Learning Critical Thinking Through Astronomy, Week 16

What time is it anyway, and what does that question even mean?

I want to describe a classroom activity that is the culmination of our discussion of time. I’ll start with a brief description of the background leading up to this activity and then describe the activity itself.

If there is any aspect of astronomy that is directly relevant to all of our lives, it is the measurement of time, which, ultimately began with watching the sky. I begin by defining the concept of a prime mover, a term I shamelessly stole from somewhere, I think a discussion of electric circuits, because it seems like the most appropriate term for this purpose. I will happily acknowledge the source if anyone can jar my memory.

A prime mover is any celestial object whose motion we observe for the purposes of measuring time.

Any celestial object can be a prime mover, but there are two main ones we usually adopt: Sun, and the point on the celestial sphere where the ecliptic crosses the celestial equator during the month of March (aka the vernal equinox). In this elementary discussion, we neglect all mention of precession and nutation. To complicate matters, there are actually two (yes, two!) solar prime movers. The Sun we’re all familar with constitutes one of them, and in this context we call it the apparent Sun. The apparent Sun moves along the ecliptic during the year, but does so at a variable rate, moving fastest in January (near perihelion) and solwest in July (near aphelion). This variability is caused by Earth’s non-zero obliquity and non-zero orbital eccentricity. I usually don’t discuss these causes, but sometimes I do depending on how much the class wants to get into it.  If they drag me there (and I secretly always hope they do!), I feel obligated to follow their lead. There’s a second Sun, though, and it’s the one by which we’ve ALL lived our entire lives, at least within the context of telling time. This Sun is called the mean Sun. It’s not a physical entity that gives off light. Unlike the apparent Sun, the mean Sun moves around the celestial equator at a uniform rate. Both the apparent and mean Sun take one year, by definition, to go around their respective celestial great circles.

Any given prime mover defines a unique timescale inherent to that prime mover’s motions.

Two consecutive passages of the prime mover over an observer’s local celestial meridian define a day on that prime mover’s timescale.

That interval can then be subdivided into twenty-four hours (Why twenty-four? Probably because it is divisible by so many small integers.)  of time on that timescale. Using the apparent Sun defines a timescale called Local Apparent Solar Time (LAST). The interval between two consecutive meridian passages of the apparent Sun defines an apparent solar day and by definition, it is subdivided into twenty-four hours of apparent solar time. LAST is embodied by a sundial (or simply a stick in the ground), which uses a shadow to track the apparent Sun’s diurnal motion across the sky. Using the mean Sun defines a timescale called Local Mean Solar Time (LMST). The interval between two consecutive meridian passages of the mean Sun defines a mean solar day and by definition, it is subdivided into twenty-four hours of mean solar time. LMST is embodied by a mechanical clock designed to track the otherwise invisible mean Sun. Because the two prime movers move at different rates around the sky, an interval or mean solar time isn’t the same as an interval of apparent solar time. Sundial users know this. The discrepancy between the two is called the equation of time.

Fundamentally, we must observe a prime mover’s hour angle (the angle between the hour circle passing through the object extended to the celestial equator and the celestial meridian, measured along the celestial equator) and operationally turn that into something that we call “time.” We could operationally define “time kept by a prime mover” as its hour angle, but there’s a problem with that. That would mean that 00:00 (hh:mm) on that prime mover’s timescale would happen when the prime mover is on the celestial meridian. So what? Well, calendar makers like to have the date rollover at 00:00 and having this happen during the middle of daylight would complicate our daily lives. Imagine waking up on one date and coming home from work on another date. Yuck! So, let’s add a twelve hour offset to put the calendar rollover in the middle of nighttime, when humans are ostensibly the least active.

Time kept by a prime mover = prime mover’s hour angle + twelve hours

LAST and LMST have the “local” attribute because hour angle inherently depends on one’s local sky. All locations on the same north-south line have share a celestial meridian, but if you move east of west, you have a new celestial meridian.

Finally, the question “What time is it?” seems so simple, but to an astronomer it really means, “Where is the prime mover relative to the celestial meridian?”

Okay, that’s all the background (except for a few minor details); now for the actual activity. I have a stack of index cards, each of which has the name of a city and its longitude on it. Usually I only use cities in North America but I have several cards with the names and longitudes of cities on other continents for occasional use.

Each student group (a group of two or a group of four) picks a random card and writes its chosen city and longitude on its whiteboard. I also randomly pick a card and put my chosen location’s name and longitude on the class whiteboard. I ask a student to randomly pick a date, and I put the date and the corresponding value of the equation of time on the class whiteboard. Then I randomly choose whether to give a LMST or LAST, and write my choice on the whiteboard (e.g. a LAST of 13:50, a LMST of 05:14, etc.). So basically, I specify a reference location, a prime mover, a time measured by that prime mover, and the equation of time on the chosen date. The object is for each student group to get the LMST, LAST, STDT (standard time), and UT (Universal Time) at the same moment at its chosen location. Everything boils down to three basic rules:

  1. Given time kept by a prime mover at one location, to find the time kept by the same prime mover at a DIFFERENT location, the difference in time is the difference in longitude expressed in time units.
  2. Given time kept by a prime mover at one location, to find the time kept by the other prime mover at the SAME location, the difference in time is the equation of time.
  3. The STDT is the LMST at the nearest time zone center, so this is just a special case of the first rule. For our purposes, we don’t account for irregular time zone boundaries.
  4. The UT is the LMST at a longitude of zero degrees (the prime meridian), so once again this is a special case of the first rule. For our purposes, we don’t distinguish among GMT and the various flavors of UT (UTC, UT0, UT1, and UT2).

Since the various groups will have sometimes wildly different longitudes, there’s no way to know whose results are “right” except by doing the necessary calculuations. It is important to do the STDT and UT last, and in that order, becuase it will always be the case that the STDT’s for the various locations will all have the same number of minutes and will have hours differing by integer amounts. These two requirements serve as a sanity check on our calculuations. I tell students that if everyone ends up with STDTs that have the same minutes and hours differing by whole numbers, their results are probably correct. A final sanity check is that the UT must be the same for everyone; that’s one reason it’s called “universal” time.

An interesting extension of this activity is to have everyone determine, for their location, the STDT at noon, where “noon” refers to 12:00 LAST. This multistep calculuation requires setting the LAST to 12:00 (i.e. noon), applying the equation of time to get the LMST, and then applying the necessary longitude correction to get the STDT. Becuase of the equation of time, “noon” doesn’t always happen when a clock reads 12:00 and therefore the phrase “twelve noon” has no practical meaning.

Another useful variation is to specify not the LAST or LMST for the instructor’s reference location, but to cite the hour angle of the apparent Sun and let students express that as a measure of time and then proceed as usual. This is actually a special case of the second rule with a subsequent application of the first rule if necessary. If a location is on a time zone’s center, then no difference in longitude exists.

I may consider writing this activity up for AstroNotes in The Physics Teacher.

As always, feedback is welcome!


Planning the Perfect Date…Astronomy Style

Relax…I promise there’s an astronomical connection here! Every semester just after the activity on lunar illumination (my way of saying lunar phases), I give a short lecture on eclipses and then ask the class if they would like to know how to plan the perfect date. This surprising question gets a lot of interested looks, and they sometimes as if I’m serious. Of course I’m serious, and of course I’m setting up something interesting as well. The target audience is an introductory general astronomy class. Here’s what I do.

To plan the perfect date you need a clear night, some wine (substitute the beverage of your choice to suit your audience, but I work at a college so…), a blanket, some music (I usually suggest smooth jazz but again, substitute for your audience), and…this is the most important thing…a total lunar eclipse.

Having assembled the necessary components, you then call your significant other or intended companion and ask, “What if I told you I could show you all the world’s sunrises and sunsets in just one night?” The spectrum of responses ranges from “no” to “I wanna see you try!”. Fair enough, but it’s totally possible.

Now the science. We’ve been using our head to represent Earth, a styrofoam ball to represent Moon, and a light bulb to represent Sun in our recent classroom activities. Earth, like me, has hair (I slowly run my fingers through my hair when I say this…long hair helps with the theatrics). We don’t call it hair; we call it Earth’s atmosphere. Our atmosphere is composed mainly of nitrogen, and nitrogen likes to scatter blue light out of Sun’s otherwise white light. (At this point, I sometimes digress into a preview of second semester astronomy by introducing photons, light’s spectrum, Rayleigh scattering, etc.) So when Sun’s white light (beware of astronomy books claiming Sun is a yellow star…”color” has multiple meanings in astronomy) passes through Earth’s “hair” that light has the blue component scattered out of the way, and we see that scattered blue light as the blueness of the sky. (Incidentally, I’ve always wondered whether or not the sky looks blue looking down to Earth’s surface from above the atmosphere and I believe sometime in the past year I saw a discussion about this somewhere online but can’t remember where. Also, this brings up the distinction, to me, between the “atmosphere” and the “sky.” I don’t think they’re the same thing any more. The sky appears blue when seen from Earth’s surface or just above Earth’s surface, but the atmosphere is transparent. This may be the basis for some good discussion at another time.) White light minus its blue component leaves the light looking reddened.

So now I stand between the light bulb and the white board (now representing Moon’s surface) so “Earth’s” shadow is projected onto the white board. I ask the class to tell me where Earth’s terminator is as I stand there, and they correctly point out that it’s in the plane perpendicular to the line connecting Sun, Earth, and Moon. Sunlight reaching Moon’s surface to illuminate it has to pass through Earth’s “hair” (I run my fingers through my hair again here) or atmosphere, and this does two things. First, as described above it reddens the light. Second, it refracts the light slightly toward the interior of Earth’s shadow, somewhat concentrating it on the lunar disk. This reddend light accounts for Moon’s color during a total lunar eclipse.

Now here’s the big reveal. At any given moment on Earth, sunrises and sunsets are happening along the terminator (the boundary between Earth’s illuminated side and unilluminated side) and as we generally know, sunlight is usually reddened at sunrise and sunset. Those sunrises and sunsets contain light that is reddened by Earth’s atmosphere as described above. That same reddened light, having passed tangentially through Earth’s atmosphere along the terminator, is the combined light of all the sunrises and sunsets happening on Earth at that very moment. Yep! You’re seeing all of the sunrises and sunsets on Earth illuminating Moon’s disk during totality! How romantic is that? In my opinion, very!

So there you go. It is indeed possible to make good on the initial promise, and I think this constitutes the perfect date…astronomy style. Your mileage may vary.

Comments and feedback are welcome.

The Learning Critical Thinking Through Astronomy Activities

I am somewhat limited in what I say in this post because I don’t want to give anything important away to students who find my blog (not that I care if they do).

It occurred to me today that a post describing my astronomy course materials might be helpful for anyone considering using them or wanting to know the underlying philosophy and purpose for the materials. When I formally began this project in 2007, I was frustrated by several aspects of the traditional approach to teaching astronomy. First and foremost was the textbook. I just don’t understand why astronomy textbooks, especially those used in introductory astronomy courses, must be updated so frequently when the updated content is far above the level of an introductory course. Knowing that a new black hole has been found isn’t equivalent to communicating any understanding about black holes any more than announcing the discovery of new exoplanets communicates any deeper understanding about them It is difficult for me to convey this to students who think they understand something just by hearing about it, so I have taken a rather hard line approach to avoiding this pitfall. I had become disappointed with every astronomy textbook I used because the content seemed watered down in every revision. I some cases, I contacted the author and was told that pressure to reduce content came from the publisher and they had to play ball. Nothing distinguished one textbook from another.

However, a bigger source of frustration was course content. My observation skills (I identified as an astronomer back then.) told me that my course, and indeed many similar introductory astronomy courses, don’t really live up to their marketing goals. They focus too much on shoveling content to students and then students regurgitating that content back on tests with no understanding. More seriously, the traditional approach gave inadequate, if any at all, treatment of the very foundational content that introductory liberal arts science courses are supposed to provide. I’m talking about among other things, you know, understanding the difference between science and belief systems, understanding testability and falsifiability, detecting faulty logic, discussing the roles of science and pseudoscience in contemporary American society, and carrying out simple investigations. These are more important than simply memorizing things that can be found in Wikipedia or by using a search engine. Still, there is more or less traditional astronomy content in the form of the geometry of the sky and correlating that with shadow behavior, understanding the periodic behavior of the most easily observed celestial bodies, and understanding gravitational attraction and its role in orbital motion. This meant discarding the majority of what is usually containd in a traditional introductory astronomy textbook and I have never looked back. I have years of course evaluations from students telling me how much they prefer the new approach to the traditional textbook approach. Eventually, I hope to be able to quantify that feedback in some way.

Anyway, I decided to leave the game and start using my own materials that I had been toying with. The course is organized around five broad topics (or chapters): critical thinking, observation, the celestial sphere, time, and gravitation. My semester has sixteen weeks, so that’s approximately three weeks per unit. However, I have always found that such attempts at pacing are highly artificial. I can’t control the rate at which students choose to engage in their own education; I can only try to do that. Therefore, I focus on deep understanding rather than breakneck pace as “covering material” because, well, that’s the way it should be.

Chapter 1 currently has six activities. They begin with an attempt at instilling the importance of collaboration (I have eight versions of this activity), and then move to science terminology, declarative vs. operative knowledge and operational definitions, detecting arguments and logical consistency, scientific frameworks, detecting logical fallacies and distraction techniques, and finally, scientific validity. I have tried to target each activity toward one of two very specific goals and accompanying standard on which students are assessed. Standards-based grading is employed. Woven into this first chapter is a thorough introduction to, and practice with, the elements of thought. Using them is a standard on which students are assessed.

Chapter 2 currently has six activities. The first five focus on observing a stick’s shadow from various places on Earth. Students are invited to look for patterns and draw conclusions about a shadow’s behavior. They have to get used to just getting the facts without trying to explain them. It’s difficult! Only two of the first five activities are required; the rest are optional. The sixth activity in this chapter invites students to use their shadow observations to work through Eratosthenes’ historical work on Earth’s shape, hopefully learning how to distinguish among various plausible scientific models in the process.

In chapter 3, the activities begin to resemble what students traditionally think of as astronomy. By this point in the course, they usually complain that they’re not “doing astronomy” when in fact, they are and just don’t understand that science doesn’t look like it does in traditional sanitized textbooks. Astute students will have understood this before this point, but most take a while longer to get there. In this series, students make simple nighttime observations to establish the main movements of celestial objects against the sky. They build a celestial sphere kit they purchased in lieu of a textbook and use it to relate a shadow’s behavior to Sun’s diurnal and annual motions. They see the same observations from a different point of view and hopefully can now connect sky observations to shadow behavior. They also model lunar phases in this chapter.

Chapter 4 is on time. If there is any astronomical topic that is relevant to everyone, it is the measurement of time. I currently have no activities in this chapter in a presentable form, but I hope to change that this fall. Students connect sky observations with the measurement of time as operationally defined by clocks. This builds up to a grea mystery, namely the mystery of why the dates or earliest sunrise, latest sunset, latest sunrise. and earliest sunset don’t coincide with the solstices. The answer lies in the analemma, and solving this mystery is the goal of this chapter.

My goal for chapter 5 is to understand the basics of orbital motion as a consequence of gravitational attraction. My approach is heavily influenced by Paul Hewitt’s approach in his conceptual physics text and I freely borrow from that source. If time permits, I take the opportunity in this chapter to lead students through the work of Galileo and Newton, again at about the level of Hewitt. In the past, I have even incorporated basic vector analysis in this chapter. Yes, vectors.

All of the completed LCTTA activities are available on the download page of my website. The latest versions will always be there. Last year, I figured out how to create PDF documents with editable text entry fields with LaTeX. I want students to quite literally create their own materials and having them type their responses directly into their activity documents helps get them there. They are asked to keep their activities with them and are free to revise their resonses as they go through the course. After all, scientists revise models all the time so why not ask students to do the same as their learning progresses? They get frustrated, though, because there are not necessarily definitive answers to every question. Unfortunately, they somsetimes bypass me with thier concerns and go over my head and I then have to, once again, explain everything to a host of administrators who claim to understand until the following semester when it happens all over again. Students are respoinsible for recording and tracking their own assessment progress. I secretly do it too, but I don’t tell them that. I want them to be invested in keeping up with their learning.

If you are interested in taking a shot at a very nontraditional way ot treating introductory astronomy, feel free to try these activities. All I ask is that you give me feedback on how well they worked or didn’t work. Most of the recent revisions have come directly from conscientious students who appreciate this approach. I also got valuable feedback from colleagues at AAPT meetings where I presented these activities as a workshop. Also feel free to post questions in the comments section below, but note that I won’t divulge answers to questions in the activities here for obvious reasons. Students are good at using search engines.

A Diversity Experiment

Today I decided to try something new in my introductory astronomy class. I wanted to get a quick insight into what students think about diversity while in the safe haven of the classroom.

Our classroom activities have “checkpoints” interspersed throughout so students can, quite literally, have a meeting of the minds in the center of the room. Since this was the first time they encountered such a checkpoint, I set ground rules for discussions. The most important rule is that the person holding my red stuffed Angry Bird is the speaker and others in the classroom are listeners and must not interrupt the speaker. When the speaker is finished (and the speaker decides when he/she is finished), the bird is thrown to someone else. I usually ask that the speaker toss the bird to whom they think is the quietest person in the class, which leads to some interesting results. Everyone must catch the bird at least once during the checkpoint.

The first activity is designed to justify working in groups and there are specific questions from the activity I want students to discuss, but this time I interjected and asked them to discuss “diversity” and intentionally left it open ended. They assumed that I wanted them to discuss diversity within the class, and specifically working in groups on this activity. This generated some good discussion. After several speakers, I then interjected again and asked them to now address “gender diversity.” Again, this generated some excellent discussion and once again, after a few speakers, I again interjected and asked them to address “ethnic diversity.” Almost all of the discussion consisted of students stating what these terms mean to them and how they saw diversity as helpful within the context of the activity. This was all great, and I never really got the impression that students were trying to read my mind and coax them into saying anything particular and I caught myself realizing this and I thought, “This is refreshing.”

So then I decided to go further and said, “New game. The holder of the bird must call out the name of a famous scientist and throw the bird to someone else.” My prediction was that they would name white men, and unfortunately that’s exactly what happened. Einstein and Darwin were named immediately and then the pace slowed almost exponentially, with each student taking longer and longer to name someone. I was pretty shocked when two students thought and thought and said they couldn’t name ANY famous scientist. Two other students named people they thought of as social scientists (one student named Karl Marx and another named Martin Luther King, Jr.). After about six scientists had been named, I said, “Okay I’ll make this easier for you. Name a famous black scientist.” The next student immediately named Neil deGrasse Tyson and another one named George Washington Carver. Sadly, too many students caught the bird, thought for a few seconds, and threw it to someone else and admitted they didn’t know any famous black scientists. I said, “Okay I’ll make it easier for you. Name a famous woman scientist.” The next student immediately named Marie Curie, and another named Jane Goodall. Once again, the pace slowed with students taking longer and longer to name someone. More students tossed the bird to someone else this time too, admitting they couldn’t name anyone. Students began to laugh when I once again interjected. I said, “Okay let me make this really simple. Name the first African American woman in space.” One student immediately shouted “Sally Ride” with little to no reaction from anyone else. I said that Ride was white and most students seemed a bit shocked. I didn’t see that coming. Two students finally did a (clandestine) Google search and shouted “Mae Jemison.” They were surprised to learn that Jemison had actually visited Hickory several years ago.

I hope to do this activity again later in the semester to see if students can more easily name scientists in general and scientists who are not white men. Fortunately, astronomy is a nearly perfect vehicle for telling the stories of the people who got us to where we now are.

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?