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

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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.

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Here then is the recipe of learning. Take a student, place him in a situation of moderate uncertainty about some topic and get out of his way while he gets excited and attentive and directs his exploration to the source of his uncertainty. Moreover, research has demonstrated that he will enjoy his exploration and the accumulation of knowledge.

This class is not easy by any means but I like that it does inspire curiosity by encouraging me to think on my own and explore math. For once I feel like I am actually utilizing my curiosity and the ideas I develop about math.

We have been able to explore mathematics with pure creativity and imagination.

After being taught wrong math for the last 12 years of schooling it’s cool to see real mathematics. With the problems that we get in class I become curious about them. I want to figure them out and I like that the professor doesn’t tell us how to do the problems.

Overall, I believe that this class has made me more curious as a person in general, and more willing to be a problem solver. I often find myself using some of the skills that we learn in class in other classes, such as being very precise when explaining things, because that is how we must be when writing up proofs. Mathematical explorations has been very beneficial to me this year.

Mathematical explorations has helped me with critical thinking problems because in class nothing is handed to us, we work for days on end sometimes on one problem, but it is worth it because it is gratifying to be able to solve a complex problem on my own.

I learned that math is a much more open topic and there is room for new ideas and new discoveries. Since there are other ways to do math than just memorizing solutions, it has made me more curious in thinking about how there is much more to mathematics that I can still learn.

I think so far the thing I enjoy most is the fact that no answer is actually given to us. We ultimately have to find it, and this is definitely changing my curiosity about mathematics. I do not dread walking into your class, I almost look forward to it so I can see my group mates and… begin working on the problems. We all constantly have a different view of the problem, and the curiosity that this drives in me has come like no other. This is our way of going about the problem. We see something that seems interesting and we explore it. None of us know the answer, but when we think we’re on to something we’re curious to find out where it leads; maybe to the answer, or maybe to nowhere, but no lead is something to ignore…

As we sit in class making slice forms for our projects I always wonder why no one has ever taught this to me before, and that is what I like about this class. I am learning things that I wish I had known earlier but enjoy the fact that I have the opportunity to learn them now and change my outlook on mathematics as a whole.

This class has made me wonder what if other classes I’ve been in did stuff like this. I wonder if I would have liked school a lot better.

Everything is math. But what [is it] about math that makes me curious? It’s the fact that because of math there are so many hidden secrets to everything and they are just waiting to be solved.

The way that the class is being taught is like no other classroom that I have participated in, and this is due to the fact that I am learning and not being taught. Through the use of the book… me and the rest of the students in my class have been immersed into a classroom in which math is learned instead of trained.

Never before had math every made me curious, there was never anything to wonder or think about, it was always just about finding a solution to a problem.

I don’t dread walking into this class in the morning nor do I refuse to do the work out of spite for the subject. I’m more interested in math than every before. This class makes me think. It doesn’t just drill information into my head causing an overload. These bit and pieces of math we’re doing are intriguing and make me want to continue with the work.

None of my other math classes actually made me think about math. They only made me think of the problem in front of me, and how I am going to pass the next test. They never made me want to think more about math. On the contrary, they made me try to shut it out most of the time.

I am astounded by the number of patterns that have been found in everyday phenomena. I am constantly looking at things and wondering if there is a pattern to it. This class has my curiosity increasing at a super nerdy level, and that, is pretty cool.

Here in this class you are not told anything, you are expected to use math by diving into the world around you. This leaves all of us sitting here with a million questions which surprisingly enough we answer ourselves by being creative and using our own ideas to understand what we are learning… The idea of math has always frightened me because I never understood the why of it all. Now I have to ask why and then answer it.

I enjoyed the comparison between art, music and math because it gave me a glimpse at how mathematicians see math. It took music, which is something I can closely relate to, and math which is something I am not passionate about and it allowed me to see what it would be like if music were taught in the same fashion as math is.

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.

Usually I absolutely dread going to math class. I think it is because I have the negative mindset that I am just going to fail. This class has helped me widen my horizons and see that I do have the ability to succeed in math.

One of my favorite projects in this course was the string art project. I believed this assignment to be very creative and quite contrary to any of my previous history in a math course.

This course is a breath of fresh air. It helps me understand why math professors enjoy math so much. I see the fun in math now and how beautiful it can be. I had a lot of fun with the projects we created. I also enjoyed the amount of help I received from working in small groups with my classmates. I would recommend this class to anyone who currently has a negative attitude towards mathematics. This class could change their opinion, just like it did for me.

In my previous math classes, proofs were something I memorized and forgot because my teachers didn’t give me the chance to experience the fun in math. In my high school math classes, I felt dumb because I couldn’t memorize all the equations for the tedious amount of textbook problems we were assigned. When I came up with my own proofs to math problems in my math explorations class, it made me feel smart and important.

Math Explorations taught me what mathematics really is. It is an art form.

This class taught me how to think independently about not only math but other subjects and everyday problem solving.

I went in to this class saying, “I’m not good at math,” and I left this class saying, “I can do math.”

When I figured out how to cut my own shape out I was actually excited. I actually had fun in doing it and was really proud of myself for being able to do it.

I never would have discovered .999… and 1 are equal to one other, I would have never made such beautiful string art and learned the mathematical principles considered with it, I would have never been able to study a famous mathematician, and learned so many essential things about life in general, without this class. The ability to do conjectures and proofs, to discuss things in a more expanded light, to be able to write out how I got something and truly know why, are qualities I didn’t have before this class.

I always knew that I would see science or art or literature in real life, but no math class ever taught me that I would see mathematics in real life. Even now I catch myself seeing fractals or hearing about math things in songs.

As a student at Westfield State University, this course has had a profound impact on me. I have always had trouble with math and it has helped me look at it in a different light. It is now possible for me to look at mathematics as an art and something that is worth to explore.

Instead of falling asleep listening to lectures I was able to solve problems and make conjectures.

Math Explorations was undoubtedly the strangest and most enjoyable math class I have ever been enrolled in. I was engaged throughout the entire class and for once in my life not dreading math class when it came around.

Like I said before, I have hated math my entire life. I never had a love of math, not once in my life. I can sit here and say taking Mathematical Explorations, and the proofs we learned and the concrete material may never change my life. However, the **way** I was taught will affect me for the rest of my life.

…We were never told the cookie cutter way of finding an answer. We struggled, using trial and error and our own thoughts to find the answer to the problem. Yes, it was insanely frustrating. I cannot tell you how many times I wanted to throw my notebook out the window, or at my teacher (sorry Professor, it’s the truth). However, when I finally figured out the answer to the problem, I have never felt smarter. Here I was in math class, a subject I feared my entire life, feeling invincible. That’s right, invincible. For the first time in my life, I felt like I was fighting for my education. That is how you feel when you’re really learning.

In mathematics in the United States today, the focus is more of throwing a ton of information of steps and procedures for solving certain math problems and, completely disengages students from exploring solutions to problems on their own through experimentation and observing data to independently create a solution… School boards have completely made math a subject of throwing information at students, using the information to solve a problem step by step, expecting students to memorize the given steps for every type of possible problem for multiple subjects, and then determine a student’s future of continuing education on their ability to memorize all the tedious steps they were told.

…Math is no longer a student-engagement subject. Students are not give the time or encouraged to experiment with a math problem and find patterns for solving it.

…It is very difficult for me to look at a non-basic math problem and be able to know exactly what I have to do to solve it because, I have no personal experience of figuring out how to handle the type of problem through my own understanding.

…[In this course] I was encouraged and guided to engage in making the discoveries and understandings for myself. The techniques for how I found my solutions were very clear to me because, I was the one who found a pattern and applied the pattern to generate the solution. I have gained my own personal experience with mathematics that has changed my view on the subject completely.

I came home one weekend and I was discussing with my mother (She's and elementary teacher.) about your rule of answering a math problem in a complete sentence. I thought that was ridiculous and tedious. My mother then went on to explain that that is how math is being taught today. I finally understood your expectations from that point on.

From Math Is Still and Unnecessary Evil: "As you can see by the title of my paper, my feelings towards math have not really changed since the beginning of the semester...Maybe I'll try being a little more open-minded when it comes to my next math class.

When this class first started I thought I was going to run into the same problem I've always had with college math courses. I have gone into my past two math courses with a good attitude each time. By the end of a month each of the classes I had practically given up in...I love this class, I can't say enough about it.

This course challenged me to think in ways that I don't usually think, and it helped open my eyes to the world of math. It actually seems interesting.

I will sort of miss this class -- most of the chapters of the book are interesting -- I'll keep the book for the fun of it.

Math intimidates so many people, including myself, because of the way it has been taught in previous years and not one of my math teachers before now has asserted that it's not that hard if you take your time and not to move on until you're sure you understand. In my mind, the mathematical dogma and attitude from teachers that didn't seem to care if I really understood killed it for me.

I feel (although I shouldn't) that I have to prove to you and myself that I can do this and that I'm as smart as we think I am...This is what the class is about; writing math out and even if Einstein was in this class he'd be in the library with me for three hours so I'm going to get back to it now.

I've never had an affinity for math and I was so surprised by how easily I understood all the concepts we went over during the semester. I think that if all math classes were taught in the systematic and comprehensible way that this one was there wouldn't be so many people that don't like math...What I thought was really great was the fact that my mother, who teaches fourth grade, got some ideas from the text how to teach certain themes to her students in concrete terms they could understand.

I have to admit, I have never had a class like this, where learning is the most important factor.

I've done more math in this class than all of high school.

Deep down inside I know that I am a math struggler and this was by far the most that I have ever comprehended. Thank you, I had fun!

I described math to my family and they could not believe that I was actually enjoying the class. Because they know that I have had math phobia since sixth grade.

As the semester is about to end, the many weeks that I spent in your class were very educational, yet stress free. I have worked hard, like you said we would, but I feel that I really do understand the material and think that maybe all these years that I dreaded math and thought I was not good in it have changed.

No one had to be shy about asking questions and if you helped out a fellow student you felt smart, like you had learned something in a math class.

This course has gotten rid of my cynical point of view about math and taught me that it can be a beautiful subject.

I used to think that mathematics was all about adding, subtracting, multiplying and dividing numbers and letters and that I would never have any real use for it at all. This class has really made me more optimistic towards mathematics.