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.
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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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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.
The general approach is amazing. I love how you get to touch upon and substantially develop so much modern analysis, in this unassuming volume for liberal arts students. I also found the way you introduced the concepts very well thought-out.
…The one guidebook I read does a great job at putting the student at the center of the action, and encourages independent inquiry… I do think you require a lot from the students, but they should be able to follow the text to wherever they like as long as they commit to keeping their intellectual spirit and curiosity alive.
…I was impressed by how much actual mathematics there was in the whole volume, as opposed to some other math for liberal arts texts that are a lot less substantial in level and content… I think the intellectual merit of this work is very high, and it is a very valuable endeavor. Liberal arts students should not be left to only learn about watered-down pizza math. They should be introduced to the most beautiful and powerful ideas of mathematics; this project is attempting to do precisely that.
…By bringing in the human components of mathematics, the projects actually help students from underprivileged and underrepresented groups feel more welcome in what is generally regarded as a cold, austere science.
I reviewed parts of 5 books being produced for this project.
…I find the material to be excellent in all respects. This is a great idea for a project, and the people involved are clearly able to execute their vision. I really hope that enough resources are available to complete all the books.
…I strongly encourage the project principles to seek another, larger, grant to enable them to complete this project. I am impressed by the amount of work which has been done on a very small budget, but it will take a lot more.
…I think that the final collection of Discovering the Art of Mathematics books will be a resource that instructors have needed for a long time. These books will have a strong positive impact on the way mathematics is taught to non-science majors at a wide variety of colleges.
Your materials demand more active involvement on the part of the students. No “correct” answers or interpretations are given for much of it. This forces the students to make sense of it, come up with their own meaning and communicate with classmates, and get confirmation/support or other ideas from peers. The experience of doing just that will stick with them much longer than material from a traditional liberal arts math class. Your topics are less common but not less mathematical. They are a refreshing change.
As well as learning how a mathematician works, and how conjecture and hypothesis becomes theory, I think that students will get an experience of learning how they can go about learning something new, and maybe in the future, they might tackle something brand new that they might not have had the courage to do without the experience of working through some of your materials.
…The “hands-on” nature of your treatment is the real thing that makes it alluring and engaging. This is the big difference between what you are creating and what is available currently.
I like the wide variety of topics, and equally diverse examples that illustrate what mathematicians find so appealing in numbers, patterns, and space. The text and exercises make connections between theory and applications (e.g. primes, congruences, and encryption) and also between the ancient and the quite recent (e.g. the Pythagorean Theorem and the conquest of Fermat’s Last Theorem). I have no doubt that many students, previously bored with mathematics, will find the explorations in these tests to be engaging.
…I am particularly impressed with his coverage of significant ideas and their development through a historical context.
I have reviewed several of Professor Fleron’s manuscripts, and I must say that they are quite fun to read. I can imagine using them for some of my mathematical outreach activities. The samples he has produced are a very good indication of his commitment to education.
I am deeply impressed by the work you have done, both in its quality and in the sheer magnitude of the project.
…[Discovering the Art of Mathematics – Geometry] is a fresh approach to geometry for undergraduates who are not mathematics majors. The choice of material is excellent, as is the idea of linking it to visual art. The overall result is a very original and appealing idea for a course. This first draft is very promising, even though it is still in very rough form and needs a lot of polishing.
…the idea of dimension, and, in particular, the geometry of four dimensions and fractals, is a particularly interesting choice in a course for liberal arts undergraduates, because of the rich connections with other aspects of culture, and with art in particular. The students may know little about art, but the authors have realized, correctly, that art can motivate the study of mathematics. They exploit this idea boldly and on the whole successfully to produce, in effect, a multidimensional approach to the study of dimension. I support this idea enthusiastically.
For me, the main purpose of a course in mathematics in the liberal arts (an MLA course) is to get the students in it to use their heads… MLA courses give them other benefits as well—widening their cultural horizons, improving their writing, and so on—but getting them to think is the main one.
The material in Discovering the Art of Number Theory (ANT for short) will do the job.
…When in its final form, ANT will admirably serve to meet the goals on MLA courses. I wish it every success.
We work in groups every day on chapters the Professor and his colleagues have put together just for us. With the lessons, it feels like he purposely starts off with the irrelevant, basic steps then the problems become a little more difficult as the packet goes on. By the end of the chapter, there are several small pictures that we have discovered that suddenly all connect to make one astounding realization. It's almost as though specific problems were chosen with the direct intention of blowing our minds every time and I love it.
The structure of the packets [chapters] and the variety of exercises was very good. The structure lent itself to a class in which students take primary responsibility and engage in the mathematical process. As this was a primary goal for me, the packets [chapters] were ideal.
…I was surprised by how into it some of the students got and how they were much more interested in developing their own solutions than in following the path laid out for them.
…I would say [the materials influenced my way of teaching this class] dramatically. It made it possible for me to design an inquiry course since the question sequences were already done. It also provided a good model so that I was able to write my own packet [chapter] on Impartial Games, extending the ideas from Nim.
Discovering the Art of Number Theory is delightful… When I was teaching an MLA courses here at Penn State… I never found a text that was close to what I needed. Prof. Fleron’s text fits the bill… [And] what is so nice is the fact that this is very much in the R.L. Moore style.