Sheyenne National Grassland in North Dakota

Should We Go With the 8-Quart Pot or the 12-Quart Pot? 

Date: 10/01/2026

Level of difficulty: Middle School/High School

Recently, I was asked this question. A family member was debating which size to get for cooking soups, preparing corn on the cob, and so on. Naturally, the first question that popped up in my head, and I suspect most others, was - well, what is a quart? Like anyone reading this right now, you may start trying to remember - is 1 quart equal to 4 cups, or is it 8 cups? Or maybe 1 quart is 16 ounces, yeah that sounds right, maybe? No matter what chain of thoughts were triggered in your head, inevitably your mind likely defined a quart in terms of some other unit (for example, cups or ounces or something else). But then that begs the question, what exactly is 1 "cup", or 1 "ounce"? Is it even possible to make sense of this made-up word "quart" without referring to some other unit of measurement? Hmm...These are interesting questions, assuming we don't run to ChatGPT or Gemini or some other robot. So let's explore.

The first one I'm curious to ponder is could I explain to an alien what a "quart" is without using any other units of measurement as a reference point. It would be difficult right? Imagine on our planet, thousands of years ago, or seconds ago but many many seconds ago, human apes roaming around, without the English language, I can't imagine them stumbling across anything in the natural world begging to be called "quart". But maybe, over time, one of these human apes became the chef of the community, and learned to perfect certain dishes. And for one of his recipes, there was the "perfect" amount of water needed for preparing his most beloved food. He knew the exact amount from repeatedly preparing the dish using a wooden pot the community had carved out for him. But when training others on how to prepare his dish, he couldn't just point at water and gesture to add it. Maybe one human ape would have added a drop, but another one might have interpreted "add water" and dumped in a whole bathtub of water. So the master chef knew he had to be more precise. He took his wooden pot, carved a little mark near the top, and communicated to his students to use "this" amount of water. If we pause this human ape story for a moment, we now at least know the purpose behind creating a word such as "quart". It helps us to describe an amount, or a volume, of something. So instead of the ape just pointing repeatedly to his carved wooden pot, for simplicity and efficiency, he could utter the word "quart" and other apes would immediately have a rough sense of what amount of liquid he was referring to. To be fair, he could utter any word, or sound for that matter, as long as that word/sound was the sole one designated for that specific amount of liquid.

With this in mind, maybe the best we can do then in our alien hypothetical where we are tasked to explain what a quart is without referencing any other units is precisely what our head master human ape chef did. Better yet, maybe beforehand, we could pour water from our chef's carved pot with the little mark on it into 100 containers all shaped differently. Some narrow and tall, others long but short, whatever weird shape we can think of. Then we tell the aliens this amount of water is what we call a "quart", and we have them pour the liquid from the 100 different containers into the chef's carved wooden pot, one at a time then emptying then repeating, so the aliens got a clear visual sense that a "quart" is this amount of water. No matter the shape of the container, once poured into the chef's wooden pot, the amount of water would always be the same and fill up right to the same mark. 

This does beg the question - can we be more precise then just referencing the human ape's carved wooden pot with a little mark on it? If I google "How much is a quart" what comes back is: 1 quart = 4 cups; 1 quart = 2 pints; 1 quart = .25 gallons. But than we are stuck in a recursive loop, because what is a cup, or a pint, or a gallon. More words, same problem. However, there is one definition, which yes, does use another unit for a reference point, but this time I can actually visualize and imagine what it physically looks like (whereas, I have no clear idea what a gallon is for example). That definition is, 1 liquid quart = 57.75 cubic inches. And as side-note, yes, I did have to google this, there's no way to deduce this on one's own, units of measurement are arbitrary and defined by humans. Now let's try and make sense of this: 1 liquid quart = 57.75 cubic inches. At first glance, we are stuck in the same predicament - what is a cubic inch. However, unlike the words gallon, cup, liter, and so on, a cubic inch is something easily imaginable by most people immediately. It's a three-dimensional cube that measures 1 inch tall by 1 inch wide by 1 inch long. Therefore, a "quart" is simply the amount of space that 57.75 of these cubes take up. Two liquid quarts would take up 115.5 of these cubes, and so on. Now I can confidently think about and imagine what a quart is because it is clearly defined in terms of a well-defined visual object. That's the difference between our ape's wooden pot with the mark on it, and a cubic inch, the cubic inch is a well-defined universal size that people on different sides of the planet can immediately agree upon and imagine in conversation, whereas the ape's wooden pot with the mark on it is not universally defined. Imagine the difficulty the ape would have on the phone trying to describe what a quart is to another ape on the other side of the planet using his wooden pot with the mark on it, it could be done in a sense, it would just be more challenging and require more time when compared to using the cubic inch definition. 

Someone reading this might be thinking "That's all well and good, but what's an 'inch'?". Fair enough. Does it ever end? I'll leave that for you to ponder...

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Appalachian Trail on Whitetop Mountain in Virginia

The Village Kid Who Needed Help With Math. Personal Struggles With Modern-Day Education

Date: 8/28/2026

Over the years, I have had this one particular imagery repeatedly come into my mind that seems to encapsulate what I am looking for when it comes to teaching math and helping others. The scene is relatively simple – I am sitting on a bench in some remote village, I overhear a child walking by who is struggling with some math question from school, in that moment I offer whatever I can to help the child (with his or her homework, confidence, understanding, etc.), then we part ways and continue on our day. For some reason, this scene tends to play over in my mind frequently, and I often find myself describing it to others when asked about how I want to offer my math skills to others in need.

As someone who has worked in a traditional classroom, I have encountered every frustration and limitation that most educators inevitably face teaching a room full of different students, with different interests, different abilities, different home situations, and with a curriculum that is artificially imposed on them that must adhere to a contrived timeline to fulfill certain educational “requirements”. It’s a nightmare right? How did we get into this situation? What meeting took place to decide what’s considered “important” for students to learn and who are these people making decisions for billions of the rest of us? Are the flaws of the system buried too deep that real change is next to impossible to achieve? When I first started teaching, like many I suspect, I was filled with energy and determined to make a difference and offer my best to my students. I wanted to share what I personally enjoyed about math and tried everything I could to bring the subject to life for others. I was particularly interested in challenging the perceptions of those who had preconceived notions of not liking math or rigidly assuming they’re bad at math, I wanted to show another side of math that perhaps they could actually connect with. And while, overall, I did have the support and encouragement from my department heads to “do my thing”, I could not help shake this feeling that the momentum of the current trend of math was so strong that my efforts to try something different felt inconsequential. It was as though there’s this raging river going downstream, and I was just this one lone person trying to go upstream, I had no chance (so it seemed). Inevitably, the weight of the current got to me and I became tired, but rather than go down with it and sell my soul so to speak, I got out completely and took a break from teaching math. It’s not easy, when the world around you is doing one thing, to go against it and do what feels true and right for you.

So what exactly didn’t feel right. Before I attempt to answer that, I need to be clear that I am not assuming my personal thoughts and feelings on the subject are undeniably correct while others’ views are absolutely wrong. Furthermore, my feelings quite naturally evolve and change over time, so these reflections are less about some absolute statement set in stone on what’s right and wrong in math education, but more about one human being sharing his personal experience and feelings in a particular moment. So, with that in mind, let’s start with my perspective on where was this current of math education going and what was driving it. As I sit here reflecting, I notice one word that keeps popping into my head “answers”. In the most simple terms, this raging current of math seemed to be driven ultimately by one thing – getting answers. Not understanding concepts, not cultivating creativity, not asking deep questions, not persevering despite struggling, but simply getting answers. As long as the correct answer was produced, we were happy and conditioned to think we were doing a proper job, whether or not the student actually understood the meaning behind the numerical derivation to the problem was largely ignored. And it almost didn’t even matter if the student truly understood the concept, as long as the answer was reached, we rushed onto to the next thing, and if that answer was reached, again with or without understanding, we rushed to the next question and so on.

Snowy Day At Doctor's Park

Something about that culture didn’t feel right to me. Was there much value in someone being able to produce a numerical answer but be unable to truly understand the meaning behind the answer and how they got there? Some of the most memorable research I encountered in graduate school was from an experiment where it was clearly demonstrated that there was not necessarily a direct correlation between getting A’s on tests and understanding the concepts tested. In other words, many students were getting A’s on their exams, but when pressed about the concepts they had supposedly mastered and asked questions that tested more of their actual understanding they struggled. To be honest, I was one of those students. I could get A’s on most my math tests, I could produce answers in algebra and trig and calculus in machine like fashion, but what my correct answers and high grades weren’t revealing was that in a certain sense I actually had no idea what was going on. That would change in graduate school when I encountered two of the most terrifying professors I’ve ever had, Dr. Thompson and Dr. Saldanha. I still remember, every night before class, being scared and anxious as I walked into the classroom because for the first time in my life, producing answers wasn’t good enough, I had to understand the math and be able to talk about it and explain it in plain English, let alone in front of experts who understood it deeper than I did. It was humbling, terrifying, and ultimately the most meaningful and influential time in all of my educational journey. I finally became confident in math on a deeper level, because not only could I derive answers, but for the first time ever I actually understood what was going on. Furthermore, and crucially, I also became more comfortable not knowing things at first and was no longer ashamed to admit I didn’t understand something because overtime I learned to trust my ability to patiently question things and reason my way through to some level of understanding, no matter how long it took.

This leads us to the other dominant force in this current of present-day math culture – the curriculum and its artificially prescribed timelines for understanding concepts at a given age/grade level. I wonder if this is where eventually teachers recognize the ship is sinking in the middle of the quarter/semester but simply go down with it. You know for this kid the material is way too easy, for that kid it’s just right, and for the kid in the back it’s hopeless. Yet, we continue to march forward because….the curriculum says we must. “This many weeks in, students should master concept x, that many weeks in, students should master concept y” and so on. Imagine someone walking up to Albert Einstein “Hey, the board had a meeting and we’re going to need your E = mc2 equation by the end of October”. That’s clearly not how meaningful insights and understandings arise, they don’t adhere to some artificial child-like timeline, they arise organically when the time is right. And when they finally do spontaneously come, they really stick and are of lasting value. I wish in some world students could have their own mini-Einstein experience with whatever concept interests them. They could “chew” on some concept again and again, keep pressing and questioning, keeping getting close but not quite there, and then, one day, the “Ah ha” moment finally hits them and they feel a tremendous sense of satisfaction because they persevered and developed their own unshakable understanding through their own efforts. That’s powerful, to me anyway. And it’s not to dismiss the curriculum entirely or say it’s futile, but just me imagining the best learning experience for a student to have. Of course, a curriculum can provide some general necessary direction for a quarter/semester. I only think we should be more flexible with it and use it to steer students into experiences that provide satisfying and long-lasting insights. Imagine someone walking up to Einstein in the middle of his E = mc2 struggle and saying “Sorry sir, we’re going to have to move onto the next topic to keep things moving”, completely depriving him of the powerful experience that awaited if he only was allowed the time to persist a little bit further. Curriculums can strip away that extra time, leaving students in the middle of conundrum that potentially could yield a lot of growth, to simply plow ahead to the next concept. No wonder so many students are lost in math, we keep marching forward compounding their sense of confusion. But if we can somehow give students the time they need to develop their understanding, then I would argue something meaningful is happening. But the trick is it has to be on their time. Einstein could not force himself to have figured out E = mc2 by a certain deadline, just as kids should not be expected to have a deep understanding of whatever it is their learning by a specific deadline. As I have repeated many times before, we are all different and learn at different rates. That shouldn’t be a problem but rather seen as completely normal and understandable. My hope is that educational systems somehow find ways to give students the time and space they need to work hard, persevere, and reach their own understandings.

So what does any of this have to do with a village kid needing help in math? I think the reason that imagery pops into my head frequently is because in that world I am removed from the weight of a curriculum and am simply responding to a real need in the present moment without any pressure to move onto the next artificially imposed thing. I simply get to help a student who has a pressing need, offer them my best, and once the problem is resolved, whether it took a few minutes or a few hours or a few days or a few months, we go our separate ways with no forced need to cover the next thing. That’s what I am ultimately looking for, responding to real sincere needs and offering whatever I can to help.

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Climbing in Canyonlands National Park in Utah

What Climbing Taught Me About Learning Math

Date: 7/22/2026

I wonder how often, while a teacher is explaining something out on a whiteboard, there is a quiet student in back, silently thinking sincerely about the question at hand, reasoning in their own organic way in a manner different, maybe completely different, than the “expert” in front, only to ultimately lose interest and give up as the teacher continues to talk, more or less to themselves, and demonstrate their approach to the class. The quiet kid then resumes doodling in his or her journal, perhaps sneaks in a glance or two at their smartphone, and all the while the deep-rooted thoughts “I am bad at math” “I am never going to understand this” “I am stupid” remain in the back of the child’s mind. The student’s unique thoughts are never given the light of day, lost possibly forever. What a tragedy. I am convinced, if we gave students a chance to share and explain their sincere thoughts on how they are thinking about something, we might be in more awe than we anticipate. I would even argue that regardless of whether or not their thinking is 100% logically correct, the mere honesty and purity of their thoughts is worthwhile and likely has some truths in them worth exploring. And how much more richer and fun would the classroom be, instead of just one approach towards math, now you have some variety. Maybe the teacher’s approach for one particular concept works for 30% of the students, the quiet kid’s approach makes sense to 50%, and some other kid’s unspoken method works for the remaining 20%. We are all different. We have different backgrounds, come from different cultures, have different interests, have different priorities in our family, why on earth should we all be expected to think alike?

So what does all this have to do with climbing? Climbing, more specifically bouldering, is a discipline where one is presented a “problem” involving some combination of “holds” for your hands and feet on a wall that is roughly 12-16 feet tall where the sole purpose is for the individual to reach the top hold. Side note, if you have never tried climbing/bouldering, and you have the faintest curiosity to do so, I highly recommend it. Similar to problems in math, there are thousands, truthfully endless, amounts of boulder problems with a wide range of difficulty. Also similar to math, in bouldering you are essentially provided with a set of rules you must operate from, rules of gravity and physics, and within those boundaries, you try your best to find a way to transport your body to the last hold. The basic nature of solving a boulder problem is very similar to solving a math problem: you start with an uncertainty, you apply reasoning and perhaps trial and error in an attempt to solve the problem, you fail, you try again, you get help from others if possible, until eventually you “solve” the problem on your own. From personal experience, the satisfaction of solving a challenging math problem is similar to the feeling of solving a boulder problem. There’s a child like sense of “I did it”.

Me Climbing at Balance Prime Boulder Gym

What I think is missing in the math world, which is encouraged in the climbing world, is “finding your own way” when trying to solve an interesting problem. In fact, in the climbing world the more interesting, unique, and weird the approach towards the problem, often the more admiration and respect the climber receives from others. And from the climber’s perspective, there is something incredibly joyful about solving a problem in a unique and unexpected way that others doubted was even possible. Whereas in the math world, many of us may ignorantly, albeit understandably, assume that the problems presented to us have only one way to solve them. “Slope is change in y over change in x, that’s all there is to it”, and the concept of “slope” then becomes dead forever. “For, sine, cosine, and tangent, just remember SOH CAH TOA, that’s all you need”, and the potential richness of sine, cosine, and tangent have now lost their life. They imagine math as some static thing, devoid of creativity and new potentials – this is how the teacher did it, so this is how it must be done. If only we could start approaching math problems with a similar attitude of how climbers approach bouldering problems, then maybe the discipline of math may start to become more alive to others.

Recently, I attended a bouldering competition at my local climbing gym where eight climbers attempted to solve four different boulder problems. It was while observing the climbers on the first problem that the idea for writing this article was born. As I watched the athletes attempt to solve the problem, I noticed ALL eight of them tried to solve the problem in a unique manner. And the more unconventional and surprising the attempt, the louder the response from the crowd. “This is exactly how math should be”, I thought, “people exploring and trying out what makes sense to them, and refining it if it does not ultimately work”. Some climbers would try one method for a few attempts, then play around and try a new method, only to go back to the original method. They were given the freedom to find their way to explore what works and what does not. Some made it to the top, some did not. Regardless, it was clear all learned something from the problem and got stronger as a result of their sincere effort.

Afterwards, I reflected on how the uniqueness of each climber’s “beta” (climbing lingo for one’s particular method of getting to the top) for certain problems made perfect sense given their different heights, arm spans, flexibility or lack thereof, preferred styles, fear levels, and so on. Isn’t it not similar in the math world? Don’t we all have different learning styles, strengths, interests, and ways of thinking? Shouldn’t students be allowed to explore, even if it ultimately does not work, what their intuition is telling them rather than be put down a narrow track that only works well for some? I hope our math culture one day can shift more in the direction of our climbing culture. One that thrives on interesting problems, encourages and rewards unique problem solving, and ultimately is about trying hard and having fun...

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Yangmingshan National Park in Taiwan

COVID, A Compost Bin, & My Approach to Math 

Date: 7/6/2020

Recently I was building my own compost bin for a garden, one of my many pandemic projects, and in this process I reached a point where I needed to determine how much paint would be required to cover a cylindrical 55 gallon plastic barrel drum, in other words a fun math problem. I researched the type of paint needed for exterior plastic, settled on the color “Golden Sunset”, and upon reading the label discovered one can of paint could cover 12 square feet. The stage was set, I had a fun and practical math problem.

My main task was in determining the surface area of this cylindrical object. Hmm, that is an interesting question I thought. Now of course, I could have used the surface area formula for cylinders, A = 2π *r*h, or try to Google my way to an answer, but I question whether or not that would have satisfied me or been all that rewarding. I much preferred to look at this afresh, to experiment, and most importantly, to think for myself and rely on my own curiosity and creativity. That seemed much more fulfilling and valuable, there was a certain freedom and excitement to think for myself, and I was confident that whatever I happened to learn along the way would be that much more likely to stick afterwards.

So, with a playful attitude, I started to ponder different ways of determining the surface area of this barrel. Initially I thought, if this were a rectangular sheet of plastic, this would be a straightforward problem. I could measure the length, then the width, multiply the two, and voila, I would have the area. But this barrel was cylindrical, not rectangular. However, I thought it is simple enough to measure the length of the barrel, and I sensed having that information would be helpful, so with measuring tape in hand I determined the length to be approximately 3 feet. Halfway there I thought! But I was still looking at a cylinder, not a rectangle. Thinking, thinking, thinking. Then, quite organically, I started to imagine taking a rectangular sheet of paper and rolling one end of it to the other, in effect creating a cylinder. Then if I released my fingers, that paper would return to its rectangular form. Then it dawned on me, the area of the paper has not changed! Whether I rolled it up from one end to the other creating a cylinder, or laid it out flat as a rectangle, it was the same area. Furthermore, and crucially, I realized the distance around the top of the paper cylinder was the exact same distance of the width of the paper when laid out flat (try it for yourself!). 

With confidence, and excitement, I grabbed some string, wrapped it around the barrel, then laid the string out flat and measured it to be roughly 5 ½ feet. I now had, in a sense, the width of the barrel (if it were to be rolled out flat like the paper). I grabbed a calculator, multiplied this width by the barrel’s length, and with a sense of satisfaction determined the surface area to be 16 square feet, no Google necessary. Feeling that healthy sense of accomplishment that comes through preserving and discovering something through one’s own efforts, I finally knew how much paint I would need for my compost bin.

Now the reason I went through the trouble to describe that experience in such detail is that it quite accurately captures my spirit when it comes to math, and also the type of learning experience I aspire for students to have for themselves. Math, above all, is a medium. A medium for students to be curious, creative, ask good questions, and persevere. The more I see a student engaged in and growing in any of these areas, the more I know I am doing a good job. In fact, the most memorable experiences I have had as an educator are the ones where I am working with a student, typically low in confidence, and am able to guide and witness them persevere through their doubts, tap into their own curiosity and creativity, and ultimately reach the understanding and answer they hunger for. And most importantly, they did it themselves. I was merely a guide providing encouragement and support. Personally, it is these types of experiences I find most rewarding for both student and teacher. 

My interest in teaching math in a more enriching way began in graduate school. I was fortunate enough to take a year-long sequence in Math Education with some great professors and researchers, including the well-known math education expert Dr. Pat Thompson. Through many long hours of discussion, countless readings of a diverse body of math education research papers, and my own experimentation with teaching as a Teaching Assistant, a new door was beginning to open for me as far as what I saw possible with teaching math. Gradually, I began to see math as much more than simply reaching a final answer. It was becoming an opportunity for students, every type of student, to really think,explore, get creative, and ultimately come to realize they too can genuinely understand concepts and abstract ideas (it was not limited to just the math teachers). Math was beginning to become more alive for me and I found myself motivated to bring it to life for others.

Sharing this newfound appreciation for math has evolved over many years. I have had extensive opportunity and hands-on experience teaching math to a diverse body of students and throughout this ever-evolving process, I have learned a lot, failed a lot, tried and tried again, and ever-so gradually have fine-tuned my craft to create, as best I can, the most meaningful experience possible for students. Over time, learning to work with each student’s unique way of thinking and striving to steer it in positive directions, rather than rigidly imposing my own thinking or a textbook’s onto them, which may simply not be a natural fit depending on the student. It is a skill that has matured over countless hours of hard committed work, and it is something I am personally quite proud of. 

Finally, back to the compost bin. My whole inspiration for teaching math again was born from sharing this compost bin math experience with a former 3rd grade teacher, who is self-proclaimed “bad at math”, and watching her reach the same insights that I had, and doing so with confidence. There and then I thought my uniquely developed skill set, coupled with my joy for working with struggling students, may be of value to others and serve a genuine need...

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My dog Red on the Ice Age Trail

When Dividing Two Fractions, Why Do We Actually Multiply by the Reciprocal? 

Date: 5/16/2025

Level of difficulty: High School/College

Recently, I was watching the Studio Ghibli movie Only Yesterday and there was a scene that aroused my curiosity regarding a common math “trick” for dividing two fractions. In the scene, a young girl had failed a math quiz and was being scolded by her family, resulting quite naturally in her feeling lots of shame, and perhaps even worse the beginnings of the idea that she was stupid was beginning to take root in her mind. However, as the older and “smarter” sister of this young girl sits down and tries to explain how to divide two fractions, it becomes clear, to me anyway, who is actually the smart one.

Repeatedly, the older sister confidently exclaims “when dividing two fractions, just flip the top and bottom of the second fraction and then multiply”. The younger curious girl sits there, silently wanting to know - but why. The older one proceeds to do dozens of examples, executing the procedure to perfection, but the younger one just sits there even more lost and withdrawn. Eventually the younger one sincerely asks “Why would you divide a fraction by a fraction?”. Grabbing pencil and paper she starts drawing a picture of an apple and says “dividing two thirds of an apple into quarters means you take two thirds of the apple and split it into four ways…how much apple does each person get?”. She concludes 1/6 which in a sense is understandable given how she has been taught division. But the actual answer, 8/3, does not make ANY sense to her, especially since division has always been associated for her with resulting in a smaller number, not bigger.

The scene affected me and highlights the difference between certain types of people. Some learners are content to execute a procedure without much understanding as to how it works and why (which is fine if that is your style). Others seem to have a thirst for genuine understanding and need to understand clearly why something works. I tend to fall in the latter camp, so I paused the movie and, like the young girl, grabbed pencil and paper and started thinking deeply.

Before sharing how I think about it, I am curious how many people reading this can honestly say they understand why you actually multiply by the reciprocal of the divisor when dividing two fractions. Everyone of my friends who I posed this question to eventually admitted they had no idea (just like me at first). So, let’s dive in.

The first question I asked myself is - what does it mean to “divide” two numbers. To begin this inquiry, I wrote down a/b = c. In words, we can interpret this as there is a number a such that when you “divide” it by b it produces the number c. Working from a/b = c, I can multiply b to both sides of this equation resulting in a = c*b. Therefore, one, emphasis on one, interpretation of division is that when we divide two numbers it produces a number such that when I multiply that number by the divisor (in kid language, the “bottom” number) it equals the dividend (the “top” number). By means of a simple example, if we are trying to solve 12/4 we must determine a number such that when I multiply that number by 4, the result is 12. What number multiplied by 4 results in 12? With a little thought, we see the answer is 3. So, 12/4 = 3 because 3*4 = 12.

With this in mind, let’s finally explore dividing two general fractions a/b ÷ c/d. We know the result of this division operation is some number, let’s call it e, such that when I multiply that number by the divisor, c/d, it equals the dividend, a/b. In symbols, this translates to a/b = e*c/d. Since we are interested in determining the number e by itself, we can multiply both sides of the previous equation by d/c to isolate e. This results in, a/b * d/c = e. In other words, we are multiplying by the reciprocal. But rather than starting blindly from that place and simply obeying orders, we are actually ending there as a result of a deeper understanding of what it means to divide two numbers...

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Sunset along the Milwaukee River

Degrees Versus Radians. What's the Difference? Why Learn Both? What About Dungs Dungs?

Date: 7/15/2026

Level of difficulty: High School/College

First question, what is a degree? We know a degree is a unit of measurement, but what exactly does a degree measure? Just now to help me try and make sense of it, I held up my pen parallel with the floor, kept the left end centered (it did not move), and started rotating the right end up and around. Each time I stop somewhere in this process, the amount of rotation is different. For example, when I am simply holding the pen in my hand parallel to the floor, there is zero rotation, but the moment the right end of the pen starts to travel up and around, there is clearly an amount of rotation occurring, and that is precisely why we invented this unit of degrees to try and capture that amount.

For units of measurement, such as degrees, to make any practical sense to us, they need useful physical reference points which we can then assign numbers to (numbers of our own choosing by the way). To that end, let us consider two amounts of rotation. One good place to start is an object that does not rotate (in our example, this would be the amount where the right end of the pen does not move at all, it simply stays parallel with the ground). We need to assign a number to capture this amount of rotation. Technically, there is not a right or wrong answer, we would just need to remember that whatever number we assign refers to the physical reality of zero rotation. However, practically speaking, most humans would agree and settle on the number zero to describe an object which exhibits no rotation (in our case, the pen did not rotate so we can call that amount “0 degrees”). But now we need a second physical reference point that captures another amount of rotation. One natural amount is the amount where an object rotates all the way around completing one full rotation (in our case, that would correspond to rotating the right end of the pen all the way around until it is back where it started). Now we need another number to capture this specific amount of rotation. Unlike the first amount that had zero rotation and more or less is universally agreed upon to be called “0 degrees”, the number we assign to one full rotation is not as clear. If we surveyed thousands of kids who know nothing about trigonometry what number to assign to one full rotation, I wonder how varied, understandably, the responses would be. I suspect we may see answers like 1 or 100, and of course random numbers like 423342. And importantly, none of their answers would technically be wrong, because it is we humans who invent units of measurement to make sense of the world. They are not things set in stone. The ones we see today are the ones we just agreed to, but maybe on some other planet, the alien adults agreed on other definitions of units to make sense of measurable things in their world (hotness/coldness, heaviness/lightness, tall/short, etc.).

Before assigning a number to this amount of rotation, notice how these two reference points (zero rotation and a complete rotation), and everything in between them, captures the entire spectrum of how much something can rotate. We could think in these terms,

1 full rotation = some number x

½ of a rotation = ½ of that number x

¼ of a rotation = ¼ of that number x

1/100 of a rotation = 1/100 of that number x

34/98 of a rotation = 34/98 of that number x

and so on…

Now most of us, without probably ever questioning why, just automatically assume 1 full rotation = 360 degrees. It’s not right or wrong, it’s just the number humans agreed upon for this made up unit “degree” to capture 1 full rotation. Maybe at the dinner discussion years ago while debating what number to assign for this unit of degrees, instead of someone shouting 360, someone could have shouted 500 and the math would be equally as sound. For example, “250 degrees” would correspond to half of a rotation (instead of 180 degrees), “125 degrees” would correspond to a quarter of a rotation (instead of 90 degrees), and so on. The main thing to appreciate is that the number we traditionally use, 360, to designate a full rotation is completely arbitrary. It comes from ancient Babylonian mathematics and astronomy where they approximated 1 year to take 360 days (ie 360 days for the earth to complete 1 rotation around the sun).

So we know 0 degrees corresponds to no rotation, and 360 degrees corresponds to 1 full rotation. Now a good question is, what does “1 degree” actually mean in terms of rotation? To make it concrete, what would “1 degree” of rotation mean in terms of our pen rotating, what would that actually look like? One way to think about it is that it’s the amount such that if rotated that same amount 359 more times, the pen would complete 1 full rotation. Take a moment to really think about this until you’re convinced of its truth. Admittedly, 1 degree is a bit hard to mimic with extreme precision on a pen, but what about trying to to make sense of “90 degrees”, what would the pen’s rotation look like then? Well, one way to think about “90 degrees” is that you have ninety 1 degrees, in other words, we repeat the amount of rotation for 1 degree ninety times (still a bit hard to be precise). Another interpretation of what “90 degrees” looks like, is to consider what amount 90 is of 360 (ie 90/360). The reason behind asking this is that I know 360 degrees = 1 full rotation, now if I can make the left hand-side of that equation equal 90 degrees somehow, I will simultaneously know the amount of rotation that number corresponds to. Using basic algebra, 360*x = 90 => x = ¼; to maintain the balance of the equation we have: 360*¼ degrees = ¼ of 1 full rotation (ie 90 degrees = ¼ of 1 full rotation). As one last example, let’s try to make sense of what “120 degrees” looks like in terms of the amount my pen should rotate. Using the same strategy, I know 360 degrees = 1 full rotation, but I am investigating 120 degrees and need to know what amount that is of 360, therefore we have 360*x = 120 => x = 1/3; maintaining the balance of that core relationship we have between degrees and rotation we have: 360*1/3 = 1/3 of 1 full rotation (ie 120 degrees = 1/3 of 1 full rotation). We can apply this understanding to make sense of any amount of degrees and the corresponding amount of rotation it represents.

Now let’s move onto the dinner table next door where instead of someone shouting “360 degrees should represent 1 full rotation”, at this particular gathering someone shouted “2π radians should represent 1 full rotation”. What on earth is a radian and why 2π? Hopefully by now you know the term “radian” is completely made up and the choice of the number 2π is completely arbitrary (it is related to the fact that the circumference of a unit circle is 2π units). What we need to keep coming back to is the physical interpretation of what 1 full rotation looks like regardless of where you are in the universe, the number and made-up word we assign to that universal physical reality are arbitrary afterthoughts. In fact, our reasoning above about “degrees” will be the exact same when we reason about “radians”. Instead of reasoning with the relationship 360 degrees = 1 full rotation, we’ll instead use this newly defined relationship 2π radians = 1 full rotation. So what does it look like if my pen rotates “π radians” for example? Using the same reasoning we employed with degrees, we need to determine what amount π  radians is of 2π radians. Using basic algebra, 2π* x = π => x = ½; maintaining the balance of that core relationship we have between radians and rotation we have: 2π * ½ = ½ of 1 full rotation (ie π radians = ½ of 1 full rotation).

What about “1 radian”, what does that look like in terms of rotation? Well, we need to determine the amount of 1 radians in 2π radians => 2π * x = 1 => x = 1/2π. Maintaining the balance of that core relationship we have between radians and rotation we have: 2π * 1/2π = 1/2π of 1 full rotation (ie 1 radian = 1/2π of 1 full rotation). This may seem a touch complicated, but if we consider that the number 1/2π, is in fact just a number that roughly equals .16, this .16 can simply be interpreted as a 16% rotation (which most of us can have a rough estimate of what that would actually look like).

So is there anything special about “degrees” or “radians”? Short answer is no. At a third dinner table, someone could’ve shouted “2880 dungs dungs should represent 1 full rotation”, and generations to follow would unquestionably answer questions about dungs dungs (that logically are equivalent to degrees and radians). What personally helps me make sense of these units is to continually remind myself that these made up terms and arbitrarily assigned numbers refer to some concrete physical reality. With all that said, hopefully you are thoroughly convinced that sin(240 dungs dungs) = ½…

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