between lecture and lab, even if the course had no separate lab listed in the CCL. So for PHY110 and PHY110A the former must be scheduled for a total of three lecture hours and two lab hours per week with no deviations permitted. This means the course could easily be scheduled traditionally as MWF lecture and either a Tu or Th lab. As I recall, we also did things like MWF lecture and M afternoon lab or W afternoon lab. Also permissible was MTuW lecture with lab on Th. The point is that the three lecture contact hours were considered one chunk of time as were the two lab contact hours. The physics courses that were eventually assigned primarily to me, PHY251 and PHY252, could be schedule as MWF lecture with either Tu or Th lab, which was how we did it. However, in fall 1999, I switched to Matter & Interactions and I quickly discovered that the traditional scheduling approach wasn’t working. Students’ contact time in class was too chopped up and I wanted a more evenly distributed numbers of hours per day. I needed to blur the lines between formal lecture and formal lab and begin the transition to a more studio-type environment. My then chair was fine with this, but warned me that the scheduling could get complicated because of the imposed restrictions. We explored things like two contact hours each on MWF, but were required to schedule three of those hours as lab hours and three as lecture hours even though that distinction no longer formally existed. This meant that for each day, I would have to mark each attendance roster twice each day, once for the lecture designated hours and once for the lab designated hours. The split with lecture on MWF and lab on either Tu or Th was easier because each day required only one marking of attendance. Because the labs for PHY251 and PHY252 were built in, my attendance rosters would have two, yes two, entries for each date, one for lecture and one for lab. Pause here to think about that for a minute. Do a think/pair/share if you wish.

Now, I must digress to explain something else. You see, the NCCCS funds its campuses almost entirely based on attendance. Attendance determines FTEs and FTEs determine hiring capability and is the first thing the NC General Assembly (NCGA) in Raleigh looks at every summer in pretending to formulate a budget for the coming fiscal year (…which always rolls over before the budget is ready…that sounded ranty too didn’t it…) . NCCCS mandated that attendance must be taken at every class meeting. Our attendance sheets were subject to onsite auditing by auditors sent at random from Raleigh. In my tewnty-one (just began number twenty-two) years, I have only seen one of these auditors ones. They’re rather like Bigfoot in that I’ve heard of sightings, and even though I think I saw one I can’t be sure. There are many apocryphal stories of community colleges registering fake students for fake courses, but honestly, I think these stories are propagated to instill fear. None of my colleagues at other community colleges knows anything about this every happening. It’s a lot like voter fraud (..that sounded ranty too…). Anyway, there must be three designated lecture hours per week and three designated lab hours per week, even if a given calendar day contains some of each. Make sense? Oh, and yes, all attendance tracking was done with pens on official computer printouts. Rosters were turned in on each semester’s census date (the date on which the State counted heads for funding purposes) and again at each semester’s end.

I continue this not-so-brief digression to state categorically that I don’t mind tracking student attendance. I think it’s a good thing. The vase majority of students we get do not yet understand the importance of class attendance. One of the reasons is that they’ve not had any very good reasons to appreciate the importance of going to class. Honestly, who wants to sit through boring lectures and obtuse and disconnected labs? In a way, I can’t blame them. Nevertheless, we’re funded based on attendance and we must document our numbers. I’m totally, completely, fantastically in favor of this. Honest. I make coming to class worth my students’ time. We now rejoin our previous non-rant, already in progress.

Now, innovative teaching almost certainly requires innovative scheduling Several years ago, I decided a good schedule for PHY251 or PHY252 was to meet for one contact hour on each of MW and two contact hours each on TuTh. This obviously requires TuTh to have both lecture and lab designated hours. Here’s how it breaks down into what the State calls meeting patterns (there’s a new term…): MW are considered two hours of lecture and are thus treated as one meeting pattern, Tu 8:00-8:50 is treated as one meeting pattern consisting of one hour of lecture, Th 8:00-8:50 is treated as one meeting pattern consisting of one hour of lab, and TuTh 9:00-9:50 is treated as one meeting pattern consisting of two hours of lab. This gives a total of three lecture hours and three lab hours, six contact hours, per week as prescribed. Okay? Well, yeah in the days of paper attendance rosters because each day I would only mark attendance once for all meeting patterns that particular day.

Then, beginning this past summer (2014), NCCCS came into the 20th century with online attendance tracking, finally! I thought this would be made very simple. Nope, I was wrong. It turns out that each meeting pattern (remember that term from above?) has its own workflow and must be marked online separately from all other attendance patterns. We use E to indicate a student’s entry into a course, ostensibly on the first class day. However, my innovative scheduling means that this week (our first week of classes), I had to mark students’ entries for the MW meeting pattern, entries for the Tu 8:00-8:50 meeting pattern, entries for the Th 8:00-8:50 meeting pattern, entries for the TuTh 9:00-9:50 meeting patterns. That’s a total of four separate attendance workflows for one course! The web-based software is smart enough to know that you can’t enter attendance for the 9:00-9:50 hour until after 9:00, so on TuTh I must enter attendance twice on each of those days. Are you beginning to get a sense for the utter nonsense community college faculty are subjected to if we want to make class schedules another other than the traditional (and largely ineffective) way? It’s very frustrating. Unfortunately, my former and current chairs react by telling me it’s my fault for insisting on non-traditional scheduling and throw it back in my lap (…not ranting…only relating…), except when we get together to do the schedules basically a year in advance. Then we’re urged to continue the innovating. Sheesh!

I have a similar situation with my astronomy courses, AST151/151A and AST152/152A. Note that the UNC system dictated that astronomy labs be separated out because THEY, the UNC campuses, don’t always require a lab for their astronomy courses but the little community colleges must have a required lab (…that sounded a bit ranty too…).

So, I want to look at a way of scheduling my PHY251 and PHY252 courses that hopefully minimized the number of attendance patterns I have to deal with. I’m strongly considering doing it as two hours each on MWF, with the first hour being designated as lecture and the second hour being designated as lab. This will give two meeting patterns, MWF at some hour and MWF at an adjacent hour, and it will also give larger chunks of time at each class meeting, which I think is a good fit for classes using Matter & Interactions.

Among my scheduling constraints are the calculus sequence (three courses) and, in the spring, differential equations. The same student take these courses and my physics courses and we are always mindful of other department’s schedules. We want our students to be as close as possible to cohorts. Another constraint is that I have a MW evening astronomy class in the fall that meets until 8:20 pm. 8:00 morning classes are tough after that, but that’s the way it’s worked out for years. I could just make MW thirteen hours days, which they usually are anyway so why stop now, right?

This semester, I have a total of twelve (12) separate attendance meeting patterns among three sections of AST151/151A and one section of PHY251. Ridiculous maybe? Yes, definitely.

Oh, and before I leave, I’ll also add that our faculty are also responsible for first line prevention of financial fraud. Yep. See, some students game the system by taking classes, waiting till after the census date, then stop attending and pocketing their financial aid money. The school audited at the federal level every year and is penalized financially (four and five figures) for each such incident that occurs. Ia has fallen on faculty to be the first to detect this by noting suspicious strings of absences and promptly withdrawing those students from courses. If we miss one, we are singled out by administration for costing the institution money and are calmly warned that if we lose our privilege of offering federal financial aid, we must close our doors and everyone will be unemployed. Exaggeration? Probably so, but that’s what we’re told. I’m also told that somewhere on campus, there’s a list of such incidents for each faculty member. I’m not convinced it exists, but gosh who knows? I don’t mind tracking attendance, but I’m not at all comfortable being responsible for financial aid infractions. It’s not what I’m trained to do and certainly not what I was hired to do. Then again, “…other duties as assigned…” is right there in our faces. Honestly, I don’t think about it much because I warn my students that I enforce our attendance policy mercilessly. BTW, the policy is that if students miss 10% of the course’s contact hours, they’re withdrawn…no exceptions…except when attendance is down, resulting in slashed budgets, causing us to “have compassion” for students’ situations. Well, I’ve always done that. As long as students stay in touch with me and let me know what’s going on, I don’t drop them. See, I really am a sweetie. I feel better now.

So, any other community college brethren out there have to deal with all this? Four year colleagues? I think I already know the answers, but I’d kinda like to be surprised.


Modeling Reading for Depth and Understanding

Critical thinking is something we all say we want to impart to our students but something for which we frequently see very little explicit classroom modeling. This is one of the reasons the LCTTA (Learning Critical Thinking Through Astronomy) Project was started. This post will provide a very explicit way to introduce one aspect of critical thinking, namely reading for depth, into an introductory calculus-based physics course. Obviously the course doesn’t have to be calculus-based, but that the context in which I currently work.

According to Arnold Arons, we must explicitly model what we want students to learn. If we want students to read our textbooks in anything other than a superficial way, we must show them precisely how to do that. We most often assume that students, especially college students, enter our classrooms already knowing to do this. My experience has been that the students I serve do not understand how to do this, or at least if they do they don’t do it for some reason. I feel it is the former rather than the latter. So, here’s what I do.

I use the critical framework developed by Linda Elder and Richard Paul of the Foundation for Critical Thinking. This framework is based on the Elements of Thought, eight components into which everything from specific concepts to very broad concepts can be mapped. This mapping process operationally defines (again, in the spirit of Arnold Arons and Percy Bridgman) what we mean by reading for depth or reading for understanding, the assumption being that if one can perform this mapping then one understand the thing being mapped. I provide all of my students with a copy of this booklet explaining both the Elements of Thought and the accompanying Intellectual Standards. Incidentally, the latter are a good place for formulating standards for standards-based grading.

I also provide them with a copy of this chapter of a 1965 introductory calculus-based physics textbook by Arnold Arons, which, in my opinion, is a model of clarity. Although this is chapter 36 of the book, I have used it as an introduction to special relativity for several years. Arons hits the history that led Einstein to articulate what we usually think of as the embodiment of special relativity with an emphasis on the Michelson-Morley experiment. Beyond that, Arons takes a deliberate and very measured approach to build up to the concepts of time dilation, length contraction, and Lorentz transformation. The chapter goes beyond that but that’s as far as we go.

So yesterday in class, students took a few minute to read section 36.1 and I asked them in their groups (four groups of three) to address the element of “question at hand” by making a list of every question that came to mind as they read through the section. The results were, expectedly, quite varied and this is what I hoped for. This gave us a reason to discuss what constitutes a “good” question in science. One student had “Does time exist?” on his list along with other rather philosophical questions. I then led them to, hopefully, see that in science we need questions which are more easily approached, especially by students. This led them to refine their list by eliminating questions that were mostly philosophical, leaving them with more concrete questions. So I assigned section 36.2 for them to read outside of class and this morning, I asked them to once again put their questions on whiteboards, do a mock poster session in which they look at all the whiteboards, question the authors (questioning about questions!), offer constructive criticisms, and refine their lists. They eventually saw much commonality in their questions. Then, I asked them to address the element of “purpose” by articulating the purpose of section 36.2. There was wide variation, but I asked them to look for any commonality, and they found it. They all realized that they has all mentioned “simultaneity” in their articulation. I asked them to condense their articulations down to one sentence, and then finally to one word. That word just happened to be “simultaneity.” So then I asked them what the author’s purpose for this section is, and they all clearly agreed that it was to introduce the notion of simultaneity.

What I didn’t expect, and had not noticed before in similar activities, was that having them write broadly at first, then narrow down to one sentence, and finally narrow down to one word allowed them to quite literally see the author’s intended purpose appear right off the printed page into their minds. I always told students that they should think this way, but the explicit action of writing it all down proved beneficial. Even more amazing was that this happened despite two or three of the students admitting they didn’t know what “simultaneity” even means. Then, and only then, did the focus turned to actually defining that term after they realized that it was THE most important word in this section. Of course, they knew more about what simultaneity is than they originally thought. Once they saw that, as one student read a dictionary definition out loud, the entire section made sense to them by their own admission. This more than backed up my statement to them at the beginning of the class that Arons’ writing is almost always exceptionally clear. Oh, and all of this happened in a standards fifty minute class period.

This is precisely how reading for understanding is supposed to work. This was for only one of the eight elements of thought, and only for one section of one chapter. Tomorrow, we will address another one and will also introduce the elements’ accompanying intellectual standards, specifically the standards of “clarity.” My experience has been that students complain most about textbooks’ clarity. They tell us, “I read the chapter but didn’t understand what the author was saying.” The Elder/Paul framework directly addresses this issue.

So, the operational definition (again channeling Arons) of “reading for understanding” as my students now see it is to “map the passage (paragraph, section, chapter, book,…) into the elements of thought while using the intellectual standards for metacognitive assessment” and suddenly (okay, not so suddenly because this is an iterative process) physics begins to make sense. I also offered this as an attempt at an operational definition of “learning” in that learning is the outcome of going through this process. The entire class bought it. One student even stated that this was difficult for him, but also that he wished he had known about this framework before now. I can ask for more than that.


Conceptual Understanding in Introductory Physics III

This series continues with a question which, I hope, causes readers and students to reflect on something that is frequently omitted from traditional introductory physics courses. I contend that words are all we have to convey conceptual understanding in physics or any other topic. Yet, in science courses, and especially in physics courses, we tend to overlook the deep meanings of the terms we use and expect students to use. The mantra “Shut up and calculate!” comes to mind. Problem solving and calculation are important, but I argue they are not at the heart of physics. Without deep understanding of what it all means, meaningful problem solving can’t happen. We should periodically assess students’ understanding of what terms really mean by just asking them. It doesn’t get any more straightforward.

So without further ado, here is the third question in this series.

Using only words (no mathematical symbols, equations, numbers, or textbook definitions), articulate the most fundamental meaning of each of the following physical quantities: position, velocity, mass, momentum, force, energy, and angular momentum, gravitational field, electric charge, electric field, magnetic field. Try very hard to restrict yourself to one complete sentence for each quantity.

Note that this question can be asked at any time during the course, and it need not cover only the terms I include here. I chose these because they, in my opinion, are probably the most fundamental.


Conceptual Understanding in Introductory Phyiscs II

This post is the second in this series. You can read the background on this series in my previous post. I hope the questions in this series promote some discussion of what the goals of contemporary introductory physics should be these days.

Here is the next question. It has multiple parts.

In one sentence, explain what we mean when we say a physical quantity is constant. In one sentence, explain what we mean when we say a physical quantity is invariant. In one sentence, explain what we mean when we say a physical quantity is conserved. Give an example of a physical quantity that is constant, invariant, and conserved. Give an example of a physical quantity that is invariant, but not necessarily conserved. Finally, give an example of a physical quantity that is conserved, but not necessarily invariant.

It is important for students to understand the distinctions among these three terms (constant, invariant, and conserved). They are not interchangeable and using them correctly requires a rather deeper understanding than students usually get. As I said last time, I won’t disclose answers here but feel free to kick things around in the discussion area.