The images and videos in the Art of Mathematics media library showcase the active student involvement in our classrooms and the decentralized role of the instructor. You can browse the most recent images and videos shown below or search for specific examples of student activities, *e.g.* search for "Pick's Theorem" in videos to watch a small group of students investigating the areas of polygons on a geoboard.

# Media Library

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## Pages

Perhaps the best way to understand the depth and powerful impact of our project is to read what the students have to say about their experiences. The following student quotes, collected during the project, are typical responses received as part of student journals, essays and reflections.

Multiple search terms will be joined with AND by default. You can also enter OR to widen your search, *e.g.* searching for maypole OR salsa will return all results containing either maypole or salsa. To reset or clear the search, delete all terms and click search again.

I guess I have an appreciation towards mathematics. I understand people devote their lives to finding the contrasting rule to a law or formula and I think that's cool.

My attitude toward math hasn't changed that much except for the fact that know I can appreciate how some people, like yourself can have such a passion for mathematics.

Hopefully I'll be able to tell my children how fun and interesting this class was and more importantly how much I learned about what mathematics really is.

When I first heard you say that math "was beautiful" I thought you were a little strange. However, now after experiencing what math can really do mentally and physically, I can say that math is beautiful when you are taught right.

These ideas of math are part of everyday life, but society doesn't take in the true meaning and significance of math.

Math can be fun and exciting if you teach students to "understand" something rather than just memorizing it.

Thank you for this wonderful class and experience.

The first thing I noticed as I walked into room were the benches. Already I was put at ease because the traditional '5th grade' desks were not there. Then when we were handed the survey and I was some of the questions regarding our feelings about math, I was excited because I knew we were having a teacher who cared as much about our feelings as he cared about numbers and equations.

But like I have told you I would have rather taken tests. Nothing against you or the class that is just my preference.

I really like the way this class is set up this semester. I like working things out with people at my table. We work well together which helps. I do wish that the Professor would teach a little bit more, but then again, I don't know what he could because it's all in the book. I do like it when he'd read things as passages from books, articles, etc.

This class reminds me somewhat of my high school math classes. In high school I never understood what the teacher was talking about. So during class I would look at the lesson in the book and teach myself. I didn't mind doing this. That's when I liked math. In college I've taken to classes and had trouble with them. All they did was lecture. This class brings me back to the ways when I liked math.

The Math in this class is something I've never experienced in any other Math class. Most of my Math classes were always problems with numbers that a calculator could figure out. There were never problems that involved much thinking.

This math class, much to my surprise is very different. My views and hopes are slowly changing about math. I now feel a little more confident in my abilities.

For a guy who is reluctant to give math a chance I must admit that classes like Mathematical Explorations help broaden the minds of students like myself.

The lack of tests and homework in this class is a refreshing change. I am used to studying page after page of formulas and tables in math class. But this class is different and I like it. When math is shoved down my throat I cannot learn. The "pressure thing" again. I learn better when I can see the process step by step.

It was a nice break from the do-it-know-it-or-fail type math course in high school or any other arithmetic course. Granted, I have to know these skills and be able to do them without much difficulty. But it was good to be able to take a course that really just shows math as a form of art and (dare I say it?) philosophy.

You've shown how someone can see beauty and passion in mathematics. Your passion for mathematics really is inspiring considering the fact that most of my past math teachers didn't really care. They didn't have that fire in their eyes that you have. You want to share what you know and expand on it.

I now appreciate my father's intellect for being an engineer. I used to think he was boring for knowing all this math, but I now see that his thought process goes well beyond just numbers.

When I first came to class in the beginning of the year you were always talking about how math was so beautiful. When I heard you say this I was like, what is this guy talking about? But as the year went on I started to think about the beauty of math more and more. Math is pretty beautiful! Math is everywhere, it is in all art work and designs. I love art and I love to draw and paint, so I guess math is beautiful to me also.

When our class began doing work in Chapter 11 [Discovering the Art of Mathematics - The Infinite], I thought I was in hell. Your handout was much more difficult than the book [Mathematics: A Human Endeavor by Harold Jacobs]. At that point, although it was a good thing, I thought it was going to be too hard and wanted to go back to the book. However, after spending time on Chapter 11, that challenge is exactly what I needed. It was in Chapter 11 that I gained my appreciation for the differing levels of infinity and some of Zeno's paradoxes. It was not until we went back to the book at the end of class that I realized that the book was no challenge at all. It was the challenge that I received in Chapter 11 that helped me to appreciate mathematics.

One of the most surprising things about this class is that I actually like being here, which is weird because every other math class that I have ever had has sucked and been boring.

I do like the fact that this class isn't boring. I tend to want to understand the work because I can't get into something if I am not actually doing it. When professors teach math generally they show you how to do something on the board when half the class is doodling in their notebooks and then they expect you to be masterful at it on a test.

This is who I was meant to be

It all started in the beginning with Mobius bands,

When we began using everyday math with our hands.

It instantly opened up my perspectives on how I view life,

For the first time I felt a sense of sharpness, like a knife.

I honestly started to view things differently,

Or at least, with a sense of pure, complete decency.

But that... that, was just the beginning.

As we moved forward with our assignments in class,

I began looking at everyday things with an open mind, at last.

When we faced those difficult challenges every day,

"I can do it. I won't give up. I'll expand my mind!" I would say.

I began to realize that maybe math was an interest of mine.

For the first time, this class literally made me shine.

I would get so mad sometimes, I would want to give up.

I never did though; I used the phrase that relates to a cup.

"It's a glass half full, not half empty" I would tell myself.

Eventually, I was able to put my worries up high away on a shelf.

I swear it was like a breathe of fresh air.

That moment I realized this class and I were a perfect pair.

No matter how much of a struggle it was,

I will never regret the mistakes I made... just, because.

Because I now know how to face academic challenges,

The class literally took away all of my regrets and salvages.

In the beginning I thought I was going to fail,

But that didn't stop me... I decided I was going to set sail.

This class was one hundred percent a turning point,

I can't stress enough, I'm happy I wound up in this joint.

I never would have expected this from a class,

But wow... I finally feel a sense of being free, at last!

Everything from the Mobius bands, string art and Maypole dancing,

Each topic made me stronger, and now I feel like prancing.

I know how to look at obstacles with a different state of mind,

It really is something that's truly one of a kind.

Thank you, Dr. for never giving up on me,

One day you will see, this is who I was meant to be.

If ever in the future I meet a mathematician I won't say something corny like I can't balance my checkbook.

I think the class was really set up for encouraging learning.

Math was never exactly my favorite class,

The homework, tests, and exams seemed to always kick my ass.

I understood the multiplying, dividing, adding and subtracting

But all the formulas have never been too attracting.

Cram as much into your brain the night before the test,

But then a week later I'd remember half of it at best.

What was the point of that in the first place?

In my opinion, math class should never be a race.

However, I quickly learned this math class wasn't like the rest

And because I could somewhat enjoy it, I wasn't as stressed.

I was never put on the spot to answer a question,

And that really did make a big difference it's my confession.

I was able to work with my peers without feeling behind

And doing so, I actually liked math and what it did to my mind.

I learned to question things that I never have before

Like infinite numbers, theories, and much more.

Instead of falling asleep listening to lectures

I was able to solve problems and make conjectures.

I wasn't forced to sit without talking

But instead it was encouraged which was at first shocking.

I do hope that my next math class will be like this one

I worked my mind, which was sometimes even fun.

The feeling you get after solving a tricky equation

Makes you feel great, like it's a special occasion.

I no longer have such a stereotypical idea of math

And I see now how I will use it throughout my life path.

There is math all around us everyday

We just don't always take the time to notice it that way.

...To be completely honest I wish my other classes challenged me as much as this course did. Although I did have certain courses challenge me at different times, this was the only course where I was being consistently challenged... I'm really thankful to have had the opportunity for someone to show me how different and truly astounding mathematics and art can be.

I remember coming home for the first time since moving in to college and my mom asked me how classes were going, and she asked about Math Explorations in particular. I remember I had such a hard time explaining it to her not only because the structure of the class is vastly different than anything she's heard of, but because I was so excited about how different it was that I was speaking gibberish. I remember her asking me exactly what we do, and the only thing I could say that made sense was "Mom... we just explore math. We touch it and manipulate it and make the math for ourselves.

This course is very different from any other math course that I've ever taken. It has a lot more character.

When I received my schedule for my first semester of college, I was surprised to see a class such as Mathematical Explorations on my paper. The last class I took in high school was pre-calculus, so I had automatically assumed that I would be put in whatever calculus class that comes next after pre-calculus. I was super skeptical when I saw that my math book was only 12 dollars. I honestly thought it was a joke because the cheapest math book I had in high school was 80 dollars, and it was a 20 year old book that was falling apart and to the math department seemed like the holy grail of mathematics. I laughed even harder when I actually purchased the book and opened it to only see text. I thought it was absolutely crazy to have a math book that had no formulas, diagrams, or graphs. But the book "A Mathematician's Lament" had actually taught me more in 12 pages than I had learned in 12 years of mathematics. I think the idea that struck me the most was the section that explained that a log of high school students are good at math just because they have become really good at following directions. Which I realized was totally true. I mean, I could learn formulas and apply them to hypothetical situations, but I learned that the reason I got the A in the class was because I learned what the teacher wanted and gave it to them. I didn't always worry about the process of learning mathematics as I was with producing something that in turn produced a grade that would benefit me.

I truly believe that this class has helped me to grow as a person and not just a math student. When describing the class to my friends it was almost difficult to describe it as a math class because it was more of a learning class that we happened to do math in.

All in all, this class was incredible. It was amazing how much I learned about myself and education. I will never see mathematics the same ever again because now I know what truly lies in the subject. It is beautiful. And more people should be exposed to the truth of math and know what it is like to be able to create with your mind.

On the first day of Math Explorations... [our teacher] told his students that he was giving us a mathematical paintbrush, and that he wants us to paint. That he was giving us a mathematical tambourine; to make our own music. And this is exactly how class should be taught... He gave us the opportunity and it was up to us to create something amazing and beautiful with what he had given us. And that is exactly what we did. The fact that it was never easy to find an answer to the problem made me want to find it so much more. I would sit at my table with my three classmates and work so hard on one problem until we got the guts to ask for help. But we were so personally determined to get the answer ourselves that we almost did not want to hear what we were doing wrong. And even when we asked for help, we were never *told* the answer. Rather we were given suggestions to turn different corners to possibly find the answers there. It taught us to think for ourselves. And it taught us so much more than just the way to get the answer to certain problems. The worst feeling in the world was to look up at the clock and realize that class was over seven minutes ago and you still have not figured out the solution. Yet the best feeling was when you figured out a solution and could not stop smiling for the rest of class because you had figured out that one question that had been pushing on your shoulders since the first time you read it.

I would definitely recommend this class to anyone who needs a math requirement. It's fun and it's exciting, it's different and most importantly, it's not a boring lecture class. We do the work, we ask the questions and we learn.

This math class was different from math classes I have taken in the past. We actually covered material that was "fun." Yes, I said it. Math is fun.

"This course changed how I speak up and speak out about what I'm being taught. I have realized that this really is MY education and I shouldn't settle for being force fed information and taking short cuts. Although the short cuts are easier and *everyone loves the easier route*, it takes deep thought and actually learning to finally grasp something you never knew and **you feel proud of yourself for it.** I felt proud of myself so many times in this class when I finally had a problem click in my head that we had been working on for days and realizing how simple it was, but it was exactly the process that I needed to **understand** it. No matter how lazy I was feeling and how I wished I could just take the short cut and get the answer, I now know that isn't learning, that is just getting by. I don't just want to get by in my classes, I want to learn and grow as a person, so the only option from now on is putting in the effort for the hard road to the answers.

Another odd, but unique aspect of this class was the write up and/or reflections that were required after finishing up a particular unit. Confucius, a great Chinese philosopher, once said, ``he who learns but does not think, is lost. He who thinks but does not learn is in great danger.’’ These words of advice explain Confucius’ idea that teachers one to learn the importance for studying and reflecting on what one has learned and also understanding that learning is a process of observation and study and that should be followed by reflection of the knowledge that was gained. This was exactly what was done in this class. We learn multiple mathematical concepts through witnessed actions; whether it was through playing with blocks, tessellations, etc. After observing these actions, we had to reflect on what we did OURSELVES. This was the unique way of learning during this class and it was for the benefit of us; the students.

This math class is one of the first times I actually enjoyed being in a math class. It wasn't necessarily the material that made me like the class, but the overall experience. There were many times throughout the semester where I did not even feel like I was in a math class, but more of like a problem solving class.

Through reading Paul Lockhart's A Mathematician's Lament I have taken away a major lesson that only helped to highlight the importance of the work we did in the classroom every day. That lesson is that "There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics" (Lockhart, p. 23) So why are people's work like George Washington, Shakespeare, and Van Gogh easily and understandably celebrated when mathematicians' are not? It is because people are unwilling to take the time to understand. People have trouble celebrating what they do not understand, and math often falls underneath this veil.

In my future classroom, while math will not be the primary subject taught, it will be celebrated in the ways that it can. Along with wars and economic booms, my students will learn about how math has helped lead us to the place we are today.

I was just taking this class to get the grade and to get another chunk of my core done and over with.

Looking back I realize how foolish my thinking was. I did not sign up for a typical math class, I signed up for a thinking and reasoning class that utilized mathematics. This class has completely revolutionized the way that I have thought about not only my math education, but education as a whole. As a person that wants to become a teacher, this class has taught me lessons that I will be able to apply. People always complain in math class, when am I going to need to know how to take the anti-derivative in my life? The answer is probably never, but if more classes were taught like this one way, then people would not even question the value.

People hate math because they are not give the opportunity to understand it on their own, they are just given the facts and expected to regurgitate it, but not this class.

I am an English major. I know I’ve said this before, but that’s relevant to what I have to say. While being interviewed for the video, I said that I had re-thought my ideas about Math majors; that I hadn’t been completely won over by the mathematicians, but the major had advantages. That was not a lie - and you have to trust me on that. Unlike mathematicians, we writers can’t always prove our words; there will always be someone trying to refute them…

I will calm down now on my rant. But first, a small quote from Albert Einstein… ``I want to know God’s thoughts. The rest are details.’’ As a writer I can show you

So don’t be surprised if I reference some of them in a novel someday.

This course changed how I speak up and speak out about what I'm being taught. I have realized that this really is MY education and I shouldn't settle for being force fed information and taking short cuts. Although the short cuts are easier and *everyone loves the easier route*, it takes deep thought and actually learning to finally grasp something you never knew and you **feel proud of yourself for it**. I felt proud of myself so many times in this class when I finally had a problem click in my head that we had been working on for days and realizing how simple it was, but it was exactly the process that I needed to **understand it**. No matter how lazy I was feeling and how I wished I could just take the short cut and get the answer, I now know that isn't learning, that is just getting by. I don't just want to get by in my classes, I want to learn and grow as a person, so the only option from now on is putting in the effort for the hard road to the answers.

My thoughts and feelings on this class? I enjoyed this class thoroughly. I didn't mind getting up early three days a week to come here. That is because for the first time in four years I didn't feel threatened, but rather embraced by mathematics.

There have been so many things that this course has changed in my mind. Compared to all of my other math classes I have ever taken, this course has actually felt **beneficial**. Not only did I learn more about actual math functions, I learned to ask why and how. I learned to *demand* further knowledge on a subject instead of just taking information and writing it down and doing repetitive problems. I now truly and deeply understand certain things we use and do in math that I have never questioned for years. I have never sat around in a math classroom and worked out problems and learned things on my own, even though that is what math is supposed to be about. When you think about how different this class has been, you really want to question why all high school classes aren't like this. The environment was so beneficial and created so many mathematical epiphanies in my head when it came to math problems I had learned years ago. It is so much better to finally understand why I do certain math steps instead of just memorizing what to do and it helps gain a deeper understanding for math that all my other math courses didn't achieve. It may be that all my other classes were so consumed with preparing us for standardized tests and always being on a deadline to reach certain material in order for us to be on time that we skip over the truly important and deeper understanding of math and instead just touch the surface by using memorization.

Math Explorations was meant to be a Liberal Arts requirement because it is a thinking course that helps a person expand their thinking...One of the greatest things about going to college is that you learn to look at things from a different perspective and change your way of thinking. This class is a wonderful example of how college can do that to a person. A lot of learning went on in this class this semester.

This course in general has been a complete paradigm shift. It has absolutely blown my mind and I am sure that it has blown the minds of other students as well. Most students never get the chance to actually understand the world of mathematics - they are just shown what to do and simply repeat the process until it is drilled into their minds enough to memorize it for a test, and then it is all simply forgotten. In this class I have been shown that there is more to math, there is a deeper understanding that other people are not fortunate enough to see. I now see mathematics in a completely different way and I want to learn more about it. I do not want to learn, I want to teach myself. Because from taking this class, I have realized that teaching yourself is the only way to get a deeper understanding of a single subject area. While this does not directly relate to my paradigm shift of the Heliocentric Theory, I believe that the concepts are very relatable. I see this because at first, people did not want to believe that the Earth was no longer the center of the universe. But once this theory was proven, people believe in it and wanted to discover more about our universe and what is out there. And when looking into this class, at first, no one wanted to believe that there was any more to mathematics than what we have previously been taught. And once these students' minds had been opened, they are all yearning for knowledge. They all want to see what more they can figure out about not only the world of mathematics, but the deeper meaning of other subjects as well. And to see what we have all been deprived of since we had started school. No one would ever think that these two things would be relatable but when looking at the scope of both, they are extremely similar in their meanings to the world or to this new generation at Westfield State.

You know something that I find strange? Sometimes I actually feel kind of smart mathematically. I'm not used to being able to comprehend or even work out problems in mathematics, and since this course, this new and strange phenomena has occurred.

Opening up the spectrum of math is how this course cleared up the misconception that math is simply computations and solving equations... Math became a wider subject in this course.

It's nice to have a class in a relaxed area/environment for a change. Being treated like a person and not an inferior 'student.'

This material was helpful in creating an alternative to lecture-dominated classes. I used some of these materials and modified some of them to fit my own purposes in the course. They also helped me by providing a structure which I mimicked in creating my own materials… [The students] were certainly actively engaged.

… I think the hands-on approach is very useful, in engaging students who might have the idea that math is a dry, boring subject.

…The concept of dimension is explained well, beginning with an intuitive notion and then proceeding to a more precisely defined topological definition of dimension. I really wish I had time to go through the section on fractal dimension, which I had to cut due to time constraints.

… I really liked the intuitive introduction to topology, as well as the intuitive working definition of topological invariance. The maps were good starting examples of topologically deformed objects.