SMT3100

Knowing and Learning in Mathematics and Science

Spring 2025, University of Florida

Table of Contents

Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 12

Week 1: Introductions

by @aidanhurwitz

Last updated 2025-01-26 13:41

 I really enjoyed my first class of SMT3100. The style was much different than my usually classes (I actually learned some of my classmates' names!). I am used to sitting down for lecture and taking notes, having little interaction with the professor and other colleagues, and occasionally asking/answering questions. This class was very different and a nice refresh on what knowing and learning can be.

Also, I work at the UF Academic Resources center as a math tutor and I didn't realize how much of this class applies to what I do there. It can be a struggle to convey concepts to students sometimes, and understanding different methods of teaching as well as the some ways that students can learn STEM topics in a fun and meaningful way are great skills to have. I found the cup staking game particularly fun and insightful. Knowledge and learning should look, sound, and feel fun. Group learning is usually dreaded by students because old-school methods of teaching are meant for individual and self-paced learning. Teaching is almost always in a group setting, so I think it is really important to understand how students can learn in groups and for the teacher to understand what makes a good classroom environment.

In response to the prompts, my favorite teacher I have had was my high school AP Calc BC teacher. She was extremely tough- and at times we were quite scared of her- but I learned to love math and discover my passion from her. By the end of my two years in her classes, I was more than ready of college-level math and still find myself relying on things I took away from that class in my current upper-level math courses. On the other hand, my least favorite teacher was a history teacher in high school. We were given textbook chapters to read and busy work to complete, and rarely did we have any discussion or lectures in class. Unfortunately, this turned me off of history and I have not had an interest in it since. Outside of the classroom, I have to give credit to my brothers and sisters who set extremely high standards for me and encourage me to explore my interests, even if I do not know where I will end up.

I am really excited to continue in this class, not only learning the content for interest but also applying it to my student and teaching life!

~410 words.

Week 2: Being a Good Teacher

by @aidanhurwitz

Last updated 2025-01-27 18:00

 To me, being a good teacher means providing the structure and environment to inspire students to want to learn on their own. I think that the most important trait a student can have is curiosity, and this often manifests itself in accelerated growth and intrigue. I know for me personally, some of my strictest teachers were the ones that I learned the most from, simply because I found the hard content and grading pushed me to strive for perfection and really understand the nuances of the topics. However, there is definitely a fine line between strictness and harshness. I think a bad teacher is one who forces their own method of learning onto others and their students, discouraging self-exploration and curiosity and instead gamifying grades and school. Students may try just to get the most points they can, rather than actually learn the content.

Being a good student follows a similar pattern in my opinion. Good students should want to learn on their own, and use school as resource to acquire knowledge and experience in collaboration. I think a crucial part of public schooling is the opportunity it gives students and teachers to learn from each other, not just from a textbook. The first step for a student is finding interest in a subject, but then they need to learn how to communicate in that subject and listen to other people's opinions and knowledge to better their own. In this way, I think all of the students and the teachers learn more about the subjects they are involving and the people they are with.

I think people learn when they are curious. I know, for me personally, it is necessary that I have an interest in the classes I am taking for me to retain any information after the semester ends. Sure, I could get an A in a class if I don't care but simply do the work. But, for the content to actually be meaningful to me, I need to have an active interest in what I am being taught/discovering. I think that this is where good teachers and bad teachers are separated. A bad teacher may have all the knowledge and teaching tools to help their students get through a course, but if they do not foster curiosity and create an engaging learning environment, then their students will not develop interest in the class. When I try and tutor or explain a concept to a friend, I always try to think about why they should care and how this topic is "cool". This does not require application, but rather a beauty in the topic for the student. This is the same reason we explore poetry and higher-level math. The curiosity and knowledge of the natural beauty of these topics is enough to keep us engaged and learning.

~471 words.

Week 3: Educational Philosophies

by @aidanhurwitz

Last updated 2025-01-27 19:09

 After the educational philosophy test, my highest scoring result ended up being Humanism. I do not disagree with this result, but I did expect some of the others to be higher on my list than not. I think of myself as agreeing with the information processing and logical deduction side of education; I would rather be able to derive facts than memorize them. However, I think that as teachers we have to understand that everyone's brain works differently and some logical steps may be obvious to some but not to others. I think the Humanism comes into play when we accept that humans should want to learn out of curiosity and not for application necessarily. While it is crucial that we apply the knowledge we gain, I think it is important to remember that learning is not always about results. Sometimes, learning should be for the sake of learning and it should be rewarding enough to discover some beauty in a topic than to get good grades or make something useful. Obviously it is easier in some subjects than others- but I think the principle still applies.

Learning how my students learn means interacting with students at a personal level. It is not enough, in my opinion, for a teacher to lecture and quiz and test to be considered a "good" teacher. Teachers should want to get to know their students, listen to their ideas, and constantly redefine their methods of explanation to fit their students best. I think testing is certainly a strong method of making sure students can learn and apply their knowledge, but also it can mislead students into thinking the purpose of their learning is to perform well on tests. In contrast, discussion in class and engaging questions during lecture can motivate students to learn and discover on their own accord, but can also lead to some misinterpretations or incorrect understandings that go uncaught. I think a good mix between these two can motivate students and also keep them on track. We can use the information gained from this to design curriculum around students and not for students.

~353 words.

Week 4: Fish is Fish

by @aidanhurwitz

Last updated 2025-02-04 20:39

 The short story Fish is Fish provides a surprisingly meta and deep metaphor for human learning. In the story, when the frog tells the fish all about the world above the water, the fish transforms ideas about cows and birds into ones revolving around fish. This is a silly but important example of constructivism, the philosophy that we learn based on our own constructions of reality, and that there is no one objective truth to be taught to everyone. In this scenario, the fish's objective reality is its own perception of what cows and birds are, and there is nothing in the story that suggests that the fish could be proven wrong. The audience gets no glimpse of these cows and birds, so we too only create the image of what we think they looked like in our minds. It is tempting to say that the fish is the one who is misinformed or unaware, however, we too are simply assuming this based on our own knowledge.

I think it is very important to translate the morals of this story into everyday life. A prominent issue in our society is overcorrection, or the action of assuming someone else is wrong without actually being able to prove it. We experience this everyday; someone who does not know the full story may explain something incorrectly to you or even call you wrong. A good example of overcorrection occurs in religion. Someone who is very religious may swear by belief that their faith is the correct one, however, it is by nature impossible to prove or disprove that thought. Like the fish, we only believe and know something based on our own preconceived ideas of it. With this in mind, we never truly understand how someone else views a topic or sees the world, since this would contradict the idea that it is self-constructed. Of course, we have to agree on somethings, after all, it is impossible to share knowledge if nothing can be proven or disproven. However, I think it is important to always consider the notion that other people do not think the same way as you, and it is not anyone's fault. It is simply a matter of being human and individualist which makes us unique, in body, spirit, and mind.

~382 words.

Week 5: Culture and Learning

by @aidanhurwitz

Last updated 2025-02-11 23:17

 I cannot say I have experiences an overwhelming amount of diversity in the teachers thus far. For the first time that I can recall, this semester I have a black teacher- although he is a PhD student for a lab. I find it interesting that I have never had a black teacher before, and I was surprised to learn that I was kind of the odd one out of my friends as most of them had. Even more surprising, perhaps, is the fact that until college I do not believe I have ever had a latino/a teacher outside of a Spanish classroom. When I think about this, I find it really strange considering the latino population of Florida- especially southwest Florida where I am from. I think the lack of diversity in educators throughout public school is a reflection of both the culture and environment of public education in the US. We often hire and want our children to be around those who share values as us, and it could be that the racial and ethnic differences between cultures means that public schools do not often include diverse staff. Of course, this is just an interpretation and by no means fact, but it is possible to think.

Additionally, I have never truly had an openly Jewish teacher- also until college. This is also a reflection on our public school system, especially in Florida. One attempt my high school made to include nonwestern culture into education was to say mantras- Hindu spiritual sayings/exercises- at the beginning of every school day. As expected, this was promptly shut down. In fact, the next year they implemented prayer time at the beginning of each day to emphasize the traditional cultural faith and understandings that the government thought was important for all Florida students to share. These types of conflicts in thought are why many believe it to be easier to just secularize school and not teach as holistically, at the cost of losing some vital culture and perspective. Unfortunately, until we can all accept a constructivist outlook on faith and life (which will probably never happen), then we will continue to deny teaching some values of life in our schools and place emphasis on more traditional ones instead.

~375 words.

Week 6: Mental Activity Forms

by @aidanhurwitz

Last updated 2025-02-24 18:38

 From my own experiences, I can provide examples to the five types of mental activity as described by Carpenter and Lehrer, which are:

1. Constructing Relationships
Last semester, I started showing up to office hours for one of my math classes. At first, this was because I was struggling in the class and needed extra help on some of the homework. However, it quickly developed into a routine and I started enjoying going to office hours, even if I did not have any pressing questions at the time. Not only did this help me form a relationship with the professor and make class more enjoyable and engaging, but it also led to some fun discussions about topics I would not have explored on my own without his guidance.

2. Extending and Applying Mathematical Knowledge
Since I work for the College of Liberal Arts and Sciences' Academic Resources center, it is extremely common for me to engage with a student who has a question and I have not an answer. This forces me to work through new questions with students, and show them that we all struggle sometimes with coming up with solutions. I like this process, not only because it is interesting and helps me develop better intuition and problem solving skills, but also because it humanizes those of us who are "talented" at math and puts us on the same level as the students.

3. Reflecting About Experiences
Not to point out the obvious, but this class has forced me to do this for the first time. I had not really reflected on how I and other students learn math before, but this class has certainly engaged me to think about different learning styles and which ones I vibe with the most. Just writing these journal entries has made me more aware of my internal thoughts and the opinions I have formed about educational philosophies in this class.

4. Articulating What One Knows
I find that explaining information to others is one of the best ways to study. If we can actively and accurately explain concepts to other people- especially to those who know very little to nothing about the topic- then we gain a more concrete understanding of what we know and why we know it. Sometimes, I will call my Mom and explain to her what I learned in math this week. It is highly unlikely that anything I say is of use to her, but just the process of me trying to explain it to her makes me understand it better.

5. Making Mathematical Knowledge One's Own
Rewriting notes from lecture is a simple, yet effective way to translate a professor's knowledge into my own. I try to act as if I am writing a textbook instead of notes. This goes back to number 3 and 4, but I think it is super important, especially in math to be precise nonetheless. By rewriting notes and trying to be as official and "professor like" as possible, I transform knowledge from lecture into actual knowledge in my brain. Additionally, I think it is important to "practice as you play" per se. If I take notes like I am taking an exam, then when the exam comes I am prepared and not confused by notation or complicated concepts I might have previously ignored/simplified.

~559 words.

Week 7: Advances in Neuroscience and Education

by @aidanhurwitz

Last updated 2025-02-24 19:10

 Learning about the brain in medical and neuroscientific ways has drastically changed our approaches to education and teaching learning. In particular, and ironically, I think we have become more unsure about learning and less objective about the true way to learn. As we learn more and more about neuroscience and the medical brain, we also know we understand far, far less about the brain than we previously thought. As the videos prove, there are a multitude of different things that can affect our brain development throughout childhood and adolescence. It is impossible to pin these developments down to just one unique educational philosophy. Instead, over time we have realized that we require all of these theories- behaviorism, cognitivism, constructivism, and ethnosocialism- to explain the development and learning patterns of humans. Unfortunately (or fortunately depending on your take), the human brain is extremely complex. We have learned through studies that there are no significant patterns in learning that can be attributed to simply a person's sex, race, ethnicity, learning style, country of origin, etc. Rather, there is a complex mix of all of these factors that contribute to a student's success. What we can study, however, is the types of learning methods and philosophies that should be applied to which students and in what context. I think it is really important not only to continue to study the physical brain, but also the metacognitive aspects of learning and what it means to be a knower. It may be that we never develop a chemical pathway or mathematic formula for knowledge, but we can certainly identify it when we see it and apply how we learn to how we teach.

~278 words.

Week 12: Perspectives on Intelligence

by @aidanhurwitz

Last updated 2025-04-01 17:30

 People transfer skills and knowledge from themselves to others through a variety of social and academic practices. In the most simplest form, knowledge of the past and cultural intuition is passed by oral tradition. Humans have been telling stories to one another since the dawn of our time, and it is undoubtably one of the most important traits of humanity. Oral tradition not only defines our culture and our knowledge, but also connects family and friends in ways we cannot connect in any other way. Furthermore, written works open up the possibility of knowledge being shared beyond families and communities, but rather to entire civilizations. Books and articles transcend time and space, allowing for knowledge to pass from generation to generation consistently and cheaply. We use knowledge every day to transfer our position in some situation to another. For example, knowledge of how one should act at home informs their knowledge of how to act in public, even if these two behaviors are not identical. We transfer, interpret, and morph our knowledge of one situation to another by adopting cultural norms as well as using our own personalities and preferences to make decisions about the world around us.

We can "teach for transfer" by showing and encouraging students to fail. It is extremely important that students understand the need for failure to inspire growth and understanding. Students who are fed knowledge and fact will not know how to apply such knowledge in new situations. For example, in mathematics, students who memorize formulas will not be able to derive new formulas in unknown circumstances. On the contrary, students who derive formulas we know to be true may be able to understand new situations from a fresh perspective and derive new formulas, making progress in the field. In the same way students must struggle and fail to develop ideas we know to be correct, they will be able to struggle and fail in new situations with the foresight that they have the ability to solve new problems. Academia takes grit and determination, and students who understand this will be able to adapt to new situations in both school and for the rest of their lives.

~364 words.