Full Calculus Reflection:
Winter Math Reflection – Calculus
Throughout not only the course of this year, but throughout the course of high school, I’ve had a tough time with math as a subject in whole. Not only in the fact that it takes me a while to understand how to do certain problems, but also in the essence of the conceptual portions of each class I’ve struggled to grasp. I was able to understand to a credible degree how to get through geometry. Algebra 2 wasn’t a class that I ultimately had an issue getting good grades in, but there was a degree of stress I can easily recall through the process of interpreting the content and completing calculations that made my sophomore year pretty tough. Even tougher so, however, was pre calculus; a class I still didn’t have even a substantial understanding of at the end of my junior year. I had barely scraped by. I soon found myself running into the same wall during this first semester of calculus, but this year something changed. Perhaps it was because I was granted an ultimatum: either come to an understanding of the content, or don’t graduate. Still, regardless of the reason I came to a beautiful understanding of the math we’re doing in calculus now, I may confidently confirm, at this point, that I do, in fact, understand the content thrown in my direction.
I believe that the wall I hit when I began this class, serving as an obstacle to me getting through calculus, was in majority due to my lack of understanding of the content I had learned in pre calculus. This is because a lot of pre calculus concepts are applicable in the process of completing calculus. Who would’ve thought? Well, I knew I was going to need some help if I was going to survive the year and go on with my class to graduate and go beyond from there. I was lucky enough to be signed up with a tutor weekly, and I’ve been meeting up with my tutor every Tuesday of every week for about 6 or 7 weeks now. I can’t say that Kyle’s teachings were entirely compatible with my learning ability, but I can’t say the same about my tutor, either. I can’t really say that about any of the teachers I’ve had that I can recall. I can, on the other hand, say that the teachings of both Kyle and my tutor have helped me to reach abstract understandings of calculus. When I say this, I mean learning from two different teachers has helped me look at some of the aspects of calculus in more than one different way. More than two, even, sometimes, because I’ve found that when I learn something in more than one different way, I’m able to diversify my own thoughts and form my own perspectives and ideas based on the math. I never remember being able to do this in the past.
Derivatives were a huge piece of the wall I hit this year that made me realize getting through calculus was going to be a big challenge. Immediately I started stressing about what this would mean for my grade, and about whether or not I was going to be able to know the ideas surrounding derivatives. Already, I wasn’t turning in worksheets, like the Derivative Three Steps worksheet, which still hasn’t been turned in. I wasn’t doing too well on the quizzes or homework, and it was obvious that derivatives were something I was struggling hard with. Of course this was a concern, so I eventually got signed up with a tutor. The teachings I got from her, along with the teachings I have received from Kyle, lead me down the path toward conceptualizing the content and ideologies of derivatives.
A derivative relates to the slope of a function, or even series of functions. I’ve known for a long time now what a slope is, and understand that it’s the direction of a function at a point on a graph. But derivatives aren’t just the slope of a graph. They deal with the change in y given the independent variable (x) throughout a function, not at just one point on the function. This portion about derivatives I understood pretty well off the bat, but when it came to being able to apply the definition of a derivative (), I was almost entirely at loss. I say almost because while I didn’t understand how I was going to apply this to the math at that time, I did see how the equation inside the parentheses, after the limit related to that of a slope function. And while we were also taught a shortcut for finding derivatives later on, this method for finding the derivative of a function was critical to the curriculum, as some of the work we’ve done required we use the definition of a derivative, instead of the shortcut. In this equation, the first thing to be understood is what a limit is, and what it does to a function. Even after all the work we had previously done in order to learn how to use limits in equations, I still had just a vague conceptualization of what limits were. Therefore, in learning how to deal with the definition of a derivative, learning how limits affected the function was crucial for me. Now, of course, I’m able to tell you that the limit of a function simply takes a point on that function, or all the points, rather, and tells you what the slope of the function is at that point, because limits require you look at the trend in the graph close to that point by using other points on that function, and seeing where the graph is leading from the other points to that point. The distance between the two points is considered , and the point itself is considered x. As a point further from the point in question moves closer to the point in question along the function, change of x ( ) gets smaller and smaller, and “approaches zero”. But we don’t want to just plug in change of x as zero, because we want to know the slope of the graph at all the points leading up to the point where change of x is zero. This method for finding derivatives and limits allows us to sketch graphs and go on from there.
I think I naturally have a tough time, often, conceiving concepts mathematically, and that has proven to provide some obstacles this semester, but I was lucky to obtain the help of my math teacher, as well as my tutor in order to gain critical ideas that give me the ability to push further on as graduation comes closer. I’m still signed up with my tutor next semester to meet one day a week, as usual, to give me the extra boost I’ll need in order to pass calculus while taking with me more than just what is needed to scrape by. It was with my tutor that I was able to recognize how the definition equation of a derivative is applicable in a lot of problems, because I had seen it used already in what we were learning with Kyle. I understand that for most students, that extra time isn’t necessary to learn the content, but this system is the first thing that’s worked for me in several years, and I am confident it will work for me going into second semester.
Throughout not only the course of this year, but throughout the course of high school, I’ve had a tough time with math as a subject in whole. Not only in the fact that it takes me a while to understand how to do certain problems, but also in the essence of the conceptual portions of each class I’ve struggled to grasp. I was able to understand to a credible degree how to get through geometry. Algebra 2 wasn’t a class that I ultimately had an issue getting good grades in, but there was a degree of stress I can easily recall through the process of interpreting the content and completing calculations that made my sophomore year pretty tough. Even tougher so, however, was pre calculus; a class I still didn’t have even a substantial understanding of at the end of my junior year. I had barely scraped by. I soon found myself running into the same wall during this first semester of calculus, but this year something changed. Perhaps it was because I was granted an ultimatum: either come to an understanding of the content, or don’t graduate. Still, regardless of the reason I came to a beautiful understanding of the math we’re doing in calculus now, I may confidently confirm, at this point, that I do, in fact, understand the content thrown in my direction.
I believe that the wall I hit when I began this class, serving as an obstacle to me getting through calculus, was in majority due to my lack of understanding of the content I had learned in pre calculus. This is because a lot of pre calculus concepts are applicable in the process of completing calculus. Who would’ve thought? Well, I knew I was going to need some help if I was going to survive the year and go on with my class to graduate and go beyond from there. I was lucky enough to be signed up with a tutor weekly, and I’ve been meeting up with my tutor every Tuesday of every week for about 6 or 7 weeks now. I can’t say that Kyle’s teachings were entirely compatible with my learning ability, but I can’t say the same about my tutor, either. I can’t really say that about any of the teachers I’ve had that I can recall. I can, on the other hand, say that the teachings of both Kyle and my tutor have helped me to reach abstract understandings of calculus. When I say this, I mean learning from two different teachers has helped me look at some of the aspects of calculus in more than one different way. More than two, even, sometimes, because I’ve found that when I learn something in more than one different way, I’m able to diversify my own thoughts and form my own perspectives and ideas based on the math. I never remember being able to do this in the past.
Derivatives were a huge piece of the wall I hit this year that made me realize getting through calculus was going to be a big challenge. Immediately I started stressing about what this would mean for my grade, and about whether or not I was going to be able to know the ideas surrounding derivatives. Already, I wasn’t turning in worksheets, like the Derivative Three Steps worksheet, which still hasn’t been turned in. I wasn’t doing too well on the quizzes or homework, and it was obvious that derivatives were something I was struggling hard with. Of course this was a concern, so I eventually got signed up with a tutor. The teachings I got from her, along with the teachings I have received from Kyle, lead me down the path toward conceptualizing the content and ideologies of derivatives.
A derivative relates to the slope of a function, or even series of functions. I’ve known for a long time now what a slope is, and understand that it’s the direction of a function at a point on a graph. But derivatives aren’t just the slope of a graph. They deal with the change in y given the independent variable (x) throughout a function, not at just one point on the function. This portion about derivatives I understood pretty well off the bat, but when it came to being able to apply the definition of a derivative (), I was almost entirely at loss. I say almost because while I didn’t understand how I was going to apply this to the math at that time, I did see how the equation inside the parentheses, after the limit related to that of a slope function. And while we were also taught a shortcut for finding derivatives later on, this method for finding the derivative of a function was critical to the curriculum, as some of the work we’ve done required we use the definition of a derivative, instead of the shortcut. In this equation, the first thing to be understood is what a limit is, and what it does to a function. Even after all the work we had previously done in order to learn how to use limits in equations, I still had just a vague conceptualization of what limits were. Therefore, in learning how to deal with the definition of a derivative, learning how limits affected the function was crucial for me. Now, of course, I’m able to tell you that the limit of a function simply takes a point on that function, or all the points, rather, and tells you what the slope of the function is at that point, because limits require you look at the trend in the graph close to that point by using other points on that function, and seeing where the graph is leading from the other points to that point. The distance between the two points is considered , and the point itself is considered x. As a point further from the point in question moves closer to the point in question along the function, change of x ( ) gets smaller and smaller, and “approaches zero”. But we don’t want to just plug in change of x as zero, because we want to know the slope of the graph at all the points leading up to the point where change of x is zero. This method for finding derivatives and limits allows us to sketch graphs and go on from there.
I think I naturally have a tough time, often, conceiving concepts mathematically, and that has proven to provide some obstacles this semester, but I was lucky to obtain the help of my math teacher, as well as my tutor in order to gain critical ideas that give me the ability to push further on as graduation comes closer. I’m still signed up with my tutor next semester to meet one day a week, as usual, to give me the extra boost I’ll need in order to pass calculus while taking with me more than just what is needed to scrape by. It was with my tutor that I was able to recognize how the definition equation of a derivative is applicable in a lot of problems, because I had seen it used already in what we were learning with Kyle. I understand that for most students, that extra time isn’t necessary to learn the content, but this system is the first thing that’s worked for me in several years, and I am confident it will work for me going into second semester.
"...I was lucky to obtain the help of my math teacher, as well as my tutor in order to gain critical ideas that give me the ability to push further on as graduation comes closer. I’m still signed up with my tutor next semester to meet one day a week, as usual, to give me the extra boost I’ll need in order to pass calculus while taking with me more than just what is needed to scrape by."
