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...

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...

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

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. TO BE CONTINUED...

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

Date: 

Coming soon...

What on Earth are "Sine" & "Cosine"?

Date:

Level of difficulty: High School/College

Coming soon.....