Predicate Logic and Popular Culture (Part 226): Wicked

Let $C(t)$ be the statement “On day $t$ , there’ll be a celebration throughout Oz that’s all to do with me,” and let $T$ be the set of all times. Translate the logical statement

$\exists t \in T(C(t))$ .

This matches a line from “The Wizard and I” from the Broadway production of Wicked.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Engaging students: Introducing the parallel postulate

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Enrique Alegria. His topic, from Geometry: introducing the parallel postulate.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The parallel postulate dates back to a man named Pythagoras of Samos. Pythagoras was a Greek philosopher that created a mysterious cult, the Pythagoreans. The purpose of the cult was to seek out a universal truth about numbers and shapes and became the foundation for Geometry. “The Pythagoreans concluded that the one universal quality of all things in the universe, the one thing that everything had in common, was that it was numerable and could be counted.” (Bryan 2014). Improving the work of Pythagoras and other mathematician predecessors was a man named Euclid who originated from ancient Greece. It was through Pythagoras’s key teachings, such as the Pythagorean Theorem, that began the fundamentals of Geometry.

Euclid wrote thirteen books named the Elements. These books were the entirety of Geometry. The Elements starts with a few simple definitions and postulates that were to be built off of each other to prove propositions. Through that work, Euclid changed the world. A masterpiece of logical thought and deductive reasoning.

Euclid caused controversy for years and years to come due to a specific part from the Elements. The parallel postulate which states, “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” Because this postulate makes drastic assumptions it is almost impossible to be proven. For that reason, the parallel postulate has caused so much controversy over the years. Euclid tried to prove all that he could without the parallel postulate and reached Proposition 29 of Book I. This topic further developed as mathematicians believed that the statement could not hold true. From there, several mathematicians are to follow on proving the Parallel Postulate.

How did people’s conception of this topic change over time?

Over time the conception of the parallel postulate changed as many mathematicians tried to prove the postulate. Mathematicians wanted to prove that the postulate was not so much a postulate but a theorem. Several proofs were created, but none had succeeded in proving the postulate from the plane in Euclidean Geometry. As no mathematicians were able to do so they moved towards other dimensions or geometries.

The beginning of Non-Euclidean Geometries. Using the first four postulates of Euclid but create a new definition for the parallel postulate. For example, Nikolay Ivanovich Lobachevsky and János Bolyai were two mathematicians that held all postulates true but the parallel postulate true when discovering Hyperbolic Geometry. The parallel postulate has been modified as such, “For any infinite straight line and any point not on it, there are many other infinitely extending straight lines that pass through and which do not intersect .” (Weisstein) This also led French mathematician Henri Poincaré to show the Hyperbolic Geometry was consistent through the half-plane model.

Many more geometries were able to follow a similar format of creating a parallel postulate equivalent to Euclid’s parallel postulate. “The parallel postulate is equivalent to the equidistance postulate, Playfair’s axiom, Proclus’ axiom, the triangle postulate, and the Pythagorean theorem.” (Szudzik). Despite the many trial and errors of trying to prove the parallel postulate, peoples’ conception of the topic was able to transform and discover new geometries where the respective parallel postulate can hold to be true.

How can technology be used to effectively engage students with this topic?

Technology can be used to effectively engage students with the parallel postulate through a short series of YouTube videos by the channel Extra Credits. The five-part video series is called “Extra History: History of Non-Euclidean Geometry” with short seven to eight-minute videos which goes through the history of the parallel postulate. The video not only explicitly states what the parallel postulate is, but it goes through the history of how peoples’ conception has changed over time and how it has applied to today’s world and expands into physics.

The video series is produced with high-quality animation and narration. An engaging visual representation of the history of geometry that mathematicians have gone through to prove Euclid’s parallel postulate. Engaging in the countless trials and the amount of time that it has taken to go through this proof. Showcasing other discoveries that Euclidean Geometry has led to being Non-Euclidean Geometry. Lastly, the discoveries that Non-Euclidean Geometries will further lead to. Allowing students to join in on the questioning of the world as we know it.

Citations

Bryan, V., 2014. The Cult Of Pythagoras. [online] Classical Wisdom Weekly. https://classicalwisdom.com/philosophy/cult-of-pythagoras/

Szudzik, Matthew and Weisstein, Eric W. “Parallel Postulate.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/ParallelPostulate.html

Weisstein, Eric W. “Non-Euclidean Geometry.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/Non-EuclideanGeometry.html

https://mathcs.clarku.edu/~djoyce/java/elements/bookI/post5.html

https://www.youtube.com/watch?v=nkvVR-sKJT8&list=PLhyKYa0YJ_5Dj3ZG-Qk9VfaCfo-Nh9S-2

Predicate Logic and Popular Culture (Part 225): George Jones

Let $D(t)$ be the statement “I am dead at time $t$ ,” let $L(t)$ be the statement “I love you at time $t$ ,” and let $T$ be the set of all times. Translate the logical statement

$\forall t \in T(\lnot D(t) \Rightarrow L(t))$ .

This matches the opening line of arguably the greatest country song ever, “He Stopped Loving Her Today” by George Jones.

Here’s how to get rid of WordPress’s Block Editor and get the good editor back — nebusresearch

Many thanks to Joseph Nebus for publishing this very helpful post for all of frustrated by WordPress’s latest “improvements.”

So I have to skip my planned post for right now, in favor of good news for WordPress bloggers. I apologize for the insular nature of this, but, it’s news worth sharing. This is how to dump the Block Editor and get the classic, or ‘good’, editor back. WordPress’s ‘Classic Editor Guide’ explains that you […]
Here’s how to get rid of WordPress’s Block Editor and get the good editor back — nebusresearch

Predicate Logic and Popular Culture (Part 224): Robert Frost

Let $G(x)$ be the statement “ $x$ is gold,” let $S(x)$ be the statement “ $x$ can stay,” and let $H$ be the set of all things. Translate the logical statement

$\forall x \in H(G(x) \Rightarrow \lnot S(x))$ .

This matches the title of a Robert Frost poem, shown below recited in the movie “The Outsiders.”

HyFlex Teaching During the Pandemic (and Beyond?)

My university’s Vice President for Digital Strategy and Innovation asked me to write an essay on my experience with HyFlex teaching: simultaneously teaching students who are physically present in class while other students are at home (including those in quarantine). My essay was recently published on Medium, and I thought I’d place it here as well.

In Spring 2021, I was one of the relatively few instructors at my institution, the University of North Texas, who taught face-to-face classes. Several adjustments to face-to-face teaching were mandated by the university to mitigate the spread of COVID-19. Some of these requirements — for example, daily temperature checks with an oral thermometer, biweekly nasal COVID tests, mandatory masks for both instructors and students, spreading out 50 students in a lecture hall designed for over 300, and a prohibition against traditional small-group work — were prudent given the circumstances but will nevertheless be happily forgotten someday.

My university also required me to record my lectures so that any students in quarantine wouldn’t fall behind. Accordingly, I used Zoom and a document camera instead of my normal in-person method of using the chalkboard or whiteboard. Unwittingly, I found myself teaching in a hybrid flexible (HyFlex) style, a term I had heard of but had not experienced firsthand.

To my great surprise, this part of the COVID method of teaching an in-person class was warmly embraced by my students. Indeed, I’m deliberating how much of this COVID method to retain in my teaching practices long after the pandemic has passed. In this essay, I share some mechanics on these adjustments to my usual teaching style that I made as well as some thoughts on the efficacy of these methods.

While my own experience is limited to teaching mathematics courses, I hope that at least some of these thoughts might help instructors in other disciplines.

Zoom. My use of Zoom for in-person classes wasn’t that different from how I taught from home in 2020. Before the semester, I created a dedicated Zoom ID for each course and posted it on Canvas. In this way, students in quarantine could attend class from home or else watch a recording on their own time. (More on this later.)

In class, I displayed the image from the document camera onto the in-class screen. I “pinned” the image from the document camera at full screen, so that only the projection from the document camera appeared on the main screen at its maximum possible size. (Before doing this, I had to wait until a student joined from home.) I also used Zoom to record my lectures in the cloud.

On the document camera, I adjusted the magnification so that one page-width was projected. I used the auto-focus feature to ensure that my handwriting was clear and then turned off the auto-focus before class started.

Even though I consider myself tech-savvy, I went to the classroom early to check for possible technological problems. Last semester, I found that I’d have to reset something about half the time — usually flipping the screen video, choosing the appropriate audio channel, or increasing the volume so that I could hear any students asking questions from home.

Colored pens. I splurged and spent about $40 on a nice set of a dozen pens of assorted colors. I wanted to make teaching with a document camera kind of like teaching with a white board, where I’d have about four different colored dry-erase markers at my disposal. By rotating pens and not overusing any one color, all of my pens lasted the entire semester even though I taught two classes with them.

I didn’t use a strict color-coding system, but I tried to be somewhat strategic about switching pen colors. I changed pens every time I started a new example. When appropriate, I switched pens in the middle of an example to do a side calculation or to break up a long calculation into smaller pieces.

It seems insubstantial, but my students absolutely loved the colored pens. In my course evaluations, they emphasized that the use of different colors helped them organize their thought processes.

To give a concrete example, below are some representative notes that I made during one of my lectures on differential equations. (I promise there won’t be a quiz at the end of this essay!) Most of the black handwriting came from my prepared notes; everything else was written during class.

Math 3410 Lecture 20210223 Download

Student interaction. As a precaution against unfortunate oversights, I told my students on the first day of class to interrupt me — mid-sentence, if necessary — if I ever forgot to start the Zoom recording or else take attendance. Predictably, I made this mistake a couple of times during the semester, and I’m glad that my students pointed this out to me when this happened.

I’ve always been a walker when teaching my face-to-face classes, and I made a point of walking around the classroom as much as possible despite teaching with Zoom. I don’t like the feeling of being tethered to the classroom computer like I was when I taught from home in 2020. However, I necessarily had to stay close enough to whatever microphone I was using so that the Zoom recording could clearly capture my voice. For one class, this was a non-issue since I was in a large lecture hall and used a lapel mic to amplify my voice. For my other class, however, I was in a smaller classroom and did not need to use a microphone for my in-person students to hear me clearly. For this smaller class, I walked around less so that I stayed close enough to the classroom microphone and my voice could be recorded properly.

Because I was mostly looking at my paper under the document camera (when writing) or my students (when not writing), I often did not immediately see when a student from home posted a question using the Zoom chat feature. When this happened, my in-person students, to be helpful, would instinctively start pointing to the screen to alert me that somebody from home had a question. I must admit that I was a little unsettled the first couple of times that my students started pointing at the screen behind me — I thought something on the computer had crashed! In time, I eventually got used to my in-person students pointing at the screen at unexpected moments.

Dual screens. In Spring 2021, I taught in one classroom that had a single instructor monitor and other classroom that had dual monitors. For sure, the dual-monitor set-up was better for teaching my class with Zoom. As mentioned earlier, I chose the settings on Zoom so that the image from the document camera occupied an entire monitor which was then projected onto the classroom screen. I put all other windows on the second monitor — the Zoom participants, the Zoom chat window, the classroom clock, the Canvas attendance sheet, and/or any other webpage or application that I planned to show my students during that particular lecture (of course, I had to use screen-share so that students from home also could see these). I did not project the contents of this second monitor onto the classroom screens; I only projected one screen and kept the second screen for my own private use (unless I was screen-sharing something that day).

All of the above can be done in classrooms with a single instructor monitor, but it’s more difficult. To screen-share something with my class, I had to deactivate the full-screen option, pull up the screen that I wanted to share, screen-share, show the application, stop screen-share, and then restart the full-screen option.

Canvas as a repository. After class, I would return to the office, use the office copier to scan that day’s lecture notes in color and email the PDF file to myself, return to my office, adjust the order of the pages in the PDF document if necessary, and save the PDF file to my computer. (The above hand-written lecture notes were produced in this way.) By then, usually enough time had passed to receive the Zoom e-mail me with the link of that day’s Zoom recording, and then I would post both the Zoom recording and the PDF file to Canvas for my students. This process usually took about 10 minutes per class period.

Student perceptions. At the end of the semester, I surveyed my students about whether I should return to my usual teaching style (writing on a large chalkboard but unrecorded) or keep my Spring 2021 style (document camera and recorded) after COVID-19 becomes a distant memory. I fully expected my students to recommend using the chalkboard since much more information is visible at any one time on a chalkboard than on a document camera. There were plenty of times last semester that I had to creatively fold my papers in order to get information from two different sheets of paper onto the same screen, and that was a bit of a hassle.

However, to my surprise, my students said my use of the document camera was no big deal, and they absolutely loved having both the full recording of the lecture and also the PDF file of the hand-written notes that I made in lecture. I am definitely considering their advice for my future classes.

An unexpected problem: absenteeism. With 20/20 hindsight, I should have seen this coming before the start of the semester, but I’m afraid I didn’t. In both of my classes — and especially my early morning class — an appreciable percentage of my students rarely came to class when I wasn’t conducting an exam. A lot of these students didn’t even participate in the live Zoom sessions and presumably only watched the recorded lectures on their own time.

Because I took daily attendance of in-person students for contact tracing, I could correlate in-person attendance with final grades. The results were predictable. I did have a handful of high-flying students who attended class in-person less than half the time who nevertheless earned As for the semester. However, and unfortunately, the vast majority of students who chose this approach flunked my course.

As a teacher who cares for the success of my students (and who selfishly doesn’t want to be known for flunking lots of students), I have a dilemma. There certainly are legitimate reasons for a conscientious student to miss class on any given day — illness, family emergencies, car problems, unexpected heavy traffic, etc. When life throws such obstacles in my students’ way, being able to watch that day’s class asynchronously is a wonderful back-up plan. My students also told me that they appreciated being able to re-watch the lectures that they had attended in-person to remind themselves of how to do homework problems or to study for my exams. So there are legitimate ways that class recordings can be beneficial to students.

It’s also an unfortunate fact of life that recordings that could be legitimately used can also be abused to the detriment of my students. I don’t yet have a good answer for how to best prevent abuse of course recordings. I have dismissed a couple of options that I won’t be enacting, like giving the Zoom links only to students who I deem conscientious (which would be inequitable) or else only to students who ask for them (which would create many extra e-mails for me to answer).

For the moment, my resolution of my dilemma is that poor decisions by some students should not deter me from doing something that could greatly benefit others. My students are adults, and, like every other piece of technology that they have, it’s up to them to not abuse the class recordings that I’m providing to them.

Postscript: Student responses. Below are representative comments that I received from my Spring 2021 students about whether I should continue to use Zoom and a document camera to record my classes in the future:

I personally prefer COVID way of teaching because I was able to learn clearly using the document camera. In my past math courses if I did not have a great seat to see the board, or if the instructor’s handwriting was not the best, reading a chalkboard or whiteboard was super annoying and frustrating, especially during hard topics. Therefore, the document camera and using various of colored pens to distinguish the notes helped me follow along. I think that you should continue to use the document camera method because it can make it easier to read than in chalkboard or whiteboard, unless if the physical classroom restricts the ability to use that method.
Recording lecture videos and posting them could help students who may have to miss class due to appointments or other legitimate reasons. However, I see that students may abuse this. Making attendance mandatory is the only way I see that recorded lecture videos can be maintained. I still think that having recorded lecture videos will definitely help students who have to miss class for legitimate reasons, thus it will be easier for them to catch up on the learning material.
There were several times I was stuck on the homework but having your lectures recorded was a life saver because I could go over a certain example as many times as I needed. You would not be able to do this with the chalkboard method, unless you somehow recorded your lecture with a camera. However, if you were to do that I feel you might as well just do the zoom method. Also, I believe the zoom method actually saves you time in your lectures allowing for more examples or discussion. With the chalkboard you must erase all your work at a certain point and with the zoom method you can just pull up another piece of paper and continue seamlessly. Also the zoom method allows us to see the whole work involved with a problem from start to finish without you having to erase a certain part so you can write the next part.
The most helpful change was definitely making your scanned notes available to students. They allow me to review the material much more quickly and efficiently than if I was trying to find the information in a book or online, which allows me to spend more of my time actually learning the information and less time chasing it down. The recorded lectures are definitely a bonus, particularly if I had to miss class for some reason, but I wouldn’t say that they necessarily added to the learning experience. To be fair however, I am definitely more of a text-based learner than some of my peers and I rarely have patience for long videos, which definitely affects my opinion. All in all, I would say that having hard records of the materials covered in class definitely eases my mind, as I spend less time worrying that I missed some important detail in the lecture.
I have difficulty focusing sometimes so having access to recordings of the lectures and completed notes helps me make sure to get as much out of your lessons as possible. The recordings were also convenient for studying purposes when I needed a refresher on how to solve certain problems. I’m also a big fan of how you used multiple colored pens for your notes which made steps easier to track as well as remember. I also liked how you would take breaks away from the projector to move around in front of the class for explanations or anecdotes, the change of pace helped lectures not seem so monotonous. While I don’t think you’d want to teach your classes exactly the same next semester, I do think you should carry over some of the methods you used this semester that were helpful to students and had positive impacts on their grades.

In the interest of full disclosure, there were a few dissenting comments encouraging me to instead return to using the chalkboard or whiteboard after COVID, but even these students encouraged me to figure out a way to record myself when writing on the board.

Predicate Logic and Popular Culture (Part 223): Daniel Caesar

Let $N(x)$ be the statement “You need $x$ ,” let $G(x)$ be the statement “I will give you $x$ ,” and let $H$ be the set of all things. Translate the logical statement

$\forall x \in H(N(x) \Rightarrow G(x))$ .

This matches a line from the song “Too Deep to Turn Back” by Daniel Caesar.

Engaging students: Finding the area of a square or rectangle

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Austin Stone. His topic, from Geometry: finding the area of a square or rectangle.

How could you as a teacher create an activity or project that involves your topic?

There are many applications to the real world that involves geometry and specifically area of squares and rectangles. Students could use this topic to find the cheapest cost of tiling the floor of a bathroom. Giving them the dimensions of the different tiles and the cost of each tile, students would have to find the area of the bathroom floor and then be able to pick the set of tiles that would be the most efficient and cheapest. This gives students a real world application to what they are learning while also giving them practice in finding the area given dimensions of a square and/or rectangle. This project also calls back to prior knowledge such as perimeter of rectangles and multiplying cost of one tile with the number of tiles used to get to total price. This project could also be a small part of a bigger PBL using area and perimeter of multiple polygons.

How does this topic extend what your students should have learned in previous courses?

The obvious prior knowledge to finding the area of a square of rectangle is being able multiply two numbers which is learned back in grade school. If the students are given the area of the square or rectangle and labeling the sides with a variable, the students would have to be able to solve for the variable. By doing this they would have to be able to multiply binomials (or polynomials if you want students to have more of a challenge). Once they multiply the two binomials and set the equation equal to the area given, they would then have to use the quadratic formula or factor which is learned in Algebra I. If students are given one side and the area, then they would have to solve for a variable with degree one which is used continually in all math classes. Depending on what information is given in the area problem, students will have to use prior knowledge to determine the answer.

How have different cultures throughout time used this topic in their society?

In East Asian mathematics during the 1^st-7^th centuries, a book called The Nine Chapters gives formulas for solid figures including squares and rectangles. The formulas are given as series of operations to get the result, called algorithms. Instead of variable and symbols, the formulas are given in sentences as in, “multiply the length of the rectangle by the width.” This puts the regular A=lw into words so that if someone who had no idea how to compute the area, they would be able to understand by the sentence given. This undoubtably was much more difficult to follow and became too long of descriptions for more complex figures, as this way of mathematics ended in Eastern Asian in the 7^th century. That does not mean that this way of math was not important. This put words into formulas instead of symbols which made it easier to understand for those that are learning it for the first time.

References

https://www.britannica.com/science/East-Asian-mathematics/The-great-early-period-1st-7th-centuries

Predicate Logic and Popular Culture (Part 222): The Notebook

Let $B(x)$ be the statement “ $x$ is a bird.” Translate the logical statement

$B(you) \Rightarrow B(I)$ .

This matches a line from the movie “The Notebook.”