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

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 update again.

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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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The explorations I liked the most are the ones like Pascal's triangle. That took me the most time but it really got me thinking about how math works as a whole. I think it challenged me into wanting to know why it worked.

I have noticed I have a hard time explaining why patterns, equations, and principles work. I know how they work, I just can’t seem to explain why. That is my weakness in the class because that is how I lose the most points on the write ups. I am curious to figure out why all these explorations work, and to further my knowledge and apply it to other concepts. I really like how this course expands your way of thinking, and shows different ways to complete a task whether it is drawing it out, making a table or listing different strategies/patterns. So far this class is one of my favorite classes, and I can’t wait to see what the rest of the semester brings.

My favorite exploration so far is all of the Rubik’s cube work that we have been doing. Having a structured way of slowly learning how it works and how to solve it is very rewarding. Being able to solve a Rubik’s cube is often associated with being smart and is an accomplishment I am excited to ﬁnish. With each move that I feel conﬁdent with the closer I become to solving the entire cube. This type of work is something that I enjoy doing and have immediate gratiﬁcation as I slowly watch pieces move into the correct place.

I like working with the Rubik’s cube more than anything we have done in the class just because it is more entertaining and I get really into it. Although I am getting the hang of many of the strategies with the Rubik’s cube, I am still very curious about it. I am so eager to know what certain moves really control everything else around it.

The explorations I enjoy most are the ones with the Rubik’s cube. I never would have believed I could solve any part of the cube. At the start of the class I would’ve been happy to get one row filled with the same color. Now, I am getting closer and closer to solving the cube. It is really enjoyable to me, and I am really proud of myself with each week and with the more progress I see. I think inquiry based learning is definitely scary at first, and I never thought I would find the answer to the first problem you told us, but I did and understood it completely. I also found the answer all on my own and I found that really impressive. I am honestly really enjoying the class and I really appreciate this learning style.

As we progressed further throughout the math portion of the class, I always was learning new things. Professor Von and myself noted my improvement and confidence growth during the semester. At the beginning, I was unsure and hesitant about what I was doing. When the class was asked how confident we were with our proofs, the responses were not great. Throughout the past few weeks, I have begun to quickly understand the theories, to some degree, fairly quickly. I have learned new ways of thinking such as deductively and inductively, and problem solving skills. I have even gained a liking for talking in front of people and I am more comfortable with standing in front of a class. I have even gained enough confidence in my answers to present them to the class and explain them while answering questions. This is a stage I never saw myself at, during the beginning of the class. I find myself excited when I figure out why the theory mathematically works and I make the connection and solve it. I look forward to my math class every day I attend; this is quite surprising compared with my mindset from high school. If in the past, I saw where I was at today, I would be quite surprised that I could ever be excited for a math class.

From the very first moment we started the Stone Game, I was able to understand it and was actually excited to discover more ways in which a player could win. After the class ended, I remember I went back to my dorm room, tore up a piece of paper into little squares, and began to search for more ways that a player could win. After I was finished working through some of my ideas, I looked around at all the little scrapes of paper that were covering my desk, and I was amazed with myself. That had been the first time I had ever gotten so excited about working on a math problem outside of the classroom.

Math had always been my favorite subject because of the equations and typically straight-forward approach to solving a problem, but now it still is my favorite subject but for a different reason. Now I enjoy math because it can be seen everywhere and that means there is always another puzzle to be uncovered. As much as I love straightforward answers, I also love mysteries and understanding the problem along with solving it.

The original goal of math explorations was to provide an opportunity to increase our appreciation of mathematics and view it through a deeper lens of a proof rather than superficial numbers and equations. Through both Lockhart’s book, A Mathematicians Lament, and the work done with the proofs, I think I was finally able to see past all the numbers and equations in math to the art of it.

I agree with Lockhart that student’s should explore math for themselves and discover equations so that they understand it better than being given a formula sheet and told to memorize them. In our class we did not use any equations or formulas and I feel that I learned more in this class than any other math class before.

I did not consider myself a math person. Due to this, I felt as if my ideas did not have a lot of significance. I felt as if other people did not really care about what I had to say during the class. However, this was a very wrong assumption. Even when I am wrong about a proof, sharing the ideas in the groups helps to see each other’s different thought processes. The realization that other people in the class actually did value my work, and I valued theirs is what ultimately helped me “broaden my horizons.” Instead of shutting down when I get a wrong answer on a proof, I now find it easier to accept the fact and try again.

Forbidding me to say that I dislike math has actually been very beneficial. I have censored myself from expressing hatred of math even outside of the hearing range of Professor Von. I do not actually hate math—something I have always known—I simply hate not being able to understand it.

I do not hate math anymore. My enthusiasm for the subject, and the mystery of numbers, has risen throughout the semester. While writing the first proof, I was incredibly reluctant. I had no idea what I was doing, and I complained almost constantly because I did not know what was going on. The same is not true now. I want to learn why aspects of mathematics work. I still sometimes balk at writing proofs, but that is due to my perfectionism rather than my lack of knowledge on the subject. I can actually understand why the pieces of math we have discussed work. I can explain it in my own words.

I remember being in sixth grade and completing sixty multiplication problems in under three minutes and how proud I felt that it came so easily to me. That is when I fell in love with math. I fell in love with its simplicity in regards to being so concrete and firm in its answers. So the experiences I have had in Math 110 have been so vastly different from every thing I was taught to think about math, that having someone always challenging the answers I present was a nice change of pace. It was frustrating, but it was new and intriguing, and in a way, I fell back in love with math. It wasn’t even that I ever fell out of love with math; it was just a different love this time. It was the kind of love that Paul Lockhart would encourage; the kind that makes you see mathematics as a work of art with so many possibilities and with endless ways to think about it.

As I look back on the semester, I noticed how my effort and confidence has changed. I have gone from a student who was never confident in anything I did and would not put in any effort because of this, to a student who now believes he can accomplish any assignment he has because of his work ethic and mindset.

In Math 110, we are not able to say that we hate Math, or that we aren’t good at Math. Now, I see why. In “Carol Dweck Revisits the ‘Growth Mindset’, published in 2015, Dweck addresses educators about the misuse of her previous work on the growth mindset. She does this because if she does not amend how people apply her work then there will be no change in mindsets, which would nullify her life’s research. By reading that article I learned more about the impact of words putting you into a mindset. I now see that if we had said that we are not good at math or the simple fact that we just do not like it, then it would have put us in more of a fixed mindset than we already are about Math, which then hinders our ability to actually be able to do the subject.

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.