# Matter & Interactions I, Week 7

**Posted:**September 30, 2016

**Filed under:**Course Log, Goals, Motivations |

**Tags:**free body diagram, introductory physics, momentum principle, system schema, teaching, vectors Leave a comment

I will edit this post at a later date to include diagrams.

This was a pretty good week. Students are beginning to ask deeper, more insightful questions and that tells me they’re engaging in the material more than before. We formally began chapter 2 on the momentum principle, but I did something quite different from the textbook.

I took the opportunity to introduce system schemas (I’ll add a link later). I think system schemas are important because they allow for some foundational physics reasoning to happen without the burden of geometry contained in free body diagrams. System schemas allow students to think about systems and interactions that cross system boundaries, and this is important because the momentum principle and (all?) other physical principles apply to systems.

I introduced one slight change to the schema. Sometimes, an interaction only exists for a brief duration when analyzing a problem. As an example, chapter 2 presents the problem of two students running down the hallway toward each other, and they eventually collide. There is a contact interaction between them, but it only exists while they’re in contact with each other and not before. Several years ago, a student of mine and I decided to indicate such “temporary” interactions with a dotted line rather than a solid line.

I also introduced free body diagrams. These of course contain the geometry in a given problem, but I had an insight this week that to my knowledge had never occurred to me before. Free body diagrams are coordinate free. One can draw such a diagram for a given physical situation without introducing a coordinate system. Why? Because vectors are coordinate free. So draw a free body diagram to reflect the physics. Then to solve the problem in the traditional ways, project that diagram onto any convenient coordinate basis. We did this in class for a particularly simple situation: a block at rest on a surface. Let the system consist of only the block. There are only two interactions with the block: one shared with Earth (a gravitational interaction) and one shared with the surface (a contact interaction). We can project this situation onto an orthogonal coordinate basis that is rotated with respect to the standard basis (+*x* to the right of this page, +*y* to the top of this page, +*z* out of this page) and when we do, we introduce components that weren’t necessarily there before, but they’re there now and they always null each other out (think of the block sitting motionless on an inclined plane tilted from upper left to lower right).

But now the class has a mystery to figure out. In the rotated basis, the components are easily understood as gravitational (the components of the force due to Earth parallel to, and perpendicular to, the surface), contact (the component of the force due to the surface perpendicular to the surface…the normal force), and friction (the component of the force due to the surface parallel to the surface). The normal force and friction force are components of the force due to the surface. But where do these components go when the surface is horizontal?

We ended the week with a computational problem from chapter 2.

As always questions, comments, and constructive criticism are welcome.

# Matter & Interactions I, Week 6

**Posted:**September 30, 2016

**Filed under:**Course Log, Goals, Motivations |

**Tags:**GlowScript, introductory physics, programming, teaching Leave a comment

Note that I’m writing this one week late.

This week we focused on getting used to GlowScript by doing all of the computational activities in chapter 1 of the textbook. Most everyone did fine. One student in particular is having significant difficulty because he is not a native English speaker. He tells me he wants to stay in the course though, and I certainly encouraged him to do so for as long as he feels comfortable.

Students continued to work their way through the chapter 1 WebAssign problem sets. It’s slow going though, because they, like my astronomy students, have been conditioned to do something for a teacher’s mark and not for the benefit of learning it as a foundation for future things. I wish there were a less painful way of undoing that conditioning. As usual, students are waiting till the weekends to work on the WebAssign sets and that’s generating a lot of frustration. On the other had, it also causes the to see what they don’t yet understand that in that respect, it’s meeting my goals.

Although nothing really exciting happened this week feedback, questions, and constructive criticism are welcome.

# Learning Critical Thinking Through Astronomy, Week 7

**Posted:**September 30, 2016

**Filed under:**Course Log, Goals, Motivations |

**Tags:**critical thinking, Eratosthenes, falsifiability, introductory astronomy, shadows, teaching, testability Leave a comment

The week began with my expectation that students had watched a YouTube clip showing an excerpt from Carl Sagan’s COSMOS series in which Sagan explains Eratosthenes’ work with shadows. Students don’t realize it yet, but this topic is the culmination of all of the previous activities. It’s the reason for the critical thinking activities. It’s the reason for the shadow activities. It’s the reason for everything they’ve done so far in the course, all embodied in Activit0206.

The first class day was spend working on some WebAssign content that provides formative assessment related to the first three shadow activities. I use WebAssign content in this course as as rough indicator of engagement. Two sections show nearly one hundred class participation, but the third section shows only about twenty-two percent participation. I said enough about my problems with motivating students in the last post so I’ll dispense with that here.

One daytime section and the evening section began Activity0206 on the second class day; each of these sections only meets twice each week. The other daytime section had to wait till its third class day to begin (today, actually), but that’s good because that was the only day that was sunny.

For this activity, students compared four models (flat Earth, nearby Sun; flat Earth, faraway Sun; curved Earth, nearby Sun; and curved Earth, faraway Sun) to actual data (most of which is simulated, but it’s still accurate) and eliminate the models that don’t support their observations. We used the whiteboards for flat Earth, yoga balls for curved Earths, and Nerf darts with suction cups for sticks. This activity is supposed to be the most scientific thing students have done up to this point and hopefully they will now see many connections to previous activities that built up the process they’re expected to go through here. Of course, many will still not be able to see all the pieces, but that’s to be expected. Most eventually will.

Part of the activity must be done outside with the best faraway light source we have available. However, the week was cloudy with the exception of today (Friday) so the other two sections has to improvise inside the classroom. That’s okay, and we made do with a “distant” light source in a far corner of the room and the various test Earth’s in the diagonally opposite corner.

It’s interesting that this has historically been one of the more difficult activities, at least according to students. This semester, I was pleasantly surprised to hear some students in one section (the section that seems the least engaged to me) to say they thought this was the best activitity so far becuase it was something they could immediately relate to. That puzzles me a bit, but I’ll take it!

As always questions, comments, and constructive feedback are welcome.

# Learning Critical Thinking Through Astronomy, Week 6

**Posted:**September 30, 2016

**Filed under:**Course Log, Goals, Motivations |

**Tags:**activities, course pacing, introductory astronomy, learning, student motivation Leave a comment

Note that I’m posting this one week late.

This was an interesting week in that there were both highs and lows.

I’ll start with the lows. We are now a week behind where we need to be. On Monday of this week, my expectation was that students would have completed the first three shadow activities, but those expectations were not met. I have been stymied over the years by how students justify not doing what is asked of them. On one hand, it seems simple. Not being students on a residential campus, they have off campus obligations ranging from raising money for transportation (or even not having transportation) to parental and family obligations, and I totally get that. I really do. Still, many students still rather glibly do not expect to have any (as in at all) outside encounters with course material once they leave the confines of the barrier-free environment I try so hard to provide for them. So I then find myself once again in an ethical dilemma. Do I decide on the fly to flip the class and have them do the things they were expected to do out of class in class that day or week, or do I keep going ahead with students who are lost becuase they haven’t done the on activities and will be even more lost if I keep pushing them forward into new territory. Either way, I feel guilty. If I flip the class, we get another week behind and that, in turn, induces more guilt becuase we’re further behind and the perception is that I’m not doing my job (as some people define it for me). If I push on, students get no benefit because they can’t understand the newer material because they didn’t understand the previous material. For better or worse, my professional conscience almost always leans toward flipping on the fly and letting students catch up. After all, the classroom is supposed to be a nearly ideal environment for doing that so it would be hypocritical for me to deny them that opportunity. I hope I’m right. Am I?

But what about motivation? I think my greatest shortcoming is that I just can’t figure out how to motivate students. My point of view, however unrealistic, is that merely registering for a course indicates a willingness to put forth whatever effort the instructor deems essential. I openly admit that I probably didn’t see it that way as an undergraduate (Why should an astronomy major spend so many hours on a western civilization course…and I honestly still wonder about that…) so yeah, I shouldn’t be surprised but it’s not like I’m asking students to do anything unreasonable. If they don’t finish something in class, it really should be finished outside of class and I don’t think that’s asking too much. I’ve tried everything I can think of to instill motivation where it seems to be lacking but apparently I’m not yet to the point where I can do that successfully. Right now, I’m at a point where I don’t think I should have to provide ALL of a student’s motivation. If there’s none there to begin with, then the student should rethink his/her reason for being in college. I understand that that’s not a politically acceptable stance, but it is indeed my stance at present.

Now the highs. I got some really good feedback on how to improve the shadow activities from the few students who did engage with them. The most drastic change is that Activity0203 needs to be completely rewritten to match the format of the previous two activities. That way, it will be more independent of the other two whereas now it relies on one or the other of them.

I’m always in favor of anything that lets students understand things they never thought they could understand before, and that happened this week so I must give a slight edge to the highs over the lows. Motivation is still an obstacle for too many students, and I just don’t know that I can fix that in any realistic way.

As always, I welcome feedback, questions, and constructive criticism.

# Are we making teaching mills?

**Posted:**September 30, 2016

**Filed under:**Workplace |

**Tags:**faculty concerns Leave a comment

**Warning! Many people will not like this post. It strays from my usual fare so unless you are interested in faculty work environments, please don’t read any further and just move on.**

I am sure you have heard of diploma mills. Basically, a diploma mill is an entity disguised as an institution of higher learning whose real purpose is to sell an academic credential in exchange for money with only a thin, if any at all, veil of legitimacy. The credential may or may not be recognized by credible agencies.

I am questioning whether or not we have now created a new thing that I will call a **teaching mill**. I define a teaching mill as an otherwise legitimate academic institution that claims to value teaching over everything else, but operates with such heavy intrusion into teaching in the the forms of micromanagement of faculty, businesslike mentality, and general administrative incompetence that effective teaching cannot be carried out.

Here is a (not necessarily comprehensive, not necessarily in any particular order) list of (not necessarily hypothetical) properties I would consider associated with operationally defining a teaching list.

- Administrators lay off non-faculty support employees, citing budget shortages, and then unilaterally reassign those duties to faculty under the guise of “pitching in to help.” Faculty are not qualified to perform those duties because they require close familiarity with state and federal laws and are outside the purview of the classroom and the academic nature of faculty hiring.
- Administrators unilaterally rewrite faculty job descriptions, with no input from faculty, when necessary to justify the addition of inappropriate non-faculty responsibilities to their positions.
- Administrators make funding availble to select faculty to promote pet projects and bad policy under the guise of professional development.
- Administrators cite “state laws” or “systemwide regulations” to justify otherwise arbitrary policies that impose on faculty’s true responsibilities when the same laws or policies cited are unknown to, or sometimes interpreted differently by, colleagues at other instituions within the same system. Of course, ignorance of the law doesn’t exempt anyone from upholding said law, but when that law is interpreted differently by the vast majority of other institutions, I tend to think exceptions are in the wrong.
- Administrators willfully violate internal policies that faculty are not permitted to violate. For example, suppose internal policy says that students must initiate all academic concerns and complaint with faculty. Further suppose that a student ignores this policy and goes to, say, a department chair or dean to complain about a faculty member for one reason or another. The written and duly approved and published policy says that the department chair or dean cannot do anything unless the student has first discussed the matter with the faculty member. However, the faculty member is unexpectedly called into conference with the chair or dean to discuss the matter, which has never even been previously brought to his/her attention before now. Administrators justify this by saying that they are ALWAYS obligated to hear student concerns and complaints, and threaten the faculty member with insubordination when the published policy is displayed.
- Faculty are evaluated by persons with no background in evaluating academic faculty and thus apply inappropriate metrics, usually from the business world.
- Faculty are required to microdocument (is that a word?) their professional development efforts despite a heavy teaching load, with no such requirement for administrators.
- Faculty are required to participate in professional development that is not discipline specific and is of otherwise dubious value (e.g. learning to use Microsoft Office, which is not appropriate professional development for college faculty) except for documentation required by accrediting agencies.
- Faculty are required to participate in professional development despite having no institutional funding for true, discipline-specific, off campus professional development.
- Faculty are threatened with insubordination when they question any aspect of poor administrative policy (including policy that is in questionable legal territory) or anything related to imposing upon faculty’s true institutional purpose, teaching.
- Faculty are punished for being the subject of unsubstantiated damaging rumors propagated by students while the students propagating the rumors are in no way held responsible.
- Faculty are quietly told to keep high school students in mind when selecting textbooks in order to keep costs to them as low as possible. In theory, this sounds fine, but being told to cater to an audience that should not be our audience but is now our audience merely in order to raise FTE count is inappropriate. Meanwhile publicly, administrators tell us that no special accommodations are made to our growing ranks of high school juniors and seniors.
- All, and I do indeed literally mean ALL, resistance to any new or established policy is ALWAYS ultimately tied to the threat of losing faculty jobs…NEVER administrative jobs…ALWAYS faculty jobs.
- In short, faculty are treated as just another expendable resource much as the way front line workers (e.g. bank tellers) are treated in a corporate environment.

I could, and probably will, add to this list over time. I have great concerns that here in North Carolina, this is the environment in which many, if not most, community college faculty must try to do what they are actually hired to do (deliver high quality classroom instruction). We are not here to advance an administrator’s political agenda. We are not here to do administrative work. We are not here to take up the slack created by administrators’ failures to secure adequate institutional funding (which, by the way, administrators proudly crow about being THEIR main responsibilty and yet they’re never actually held accountable for THEIR failure to follow through…always blaming state legislatures and politicians…). We are not here to be surrogate parents to unprepared students. We are not here to do the work of public high school teachers. We are not here to be cheerleaders for an institution (although that is usually a welcome consequence of having high quality students). We are not here to be office workers. We are not here, I assert, for anything other than what we are qualified to do, and that is not up for negotiation.

Faculty must start asserting more authority and power against incompetent administration in order to prevent our institutions from becoming teaching mills.

# Learning Critical Thinking Through Astronomy, Week 5

**Posted:**September 19, 2016

**Filed under:**Course Log, Goals, Motivations |

**Tags:**concepts, critical thinking, introductory astronomy, learning, misconceptions, observation, shadows, teaching Leave a comment

I originally had one set of plans for this week but as sometimes happens, those plans changed radically. Instead of starting in on the next series of activities, we spent some time doing some formative assessment (and I’m really beginning to hate the word “assessment” because of the current political climate in which it’s (mis)used but I will continue to use it anyway) and the first large summative assessment. These assessments cover the chapter 1 (01xx) activities, of which there were seven. As usual, my suspicions were that most of the students had not dug deeply enough into the activities, if at all, outside of class. They historically tend to underestimate the amount of engagement I expect despite having articulated this on the syllabus (I’m assuming they read the syllabus). I’m still evaluating the assessments (my new term for grading them).

I did something new with the first three activities (Activity0201-Activity0103) and in the process, made a big mistake that I need to correct befor next semester. The activities in this series focus on observing shadows, shadows of identical sticks at three strategic locations (Hickory, NC in Activity0201; Cape Town, South Africa in Activity0202; and Quito, Ecuador in Activity0203). For efficiency and to promote collaboration, I assigned each activity to one third of the class with the expectation that each group will collaborate to get the necessary information for the remaining two activities from the other two groups. My big mistake was forgetting that Activity0203 requires information from the either of the first two activities. To compensate, I put the students doing Activity0203 in the somewhat unfortunate, but all to real in the actual scientific world, situation of having to rely on other “research groups” for information before they can complete their assigned tasks. Alternatively, they can just go ahead and do do the other two activities since I expect them to have that information eventually anyway. I’m just trying to make the information gathering more active, more realistic, and more fun. Seriously, though, I need to revise Activity0203 to reflect this way of doing things.

Historically, again, these initial chapter 2 activities are very slow going. All students are doing is getting the basic facts from simple observation. The facts consist of a complete description of the stick’s shadow from the various locations and detailed operational definitions of the four cardinal directions (north, south, west, and east) based on the shadow’s behavior. Sounds easy, right? It isn’t. Despite my warnings, students fail to read the directions carefully enough to get what they need from them. They sometimes agonize over words they should already know (e.g. convex, concave, etc.) and fret when I ask them to look up their meanings rather than just have me regurgitate the definitions. Most significantly, they rely on things they have been told previously in other courses at other institutions (sometimes colleges, sometimes high school) and get angry when I don’t allow them to cite on these things. I’m speaking of things like “Earth orbits Sun” and “Earth is spherical, not flat” and “Sun always rises in the east and sets in the west” and “noon is equivalent to twelve o’clock” and “a stick’s shadow always vanishes at noon every day of the year.” The first two are not allowed because we have done nothing at this point in the course to establish these “facts.” All students have and operationally “know” about at this point is what a stick’s shadow does between sunrise and sunset. **I impose the following rule: If you haven’t operationally defined a given concept at this point, you are not allowed to rely on it as part of an explanation.** The third is patently false by all definitions, yet it lingers as a misconception that students stubbornly refuse to shed. The fourth is subtle, and is the subject of timekeeping, which is in turn the theme of chapter 4 (for which only rough draft activities exist at this point and they’re not ready to distribute). The fifth is also patently false because if it were true, noon would never happen here in Hickory. We really are, quite literally, strating from the ground up with this material, all within the previously established framework of critical thinking from chapter 1 and the previous series of activities.

For this series of activities, students use an Easy Java Simulation app created by Todd Timberlake (Berry College) modified by me for my needs. I will eventually make it available on ComPADRE once I properly document its features.

I welcome questions, comments, and constructive feedback.

# Matter & Interactions I, Week 5

**Posted:**September 16, 2016

**Filed under:**Course Log, Goals, Motivations |

**Tags:**component, concepts, geometry, introductory physics, mathematics, projection, symmetry, tensors, vectors 1 Comment

This week, we transitioned to chapter 1 of the Matter & Interactions textbook (fourth edition). I have WebAssign problem sets for each chapter available for formative assessment and practice while working their way through the reading. I encouraged them to use the book the way it was intended to be used, specifically by stopping and doing the checkpoints in situ and NOT going forward until they can get the correct answers by working them out (checkpoint answers are provided at the end of every chapter). My expectation is for them to have worked their way through chapter 1 by Monday of next week.

In class, I spent two days introducing vectors. I have never been happy with the introductory course’s treatment of vectors, mainly because of inconsistencies in terminology and notation (sometimes within a given textbook). More importantly, vectors are almost always presented in terms of their coordinate representations and not as the inherent geometric entities they are, and this completely undermines the main reason vectors are used in the first place: to describe physics without the need for coordinates. If we want students to understand that physics is independent of both reference frame and coordinate system, then why not present it that way from the beginning? I think it might pave the way to tensors, which are nothing but an extention of the vector concept to slightly more complicated quantities.

So, I began by introducing a vector as a quantity which can be represented with an arrow. The arrow’s length encodes the quantity’s magnitude (i.e. how much? or how many?) while the arrow’s direction encodes the quantity’s direction. I don’t like saying that a vector is “a quantity with magnitude and direction” because there are quantities that have magnitude and direction but are not vectors (e.g. finite angular displacements). Equally troublesome is saying that a vector is “a list of three numbers that transform a certain way from one frame to another” because, well, HUH? Defining a vector as an “element of a vector space” is rare in physics contexts (I don’t think I’ve seen it done that way, at least not in the intro course) and it’s really no less circular than speaking of mysterious transformation properties. Saying a vector is “something which can be represented with an arrow” seemed a relatively decent compromise. I don’t know…maybe I could do better.

As I introduced these ideas, I “invented” symbols on the board for them and gave the appropriate LaTeX commands (defined in my mandi package) to get these symbols. I want students to start associating LaTeX with material from the textbook so we can be consistent with notation from the beginning. They know to use **\vect{}** to indicate the symbol for a vector quantity, **\magvect{}** (think “magnitude of vector”) to get the symbol for a vector’s magnitude (always with the appropriate unit…mandi knows every quantity’s SI unit in as many as three different formats: base, derived, and one I made up called traditional…see the mandi documentation for details), and **\dirvect{}** (think “direction of vector”) to get the symbol for a vector’s direction.

The arrow, and the vector it represents, has inherent properties that don’t change from one coordinate system to another, and some properties that do indeed change from one coordinate system to another. I want to do two things: 1) make a solid connection between what I’m saying and what students see in the textbook and 2) get them to think deeply about geometry. Where do the numbers associated with vectors in the textbook comefrom? **They come from projecting a vector onto a coordinate basis.** What does THIS mean? Operationally, it means the following:

- Establish an otherwise arbitrary orthogonal coordinate system, which I drew on the board apparently arbitrarily oriented around the arrow representing our velocity vector. I intentionally kept the arrow in the first quadrant though. The arrow’s tail need not be at the origin.
- Place an imaginary light source “far away” above the x-axis and let it shine down on the arrow with light paths that are parallel to the y-axis so that the arrow casts a shadow onto the x-axis. The length of this shadow, along with this shadow’s orientation, tell us “how much of the arrow lies in the x direction.”
- Place an imaginary light source “far away” above the y-axis and let it shine down on the arrow with light paths that are parallel to the x-axis so that the arrow casts a shadow onto the y-axis. The length of this shadow, along with this shadow’s orientation, tells us “how much of the arrow lies in the y direction.”
- Do a similar procedure to figure out “how much of the arrow lies in the z direction” but understand that it will be zero in this particular case because our arrow lies in the xy-plane.
- These three “how muches” are the numbers that go into the slots in the notation M&I uses to denote the coordinate representation of a vector.

Now, when I then said that each of these “how muches” has a techinical name, and that this name is “the projection of the arrow onto an axis” there were audible “Oooos” and at least one “AHA!” because they said they’d heard this term before but didn’t truly understand what it meant until just now. I assume they were telling the truth, and I was happy. At this point, I did not distinguish between “component” and “projection” as the math textbooks do, and I will return to that later.

Next, we did a calculuation to get algebraic expressions for the three “how muches” and noted a pattern: **the “how much” along a particular coordinate axis always works out to be the arrow’s length scaled by the cosine of the angle between the arrow and the axis in question.** I introduced the notation with obvious extension to the other two coordinate directions. **Note the use of double bars for magnitude, consistent with the students’ calculus textbook.** One question that was raised was why is cosine, rather than sine, used. We then discussed how this relates to the geometry of the problem, which is how much of the arrow lies along a chosen direction, so we need the side of the relevant right triangle that is adjacent to the angle we know, and thus we need cosine. Yes, you could use sine but then you’d have to refer to the complement of the obviously relevant angle and we want to be conceptually consistent throughout. So, we will always use the angle between the arrow and the chosen coordinate direction and we will always want the part of the arrow parallel to this direction and thus we will always use cosine. Students accepted this. They took the bait, because consistent use of cosine here leads to the next important idea: dot products.

Next, I explained that this process of getting “how much of a vector is parallel to a chosen direction” can be framed as a coordinate-free geometry issue. Given any two arbitrary vectors and (represented by arrows of course, but now I’m referring to the actual vectors rather than their represtational arrows), and without introducing a coordinate system, we can answer the question of “how much of one is parallel to the other” in an elegant way. I introduced the new symbol as the symbol for “how much of is parallel to with no regard for orientation.” Note that’s one symbol! Specifically, it’s just a symbol for a real number, an element of . The actual number is merely the multiplicative product of the two vector magnitudes scaled by the cosine of the angle between them or just , which requires visualizing them as arrows for simplicity. Think of the symbol as having two slots, each of which takes a vector as input and the complete symbol represents the resulting real number. Okay here’s where I’m setting the stage for something huge later.

This leads to three obvious extensions: 1) filling both slots with the same vector and 2) filling one slot with a unit vector (which we call simply a direction or, now, a basic vector) and 3) reversing the order of the two slots. Well, the third is interesting because it doesn’t change anything! Yes, reversing the two slots changes the way the overall symbol looks, but we get the same real number out. BOOM! We have discovered an elementary example of a symmetric tensor! The first is interesting because is gives us a geometric way of calculating the magintude of a vector…or does it? It’s not incredibly useful at this point, but we’ll make note of it and file it away nonetheless. The second is the important one for the moment, because by dropping a basis vector into one of the slots (doesn’t matter which one, remember) we get the same “how much” number we got earlier from other considerations! **Now we have a solid way to quantity what we mean by projecting a vector onto a coordinate basis.** It’s just a dot product!

To follow up this discussion, students did a whiteboard exercise. I asked them to draw an arrow represent a velocity vector with an arbitrary magnitude of five units and a direction to the right on their boards. They quickly picked up on the fact that the actual length of their arrow didn’t matter as long as they labeled as having five units. Next, I came around to each group’s whiteboard and drew a new orthogonal coordinate basis on it and asked each group to “project the velocity vector onto the coordinate basis that I drew.” They got it! I mean they totally got it! The only difficulty was, as I have come to expect, someone’s calculator being in radian mode and not in degree mode. Using VPython for computations will permanently fix this problem by eliminating it entirely. Anyway, the amazing moment came when I asked them to calculate the vector’s magnitude in terms of the components they just derived and they all got five units back! Some were amazed. Some expected this. I was happy either way. The takeaway? **Projecting a vector onto a coordinate basis changes the numbers we use to represent the vector the way the textbook does, but the actual vector itself doesn’t change.** **That’s geometry! That’s physics! That’s cool!**

That huge thing I mentioned above? I didn’t explicitly go into this in class this week, but here it is. **We can also think of a vector as a function (a linear function, to be precise) that takes as an input another vector and outputs a real number by doing a dot product.** Misner, Thorne, and Wheeler operationally describe is as a machine with an input slot that takes a vector and an output slot what spits out a real number. Yep…a vector framed as a function that takes another vector and outputs a real number, the dot product. This is conceputally very simple, but not at all anything close to what is seen in introductory physics texts. I’m trying to change that. Why? Because by stringing together such vectors-treated-as-functions in a certain prescribed way, we can build objects called tensors. Tensors appear in introductory physics, but we rarely point them out and expliot them. I’m trying to change that. At some point, I will revisit our class discussion from this week and present a vector as one of these functions that takes another vector and spits back a real number.

I welcome questions, feedback, and constructive criticism.