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About

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Welcome

Welcome to **SolidAngl.es**! My name is Bill Shillito, and I'm a math educator living in Marietta, Georgia. I've been helping students learn mathematics since 2009, and every day that I teach, I learn as well. I decided to start this site to share what I've learned and my thoughts on how to make math education even better.

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My Philosophy of Teaching

#### Teaching Techniques

#### There’s a lot of debate these days on the “right” way to teach. Should one be a “sage on the stage” or a “guide on the side”? My answer is “yes!” I engage my students with multiple modes of instruction — one minute they may be learning definitions and theorems involving matrix multiplication through a mini-lecture, but the next they may be moving around the room and rolling dice to simulate a real-life Markov chain. They engage in productive struggle as they investigate the world of math, and they experience the thrill of discovery as they see why math works the way it does. Each student is unique in their abilities and experiences, and I differentiate my instruction accordingly, designing my lessons and activities so that every student has the opportunity to attain understanding and succeed. And, of course, it’s no secret that I get excited about mathematics — my voice and my motions convey the enthusiasm I have for teaching math as well as math itself. And that enthusiasm is often contagious!

#### Playing with Math

#### The technology available today provides me with a unique opportunity to make math come alive for my students. Through computer programming, students can come to understand mathematics as not just something they learn, but something they interact with, even something they can create. Gone are the days when students could only imagine what a given curve or surface might look like. With software like GeoGebra and Desmos, my students can see it with their own eyes, but even more importantly, they can play with it, explore it, tweak it — really get to know it. Figures are no longer confined to a textbook, but instead gain a life of their own. A trapezoid camouflages itself as a triangle; an ellipse’s focus goes to infinity and comes out the other side; a curved line zooms in until it appears virtually straight. I encourage my students to uncover the relationships between seemingly disparate objects, and in doing so, they create knowledge that is meaningful, knowledge that is their own.

#### Solve for “Why”

#### Give a student a formula, and they can do math. Teach a student where the formula comes from, and they can understand math. Never do I teach a topic without delving into exactly why it works. Rote memorization and mnemonic tricks can only get students so far — to excel in mathematics, students need to truly understand the concepts and procedures. I’m not interested in having my students regurgitate the sine of an angle. I’d much rather have them know what the sine of an angle is and be able to visualize the location of that angle — that way, not only are they more likely to correctly compute the value, but they know what it means and can see how it relates to other problems and real-world situations. It’s even more important that students learn to connect the dots, to approach a problem from multiple viewpoints. The distance formula, the absolute value function, the Pythagorean theorem, and the graphs of a circle and of a sphere may seem on the surface like a multitude of things to have to remember, but when students realize that they’re all manifestations of the same concept, each one explains and reinforces the others, ensuring that students never forget any of them.

#### The Faces behind the Formulas

#### All too often, math is viewed as a collection of expressions and equations, facts and formulas. In my classroom, I emphasize an all-too-often-forgotten element of mathematics: the people who discover and create it. Behind every formula and equation is a story, and no story can be told without its cast of characters. Students don’t just learn to factor polynomials. They feel the excitement when great Italian minds engage in a battle of wits over who can solve the cubic and the quartic first. They feel the pain of a young French romantic who can prove that the quintic can’t be solved with basic operations, yet dies in a duel at only twenty years old. The friendships and the rivalries, the failures and the triumphs — they all show students that mathematics is a fundamentally human endeavor, as much a liberal art as it is a science. I make sure to emphasize not only the contributions of Europeans and Americans, but also those of the Chinese, Indians, Mayans, Persians, Egyptians, and more. Students must know that mathematics has been pursued by men and women from every part of the world, from the beginning of recorded history to the present day. Perhaps one of my students will even write the next chapter in the story.

#### Communication is Key

#### There’s another important point to take away from the human aspect of mathematics: the fact that, since the beginning of recorded history, mathematics has been a social activity. This is a truth that many classrooms seem to have forgotten, as many people equate “math class” with students listening quietly and taking notes while the teacher talks. Real mathematicians don’t do math this way — so why should students? I treat math more like a conversation. My classroom is a place of inquiry and discussion, of conjectures and justifications, of points and counterpoints. Every student brings a different perspective to the table, and I challenge them to share those perspectives for the better understanding of everyone. In an increasingly connected world, the ability to communicate ideas is essential, and so this is a key skill that needs to be emphasized in education.

#### Reassessing Assessment

#### At some point, the chips go down, students have to prove what they know, and that’s where math anxiety most often rears its ugly head. I see this as an opportunity to rethink just what we mean by “assessment.” In my classes, students have multiple opportunities to show their mastery, and multiple modes through which to do so — not only summative tests but also low-stakes formative assessments or even learning portfolios and personal reflections. Even more important is the opportunity for students to show growth through reassessment and revision. Students are often told that failure is an important part of learning mathematics, but in order for those words to have meaning, both the classroom environment and the grading system must be set up in such a way that students have the ability to learn from those failures without permanent, demoralizing black marks on their grades. I have seen the effect that changing the conversation from “points” to progress has had on my students, and I believe this is the key to encouraging and empowering each student to reach their highest potential.

#### So What’s the Point of it All?

#### Everything above has explained *how* I teach mathematics, but *why* do I teach mathematics?

- I teach because I love to teach, and I love what I teach.
- I teach because I live for that fabled “light-bulb moment,” which I know that every student has the capability to reach.
- I teach because mathematics is everywhere, more important than ever in our increasingly complex and connected world.
- Most importantly, I teach because I believe in my students and do everything I can to inspire them and prepare them to succeed, not just in my classroom but also in life.